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Physics-Related Articles
This page contains papers or articles in Mathematical Physics that I have produced — most written recently. In a few cases they have been derived from USENET postings (some rather lengthy). In other cases (where indicated), they are notes taken from or redactions or edits of various sources. In the latter cases, the notes have turned into full-scale simplifications, generalizations and rewritings of the originals.
Comments and feedback.
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Contents
The 5-Dimensional Black Hole
N-Dimensional Black Hole
N-Dimensional Monopole Solution
5-Dimensional Relativity
The N-Dimensional Gravitational Coupling Constant
Derivation of the Raychaudhuri Equation — Dadhich
Thermodynamics of Spacetime: The Einstein Equation of State — Jacobson
The Unruh Effect
Notes on Thermodynamics — derived from Plischke & Bergersen
Derivation of the Maxwell Distribution
Newton-Cartan Spacetimes
Newton and the Scholium
| Supplement | Dimension & Signature |
|---|---|
| The Inverse Metric Problem | The Inverse Metric Problem — Take #1 |
| The Inverse Metric Problem — Take #2 | |
| Spectral Decomposition for the Stress Tensor | Gauge Field |
| Combined Scalar and Gauge Field | |
| Fluid Dynamics Formalisms | The 5-D Representation of Fluid Dynamics & The Unified Group |
| Fluid Mechanics for the Schrödinger Equation |
| Time in Quantum Theory and General Relativity |
| Curved Spacetime Local Quantum Field Theory Meets Epstein-Glaser |
| Stress Tensors in Gauge Theory and Gravity |
| Articles | The Classical Renormalization Group |
|---|---|
| Renormalization of the Classical Scalar Field | |
| Scale Invariance and the Gauge Group |
The 5-Dimensional Black Hole
(PDF, 101k)
Geometric Renormalization and Charge Screening
In the late 1990's, the general 5-dimensional black hole solution was presented in a series of articles in sci.physics.research. This result is expanded on here, with a generalization that demonstrates that the general solution will split into 2 families, each corresponding to a different variety of Kaluza-Klein and different approach to gauge theory. One will correspond to the classical electrovac solution, the other (significantly) to a charge source that exhibits the classical variant of charge screening and the running of the couplings seen in quantum field theory.
Contents
1. Introduction and Background
2. Symmetries
3. Metric and Frame
4. Connection and Curvature
5. Riemann and Ricci Tensors
6. Field Equations
7. Schwarzschild Solutions
8. Electrovac Solutions
9. Gravito-Electric Solutions
Reference
"A Black Hole Solution in 5 Dimensions",
(Thread 1,
Thread 2);
1997 March 31 - April 16; sci.physics.research
N-Dimensional Black Hole
(PDF, 103k)
Originally posted in sci.physics.research
Following up on a previous development of the 5-dimensional black hole solution, the general N-dimensional solution spherically symmetric with respect to 3 spatial dimensions and symmetric with respect to the remaining dimensions is derived.
Contents
0. Introduction
1. Ansatz
2. The Moving Frame
3. Connection and Curvature Components
4. Ricci Coefficients
5. Coordinate Gauges
6. Field Equations
Reference
"N-Dimensional Black Hole";
2003 June 1 - 12; sci.physics.research
Supplement: N-Dimensional Monopole Solution (PDF, 61k)
This is the computation of the Einstein-Hilbert action for a geometry that possesses a combination of spherical symmetry and a symmetry given by a Lie group. This includes, as a special case, stationary 3+1 solutions, since time translation can be incorporated with the rest of whatever Lie group is present. Thus, the development is general to 2+1 dimensions or more and is substantially cleaned up.
5 Dimensional Relativity
(PDF, 167k)
This is derived from a presentation given at UW-Milwaukee, Department of
Physics, in 1999 of the 5-dimensional black hole solution. Much of the
material has now been moved to or superseded by the articles in the 5- and
N-dimensional black hole.
After briefly surveying the background and recent developments in
higher-dimensional relativity, we will derive the general spherically
symmetric, stationary, cylindrical 5-dimensional solution to Einstein's
Vacuuum Equations — the 5-dimensional Black Hole. This
will be followed by a discussion of the general N-dimensional solution
(N ≥ 3), which is spherically symmetric in 3 spatial dimensions and
symmetric with respect to the remaining dimensions.
Contents
1. Introduction
2. Campbell's Theorem
3. Dimensional Reduction and Space-Time-Matter Theory
4. Kaluza-Klein versus Gauge Gravity
5. Spherical Symmetry in 4-Dimensions
6. Spherical Symmetry in 5-Dimensions
7. The Connection and Curvature Forms
8. The Field Equations and Solution
The N-Dimensional Gravitational Coupling Constant
(PDF, 49k)
Where the Gravitational Coupling Constant Comes From
The coupling constant for gravity, when generalizing to higher dimensions, is frequently taken to be a multiple by 8π of the “Newton” constant G, forgetting that where the coefficient came from relies on an argument that depends on dimension. This analysis shows the correct relation between the "Newton" constant and coupling coefficients for higher dimensions.
An application of this result is the formulation of the correct values for the coefficients in the N-dimensional generalization of the Raychaudhuri equation.
Derivation of the Raychaudhuri Equation — Dadhich;
(PDF, 73k)
Derived from an article by Dadhich, with the derivation cleaned up and generalized to N dimensions and generic fluids.
The Raychaudhuri equation described the volume expansion and contraction of a system whose components are all moving under the influence of gravity. One of the simplest, and easiest-to-describe special cases of this equation is as follows: an initially stationary spherical shell of mass surrounding a gravitational source of mass M (e.g., the Earth) will occupy a volume that will begin to contract with an acceleration equal to GM, where G is Newton's gravitational constant.
A straightforward derivation of the Raychaudhuri equations is presented here, followed by a summary of the main issues of concern to Raychaudhuri at the time of his death. The derivation here is in the setting of a Lorentzian spacetime of N+1 dimensions. The derivation, itself, is factored into a purely geometrical part and a dynamic part.
The geometrical part is not specific to the spacetime geometry of General Relativity and, in fact, can be formulated generically with respect to spacetimes possessing a connection. It directly relates to, and provides a geometric interpretation for, the Ricci tensor.
The dynamic part applies the field law of Einstein to translate the geometric result into one involving the properties of gravitating matter. The result is normally stated in terms of perfect fluids, can be interpreted in terms of more general fluids with a suitable definition of the fluid's density and pressure. In 3 dimensions, reflecting on the observation made above about the volume contraction of a spherical shell, the volume contraction in the more general case involves a multiple, by 4πG, of the effective density of matter. The latter adds to the density of matter a contribution consisting of three times its pressure (in units where c = 1).
In N+1 dimensions, the multiple, in place of 4πG, is the coupling constant (discussed in a nearby article), multiplied by (N-2)/(N-1). The effective density, this time, adds in a multiple of the pressure by N/(N-2).
Thermodynamics of Spacetime: The Einstein Equation of State — Jacobson;
(PDF, 93k)
Derived from Jacobson (1995), the sections have been relabelled, graphics enhanced and a new section on volume contraction added.
The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation DQ = T dS connecting heat, entropy and temperature. The link between gravity/acceleration and its (Unruh) temperature is also being postulated.
The requirement that the fixed proportion between entropy and horizon area hold for all local Rindler causal horizons through each spacetime point, with DQ and T intepreted as the energy flux and Unruh temperature seen by an accelerating observer just inside the horizon, leads to the very kind of gravitational lensing by matter and energy that distorts the causal structure of spacetime in just the right way to yield Einstein's equations. In addition, the proportionality constant is determined to be such that one bit of information is equated to 1/4 of a Planck area.
From this perspective, Einstein's equation is an equation of state, suggesting that it may be no more appropriate to canonically quantize the Einstein equation than it would be to quantize the wave equation for sound in air.
Of necessity, this entails that any quantum theory that incorporates gravity can only be semi-classical (i.e. hybrid classico-quantum), and can not be purely quantum.
Contents
1. Introduction
2. Local Rindler Horizons and Horizon Entropy
3. Volume Contraction and Gravitational Mass
4. Volume Contraction and Energy Flow
5. Concluding Generalizations
The Unruh Effect
(PDF, 35k)
This is a short writeup, derived from an article on the Wikipedia, describing the Unruh effect.
Notes on Thermodynamics
(PDF, 187k)
Based on Equilibrium Statistical Mechanics (2nd edition, Plischke & Bergersen, World Scientific), these are some notes I took on my reading of the first chapter.
Everything is stripped down to the bare essentials, and the graphs are colorized, because the absence of color really obscures what's actually being shown. If you know anything about Rene Thom, the first think you'll think of the instant you see the 2 pictures for the PVT phase diagram is the Cusp of Thom's “Catastrophe Theory”.
The exercises are left undone, but in time they will each comprise the core of additional sections. The above reference (by the authors' own account) is a hand-me-down of Reichl. With time, some of Reichl will be integrated into this, since Reichl's coverage is more comprehensive in most areas.
| 1.1. State Variables and Equations of State | |
|---|---|
| 1.2. Law of Thermodynamics | 1.2.0. Zeroth Law |
| 1.2.1. First Law | |
| 1.2.2. Second Law | |
| 1.3. Thermodynamic Potentials | |
| 1.4. Gibbs-Duhem and Maxwell Relations | |
| 1.5. Response Functions | |
| 1.6. Conditions for Equilibrium and Stability | |
| 1.7. Thermodynamics of Phase Stability | |
| 1.8. Problems |
| 2.1. Isolated Systems: Microcanonical Ensemble |
|---|
| 2.2. Systems at Fixed Temperature: Canonical Ensemble |
Derivation of the Maxwell Distribution
(PDF, 33k)
The Maxwell Distribution is derived from the canonical distribution. In turn, a short heuristic derivation of the canonical distribution is presented.
Newton-Cartan Spacetimes
(PDF, 82k)
Though it is not generally realized or appreciated as such, the distinctive feature of General Relativity versus the Newtonian theory of gravity is not the notion of curved spacetimes, but rather that of the causal structure. In essence, General Relativity represents the adaptation or retrofitting of the law of gravity from Galilean spacetimes to Lorentzian spacetimes. The distinctive feature of the former is the existence of hyperplanes of absolute simultaneity. An equivalent way of saying this is that in Galilean spacetimes, the invariant velocity is infinite. In contrast, in Lorentzian spacetimes, the invariant velocity is finite and the plane of simultaneity bifurcates into a past and future light cone, which represents, respectively, the locus of all motions at invariant velocity emanating to or from a given spacetime event. This characterizaiton is the same for Galilean spacetimes, but since the invariant velocities are infinite, there is no concept of to or from, so the past and future light cones merge into a single structure — the plane of simultaneity.
The flip side of all this is that the notion of curved spacetimes does not originate with the departure from Newtonian theory to that of General Relativity. In fact, what justifies it is the Equivalence Principle, and this principle could just as well have been applied to Newtonian theory, yielding the same result: that gravity there, too, arises as a result of curvature in spacetime.
Under the equvialence principle, the distinction between inertial and gravitational mass is lost, so that one may regard free fall as simply being inertial motion — but in an arena where the underlying spacetime is now curved. So, the question naturally arises: just as Minkowski space generalizes to a Lorentzian spacetime, and a Euclidean space would have generalized to a Riemannian spacetime, what does the Galilean space of Newtonian Physics generalize to? This is what we will answer here.
| Eternal | duration spanning all time |
| Infinite | presence spanning all space |
| Omnipotent | governor of all things |
| Omniscient | knower of all things that are or can be done |
Essentially identical with the listing of attributes to what the character Ellie recites in in The Untold Story of Contact, Newton's listing differs in certain key aspects which can only be regarded as mistakes by Newton. One, his definition of Omnipotence is wrong because it is self-contradictory — particularly the can part of this definition. The use of any modal, such as “can”, in this context also contradicts the Omniscience attribute (there is no such thing as modality to an Omniscient being). The contrasts to the listing used by Ellie were:
| Eternal | does not exist at “all times” but has a timeless existence over and above time, itself. |
| Omnipresent | does not exist at “all places” but has an existence that transcends space-time, itself. |
| Omnipotent | does not mean “one who is able to make anything happen”, but simply “one who makes all that will happen, happen; and makes all that will not happen, not happen”. |
| Omniscient | contains an account of every outcome. |
Newton's space-time view is firmly grounded in the notion that reality is segmented into what are effectively instantaneous 3-dimensional snapshots of space. In each one, every point is simultaneous with every other point. The discussion of God was partly a cloak under which this world view was to be laid out. Newton disagreed on the point about equating God to Natural Law. Instead, he insisted on going against his own creed and asserted a divine source that has neither action nor reaction with tangible matter. Newton also expressed the notion that the initial configuration, itself, had to be the product of direct divine intervention. This point of view should be contrasted with the countervailing point of view expressed here, in The Secret of Creation.
Maxwell's Original Equations
(PDF, 233k)
| Supplements | 1 | The Curl of the Curl |
|---|---|---|
| 2 | On Physical Lines of Force — Maxwell | |
| 3 | A Dynamical Theory of the Electromagnetic Field — Maxwell | |
| 4 | The Elements of Maxwell's Renormalization Theory |
It may surprise the reader to learn that the formulation of electrodynamics, Maxwell's equations, has evolved through inequivalent states from the time of Maxwell onward — and even during the time of Maxwell. The original forms are, themselves, barely recognizable to modern eyes, are not synonymous with what are presently known as “Maxwell's equations”, and mix in a hodge-podge of elements that are now understood to be naturally segregated both to different levels of fundamentality and to different levels of abstraction.
An historical account is provided of some of these states, and of later developments.
A point of departure from the original, which this article is based on, is to dispel the notion that the natural appearance of 4-dimensional notation has anything to do with Minkowski geometry or Special Relativity. This is, in fact, red herring. Rather, what it has to do with is that a signifcant part of the mathematical infrastructure identified by Maxwell's equations resides at the level of a topological or diffeomorphism-invariant field theory and is generic to any and all 4-dimensional continuua, regardless of whether it be Euclidean, Galilean/Non-Relativistic, Lorentzian/Relativistic (or even Platonic/Archimedean); flat or curved; or any combination of the above. This discussion is expanded to the “Maxwell equations” form of non-Abelian gauge theory in the supplement The Maxwell Gauge Field Equations in 4 Dimensional Form.
The Galilei limit of electromagnetism, following Maxwell, requires the introduction of a velocity vector G. Along the way, clarification may also provided of the so-called “one-way” speeds and their relation to G.
The spinor notation is developed in invariant form; and the closely-related “imaginary quaternion” form of Maxwell's equations is developed (i.e., the form within the algebra C×H, which is just another name for the Pauli matrix algebra).
| 1. Introduction | |
|---|---|
| 2. Maxwell's Equations | 2.1. The Original Equations |
| 2.2. Basic Equations | |
| 2.3. The Quaternion Form of Maxwell's Equations | |
| 2.4. Today's Vector Notation of Maxwell's Equations | |
| 3. Real Expansions of Maxwell's Equations | 3.1. The Hertz-Ansatz |
| 3.2. The Dirac-Ansatz | |
| 3.3. The Harmuth Ansatz | |
| 3.4. The Múnera-Guzmán Ansatz | |
| 4. Imaginary Expansions of Maxwell's Equations | 4.1. Four-Dimensional Notation |
| 4.2. The Inverse Correspondence Limit — What it's Like to Travel Alongside a Light Beam | |
| 4.3. The “One-Way” Speeds | |
| 4.4. Duality, Minkowski Space and Spinor Notations | |
| 4.5. The Imaginary Quaternion Notations |
Supplement 1: The Curl of the Curl
(PDF, 89k)
This expands on the notations given in section 4 of the above article Maxwell's Original Equations, dicussing the origin of the curl-of-the-curl identitity and its relation to the Hodge star operator. A simple account of the Hodge star operator is given. Then it is generalized so that it can encapsulate the most general constitutive law.
Supplement 2: On Physical Lines of Force
(PDF, 412k)
This is an annotated reproduction of the original, from Maxwell, published in 4 parts in 1861-1862 in Philosophical Magazine. The verbal text has not yet been double-entry validated.
The modern notational equivalents of key equations and quantities are listed in footnotes, along with further descriptive commentary.
Supplement 3: A Dynamical Theory of the Electromagnetic Field (PDF, 410k)
(The 1864 paper (Wikipedia clone page))
This is an annotated reproduction of the original, from Maxwell, published by the Royal Society in 1864-1865. The verbal text has been independently double-entry validated.
The modern notational equivalents of the key equations and quantities are also listed in footnotes. The table in section 70 is expanded to include the implicit 21st-23rd equations and to list, alongside, the modern renditions of the key items.
Supplement 4: The Elements of Maxwell's Renormalization Theory (PDF, 57k)
This was originally included as a part of the Standard Model Lagrangian article (which it still refers to), and describes in some detail the key points, outlined by Maxwell, of his resolution of the self-force and self-energy problems of classical field theory. In today's language, we would recognize this as the 19th century version of the very renormalization programme (re-)discovered by Feynman, Tomonaga and Dyson in the 1940; which, indeed, it may claim direct ancestry (by way of the Heisenberg-Euler Lagrangian).
Key points worth mentioning are the thought experiments meant to explain the necessity of an active dielectric medium in order to smear out the self-force infinities for point-like and line-like distributions and yield finite values for E in the vicinity of such sources. Also introduced are the concepts of what we now call bare vs. renormalized charges, charge screening around concentrated sources and vacuum polarization.
Galilean-Invariant Maxwell Equations
(PDF, 57k)
It is a little-known fact that the description of the electromagnetic field provided by Maxwell's equations — the equations, that is, which were actually posed by Maxwell in his treatise — were not Poincaré invariant, but Galilean invariant ... inasfar as their invariance properties were studied in the treatise.
This article takes the process to its conclusion, showing what the full set of equations look like under the assumption of Galilean invariance. Both sets of Maxwell equations; the set for (E, B) and the set for (D, H) are actually invariant under a proper superset of the full diffeomorphism group. The distinction between the different causal structures corresponding to Poincaré or Galilean relativity lies entirely within the constitutive law that relates the (E, B) and (D, H) fields.
These equations serve as a point of comparison with the formalism (originally developed by Lorentz) now known as “Maxwell's equations”. A particular feature of the constitutive law in the Galilean variant of Maxwell's theory, that serves as a basis for empirical tests distinguishing Poincaré from Galilean relativity, is the dependence of the electric induction, D, on the magnetic field, B.
Relativity with an Aether Frame and Absolute Time
(PDF, 184k)
These are a series of articles that haven't yet been fully sorted out). They are mostly derived from several posted in sci.physics.research (one as far back as 2002, most others in 2008), that combines the 5-dimensional projective representation of the Unifed Group articles with Maxwell's “aether velocity” vector G in a geometric formulation that tries to bridge the gap between general relativity and gravity in a Newton-Cartan space. The general formalism includes Einstein-Aether models, such as the recent ones posed by Jacobson, as special cases.
