# Developed method: interactions and their quantum picture

- Kozminski University, Warsaw, Poland

By developing the previously proposed method of combining continuum mechanics with Einstein’s field equations, it has been shown that the classic relativistic description, curvilinear description, and quantum description of the physical system may be reconciled using the proposed Alena Tensor. For a system with an electromagnetic field, the Lagrangian density equal to the invariant of the electromagnetic field was obtained, the vanishing four-divergence of canonical four-momentum appears to be the consequence of the Poynting theorem, and the explicit form of one of the electromagnetic four-potential gauges was introduced. The proposed method allows for further development with additional fields.

## 1 Introduction

Over the past decades, great strides have been made in attempts to combine the quantum description of interactions with general relativity [1]. There are currently many promising approaches to connecting the quantum mechanics and general relativity, including perhaps the most promising ones: loop quantum gravity [2–4], string theory [5–7], and noncommutative spacetime theory [8, 9].

There are also attempts made to modify general relativity or find an equally good alternative theory [10–12] that would provide a more general description or would allow for the inclusion of other interactions. A significant amount of work has also been carried out to clear up some challenges related to general relativity and the Λ*CMD* model [13]. An explanation for the problem of dark energy [14] and dark matter [15] is still being sought, and efforts are still being made to explain the origin of the cosmological constant [16–18].

The author also attempts to bring his own contribution to the explanation of the above physics challenges based on a recently discovered method, as described in [19]. In this article, this method seems very promising and may help clarify at least some of the issues mentioned above. The author’s method, which is similar to the approach presented in [20–22], also points to the essential connections between electromagnetism and general relativity; however, the postulated relationship is of a different nature and can be perceived as some generalization of the direction of research proposed in [23–26].

According to the conclusion obtained from [19], the description of motion in curved spacetime and its description in flat Minkowski spacetime with fields are equivalent, and the transformation between curved spacetime and Minkowski spacetime is known because the geometry of curved spacetime depends on the field tensor. This transformation allows for a significant simplification of research because the results obtained in flat Minkowski spacetime can easily be transformed into curved spacetime. The last missing link seems to be the quantum description.

In this article, the author will focus on developing the proposed method for a system with an electromagnetic field in such a manner, as to obtain the convergence with the description of QED and quantum mechanics. In the first section, the Lagrangian density for the system will be derived, allowing to obtain the tensor, as described in [19]. These conclusions will later be used in the article to propose the possible directions of research on combining the GR description with QFT and QM.

The author uses Einstein’s summation convention, metric signature (+, −, −, −), and commonly used notations. In order to facilitate the analysis of the article, the key conclusion from [19] is quoted in the subsection below.

### 1.1 Short summary of the method

According to [19], the stress-energy tensor *T*^{αβ} for a system with an electromagnetic field in a given spacetime, described by a metric tensor *g*^{αβ}, is equal to

where *ϱ*_{o} is for remaining mass density, *γ* is the Lorentz gamma factor, and

In the above equations, ^{αβ}, may be represented as follows:

Thanks to the proposed amendment toward the continuum mechanics, in the flat Minkowski spacetime occurs

Thus, denoting four-momentum density as *ϱU*^{μ} = *ϱ*_{o}*γ U*^{μ}, the total four-force density *f*^{μ} operating in the system is

Denoting the remaining charge density in the system as *ρ*_{o} and

the electromagnetic four-current *J*^{α} is equal to

The pressure p in the system is equal to

In the flat Minkowski spacetime, the total four-force density *f*^{α} operating in the system calculated from vanishing *∂*_{β} *T*^{αβ} is the sum of electromagnetic

As shown in [19], in curved spacetime (*g*_{αβ} = *h*_{αβ}), presented method reproduces Einstein’s field equations with an accuracy of

where *h*_{αβ} appears to be the metric tensor of the spacetime in which all motion occurs along geodesics and where Λ_{ρ} describes the vacuum energy density.

It is worth noting that although in flat Minkowski spacetime Λ_{ρ} has a negative value due to the adopted metric signature, this does not determine its value in curved spacetime. Therefore, solutions with a negative cosmological constant are also possible, which is an issue discussed in the literature [27–29].

