In this paper, the mechanics of Eli Cartan is used, which is an alternative to the Lagrange-Hamiltonian formalism, has certain advantages in the formulation of quantum electrodynamics. To demonstrate this fact, it was described as the interaction of fermions with an electromagnetic field. We demonstrated the possibility of using the mechanics of E. Cartan in quantum field theory. Based on the use of these mechanics, additional conditions can be introduced directly into the Cartan equations. Such conditions include, for example, switching conditions between pulses and coordinates, as well as Lorentz calibration conditions
The mechanics of Eli Cartan, which is an alternative to the Lagrange-Hamiltonian formalism, has certain advantages in the formulation of quantum electrodynamics. To demonstrate this fact, we describe the interaction of fermions with an electromagnetic field.
All particles that make up the Universe fall into two groups: fermions and bosons.
Graduate students of Leiden University (Holland) Samuel Gaudsmith and George
Uhlenbeck introduced this distinction. Gaudsmith, who was more engaged in research,
noticed an additional splitting of the emission spectrum of helium atoms. Uhlenbeck,
who knew better classical physics, saw the reason for this splitting in some internal
property of the electron. Together they concluded that the electron initially has a certain
angular momentum - spin [
The foundations of quantum mechanics were only then laid, so this idea led to the addition of a fourth quantum number (in addition to the main, orbital, and magnetic), called the spin quantum. The electron is depicted as a tiny, rapidly spinning top, but *
Copyright 2021 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
such a description does not need to be taken literally. In 1928, the development by the
British physicist P. Dirac of relativistic quantum mechanics created a theoretical basis
for the spin of an electron; the guess of Gaudsmith and Uhlenbeck turned out to be very
successful [
Main content. Interaction of fermions with an electromagnetic field based on Cartan mechanics In 1925, the Austrian physicist Wolfgang Pauli concluded that two electrons couldn’t be in the same quantum state in the same place. This principle of Pauli's prohibition lies at the heart of the Periodic Table of Chemical Elements.
In studying the statistical behavior of electrons, the Italian-American physicist
Enrico Fermi and Dirac developed the Fermi-Dirac statistics theory. Its provisions were
subsequently extended to other particles with a half-integer spin. These particles, called
fermions, encompass all leptons and quarks. Thus, the mass of the universe is made up
of fermions [
The study of particles with zero or integer spin in 1924 was carried out by the Indian
physicist Chatyatranat Bose. While working at the University of Dhaka (Bangladesh),
Bose sent the results of his research for review to Einstein. He translated his work into
German and strongly advised him to publish it. The following year, Einstein expanded
the Bose results to include all particles that are not fermions. The statistical behavior of
such particles came to be called Bose-Einstein statistics. Particles obeying these
statistics, Dirac called bosons [
If two fermions cannot be in the same quantum state, then there is no such restriction
for bosons. Indeed, the more bosons are in a certain energy state, the greater the
likelihood that all other bosons will be in this state. This phenomenon underlies stimulated
emission in lasers when photons are brought into the same energy state. This kind of
"herd" helps to explain the superfluidity of helium and even superconductivity when
the electrons collide in pairs and act like bosons. In 1995, it was possible to reduce the
temperature of gaseous rubidium in such a way that all atoms found the same quantum
state. Such a cluster is called the Bose-Einstein condensate [
The tendency to “loneliness” in fermions and the “sociability” of bosons make them so dissimilar. However, this difference turns out to be decisive for the nature of the universe. For example, if fermions united like bosons, all the electrons in the atom would collect at the lowest energy level, and then there could be no talk of chemical reactions, and therefore, of life.
The electromagnetic interaction is one of four fundamental interactions. It exists
between particles with an electric charge [
In the framework of quantum field theory [
Fermions are among the fundamental particles that have an electric charge and participate in electromagnetic interaction.