Contents
1. Galilean-Invariant Maxwell Equations
2. What it's Like to Travel Alongside a Light Beam
3. An Oversight in the Inverse Correspondence Limit for Gravity
4. General Relativity with an Absolute Time
5. Combining Aether Velocity with Absolute Time
6. Aether Velocity as a Covariant Derivative
7. 5-D Representation
8. Gauging the Extended Group
The Equivalence of Maxwell's Equations and Kirchhoff's Laws
(PDF, 42k)
Kirchhoff's laws, in differential form, state that the divergence of the total current K and the curl of the potential drop F are both zero. Though it is common to see these derived from Maxwell's equations, it is remarkable to notice that Maxwell's equations, themselves, may be derived from these two laws.
More precisely, there is a one-to-one, continuous, correspondence of solutions K and F to "Kirchhoff's laws" to each solution (E, B, J, ρ) of Maxwell's equations, where E and B are the electric and magnetic fields, respectively, and J and ρ, respectively, the current and charge density.
Thus, the total current and potental drop serve simultaneously as generating functions for both the sources and fields, ranging over all combinations of the two that solve Maxwell's equations.
Interestingly: the treatment given here essentially matches one of the LeBellac/Lévy-Leblond “Galilean limits”, and so this article is being included here.
Towards a General Theory of Signature and Signature Change
(PDF, 120k)
| Supplement | Dimension & Signature (PDF, 60k) |
|---|---|
| The Inverse Metric Problem | The Inverse Metric Problem — Take #1 |
| The Inverse Metric Problem — Take #2 | |
| Spectral Decomposition for the Stress Tensor | Gauge Field |
| Combined Scalar and Gauge Field | |
| Fluid Dynamics Formalisms | The 5-D Representation of Fluid Dynamics & The Unified Group |
| Fluid Mechanics for the Schrödinger Equation |
A surprising, but rarely cited, fact is that the stress tensor determines most of the metric. This is because the stress tensor in its “pristine” form — the form that it assumes before any metric is used to raise or lower its indices — is as a rank (1, 1) tensor density. This has two consequences.
First, the eigenvalue-eigenvector problem can be formulated for the stress tensor, independent of any considerations of any metrics. In the generic case, one has 4 eigenvectors with 4 separate eigenvalues, thus determining a natural orientation for a coordinate axis. Second, the “symmetry” property normally ascribed to stress tensors is actually seen to be a property of both the covariant metric gμν and contravariant metric gμν. As restrictions on the metrics, the conditions will generally determine 6 out of 10 components of the metrics, and will only permit metrics with respect to which the 4 orientations are orthogonal. The remaining 4 components of each metric give us scaling and signature parameters for the 4 principal directions.
This serves as the basis for a general theory of signature. A key consideration we will address is (1) how to derive the condition that the two metrics should be inverses of one another for non-singular signatures and (2) how to deal with singular signatures. Item (2) is resolved by requiring that the symmetry group which preserves both the metric and dual metric be rank 6 and effective. Though the analysis is not fully carried out in the article, it turns out that this requires the two metrics to be of the form
| (gμν) = Aw×w + Bx×x + Cy×y + Dz×z, |
| (gμν) = aW×W + bX×X + cY×Y + dZ×Z, |
| Aa = Bb = Cc = Dd, | (A, a), (B, b), (C, c), (D, d) ≠ (0, 0) |
This article also addresses the issue of what happens to field theory across a signature boundary. This question is closely tied to the question of the Galilean Limit of field theories. The two questions are really just different takes on the same issue.
Contents
1. Stress Tensors and Diffeomorphism Invariance
2. The Symmetry and Orthogonality Conditions
3. The Spectral Decomposition and Derivation of the Metric
4. The “Garden Variety” Solution
5. Field Theory Across a Signature Boundary — The “Galilean Limit” Question
These resolutions lead to the following observation:
|
13 1/2 billion years worth of light's travel in every direction into the sky lands you back — in direct line of sight — to the very first moment of time in the Universe. But it's not a Beginning, but just a boundary, on the other side of which lies another domain. Scientists presently call it the Big Bang. Yet, not even they know or fully realize that it is only a boundary and not a wall. They think of it as the Cosmic Singularity — the Creation Event, out of which all that exists came into being. Only a select few dare to imagine the possibility that there is a flip side to this boundary. And one of those people is none other than the recently retired Steven Hawking. Unlike our side, which has 3 spatial dimensions and one dimension of time, the other has only spatial dimensions — 4 of them — but none of time. If you make even one exception to the Laws of Nature, if you allow for even one event in spacetime where something singular happens, like a Creation Event, then Hawking argues that you would have to allow for the possibility of singular exceptions at all events in the Universe. But then, the Laws of Nature go from being something that applies “everywhere, without exception”, to something that only applies “everywhere, except where they don't feel like it”; which misses the point of having any laws at all. And that's not science. And, so Natural Law demands that there be no such thing as a singular event of Creation! Therefore, the “boundary” of the Big Bang is nothing more than a barrier to Another Side. Not a Wall to Nothing. But no physicist knows how to stitch together a domain with 3 space and 1 time dimensions to one that has 4 space and no time dimensions; without a suture that is singular. But there is a secret that nobody yet knows, not even Hawking, himself. It can be done, and can be done without the need for anything as exotic as quantum gravity or quantum-anything else. Few who hear this secret will understand; but those who understand it will, upon hearing it, never see or think of their world or outer space the same again: It can be done by nullifying the 100-year-old marriage between the covariant and contravariant metric. |
The Inverse Metric Problem
The Inverse Metric Problem — Take #1 (PDF, 39k)
These are the basic computations — done early on in my investigations of these issues a few years ago — that attempt to directly solve the “symmetry” condition on the metric, and derive the general form of the metric from the stress tensor based on this condition.
The Inverse Metric Problem — Take #2 (PDF, 103k)
This is a compilation of 3 related articles taken from my notes or posted in sci.physics.research. Here, the problem is posed in a more general context — for geometrical formalisms based on the Riemann-Cartan geometry. A key point made is that the “gravitational stress tensor” is simply the Einstein tensor, up to sign, and that the fact that this combines with the matter stress tensor to yield a total of 0 (on shell) may be seen as a generalized form of the Vis Viva theorem; not as an “no go” negative result for the gravitational stress tensor.
In the third part, the Jordan classification of the stress tensor is used to produce a generalized Segre classification, combined, for the metric, the inverse metric and stress tensor.
Contents
1. Lorentz & Diffeomorphism Invariance and the Stress Tensor
2. Vis Viva and the Gravitational Stress Tensor
3. The Generalized Segre Classification Problem
Spectral Decomposition for the Stress Tensor
Gauge Field (PDF, 111k)
The spectral decomposition is carried out for all 1-component gauge fields (i.e. all dynamic models for a Maxwell field). The general result is that the eigenvalues occur either with a 4-fold degeneracy (for the null field) or a 2+2-fold degeneracy. Since these computations are done in the general setting of isotropic media for arbitrary signature (Lorentz, Galilei, Euclid), the term “null” is defined in a medium-dependent way and only coincides with the term for the vacuum of the Lorentz signature.
The computation assumes only that the field dynamics are given by a Lagrangian which does not explicitly depend on the potentials. It is carried out in a way that is independent of the Lagrangian. We then proceed to examine the case where the Lagrangian is invariant under the members of the “Unified Group” 1-parameter family of groups that includes Lorentz, Galilei, Euclid signatures.
An important conclusion drawn here is that a distinction may be made between the Einstein form of E = mc², originally published in 1905; and the Poincaré form of E = mc², published by Poincaré in 1900. Though these two equations coincide in the vacuum of a Lorentz-signature space-time, they become distinct in all other signatures, and even in a Lorentz space in an isotropic medium. Hence, both men should be credited with the discovery of separate forms of the mass-energy conversion formulae, that just happen to look and sound the same, but are not.
Contents
1. Principal Components of the Stress Tensor
2. Constitutive Relations
3. Reduction of the Stress Tensor
4. Orthogonal Reduction of the Fields
Combined Scalar and Gauge Field (PDF, 51k)
The classification is extended to the case of the combined gauge and scalar field. The 2+2-fold degeneracy generally splits into (1+1)+(1+1). In special cases, the 1's may line up askew to produce a 2+2 degeneracy; or one pair may meet in the center to produce a 1+2+1-fold degeneracy.
Contents
1. Background
2. The Eigenvalue Problem
3. The Pure Scalar Field
Fluid Dynamics Formalisms
The 5-D Representation of Fluid Dynamics & The Unified Group (PDF, 134k)
These are a series of sections originating from a sci.physics.research article which use the 5-dimensional representation of the Galilei group to answer the question of where the 5th fluid dynamics equation went. This may be considered as a sequel to the Schrödinger, Nelson and Born-Sommerfeld article.
The 5 equations are for mass conservation, momentum conservation and the conservation of kinetic energy. What distinguishes this analysis is that since the 5-dimensional representation can be “relativized” (as was one of the main points of the unified group articles), then all the foregoing continues to hold in the relativistic setting. This, provides an answer to where the 5th equation goes.
A key point is that the corresponding stress tensor — now a 5 × 5 stress tensor — is symmetric with respect to the extended 5-metric.
The Schröinger equation may be cast in fluid dynamics form as a set of 5 fluid dynamics of 5 equations and fit within this framework, as may the Klein-Gordon equation. Because of the extra room afforded by the 5-dimensional representation in relation to what can be provided by the Poincaré representations (a point discussed in the “incompleteness of Relativity” articles above), it might be possible to avoid the usual energy-bound “no-go” arguments that are normally employed to block the interpretation of the fluid dynamic intepretation of the Klein-Gordon equation.
Contents
1. What Happened to the Extra Fluid Dynamic Equation and Stress Tensor Symmetry?
2. T44 & the Mysterious Density Z; Deformation to Relativistic Fluid Dynamics.
3. Fluid Dynamics Equations from Symmetry and the Relativistic Formulae.
4. The Correct form of the “General Relativistic” Newtonian Gravity Law. (c.f. Newton-Cartan Spacetimes)
5. Generalization to Relativistic Form
6. The Schrödinger Equation as a Fluid Dynamics System
7. Klein-Gordon and Dirac as Fluid Dynamics
Fluid Mechanics for the Schrödinger Equation (PDF, 44k)
Here, the computation of the combined Schrödinger/Klein-Gordon equation is done coupled to a 5-component gauge field. Though not discussed in detail here: the 5th component is naturally identified as the B component of the B-field formalism, which is used to represent massive gauge fields.
Unification of Galilei, Poincaré, and Euclid
(PDF, 214k)
The passage of 100 years has not uncovered all that can be said about Relativistic Physics and its relation to Newtonian Physics and Galilean Relativity.
Defining the Poincaré Group → Galilei Group correspondence limit is not a trivial exercise. Since the Galilei group has a central charge (mass), then it is necessary to include an 11th generator into the Poincaré group and a 5th coordinate to the underlying geometry. The result is a unification of Galilei and Poincaré into a continuous 1-parameter family of gauge groups that also happens to include, for free, the 4-dimensional Euclidean group and a definition of absolute time.
All members of the family are embedded in the 4+1 Poincaré group in a way analogous to the unification of Euclidean, Hyperbolic and Spherical geometries into Projective geometry.
We will review the ramifications of the unification; revisiting such questions, along the way, as: “what is the meaning of Euclidean time?” and “what is the meaning of imaginary mass for tachyons?”
This analysis is a follow-up of the earlier analysis carried out under The Wigner Classification for Galilei/Poincaré/Euclid.
| 1. Background | 1.1. The Non-Triviality of the Correspondence Limit |
|---|---|
| 1.2. Generalized Mass Shell Invariants | |
| 1.3. Representation Classes | |
| 1.4. Generalized Metric | |
| 1.5. The Emergence of an Absolute Time | |
| 2. The Unified Group | 2.1. Definition |
| 2.2. Adjoint Action | |
| 3. The Invariants of the Unified Group | 3.1. Classical vs. Quantum Invariants |
| 3.2. Translation Invariants | |
| 3.3. Rotation and Boost Invariants | |
| 3.4. The Luxon Sector | |
| 4. Position and Time Operators in the Unified Group | 4.1. An Enveloping Formula for the Position Vector |
| 4.2. General Solution | |
| 4.3. Time Operators |
Supplements
What it's Like to Travel on a Light Beam (PDF, 106k)
Extending the analysis already carried out under Galilean-Invariant Maxwell Equations), we will also see that the long-forgotten G field of Maxwell may be recovered, as we expand the causal background from the 1-parameter family of the Unified Group to a 2-parameter family that also includes both an invariant velocity (c) and a non-invariant light speed (V). In the limit as V → c, the aether velocity G drops out and becomes “superfluous”. However, there will turn out to be one significant exception: namely, the case where G = V. Thus, an answer can actually be given to the question that served as a motivation for Einstein's formulation of relativity theory: “what's it like to travel alongside a light beam?” even in the limit V = c.
Einstein's Big Idea, PBS, NOVA (2005 October)
| Einstein's Big Idea & Einstein Revealed |
| Einstein's Big Idea (annotated transcript) |
| Einstein Revealed (annotated transcript) |
In part. this is a critical review in which some of the key misconceptions, or myths, that pervade present-day folklore are straightened out (e.g. that Maxwell's equations, as conveyed by Maxwell, were Poincaré invariant; that energy and matter are fully equivalent; etc.; the account of how and why the square in c² comes about, etc.)
The Return of Relativistic Mass and Relativistic Kinetic Energy (PDF, 69k)
(See also Definition of Mass).
 |
| Einstein (1912) — “L”: a trace of hidden thoughts exposed in a Freudian slip. Expect to find “H” in his early papers too. |
| L = mv²/(1 + √(1 - αv²)) | (α = 1/c²) |
| H = Mv²/(1 + √(1 - αv²)) | (M = m/√(1 - αv²)) |
A distinguishing feature of this analysis, which is understood to apply within the framework of the Unified Group where the mass-shell constraint is relaxed, is that it may be applied to Wigner sectors other than the Tardion sector; e.g., to Tachyons. Here, the invariant mass “m” is no longer interpreted as the rest mass, but rather as a function m = -U/c² of the “intrinsic” part of the kinetic energy (i.e., the 11th parameter).
As a precursor to this analysis, we will also provide an account of mass-energy conversion that is also more firmly grounded in concepts arising from pre-relativistic 19th century physics.
On the Incompleteness of Relativity
(PDF, 31k)
Originally posted in sci.physics
A transition A → B between two paradigms, A and B, involves a continuous transformation of all the formal language of A into the formal language of B, while B → A carries out the transformation in reverse. The B → A arrow is generally called the “classical limit” or the “correspondence limit”. However, the A → B arrow does not have a general name.
If B is quantum theory, the process is called “quantization” and is well-understood. However, for the case where B is relativity, there is no standard name, though an obvious choice would be to refer to it as “relativization”
While quantization is well-understood and has a well-developed body of literature associated with it, relativization is not and does not. This is a significant oversight in foundational physics which, even to the present-day, has not been properly addressed or resolved.
The two key issues surrounding the correspondence limit are completeness and soundness. Soundness guarantees you that the statements of the newer paradigm B either correspond to empirically valid statements in A or present corrections to statements in A formerly thought to be empirically valid. The corrections are what experimental tests then seek to empirically test for, and it is those tests which determine the validity of the transition A → B.
However, completeness is by far the more important property. Completeness is what justifies using B in place of A. It is what enables one to grandfather the entire history of empirical success of the paradigm A, going all the way back to the Stone Age onwards, into paradigm B. Without completeness, it becomes necessary to go back and retest everything with respect to B.
Quantization is complete and has been well-studied from many different perspectives; whether it be the process of quantizing a state space (coherent state quantization) or the variables describing systems (operator quantization). In contrast, there is no analogous formalism for representing the “relativization” transition.
The closest version we have of such a transition is the Inönü-Wigner contraction. However, though sound, the contraction is woefully incomplete! In the most general case, one may define relativization as the process of effecting a continuous transition of the entire representation space for the Galilei group — which is the symmetry group governing non-relativistic theory. However, this space is far more complex than the representation space for the relativistic symmetry group: the Poincaré group. An important case in point, which has provided the central focus of the Unification of Galilei, Poincaré, and Euclid article, are the 5-dimensional projective representations of the Galilei group.
The general problem of defining a complete A → B transition is, thus, far less trivial than a naive account may lead one to believe. However, the situation is much more serious than even this implies: there may not actually exist any finite-dimensional group that contains the Poincaré group which is large enough to make the A → B arrow complete, while containing the Inönü contraction and projective representations. Thus, the completeness problem may actually be intractable.
The Maxwell Equations for Gauge & Proca Fields with a B Field
(PDF, 106k)
The B field formalism is an extension of the 5-dimensional representation of the Galilei group to relativistic settings. Despite the simplicity of this characterization, it is still largely unknown in the literature — even by those who originally developed the idea and by those who use it.
In this article, some of the basic elements are laid out. This will be expanded on in the near future.
| 1. Background | |
|---|---|
| 2. The Coordinate Representation | |
| 3. The B field as a 5-Gauge Field | 3.1. Field Kinematics |
| 3.2. Proca Fields with B-Fields | |
| 3.3. Lorentz Invariants | |
| 4. Dynamics | |
| 5. Constitutive Laws |
Dirac, Gamma5 and SO(4,1)
(PDF, 51k)
As originally suggested by the convention Dirac adopted in naming γ5, the extra matrix has something to do with a 5-dimensional representation — it is none other than the “relativization” of the 5-dimensional representation of the Galilei group that we've been discussing in the other supplementary articles.
This may be considered as an expansion on section 1.4 of the Unified Group article and section 1.2 of the Generalized Wigner article. The discussion is generic with respect to signature and applies to the Poincaré, Galilei and Eulidean sectors.
The particular insight conveyed here involves a subtlety which is rarely discussed regarding the Dirac algebra — the distinction between the real and complex algebra. Only here, the explanation of what lies behind the “complexification” departs company from the usual textbook account of what's going on here. We relate it, instead, directly to the emergence of the 5-dimensional representation associated with mass. This is a second movement of the theme already played out with the Proca Fields with a B Field article.
The “Dirac algebra”, when meant in the sense of the “real Clifford algebra assoiated with the signature (+, -, -, -)”, is generated as a real algebra from (γ0, γ¹, γ², γ³). However, this is not the algebra of 4×4 complex matrices the Dirac algebra is usually represented as. Rather, it is the 2 × 2 quaternion matrices!
To get the 4 × 4 complex matrices requires “complexifying” the Dirac algebra. This amounts to adding another imaginary unit i. Equivalently, since γ5 is a product of the other γ's with i, then it amounts to expanding the algebra with to the real linear algebra generated by the following five matrices (γ0, γ¹, γ², γ³, γ5). This is the Dirac algebra, as it is usually known, and is equivalent to the complex 4 × 4 matrices.
