# Topological Dipole Field Theory

### Abstract

Quantum theory has found that elementary particles in addition to the classic field quantity have also quantum-mechanical degree of freedom. This research paper defines another hypothetical intrinsic degree of freedom which has a topological nature.  A topological quantum field theory is constructed to this hypothetical degree of freedom.

### Introduction

A well-known topological quantum field theory is the Chern-Simons theory which is strongly related to knot theory (Chern et al. 1974). This theory is applied in various disciplines of theoretical physics. The governing field of the theory is the non-abelian 1-form geometric connection  where the action of the Chern-Simons theory does not change when it is varied by . Therefore the field theory is topological. Observables of this theory are given by knot invariants. Chern-Simons theory is a Schwarz-type topological quantum field theory in which the whole action is independent on variations in geometric quantities. There were postulated a couple of other topological quantum field theories in literature (Atiyah 1989).

In physics, the occurrence of topological defects are also well-known.  As an example, ordered media can have topological defects (Mermin 1979). Topological defects can affect electromagnetic interactions taking place in a physical system. Electromagnetic systems with topological defects are also studied in research literature (Bakke et al. 2010).

This research paper shows how it is possible to generalize the concept of topological defects to elementary particles. Since quantum physics has found out that elementary particles have properties that are not predicted by classical physics

(e.g. the spin of a particle) it can be assumed that some other microscopic properties of particles are present but not predicted yet. The main purpose of this paper is to show that charged elementary particles like electrons can possess additional internal degree of freedom. This is performed by regarding a topological quantum field theory which is able to take topological defects into account. Primarily, particles with electric charge are described by Quantum electrodynamics. Quantum electrodynamics is a physical theory with a very high agreement with experiments. However, there can be a difference between the real behavior of charged particles and the predictions of Quantum electrodynamics. Some more detailed experimental tests for Quantum electrodynamics are the measurement of the anomalous dipole moment of the muon (Hagiwara et al. 2007).

For the derivation of a quantum field theory which includes topological corrections to ordinary quantum electrodynamics a Witten-type topological quantum field theory is proposed (Witten 1988). The basic quantum field is assumed as a dipole field strength tensor that arises from topological defects. This field is regarded as the geometric quantity of the theory. An additional dipole field will generate a proper generalization of the electromagnetic field strength tensor in quantum electrodynamics. It is shown, how quantum observables will be independent on the dipole field strength tensor. This ensures that the quantum field theory is a topological quantum field theory.

### Theory

Witten-type topological quantum field  theories are based on cohomology theories. The action  of such theories must contain a symmetry. More precisely, there must exist a differential operator  such that . This differential operator is similar to a Lie derivative and must satisfy the exactness condition

(1)

The theory treated in this paper is assumed to be a 4-dimensional theory like ordinary quantum electrodynamics. It is assumed that the operator  is the Čech coboundary map. Assuming that geometric fields are defined on a set  with arbitrary sets  and the condition that  is a point in spacetime. Then it is easy to compute the action of the Čech coboundary operator on a function :

.     (2)

Here, the hat denotes that the set is omitted. It is easy to show that the definition (2) satisfies the exactness condition (1). The number of intersecting sets  where the intersection of these sets generates the spacetime point can be chosen arbitrary. A simple case is given by the choice . Let be  a 2-form field which is the dipole field strength tensor of the charged elementary particle. Because this field is induced by hypothetical topological defects in the elementary particle, this field must be the geometrical quantity of the action. When assuming that the generalized electromagnetic field strength tensor is given by

(3)

with the electromagnetic 1-form gauge connection , the field  can be interpreted as the intrinsic curvature. In the model treated in this paper quantum electrodynamics is replaced by the generalized field strength tensor (3) where the field of the intrinsic degree of freedom  induces additional topological interactions. The fields  are the observables of the topological quantum field theory. Therefore,  must lie in the cohomology classes of the Čech cohomology, i.e.  but  with an arbitrary 2-form field .

A suitable action for the generalized quantum electrodynamics has the form:

.     (4)

The field  is the fermion field. Assuming that  and  always satisfying the Čech cocycle condition , the action of  on the first term of (4) vanishes. Considering the action functional existing on the spacetime manifold

(5)

then it can be easily shown that  if it holds  for all .