Along with electromagnetic, there are also weak [
According to the classical (non-quantum) approach, electromagnetic interaction is
described by classical electrodynamics [
First, we quantize the electromagnetic field.
Consider a 2-form Ω of the form: = ∫ ′ { ( 1 2 ′ ′ ) ∧ } = ∫ ′ 2 ′ ′
= 0 – is the equation of the dynamics of the electromagnetic field vector potential.
We introduce the vectors ⃗⃗ and ⃗⃗ , which also describe the electromagnetic field: ⃗⃗ = −∇⃗⃗ 0 − 1 ⃗ + ∇⃗⃗ ∙ ⃗ = 0 = ∇⃗⃗ × ⃗⃗ = − ∇⃗⃗ ∙ ⃗⃗ = ∇⃗⃗ × (∇⃗⃗ × ⃗) = ∇⃗⃗ ∇⃗⃗ ∙ ⃗ − ∆ ⃗ = −∇⃗⃗ 1 0 − ∆ ⃗ = (1) (2) (3) (4) (5) (6)
.
This is the continuity equation for the energy of an electromagnetic field [
To record the energy of the electromagnetic field in the secondary quantization representation, we express ⃗⃗ and ⃗⃗ through the generalized coordinates and momenta of the electromagnetic field:
The energy of the electromagnetic field is: ℋ = 81 ∫
( ⃗⃗ 2 + ⃗⃗2).
Really: ℋ = 41 ∫ ( ⃗⃗ ∙ ⃗ + ⃗⃗ ∙ ⃗⃗ ) = 1 ( ⃗E⃗ ∙ (∇⃗⃗ × ⃗⃗ ) − ⃗H⃗⃗ ∙ (∇⃗⃗ × ⃗⃗ )) = 4 4 −∇⃗⃗ ∙ ⃗⃗ × ⃗⃗ ∫ ℋ = ∫ − 4
(⃗E⃗ × ⃗⃗ , ⋯ , ⋯ )
This is the continuity equation for the energy of an electromagnetic field. To record the energy of the electromagnetic field in the secondary quantization representation, we express ⃗⃗ and ⃗⃗ through the generalized coordinates and momenta of the electromagnetic field: ⃗⃗ = ∫ ⃗⃗ ∑ ⃗⃗ ( )ℎ⃗⃗ ⃗⃗ ( ⃗ )√4 ; ⃗⃗ = ∫ ⃗⃗ ∑ ⃗⃗ ⃗ we get ∇⃗⃗ ∙ ℎ⃗⃗ ⃗⃗ ( ⃗ ) = ∇⃗⃗ ∙ ⃗
⃗⃗ ( ⃗ ) = 0. ∇⃗⃗ × ⃗⃗ = ∫ ⃗⃗ ∑ ⃗⃗ ⃗⃗ ( )⃗∇⃗ × ⃗ ⃗⃗ ( ⃗ ) = − 1 ⃗⃗ and ⃗⃗ ( ) = ̇ ⃗⃗ ( ).
⃗⃗⃗ ⃗⃗
ℎ ⃗⃗ ( ⃗ )
∇⃗⃗ × ⃗⃗ = ∫ ⃗⃗ ∑
⃗⃗ ( )⃗∇⃗ × ℎ⃗⃗ ⃗⃗ ( ⃗ ) =
1 ⃗
It describes a mathematical pendulum for each momentum ћ ⃗⃗ and polarization .