It was also shown that in this solution Einstein’s tensor describes the spacetime curvature related to vanishing in curved spacetime four-force densities

The proposed method allows adding additional fields while maintaining its properties. One may define another stress–energy tensor describing the field (e.g., describing the sum of several fields) instead of ϒ^{αβ} and insert it into the stress–energy tensor *T*^{αβ} in a manner that is analogous to that presented above. As a result of the vanishing four-divergence of *T*^{αβ}, one will obtain in the flat spacetime four-force densities related to the new field and in curved spacetime, the equations will transform into EFE with the cosmological constant depending on the invariant of the considered new field strength tensor.

## 2 Lagrangian density for the system

Since the transition to curved spacetime is known for the considered method, the rest of the article will focus on the calculations in Minkowski spacetime with the presence of an electromagnetic field, where *η*^{αβ} represents the Minkowski metric tensor.

Using a simplified notation

it can be seen that the four-force densities resulting from the obtained stress–energy tensor (12) in flat Minkowski spacetime can be written as follows:

where

Referring to definitions from section 1.1, one may notice that by proposing the following electromagnetic four-potential

one obtains the electromagnetic four-force density

where *J*_{β} is the electromagnetic four-current and where the Minkowski metric property was utilized:

Four-force densities operating in the system may now be described by the following equality:

Comparing equations (15) and (17), it is seen that the introduced electromagnetic four-potential yields

which is equivalent to imposing the following condition on the normalized stress–energy tensor

and which yields the gravitational four-force density in Minkowski spacetime in the form of

Now, one may show that the proposed electromagnetic four-potential leads to correct solutions.

At first, recalling the classical Lagrangian density [30] for electromagnetism, one may show why, in the light of the conclusions from [19] and above, it does not seem to be correct and thus makes it difficult to create a symmetric stress–energy tensor [31]. The classical value of the Lagrangian density for the electromagnetic field, written with the notation used in the article, is

In addition to the obvious doubt that is observed by taking the different gauge of the four-potential

It is observed that the above Lagrangian density is not invariant under gradients over four-positions, and *J*_{μ} are dependent, what is not taken into account in classical calculation:

The above analysis yields

One may decompose

and simplify (26) to

where the above equation yields

which leads to the conclusion that Λ_{ρ} acts as the Lagrangian density for the system

which would support the conclusion from [32] and what yields the stress–energy tensor for the system in the form of

The proof of correctness for the electromagnetic field tensor (noted as ϒ^{αβ}) allows seeing this solution as follows:

and what yields following property of the electromagnetic field tensor:

Using the above substitution, one may note that

Therefore, the invariance of Λ_{ρ} with respect to _{ρ} in the role of Lagrangian density

what yields for (34) that

Equations (1), (6), and (31) yield

and what yields the second representation of the stress–energy tensor is

After four-divergence, it provides additional expression for relation between forces and provides useful clues about the behavior of the system when transitioning to the description in curved spacetime.

## 3 Hamiltonian density and energy transmission

By observing Hamiltonian density as

The above Hamiltonian density agrees with the classical Hamiltonian density for the electromagnetic field [33] except that this Hamiltonian density was currently considered for sourceless regions. According to conclusions from previous sections, this Hamiltonian density describes the whole system with an electromagnetic field, including gravity and other four-force densities resulting from the considered stress–energy tensor. Therefore, the above equations may significantly simplify quantum field theory equations ([34)–(36)], which will be shown in this section for the purposes of QED.

