#  Weyl geometry, topology of space-time and reality of electromagnetic
potentials, and new perspective on particle physics

S. C. Tiwari
Department of Physics, Institute of Science, Banaras Hindu University, and
Institute of Natural Philosophy,
Varanasi 221005, India

###### Abstract

Consistency of Weyl natural gauge, Lorentz gauge and nonlinear gauge is
studied in Weyl geometry. Field equations in generalized Weyl-Dirac theory
show that spinless electron and photon are topological defects. Statistical
metric and fluctuating metric in 3D space with time as a measure of spatial
relations are discussed to propose a statistical interpretation of Maxwell
field equations. It is argued that physical geometry is an approximation to
mathematical geometry, and 4D relativistic spacetime is essentially 3D space
with changing spatial relations. The present work is suggested to have radical
new outlook on elementary particle physics.

###### pacs:

12.10.-g, 12.90.+b

##  I Introduction

Riemann, way back in 1861, had speculated on the physical significance of the
electromagnetic (EM) potentials 1 . Historical evolution of the concept of EM
potentials has witnessed landmark achievements 2 ; 3 , yet their physical
reality remains unsettled. Aharonov and Bohm in a comprehensive article 4
present the role of geometry, topology, and measurability in connection with
the physical reality of the potentials. Unfortunately the manner in which this
question was introduced underlining classical versus quantum has become a
source of confusion in the literature. Aharonov-Bohm (AB) effect has been
demonstrated experimentally many times 5 , however multiple shades of
philosophical/mathematical arguments tend to obscure the nature of the EM
potentials 6 .

The issue has also acquired great significance in the context of the standard
model (SM) of particle physics. A crucial element in SM and modern gauge
theories is the postulated concept of hypothetical internal spaces for the
gauge symmetry. For example, in SU(3) gauge group the set of parameters
$\theta^{a}$ are Lorentz scalars and the index $a=1,2,...8$ for 8 generators
of the group, and the unitary gauge transformation is
$U_{SU(3)}=e^{i\theta^{a}(x)\lambda^{a}/2}$ where $\lambda^{a}$ are the Gell-
Mann matrices. In quantum electrodynamics (QED) there is just one parameter
for the gauge transformation $U(1)=e^{i\theta(x)}$. If the parameters do not
depend on space-time coordinate $x$ then we have global gauge transformation.
Gauge invariance of the action function or the Lagrangian density leads to
conserved Noether currents and corresponding gauge charges, for example, the
electric charge is interpreted gauge charge for U(1) gauge symmetry in QED.
Note that QED is a paradigm for SM and modern gauge theories. However,
originally the idea of gauge symmetry proposed by Weyl 7  represented scale
transformation of the line element in the pseudo-Riemannian spacetime of
general relativity resulting into the change in the length of a vector under
parallel transport. Thus gauge symmetry was a spacetime symmetry
transformation with a wider group of transformations than the general
coordinate transformations of general relativity 7 . We may now have two view-
points. First, the empirical success of SM in precision experiments, and the
detection of the gauge bosons of the electroweak unified theory could be taken
to imply that the question of the reality of potentials has been rendered
vacuous: the dynamical gauge potentials in this theory represent the gauge
bosons i. e. photon and massive weak gauge bosons $W^{\pm};Z$, and they have
physical reality. This point is further elaborated in the next section. The
second view could be as follows. The fundamental role of the internal spaces
in modern theories, e. g. SM and superstrings, departs from the geometrization
of physics based on the intuitive reality of space and time speculated by
Riemann, Clifford, Einstein and Weyl 1 ; 7 . Could one seek alternative
scenario in that tradition?

The present work is focused on this question. Obviously the concepts of space
and time would need radical revision in which nontrivial geometry and topology
acquire central role 8 . In the new perspective it emerges that classical Weyl
geometry of space-time has not only a nontrivial topological structure it also
admits statistical interpretation of geometric quantities. Appraisal of a
comprehensive review on Weyl geometry 9  and a critical reading of Dirac’s
paper 10  show that most of the present work is a new contribution. The
significance of our work lies in the fact that rather than discarding the
reality of space and time we articulate a framework in which space and time
constitute fundamental physical reality.

The paper is organized as follows. In the next section we explain the
difference between geometry and topology; and discuss physical geometry for
physical phenomena and its description. The conception of physical reality
depends in a fundamental way on the approximation and/or limitation of the
mathematical reality. The role of language in human affairs is primarily to
express the thoughts and the feelings in a tangible form; this process has
intrinsic limitation in the exact reproducibility of the subtle thoughts. The
origin of thoughts, mind-brain duality, and the Platonic reality of
mathematics invite attention to deep questions 8 . Note that the symbolic
logic had origin in Boole’s work “The Laws of Thought”. Our proposition that
physics is the language of mathematics echoes the commonplace understanding
regarding the expression of subtle thoughts in tangible form of the language.
The philosophical discussions could be found in 8  and references cited
therein. Here the considerations are primarily concerned with electromagnetism
and Weyl geometry leading to the important result that Weyl geometry could be
endowed with a nontrivial topology and statistical nature.

In section III we investigate various kinds of gauge conditions, i. e. Weyl’s
natural gauge, Lorentz gauge and nonlinear gauge, and their consistency in
Weyl 7  and Weyl-Dirac 10  theories. The main result of this section is that
the principle of gauge invariance may have statistical interpretation, and the
elementary geometrical objects defined by pure gauge potentials may have a
topology akin to that of AB effect 4 . The field equations derived in a
generalized Weyl-Dirac framework 11  are discussed in section IV. Three
important results are: electron could be decoupled from EM field and
potentials, single electron could be visualized as a propagating topological
defect, and new insight is gained on the topological model of photon 12 .

The physical significance of the nonlinear term in scalar curvature in Weyl
geometry remains obscure 7 ; 13 . In flat spacetime geometry Dirac in 1951 14
investigated the nonlinear gauge to develop a new theory of electron. On the
other hand, Gubarev et al 15  discussed topological structures for the minimum
value of the volume integral of the squared potentials. The role of nonlinear
gauge in the generalized Weyl-Dirac theory in connection with 14 ; 15  is
discussed in section V. A logically consistent approach is proposed for
macroscopic system drawing analogy to fluid dynamics. In the last section a
brief discussion on the implications of the present work on modern gauge field
theories and the current ideas on emergent spacetime is presented followed by
concluding remarks.

##  II Geometry and physics

Classical physics comprises of a classical system or object,
observable/measurable physical quantities, and a theoretical description quite
often having a mathematical formalism. A simple example is that of a
macroscopic body that has well defined macroscopic variables determined by
local differential equations. One may also consider a microscopic system and
try to develop a macroscopic system from it using the methods of statistical
mechanics. The intricate question is that of the necessary criteria
characterizing a system macroscopic or microscopic.

Majority of physicists follow Newton’s experimental natural philosophy:
experimentally measured quantities are defined in terms of mathematical
variables and concepts, and depending on the physical laws or hypotheses the
mathematical formalism is developed. In this method abstract mathematics is a
convenient tool. It seems to have given rise to the belief that mathematics is
a language of physics. In CED the experimental force laws of Coulomb and
Ampere, and Faraday’s law of induction serve the basis for Maxwell equations

${\bf\nabla}.{\bf E}=\rho$ (1) ${\bf\nabla}\times{\bf B}={\bf
J}+\frac{1}{c}\frac{\partial{\bf E}}{\partial t}$ (2) ${\bf\nabla}\times{\bf
E}=-\frac{1}{c}\frac{\partial{\bf B}}{\partial t}$ (3) ${\bf\nabla}.{\bf B}=0$
(4)

EM fields are mathematical variables defined from the force laws using a
limiting procedure. First Chapter in the textbook 16  rightly notes that
independent physical reality of EM fields emerges when energy, momentum, and
angular momentum associated with them are defined. We may add that only after
the advent of special relativity and the observed mechanical effects of the
radiation field separated from the sources $\rho$ and ${\bf J}$ the true
physical significance of the EM fields was recognized and accepted.

Mathematically one may proceed defining a second-rank antisymmetric tensor
$F_{\mu\nu}$ in 4D space

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ (5)

and define a scalar

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ (6)

that gives the Euler-Lagrange equation from the action principle

$\partial_{\mu}F^{\mu\nu}=0$ (7)

Definition (5) gives

$\partial_{\mu}F_{\nu\sigma}+\partial_{\nu}F_{\sigma\mu}+\partial_{\sigma}F_{\mu\nu}=0$
(8)

The set of equations (5) to (8) constitutes a mathematical system. Formally
they are equivalent to the source-free Maxwell equations if 4D space is
identified with spacetime: Eqs. (1) and (2) $\rightarrow$ Eq.(7); Eqs. (3) and
(4) $\rightarrow$ Eq.(8). Physical interpretation of the tensor $F_{\mu\nu}$
and the 4-vector $A_{\mu}$ representing EM phenomena is essential to identify
Eqs. (7) and (8) as Maxwell equations. Historically physics laws preceded
Maxwell equations and it appeared as if the role of mathematics was just that
of a convenient description tool.