Thus, the reason the algebra is complexified actually has nothing per se to do with complexification, itself. Rather, what it has to do with is extending the real linear algebra to one that is associated with a five dimensional signature — namely, the signature (+. +, +, +, -) of SO(4, 1).
The ostensible reason the extra generator is required is to account for “chirality”. In turn, this is necessary to include mass in the Dirac equation. However, the root of the matter simply boils down to the fact that with the inclusion of mass, we're back to the theme of the Unified Group article — the generalized momentum
| P = (P1, P2, P3), | P4 = -H, | P5 = M |
The Jordan Decomposition in the Unified Group
(PDF, 72k)
Expanding on section 4B of Wigner [1], the Jordan decomposition of the orthogonal transformations is expanded to the Unified Group. Here, the “Platonic” signature is also included.
The Jordan classes are [(1111)] for the Identity Transformation, [(211)] for the Galilean Boost and Poincaré Shift. (This is for the Platonic signature. The transformation is r → r, t → t - λ·r). The Jordan class [(31)] includes the “Null Boost” (in null coordinates, this has the same form as uniform acceleration does in 3-space).
The Jordan class [(11)11] includes the Lorentz Boost, while [(11)zz*] includes rotations for all signatures.
Finally, the Jordan classe [2zz*] gives us the generic Galilean and Platonic transformations, which combine a boost/shift with a rotation, while [11zz*] gives us the generic Lorentz, and [zz*zz*] the generic Euclidean transformation.
Reference
| [1] | E. Wigner, “On Unitary Representations of the Inhomogeneous Lorentz Group”, The Annals of Mathematics 40(1), 149-204 (1939). |
Gravity as a Gauge Theory in the Unified Group
These articles attempt to deal with the issue of formulating a continuous transition for gravity crossing the Lorentzian → Galilei → Euclidean sectors.
1. Quasi-Galilean Tetrads: An Alternative to Plebanski
(PDF, 80k)
This was originally part 3 of Basic Topics in the Mathematics of General Relativity and is the earliest attempt to deal with the above issue.
This lays out the basic elements that are universal to any formulation of gravitational dynamics over all the signatures (Poincaré, Galilei, Lorentzian) for any dyanamics in a Riemann-Cartan space based on a Lagrangian formulation.
Contents
1. The Metric Tensor
2. The Quasi-Galilean Tetrad
3. The First Structure Equation and Torsion
4. The Second Structure Equations and Curvature
5. The Bianchi Identities
6. Complexified Bases
7. Action Principle
8. The Field Equations and Conjugate Fields
2. Gravity with Signature Change (PDF, 135k)
Metric-Affine Gravity for Poincaré/Galilei
This article attempts to formulate gravity as a gauge theory over the one-parameter family of groups comprising the “Unified Group”, thus partially abstracting out signature. This predates, by a couple years, the other articles in this group and is where I first encountered the problem alluded to in [1,2,3].
The distinguishing feature of this approach (which actually supersedes the treatment in [1,2,3], but has not yet been collated with that material) is that everything is done within both the 5-dimensional “projective” representation of the extended Galilei group and its “relativization” to the 5-dimensional projective representation of the 11-parameter extended Poincaré group. A distinguishing feature of this geometry is the retention of absolute time from the Galilei sector over into the Poincaré sector and the retention of the non-absolute time from the Poincaré sector over into the Galilei sector.
There is significant simplification of the treatment in [1,2,3] to be had by directly incorporating the projective 5-D representation and its relativization. This analysis is still pending. But synergistic merging of the two approaches may help resolve the issues that plagued this article, as well as reducing the number of degrees of freedom of the gauge theory in [1,2,3] from 27 down to a more reasonable number (15 may be possible).
Contents
1. The Galilean Limit Revisited
2. The Generalized Lorentz Group
3. The Quasi-Galilean Pentad
3. On the Unified Group and Space-Time Geometry
(PDF, 90k)
This is partly a followup to the Gravity with Signature Change article. This is a compilation of several short articles which try to work out some of the consequences of the problems discussed in [1,2,3]; all revolving around the absence of a continuous bridging between the Galilei and Lorentzian sectors in gravitational dynamics. These will all be expanded on, time permitting.
Contents
1. Proper Time as the Lagrangian for both Relativistic and Non-Relativistic Mechanics
2. Schwarzschild & Newton Bridged Together.
3. ADM in Invariant Form
References
| [1] | R. De Pietri, L. Lusanna, M. Pauri, “Galilean Theories of Gravitation” (arXiv:gr-qc/9212002v2 13 Dec 1992) |
| [2] | R. De Pietri, L. Lusanna, M. Pauri, “Newtonian Gravity as a Gauge Theory — I: The Standard Theory” (arXiv:gr-qc/9405046v1 22 May 94) |
| [3] | R. De Pietri, L. Lusanna, M. Pauri, “Newtonian Gravity as a Gauge Theory — II: Dynamical Three-Space Theories” (arXiv:gr-qc/9405047v1 22 May 1994) |
Einstein's Early Work
On the Electrodynamics of Moving Bodies (PDF, 204k)
This is an English translation of the 1905 article that originated from The Principle of Relativity (1923), was adapted as an on-line version prepared by John Walker (1999), and was updated by me with the addition of extra commentary, and a change in notation. This article is provisional pending the acquisition of and close study the German original.
The opening section has been redone (PDF, 45k) with additional commentary embedded to put back into context and make more explicit what Einstein was referring to and what the 1905 paper was actually in reference to.
A transcription of the German original is also included here: Zur Elektrodynamik bewegter Körper (PDF, 181k). Some of the explanatory footnotes contained in The Collected Papers of Albert Einstein will also be added later.
Energy and Mass
Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? (PDF, 47k)
This is the first of the E = mc² papers. In it, the equation was originally expressed in terms of m: m = (1/c²) E, consistent with the frame of mind purveyed by the supplements above. Time permitting, this will be translated in full.
The Electrodynamics of Moving Media (PDF, 220k)
This is a compilation of the following articles:
| Annalen der Physik 26 (1908), 532-540 |
| Annalen der Physik 26 (1908), 541-550 |
| Annalen der Physik 27 (1908), 232 |
| Annalen der Physik 28 (1909), 445-447 |
| Annalen der Physik 28 (1909), 885-888 |
A key insight behind this approach involves recovering Maxwell's original approach, which was to base the field dynamics on an isotropic medium. In a Galilean setting, this is a necessity since there is no way to get a finite wave speed for light since Galilean relativity does not permit any finite invariant velocities. Hence, a Galilean-based formulation of electrodynamics must be founded on an isotropic medium. That is: there is no vacuum in the Galilean-invariant formulation, if by “vacuum” one means a medium which is invariant with respect to change in velocity.
This means that in order to bridge the gap between the relativistic and non-relativistic theories, one must first broaden the scope of the relativistic theory to cover isotropic media. Once this is done, then one can effect a continuous transition from one to the other.
The distinguishing feature of isotropy is that the property only applies in a single frame of reference (unless the medium is also invariant under change of velocity — in which case we'd call it a “vacuum”) In the medium's frame of reference, the constitutive laws relating the force fields E and B with their conjugate fields D and H is given by
| D = εE, | B = μH. |
| D = ∂L/∂E, | H = -∂L/∂B. |
| I1 = E·E/2, | I2 = E·B, | I3 = B·B/2. |
| ε = ∂L/∂I1, | θ = ∂L/∂I2, | -1/μ = ∂L/∂I3. |
| D = εE + θB, | H = B/μ - θE. |
| ∂ε/∂I2 = ∂θ/∂I1, | ∂ε/∂I3 = ∂(-1/μ)/∂I1, | ∂θ/∂I3 = ∂(-1/μ)/∂I2. |
So, at this point, both the relativistic and non-relativistic relations are the same since we're in the stationary frame. The distinction between the relativistic and non-relativistic forms occurs only when transforming over to a moving frame. This first requires introducing Maxwell's “medium velocity” vector G. Then, excluding θ under a boost transformation, the relations become
| D + αG×H = ε(E + G×B), | B - αG×E = μ(H - G×D). |
A key feature of the analysis is that a distinction is drawn between the wave velocity V = √(1/με) and the invariant velocity c. Thus, one can actually examine the question of just how the vector G becomes “superfluous” in the limit α → με. This, then, leads to a HUGE oversight of Einstein-Laub.
That oversight is this: when they solve the equations, they only solve for (B and D). This solution has a branch point at G = V — the “Cherenkov limit”. Even then, they drew the wrong conclusion that the Cherenkov limit is ill-defined. The correct conclusion can be seen by solving for any of the other modes (B,E), (D,H) (which have branch points only at G = c), or (E,H) (which has a branch point only at G = c²/V). Each of these three modes yield results that pass through the Cherenkov limit without any problem. Significantly, the last mode yields a regular transition even over the “causal barrier” and permits one to consistently treat tachyonic fluids (G > c). Another way of looking at it is that the (E, H) solution yields a regular transition across the “light barrier”. Neither this, nor the regularity of the other two modes was (or is yet) known. In effect, it answers Einstein's question “what is it like to travel alongside a light beam?”; and with the (E, H) solution — “what it's like to overtake the light beam?!”
The second part of the Einstein-Laub treatment tried to grapple with the issue of the stress tensor. This was bungled up and is widely regarded nowadays in this way. Rather, Minkowski's 1908 treatment is given priority. However, both treatments suffer from the same problem as above — not proceeding from fundamental principles in general terms and not mixing in extraneous assumptions. The appropriate way to handle this issue is discussed in commentary added to the Einstein-Laub papers. A more detailed treatment of the stress tensor will also be provided in a further supplementary article in the near future.
Before Relativity
Pre-Einstein Special Relativity (Larmor's approach) (PDF, 61k)
Based on an article by M. N. Macrossan, this is a brief discussion of Larmor's approach, which foreshadowed Special Relativity. Much more will be said on this, and on Hertz's treatment (also) in a supplement to be added later below in the larger context of discussing the “Galilean limit” of electromagnetism.
Simplified Theory of Electrical and Optical Phenomena in Moving Systems, Lorentz (1899)
(PDF, 120k)
This is a transcription of the 1899 article by Lorentz, which (along with the 1895 article by Lorentz) shows a clear affinity to the notation Einstein employed in 1908 with Laub. So, it may be viewed as a point of reference for the later developments in 1908. There will be further commentary added on this and the 1895 and 1904 Lorentz articles in the future.
The Wigner Classification for Galilei/Poincaré/Euclid
(PDF, 217k)
Supplements:
| • |
The Wigner Classification for General 4-Space Signatures Originally slated for “Unitary Representation of the Poincaré Group” thread in the n-Category Café, this is an expansion both of Wigner's 1937 original paper and of the main article, in which the Archimedean signature is included alongside the other three signatures. The problem is recast as a problem in non-linear representation theory (as in the original article), with decomposition into “irreducible representations” replaced by decomposition of the Poisson-Lie manifold of the Unified Group into its symplectic leaves. All the important information is contained within the geometric structure of the individual symplectic leaves. This supplement is not yet complete. This will ultimately include a revised version of Wigner's 1937 original paper, substantially cleaned up & simplified. |
| • |
Poincaré Representations — n-Category Café The entire thread — up to the time of this reply — from the n-Category Café on this topic, translated into HTML. |
The Poincaré, Euclidean and Galilei symmetry groups for 4-dimensional spacetime can be consistently unified into a 1-parameter family of algebras. To do so, however, requires introducing an 11th parameter, M, identified as the relativistic mass; to regard the kinetic energy, H (rather than the total energy E) as the conjugate to time translations; and to generalize the mass shell constraint to that of only requiring the constancy of P2 - 2MH + αH2, where α is the family parameter. A 3rd Casimir invariant, M - αH also emerges as a result of the introduction of the 11th parameter.
The resulting classification yields, in addition to the tachyon, luxon and tardion sectors and the heretofore unnamed vacuon sector, an additional sector: the synchons (or: massless “action-at-a-distance” Galilei states). Furthermore, the generalization of the mass shell constraint makes it possible to define a consistent set of mass, momentum and energy relations for all sectors, including the tachyons and synchrons — thus resolving a long-standing problem.
With the generalized constraint available, it becomes possible to reinterpret the Schrödinger equation as a 2nd-order equation subject to the additional constraint provided by the 3rd invariant, rather than a 1st-order equation. It will become possible to generalize the Dirac-Kemmer, Klein-Gordon and Schrödinger equations across the board to all sectors and all values of the family parameter, α. Notably, this includes Galilean and tachyonic versions of all three equations.
| 1. Introduction | 1.1. The Unification of Galilei, Poincaré and Euclidean Symmetry |
|---|---|
| 1.2. The Generalized Mass Shell Condition | |
| 1.3. The Generalized Schrödinger and Dirac Equations | |
| 2. The Unified Group | 2.1. The Enveloping Algebra |
| 2.2. The Poisson-Lie Manifold | |
| 2.3. Quantization & Weil Operator Ordering | |
| 2.4. Symplectic Reduction & Invariants | |
| 2.5. Constraints & Constraint Propagation | |
| 3. The Translation-Invariant Sectors: Vacuons | 3.1. The Homogeneous Unified Group and Vacuon Energy |
| 3.2. Interpretation as Degenerate Gravitational Vacuua | |
| 3.3. The Vacuum | |
| 3.4. The Quasi-Vacuum | |
| 3.5. The Generic Vacuon | |
| 4. The Translation Non-Invariant Sectors | 4.1. Translation Invariants |
| 4.2. The Helical Sectors | |
| 4.3. The Non-Helical Sectors | |
| 4.4. Generalized Wigner Classification | |
| 4.5. Coordinatization of the Helical Sectors | |
| 4.6. Toward a Coordinatization of the Non-Helical Sectors | |
| 5. Coordinatization of the Translation Non-Invariant Sectors | |
| 6. Position-Space Representation | |
| 7. The Synchron Sector | |
| 8. The Spin/Helicity Subalgebra | |
| 9. The Mass/Energy/Momentum Relations | 9.1. The Synchron Sector |
| 9.2. The Luxon Sector | |
| 9.3. The Tachyon Sector | |
| 9.4. The Tardion Sectors |
The Newton-Wigner Position Operator
(PDF, 180k)
Based on [1,2], this is an reworking of the analysis carried out by T. F. Jordan; both to make it notationally consistent with Unification of Galilei, Poincaré, and Euclid (which contains a more general analysis that includes the cases discussed here as special cases), but also to redo it in a classical setting. The essential problem here (of defining a position operator for representation families of a space-time symmetry group) has nothing, per se, to do with quantum theory; and its inclusion only clouds the issue, by making it seem like it's a matter of quantum theory!
The quantum arguments in 1.4, however, have been largely left intact. But the analysis is already superseded by part 4 of the Unified Group article — which is entirely reduced to classical theory.
The first part, drawing from [1], derives the Newton-Wigner position operator for the “Tardion” sector. This has been substantially cleaned up. The author (by his own acknowledgement) complicated the picture by bringing in the Moses form of the various generators. It's not needed and can be subsumed; the notation is the same as in the Generalized Wigner article.
The second part, drawing from [2], shows the impossibility of a position operator for the “Luxon” sector — except for spin 0. Though not explicitly mentioned as such in Unification of Galilei, Poincaré, and Euclid, that was one of results of the analysis done there.
Stepping back, however, one sees that the conclusion is somewhat misleading. Basically, the assumptions made amount to forcing SO(3) as the little group. This is fine, as long as one as dealing with timelike worldlines. As discussed in the The Missing Heisenberg Relation article, the “mass shell” constraint for tardions is naturally paired off with the “classicalization” of the time-like coordinate. The Heisenberg relation [H,t] = 1, then reduces to [H,t] = 0. However, for the luxon sector, the worldlines are not timelike (and for tachyons or “synchrons” in the Galilei sector the worldlines are spacelike). Thus, what pairs off with the mass shell constraint should not be a classicalization of the time-like coordinate, but of a light-like coordinate (for luxons) or space-like coordinate (for tachyons and synchrons).
What happens with the Jordan analysis is that the conditions imposed on a prospective r vector are strong enough to define an SO(3) spin operator S, simply by taking the rotation generator J, translation generator P and subtracting off S = J - r × P. Once this happens, the deal is clinched, and we're stuck with SO(3) as the little-group — which specifically rules out luxons (unless they have S = 0) and tachyons. (The author makes brief mention of this issue, in passing, in a footnote; kept intact from the original).
The question, however, of how to generalize the analysis so as to include the other sectors is left unanswered. But a significant part of the answer is already contained in part 8 of the Generalized Wigner article. Here, the spin operator S for the tardion sector is given as:
| 1. The Tardion Sector | 1.1. Introduction |
|---|---|
| 1.2. Foldy Form | |
| 1.3. Moses Form | |
| 1.4. Uniqueness Proof | |
| 1.5. Non-Relativistic Limit | |
| 2. The Luxon Sector | 2.1. Introduction |
| 2.2. Generators | |
| 2.3. Fixed Helicity | |
| 2.4. Variable Helicity |
References
| [1] | T.F. Jordan, “Simple Derivation of the Newton-Wigner Position Operator”, Journal of Mathematical Physics 21 (8), 2028-2031. |
| [2] | T.F. Jordan, “Simple Proof of No Position Operator for Quanta with Zero Mass and Non-Zero Helicity”, Journal of Mathematical Physics 19 (6), 1382-1385. |
The Missing Heisenberg Relation
(PDF, 125k)
Where did the fourth Heisenberg relation go? This is a compilation of related articles originally posted from around the same time to sci.physics.research. Some of these may be separated out and placed elsewhere (e.g. section 4 moved over to the Definition of Mass article).
An important point made in one of the sections is to dispel the notion that the absence of a fourth relation is somehow a lingering after-effect of non-relativistic theory (particularly since the absence is of the energy-time relation). This is a red-herring in two ways: first, the reduction to 3 relations actually comes about because of the mass shell constraint, whose appearance is independent of whether one is dealing with relativistic or non-relativistic theory (or within any other signature; such as Euclidean or even “Platonic”). Second, the “synchron” sector in non-relativistic theory actually does involve a time-energy uncertainty relation!
Contents
1. Where Did the Fourth Heisenberg Relation Go?
2. Commutation Relations between J, r, p
3. Commutators in 4-D
4. Schrödinger and Klein-Gordon in both Galilean and Relativistic Case
Definition of Mass
(PDF, 68k)
It is a little-known fact that mass does not need to be postulated as a “undefined” primitive in the formalism of non-relativistic theory. It emerges naturally as a consequence of more basic consideration centering around Galilean group and its representations.
This article is derived from an article in sci.physics which directly addressed the challenge “I challenge anyone to define mass.” This account expands on Mass-Energy equivalence article.