At first, one may notice that in transformed (31)

the first row of the electromagnetic stress–energy tensor ϒ^{α0} is a four-vector, representing the energy density of an electromagnetic field and Poynting vector [37]—the Poynting four-vector. Therefore, the vanishing four-divergence of *T*^{α0} must represent the Poynting theorem. Indeed, properties (33) and (35) provide such an equality

Next, one may introduce the auxiliary variable *ɛ* with the energy density dimension defined as follows:

and comparing the result

between the two tensor definitions (31) and (38), one may notice that it must hold

because the second component of above vanishes contracted with *U*_{β}, due to the property of the Minkowski metric (18). Therefore, (31) and (38) also yield the following:

where *ϵ*_{o} is electric vacuum permittivity, and

Since *∂*^{μ}*p* = *∂*^{μ}*ϱc*^{2}, thus from (44), one obtains

and thanks to (44) that was substituted to (38), one also obtains

Since from (1) and (6) for *T*^{00}, one obtains

Therefore, comparing the zero-component of (45)

to (49) and comparing that to (48)

one may notice that

is a valid solution of the system, which yields

There is a whole class of solutions (52) in the form

From the analysis of Eq. 45, it may then be concluded that after the integration of

Therefore, by analogy with the Poynting four-vector *Z*^{β} that is understood as its equivalent for the remaining interactions and rewrite (45) as

where

The above result ensures that the canonical four-momentum density for the system with the electromagnetic field depends on the four-potential and charge density as expected. This supports the earlier statement about the need to set

One may also note that the above solution yields *p* < 0 since the energy density of the electromagnetic field is

where Λ_{ρ} < 0 in flat spacetime, due to the adopted metric signature. Thus, *Z*^{β} may also be simplified to

Finally, one may define another gauge

and it is to be noted that

The four-divergence of *T ^{0β}* vanishes, and therefore, (53) indicates that

which yields

The above equation brings two more important insights as follows:

•

• Hamilton’s principal function may be expressed based on the electromagnetic field only, so in the absence of the electromagnetic field it disappears.

All the above equations also lead to this conclusion that (54) may also act as Lagrangian density in the classic relativistic description based on four-vectors.

One may, thus, summarize all of the above findings and propose a method for the description of the system with the use of classical field theory for point-like particles.

## 4 Point-like particles and their quantum picture

Initially, it should be noted that the reasoning presented in Section 3 changes the interpretation of what the relativistic principle of least action means. As one may conclude from above, there is no inertial system in which no fields act, and in the absence of fields, the Lagrangian, the Hamiltonian, and Hamilton’s principal function vanish. Since the metric tensor (5) for description in curved spacetime depends on the electromagnetic field tensor only, it seems clear that in the considered system, the absence of the electromagnetic field means the actual disappearance of spacetime and the absence of any action.

Then, one may introduce generalized, canonical four-momentum *H*^{μ} as four-gradient on Hamilton’s principal function S

where

One may also conclude from previous sections that the canonical four-momentum should be in the form of

where

and where four-momentum *P*^{μ} may now be considered just “another gauge” of −*V*^{μ}:

Since in the limit of the inertial system, one obtains *P*^{μ}*X*_{μ} = *mc*^{2}*τ*, and therefore, to ensure vanishing Hamilton’s principal function in the inertial system, one may expect that

which would also yield vanishing in the inertial system Lagrangian *L* for the point-like particle in the form of

where *F*^{μ} is the four-force. Equation 48 yields

where

and where

In the above equation, the Hamilton’s principal function, generalized canonical four-momentum, and Lagrangian vanish for the inertial system, as expected.

Since

therefore, to ensure compatibility with the equations of quantum mechanics, it suffices to consider the properties of

then, by introducing quantum wave function Ψ in the form of

where *K*^{μ} is the wave four-vector related to the canonical four-momentum

from (73), one obtains the Klein–Gordon equation as follows:

This shows that in addition to the alignment with QFT (39), the first quantization also seems possible which allows for further analysis of the system from the perspective of the quantum mechanics, eliminating the problem of negative energy appearing in solutions [39].

The above representation allows the analysis of the system in the quantum approach, classical approach based on (40), and the introduction of a field-dependent metric in (5) for curved spacetime, which connects the previously divergent descriptions of physical systems.