The constructs (5) and (6) look artificial because we already have the EM
field tensor in mind, however Weyl theory 7  presents a new picture not
recognized adequately. Spacetime metric geometry

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$ (9)

is generalized postulating a linear groundform $W_{\mu}dx^{\mu}$ such that
under gauge transformation

$ds\rightarrow ds^{\prime}=\lambda~{}ds$ (10) $W_{\mu}\rightarrow
W_{\mu}^{\prime}=W_{\mu}+\phi_{,\mu}$ (11)

where $\phi=\log\lambda$ and $\lambda$ is an arbitrary function of
coordinates. Notation and metric convention are those of 10 ; 13 . In Weyl
geometry both length and direction of a vector undergo changes under parallel
transport from spacetime point $x^{\mu}$ to $x^{\mu}+dx^{\mu}$ and the total
change in the length of a vector round a small closed loop is

$\oint\delta l=lW_{\mu\nu}\delta S^{\mu\nu}$ (12)

Here $\delta S^{\mu\nu}$ is the area enclosed by the loop and $W_{\mu\nu}$ is
a geometric quantity termed distance curvature, see section 15 in 7

$W_{\mu\nu}=W_{\mu,\nu}-W_{\nu,\mu}$ (13)

An invariant integral $\int W_{\mu\nu}W^{\mu\nu}~{}\sqrt{-g}~{}d^{4}x$, and
intrinsic geometric property, i. e. Bianchi identity

$\partial_{\mu}W_{\nu\sigma}+\partial_{\nu}W_{\sigma\mu}+\partial_{\sigma}W_{\mu\nu}=0$
(14)

could also be given in Weyl geometry.

Dirac notes that physicists rejected Weyl unified theory 10 , Weyl himself
abandoned it 3 ; 7 , and Einstein objected to it on physical grounds 17 .
Dirac was, however fascinated with its simplicity and beauty. Eddington 13
makes a distinction between natural geometry and world geometry: exact natural
geometry applies to real physical world, and world geometry to conceptual or
mathematical space. Eddington is critical of Weyl geometry on mathematical
ground 13 ; 17 . In our view the functions $g_{\mu\nu},~{}W_{\mu}$
characterize the abstract Weyl space, and it cannot be rejected on either
physical or mathematical grounds. The reason is as follows. The Maxwell-like
equations for the distance curvature $W_{\mu\nu}$ on their own do not
represent physics and the gauge-invariant zero length need not be related to
light propagation, in that case Eddington’s geometry 13  is a natural
mathematical generalization of Weyl geometry, and obviously it is not a
refutation of Weyl geometry. The question now becomes whether Weyl space has a
physical realization, that is, does there exist a physical interpretation of
the geometrical quantities $g_{\mu\nu},~{}W_{\mu}$? In analogy to Einstein’s
interpretation of the coefficiants $g_{\mu\nu}$ of the quadratic groundform
(9) representing the gravitational potentials, the coefficients of the linear
groundform $W_{\mu}$ are interpreted as EM potentials by Weyl. It seems deep
influence of Einstein’s relativity on Weyl prevented him from propounding the
true mathematical significance of his generalized Riemannian geometry. In
fact, Weyl states that a truly realistic mathematics should be conceived in
line with physics. It may be of interest to reproduce here intriguing
statements due to Weizsaecker in this context 18 : “A mathematical formalism
like, e. g. Hamilton’s principle with its mathematical consequences, Maxwell’s
equations with their solutions, or Hilbert space and the Schroedinger
equation, is not eo ipso a part of physics. It becomes physics by an
interpretation of the mathematical quantities used in it, an interpretation
which one may call physical semantics.” It is true that on its own account
Maxwell-like equations for $W_{\mu\nu}$ in Weyl geometry do not represent EM
phenomena. At the end of the book 18  an interesting discussion, mainly
between Linney and Weizsaecker, mentions classical and quantum languages, and
the problem of continuity, but unfortunately remains confusing and
inconclusive. What is the significance of physical semantics? Does it imply
that physics is a language of mathematics?

Translating abstract mathematical reality to physics necessarily involves
metaphysical elements, and what one does in practice is to use the probabilty
rules/statistical quantities to get physical quantities. In the Newtonian
mechanics the absolute time is a metaphysical concept without which the role
of relative time defined by Newton to explain the observed phenomena would
make no physical sense 8 . Mathematically a point, a circle, 2D spherical
surface and a punctured 2D Euclidean plane $R^{2}-\\{0\\}$ are well defined. A
point is dimensionless, a circle with unit diameter has the circumference
$\pi$, and 2D spherical surface has the direction holonomy, i. e. the
direction of a vector parallel transported on the surface after the completion
of a cycle is changed. Removing the origin in 2D plane results into a
topological defect in $R^{2}-\\{0\\}$. All the four objects find numerous
physics applications, in fact, a point particle is ubiquitous: classical
mechanics, CED, QED and SM. Physical realizations must be approximate, a
physical particle is not a point and a hole in $R^{2}-\\{0\\}$ is not an exact
physical void in physical systems like singular vortices in fluid and
optics/light beams. According to Poincare 8  the rule of induction is a mental
act exact in mathematics but approximate in physics. Present arguments suggest
that geometry and topology too are approximate in physical world.

The most radical change that we propound is on the nature of time 8 ; 19 .
Absolute time $T$ has a unidirectional flow in discrete steps and creates the
sequence of natural numbers intuitively experienced by mind; the discrete step
has an extension that presents itself as space with each successive step
acquiring the property of wholeness as a result of mergence. Real numbers are
secondary logical and random constructs. Spatial regions of different
numerical magnitudes have random flow in which coalescence and decomposition
create connected quasi-permanent structures that we call matter. The flow has
meaning with reference to the absolute time and a common or relative time
variable $t$ can be introduced that measures the changes in the flux of the
space. Thus matter has no intrinsic existence but a form of spatio-temporal
bounded structure. In physical geometry a point has a spread and distance
between two points has a fluctuating statistical character. Geometrization of
physics becomes a unified picture in which space and time are fundamental.

The ideas of Riemann, Einstein and Weyl on geometrization of physics are
entirely different than our ideas. Riemann’s infinitesimal geometry is a great
advancement over the finite Euclidean geometry; Weyl terms the idea to gain
knowledge from infinitesimal as the mainspring of the theory of knowledge,
page 92 in 7 . Riemann visualizes metric space only due to the presence of
matter, page 98 in 7 . Einstein’s geometrization of macroscopic world is based
on the continuation of Riemann’s philosophy assuming infinitesimal pseudo-
Riemannian spacetime geometry for gravitation 1 ; 7 . Weyl follows this line
of thinking in the unified theory of gravitation and electromagnetism.

Clifford’s speculation differs from that of Riemann, Einstein and Weyl: matter
is a kind of crinkles of space, footnote 2, page 156 in 1 . Menger’s
statistical metric space 20  endows statistical distribution function for the
distance between any pair of points. Poincare’s observation, see 8 , that
$A=B,~{}B=C\Rightarrow A=C$ is mathematical while physical experiences show
that $A=B,~{}B=C\Rightarrow A<C$ has a probability description in the Menger
space. Menger suggests that the conceptual problems of microphysics could find
resolution in this underlying geometry. The geometric ideas of Clifford and
Menger are of interest in the proposed space and time reality 8 ; 19 ; 21 ,
however they have to be substantially generalized.

Critique on the relativistic time 8 ; 19 ; 21  shows that time in relativity
is just a convenient parameter to label/order changes in spatial lengths
between two points in 3D space, and the role of vacuum velocity of light is
that of a unit conversion factor. Now a physical point in our approach has a
statistical spread and fluctuations, therefore, Menger’s statistical metric
cannot be used; for the distance between two points also we need a suitable
description. Earlier assuming stochastic electron motion 21  it was suggested
that the line element of special relativity represents standard deviation. The
proposition that spatio-temporal objects in the connected whole constitutes
physical reality at a fundamental level liberates physics from extraneous
matter 8 ; 22 . In the stochastic theory 23 ; 24  one follows the legacy of
the Newtonian point particle picture and introduces fluctuations invoking some
kind of hidden thermostat a la de Broglie or Nernst’s zero point field (ZPF).
The present approach dispenses with them since spatial regions are themselves
in random flux, and spatio-temporal objects have fluctuating spread intrinsic
to them. As a consequence point particle, instantaneous velocity and potential
functions are mathematical idealizations. We seek approximations to define
physical properties from the statistical averages depending on length and time
scales.