Poincaré Representation for Arbitrary Spin: Dirac-Kemmer Equation
(PDF, 55k)
Performing an analysis analogous to Dirac's derivation of the Dirac equation,
the most general form for the matrices α, β, γ, δ, σ are determined such that
the relations
| Ea = αma + γp·b, | Eb = βmb + δpa + σp × b, |
| p2a + m2a = E2a, | p2b + m2b = E2b. |
Dirac with Chiral Gauge Fields
(PDF, 56k)
Expanding on an article in sci.physics.particle from 2008 July 28, the question was raised on whether the generators of a gauge group commute with the matrices (γμ: μ=0,1,2,3) of the Dirac algebra. In fact, this condition is precisely what distinguishes the feature of “chirality”. In fact, it is possible to restate the condition in invariant form in the language of differential forms in a way that completely separates out the question from the particulars of the Dirac field ψ or Dirac algebra.
The discussion here is generic to all 0-form fields coupled to gauge fields (which is covered more generally in The Gauge-Scalar Fields in Maxwell Form). The fields are represented in an involutive covering algebra. This also provides a natural venue for incorporating chiral fields (and, indeed, gives us a transparent account of what lies at the root of the matter with chirality). With respect to the algebraic and notational conventions adopted, the relevant condition is directly related to the involute A− of the gauge field A.
A Dirac Lagrangian Quadratic in the Velocities
(PDF, 62k)
Applying the same idea as in The Electric Field is the Curvature, not the Area, the Lagrangian for the Dirac field is made quadratic in the field velocities by adding in co-boundary terms. This results in a Lagrangian whose Legendre transform is non-singular. As a result, a unique de Donder-Weyl Hamiltonian can generally be defined. The form of the ordinary Hamiltonian is only briefly discussed at the end. This will need to be expanded on in a future addition.
As with the previous article Dirac with Chiral Gauge Fields, the discussion is actually generic to all 0-form fields coupled to gauge fields; and makes use of the same representation within an involutive covering algebra.
The Quantization of Dirac Fields as Gauge Fields
(PDF, 70k)
The spin 1/2 field is quantized by a method whose novel element consists in treating the Dirac spinor as a fermionic gauge field. A significant feature of the Lagrangian for the corresponding fermionic potential is that it splits down the middle into a “square root”.
The novelty is introduced by first observing that the Lagrangian for the classical Dirac field is completely singular. Its kinetic part is linear, the field equations are first order and under their action, the Lagrangian reduces to 0. When such a situation occurs, it is an indication that coordinates and velocities have been mixed together.
What we will find is that the spinor plays the analogous role for spin 1/2 that the electromagnetic field strengths play for spin 1. This is to be expected, since the electromagnetic field satisfies the Dirac-Kemmer equations, so that the converse should be true: the Dirac equation being reducible to a form similar to Maxwell's equations. This has been shown in a related article to be the case.
In the case of the electromagnetic field, though a Hamiltonian can be constructed from the field strengths alone (at least for the free field), what we eventually find is that there must be a deeper level: the potentials. This is particular so, when considering interactions.
Here, an analogous role is played by a second spinor field, which may be regarded as the fermionic potential. The only equation required for it is the Klein-Gordon equation and a Lagrangian can be posed which has it as its field equation. The Dirac equation is then automatically satisfied for the "field strength" spinor field, as a result. With the alternate formulation, the Lagrangian is no longer singular, though one now has an analogue of the gauge invariance seen in the Maxwell field.
The spin 1/2 Dirac field is, thus, treated here as a gauge field in its own right, the gauge potential simply being a spin 1/2 Klein-Gordon field.
Contents
1. Solution of the Dirac Equation
2. The Lagrangian and Hamiltonian Formulations
3. Quantization of the Fermionic Gauge Fields
The Dirac Equation in Maxwell Form
(PDF, 118k)
An apparent gap exists in the treatment of fields for different spins. Whereas the field equations for arbitrary spin may be written in a form that generalizes the Dirac equation to the Dirac-Kemmer equation, there seems to be no corresponding form, analogous to Maxwell's equations, for fields of odd half-integer spin. We will rectify that situation here, showing the results for spin 1/2 fields. Within this general setting, the Physics associated with spin 1/2 field will be worked out in detail, including a development of its invariance properties and conservation laws.
Contents
1. The Dirac-Maxwell Equations
2. Potentials and Gauge Invariance
3. Transformation Properties
4. The Multilinear Invariants and Charge/Current Conservation
31 Lessons in the Mathematics of Curved Space Geometry
(PDF, 142k)
The purpose of these "lessons" is to provide an introduction to the
mathematical background underlying modern geometrical theories of gravity,
with application specifically to the theory of General Relativity in mind.
Coverage includes material on the torsion, curvature, connection, Bianchi identities, metric, vector fields and derivations, tangent space, exterior differentials, differential forms, tensors, the Cartan structure equations and their generalization.
Background material is provided on manifolds; contractions, differentials and differential forms; connections. In addition, a novel element — the generalized index notation is also introduced here.
This article is slated for expansion to include material on the Hodge-de Rham operator, particularly featuring its close relation to the Ricci tensor; on the mathematical background concerning the geodesic deviation, the Raychaudhuri equation, the Weyl tensor and scale invariance; on the Einstein tensor and conservation law, and the closely related Carmeli coordinate gauge.
Basic Topics in the Mathematics of General Relativity
(PDF, 279k)
Based on, and closely following, the first two chapters of [1], this offers
substantial elaboration on and additions to the original material covering the
mathematical formalism of differential and Riemannian geometry.
In place of Chapter 3 of [1], a new concept of a Quasi-Galilean frame is developed. The notable features of this formalism are that:
(1) it is connection-based, suitable for Einstein-Cartan gravity
(2) it applies to both Lorentzian and Newton-Cartan spacetimes
(3) it is closely linked to the fluid dynamic interpretation and the Raychaudhuri equation
(4) it generalizes and contains, as a special case, the ADM decomposition
(5) it bears a close relation to the Ashtekar formalism, but eliminates the use of complex numbers.
This is being broken out into a separate article Quasi-Galiean Tetrads: An Alternative to Plebanski.
In a similar vein, the null-tetrad formalism is developed in essentially tensor-vector based form. The well-known spinor decompositions relating to various symmetric and anti-symmetric combinations of vector and tensor products are tied directly to the Klebsch-Gordon decompositions of SU(2) representations.
A significant novelty associated with 3+1 Quasi-Galilean tetrads has not yet been incorporated here, but is implicit in the Generalized Wigner article. Since the Galilei group requires 11 parameters, rather than 10; as does the enveloping generalization of Galilei and Poincaré (and Euclid), then a generalization of the Einstein-Cartan theory (which is normally cast as a gauge theory for the Poincaré group) must embody an 11th field component. This extra component corresponds to the generator for mass and may be regarded as a modern treatment of the old Weyl gauge symmetry. In turn, the extension to the 5-metric associated with the generalized mass shell invariant may lead to a further generalization that equivalently captures the Doubly Affine gauge formulation of gravity.
Reference:
[1] D. Kramer, H. Stephani, E. Herlt, M. MacCallum, E. Schmutzer, Exact Solutions of Einstein's Field Equations, (1980), Cambridge University Press.
The Development of Gravity in a Riemann-Cartan Geometry
(PDF, 350k)
This is a rewriting of parts of chapters 5 and 6 from Göckeler and Schücker's Differential Geometry, Gauge Theories and Gravity. The field kinematics and field dynamics for a gravitational dynamics in a Riemann-Cartan geometry are worked out in the most general form. Then these results are applied to derive an enveloping generalization of the pure field Lagrangian that includes Einstein-Cartan Lagrangian as a special case.
Part of this started out as a discussion with a colleague at UI-Chicago on the importance of language — a topic that had been brought up in my discussions with one of the grad students and his wife (who studied language) at a gathering in the summer of 2008 held by the UW-Milwaukee Physics department. Here, as I was pointing out, the difficulty in reading G & S lay primarily in the awkward language used — particularly in chapter 5. In turn, this is because the authors' native language is German and not English. One can actually see the German in the English, if one knows German. There is significant clarification to be had by expressing the same material in idiomatic English. That's partly what was done here.
On top of that, there was also a significant rewrite on everything else. This includes simplification & generalization, and an expansion on the original material. Solutions to some of the problems and proofs of some of the theorems, left untended, have also been added.
There are a couple sections added which carry out a blind derivation of both the Yang-Mills and (generalized) Einstein-Hilbert Lagrangian from first principles. The respective structures emerge from general assumptions. It's important to see these assumptions laid out explicitly, to see not just where everything comes from but where all the alternatives fit in.
The general form of all Lagrangians for gauge fields (not just “Yang-Mills” fields) and all gravitational fields, whose Lagrangians satisfy certain general properties is derived.
Part of the reason section 5.6 of the original is left out (the discussion on the stress tensor for the Maxwell-Lorentz Lagrangian) is that the authors' analysis specifically requires the Lagrangian to be homogeneos to the second degree in the field strength. The most general Lagrangian (that which comes out of the Noether current associated with the diffeomorphism group) yields a stress tensor which does not match the stress tensor associated with the Einstein-Hilbert Lagrangian.
The homogeneity assumption comes out of the assumption of scale invariance. In turn, the difference between the two stress tensors — for a general Lagrangian — amounts to the Noether current associated with dilation symmetry. Though the far-field shows the appearance of scale symmetry, it is wrong to conclude the field theory, itself, is also scale symmetric. This is because the far field reduces to the null field, and the null field is scale-symmetric regardless of whether the field theory and its Lagrangian are scale-symmetric or not.
As has been discussed elsewhere (e.g. Curing the Ultraviolet Divergence), a classical field theory with scale symmetry cannot be consistently quantized because it entails the self-energy and self-force divergences; nor is the classical theory, itself, consistent because of these issues. So, a more detailed discussion of section 5.6 will be deferred to another time.
Supplement:
Algebraic Invariants (PDF, 25k)
This is a list of the algebraic invariants formed frame 1-form, torsion 2-form and curvature 2-form, generalized to arbitrary dimensions. In addition, the GL(4) invariant combinations of these fields are listed.
Reference:
[1] M. Göckeler, T. Schücker, Differential Geometry, Gauge Theories and Gravity, (1987), Cambridge University Press
Diffeomorphism and Local Frame Covariance
(PDF, 43k)
This is a brief article that tries to make sense of the relation between a localized Poincaré group as a local symmetry on an affine bundle versus the diffeomorphism group. A bundle morphism on an affine bundle, each of whose fibres is a Minkowski space, yields a local affine transformation. Given a continuous choice of the “origin” on each fibre, this leads to the conversion between a diffeomorphism and a local “translation”. In this way, a semblance of translation operators may be recovered and directly related to diffeomorhisms.
Contents
Minkowski Space and the Poincaré Group
Diffeomorphism and Local Lorentz Group
Gravity in a Riemann-Cartan Setting
(PDF, 242k)
Derived, in part, from a presentation given at UW-Milwaukee.
This is an expansion of the short introductory treatise, by Trautman [1], on the Einstein-Cartan theory. Material from the Wikipedia has been added, as well as additional sections dealing with the teleparallel geometries, and the reduction of the field equations in 3-vector form (the Maxwell equations of gravity).
Some of this material (partiularly the 6-parameter model) is also covered in The Electric Field is the Curvature, not the Area
This article is undergoing serious revision; in part, because this was something I wrote, a long while back, as an early exercise to get acquainted with the field and introduce myself to key concepts. An on-line version of the talk, itself, will also be inluded.
| 1. Introduction | 1.1. Notation |
|---|---|
| 1.2. Historical Remarks | |
| 1.3. Physical Motivation | |
| 2. Geometric Preliminaries | 2.1. Tensor-Valued Differential Forms |
| 2.2. Hodge Duals | |
| 2.3. Linear Connection, its Curvature and Torsion | |
| 2.4. Metric-Affine Geometry | |
| 2.5. Riemann-Cartan Geometry | |
| 3. The Einstein-Cartan Theory of Gravitation | 3.1. An Equation of State Resulting from Local Invariance |
| 3.2. Projective Transformations and Metric Connections | |
| 3.3. The Sciama-Kibble Field Equations | |
| 3.4. The Bianchi Identities and Conservation Laws | |
| 3.5. Spinning FLuid and Generalied Mathisson-Papapetrou Equation of Motion | |
| 3.6. From Einstein-Cartan to Riemann: the Effetive Energy-Momentum Tensor | |
| 3.7. Cosmology with Spin and Torsion | |
| 3.8. An Effective Einstein-Cartan Theory in General Relativity | |
| 4. Summary | |
| A. The Einstein and Spin Tensors as Differential Forms | |
| B. Generalized Dynamics for Metric-Affine Theories | B.1. Kinematic and Dynamic Laws |
| B.2. Local Lorentz Invariance | |
| C. A 6 Parameter Dynamics | |
| D. Ashtekar, New Variables and Immirzi | |
| E. Teleparallel Gravity | |
| F. 3-Vector Formulation: Key Equations and Notation | |
| G. Gravity in 2+1 Dimensions | G.1. Kinematics |
| G.2. Lorentz Variational | |
| G.3. Dynamis | |
| G.4. Constitutive Model | |
| G.5. Chern-Simons Form | |
| H. Four Dimensional Formulation | H.1. General Formulation |
| H.2. Constitutive Model | |
| H.3. First Order Model | |
| H.4. Complexification | |
| H.5. Quaternification |
Non-Symmetric Stress Tensors & the Localization of Angular Momentum
(PDF, 76k)
This is a discussion, originating from sci.physics.research, on the meaning of non-symmetric stress tensors and relation to angular momentum conservation.
The key point raised is that the first-order moments of energy-momentum currents in flat-spacetime can be rewritten in local form and pulled back to curvilinear coordinates and curved spacetimes by exploiting the conservation law for momentum.
In particular, if Pa represents the current 3-form for energy & momentum (with “a” representing an index referred to a basis of local frame fields), then using its conservation law DPa = 0, one may reduce the first moments involving coordinates xa Pb. First, one derives a “superpotential” pa for Pa, such that dpa = Pa. Then, one “borrows” a “d” from P and places it on x:
Though the first moments can only be written in flat-space Cartesian form, because of the explicit appearance of the local frame index on the coordinates xa, the revised form is generic to curvilinear coordinates. By replacing “d” with the covariant derivative operator “D”, one generalizes further to curved spacetimes, thus introducing the connection 1-forms ωab and their associated curvature 2-forms Ωab.
The conservation of angular momentum dJab = 0 may then be stated in local form, upon making the conversion
From this, an important observation emerges: if the relevant conservation laws (DPa = 0, DJab = 0) are the identities derived from the “potential” relations (Dpa = Pa, Dsab = Jab); and these, in turn, are Euler-Lagrange equations of a variational principle, then the Lagrangian involved will be a function L(θ, Θ, ω, Ω) of the frame 1-forms θ, connection 1-forms ω, curvature 2-forms Ω, and ... the torsion 2-forms Θa = Dθa. In particular, one finds the following variational of the Lagrangian 4-form L (up to constant multiple):
Comment on Teleparallel Gravity and Metric Affine Geometries
(PDF, 86k)
Clarification of all matters related to Metric Affine, Einstein-Cartan, Teleparallel and Riemannian geometry. Brief review of the various dynamics for gravity that can be posed on each ... and their equivalences with one another.
The Development of the Newman-Penrose Formalism
(PDF, 281k)
Outline of the Newman-Penrose formalism, developed in invariant algebraic form suitable for generalization to Riemann-Cartan spacetimes. The original papers by Newman and Penrose have been grafted into this newer treatment.
Supplement:
Null tetrads and spinor notation for differential geometry (PDF, 191k)
This expands on the treatment of the Newman-Penrose equations; particularly, filling in the details of the set of equations overlooked by Newman-Penrose — the compatibility relations.
Desperately Seeking Spinors
(PDF, 54k)
Derived from a sci.physics.research article, under the same title, from 2007 May 26.
This develops the spinor algebra in algebraic form in the language of invariants. The well-known reduction of 2-forms to spinor form is easily captured within this setting and its relation to the Klebsch-Gordon decomposition 2 × 2 = 3 + 1 becomes particularly clear. There is also a spinor decomposition for symmetric tensor products of vectors, and other combinations from vector and tensor analysis that are not well-known. The reduction for symmetric tensor products is worked out, as well.
This material has, effectively, been incorporated into the Penrose-Newman article.
Frames and Lorentz Invariance in General Relativity
(PDF, 247k); revised form (PDF, 277k)
These are unpublished notes, distributed in 1961 April 19 as a technical report at Lawrence Radiation Laboratory by a long-time associate, Elihu Lubkin; who is known as one of the pioneers in the development of the geometric approach to gauge theory.
Though written before the publicaton of the Penrose-Newman formalism, this contains all the key elements of the latter (and more). An explicit tie-in to the Penrose-Newman formalism is made in the revision. The notation has also been adapted for use in the Penrose-Newman article.
The approach adopted in the Lubkin paper is to essentially treat gravitational dynamics in a Riemann-Cartan setting. In this geometry, the metric can be eliminated in favor of a connection that, itself, becomes the gauge field for the Lorentz group.
Here, the process is taken on step further, with the representation in terms of the Lorentz group pulled back, via a reduction to spinor form, to a gauge field for the double-covering SL(2,C). The key result proven is to show that the most general connection respecting the soldering form
This leads to a more refined classification of geometries along the following lines
| Metric-Affine | — | Metric & connection independent, metric has a non-zero covariant derivative |
|---|---|---|
| “Spinor-Affine” | — | Metric & connection independent, σ is covariantly constant |
| “Causal” | — | Metric and σ covariantly constant, but not ε or ε* |
| Riemann-Cartan | — | Metric, σ, ε and ε* all covariantly constant. Connection reduces to gauge potential for the Lorentz group (and is anti-symmetric in its two frame indices) |
| Riemann | — | Riemann-Cartan with zero torsion, but generally non-zero curvature |
| Teleparallel | — | Riemann-Cartan with zero curvature, but generally non-zero torsion |
| Flat | — | Riemann-Cartan with zero curvature and zero torsion |
| I. Introduction | |
|---|---|
| II. Skew Frames, Inertial Frames, and Parallel Displacement | 1. Skew Frames |
| 2. Frame Transformations | |
| 3. Parallel Displacements and Inertial Frames | |
| 4. Lorentz Transformations and Inertial Frames | |
| 5. Antisymmetric Connections | |
| 6. Christoffel Symbols | |
| 7. Curvature and Torsion | |
| III. Spin 1/2 | 1. Spinor Algebra |
| 2. The Parallel Displacement and Covariant Derivative of a Spinor | |
| 3. Relation to the Parallel Displacement of a Vector | |
| 4. Dirac Equations | |
| Appendices | 1. Non-Invariance of a Quantity in the Christoffel Symbol Derivation |
| 2. Form of Connections Compatible with a Spinor Connection | |
| 3. Existence of a Spinor Connection Compatible with an Antisymmetric Connection | |
| 4. The Vector Nature of the Extra Degree of Freedom |
Revised Version
This dates from 2007 November 26. In it, I've also added material on the Penrose-Newman coefficients, updated some of the notation from the original and added a section dealing with fixed non-orthogonal frames (e.g. the null tetrad). The Hybrid Index Notation is introduced and used in parts of the revision to make the presentation cleaner, more general and efficient.
| 1. Introduction | |
|---|---|
| 2. Skew Frames, Inertial Frames, and Parallel Displacement | 2.1. Skew Frames |
| 2.2. Frame Transformations | 2.2.1. Hybrid Index Notation |
| 2.3. Parallel Displacements and Inertial Frames | |
| 2.4. Lorentz Transformations and Inertial Frames | |
| 2.5. Antisymmetric Connections | 2.5.1. Fixed Non-Orthogonal Frames |
| 2.6. Christoffel Symbols | |
| 2.7. Curvature and Torsion | |
| 3. Spin 1/2 | 3.1. Spinor Algebra |
| 3.2. The Parallel Displacement and Covariant Derivative of a Spinor | |
| 3.3. Relation to the Parallel Displacement of a Vector | 3.3.1. Form of Connections Compatible with a Spinor Connection 3.3.2. Existence of a Spinor Connection Compatible with an Antisymmetric Connection 3.3.3. The Vector Nature of the Extra Degree of Freedom |
| 3.4. Dirac Equations | |
| 3.5. Penrose-Newman Coefficients |
The Kerr Solution in Tetrad Form
(PDF, 39k)
As the title indicates, this expresses the Kerr(-Newman) solution in tetrad form. What this involves is the extraction of the principal components of the metric (defining the members of the co-frame, or tetrad) and the reduction of the metric to a sum and difference of the (tensor product-)squares of the 4 vectors in the co-frame.