## 5 Conclusion and discussion

As shown above, the proposed method summarized in Section 1.1 seems to be a very promising area of further research. In addition to the previous agreement with Einstein’s field equations in curved spacetime, by imposing condition (21) on the normalized stress–energy tensor (1) (hereinafter, referred to as the Alena Tensor) in flat Minkowski spacetime, one obtains consistent results, developing the knowledge of the physical system with an electromagnetic field. Gravitational, electromagnetic, and sum of other forces operating in the system may be expressed as shown in (15), where gravitational four-force density is dependent on the pressure *p* in the system and equal to

The conclusion from the article can be divided according to their areas of application, as conclusion for QED, QM, and that regarding the combination of QFT with GR.

### 5.1 Conclusion for QED

Condition (21) leads to the electromagnetic four-potential, for which some gauge may be expressed as

The above would also simplify the Lagrangian density used in QED. Assuming that there is only the electromagnetic field in the system and substituting Λ_{ρ} for the current Lagrangian density used in QED, one should obtain equations that describe the entire system with the electromagnetic field. Interestingly, such equations would also take into account the behavior of the system related to gravity because according to the model presented here, gravity naturally arises in the system as a consequence of the existence of other fields (more precisely, existence of the energy momentum tensors associated with these fields), and the resulting Lagrangian density takes this into account.

Perhaps, this explains why it is so difficult to identify quantum gravity as a separate interaction within QFT, and it could also explain QED’s extremely accurate predictions, assuming that it actually describes the entire system with an electromagnetic field.

### 5.2 Conclusion for QM

As shown in the article, *H*^{β} turns out to be the consequence of the Poynting theorem.

The obtained canonical four-momentum *H*^{μ} should satisfy the Klein–Gordon (Eq 77), and it is equal to

where *P*^{μ} is four-momentum and L is Lagrangian for the point-like particle, *V*^{μ} describes the transport of energy due to the field. It may be calculated as

where *Y*^{μ} is the volume integral of the Poynting four-vector.

It seems that in such an approach, it would be possible to isolate gravity as a separate interaction, although this would probably require further research on the influence of individual components on the behavior of the particle. It is also not clear how to deal with the interpretation of time in the first quantization; however, a clue is to rely on the possibility of using Geroch’s splitting [40], providing (3 + 1) decomposition.

### 5.3 Conclusion regarding the combination of QFT with GR

It should be noted that the presented solution applies to a system with an electromagnetic field, but it allows for the introduction of additional fields while maintaining the properties of the considered Alena Tensor. Therefore, it appears as a natural direction for further research to verify how the system with additional fields will behave and what fields are necessary to obtain the known configuration of elementary particles and interactions.

For example, considering the previous notation, one may describe the field in the system by some generalized field tensor

where

The Alena Tensor that is defined in this manner retains most of the properties described in the previous sections; however, it now describes other four-force densities in the system, related to its vanishing four-divergence.

Further analysis of the above properties using the variational method may lead to future discoveries regarding both the theoretical description of quantum fields and elementary particles associated with them, and the possibility of experimental verification of the obtained results.

As can be seen from the above summary, the conclusion obtained cannot be treated as the final result and requires additional research. These results should be understood as another small step on the path of science, which opens up the possibility of research on the properties of the Alena Tensor in terms of its applications in quantum theories. If these results spark an interest in the scientific community, perhaps further steps can be taken through a concerted effort.

## Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

## Author contributions

PO: conceptualization, investigation, writing–original draft, and writing–review and editing.

## Funding

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

## Conflict of interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

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Keywords: field theory, Lagrangian mechanics, quantum mechanics, general relativity, unification of interactions

Citation: Ogonowski P (2023) Developed method: interactions and their quantum picture. *Front. Phys.* 11:1264925. doi: 10.3389/fphy.2023.1264925

Received: 21 July 2023; Accepted: 20 November 2023;

Published: 06 December 2023.

Edited by:

Louis H. Kauffman, University of Illinois Chicago, United StatesReviewed by:

Rami Ahmad El-Nabulsi, Chiang Mai University, ThailandPeter Rowlands, University of Liverpool, United Kingdom

Copyright © 2023 Ogonowski. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Piotr Ogonowski, piotrogonowski@kozminski.edu.pl