When the auxiliary condition

(6)

is imposed, it is straightforward to show that the only tensor invariant of the antisymmetric tensor  that is not vanishing is given by .

To show that above assumptions construct a topological quantum field theory, the action (5) must be varied by the field . If all quantum fields vanish on the boundary of it holds  for arbitrary 4-form field  . For the variation operator  it holds the functional derivative rule . From

and (6) it follows .

Hence:

(7)

The condition (7) is also a condition for a Witten-type topological quantum field theory. It ensures that the every observable that lies in a Čech cohomology class has an expectation value

(8)

is independent on variations in the geometrical quantity . If the auxiliary field  is introduced which is also involved in the functional integration, i.e. , the action  can be extended by a term that describes the auxiliary condition in the following manner:

(9)

The action in the general form (9) describes the complete quantum field theory.

### Conclusions

This generalization of quantum electrodynamics by topological quantum fields is a model for describing a hypothetical intrinsic dipole moment of a charged elementary particle. Above considerations show that the topological term of the action is similar to the abelian Chern-Simons theory. The photon field strength tensor of quantum electrodynamics is coupled to an additional dipole moment which is of purely topological nature.

## References

Chern, S.- S. & Simons, J. "Characteristic forms and geometric invariants". Annals of Mathematics, 1974, 99 (1): 48-69. doi: 10.2307/1971013

Atiyah, Michael. "Topological quantum field theories". Publications des Mathématiques de l'IHÉS, 1989, 68 (68): 175-186. doi: 10.1007/BF02698547

Mermin, N. D. "The topological theory of defects in ordered media". Reviews of Modern Physics , 1979, 51 (3): 591.

doi: 10.1103/RevModPhys.15.591

Bakke, K. & Ribeiro L.R. & Furtado C. "Landau quantization for an induced electric dipole in the presence of topological defects". Central European Journal of Physics, 2010, 8 (6): 893-899. doi: 10.2478/s11534-010-0006-z

K. Hagiwara & A.D. Martin & Daisuke Nomura & T. Teubner. "Improved predictions for g−2 of the muon and αQED(MZ²)". Phys.Lett. B, 2007, 649: 173-179. doi: 10.1016/j.physletb.2007.04.012

Witten, E. "Topological quantum field theory", Communications in Mathematical Physics, 1988, 117 (3): 353–386. doi: 10.1007/BF01223371

### Showing 10 Reviews

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3

The individual topics that are mentioned are
certainly interesting, but they appear a bit mixed up by the author. I
am not sure what he means by a dipole field strength tensor. Perhaps a
polarization tensor? Adding a (non-closed) 2-form B-field to the
electromagnetic tensor F can be useful to describe magnetic charges
(monopoles).

This review has 2 comments. Click to view.

I have posted a comment below where I explained why the theory is called "Topological Dipole Field theory" (see Review of Patrick Linker and in the last comment of it).

The topological Lagrangian density is inspired by the Chern-Simons theory for 1-form fields. As the Chern-Simons theory describes knot interlinkings, the Topological Dipole Field Theory describes topological interlinkings between 2-form intrinsic Dipole fields. Since these intrinsic Dipole fields couple on gauge bosons (see also other papers about TDFT), interlinkings (i.e. self-interactions) between these gauge bosons are present. As there is only the field B in the topological Lagrangian density (the topological Lagrangian depends on no other degree of freedom than the Dipole field and recovers ordinary Quantum electrodynamics (more General: ordinary Yang-Mills theory) in absence of the fields B), the theory has the most simple and Lorentz-invariant form (it is a generalized Gaussian integral in the B fields).

2

Additional Information for better understanding of the paper:
For computation of scattering amplitudes one can define
B = gB'+ \phi \delta B
where \phi is the 0-form indicator function defined on the support of B and g is a coupling constant. When applying the Operator \delta to B one obtains (due to \delta B' = 0):

\delta B = \delta(gB'+\phi \delta B) = g \delta B' + (\delta \phi) \delta B + \phi \delta^2 B =
(\delta \phi) \delta B.

Note that since \delta \phi is the indicator function defined on the support of the Čech coboundary and therefore on the Support of \delta B, it holds also (\delta \phi) \delta B = \delta B. Hence, above decomposition is possible.