The Electric Field is the Curvature, not the Area
(PDF, 121k)
Maxwell Equations of Gravity: A Yang-Mills Replacement for Ashtekar's New Variables
There are 6 polynomial combinations of 4-forms that can be formed from the frame 1-forms θa, the torsion 2-forms Θa and curvature 2-forms:
| Iα = εabcd Eab ^ Ωcd, | Iβ = Eab ^ Ωab, | Iγ = εabcd Eab ^ Ecd, |
| Iλ = εabcd Ωab ^ Ωcd, | Iμ = Ωab ^ Ωab, | Iν = Θa ^ Θa, |
It's at this point that we confront a mistake that is all-too-common. That mistake is to throw out all the terms that can be expressed as total differentials. The reason this is done is that no contribution from these terms enter into the field law. Here, this would be the terms
| Iλ, | Iμ, | Iβ - Iν |
The mistake is that this definition of “equivalence” does not lead to equivalent Hamiltonian formulations! In particular, some formulations will lead to de Donder-Weyl Hamiltonians whose Legendre transforms are singular, while others will not. Here, a singular Legendre transform occurs precisely when λ, μ are both 0, and when ν is 0.
If we relax these conditions, then the Legendre transform becomes non-singular and the conjugate to the curvature 2-form Ωab will be a combination of the curvature 2-form, itself, the “area” 2-form (and their duals). Thus, the “electric field” (which, by the way, should be “electric displacement” (D), not the electric field E) is not the area, but the curvature, offset by a contribution from the area.
Contents
1. Kinematics
2. A Retake on Ashtekar
3. Dynamics
4. Lorentz Invariance
5. Constitutive Laws
6. Field Laws
7. Field Equations in 3-Vector Form
8. First Order Dynamics
9. Conclusion
The Einstein-Hilbert Action in Quadratic Form
(PDF, 34k)
This is basically an extension of the The Electric Field is the Curvature, not the Area article. Here, the Einstein-Hilbert Lagrangian, combined with a non-zero cosmological term, can be written in quadratic form (up to a co-boundary term) as either
The calculations for this result are worked out.
The Differential Geometry of Gauge Theory
(PDF, 529k)
This material outlines the mathematics of differential geometry and manifold theory; based on, and closely
following section 0.2 of [1], but with substantial rewriting and additions: the Hodge deRham operator, material
on topology, a complete reformulation of the theorem on total differentials, on the generalization
of tangent vectors to tangent surface elements, the relation of the Hodge deRham operator and Ricci tensors,
a more complete working out of the example on Maxwell's equations and the expansion of the example to Yang-Mills
fields, etc.
Additional material is provided that completely reworks the theory of Lie groups in the new notation that has been used elsewhere in this archive. The additional sections corresponding to Chapter 1 is a new addition (e.g., an explicit working out of the “Maxwell equations” for the classical U(2) electroweak field) and Chapter 1 has been completely redone.
Reference:
[1] D. Bleecker, Gauge Theory and Variational Principles, (1979)
Gauge Fields as n-Forms and Duality
(PDF, 124k)
The notion that a gauge field may present itself in different ways to different types of sources was mentioned almost in passing by Zee in his 2003 Quantum Field Theory in a Nutshell book, along with the related notion of gauge fields as n-forms. This centers on the idea of generalzing currents to sources that are not just concentrated along worldlines, but also currents concentrated on subsurfaces of other dimensions (even 0, if one equates a 0-brane with an "event").
For electromagnetism, the 0-form variant of the gauge field is a scalar field. The 2-form variant is the dual magnetic field, where the Maxwell D and H fields are now the components of the (2-form) current source. The sources are generated from “superpotentials” which may be thought of as the “magnetic potentials” that generate D and H.
E and B fields components of the (2-form) potential; while the “magnetic currents” are the components of the field strength derived from this potential. This puts into perspective the fact that the same units for electric potential (Volts) are those of magnetic current (Webers/second), while those for electric current (Amps) are those of magnetic potential.
Gravity as a Gauge Theory on Doubly Affine GL(n)
(PDF, 100k)
An inhomogeneous extension of the general linear group may be defined in which the generators show a natural affinity to those of the conformal group, as well as to the Heisenberg algebra. The corresponding gauge potentials not only include fields in a non-metrical background that play the role of the connection and frame, but also a field that plays the role of the dual frame. Moreover, despite the absence of a metric at the outset, it is possible to construct one from the potentials in such a way that its symmetry and signature are no longer constraints posed at the outset. This enables us to also remove the constraints on the metric from the backgroud and to treat both the signature and symmetry as part of the dyamanics.
The resulting formalism for the gauge theory seems to simultaneously embody Weyl's old gauge theory, Einstein's old unified field theories for non-symmetric metrics and connections. In virtue of the relation of the gauge group to GL(5), it also embodies a variant of the Kaluza-Klein representation of electromagnetism. The resulting correspondence may put into context the older unified field theory formalisms, while breathing new life into each by bringing them all together.
This is basically the treatment of gravity formulated atop an SL(5) gauge theory.
Curved Spacetime Geometry with Torsion
(PDF, 72k)
This is a followup on the 31 lessons writeup, and the article by Arcos, Andrade and Pereira.
The structure equations and Bianchi identities are derived for the general case of a spacetime with a torsional connection. The issue is raised that the correct forms for the geodesic and field laws, notwithstanding the appearance of torsion, continue to remain in reference to the Levi-Civita connection given by the metric. Nonetheless, it is possible to express the Levi-Civita connection as a function of the connection native to the manifold and the torsion. From this, an intrinsic form for the geodesic equation and field equations can be written.
A notable feature is that this form is invariant with respect to a large family of transformations of the connection and torsion that include, within it, transformations to spacetimes of either zero curvature or zero torsion.
A Hybrid Mathematician's/Physicist's Notation
(PDF, 72k)
A new extension of the Einstein convention and tensor index notation is developed here which bridges the gap between the Mathematician's and Physicist's languages. It is illustrated, by writing out some of the key definitions, equations and identities encountered in differential geometry.
Mass as Gravitational Charge and Derivation of Geodesic Equation of Motion
(PDF, 85k)
The geodesic equation and, more generally, the “harmonic law” for n-branes for n > 1, can be derived, within the framework of distribution theory, as specializations of the conservation law for the stress tensor.
Torsion and Curvature: Poincaré Gauge Theory — Arcos, Andrade and Pereira;
(PDF, 102k)
Derived from Torsion and Gravitation: A New View. Arcos, Andrade, Pereira; with minor edits to smoothe out the language.
Since the time of Cartan, it has been known that a connection can be redefined in such a way as to transform a manifold with non-zero curvature or torsion to or from one which has either zero torsion or zero curvature. The former case leads to General Relativity, the latter to its purely torsional equivalent: teleparallel gravity, where torsion plays the role of a Lorentz force. For a spin 0 particle, the right-hand side of the equation of motion, normally 0, is non-zero in the teleparallel equivalent, which means the minimal coupling only occurs in the torsion-free case.
Here, a different coupling prescription is proposed which transcends the equvialence, invariant with respect to redefinition of the connection, and reducing to the usual torsion-free case when the connection is transformed to torsion-free form. According to this view, no new physics is connected with torsion. Instead, the torsion appears as a mere alternative to curvature in the description of gravitation. The formulation is then applied to the task of generating the equations of motion both for a spin 0 particle and a particle of non-zero spin.
Contents
1. Introduction
2. Gauge Potentials and Field Strengths
3. Gravitational Coupling Prescription
4. Spin 0 Particles
5. Particles with Spin
6. Conclusions
The Geometry of Lagrangian Dynamics
(PDF, 117k)
A response to a 2007 June sci.physics.research article (which, itself, is expanded on), this puts into detailed comparison the mechanics-based formulation of Lagrangian theory vs. the field-based formulation. The former treats configuration space Q as a fixed manifold with time evolution external to the actual formalism, with the central focus being the first and second order tangent spaces TQ and TTQ. The latter internalizes time evolution, generalizing to accommodate evolution in more general spaces. This treats configuration space Y as a bundle over a base space manifold M. The space Q, here, is the typical fibre. The fibration (Q_x: x in M) internally encapsulates the notion of "time-dependence" (if x = (t) and M = R1) or (more generally) "spacetime dependence" (if dim M > 1). Its central focus is the first and second jets J1Y and J2Y of Y.
Legendre Transformation
(PDF, 118k)
Based on an article by Christopher Hillman originally posted to sci.physics and sci.math in 1997, this significantly expands on Hillman's reply, as well as rewriting the reply, itself, to clarify it. As part of the expanded treatment, the issue of field theory is also addressed.
In a Lagrangian formulation of dynamics for a given system, there are 4 types of quantities that make their appearance
| • | The configuration coordinates | (qa: a = 0, 1, ..., N-1) |
| • | The configuration velocities | (va: a = 0, 1, ..., N-1) |
| • | The conjugate momenta | (pa: a = 0, 1, ..., N-1) |
| • | The conjugate forces | (fa: a = 0, 1, ..., N-1) |
The qa, va, together, comprise the kinematics of the system and are governed by a kinematic law, va = dqa/dt. The pa, fa, comprise the dynamics, and are governed by a dynamic law, fa = dpa/dt. This is the universal form which any system must possess, when presented in terms of these variables, if its dynamics are governed by a Lagrangian.
The features specific to a given system, which distinguish it from the dynamics of other systems possessing these variables, are governed by the constitutive relations — which determine how the dynamic variables are to be related to the kinematic variables. For a Lagrangian system, the constitutive relations are given by a single generating function: the Lagrangian L = L(q, v, t); and take on the form
| pa = ∂L/∂va, | fa = ∂L/∂qa. |
Other generating functions may be devised to encapsulate the same dynamics. The key feature is captured by the differential of the Lagrangian
The most important case in point replaces (q,v) by (q,p),
| ∂H/∂pa = va, | ∂H/∂qa = -fa |
The Kepler problem is used as an illustration. In addition, these concepts are generalized to field theory and the scalar field is used as an illustration.
Why the Hamiltonian and Lagrangian Formulations Look Like They Do
(PDF, 52k)
This article places the T-V and T+V forms, respectively, of the Lagrangian and Hamiltonian in their proper context, where T is the kinetic energy and V the potential energy.
An important point is made that neither of these forms is to be considered fundamental. Instead, the Lagrangian is closely associated with the differential form p·δq (called the canonical 1-form), where q represents the vector of configuration coordinates, and p the conjugate momenta. It is the action, S, rather than the Lagrangian that assumes the fundamental role. The central principle is that the action corresponding to the actual evolution of a system over an interval of time [t-, t+] will depend only on the values of q and p at the bounding times t- and t+, such that the differential of the action shall be the difference of the canonical 1-forms at the respective times, δS = p+·δq+ - p-·δq-.
For a small time difference t+ - t- = dt, the differences of the boundary values are related to the velocity v and force F by the force law p+ - p- = Fdt and the law of motion q+ - q- = vdt; and the action to the Lagrangian by δS = δL dt. From this ultimately arises the differential identity for the Lagrangian δL = F·δq + p·δv, from which the integral L = T - V emerges.
One of the more important points to come out of this is that, in fact, the Hamiltonian will generally not be of the form H = T + V. In relativity, H = TH + V > T + V. The difference of the “Hamiltonian” kinetic energy TH from the “Lagrangian” kinetic energy, T, is precisely the (Lagrangian) kinetic energy that comes from the mass equivalent of the kinetic energy, itself! In fact, if this relation is assumed at the outset, the mass-energy-momentum relations may be derived — not only for massive particles, but massless ones, as well.
Poisson Bracket
(PDF, 57k)
Supplements:
Poisson Algebras
Poisson Manifolds
Symplectic Manifolds
This is a short article, describing the Poisson Bracket in greater depth than the Wikipedia article it is based on. The exact solution of the Kepler problem is used as an illustration of the constants of motion section.
Contents
1. Canonical Coordinates
2. Equations of Motion
3. Constants of Motion
4. Definition
5. Lie Algebra
Rovelli's Polycanonical Formalism
(PDF, 53k)
The elements of the polycanonical approach to field theory adopted by Rovelli and discussed in his Quantum Gravity treatise are described here. They are applied to the general scalar field — that is, the most general scalar field whose dynamics are given by a Lagrangian. This is in contrast to Rovelli who (in his Quantum Gravity treatise) only deals with the scalar field whose dynamics are quadratic in the field velocities.
Contents
1. The Formulation
2. Scalar Field
Noether's Theorem, Space-Time Boundaries and Horizons
(PDF, 93k)
Derived partly from a sci.physics.research article in 2008 June 9, this discusses an important and often-neglected issue, as well as an important consequence that follows from it — the dependency of the “conserved charge” on the “horizon”.
In rigorous form the Noether theorem may be formulated for compact regions of space-time. The most important case is where the region is foliated into a series of layers St parametrized by t, where each region shares a common boundary ∂St = H (the “Horizon”). The conserved quantity of the Noether theorem is actually an integral J(t) of a “current” 3-form over the 3-surface St. The key point of the theorem is that J(t) is independent of t, that is: it is “conserved”.
The quantity, however, is not actually a constant — it has a dependence on the horizon H. For local symmetries — symmetries that leave points fixed — the issue of the horizon H does not come into play. One may unambiguously define the conserved quantity and ignore the horizon dependence. For more general symmetries, however, the situation is different.
In particular, for diffeomorphisms, since points are moved, then the boundary changes. In the corresponding quantized theory, this introduces an entanglement and associated entanglement entropy. Thus, the symmetry transformation will acquire an anomalous contribution directly linked to the movement of the boundary. In effect, this creates a breaking of diffeomorphism symmetry.
Contents
1. Generalized Noether Theorem
2. Conservation and Horizon-Dependence
3. Diffeomorphism Non-Conservation
4. A Fundamental Classico-Quantum Law of Gravity
5. Signature and Signature Change
Second Order Laws and Space-Time Boundaries
(PDF, 30k)
Derived from a sci.physics article from 2008 April 10, this addresses the question of why the laws of nature are second order laws.
The general principle used here is that the unfolding of a system occupying a region of space-time should generally only depend on the configuration of the system on its space-time boundary. In mechanics, systems are presented as evolving in time, so that the regions comprise a contiguous series of “equal-time” slices, while the boundary is marked by the extremal times. If one assumes that the configuration of the system at all times between is a function of the boundary values, then it follows that the unfolding of the system's configuration in time is described by a second order differential equation.
Quantizing the Variational Principle
(PDF, 39k)
If the operator correspondence h ∂/∂qa ↔ 2πi pa is substituted back into the definition of the conjugate momenta pa = ∂L/∂qa, for a system whose dynamics are given by a Lagrangian L, the result is an interesting operator form for the variational of the action principle, itself:
This correspondence is illustrated on simple harmonic oscillator and uniformly accelerating system.
The Lagrangian Method
(PDF, 66k)
An interesting numerical method for dynamics foregoes the differential equations of motion working, instead, directly with the action. Since the object of a numerical evolution is to find a sequence of points that approximates the motion of the system, the question comes down to formulating the least action principle with repsect to a path defined piecewise by a given interpolation. Assuming the interpolation is fixed, this reduces to an optimization problem over the point sequence, itself. Solving this yields a series of iteration equations which can then be used to numerically evolve the system. Since the interpolation is fixed, this produces a suboptimal solution, but one optimal with respect to the constraint.
The process is illustrated with application to the simple harmonic oscillator and the Kepler problem. A notable feature in the latter application is that the iteration involves logarithms, rather than polynomials!
Contents
1. Formulation
2. Examples
2.1. Simple Harmonic Oscillator
2.2. The Kepler Problem
Least Action and Kinetic Energy
(PDF, 66k)
Originally derived from an article in sci.physics.research.
Is there a "proof" or idea to explain how the least action principle arose? This provides a historical snapshot of the formalism, as seen in the 19th century. The treatment of the Hamiltonian and Lagrangian approach provided by Maxwell in his treatise on electromagnetism is reviewed.
No Lagrangian? No Quantization! — Hojman and Shepley;
(PDF, 131k)
Generalizing on the work by Feynman and Dyson (American Journal of Physics 58 209-211 (1990)), Hojman and Shepley show that the result of imposing general conditions on the commutation relations governing a system is a set of constraints on the system's equations of motion that imply the existence of a Lagrangian (in the classical limit). This, and the Bandyopadhyay article are two in a series of articles that expand on the Feynman-Dyson idea. There are others that have appeared before and since.
Based on the original by Hojman and Shepley, the notation has been revised to make it consistent with the Quantum Dynamics article. A few mistakes in the original have been corrected, the commutators replaced by "quantum Poisson brackets", and the underlying notions of operator ordering and the classical limit have both been developed here in detail.
The examples provided by Hojman and Shepley are reworked. In particular, the example of a system that "does not come from a Lagrangian and therefore cannot be quantized consistently" is shown to be an essentially hybrid classico-quantum system — thereby tying into one of the central elements of the Quantum Dynamics article.
The root of the matter, ultimately, is that the dynamics for many-body systems are more heavily constrained and can only take on certain familiar forms. This is closely linked to the No-Interaction Theorem.
Contents
1. Introduction
2. Quantization Requires Lagrangian
3. First-Order Systems
4. Examples
5. Conclusions
Commutator Relations of Test-Charges in Einstein-Maxwell Fields — Bandyopadhyay;
(PDF, 108k)
Expanding on the Hojman & Shepley result, this article investigates the special case of a one-particle system corresponding to a test charge. Assuming the equal-time position-velocity commutators are c-numbers and identifying them (up to proportion) with the metric of an underlying spacetime manifold; and making general assumptions concerning the nature of the momentum-momentum equal-time commutators; one may show that the most general interaction the particle can be subject to is a combination of an electromagnetic and scalar field.