In special relativity the interpretation of the line element is nicely
explained in section 7 of 13

$ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}$ (15)
$(\frac{ds}{dt})^{2}=c^{2}-v^{2}$ (16)

Time-like interval, i. e. positive value of $ds^{2}$ is physical for material
particles that cannot travel faster than light, i. e. $v<c$ in (16); and
$ds^{2}=0$ defines null-cone or light geometry. Unfortunately existence of
space-like intervals have to be inferred in a contrived way if one accepts
Eddington’s suggestion, moreover physical time is not the common sense time,
see section 8 in 13 . Statistical nature of physical points would suggest that
variance or standard deviation could be the best characterization for the
metric

$ds^{2}\rightarrow\sigma^{2}=\overline{dl^{2}}-(\overline{dl})^{2}$ (17)

where $\overline{dl}$ is the average or mean of the distance between two
points $\|{\bf r}_{2}-{\bf r}_{1}\|$. A parameter $t$ that represents time
variable with reference to the absolute time $T$ can be introduced using $c$
as a unit conversion factor $\overline{dl^{2}}=c^{2}dt^{2}$. Since variance
cannot be negative space-like interval has no meaning, and light-cone has zero
variance. In the flow of space a set of physical points having uniform
relative change could be selected, and the collection of all such sets
$\\{I\\}$ may be interpreted as the inertial frames. Assuming one of the sets
as a standard for calibration one can define expression (17); time $t$ and
light velocity $c$ are convenient parameters. Since changes in spatial
relations may occur in both directions the time variable $t$ may have
inversion $t\rightarrow-t$. There is, however no real time reversal since $t$
has meaning with reference to $T$. For more details we refer to 8 ; 19 .

We have argued that statistical theory has crucial role in translating
mathematics to physics; obviously a general and comprehensive treatment is
needed for this purpose. Here a few examples are worth mentioning. The well
known relativistic relation

$E^{2}-c^{2}{\bf p}.{\bf p}=m^{2}c^{4}$ (18)

may be interpreted statistically in which momentum has average value
$\overline{\bf p}$ and energy is $c^{2}\overline{p^{2}}$. Invariant mass turns
out to be a measure of standard deviation opposed to its intrinsic invariant
value in relativity 13 . Note that in a different way already in
nonrelativistic quantum mechanics Ehrenfest theorem makes use of the
expectation values of position, momentum and energy operators. Infinitesimal
Riemannian geometry in physical phenomena has logical interpretation in terms
of the Menger metric space 20  while pseudo-Riemannian space-time geometry may
be obtained based on the generalization of Eq.(17) such that $g_{\mu\nu}$ has
statistical significance. Stationarity of $\int ds$ in relativity corresponds
to the minimization of the variance $\sigma$ in Eq.(17). Extending the
arguments to CED, the EM fields and potentials have statistical nature. One
may envisage a true stochastic formulation without hidden thermostat or ZPF 23
; 24  for the Newton-Lorentz equation of motion in this approach.

Preceding discussion leads to a challenging question: Do elementary objects
like a single electron and a single photon make physical sense in the
statistical approach? Note that advanced technology and experimental methods
have enabled numerous experiments at a single electron and a single photon
level. Electric charge and spin of electron and spin of photon appear to have
unambiguous physical reality. Electron mass in relativity also has invariant
and intrinsic characteristic, section 13 in 13 . However, re-examining the
relativistic relation written in a different form

$m^{2}c^{2}=M^{2}c^{2}-M^{2}{\bf v}.{\bf v}$ (19)

in view of Eq.(17) statistical interpretation implies that
$\sigma^{2}=m^{2}c^{2}$ and $\overline{\bf p}=\overline{M\bf v}$. Electron
mass $m$ no longer represents intrinsic or internal physical attribute of the
electron. Thus charge and spin are the physical quantities that we have to
explain. Could their understanding come from topology?

Let us briefly explain our idea on the topological defect in 1D. A straight
line segment could be continuously deformed to an arc in which shape and size
are unimportant. Removing any one of the interior points of the arc decomposes
it inro two parts. A circle cannot be obtained from a straight segment using
topological transformation. Let us consider a real line that is a geometric
representation of the set of real numbers $(R^{+},R^{-})$: a real number $a$
is positive if $a>0$, and negative if $a<0$. In the real analysis
conventionally one divides the line such that specifying (not removing) a
point O, i. e. the origin, on the line the right is positive and the left is
negative; such a line is called a directed line. There is no continuous
transformation that maps points on $R^{+}\leftrightarrow R^{-}$. One may
introduce an imaginary axis and define a continuous transformation
$e^{i\theta}$ on the complex plane to map points on $R^{+}\leftrightarrow
R^{-}$. A departure from this construction 28  envisages the origin as a point
defect: real lines on either side have continuous transformations, however
crossing the origin is a discontinuous jump $0^{+}-0^{-}=\delta_{0}$, where
$\delta_{0}$ is an infinitesimally small real number. This point defect is in
1D and distinct from that of the punctured plane $R^{2}-\\{0\\}$. Tifold in 12
may be related with this kind of topological obstruction.

Topology has two key characteristics 25 ; 26 , namely continuity and global,
and defects and discreteness. The concepts of connectedness, adjacent points,
neighborhoods, and continuous mappings belong to the former while counting of
holes/defects to the later; these could be made technically precise 25 .
However metric-independence of topological properties does not preclude the
metric spaces from possessing nontrivial topology. We have argued that ideal
mathematical geometry is realized statistically in physics; the statistical
metric spaces may also have nontrivial topology. Holes/defects in physical
space-time, for example, vortices and tifold 12 ; 22  may be such examples.

Could topological models for single electron and single photon be related with
the standard field theory? Let us consider the photon model in which pure
gauge potential plays the role for the field description of orbifold and
tifold 12 . Now, we face two questions. Firstly the reality of a discrete
localized photon has been questioned in the literature, e. g. Lamb’s anti-
photon viewpoint 27  and in stochastic electrodynamics 24 . However,
photodetectors do show direct evidence of an observed photon 28 . Moreover, in
the SM along with weak gauge bosons photon is also a gauge boson and the
observed data in particle physics experiments 29  show the presence of photon
directly as compared to rather indirect observation of weak gauge bosons
inferred from their decay modes. Low frequence EM radiation and electron
scattering treated classically explains the measured cross sections
satisfactorily using the Thomson formula. An important concept of classical
electron charge radius $e^{2}/mc^{2}$ also follows from this. On the other
hand, at high energy $\nu\approx mc^{2}/h$ quantum theory of Compton
scattering utilizing photon picture becomes essential. For more details we
refer to 16  and review on the photon concept in 30 .

The second question is an age-long one: potentials are convenient
calculational tools or they have independent reality. Aharonov and Bohm 4
state the problem very clearly in the modern context that potentials seem to
have the role of auxiliary variables devoid of physical significance either
because they can be eliminated from the equations in classical theory (CED) or
due to the principle of gauge invariance in quantum theory. Vast literature on
this contentious issue seems to ignore the important fact that the observed
photon is a spin one gauge boson in SM 29  and spacetime symmetries associate
spin one with a vector field not a tensor field. Does it not prove the reality
of EM potentials?

Originally the Maxwell equations were obtained from the macroscopic EM
phenomena. There is no compelling reason either experimental or theoretical to
associate EM fields with a single photon; it is just a convention. Traditional
meaning of the principle of gauge invariance has to be modified. A logically
consistent approach could be to distinguish a single photon and a system of
large number of photons described by statistically averaged physical
quantities. Postulating ${\bf E}={\bf B}=0$ for a single photon has profound
implication on the physical nature of photon and the significance of EM
potentials 12 . Denoting the EM potentials for a single photon by
$\mathcal{A}_{\mu}$ we have the following equations satisfied by them

${\bf\nabla}.\mathcal{A}+\frac{1}{c}\frac{\partial\mathcal{A}_{0}}{\partial
t}=0$ (20)
${\bf\nabla}\mathcal{A}_{0}+\frac{1}{c}\frac{\partial\mathcal{A}}{\partial
t}=0$ (21) ${\bf\nabla}\times\mathcal{A}=0$ (22)

First-order partial differential Eqs. (20)-(22) may be combined to derive the
second-order wave equations

$\nabla^{2}\mathcal{A}-\frac{1}{c^{2}}\frac{\partial^{2}\mathcal{A}}{\partial
t^{2}}=0$ (23)
$\nabla^{2}\mathcal{A}_{0}-\frac{1}{c^{2}}\frac{\partial^{2}\mathcal{A}_{0}}{\partial
t^{2}}=0$ (24)

Note that factoring out electric charge unit $e$ from the EM fields and
potentials one gets the geometrical unit for them; in Weyl geometry a
geometric unit arises naturally. Multiplication with $\hbar$ makes the vector
potential $\mathcal{A}$ to have the dimension of momentum. We have proposed a
screw disclination and tifold model for photon 12  in which spin is a
topological invariant. Particle nature of photon has been suggested to be due
to the topological defect associated with the spin.