Now the Haar measure d[B] can be decomposed in a cohomological part and in the non-exact part \phi \delta B , i.e. \int d[B]= \int d[B'] \int d[\phi \delta B]. For abbreviation it is set

W = \phi \delta B = (\delta \phi) \delta B

since \phi and \delta \phi have the same support as \delta B. The partition function can be written as:

= \int d[\lambda] \int d[B'] \int [W] O exp(igB' \wedge W + i(1+\lambda)W \wedge W). (*)

Physical observables are given by products of B'. In this case the Integration over B' in partition function (*) can be performed such that delta distributions and derivatives of delta distributions in argument W occur. Finally, it can be integrated over W and \lambda which gives a number. It is important that Integration over \lambda is performed between -L and +L with L \mapsto \infty. With this regularization, the infinities can be absorbed in the coupling constant g, i.e.

g' = gL.

Now the coupling constant of Topological Dipole Field Theory is given by g'. For the computation of expectation values the division by the normalization factor

N = \int d[B'] \int d[W] \int d[\lambda] exp(igB' \wedge W + i(1+\lambda)W \wedge W) =
\int d[W] \int d[\lambda] (2 \pi)^n \delta(gW) exp(i(1+\lambda)W \wedge W) =
\int d[\lambda] (2 \pi g^(-1))^n.

with n \mapsto \infty has to be executed. Noting that \int d[\lambda] \lambda = 0 and \int d[\lambda] \lambda^2 \neq 0 it is easy to show that the first nontrivial physical expectation value that does not vanish is an expectation value over a polynomial in B which has at least the
4th degree. This expectation value is of order g'^(-4); hence if g'^(-1) is small then the correction of Topological Dipole Field theory to Quantum Electrodynamics is also small.

The differential Operator \delta maps to the Čech coboundary that must lie in an infinitesimal neighborhood of the spacetime Point to act as a differential operator.

This review has 3 comments. Click to view.

Sorry, there was a small error in this review. Instead of g' = gL it must be g' = gL^(-1).

To be correct in the calculations (with correct order exponent) it must hold:
g' = gL^(-1/2).

In general it must hold for the observable expectation value (the word "Partition function" was not the right word in this review): = \int d[\lambda] \int d[B'] \int [W] O exp(i \int_M gB' \wedge W + i \int_M (1+\lambda)W \wedge W). (integral over spacetime is included in the action). After Integration one obtains products of Delta distributions and derivatives. As an example, if O = 1 one obtains the normalization factor:
N = \int d[\lambda] \int d[W] (2 \pi g^(-1))^n \delta^n (W)exp(i \int_M (1+\lambda) \delta B \wedge \delta B).

The number n denotes the "number of integrals over B'".

In general it must hold for the observable expectation value (the word "Partition function" was not the right word in this review): = \int d[\lambda] \int d[B'] \int [W] O exp(i \int_M gB' \wedge W + i \int_M (1+\lambda)W \wedge W). (integral over spacetime is included in the action). After Integration one obtains products of Delta distributions and derivatives. As an example, if O = 1 one obtains the normalization factor:
N = \int d[\lambda] \int d[W] (2 \pi g^(-1))^n \delta^n (W)exp(i \int_M (1+\lambda) \delta B \wedge \delta B).

The number n denotes the "number of integrals over B'".

Now I explain why the theory is called "Topological Dipole Field Theory": Clearly it is a topological Quantum field theory. It is called a Dipole field theory because the 2-form generalized field strength Tensor F = dA+B' generates the Lagrangian density:

L = F \wedge *F = dA \wedge *dA + dA \wedge *B' + B' \wedge dA + B' \wedge B'.

The Terms dA \wedge *B' + B' \wedge dA remember on the coupling of the electrodynamic field to a Dipole. Here, the 2-form field B' contains Information of the Dipole e.g. the Dipole Moment.

it must hold

L = F \wedge *F = dA \wedge *dA + dA \wedge *B' + B' \wedge *dA + B' \wedge *B'

and therefore dA \wedge *B' + B' \wedge *dA remembers on the coupling of electromagnetic field to Dipole.