Based on the original (Unified Equation of Motion of a Test Charge in Electromagnetic and Gravitational Fields) by Bandyopadhyay, the notation has been revised to make it more consistent with the Quantum Dynamics article. The original article purports to pull electromagnetic theory out of the hat. This needs to be revised (as the authors implicitly acknowledged in their concluding remarks).
In this version, the needed corrections have been made. What had been posed as definitions in equations (2.7) and (2.8) of the original, in the context of their intended use, are actually assumptions. What had been posed as a definition in (2.9) is actually an assumption concerning the nature of the interaction — an assumption that excludes general Yang-Mills forces and which could have been generalized in this direction. The concluding remarks in the original about the generalization to Yang-Mills, therefore, had to be redone.
The assumption made by equation (3.5) of the original cannot be freely made, but is actually determined by (2.7) or (4.1). The most general form of (3.5) is not merely a symmetric ordering, but involves a residual term. This changes the character of the arguments following (3.5). This brings out the scalar field, which the authors missed.
More generally, the issue of "operator ordering" or quantiation, as applied here and in the various contexts in this article, had to be significantly clarified and is what leads to these and other refinements. Therefore, this article is also an exercise in the application of the formalism surrounding the notion of classical limits and operator ordering — i.e., of what is known as "quantization theory".
Contents
1. Introduction
2. Field Equations of Gravitation and Electromagnetism
3. Equation of Motion of the Test Particle
4. Remarks and Discussion
On the Quantum Dynamics of Moving Bodies
(PDF, 221k)
Between a 2nd order law of motion, q''(t) = a(q(t), q'(t)) and equal-time commutator relations [q(t), q(t)] = 0, for a system described by configuration coordinates q(t) = (q1(t), q2(t), ...), exist compatibility constraints similar to the Helmholz conditions which, in classical dynamics, determine the existence of a Lagrangian and Hamiltonian. Out of this arises an ab initio construction of a significant portion of the formal structure of an enveloping generalization of classical and quantum theory.
The consequences of the connection to the Helmholz conditions have been previously investigated by Hojman and Shepley, who demonstrated that a quantum system will have a Hamiltonian dynamics in its classical limit. In contrast, our attention is focused on the system, itself, independent of any question concerning a classical limit. To this end, we will restrict our attention to the special case where the [q, q'] equal-time commutators are c-numbers.
For closed systems with a finite number of degrees of freedom, this implies the reduction to the combination of a classical subsystem and a quantum subsystem, the latter being canonically quantized with respect to a Hamiltonian quadratic in the conjugate momenta and evolving in superselection sectors provided by the former. This includes, as limiting cases, purely classical or purely quantum systems.
In contrast to the conclusions, drawn by Feynman and (before him) Dyson, who purported to pull out the formal structure of electromagnetic theory from the consideration of the commutator algebra, here the Lorentz law and force are placed in their proper context. The "Lorentz force" resides in configuration space, not in ordinary space. The confusion with electromagnetism occurs when one restricts attention to a one-particle system (as Feynman and Dyson did), where the configuration space is three-dimensional. In fact, it is best seen as a feature generic to Hamiltonians that are quadratic functions of their momenta and is directly linked, as a special case, to the variational bicomplex that figures centrally in the more general context of the inverse Lagrangian problem.
Contents
1. Introduction
2. Compatibility Conditions
3. Invariance with Respect to Linear Transformations
4. Reduction of Closed Canoical Sectors
5. The Classical Sector
6. Generalization to a General Relativistic Context
7. Many Body Systems and Yang-Mills-Higgs Interactions
8. Toward a Generalization to Field Theory
9. Concluding Remarks
A. Appendix: The Helmholz Conditions and the Bivariational Complex
B. Appendix: Technical Assumptions
The No-Interaction Theorem for Relativistic Dynamics
(PDF, 154k)
This is also known as Leutwyler's Theorem. Based on [1], by Marmo, Mukunda and Sudarshan, this article is closely related to the No Lagrangian? No Quantization! and Helmholz Conditions articles. The theorem rules out the existence of non-trivial interactions in relativistic dynamics between systems that are described by a position vector r, velocity v; if the transformation of r and v under an infinitesimal boost υ is given by
| δr = αυ·r v, | δv = -υ + αυ·v v + αυ·r A, |
However, there are several problems with both this and the landmark result. First, the formalism is narrowed only to time-independent Lagrangian/Hamiltonian dynamics. This becomes a problem for α≠0 because the time-like coordinate transforms under boost δt = -αυ·r. A “simultaneous” interaction, following the classification given in the Unfied Group article, corresponds to a instantaneous transmission of impulse over distance: a “Synchron”. In a prospective relativistic generalization “instantaneous” is replaced by “instantaneous in at least one frame” and a transfer is seen in other frames of reference as if it were a faster-than-light transfer of impulse. This is equivalent to what's called a “Tachyon”. An explicit time-dependency then seems imperative to handle non-instantaneous transfers of impulse.
The second and more fundamental problem is that none of the analysis is properly motivated from fundamentals; and this point cuts across the paradigm divide and applies to non-relativistic dynamics, as well. The fundamentals are to treat the “bodies” as elementary systems. Instead of visualizing these as somehow being tiny ball-like objects, they are treated as nothing more than systems whose state spaces are acted on irreducibly by the space-time symmetry group — that is, as “irreducible representations”. No further hypothesis ever need be stipulated of what an elementary system “actually is” because all of its physical characteristic is locked up in the classification of the irreducible sectors.
One should then derive a position operator r and derive the transformation behavior for the system. It turns out, following the notation and results of the Generalized Wigner, Position Operator and Unified Group articles, that for the tardion sector, the rotation generator J, boost generator K, spatial translation generator P and time translation generator H decompose into
| J = Mr×v + S, | K = Mr + αP×S/(m + M), | P = Mv, | H = P²/(m + M) + U |
| δr = αυ·r v - α/(m+M) (υ×S - αM/(m+M) υ·S×v v), | δv = -υ + αυ·v v, | δS = α/(m+M) (υ×P)×S. |
This is suitable for the general non-interacting elementary system. With interaction, the total system is no longer additive, so that (in effect) interaction terms are posed not just for the H generator, but also P and possibly the other generators. The revised analysis may be included, in the future, as a further rewrite to this article, a supplementary section or a supplement.
| [1] | G. Marmo, N. Mukunda, E.C.G. Sudarshan, “Relativistic Particle Dynamics — Lagrangian Proof of the No-Interaction Theorem” |
| [2] | H. Leutwyler, Nuovo Cimento 37, 556 (1965). |
The Helmholz Conditions and Field Equations
(PDF, 79k)
An important subclass of field equations are those derivable from an action principle through a Lagrangian density. The question posed here is: assuming the field equations are given first, what conditions will ensure that such a Lagrangian formulation actually underlies the system?
The system is posed in the form va = dqa, and mab*d*vb = Qa, where Qa and qa have the same dimension. The issue, therefore, comes down to the question of what conditions need to be posed on the coefficients of inertia, mab, and on the generalized force, Qa, so that the system comprises the Euler-Lagrange equations of an action principle specified by a Lagrangian density. question, thus, comes.
A novel feature of this development is that it takes place entirely within the framework of a finite-dimensional symplectic geometry. Though this is well-known in the mathematical community, it serves as a point of contrast to point of view of the Physics community, in which field theories are generally treated as mechanical systems with an infinite number of degrees of freedom and a dynamics given by an infinite-dimensional symplectic geometry.
A second novel feature, which serves to contrast from the Helmholz conditions for mechanics, is the appearance of a non-trivial structure, requisite to the building of a Lagrangian for an already-given dynamics. This structure is known in the field as the variational bicomplex.
Contents
1. Background
2. Helmholz Conditions — Necessity
3. Helmholz Conditions — Sufficiency
Generalization of the Poisson Bracket to Field Theory
(PDF, 57k)
Field theory as field theory contrasts with the instantaneous time formalism in that it takes place over a finite dimensional phase space. This approach, which underlies the more standard mathematical treatment of field theory and figures centrally in the resolution of the inverse Lagrangian problem, treats all the dimensions of the underlying spacetime on equal par. No distinction is needed or made between spatial and temporal coordinates, and the formalism, itself, lies underneath whatever causal structure may exist on the spacetime manifold.
The Hamiltonian, instead of being generated by a Legendre transform over a single time-like variable and interpreted as the energy, is generated by a wholesale Legendre transform on all the spacetime variables. This results in what is known as the de Donder-Weyl Hamiltonian.
One of the main benefits arising from this approach is that with the recasting of field theory in this guise, and the freeing from spacetime causality, the spotlight has been put back on quantum theory. A critical element of incompleteness of quantum theory is fully brought out into the open: there is no well-established multivariate generalization of the Poisson bracket! In fact, following Kanatchikov, we see that there are numerous subtleties and opportunities opened up that would have not been otherwise visible. This review is further expanded on in Time in Quantum Gravity, below.
Reference:
Canonical Structure of Classical Field Theory in Polymomentum Phase Space, Igor V. Kanatchikov, hep-th/9709229
Review of Quantum Gravity
The following is a compilation of a set of reviews and notes taken from the 2007 Quantum Gravity compedium, by Fauser, Tolksdorf and Zeidler.
| 1. Time in Quantum Theory and General Relativity | PDF, 177k |
| 2. Curved Spacetime Local Quantum Field Theory Meets Epstein-Glaser | PDF, 96k |
| 3. Stress Tensors in Gauge Theory and Gravity | PDF, 84k |
1. Time in Quantum Theory and General Relativity
"Time Paradox in Quantum Gravity", Maci'as and Quevedo
Contents:
1. Introduction
2. Time in Canonical Quantization
2. Curved Spacetime Local Quantum Field Theory Meets Epstein-Glaser
"Background Independent Formulation of Quantum Gravity", Brunetti and Fredenhagen
Contents:
1. Perturbative Quantum Field Theory a' la Epstein-Glaser
2. Local Covariant Quantum Field Theory
3. Stress Tensors in Gauge Theory and Gravity
"Gravitational Waves and Energy Momentum Quanta", Dereli and Tucker
Contents:
1. Introduction
2. Conserved Quantities and Electromagnetism
3. Conserved Quantities and Gravitation
4. Kauffman, Spencer-Brown and Feynman-Dyson — A New Golden Braid
Review article forthcoming
The Statistical Foundation of Classico-Quantum Theory
(PDF, 144k)
Originally from chapter 0 of the Statistical Structure of Quantum Theory — Holevo.
Closely following and expanding on Holevo's treatment of the statistical foundations of quantum theory, a more broad-based view is developed here that combines classical and quantum theory into a single enveloping formalism that also covers hybrid classico-quantum systems. Most side-by-side comparisons of "classical versus quantum analogies" are, in fact, cases where the classical concept is not merely analogous to the "non-commutative" version, but already included within it as a special case. This correspondence can, indeed, by exploited to formulate a generic Correspondence Principle by which he Hilbert space superstructure of quantum theory is pulled back from the Hilbert space representation of classical physics. The section on coherent states, Wigner functions and inverse Gaussian convolutions has been added to the original tract.
The result is a generalization of classical and quantum theory, the most notable feature of which is the formulation of both the Heisenberg picture version and the classical version of the so-called Projection Postulate.
Contents
0. Introduction
1. Finite Dimensional Systems
2. General Postulates of Statistical Description
3. Classical and Quantum Systems
4. Randomization in Classical and Quantum Statistics
5. Convex Geometry and Fundamental Limits for Quantum Measurements
6. The Correspondence Problem
7. Repeated and Continuous Measurements
8. Irreversible Dynamics
9. Quantum Stochastic Processes
10. Coherent States, Wigner Functions and Inverse Gaussian Convolutions
Schrödinger, Nelson and Born-Sommerfeld
(PDF, 50k)
As is well-known, the Schrödinger equation for a system described by a general Hamiltonian will, under a polar decomposition of the wave function, yield a deformation of the Hamilton-Jacobi equations. For Hamitonians that are quadratic in the momenta, this deformation takes the form of Bohm's Quantum Potential.
A deeper analysis shows that the quantum potential is generated by the quantum deformation of a stress tensor, representing the probabilistic flow of the system in configuration space, subject to the continuity equation and a force law that expresses the flow of the momentum density for the system in terms of a configuration space Lorentz force. The deformation to the stress tensor, in turn, is generated by the gradient of Nelson's osmotic velocity, thus linking Bohm and Nelson's interpretations of quantum mechanics.
An equivalence result is stated which characterizes the Schrödinger equation in terms of the continuity equation, force law, a third equation expressing the rotation of the fluid in terms of the configuration space “magnetic” force, and an integrability constraint comprising the Born-Sommerfeld condition imposed over all loops in configuration space.
This analysis is expanded on in The 5-D Representation of Fluid Dynamics & The Unified Group, where a 5 × 5 stress tensor is derived for the Schrödinger (and Klein-Gordon) equation in the 5-dimensional representation of the Galilei group.
A Treatise on Quantum Theory — topic outline and prologue only;
(PDF, 181k)
This is an early draft the forward and prologue of a new treatise on Quantum
Theory.
The aim of this treatise is to provide a broad-based introduction to Quantum Theory that avoids, as much as possible, the now cliche' textbook treatment of the subject and provides an entirely fresh perspective to the topic. This treatise is being written much in the same spirit as Maxwell's treatment of Electromagnetic Theory.
There will be many points of departure, the most significant being the ab intio construction of the formal structure of the enveloping "classico-quantum" framework that will embody both classical and quantum theories, as well as its hybrids (i.e., quantum theories with superselection). The treatment will be primarily centered on the Heisenberg picture. Lesser-known features, such as the Hilbert space representation of classical physics, the classical version of entanglement and Naimark's theorem, the classical version and Heisenberg picture version of "wave function collapse" will be featured.
A possible outline is given as follows:
| Volume | Book 1 | Book 2 |
|---|---|---|
| 1 | Foundations | Quantum Mechanics |
| 2 | Quantum Field Theory | Particle Physics |
| 3 | Gauge Theory | Quantum Gravity |
Elements of Orbital Mechanics
(PDF, 278k)
This is a discussion, originating from a USENET article in the 1990's, of the elements of orbital mechanics (and an account of my exploration of the issue when I was in high school).
| 1. Linear Orbits | 1.1. Escape Velocity |
|---|---|
| 1.2. Generalization of Escape Velocity | |
| 1.3. The Timings | |
| 2. Elliptical Orbits | 2.1. Circular Orbits |
| 2.2. The Lighthouse Approximation | |
| 2.3. Kepler's Laws | |
| 2.4. General Orbits | |
| 3. The Two-Body Problem | 3.1. The One Body Problem |
| 3.2. The Reduction: Two-Body Problem → One-Body Problem | |
| 3.3. The N-Body Problem | |
| 4. Other Issues | 4.1. Flight Paths |
| 4.2. Determination of Orbits by Observations | |
| 4.3. Solving Kepler's Equation | |
| 4.4. Perturbations | |
| 4.5. The Delta Method | |
| Further Reading | |
| References and Notes |
Exact Equation for Orbits
(PDF, 55k)
Derived from an article in sci.physics.
The two-body problem in orbital mechanics is generally split into several cases based on whether the bodies are engaged in a circular/elliptic, a parabolic or hyperbolic orbit; thus leading to a fragmented approach to their actual computation.
This problem is remedied here, with a single formula provided for the time evolution of the orbit that applies in all cases. This requires the creation of a function to solve the Kepler equation, and the amalgamation of circular/hyperbolic functions into one. The parametrization adopted is superficially singular for circular orbits. However, a more refined error analysis demonstrates the computational viability of the formulae for near circular orbits.
The Kepler Problem
(PDF, 68k)
The action principle asserts, in effect, that the dynamics of a system enclosed within a region of spacetime may be extrapolated from the system's configuration on the boundary of the region. In the special case, where the region is a slice of spacetime wedged between two equal time Cauchy surfaces, the boundary data reduces to the initial and final values of the system's configurations; the extrapolation to that of the intermediate values.
Applying these considerations to the Kepler problem, an exact solution may be developed that expresses the evolution of the system in terms the initial and final coordinates; and an extra parameter that plays the role of time. The details are worked out, and expressions are derived for the action, for the ephemeral coordinates, etc.
Contents
1. Introduction
2. Boundary Conditions
3. Instantaneous Coordinates
4. Ephemeral Coordinates
5. Ephemeral versus Instantaneous Coordinates
The Kepler Problem in Quantum Theory
(PDF, 83k)
One of the greatest failings of a traditional course in Quantum Theory is
the nearly exclusive focus on the Schrödinger picture at the expense of
the Heisenberg picture. Indeed, bearing out the notion of wave-particle
duality is the duality of pictures, themselves, with the wave-like nature
brought out by the Schrödinger formalism, and the particle-like nature by the
Heisenberg formalism. Therefore, one is only getting half the story when
short-shrift is given to the latter.
The most significant case in point is the Hydrogen atom, itself, and more generally: the Kepler problem. Though it is not well-known, the orbital parameters that characterize the solution to the classical Kepler problem also apply (with a suitable factor ordering) to the quantum problem. The only difference is that they form a non-commutative algebra. This algebra is a constrained version of SO(4) for circular/elliptical orbits, of SE(3) for parabolic orbits, and of SO(3,1) for hyperbolic orbits. Hence, the terms Hydrogen SO(4) and H-Quarks.
Another feature, not well-known, is that though the orientation of the orbit is subject to uncertainty relations, the shape and size of the orbit are not. For circular and elliptical orbits, each energy and angular momentum eigenstate (i.e., orbital) is associated with a well-defined shape and size. Though similar to the parameters that comprised the Bohr-Sommerfeld theory, there are corrections, ultimately arising from the uncertainty relations, to the value for the minor axis. In particular, the s orbitals have a non-zero width.
The values of the parameters are tabulated for the low-lying orbitals.
Even less well-known is the relativistic extension of the foregoing. Some inroads into this issue are made, here, with respect to the Kepler problem for both massless spin 1/2 particles and massive spin 1/2 particles.
Contents
1. Orbital Mechanics in the Heisenberg Picture
2. The Relativistic Kepler Problem in the Heisenberg Picture
Generalized Gravitational Dynamics
(PDF, 65k)
In part, this is an extrapolation of a few notes taken during a recent
trip to the library of a nearby campus. The question posed is whether there
is a way to formulate an extension to the usual approach involving a
connection and frame field in which these are incorporated as the gauge
potentials of a Poincaré gauge field that, in turn, is generalized so that
the conjugate fields remain independent of the field strengths.
The basic formula for a gravitational dynamics based on Poincaré symmetry is worked out here. Since the Poincaré group is not semi-simple, the usual apparatus of gauge theory is not available. In particular, there is no non-degenerate gauge-invariant metric for the gauge group, and therefore no possibility of defining a quadratic self-action for the action.