In an abstract mathematical formalism Salingaros 31  obtains holomorphic field
equations in 4D spacetime for two types of tensors that formally resemble
Maxwell equations. For a vector field $a_{\mu}$ the holomorphy condition is
the set of equations (18) in 31 . These equations are nothing but Eqs.
(20)-(22) as above. The author emphasizes the spacetime interpretation of EM
phenomena; this interpretation is in the spirit of Weyl’s interpretation in
which EM field tensor is a spacetime distance curvature. Unfortunately, the
equations satisfied by $a_{\mu}$ have been unimaginatively treated as just the
Lorentz gauge condition and the identity that ${\bf E}={\bf B}=0$. Note that
the equations for $a_{\mu}$ in 31  are independent of Maxwell-like equations
for a tensor of type 2, and could be viewed as generalized Cauchy-Riemann
equations. For this reason, mathematically the equations for $a_{\mu}$ are as
significant in 4D as Cauchy-Riemann equations are in 2D. The photon model
based on Eqs. (20)-(22) in 12  acquires added significance.

The present perspective on mathematics and physics, specifically in the
context of electromagnetism, leads to two new directions: 1) foundations of
CED have to be re-examined interpreting EM fields as statistically averaged
macroscopic quantities for photon fluid, and 2) the fundamental role of pure
gauge potentials in the topology necessitates a thorough analysis of the gauge
conditions in Weyl 7 ; 13  and Weyl-Dirac 10  theories.

##  III Gauge invariance and Weyl geometry: new insights

Relativity and 4D spacetime geometry had profound influence on Weyl, and his
aim of a unified theory was greatly inspired by Mie’s theory 7 . Curiously
Weyl perceptively quotes Clifford’s speculation of 1875:“the theory of space
curvature hints at a possibility of describing matter and motion in terms of
extension only”. Weyl asserts that “physics does not extend beyond geometry”,
but reverts to the conventional setup of matter immersed in space in his
unified theory. The central problem in his work is that of the electron model;
according to him the Maxwell-Lorentz theory is invalid in the interior of
electron. Unfortunately the unified theory of Weyl could not make much
progress in this objective 7 . Relativity of magnitude together with the
relativity of motion resulted into a true infinitesimal geometry, namely Weyl
geometry: a point in spacetime needs specification of coordinate system and a
gauge or units of measure. Besides 7 ; 13  a self-contained necessary tensor
analysis in Weyl geometry could be found in Dirac’s paper 10 .

Following the metric and gauge conventions in 10 ; 13  in-invariants and in-
tensors are gauge invariant. Metric tensor $g_{\mu\nu}$ is a co-tensor of
power 2, and $\sqrt{-g}$ has power 4. The generalized affine connection or
Christoffel symbol invariant under gauge transformation denoted by
${}^{*}\Gamma^{\alpha}_{~{}\mu\nu}$ is

${}^{*}\Gamma^{\alpha}_{~{}\mu\nu}=\Gamma^{\alpha}_{~{}\mu\nu}-g^{\alpha}_{\mu}A_{\nu}-g^{\alpha}_{\nu}A_{\mu}+g_{\mu\nu}A^{\alpha}$
(25)

In Weyl geometry the transformation of the tensor quantities has to be
considered for both coordinate and gauge transformations, therefore, one has
to assign Weyl powers as well to them. The meaning of tensor densities and the
role of $\sqrt{-g}$ could be found in sections (16) and (17) in 7 . Following
Eddington 13  the scalar curvature ${}^{*}R$ is termed a co-invariant of power
$-2$

${}^{*}R=R-6W^{\mu}_{~{}:\mu}+6W^{\mu}W_{\mu}$ (26)

Assuming that variational principle for an appropriate action integral yields
the field equations for the unified theory the problem becomes that of
constructing $W$ in the action

$S=\int W\sqrt{-g}~{}d^{4}x$ (27)

Gauge invariance of the action $S$ demands that $W$ must be a co-scalar of
power $-4$ since $\sqrt{-g}$ has power 4. Weyl opts for the squared scalar
curvature ${}^{*}R^{2}$, however he admits that it may not be realized in
nature. Dirac 10  introduces an arbitrary scalar function $\beta$ of power
$-1$ and constructs $W$ linear in ${}^{*}R$. Their respective functions are

$W_{W}=~{}^{*}R^{2}-\alpha~{}W^{\mu\nu}W_{\mu\nu}$ (28)
$W_{D}=\beta^{2}~{}^{*}R-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$ (29)

Here our main interest is regarding the issue of gauge invariance rather than
the field equations. In Weyl geometry the fundamental postulate is that the
comparison of lengths at a distance is not possible and a unique and uniform
calibration does not exist. To reconcile this with the physical experiences
Weyl puts forward two arguments that have been critically elaborated by
Eddington. The first argument is that we may define a unit of length assuming

${}^{*}R=4\lambda$ (30)

where $\lambda$ is a constant. This gauging equation is supposed to represent
a natural gauge. The existence of the natural gauge could be justified in
analogy to the existence of a unique Galilean system of coordinates in the
pseudo-Riemannian spacetime geometry of general relativity. However, there
arises a serious difficulty in fixing a gauge since the law of parallel
displacement of a vector contradicts it. To resolve this problem the second
argument is developed based on the philosophy that there are two ways of
determining the quantities: by persistence and by adjustment. A priori one
cannot conclude that pure transference following the tendency of persistence
is integrable. Size of an electron is determined by adjustment in view of the
presence of space curvature, and not by the persistence of its time history.
Another example is that of electric charge: conservation of charge cannot
explain why electron has the same charge after a lapse of long time, and why
this charge is the same for all electrons. Thus charge is determined by
adjustment, not by persistence, i. e. at every instant of time the state of
equilibrium of negative electricity adjusts to this value. In Eddington’s view
this interpretation of the gauge principle amounts to a graphical
representation of physical facts; a measuring rod taken from one point to
another for calibration could be taken to satisfy the natural gauge condition
(30), and this process should not be viewed as a parallel displacement of a
vector.

The question is: What is the significance of the gauge condition (30) in the
unified field equations? First an important point has to be noted that even if
${}^{*}R$ is not constant one could change the measure to transform it to a
constant in a new gauge. Though unified field equations correspond to
Einstein-Maxwell like field equations; let us drop “like” and accept Weyl’s
interpretation that they represent electromagnetism and gravitation. In that
case the Maxwell equations for $W_{\mu\nu}$ depart radically from standard CED
equations since the gauge potential $W_{\mu}$ itself acquires the role of
source current density

$J_{\mu}\propto W_{\mu}$ (31)

Weyl postulates a unique calibration setting ${}^{*}R=1$ and introduces the
radius of curvature of the universe for the recalibration. This leads to the
gauge condition (30) and $\lambda$ acquires the significance as a cosmological
constant in the Einstein equation.

A remarkably unusual result (31) that charge current density is equivalent to
the potentials has intriguing aspects. First it may be noted that the current
continuity equation follows from the Maxwell equation

$J^{\mu}_{~{}:\mu}=0$ (32)

In view of Eq.(31) formally we have

$W^{\mu}_{~{}:\mu}=0$ (33)

Lorentz gauge condition of the standard CED assumes a radically new role in
Eq.(33): it is inseparably related with the charge conservation law and
according to Weyl, Eq.(31) shows that the electric charge is “diffused
throughout the world” since the gauge potential is present in all space.

Alternative derivation of Eq.(32) is based on the contraction of the Einstein
equation and the use of the gauge condition (30). Two routes to get current
continuity pose a serious problem since the Lorentz gauge condition is
inconsistent with the gauge condition (30). Weyl 7  and Eddington 13  seek a
resolution of this issue exploring the interplay between the energy tensor of
matter and that of EM field. The arguments run as follows. Total energy-
momentum tensor in the Einstein field equation with a cosmological constant
could be assumed comprising of the EM part, $E_{\mu\nu}$, and the matter one
$M_{\mu\nu}$. The contraction gives the trace equation interpreted in terms of
the proper-density of matter

$\rho_{M}\propto~{}R-4\lambda$ (34)

Using the gauge condition (30) $\rho_{M}$ becomes

$\rho_{M}\propto~{}W^{\mu}_{~{}:\mu}-W^{\mu}W_{\mu}$ (35)

Empty space is defined to be the absence of electrons, though EM field is
nonzero, then $\rho_{M}=0$, and Eq.(35) implies

$W^{\mu}W_{\mu}=W^{\mu}_{~{}:\mu}$ (36)

It is clear that the problem has become more intricate as there are now three
incompatible gauge conditions: natural gauge (30), Lorentz gauge (33), and
nonlinear gauge (36). Weyl asserts that the relation (31) has fundamental
importance, in that case, there is no empty space and the Lorentz gauge would
have general validity. However substituting (33) in Eq.(35) one has

$\rho_{M}\propto
W^{\mu}W_{\mu}~{}\Rightarrow~{}\rho_{M}\propto~{}-J^{\mu}J_{\mu}$ (37)

The positivity of mass density implies that $J_{\mu}$ is a spacelike vector,
and it cannot be true for the electric charge. A speculative idea is to
consider interior of the electron where Eq.(37) holds, and the outside region
separated by the boundary of the electron. Eddington suggests that the
internal structure of the electron may have the constituents something like
magnetic charges. In Weyl theory the Lorentz gauge condition occupies special
position, and is assumed to hold in general. On the other hand, the nonlinear
gauge (36) is interpreted to represent the vanishing of the current density
that in view of (31) becomes consistent with the Lorentz gauge. Outer region
of the electron, however has small charge and current extending to infinity.

