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Developed Method. Interactions and Their Quantum Picture

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08 June 2023

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09 June 2023

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Abstract
By developing the previously proposed method of combining continuum mechanics with Einstein Field Equations, a complete description of the physical system was obtained, which indicates that the descriptions of the system dynamics used in physics do not have to be divergent. The density of the Lagrangian for the system turns out to be equal to the invariant of the electromagnetic field tensor, vanishing four-divergence of canonical four-momentum appears to be consequence of the Poyinting theorem, and explicit form of one of gauges of the electromagnetic four-potential was introduced. A quantum picture of the system was also proposed, leading to the Klein-Gordon equation, which seems to finally reconcile the curvilinear description, flat-spacetime relativistic description and quantum description of the physical system.
Keywords: 
Subject: 
Physical Sciences  -   Theoretical Physics

1. Introduction

Over the past decades, great strides have been made in attempts to combine 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 [2,3,4], String Theory [5,6,7] and Noncommutative Spacetime Theory [8,9].
A lot of work has also been done to clear up some challenges related to General Relativity and Λ C M D model [10]. An explanation for the problem of dark energy [11] and dark matter [12] is still being sought, and efforts are still being made to explain the origin of the cosmological constant [13,14,15].
The author also tries to bring his own contribution to the explanation of the above physics challenges, based on a recently discovered method, described in [16]. As this article will show, this method seems very promising and can help clarify at least some of the issues mentioned above.
According to conclusions from [16], 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. This allows for a significant simplification of research, because the results obtained in flat Minkowski spacetime can be easily transformed into curved spacetime. The last missing link seems to be the quantum description.In this article, the author will focus on developing the method proposed in [16] in such a way, as to obtain the convergence of the description with the description of quantum mechanics. In the first chapter, the Lagrangian density for the system will be derived, allowing to obtain the tensor described in [16]. These conclusions will be used later in the article to propose quantum description of the system.
The author uses the Einstein summation convention, metric signature ( + , − , − , − ) and commonly used notations. In order to facilitate the analysis of the article, the key conclusions from [16] are quoted in the subsection below.

1.1. Short summary of the method

According to [16], stress-energy tensor T α β for a system in a given spacetime described by a metric tensor g α β is equal to
T α β = ϱ U α U β − c 2 ϱ + Λ ρ g α β − ξ h α β
where ϱ o is for rest mass density and
ϱ ≡ ϱ o γ
1 ξ ≡ 1 4 g μ ν h μ ν
Λ ρ ≡ 1 4 μ o F α μ g μ γ F β γ g α β
h α β ≡ 2 F α δ g δ γ F β γ F α δ g δ γ F β γ g μ β F α η g η ξ F ξ μ
where F α β represents electromagnetic field tensor, and where the stress–energy tensor for electromagnetic filed, denoted as Υ α β may be presented as follows
Υ α β ≡ Λ ρ g α β − ξ h α β = Λ ρ g α β − 1 μ o F α δ g δ γ F β γ
Thanks to the proposed amendment to the continuum mechanics, in flat Minkowski spacetime occurs
∂ α U α = − d γ d t → ∂ α ϱ U α = 0
thus denoting four-momentum density as ϱ U μ = ϱ o γ U μ , total four-force density f μ acting in the system is
f μ ≡ ϱ A μ = ∂ α ϱ U μ U α
Denoting rest charge density in the system as ρ o and
ρ ≡ ρ o γ
electromagnetic four-current J α is equal to
J α ≡ ρ U α = ρ o γ U α
The pressure p in the system is equal to
p ≡ c 2 ϱ + Λ ρ
Total four-force density f α acting in the system calculated from ∂ β T α β = 0 is the sum of electromagnetic ( f E M α ), gravitational ( f g r α ) and other ( f o t h α ) four-force densities
f α = f E M α ≡ − Λ ρ ∂ β ξ h α β ( e l e c t r o m a g n e t i c ) + f g r α ≡ g α β − ξ h α β ∂ β p ( g r a v i t a t i o n a l ) + f o t h α ≡ ϱ c 2 Λ ρ f E M α ( o t h e r )
As was shown in [16], in curved spacetime ( g α β = h α β ) presented method reproduces Einstein Field Equations with an accuracy of 4 π G c 4 constant and with cosmological constant Λ dependent on invariant of electromagnetic field tensor F α γ
Λ = − π G c 4 μ o F α μ h μ γ F β γ h α β = − 4 π G c 4 Λ ρ
where h α β appears to be metric tensor of the spacetime in which all motion occurs along geodesics and where Λ ρ describes vacuum energy density.It is also shown, that Einstein tensor describes the spacetime curvature related to vanishing in curved spacetime four-force densities f g r α + f o t h α .