Another reason for the absence of a natural quadratic directly relates to the nature of the field itself. Whereas in a gauge theory, the conjugate fields are built linearly from he field strengths by the duality relation (which depends on there having been a metric defined first!), a gravitational theory treats the metric, itself, as a part of the dynamics — thus precluding any notion of duality.
A step backwards is, therefore, taken, with the dual fields treated as independent; and the generic source terms for the currents associated with frame field (the stress tensor) and connection (the spin tensor). The special case is briefly examined, in which the Lagrangian is constrained so that its dependence on the curvature reduces to a dependence on the Ricci tensor.
The question of how these quantities may be matched with those in the standard theory is left wide-open. A particular feature of the generalized gauge theory approach adopted here is that, despite the relaxing of the assumption of linear duality, a stress tensor for the field can still be defined, since it is specified at the outset by a Lagrangian. In contrast, in the standard theory, there are well-known difficulties associated with localizing gravitational stress, momentum and energy whose only route of escape is the addition of new elements or a deeper reduction of the gravitational field in terms of new elements. Given the definability of the stress tensor, in this context, the relaxation of linear duality then appears to represent a bona fide extension that may not have a direct reduction to the Einstein theory.
Though there is additional room for movement, within this framework, beyond the standard theory of gravity; the related problem of defining a correspondence limit is left open here.
Standard Model Lagrangian
(PDF, 294k)
The Lagrangian for the Standard Model is written out in full, here.
The primary novelty of the approach adopted here is the deeper analysis of the fermionic space.
 |
 |
The Yukawa sector is developed from first principles, the conversion from the charge to mass eigenstates is worked out in detail; and the boson spectrum is also developed in terms of both bases.
 |
 |
A second novel feature is that the dependence on the "hidden metrics" underlying the representation spaces of the fermions, bosons and scalars is explicitly brought out in the Lagrangian. In particular, the gauge metric is simultaneously linked to both the Jordan-Brans-Dicke scalars and the classical theory of dielectric media.
Finally, the terms involving gravitational interaction are included, leading to the question of how to incorporate gravity and the Standard Model within a unified framework; particularly one that cures the problem of infinities that plagues both classical and quantum field theory. Going beyond the usual Kaluza-Klein route, we exploit the "hidden metric" issue to this end.
Classical Origin of Renormalization
Derived, in part, from articles posted to sci.physics.research.
Despite the historical appearance of methods related to renormalization alongside the appearance of quantum field theory, the link between the two notions is nothing more than a red herring. Renormalization is grounded at the classical level, in classical field theory.
The prevalent misconception in the contemporary literature is that somehow the appearance of the “qualitative” differences associated with renormalization-related matters (e.g. running of the couplings) are magical after-effects of quantizing a classical theory. In fact, what is actually going on here, is that there we've found out the hard way that we've simply been quantizing the wrong classical field theory.
Muddling over the decades, people have stumbled onto the correct classical theory that the quantum theory is actually trying to be a quantization of. It is a theory that has scale non-invariance at the classical level.
But because the muddling is still not yet fully emergent from the muddle, there's still the impression that all the particulars of the qualitative differences are somehow rooted in the quantum level. There is still not yet a full recognition that this is actually all classically grounded in origin and that the appearance the qualitative feature actually has nothing specifically to do with quantum theory!
The Classical Renormalization Group
(PDF, 57k)
Here, the emergence of the renormalization group, the renormalization coefficients, and the beta coefficients at the classical level is illustrated with the scalar field. In the process, a simple classical interpretation of these otherwise cryptic concepts emerges.
If we were to write down the most general Lorentz-invariant Lagrangian for a scalar field, what we would find is that the conjugate fields satisfy constitutive laws of a form that is independent of the Lagrangian. All the Lagrangian-dependency is locked up in a few constitutive coefficients. For the simple one-component scalar field, these coefficients (along with the field, itself), can be renormalized — thus leading directly to the appearance of the renormalization group at the classical level.
Though the analysis is not included here, it turns out that a similar analysis applies for the multi-component scalar field, though extra conditions need to be imposed on the constitutive coefficients to make the same analysis work.
Renormalization of the Classical Scalar Field
(PDF, 88k)
This is an expansion on the Classical Renormalization Group article, focusing mainly on the scalar field, but also spelling out some of the generalities for other Lagrangian field theories. It also treats the generalization of scalar field dynamics to isotropic media.
| 1. The Structure of a General Lagrangian Field Theory | |
|---|---|
| 2. Isotropic Media and Vacuua | 2.1. Constitutive Relations |
| 2.2. Isotropic Media in “Moving Frames” | |
| 3. Renormalization Group and Renormalization Coefficients | 3.1. One-Mode Scalar Field |
| 3.2. General Scalar Field | |
| 3.3. 1-Form Fields |
Scale Invariance and the Gauge Group
(PDF, 76k)
This is an elaboration on the derivation carried out under the G & S article and it is expanded on here to make a point: Yang-Mills and Gauge fields are not the same thing!. The literature seriously confuses this (on account of this, I used to, as well).
A Yang-Mills field is characterized as a gauge field whose Lagrangian is a homogeneous second order, parity-symmetric quadratic function of the field strengths; such that the quadratic coefficients are adjoint-invariant. Formally, the following conditions may be set out to characterize the conditions required for a gauge field dynamics to define a Yang-Mills theory. A Yang-Mills field is a gauge field whose dynamics are given by a Lagrangian such that
| 0 | The Lagrangian is an analytic function of the gauge potentials A and their exterior differentials dA |
|---|---|
| 1 | The Lagrangian is Lorentz-invariant |
| 2 | The Lagrangian is gauge-invariant |
| 3 | The Lagrangian is scale-invariant |
| 4 | The “permittivity” coefficients are adjoint-invariant |
The main point of this characterization is that — since the central premise of all matters related to renormalization theory is the breakdown of scale-invariance — this means that a quantized gauge field must violate condition (3) and therefore can not be a Yang-Mills field!
This feature is not one which magically emerges as a result of quantization. Rather, it must be something also present at the classical level. Quantization cannot produce new qualitative features, but only provide corrections on those already present at the classical level.
At the root of the difference lies the very distinction between the Maxwell-Lorentz Lagrangian (in which the permittivty ε is reduced to a constant conversion factor, ε = ε0) and the older theory of Maxwell, which required ε to be variable (specifically in order to cancel out the infinities of field theory — this is discussion in further detail under The Elements of Maxwell's Renormalization Theory).
More generally, a scale-invariant analytic Lagrangian also satisfying properties 1 and 2 will reduce to a quadratic function of the field strengths, with constant coefficients that generalize ε. The difference between scale invariance and scale non-invariance boils down to the question of whether ε is constant or variable — the very issue that has its roots in the difference between the Maxwell versus Lorentz views of electromagnetism.
The cost of trying to quantize the scale-invariant classical theory is that the issue of (classical) inconsistency comes to a head in the quantized theory and we're forcibly kicked out of the scale-invariant box, straight into a scale non-invariant theory. What this shows is that we've actually been quantizing the wrong classical theory and the mistake comes back as a forced “correction”, by the time we get around to quantizing the mistake. In other words, Lorentz was wrong for “shutting off” Maxwell's ε permittivity coefficient.
Curing the Ultraviolet Divergence
(PDF, 172k)
Related Articles:
Einstein-Maxwell-Scalar
The Origin of the Classical Electromagnetic Singularity (PDF, 64k)
In this article, the fundamental elements of a comprehensive resolution to the ultraviolet divergence problem in both classical and quantum field theory are presented. The central theme of this resolution is that both the problem and its resolution are firmly rooted in classical physics and do not involve the consideration of any prospective theory of "quantum gravity".
Though no longer well-known, one of the central themes of Maxwell's treatment of the electromagnetic field was a renormalization theory whose purpose was specifically to eliminate the ultraviolet divergence of classical field theory. When translated into modern language, it can be generalized to a form applicable to all Lagrangian field theories, both classical and quantum. The most significant point of contrast with modern renormalization theory is that the scale dependency of couplings is replaced by a bona fide field governing the relation between field velocities and their conjugate momenta. The kinetic terms in the Lagrangian become cubic, and are only asymptotically quadratic. The corresponding Green's functions and propagators are no longer singular on the light cone.
For gauge theories, this amounts to promoting the gauge group metric to a field. The metric is synonymous to what is referred to in renormalization theory as the so-called renormalization constant Z3. Similarly, the inclusion of a fermion metric accounts for what are referred to as Z1 and Z2.
A natural starting point for addressing the dynamics of the gauge group metric is then to treat the gauge theory as a Kaluza-Klein model on a principal bundle endowed with a non-trivial fibre metric. This means treating gmn (for m,n ≥ 5) as variable, where (gmn) is the total space metric. When applied to electromagnetism, the resulting theory gives a dynamics for the vacuum permittivity that captures, even at a classical level, the running of the couplings, (anti-)screening and Landau poles.
In the particular model investigated, the running of the fine structure constant, alpha, is governed by an equation ((log alpha)''(1/r) = 3A alpha, where A is the Planck area) that has the appearance of something seen in quantum gravity. However, it is independent of Planck's constant and resides firmly at the level of classical Physics; thus underscoring the theme of these developments.
We will address the ramifications of this approach in other Lagrangian theories and discuss its relation to the renormalization group methodology of quantum field theory.
Contents
1. Background
2. The Effective Lagrangian
Appendix: Maxwell's Renormalization Theory Revisited
Einstein-Maxwell-Scalar
(PDF, 65k)
The calculations underlying the Curing UV article are carried out in further detail, with the effective stress tensor, Riemann and Ricci tensors worked out in detail.
Hidden Metrics, Field Divergences and Quantum Field Theory
(PDF, 53k)
The argument that had been posed originally by Maxwell to eliminate the field-theoretic infinity generalizes and translates in modern form. Associated with each field is a bilinear form or "metric" through which the kinetic terms appear in the Lagrangian. When this metric is assumed constant or otherwise placed in the background, a simple linear duality relation exists between the fields "velocities" and "momenta". Out of this arises sharp, singular Green's functions and (in the quantized theory) propagators. The appearance of such terms forces the representation of the quantum field as a singular distribution, along with all the problems attendant to forming non-linear combinations, particularly the stress tensor. Hence, the ultraviolet divergence.
Near a point-like singularity, in order to recover regularity, one is therefore forced to bring the "hidden metric" back into the foreground as a dynamic variable in its own right. The linear relation between velocities and momenta and the sharpness of the propagators will therefore break down near sources. Ultimately, this is how the ultraviolet divergence is resolved.
Closely related to these developments, in [1], the hidden metric is seen to arise from the classification of the possible Yang-Mills-Higgs Lagrangians quadratic in the field velocities [1, Proposition 4.3]. The classification ultimately arises from a generalized version of Utiyama's Theorem [2]. In turn, the link to the coupling "constants" of the underlying field theory is made clear [1, Proposition 4.8-4.11]. These, too, become part of the foreground with the metric near concentrated sources — thus establishing a running of the couplings at the classical level.
References:
[1] M. Lopez, J. Masque, "Gauge-Invariant
Characterization of Yang-Mills-Higgs Equations", Ann. Henri. Poincaré
8 (2007), 203-217.
[2] J. Geom. Phys. 6 (1989) 107-125.
Hidden Metrics and Resolving the Field Infinity
(PDF, 129k)
Following up on the "Hidden Metrics" article, this delves deeper into the issue, raising an important point that leads directly to a generalized Kaluza-Klein formalism.
Field velocities reside in an affine bundle. Another way of stating this point is simply that velocities are "relative". Of necessity, this entails the appearance of a generalized connection in any velocity-momentum relation. Taken in conjunction with the underlying field metric, this is sufficient to define the bundle metric for the total space comprising the bundle in which the fields reside.
The last part of the article (not yet complete) develops the two variants of Kaluza-Klein for gauge fields and principal bundles from first principles, treating the gauge symmetries as Killing fields in the total space. The formalism adopted here generalizes to homogeneous spaces, though this generalization is not carried out here.
Dimensions and Units in Gauge Theory
(PDF, 317k)
Supplementary article: The Anatomy of the Electroweak and Color Gauge Forces (PDF, 70k)
Based on the original, “Dimensions and Units in Electrodynamics” (Hehl & Obukhov, arXiv:physics/0407022 v1 5 Jul 2004), this is a substantial rewriting and expansion of Hehl's “Dimensions and Units in Electrodynamics” (Hehl & Obukhov, arXiv:physics/0407022 v1 5 Jul 2004), in which everything is generalized to non-Abelian gauge fields.
The extra material works out the form of the stress tensor, the constitutive laws for all Lorentz-invariant & gauge-invariant Lagrangians. The relation between the permittivity, the scalar and dilaton (which Hehl began to allude to) are clarified here. The dimensional analysis in the original is nothing more than a repeat of the analysis given in section §623 of Maxwell's treatise, though Maxwell is not credited anywhere in the paper's reference list.
The original article was an axiomatization of the classical electromagnetic field. However, since the advent of GSW, the classical field (not the quantum field, but the classical field) is now understood to be described by non-linear equations that subsume Maxwell's equations. Instead, it's the hypercharge field that's described by Maxwell's equations.
Hence, the revision is meant to include electromagnetism, as it is now understood, since it is an integral part of a non-Abelian gauge field. There are major changes required in the axiomatics.
First, with the oversight, with electroweak unification, the Maxwell equations are no longer valid, but are now replaced by non-linear equations that possess non-zero magnetic currents. All 3 of the Hehl axioms are false for electromagnetism, in virtue of the unification. Moreover, the dimensional analysis is seen to be incomplete when electromagnetism is viewed in the broader context of gauge theory; in particular, the appropriate dimension for the field is [F] = 1, while Hehl's “absolute” dimension is, itself, actually a “relative” dimension: [Fa] = H/Q, where a is the gauge index
The abstract, acknowledgements and references have been kept intact from the original.
The “Maxwell Equations” for Non-Abelian Gauge Fields
(PDF, 222k)
The following is based on part 2, by Hehl, of the “Two Lectures on Fermions and Gravity” article (Hehl, Lemke and Miekle), which was cast solely in terms of the electromagnetic field, by the original author (Hehl). A large number of additions have been made; the analysis has been reworked in the more generalized context of non-Abelian gauge theory, and errors present in the original (particularly the derivation of the field Lagrangian) have been corrected.
The most significant oversight in the original has to do with electromagnetism, in the context of gauge theory. There are essential elements in gauge theory, highly relevant to the question of duality even in the more restricted context of electromagnetism, which remain hidden when focusing only on electromagnetism. Hehl has missed the full extent of the general idea he's posing here.
This goes one step further than
When seen in the full context of gauge theory it becomes immediately apparent that the fields D and E are not even the same type of object! In particular, the (D, H) and (E, B) fields cannot be related by a mere spacetime duality, since their gauge indexes are in different positions - (Da, Ha) and (Ea, Ba). The constitutive law involves a metric. That metric is none other than the permittivity, in disguise! Thus, we further the analysis of Hehl by establishing a link to a geometric interpretation. It is through this interpretation, ultimately, that one can understand how the permittivity, dilatons, and quintessence scalars are all connected to one another.
Much of sections 1 and 2 are kept intact, but with numerous additions. Section 3, however, is almost completely rewritten, with the analysis contained therein completely redone. The references of the original are kept intact, but are not listed here. They may be found in the “Two Lectures” article.
The Constitutive Law in Gauge Theory
(PDF, 166k)
Based on Goldin & Shtelen; “Generalizations of Yang-Mills Theory with Nonlinear Constitutive Equations”.
This completely reworks the analysis from scratch; first writing out the 4 families of Lorentz invariants for a general gauge field, and centering the discussion in this more general context ... and only then bringing up the SU(2) field somewhat as an afterthought.
An analysis is done to determine the most general constitutive law for a gauge theory under the assumption that it is derived from a Lagrangian possessing local Lorentz (or Galilean) invariance. Later this is expanded to Lagrangians that possess only SO(3) invariance (that is: constitutive laws for “isotropic media”)
In the first case, the resulting relations not only involve constitutive laws of the form
| D = εE + θB, | H = εc²B - θE, |
In the second case, we deal with the constitutive relations for isotropic media. In order to continuously bridge between the non-Relativistic and Relativistic theories, it is necessary to first broaden the scope of both sets of constititive laws to moving media. Then, it turns out that the most general set of constitutive laws for isotropic media involve seven families of constitutive coefficients, with the original four representing what survive the imposition of boost invariance for the Lorentzian (or Galilean) signature.
This extra element was a critical part of the non-Relativistic theory, as originally posed by Maxwell; and it is almost entirely neglected in the contemporary literature, though the necessity of broadening the scope of electrodynamics to moving media to effect a bona fide Galilean limit was recognized early on in the (widely misunderstood) treatment of the Electrodynamics of Moving Media by Einstein & Laub in 1908-9.
Therefore, the discussion by the original authors about the Galilean limit is expanded on, significantly, by re-introducing this aspect of Maxwell's original formulation, along with his G velocity field.
Without this additional element, the authors' formulation of a Galilean limit is inadequate in that it does not allow for the formulation of a non-zero permittivity — at least, not when the conjugate fields (D, H) are expressed in terms of the fields (B, E). With the expansion of the scope of the dynamics to isotropic media, however, there is sufficient additional structure to accomplish the desired result and effect a Galilean limit.
What we find is that the permittivity ε and permeability μ = 1/(εc²) separate out into independent constitutive coefficients. Thus, the concept of an invariant velocity c and wave speed V = 1/√(με) are separated from one another, in both the non-relativistic theory and the relativistic theory (in isotropic media). Even more, when generalizing from electromagnetism to gauge fields, this bridging of Lorentz → Galilei will not just involve a splitting of the notions of the two speeds V versus c, but the inclusion of yet more structure, part of which will be discussed in the analysis.
Gauge Field Equations in Maxwell Form
(PDF, 120k)
The general form for all classical Lagrangian-based gauge fields is rendered in a “Maxwell's Equations” format, as a series of equations that includes Maxwell's equations, themselves, as a special case.
The gauge group metric kab, and “coupling” coefficient g, in the process, are both identified as the generalization of the vacuum permittivity &esilon;, thus providing a direct physical interpretation of the scaling of the gauge group in terms of a non-trivial dielectric structure of the vacuum, itself, and recovering arguments posed originally by Maxwell regarding the existence of a dielectric structure within the vacuum.
This formalism generalizes Yang-Mills theory in treating the two sets of Maxwell fields (D, H) and (B, E) independently, even in vacuuo. A notable feature of the generalization is that the stress tensor for the (generalized) Maxwell field need not be trace free.
The generalization factors out the constitutive relations linking the two sets of fields from the more fundamental diffeomorphic theory. One application is that gauge theory can be formulated in non-Lorentzian manifolds, including the Galilean limit of Newton-Cartan spacetimes.