Preceding account shows that the issue of gauge invariance in the context of
the Weyl action function for the unified theory could not be settled
satisfactorily. Of course, Weyl himself states that, “I do not insist that it
is realized in nature” 7 , and Eddington 13  remarks that Weyl action “has no
deep significance”. We emphasize that questioning the specific action
function, e. g. Eq.(28), does not mean a rejection of Weyl geometry. For this
reason, Dirac’s view on Weyl geometry is very important: wider group of
spacetime transformations for physical laws noted in the conclusion of his
paper 10 . It is important to note that Eddington’s criticism is applicable to
Dirac’s action function (29): combining two in-invariants of different forms,
namely the Maxwell action $F^{\mu\nu}F_{\mu\nu}$ and $\beta^{2}~{}^{*}R$.

Dirac revived Weyl geometry 10  primarily with the motivation of his large
number hyp[othesis in which gravitational constant varies with time. He
assumes a constraint equation

${}^{*}R=0$ (38)

and using $\beta^{2}$ as a Lagrange multiplier constructs a gauge invariant
action integral (29). Dirac lists three choices of the gauge: 1) the natural
gauge; if $W_{\mu\nu}=0$ one has $W_{\mu}=0$, 2) the Einstein gauge $\beta=1$
in which Einstein field equations are obtained for the vanishing of the EM
quantities, and 3) the atomic gauge. Surprisingly Dirac makes no attempt to
explain (38) in relation to Weyl’s natural gauge (30). The issue of gauge
invariance in empty space with the contradicting gauge conditions (33) and
(36) that troubled Weyl and Eddington in connection with CED and electron
structure also did not receive Dirac’s attention. Though symmetry breakdown in
Weyl space for charge conjugation and time reversal is separately pointed out
by him, Weyl’s speculation that inequality of positive and negative
electricity may be related with the past $\rightarrow$ future asymmetry, see
pp 310-311 in 7 , is also not discussed.

Weyl in the Preface to the American edition of 7  states that his attempt of a
unified theory had failed. It is well known that Weyl’s original idea of gauge
symmetry metamorphosed into the modern gauge field theories 2 ; 3 . Revival of
the original Weyl geometry by Dirac 10  and current interest in this geometry
9  are mainly in connection with the problems in cosmology. The literature on
Weyl theory has not unraveled the deeper issues related with gauge invariance
and gauge conditions. Could there be a new strand for Weyl geometry applicable
to particle physics? This question is made precise in the light of the new
perspective emerging from our analysis.

[I] Weyl and later Eddington exclusively focused on electron and diffused
charge interpretation. They did not explore possible implications on the
nature of photon and EM fields. Note that Einstein’s light quantum hypothesis
was well known at that time though the word photon was coined later in 1926.
It is also not clear as to why Dirac in spite of his sustained engagement with
new theories of the electron did not investigate Weyl geometry from this
angle. May be it was due to his preoccupation with the large number hypothesis
in 10 . Setting an arbitrary constant equal to 6 appearing in his action
integral he obtained vacuum field equations. Unfortunately that missed
uncovering geometric origin of current density in the Maxwell equation. This
issue is of fundamental importance for physical interpretation of the EM
potentials and current density. To see this let us compare the standard
Maxwell-Lorentz theory and Weyl theory. Eq.(7) shows that vanishing of EM
field tensor does not necessarily imply vanishing of the EM potentials since
one may have $A_{\mu}=\partial_{\mu}\chi$ and $F_{\mu\nu}=0$. In contrast, in
Weyl theory the current density (31) severely restricts the potentials since
$W_{\mu\nu}=0~{}\Rightarrow~{}W_{\mu}=0$. As a consequence the freedom to
envisage multi-valued pure gauge potential in Weyl theory is lost.

[II] Relaxing Dirac’s assumption of setting the constant equal to 6 Papini and
his collaborators have extensively studied Weyl-Dirac theory 32 . However
their work does not throw new light on the gauging conditions, and merely
repeat Dirac’s argument that to reconcile Weyl gauge principle with the atomic
standrads of length one could postulate two metrics. Papini’s idea on multi-
valued scalar field is however quite interesting.

[III]  Weyl’s motivation for his geometry was unified theory for gravitation
and electromagnetism 7 . The subject as such has three aspects: mathematical
comprising of geometry, physical interpretation of geometric quantities and
field equations, and physics motivated action principle. Most of the criticism
pertain to Weyl action (28) and Dirac too modifies the action to (29) in 10
introducing an additional field variable as a Lagrange multiplier in the
assumed constraint equation (38). For physical interpretation, analogy is
drawn to Einstein’s theory of gravitation. Einstein field equation is founded
on the identification of the metric tensor as gravitational potential and the
equivalence between the Einstein tensor and energy-momentum tensor. Now,
Einstein field equation could be derived from the Hilbert-Einstein action.
But, in general, a symmetry of the action may not necessarily be a symmetry of
the field equation. For a lucid exposition on the calibration/gauge symmetry
we refer to Bock 33 .

Axiomatically the metric tensor $g_{\mu\nu}$ for the quadratic form (line
element) and $W_{\mu}$ for the linear form are the fundamental geometric
quantities in Weyl space. It is well known, see e. g. 13  that in Riemannian
geometry one can construct Galilean coordinate system for constant
$g_{\mu\nu}$, and the necessary and sufficient condition for flat geometry is
that Riemann curvature tensor is zero. Note that the Christoffel symbol
vanishes for constant metric tensor, however it is not a tensor, therefore, it
may assume a nonvanishing value in a new coordinate system. In Weyl geometry
the definition of the distance curvature $W_{\mu\nu}$ ensures that it vanishes
if $W_{\mu}=0$, however, for $W_{\mu\nu}=0$ one could have a nonvanishing pure
gradient for $W_{\mu}$. It may be argued that by a gauge transformation
$W_{\mu}$ could be made equal to zero. In a nontrivial topology of Weyl space
an interesting possibility exists: a nonzero $W_{\mu\nu}$ exists in a small
localized region and it is zero everywhere outside that region, but $W_{\mu}$
is nonzero there. This case is similar to the topology due to a confined
magnetic flux in a solenoid for AB effect 4 .

Concluding this section, a new physical manifestation of Weyl space is
indicated from our analysis where one may consider two ideas: a single
electron and a single photon could be visualized as topological defects in
Weyl space, and EM fields and current density are treated as macroscopic
quantities having statistically averaged significance. In the light of the
proposition of statistical metric tensor discussed in section II it seems
natural to associate statistical meaning to the gauge transformation.

##  IV New Field Equations in Weyl Space: Nontrivial Topology

Rather than Dirac’s emphasis on the large scale structure of the universe in
connection with the Weyl geometry, a new action function was setup to explore
the nature of electron charge in 11 . Let us call it a generalized Weyl-Dirac
theory. A beautiful feature of Weyl geometry is the gauge invariance of zero
length: massless fields become natural objects in this geometry. If one
speculates that at a fundamental level electron comprises of massless fields
then photon and electron assume special place in Weyl geometry. Could there be
a fundamental role of nontrivial topology and the gauge potentials in this
scenario? To address this question we discuss the generalized Weyl-Dirac
theory.

Introducing an in-invariant field $\Psi$ representing electron in the proposed
action integral 11  we have

$S=\int(~{}^{*}R\xi+pF^{\mu\nu}~{}^{*}R_{\mu\nu})\sqrt{-g}~{}d^{4}x$ (39)
$\xi=\Psi^{:\mu}\Psi_{,\mu}$ (40)

Field $\xi$ is a co-scalar of power -2 and it seems akin to Dirac’s
$\beta^{2}$, however unlike cosmological interpretation of $\beta$, here the
field $\xi$ is proposed to be related with the electron and photon.
Variational principle treating $g_{\mu\nu},A_{\mu},\Psi$ as independent
dynamical variables leads to a set of field equations comprising of the
gravitational field equation, modified Maxwell field equation, and a field
equation for $\Psi$, i. e. Equations (25)-(27) respectively in 11 . The
expression for ${}^{*}R_{\mu\nu}$ contains antisymmetric tensor $F_{\mu\nu}$
following 13  in contrast to the symmetrized ${}^{*}R_{\mu\nu}$ used by Dirac
10 . We have returned to the standard notation $A_{\mu},F_{\mu\nu}$ for the EM
quantities but expressed in geometrical units. The source current density in
the modified Maxwell equation, setting the arbitrary constant $p=1/4$, shows
dependence on the gauge potentials

$F^{\mu\nu}_{~{};\nu}=-3(\xi^{:\mu}+2\xi A^{\mu})$ (41)