2. Lagrangian density for the system

Since for the considered method the transition to curved spacetime is known (based on electromagnetic field tensor), the rest of the article will focus on the calculations in the Minkowski spacetime with presence of fields, where η α β represents Minkowski metric tensor.
Using a simplified notation
d ln ( p ) d τ = U μ ∂ μ ln ( p ) = U μ ∂ μ p p
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
f g r α = η α β − ξ h α β ∂ β p = d ln ( p ) d τ ϱ U α − T α β ∂ β ln ( p ) f E M α = Λ ρ p f α − f g r α f o t h α = ϱ c 2 p f α − f g r α
where f E M μ can also be represented in terms of electromagnetic four-potential and four-current. This means that to fully describe the system and derive the Lagrangian density, it is enough to find an explicit equation for the gravitational force or some gauge of electromagnetic four potential.
Referring to definitions from Section 1.1 one may notice, that by proposing following electromagnetic four-potential A μ
A μ ≡ − Λ ρ p ϱ o ρ o U μ
one obtains electromagnetic four-force density f E M α in form of
f E M α = J β ∂ α A β − ∂ β A α = Λ ρ p f α − d ln ( p ) d τ ϱ U α + ϱ c 2 ∂ α ln ( p )
where J β is electromagnetic four-current and where Minkowski metric property was utilized
U β U β = c 2 → U β ∂ α U β = 1 2 ∂ α U β U β = 0
and where the forces in the system can be described by the following equality
J β ∂ α A β − ∂ β A α + ϱ U β ∂ β ϱ c 2 p U α − ∂ α ϱ c 2 p U β = ϱ U β ∂ β U α − ∂ α U β = f α
Comparing (15) and (17) it is seen, that introduced electromagnetic four-potential yields
0 = T α β − ϱ c 2 η α β ∂ β ln ( p )
which is equivalent to imposing following condition on normalized stress-energy tensor
0 = ∂ β T α β T r ( T α β ) + ∂ α ln T r ( T α β )
and yields gravitational four-force density in Minkowski spacetime in form of
f g r μ = ϱ d ln ( p ) d τ U μ − c 2 ∂ μ ln ( p )
Now, one may show, that proposed electromagnetic four-potential leads to correct solutions.
At first, recalling the classical Lagrangian density [17] for electromagnetism one may show why, in the light of the conclusions from [16] and above, it does not seem to be correct, and thus does not allow to create a symmetric stress-energy tensor [18]. The classical value of the Lagrangian density for electromagnetic field, written with the notation used in the article, is
− L E M c l a s s i c = Λ ρ + A μ J μ