Contents
1. Review of Gauge Theory
2. Reduction To 3+1 Dimensional Form
3. Maxwell on Abelian Yang-Mills Fields; Yang-Mills Complexion
Supplement: The Maxwell Gauge Field Equations in 4 Dimensional Form (PDF, 60k)
The key equations are listed in crib-sheet form, written fully out in component form to stress the diffeomorphism-invariant underpinning (and independence of space-time signature).
Some extra material is included on the development of the stress tensor and on the issue of scale invariance and renormalization (at the classical level).
The Proca Action
(PDF, 96k)
The theory of Maxwell-Proca field is rendered in form analogous to classical electrodynamics. Particular focus is placed on the constraint dynamics in the Hamiltonian formulation; the permittivity coefficient and its link to the renormalization coefficient Z3; and on the process by which a Maxwell field may “eat” a scalar mode to become a Proca field.
The classical underpinning is also laid out to what is generally referred to as renormalization, but actually pertains to the dynamics of the permittivity coefficient.
The Gauge-Scalar Fields in Maxwell Form
(PDF, 165k)
Expanding on the “Maxwell/Gauge” article, the general form for all classical Lagrangian-based combined scalar/gauge fields is written as a series of Maxwell equations, coupled to a series of what might be described as “pre-Maxwell” equations for the scalar field. Again, this generalizes on what is normally defined as the classical field theory in that both the fields and their duals are treated independently, and the constitutive relations need not be linear with constant coefficients.
The “Maxwell's equations” formulae are for 1-form fields. For the 0-form fields, the naming of the equations “pre-Maxwell” is because in them everything slips back one notch. The fields (E, B) become identically 0, the potentials (A, φ) become the new field strengths, and the “gauge” χ becomes the field potential. The “sources” (J, ρ) become the conjugate fields, and the homogeneous “continuity” equation becomes an inhomogeneous field equation for them. (A similar slip, in the opposite direction, up the notch occurs when working with 2-form fields).
A detailed analysis is then carried out to demonstrate — within the field equations themselves — the stages that leads to a scalar mode being “eaten” up by a gauge mode. All of this is in the classical field theory. Quantum theory is not involved in this discussion anywhere. Finally, though not explicitly mentioned in the article, this general formalism is also meant to include spinor fields, since a spinor field can be treated as a 0-form field associated with a gauge field which contains the Lorentz group. This is covered in more depth in Dirac with Chiral Gauge Fields, and A Dirac Lagrangian Quadratic in the Velocities.
| 1. The Kinematic Fields | 1.1. The Field Coordinates |
|---|---|
| 1.2. The Field Velocities | |
| 2. The Kinematic Laws | 2.1. Field-Potential Relations |
| 2.2. Bianchi Identities | |
| 2.3. Gauge Invariance | |
| 3. The Dynamic Fields | |
| 4. The Dynamic Laws | 4.1. The Field Equations |
| 4.2. The Conservation Laws | |
| 5. Force, Energy and Stress | 5.1. The Force Law |
| 5.2. The Field Stress Tensor | |
| 5.3. The Combined Stress Tensor | |
| 5.4. The Motion of a Test Body |
5.4.1. Charge Precession: The Generalized Wong's Equation 5.4.2. Combined Gauge-Scalar Forces: The Generalized Lorentz Force Law 5.4.3. The Gauge-Scalar Hamiltonian and Poisson Brackets |
| 6. Scalar-Gauge Equations in Maxwell Form | 6.1. Reduction to 3+1 Form |
| 6.2. Constitutive Relations | |
| 6.3. The Field Equations | |
| 6.4. Quaternion Form |
SU(2) Yang-Mills Monopole Solution
(PDF, 78k)
A static SU(2) magnetic monopole solution of the form
Wong's Equations: General Relativistic Yang-Mills Particles
(PDF, 119k)
Derived from Duviryak's "Classical Mechanics of Relativistic Particle with Colour".
The equations of motion for a classical general relativistic Yang-Mills point particle are derived within the framework of principal fibre bundles from a generic family of Lagrangians. (The original treatment stayed within the confines of special relativity.)
The resulting dynamics splits into the external dynamics which describes the interaction of particle with gauge field in terms of Wong's equations, and the internal dynamics which yields a mass-charge relation and results in a spatial motion of the particle via integrals of motion only. The external dynamics is independent of the form of the Lagrangian; only the internal dynamics and the mass-charge relations resulting will show sensitivity to the specific structure form of the Lagrangian.
The Hamiltonian theory for the constrained system is also formulated.
The distinguishing feature of particles in the presence of a non-Abelian Yang-Mills field is that though the charge may precess (with the field, itself, now associated with a charged current), the magnitude of the charge remains a constant of motion. This leads to both a quadratic invariant for the mass, momentum and energy and for the components of the Yang-Mills charge.
The derivation is carried out within the setting of the quotient operation and new notational conventions for fibre bundles, leading to a more transparent derivation that bridges the gap between the mathematician's and physicist's languages.
Contents
1. Introduction
2. Kinematics on a Principal Fibre Bundle
3. Lagrangian Dynamics of Yang-Mills Particles
4. Examples
5. Transition to a Hamiltonian Description
6. Conclusions
The Equation of Motion for Yang-Mills Particles
(PDF, 95k)
Originally posted to sci.physics.research.
Within the setting of the quotient operation and new notation conventions for fibre bundles, the equations of motion for a General Relativistic Yang-Mills particle are derived. This time, the gauge metric is allowed to vary with position. The resulting equations include Wong's equations for the precession of the Yang-Mills charge, and the inhomogeneous geodesic equations subject to a Lorentz force and a second force quadratic in the Yang-Mills charge components related to the gradient of the gauge metric.
Contents
Abelian Case
Non-Abelian Case
Yang-Mills Fields and Mass-Charge Relations
(PDF, 109k)
Interesting regularities involving the mass spectrum in both the lepton and gauge boson sector of the Standard Model exist: (1) the sum of the lepton masses is approximately equal to the fine structure constant multiplied by the vacuum expectation value of the Higgs; while (2) the sum of the gauge boson masses approximates the vacuum expectation value, itself.
Supplementing the Wong's Equations article, and following up on speculations posted recently in sci.physics.research concening possible mass-charge relations in the Standard Model, the general issue of mass charge relations; of parity in the presence of chiral interactions; and of the origin of the generational splitting of the mass spectrum are discussed.
The Poisson bracket developed in the Wong's Equations paper is also extended to incorporate the contribution arising from a positionally dependent gauge metric; and the relation between the classical and semi-classical (1st quantized) theories is explored in greater depth.
A distinguishing feature of the dynamics (assuming the gauge metric is constant) is the conservation of charge magnitude. Though a Yang-Mills charge may precess — leading to flavor-changing interactions — the magnitude of the charge should remain constant. This leads naturally to a search for quadratic invariant in the fermion spectrum of the Standard Model. Remarkably, there are none. However, playing on the historical analogy of Maxwell's "displacement current", if the right neutrino sector is inserted into the spectrum, and an additional gauge degree of freedom posited for the quantum number "Baryon - Lepton", then two quadratic invariants emerge.
Contents
1. Overview
2. A Review of Gauge Theory and Chiral Interactions
3. On the Origin of the Generations
Unified Field
(PDF, 56k)
Since the time of Maxwell, the close relation between momentum and the magnetic potential A has been known. It is well-known that the canonical momentum and energy, respectively, acquire contributions proportinal to A and the electric potential; the proportionality factor being the electric charge. This falls in line with the interpretation of the charge, itself, as a component of momentum with respect to an unseen dimension, thus leading to the general idea of Kaluza-Klein.
The goal of unification, in the earlier part of the 20th century, was to provide a consistent geometric interpretaiton for all dynamics. The primary obstacle is that a geometry space-time interpretation, itself, requires the foundation of the Equivalence Principle. It is what justifies making gravity epiphenomenal to space-time curvature. In fact, it is what would even justify retrofittinmg Newtonian theory into a curved spacetime theory of Newton-Cartan general relativity. One could conceivably even do the same for Aristotle's spacetime physics. In all cases, the geometrization process requires the Equivalance Principle.
Yet, the principle is obviously false when taken at face value, for just about every instance one can conceive of in the real world. Two objects proceed to diverge in their subsequent motions, even when set about identically. Two lovers in an embrace eventually go their separate ways (especially in Hollywood).
So, either the Equivalence Principle is wrong or the apparent motions seen in the everyday world are just a shadow of the actual motions the objects undergo, with the divergence of subsequent motion being a consequence of the initial (unseen) parts of the motions having been initially different. It is at this point that one meets the older conception relating the electromagnetic potentials to the canonical energy and momentum. This conception is further developed within the theory of first-class constraints, showing that the requirement for consistency with the dynamics leads to poisson bracket relations that very nearly capture the Yang-Mills nature of the field and Wong's equations.
Torsors and Principal Bundles
(PDF, 65k)
Originally posted to sci.math.research.
A torsor is a group in which the identity has been "forgotten". It bears the same relation to a group that an affine space does to a vector space. Remarkably, there is a simple axiomatic characterization of these structures in terms of the ternary operation a/b.c:
| a/a.b = b, | a/b.b = a, | a/b.(c/d.e) = (a/b.c)/d.e |
Contents
1. Torsors
2. Principal Bundles
3. Connections, Lie Torsors and Tangents
Homogeneous and Affine Spaces
(PDF, 187k)
Based on the Wikipedia articles on Homogeneous and Affine spaces, an algebraic formalism for homogeneous spaces and affine spaces is developed. An affine space is a vector space in which location has become relative so that no distinguished point stands out as the 0. Consequently, in place of the vector addition u + v, one requires a variant that explicitly indicates the point of origin — the ternary operation u - O + v. In place of scalar multiplication ru, the corresponding operation is [O, r, u] = (1 - r)O + ru.
An intrinsic characterization of the vector space the affine space is "modelled on" may be provided through the definition of "formal differences". The resulting space acts on the affine space, thus producing a vector bundle structure that is one and the same as the tangent bundle of the affine space.
An algebraic axiomatization may be provided in terms of the two ternary affine operations or, remarkably, solely in terms of the operation [O, r, u] by the following three axioms:
| [u, 0, v] = u; | [u, 1, v] = v; |
| [u, rt(1-t), [v, s, w]] = [[u, rt(1-s), v], t, [u, rs(1-t), w]] | |
Finally, a route toward a synthetic axiomatization of affine spaces is outlined, in which the standard axiomatization of a projective geometry is adapted by a suitable redefinition of point and line. Out of this arises in a natural way the concepts of parallelism and planes.
This Week's Finds
The following are redactions of a select range of articles from the This Week's Finds series, originally by John Baez.
Weeks 1-10 (HTML, the original sources: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
Week 248 (PDF, 81k, the original source)
This is towards the beginning of the tale of groupoidification, where the concept of witnessed relations is briefly discussed.
In actual fact, this article serves as a secret venue where prophecies from an enigmatic source have been quietly stashed, and the place where true identity of Chris Hillman is also revealed.
Week 249 (PDF, 85k, the original source)
The concept of transitive group actions is translated into the framework of groupoids and category theory. To facilitate the discussion from the original article, the inner quotient is introduced. For instance, the equivalence class map is just the inner quotient by a fixed point. Other additions have also been incorporated into the original.
Week 250 (PDF, 208k, the original source)
This goes one step further in the Tale of Groupoidification, with the idea of a double coset raised.
There was also a brief excursion on the concept of the Relativity of Motion. This has been significantly expanded with corrections made regarding the roles Maxwell, Lorentz and Einstein played (e.g., the little-known fact that Maxwell actually repudiated the Relativity of Motion in his treatise, and that his theory was actually Galilean invariant, rather than Poincaré invariant); and on drawing out the link between relativity and affine (& projective) spaces and affine bundles.
As part of this expanded treatment, it is also shown that the ultimate origin of the space-time concept is not Minkowski, but Galileo's principle of relativity. For, one of the consequences of asserting the relativity of motion is that of asserting the relativity of the sameness of a point at different times — the Relativity of Genidentity. This means that there is no longer a cohesive concept of point-that-endures-in-time. What one finds is that, instead, one needs to go one step deeper and replace the primitive of “point” by that of a “point at a specific time”.
Week 264 (PDF, 64k, (the original source)
A brief discussion of Sidney Coleman and of homotopy groups. Somehow, in the revision I made here, a scene from the Star Trek parody The Awakening found its way into the article.
Not all of the original Week 264 has yet been included in this article.
Gauge Theory and Quotients
(PDF, 49k)
Originally sent as a letter to Dr. Lubkin, well-known for his discovery of
the geometric interpretation of gauge theory.
There is a simpler way to understand the whole apparatus for fiber bundles, connections, etc. that encompasses their formal mathematical machinery. A new notational convention is developed for the tangent space map of group actions that bridges the gap between the mathematician's and physicist's language. In this context, the quotient operation is introduced. Unlike group actions, quotients do not have a natural action under the tangent map. The extra structure of a connection is required. Remarkably, however, the relation can be inverted and a connection may be defined simply as thke derivative of the quotient.
These results lead to simple characterizations for the gauge potential, the connection, for horizontal and vertical vectors, for the adjoint and coadjoint actions, for the horizontal lift; all of which are developed here.
Finite Electromagnetism
(PDF, 337k)
Dated from 1987, and originally my Master's Thesis. Not all figures have
yet been included.
In this paper, electromagnetism is formulated entirely in integral form. The goal of this exercise is to capture an equivalent statement of the theory, which would normally otherwise be stated in the language of calculus, in integral form; in such a way that it may be used to also generalize to finite or discrete geometries.
Focus is placed on developing the underlying geometry as a cell complex, or generalized polytope. The first half of the paper reviews the relevant elements of the classical theory and the underlying geometry, while the second half translates it all into the "combinatorial" framework of the discrete geometry.
Two key issues arose that were not fully resolved. First, the duality operation (i.e., Hodge star operator), used in converting the field kinematics to its conjugate momenta, does not have an integral form. This touches on a much deeper issue: the constitutive relations in field theory cannot be rendered trivially as Lorentz relations without the resulting field law becoming ill-defined (i.e., the self-force infinity). Second, the stress tensor and dynamics of interaction with other fields could not be successfully incorporated. This, too, touches on a deeper gap in the underlying field theory — the ill-definedness of the stress tensor (i.e., the self-energy infinity) for a field theory with linear constitutive relations.
A de facto resolution of the duality question was to pose a linear, but non-local, relation between the kinematic and dynamic quantities. Ultimately, the most decisive resolution will lie in translating the action principle to the cell complex geometry.
The Path Representation of Gauge Theory
(PDF, 129k)
Derived from Gambini and Pullin's "Loops, Knots, Gauge Theories and
Quantum Gravity."
These notes are primarily intended as a review of the "universal" gauge theory formalism developed in the first chapter of Gambini and Pullin, with a mind toward extending the formalism (along with the rest of the associated mathematics) to path and gauge groupoids.
The present state, at the time of writing, in the loop approach to gravity and gauge theory focuses on the holonomies associated with the horizontal lifts of loops. At the time of its original evolution, the concept of a groupoid was not well-known. It is this missing ingredient that leads to the awkwardness in the holonomy approach, where the gauge dependency of holonomies necessitates the use of further auxillary operations (the trace).
A connection may be globally characterized as a homomorphism from the path groupoid of the base space to the gauge groupoid associated with the gauge theory. Expanding on the prior development of the new notation and quotient operation, an simple characterization of the gauge groupoid is provided in terms of the outer quotient. If r: x → y is a path linking points A to B, and A(r) is the horizontal lift of the path with respect to a connection A, then the outer quotient A(y)/A(x) defines a gauge invariant generalization of the holonomy.
These developments are all preliminary and subject to future revision, along with the contents of these notes.
Strings, Loops and Others: A Critical Survey of the Present Approaches to Quantum Gravity — Rovelli;
(PDF, 332k)
This is a copy of arXiv:gr-qc/9803024, reproduced here with the permission of the original author, Carlo Rovelli, with enhanced graphics, layout and additional commentary.
Contents
1. Introduction
2. Directions
3. Main Directions
4. Traditional Approaches
5. New Directions
6. Black Hole Entropy
7. The Problem of Quantum Gravity
8. Relations Between Quantum Gravity and Other Major Open Problems in Fundamental Physics
General Frame Covariance in Achronal General Relativity
(PDF, 63k)
Originally posted to sci.physics.research.
A 100 years past, at the time of writing, and no new Einstein appears imminent. This was the topic of the sci.physics.research thread No new Einstein.
In fact, matters are not as they seem and only the perspective of future history will lend clearer insight. The analogue, today, nearly 100 years past, is that you have a situation today just like 1905, where you have two seemingly irreconcilable theories, and people (unbeknownst to themselves in some cases) have already laid out the essential foundation of its resolution. In the meantime, the mainstream may very well be off on a modern-day analogue of the Ether Primrose Path.
This article focuses on one of the most important aspects of the newly emerging unification of these two foundations: the question of covariance with respect to arbitrary change of observer frame. The idea is raised here restricting the formulation of quantum theory over locally hyperbolic regions, such that the the patches do not necessarily consistently mesh together. Of necessity, this requires a bridging of the gap between classical and quantum theory, leading to a wider range of alternatives (the quantum theories with superselection).
The toy problem of writing the quantum field over a timelike flow restricted to an Alexandroff interval is mentioned in passing.
Deformed Poincaré and Doubly Special Relativity
(PDF, 45k)
This is a treatment whose primary purpose is to demonstrate the application of a more transparent coordinate-free treatment of the Poincaré group and its extension to a group that includes dilation.
The basic idea behind “doubly special relativity” is to deform the Lorentz operator by the addition of a momentum-dilation term, such that the resulting group still has Lorentz symmetry. The analysis may then be carried forth in a simple, elegant fashion and lead to results that recall the expressions for torsion and curvature.
Contents
Coordinate-Free Poincaré and Dilation
Poincaré with Deformation
Effective Field Theory
(PDF, 85k)
These are notes taken from a tract, Effective Lagrangian Field Theory, with the math cleaned up and simplified; particularly for the material on the Ward identities. This will be expanded on, in the future.
Contents
1. Effective Field Theory
2. Fermion Lagrangian with Gravity
The Running Coupling Constant in QCD
(PDF, 46k)
A short article discussing the running coupling constant and the renormalization group equation.
Induced Representations
(PDF, 39k)
A short article, derived from the Wikipedia, describing the method of induced representation, in which the representations for a symmetry group are constructed from those of a smaller subgroup.
A Geometric Definition of the Lie Derivative — Fatibene, Ferraris, Francaviglia and Godina;
(PDF, 124k)
Derived from Fatibene, Ferraris, Francaviglia and Godina, with additions made discussing Clifford algebras, and various simplifications made in the math and notation.
Applying the general theory of Lie derivatives, a new geometric definition is given for the Lie derivative of a spinor, generalizing the definition provided by Kossmann over 20 years ago. The latter is recovered for particular infinitesimal lifts, ie., those for Kosmann vector fields.
Contents
Introduction
Spin Structures
Lie Derivatives of Spinor Fields