The conservation law for the current density

$J^{\mu}=-3(\xi^{:\mu}+2\xi A^{\mu})$ (42)

follows from Eq.(41); it could also be obtained as an identity from the
contraction of the Einstein field equation (25) in 11 . Thus we have

$\xi^{:\mu}_{~{}:\mu}+2\xi_{,\mu}A^{\mu}+2\xi A^{\mu}_{~{}:\mu}=0$ (43)

The modified Maxwell field equation (41) differs from that of Weyl 7  and
admits nonvanishing EM potantial for $F_{\mu\nu}=0$ given by

$\xi^{:\mu}+2\xi A^{\mu}=0$ (44)

Equation (44) needs a careful analysis since $\xi$ is also related with the
postulated electron field $\Psi$ in Eq.(40). Let us consider the full set of
the field equations and assume that the EM fields are zero then the energy
tensor $E_{\mu\nu}=0$. Making use of Eq.(44) to eliminate derivatives of $\xi$
the Einstein field equation becomes

$G^{\mu\nu}=-\frac{{}^{*}R}{\xi}\Psi^{;\mu}\Psi^{;\nu}+A^{\alpha}A_{\alpha}g^{\mu\nu}+2A^{\mu}A^{\nu}+2A^{\mu;\nu}-2A^{\alpha}_{~{}:\alpha}g^{\mu\nu}$
(45)

where the Einstein tensor is

$G^{\mu\nu}=R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R$ (46)

Note that the trace of the Einstein field equation (45) gives just the
identity (26). Imposing Weyl natural gauge condition (30) the $\Psi$ field
satisfies the massless wave equation

$\Psi^{;\mu}_{~{}:\mu}=0$ (47)

Introducing the energy tensor for $\Psi$ and $A_{\mu}$

$T^{\mu\nu}_{(\Psi)}=\Psi^{;\mu}Psi^{;\nu}-\frac{1}{2}g^{\mu\nu}\Psi^{;\alpha}\Psi_{,\alpha}$
(48)
$T^{\mu\nu}_{(A_{\alpha})}=-A^{\alpha}A_{\alpha}g^{\mu\nu}-2A^{\mu}A^{\nu}-2A^{\mu;\nu}+2A^{\alpha}_{~{}:\alpha}g^{\mu\nu}$
(49)

Eq.(45) is re-written as

$G^{\mu\nu}+2\lambda
g^{\mu\nu}=-\frac{4\lambda}{\xi}T^{\mu\nu}_{(\Psi)}-T^{\mu\nu}_{(A_{\alpha})}$
(50)

A simple calculation gives the trace expressions

$T_{(\Psi)}=-\xi$ (51)
$T_{(A_{\alpha})}=-6A^{\alpha}A_{\alpha}+6A^{\alpha}_{~{}:\alpha}$ (52)

One of the important consequences of the set of the field equations is that
both the natural gauge condition (30) and the Lorentz gauge condition (33) or
(20) could be simultaneously assumed without any inconsistency. Making this
assumption we have

$T^{\mu\nu}_{(A_{\alpha})}=-A^{\alpha}A_{\alpha}g^{\mu\nu}-2A^{\mu}A^{\nu}-2A^{\mu;\nu}$
(53) $T_{(A_{\alpha})}=-6A^{\alpha}A_{\alpha}$ (54)
$T_{(A_{\alpha})}=R-4\lambda$ (55)

The expression for the energy tensor (53) contains only the vector gauge field
and logically one would expect its interpretation as that of a massless vector
field. In relativistic field theory it is known that for a massive vector
field the representation splits into a 3-vector and a scalar under rotation,
and the Lorentz gauge condition is needed to resolve the question of the
positivity of energy 34 . In our case, the interpretation of (53) as energy
tensor of $A_{\alpha}$ is inferred from the Einstein field equation (50). It
is important to note that the trace expressions (52) and (54) remain unaltered
even if EM fields are nonzero since $E_{\mu\nu}$ is traceless. On the other
hand, the energy expression (90.6) in Weyl theory 13  containing potentials is
attributed to the matter energy tensor. Recall that in the standard CED Eqs.
(20)-(24) show that for EM field free case, and the assumption of the Lorentz
gauge, each component of $A_{\mu}$ satisfies the massless wave equation. Field
theoretic arguments 34  and topological model of photon 12  suggest that pure
gauge field $A_{\mu}$ may be identified as photon field.

The presence of $\xi$ in Eq.(44) and Eq.(50) indicates subtle role of the
coupling between electron field $\Psi$ and photon field $A_{\mu}$. Is it
possible to decouple electron from the influence of the EM field as well as
pure gauge potential? Towards this aim let us examine the consequences of
setting $\xi=0$ . Eq.(40) shows that

$\Psi^{;\mu}\Psi_{,\mu}=0$ (56)

Eq.(50) reduces to

$4\lambda\Psi^{;\mu}\Psi^{;\nu}=0$ (57)

Mathematically there is no contradiction if the expression multiplied by by
$\xi$ in Eq.(50) is also equal to zero. In that case, one gets the Einstein
equation

$G^{\mu\nu}=-T^{\mu\nu}_{(A_{\alpha})}$ (58)

Physical arguments show that one should also take $\lambda=0$ as a solution of
Eq.(57) and

$\Psi^{;\mu}\Psi^{;\nu}\neq 0$ (59)

Trace of Eq.(58) shows that ${}^{*}R=0$ in agreement with $\lambda=0$ .
Eq.(55) reduces to

$R=-6A^{\alpha}A_{\alpha}$ (60)

Decoupled electron field $\Psi$ satisfies the massless wave equation (47) as
well as Eq.(56). Multiplying Eq.(47) by $\Psi$ and integrating over a compact
manifold without boundary gives an uninteresting result when use is made of
Eq.(56). Evidently $\Psi$ field cannot be a continuous field of spacetime
variables. Let us examine the possibility of a discontinuous solution. In the
following for most of the discussions on topological aspects we consider the
illustrative examples in flat spacetime geometry, and conjecture that the
topological characteristics would be carried over to the Weyl space. The
conjecture is motivated by metric-independence of topological properties.

Define a boundary surface at which the field is discontinuous

$F(x_{\mu})=L(x,y,z)-ct=0$ (61)

Instead of a plane wave solution of Eq.(47) the field on the discontinuity
surface could be assumed as

$\Psi=e^{iL-i\omega t}$ (62)

Substituting expression (62) in the wave equation, and making the
approximation that $\nabla^{2}L$ is small compared to other terms we get

${\bf\nabla}L.{\bf\nabla}L=\frac{\omega^{2}}{c^{2}}$ (63)

Using (62) in Eq.(56) we get once again Eq.(63). Following a nice discussion
in the Appendix VI of 35  we define a unit normal vector to the surface (61)

$\hat{\bf n}=\frac{{\bf\nabla}F}{|{\bf\nabla}F|}$ (64)

and the speed of the moving discontinuity by

$v_{f}=-\frac{1}{|{\bf\nabla}F|}\frac{\partial F}{\partial t}$ (65)

In the present case the discontinuity moves at the velocity of light. In
analogy to geometrical optics limit of the light wave propagation 35 , Eq.(56)
or Eq.(63) is the eikonal equation. Thus, the electron field is a defect
propagating at the speed of light.

In the Einstein field equation (45) the energy contribution of $\Psi$ field
disappears, and the decoupled electron has no self-field of the EM origin. The
significance of this field is two-fold: its existence signals a nontrivial
topology of Weyl geometry, and the possibility of nonzero $\xi$ makes it
observable through EM interaction via Eq.(42). The drawback of scalar field
representation is that the spin property is not explained, in fact, it
corresponds to spinless electron in CED; for a recent attempt to explore spin
of electron, see 22 .

Decoupled photon field satisfies Eq.(58) and Eq.(60). Assuming traceless
energy-momentum tensor for $A_{\mu}$ Eq.(58) leads to $R=0$. In this case
Eq.(60) reduces to the nonlinear gauge

$A_{\alpha}A^{\alpha}=0$ (66)

Mathematically Eq.(66) shows that $A_{\alpha}$ is self-perpendicular. The most
important result is that the gauge conditions are unambiguously consistent and
compatible: $\lambda=0~{}\Rightarrow~{}^{*}R=0$; each term in ${}^{*}R$
vanishes, therefore $R=0$, Lorentz gauge (33) and nonlinear gauge (66) are
also satisfied. Note that for a pure gauge field , $F_{\mu\nu}=0$ and using
the Lorentz condition one finds that $A_{\mu}$ satisfies the massless vector
wave equation. To unravel the topology of Weyl geometry we examine the nature
of $A_{\mu}$ in three ways.