In addition to the obvious doubt that by taking the different gauge of the four-potential A μ one changes the value of the Lagrangian density, one may notice, that with considered electromagnetic four-potential, such Lagrangian density is equal to
− L E M c l a s s i c = Λ ρ − Λ ρ p ϱ o ρ o U μ U μ ρ o γ = Λ ρ − Λ ρ ϱ c 2 p = Λ ρ 2 p
As it is seen, above Lagrangian density is not invariant under gradient over four-position and A μ and J μ are dependent, what is not taken into account in classical calculation
A α A μ A μ = J α A μ J μ
Above yields
∂ ln 1 A μ A μ ∂ A α = − J α A μ J μ = p ϱ c 2 J α Λ ρ
One may decompose
ln 1 A μ A μ = ln p ρ o Λ ρ ϱ o c = ln p − ln Λ ρ − ln ϱ o c ρ o
and since ϱ o c ρ o are constants, one may simplify (26) to
∂ ln p ∂ A α − ∂ ln Λ ρ ∂ A α = p ϱ c 2 J α Λ ρ = J α ϱ c 2 + J α Λ ρ
It is now appropriate to propose a solution to the Lagrangian density problem, allowing the stress-energy tensor under consideration to be derived, noting that in (28) occurs
∂ ln Λ ρ ∂ A α = − J α Λ ρ
which leads to the conclusion that Λ ρ acts as the Lagrangian density for the system
∂ Λ ρ ∂ A α = ∂ ν ∂ Λ ρ ∂ ( ∂ ν A α ) = − J α
thus stress-energy tensor for the system is equal to
T α β = 1 μ o F α γ ∂ β A γ − Λ ρ η α β
In fact, the proof of correctness of the electromagnetic field tensor (noted as Υ α β ) allows to see this solution
f E M β = ∂ α Υ α β = J γ F γ β − 1 μ o F α γ ∂ α F γ β
what yields following property of electromagnetic field tensor
F α γ ∂ α ∂ γ A β = F α γ ∂ β ∂ α A γ
Using the above substitution, one may note
∂ α Υ α β = ∂ α 1 μ o F α γ ∂ γ A β − ∂ α 1 μ o F α γ ∂ β A γ = 1 μ o F α γ ∂ β ∂ α A γ − J γ ∂ γ A β − ∂ α 1 μ o F α γ ∂ β A γ
Therefore the invariance of Λ ρ with respect to A α and ∂ ν A α is both a condition on the correctness of the electromagnetic stress-energy tensor and on Λ ρ in the role of Lagrangian density
0 = ∂ Λ ρ ∂ ( ∂ α A γ ) ∂ β ( ∂ α A γ ) + ∂ Λ ρ ∂ A γ ∂ β A γ = 1 μ o F α γ ∂ β ∂ α A γ − J γ ∂ β A γ = ∂ α 1 μ o F α γ ∂ β A γ
what yields for (34) that
∂ α Υ α β = J γ ∂ β A γ − J γ ∂ γ A β = f E M β
Equations (1), (6) and (31) yield
1 μ o F α γ ∂ γ A β = ϱ U α U β − c 2 ϱ Λ ρ Υ α β
what yields second representation of the stress-energy tensor
T α β = p ϱ c 2 · 1 μ o F α γ ∂ γ A β − Λ ρ c 2 U α U β = p ϱ c 2 ∂ γ 1 μ o F α γ A β
After four-divergence, it gives additional expression for relation between forces and gives useful clues about the behavior of the system when transitioning to the description in curved spacetime.