• Mohammed S. Hamed
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1

Good topic , wish you all the best

• Somak Chaudhuri
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An articulate matter indeed. However it is of slight confusion for me about the dipole field strength tensor. Overall a genuinely verdant calculation with standard tools and very appropriate use of resources and references. Introduction is way more lucid and makes it an easier approach.

This review has 1 comments. Click to view.

The Dipole field strength Tensor is not an electric Dipole; it is an observable of a topological Quantum field theory. The field B' has only the Name: "Dipole field strength tensor".

I have a comment on my Review at this paper: https://thewinnower.com/papers/2859-nonabelian-generalization-of-topological-dipole-field-theory

Here, I have explained more mathematical Details of the theory.

• Ritchie Yan
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Its a very interesting research topic. I like your idea and theory. It looks like you give a good explanation to your figures about your theory. I love subjects that deal with Topological and especially Cosmology.
I have read other research papers on the internet. I have notice some researchers use subtitles and diagrams in their research paper. You can try to use that, Its only a suggestion.

This review has 1 comments. Click to view.

Thanks for your good Review! In papers that I am writing in the future I will involve more practical calculations and therefore I will also use diagrams.

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Incredible work ever I seen in our young generation, and also my commment is on equation (1) and (2), where (1) is the exactness condition. I cant understand here, but physics and idea is really good!

This review has 1 comments. Click to view.

To understand equations (1) and (2) see:

https://en.wikipedia.org/wiki/%C4%8Cech_cohomology

This is an article where Čech cohomology is explained.

1

very good topic

• John Smith
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0

Nothing is actually acheived, despite the author claiming so. There is no connection made to the existing tested models of physics. The author uses sophisticated mathematical terminology to hide the fact that nothing is being said. On the note of mathematics, that is also incorrect. Functions/maps are not well-defined; for example, the function f acts on a set or a point depending on context. This is another article which any self-respecting editor would reject immediately, without wasting time for reviewers. A significant amount of work is required if the author would like to see this received by anyone in the physics community.

• Monsieur LeSeriös
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I have the impression this paper is serious. Not only the exactness condition is fulfilled, also the action in the general form describes the complete quantum field theory.
This theory is a vast and immane success. You can do anything. The only limit, is yourself!

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The author has written an interesting Topic. Calculation steps are Logically consistent and on a high mathematical Level. At the end of the paper a more plausible equation (without usin cohomology theory) is provided. An interesting fact is that the theory does not depend on mathematical Details of the underlying cohomology theory which makes TDFT a very interesting theory.

This review has 4 comments. Click to view.

This Review also belongs to the paper located at the following link (which is a continuation of this paper): https://thewinnower.com/papers/2859-nonabelian-generalization-of-topological-dipole-field-theory

The stars Rating (e.g. Quality of writing) corresponds only to the paper: The author has written an interesting Topic. Calculation steps are Logically consistent and on a high mathematical Level. At the end of the paper a more plausible equation (without usin cohomology theory) is provided. An interesting fact is that the theory does not depend on mathematical Details of the underlying cohomology theory which makes TDFT a very interesting theory."

There was a disaccord (concerning the rating) during the Review process.

This was a redundant error, But the Review text is also valid for this paper.

Sorry, wrong text pasted. It should be:

The stars Rating (e.g. Quality of writing) corresponds only to the paper: https://thewinnower.com/papers/2859-nonabelian-generalization-of-topological-dipole-field-theory

There was a disaccord (concerning the rating) during the Review process.

Sorry, wrong text pasted. It should be:

The stars Rating (e.g. Quality of writing) corresponds only to the paper: https://thewinnower.com/papers/2859-nonabelian-generalization-of-topological-dipole-field-theory

There was a disaccord (concerning the rating) during the Review process.

"This was a redundant error, But the Review text is also valid for this paper." means that this Review text is valid without the sentence "At the end of the paper a more plausible equation (without usin cohomology theory) is provided."

This was a redundant error, But the Review text is also valid for this paper.

IMPORTANT NOTE: The text "At the end of the paper a more plausible equation (without usin cohomology theory) is provided." was written due to an redundant error and it belongs to the paper here: https://thewinnower.com/papers/2859-nonabelian-generalization-of-topological-dipole-field-theory