I  Simplest and conventional solution is given by

$A_{\mu}=\eta_{,\mu}$ (67)

Substituting (67) in the Lorentz condition we have

$\eta^{;\mu}_{~{}:\mu}=0$ (68)

and the nonlinear gauge (66) becomes

$\eta^{;\mu}\eta_{,\mu}=0$ (69)

Eqs.(68) and (69) for $\eta$ are exactly the same as the ones for $\Psi$,
therefore, for a nonvanishing $\eta$ the propagating discontinuity
interpretation would be essential. The photon field becomes a propagating
topological defect.

It is, however not necessary to proceed with the solution (67); the field
$A_{\mu}$ itself shows the topological property if we consider the
discontinuous surface (61) and calculate $A_{\mu}$ on it following 35 . The
vector potential ${\bf A}$ and the scalar potential $\Phi$ may be defined on
either sides of the surface (61) and using the unit step or Heaviside function
we may write, for example, for the vector potential

${\bf A}=H(-F){\bf A}^{1}+H(F){\bf A}^{2}$ (70)

The derivative of the step function is the Dirac delta function, therefore
space and time derivatives of the potentials can be calculated using the
standard calculus. For example,

${\bf\nabla}\times{\bf A}=H(-F){\bf\nabla}\times{\bf
A}^{1}+H(F){\bf\nabla}\times{\bf A}^{2}+\delta(F){\bf\nabla}F\times\Delta{\bf
A}$ (71) $\frac{\partial{\bf A}}{\partial t}=H(-F)\frac{\partial{\bf
A}^{1}}{\partial t}+H(F)\frac{\partial{\bf A}^{2}}{\partial
t}+\delta(F)\frac{\partial F}{\partial t}\Delta{\bf A}$ (72) $\Delta{\bf
A}={\bf A}^{2}-{\bf A}^{1}$ (73)

Expression(73) denotes the discontinuous change. The Lorentz condition using
(64) and (65) finally leads to

$\hat{\bf n}.\Delta{\bf A}-\Delta\Phi=0$ (74)

Note that ${\bf E}=0$ and ${\bf B}=0$ also give Eq.(74). On the moving surface
the potentials are calculated using the fact that ${\bf A}^{2},\Phi^{2}$ are
zero, therefore, Eq.(74) becomes

$\hat{\bf n}.{\bf A}-\Phi=0$ (75)

It is easy to verify that in view of Eq.(75) the nonlinear gauge (66) is
satisfied. Thus the photon is a discontinuity of the potentials propagating at
the speed of light.

II  The field theoretic description of topological photon 12  is given based
on $F_{\mu\nu}=0$ and the Lorentz condition. Here in addition we have the
constraint (66). Screw disclination for the vector field ${\bf A}$ has been
interpreted in terms of 2D defect termed orbifold and 1D defect as tifold in
12 . The 4-vector field $A_{\mu}$ is split into transverse ${\bf A}_{t}$ and
longitudinal components $(A_{z},\Phi)$ choosing a propagation direction along
z-axis. Following 36  the assumed solution in complex representation is

$A_{x}=kre^{i\chi}$ (76) $A_{y}=ikre^{i\chi}$ (77)

where $\chi=\theta+kz-\omega t$. It is easy to verify that

$\frac{\partial A_{x}}{\partial x}+\frac{\partial A_{y}}{\partial y}=0$ (78)

and the Lorentz condition reduces to

$\frac{\partial A_{z}}{\partial z}+\frac{1}{c}\frac{\partial\Phi}{\partial
t}=0$ (79)

Could one find $A_{z},\Phi$ such that for the solutions (76) and (77) the
nonlinear gauge (66) is also satisfied? Since the sum of the squares of the
real parts of (76) and (77) is $k^{2}r^{2}$ a simple solution $A_{z}=kz$ and
$\Phi=-\omega t$ leads to a curious result: Eq.(66) becomes the null-cone
equation

$x^{2}+y^{2}+z^{2}-c^{2}t^{2}=0$ (80)

Note that Nye 36  considers the electric field vector to study disclination
for polarization effects of EM wave; in that case ${\bf\nabla}.{\bf E}=0$.
Here we consider $A_{\mu}$ that satisfies the Lorentz condition, and hence
follows Eq.(79).

III  Definition of $F_{\mu\nu}$ , expression (5) . in analogy to curl in 3D
could be treated as curl of $A_{\mu}$ in 4D , see section 3.2 in 13 . The
4-divergence of $A_{\mu}$ is $\partial^{\mu}A_{\mu}$. Now the curl-free and
divergence-free vector in 3D may have peculiar characteristics made
transparent in the abstract language of differential forms and de Rham
periods. An intuitive and physics-oriented account on de Rham periods on the
manifolds can be found in 37 ; also summarized in 12 . In a Euclidean domain
formally a 1-form is decomposed into three parts : an exact form (nonzero
divergence, zero curl) , a closed form (zero divergence, nonzero curl), and a
harmonic form (both divergence and curl are zero). According to de Rham
theorem the closed loop integral of the harmonic form over a non-bounding
cycle counts the number of holes/topological defects. It may be asked if de
Rham theorem could be extended to 4D spacetime.

Formally photon field, i. e. $A_{\mu}$ is curl-free $F_{\mu\nu}=0$ and
divergenceless $\partial^{\mu}A_{\mu}=0$; it implies that $A_{\mu}dx^{\mu}$ is
a harmonic form with the loop integral

$\oint A_{\mu}dx^{\mu}=2\pi N$ (81)

Note that in Weyl geometry $A_{\mu}$ has the dimension of $length^{-1}$. Here
N is an integer counting the number of topological defects, and for a single
photon $N=1$. Weyl geometry for harmonic 1-form (81) acquires a nontrivial
topology. However, the non-Euclidean spacetime geometry limits the
straightforward application of de Rham theorem. We may get some physical
insight considering the vector potential ${\bf A}$ such that ${\bf B}=0$, and
take the manifold $R^{3}-\\{0\\}$. An example of a harmonic form is given in
Appendix C of 37

${\bf A}=\frac{y}{r^{2}}\hat{\bf i}-\frac{x}{r^{2}}\hat{\bf j}$ (82)

Though ${\bf\nabla}.{\bf A}=0$ and ${\bf\nabla}\times{\bf A}=0$, the loop
integral enclosing the origin is nonzero

$\oint{\bf A}.{\bf dl}=2\pi$ (83)

The harmonic form (82) may also be given a physical realization in the form of
a singular vortex: the fluid flow in concentric circles around the origin at
which one has a singularity of the velocity field. Note that multiplying
$A_{\mu}$ by $\hbar$ its dimension is that of momentum, and Eq.(83) may be
related with the spin of the photon 30 .

In this section possible nontrivial topology of the Weyl space has been
investigated based on the generalized Weyl-Dirac theory 11 . The most
important new result is that $\Psi$ field for spinless electron and $A_{\mu}$
for photon represent topological defects in space-time of Weyl geometry.

##  V Nonlinear Gauge, Particles and Fields

In the preceding section Eq.(56) and Eq.(68) are look-a-like relativistic
Hamilton-Jacobi (HJ) equations for a massless particle. In classical mechanics
point particle dynamics may be described using the Hamilton principal function
$S(q,p,t)$ that assumes the following form if Hamiltonian is a constant of
motion

$S(q,p,t)=W(q,p)-Et$ (84)

Here $W(q,p)$ is the characteristic function, and $(q,p)$ canonical variables.
In analogy to optics , $W$ in HJ equation has the same role as the eikonal
$L$. Textbook 38  section (10-8) makes quite insightful remarks on the
geometrical optics limit of the light wave and the role of HJ equation in
understanding particle trajectory in Schroedinger wave mechanics. The main
problem that remains unsatisfactorily resolved till date is the meaning of a
localized point particle in the wave description or the continuous field
theoretic description.

In the preceding section we have suggested that particle aspect is embodied in
a topological defect. Unfortunately the continuous fields of space and time
obeying specific partial differential equations seem to have intrinsic
limitations for incorporating the topological objects as we have seen for the
$\Psi$ field or nonlinear gauge for $A_{\mu}$ that makes the necessity of the
discontinuous field natural, however the particle interpretation and the
description of the observed physical phenomena, e. g. electron and EM field
interaction and EM waves, in terms of these topological objects are not
obvious. Reflecting on his life-long efforts to understand waves and particles
de Broglie made profound remarks on them in his essay 39 . Here we note two of
them that seem to be useful for the present considerations. First one concerns
the definition of a particle as a localized object of high energy
concentration that to a first approximation is a shifting singularity. The
second point is that the principle of least action is a particular case of the
second law of thermodynamics; entropy of a particle is defined by the relation
entropy/Boltzmann constant is equal to action/Planck constant. Now
thermodynamical concept of a particle is very important: it brings the role of
statistical mechanics to the prominence even for an isolated free particle; in
de Broglie theory it is caused due to a hidden thermostat. The present work
marks a more radical departure than that of de Broglie: space and time are
endowed with nontrivial topology, and the geometric objects and fields are
proposed to have the statistical nature.