3. Hamiltonian density and quantum picture

Since
∂ Λ ρ ∂ ( ∂ A γ / ∂ x α ) = 1 μ o F α γ → ∂ Λ ρ ∂ ( ∂ A γ / ∂ x 0 ) = 1 μ o F 0 γ
thus noting Hamiltonian density in terms of the Lagrangian density [19] (and also from (31)) one gets
H = ∂ Λ ρ ∂ ( ∂ A γ / ∂ x 0 ) ∂ A γ ∂ x 0 − Λ ρ = 1 μ o c F 0 γ ∂ A γ ∂ t − Λ ρ
where conjugate momentum field Π γ is
Π γ ≡ 1 μ o F 0 γ
As it is seen, above Hamiltonian density agrees with the classical Hamiltonian density for electromagnetic field [19] except that this Hamiltonian density was currently only considered for sourceless regions and only to consider the system with electromagnetic field. According to the result above, this Hamiltonian density describes the entire physical system, taking into account all known interactions. Above, therefore, significantly simplifies quantum field theory equations [20,21,22].
At first one may notice, that in transformed (31)
T α 0 = 1 μ o F α γ ∂ γ A 0 − Υ α 0
first row of electromagnetic stress-energy tensor Υ α 0 is as four-vector, representing energy density of electromagnetic field and Poynting vector [23]. Therefore vanishing four-divergence of the T α 0 must represent Poynting theorem. Indeed, properties (33) and (35) provide such equality
0 = − ∂ α T α 0 = J γ F 0 γ + ∂ α Υ α 0
which also shows, that for the constant total energy of the system, T α 0 is independent of the four-position.
Next, one may define another gauge A ¯ γ of electromagnetic four-potential A γ in following way
A ¯ γ = A γ − ∂ γ A β X β = − X β ∂ γ A β
and note, that
X β T 0 β = 1 μ o F γ 0 A ¯ γ − X β Υ 0 β
Since for constant total energy of the system
∂ α X β T 0 β = T 0 α
this brings three more important insights:
  • X β T 0 β plays a role of the density of Himilton’s principal function,
  • Hamilton’s principal function may be expressed based on the electromagnetic field only (so in the absence of the electromagnetic field it disappears),
  • in (31) there must occur
1 γ U β T 0 β = − Λ ρ → F 0 γ d A γ d t = 0
One may now summarize the above and propose a method for quantizing the system.
At first, it should be noted, that the above reasoning 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 the absence of the electromagnetic field means actually the disappearance of spacetime and the absence of any action.
One may then introduce generalized, canonical four-momentum H μ as four-gradient on Hamilton’s principal function S, where relation with wave four-vector K μ occurs
H μ = 1 c ∫ T 0 μ d x 3 ≡ − ∂ μ S ≡ ℏ K μ
and where K μ does not depend on four-position. One may also conclude from previous findings, that it should be in form of
H μ = P μ − V μ
where V μ represents generalized, electromagnetic field related four-potential and where four-momentum P μ may be now considered as just other gauge of this four-potential
∂ α P μ = ∂ α V μ
Since in the limit of the inertial system one gets P μ X μ = m c 2 τ , therefore, to ensure the decay of Hamilton’s principal function in the inertial system, one can order that
V μ X μ ≡ m c 2 τ
and obtain Hamilton’s principal function as follows
− S ≡ H μ X μ = P μ X μ − V μ X μ
what also yields vanishing in the inertial system Lagrangian in form of
− L = 1 γ F μ X μ
and also
H μ H μ + m 2 c 2 = 2 m F μ X μ − V μ V μ
Using the above definitions, the action, the Hamiltonian and Lagrangian vanish for an inertial system (when P μ = V μ ). To relate the Lagrangian to the generalized field, and to ensure compatibility with the equations of quantum mechanics it is enough to assume the following
− γ L = X μ F μ = V μ V μ 2 m
Thanks to above, by introducing quantum wave function Ψ in form of
Ψ ≡ e ± i K μ X μ
from (54) one obtains Klein-Gordon equation
□ + m 2 c 2 ℏ 2 Ψ = 0
which allows for further analysis of the system in the quantum approach, eliminating the problem of negative energy appearing in solutions [24].
The above representation allows the analysis of the system in the quantum approach, classical approach based on (42) and the introduction of a field-dependent metric in (5) for curved spacetime, which connects previously divergent descriptions of physical systems.

4. Conclusions and Discussion

As shown above, the proposed method of physical systems analysis summarized in Section 1.1 seems to be very promising area of farther research. By imposing condition (20) on normalized stress-energy tensor, one obtains consistent results, developing the knowledge of the physical system:
  • Lagrangian density for the systems appears to be equal to L = Λ ρ = 1 4 μ o F α β F α β
  • Some gauge of electromagnetic four-potential may be expressed as A μ = − Λ ρ p ϱ o ρ o U μ
  • The vanishing four-divergence of the canonical four-momentum turns out to be the consequence of Poynting theorem
  • For-force densities acting in the system may be expressed as shown in (15) where gravitational four-force is dependent on logarithm of the pressure p in the system as shown in (22)
It also seems, that this approach allows to combine previously divergent methods of curvilinear and Minkowski spacetime description with the quantum description and simplifies further research on the quantum picture of individual fields, significantly simplifying equations of the quantum field theory. Proposed approach drives to generalized, canonical four-momentum related to four-position independent wave four-vector, described by relation between four-momentum and generalized field four-potential V μ . Such approach also drives in natural way to Klain-Gordon equation (54).
Further analysis using the variational method should also allow for an explicit derivation of the generalized four-potential V μ , which may lead to further 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.

5. Statements

Data sharing is not applicable to this article, as no datasets were generated or analyzed during the current study.
The author did not receive support from any organization for the submitted work.
The author has no relevant financial or non-financial interests to disclose.

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