We elucidate these ideas considering the significance of the nonlinear gauge
in the context of 14 ; 15 . Dirac attempts a modification of CED with the aim
to cure the problem of infinities in QED 14  proposing the nonlinear gauge

$A_{\mu}A^{\mu}=\frac{m^{2}c^{4}}{e^{2}}$ (85)

The Maxwell action is modified incorporating the constraint (85) using a
Lagrange multiplier $\Lambda$. The charge current density is identified to be

$J_{\mu}=-\Lambda A_{\mu}$ (86)

Dirac argues that if the source-free field has the potentials $A_{\mu}^{*}$
then the gauge transformed potentials $A_{\mu}^{*}+\Phi_{,\mu}$ corresponds to
the charges. Eq.(85) then becomes

$(p_{\mu}+eA_{\mu}^{*})(p^{\mu}+eA^{\mu*})=m^{2}c^{4}$ (87)

where $e\Phi{,\mu}$ is interpreted as the energy-momentum 4-vector $p_{\mu}$
in the relativistic HJ equation (87) for the electron. Note that Dirac
develops his theory in flat spacetime. The formal similarity of the charge
current density (86) with that of Weyl theory, and for constant $\xi$ with the
generalized Weyl-Dirac theory 11  is noteworthy. However in the generalized
Weyl-Dirac theory $\xi$ is a co-scalar field, therefore, it cannot be a
constant. Moreover the analogue to HJ equation for electron is Eq.(40) that
suggests that mass is a space-time dependent field variable, and $\Psi$ field
does not appear directly in the current density (42). In contrast to particle
aspect introduced through $p_{\mu}$ in Dirac theory we have the propagating
$\Psi$ field discontinuity representing the particle.

Gubarev et al 15  introduce a novel idea seeking the physical significance of
the gauge noninvariant quantity $A_{\mu}A^{\mu}$. The volume integral of this
quantity in Euclidean space is shown to have minimum value under the gauge
transformation for a specific gauge condition. For example, ${\bf
B}={\bf\nabla}\times{\bf A}$ is gauge invariant, and the volume integral
$\int{\bf A}.{\bf A}d^{3}x$ under the gauge transformation is stationary for
the Coulomb gauge ${\bf\nabla}.{\bf A}=0$. Authors emphasize the point that
their objective is to find the minimum value of this quantity itself that may
throw light on the nontrivial topology. This motivation comes from the
considerations on the vacuum condensates, for example, quark and gluon
condensates in QCD.

It would, of course, be logically more satisfying to treat $A_{\mu}$ as a
statistical field variable, and interpret the minimum value in terms of the
averages

$\sigma(A_{\mu})=\overline{A_{\mu}A^{\mu}}-\overline{A_{\mu}}~{}\overline{A^{\mu}}$
(88)

The authors 15  do consider the expectation values of the field operators in
QFT, however here we associate probability distribution function with the
potentials.

Let us recall that the pure gauge potential $\mathcal{A}_{\mu}$ satisfies the
Lorentz gauge condition (20) and the minimum value of the integral of
$\mathcal{A}_{\mu}\mathcal{A}^{\mu}$ is zero in view of Eq.(66). The potential
$\mathcal{A}_{\mu}$ is harmonic (81) for the nontrivial topology representing
a single photon. For a monochromatic photon beam of uniform photon number
density it would be natural to assume Eqs. (20) and (66) for the photon beam.
In a fluid flow with velocity ${\bf v}$ and probability density $\rho_{f}$ the
local conservation of particles satisfies the continuity equation

$\frac{\partial\rho_{f}}{\partial t}+{\bf\nabla}.\rho_{f}{\bf v}=0$ (89)

Statistical distribution function for photon fluid in analogy to this may be
introduced interpreting

${\bf A}=f_{p}(x_{\mu})\mathcal{\bf A}$ (90)

as the momentum density and the Lorentz condition as the energy-momentum
conservation equation; note that the geometric quantity $\mathcal{A}_{\mu}$ in
Weyl geometry has been multiplied by Planck constant. In 15  the minimum value
of the vacuum expectation value of $<0|A_{\mu}A^{\mu}|0>$ has been discussed
and suggested to be nonzero. In the present paper we have geometry/topology
plus statistical/thermodynamical approach: at zero temperature the minimum
value is proposed to correspond to the frozen phase of photons with zero
momenta and only spinning topological objects possessing energy
$\frac{h\nu}{2}$ per photon 12 . It may be contrasted with the assumed ZPF in
stochastic electrodynamics, see section 2.2 in 24 .

The main inference that could be drawn from these considerations is that the
EM fields in CED have to be interpreted as averaged statistical quantities
representing the photon fluid, i. e. rotating fluid of microscopic spinning
topological objects, namely the photons. Interpreting $A_{\mu}$ as averaged
energy-momentum vector would imply to interpret $F_{\mu\nu}$ in analogy to the
antisymmetric second rank angular momentum tensor $L_{\mu\nu}$: ${\bf B}$ as
averaged angular momentum (orbital plus spin) and ${\bf E}$ as the averaged
value of the energy and the momentum of the fluid as a whole about center of
energy; this interpretation is based on the classical relativistic mechanics,
see section 14 in 40 .

##  VI Discussion and Conclusion

Radically new approach to geometrize physics is articulated in the present
paper: (1) spacetime of relativity has statistical nature generalizing
Menger’s statistical metric space to Eq.(17), (2) EM potential
$\mathcal{A}_{\mu}$ has fundamental reality representing a single photon as a
topological defect in space-time, and (3) gauge conditions in original Weyl
geometry, i. e. Weyl natural gauge, Lorentz gauge and nonlinear gauge are
proved to be consistent in generalized Weyl-Dirac theory 11  necessitating the
existence of topological defects. It is argued that Maxwell field tensor and
metric tensor represent statistically averaged quantities.

Space-time itself is visualized as a fluid comprising of microscopic particles
defined by topological defects, for example, spinless electron as a moving
surface discontinuity and photon as a harmonic 1-form. EM waves would be like
sound waves in a fluid whereas photons are localized space-time topological
defects, Eq.(81). Interaction of electron with EM field in the present
framework is viewed as a random scattering of electron propagating in the
photon fluid. Momentum exchange with most probable fraction of photon momentum
given by the fine structure constant is envisaged from

${\bf p}-\frac{e}{c}{\bf A}~{}\rightarrow~{}\frac{\bf
p}{\hbar}-\frac{e^{2}}{\hbar c}{\bf A}_{g}$ (91)

where ${\bf A}_{g}$ is in geometrical units a la Weyl geometry. The most
important implication of the present work is that it paves the path for
developing an alternative theory of elementary particles and their
interactions solely on the geometry and topology of physical space and time 22
. We emphasize that the present ideas are radically new as compared to those
of Riemann-Einstein-Weyl on spacetime, however they adhere to the conventional
wisdom of space and time reality in contrast to the current radical ideas
articulating emergent spacetime and/or discarding space and time reality.

To put the speculations in perspective we present a brief discussion on
Einstein’s belief 41  and alternative ideas 23 ; 24 ; 39 . Einstein in reply
to the criticisms 41  states his belief in field theory as a program for
physics. Fields are continuous functions in 4D spacetime continuum.
Illustrating his idea taking the example of general relativity he puts forward
three points. E1: Physical things are described by continuous functions of
spacetime, E2: the fields are tensors, e. g. $g_{\mu\nu}$ for gravity, and E3:
physical measurability of the invariant line element. According to him the
construction of a mathematical theory rests exclusively on E1 and E2. If a
complete physics theory exists E3 is not required. Einstein expresses serious
reservations on radical efforts of Menger in this perspective. What is the
meaning of a mathematical theory of physics? As pointed out in section II
mathematical objects like a geometrical point or a circle are mental objects
and mathematical logic is sufficient to develop a mathematical realm, for
example, Riemannian geometry or Weyl geometry. However physical objects need
physicalization or approximation to the mathematical objects, e. g. the
fields. Therefore, the geometrization of physics as a program has to alter the
Einsteinian ideas, and adopt in some way Clifford and Menger speculations. The
present approach has sought this objective based on the Weyl geometry.

The role of probability in fundamental physics distinct than the orthodox
quantum theory has also been of interest among some physicists; unfortunately
such works have remained sidelined in the mainstream physics literature. A
common lament by de Broglie 39  , Luis de la Pena 23  and Boyer 24  is that
full potential of their ideas remains unexplored. Could the present modest
contribution provide impetus to such efforts? It seems the inclusion of
topological model of photon and associated interpretation of zero point energy
would be essential to develop a complete stochastic theory for Maxwell
equations and Newton-Lorentz equation of motion.

In conclusion, consistency of various gauge conditions in Weyl geometry is
thoroughly studied and the physical reality of EM potentials is established.
Harmonic 1-form in 4D spacetime of Weyl needs further investigation in
connection with the physical mechanism of the AB effect 6 . Novel aspects on
the topology and statistical nature of space and time geometry proposed here
would strengthen the outlook on particle physics based on the real wave
equations in 22 .

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