Let us prove that ℎ⃗⃗ ⃗⃗ are orthogonal to ℎ⃗⃗ ⃗⃗′ ′ for ⃗⃗ ≠ ⃗⃗ ′, α≠α', as well as ⃗
⃗⃗ : Using
(10) we obtain [
∫ 2 = 1 2 + ∫ ⃗⃗ ⃗⃗ ′ ∑ ′ ⃗⃗ ( )
⃗⃗′ ′ ( )∫ + 2 2 ⃗⃗
⃗⃗ ( )).
form [
The energy of the electromagnetic field ℋ is the Hamiltonian of the Hamilton equation for the electromagnetic field (14). And can be represented in a second quantized ∫ ⃗⃗ ∑ ( 2 2 ⃗⃗ Therefore: ⃗ ⃗⃗ ∙ ⃗ ⃗⃗′ ′ = ∫ ℎ ⃗⃗ ∙ ℎ⃗⃗ ⃗⃗′ ′ = ⃗⃗ ⃗⃗′ ⃗⃗
′ . (∫ ⃗⃗ ⃗⃗ ′ ∑ ′ ⃗⃗ ⃗⃗′
⃗⃗ ( ) ⃗⃗′ ′( )× ⃗ ⃗⃗ ( ⃗ )∙ ⃗ ⃗⃗′ ′ ( ⃗ )+ ∫ ⃗⃗ ⃗⃗ ′ ∑ ′ ⃗⃗ ( )
⃗⃗′ ′ ( )× ℎ⃗⃗ ⃗⃗ ( ⃗ )∙ ℎ⃗⃗ ⃗⃗′ ′ ( ⃗ )) = (∫ ⃗⃗ ⃗⃗ ′ ∑ ′ ⃗⃗ ⃗⃗′ ⃗⃗ ( ) ⃗⃗′ ′ ( )∫
⃗ ⃗⃗ ( ⃗ )∙ ⃗ ⃗⃗′ ′ ( ⃗ )+ ℎ ⃗⃗ ( ⃗ )∙ ℎ⃗⃗ ⃗⃗′ ′ ( ⃗ )) = 1 ⃗⃗ ∫ ⃗⃗ ∑
( 2 ⃗⃗ ( )+ 2 The following notation is introduced here: = ∑ ∫ ⃗⃗ ħ ⃗⃗ ( + + ⃗⃗ = √2ħ ⃗⃗⃗ (14) (15) (16) (17) (18) ⃗⃗⃗ 2ħ vector. This equation can also be obtained as the Cartan equation by taking the 2-form Ω in the form:
ħ ̅ (20) (21) (22) (23) (24) (25) (26) (27) We introduce new matrices to write these equations in covariant form: = 0 2 = ( )2 = 1 ⃗ = ⃗ = ( , ⃗ ) Multiplying equation (26) by 0 = and dividing by ħ , we obtain: ( Thus, we again obtained the Dirac equation, and, therefore, proved the effectiveness of the 2-form and Cartan equations, that is, the Mechanics of E. Cartan [27-29].
To quantize the Dirac field a second time, we consider the eigenvectors and eigenvalues of the operator H of equation (26):
We expand the solutions of equation (26) in these vectors: { ⃗ ∙ ⃗ + 2 } ( ⃗⃗ ′, ⃗ , ′) = ⃗⃗′ ′ ( ⃗⃗ ′, ⃗ , ′) ( ⃗ , ) = ∑ ⃗⃗,
( ⃗⃗ , ⃗ , ) ⃗⃗ (t) = ̅1̅,̅4̅ +( ⃗ , ) = ∑ ⃗⃗, ̃ ∗( ⃗⃗ , ⃗ , ) + ⃗⃗ (t) = ̅1̅,̅4̅ The energy of the fermion field is equal to the sum of the energies of its quanta. The momentum of the fermion field P ⃗ is equal to: – 4 component spinor, ̃ ∗- complex conjugated transposed spinor.
The operator of the number of fermions N has the form: = ∫ + = ∫ ∑⃗⃗⃗⃗, ⃗⃗′ , , ′ + ⃗⃗
⃗⃗ ̃ ∗( ⃗⃗ , , ⃗ ) ( ⃗⃗ ′, ′, ⃗ ) = = ∑ ⃗⃗, ⃗⃗′ , , ′ + ⃗⃗ ⃗⃗′ ′ ⃗⃗ ⃗⃗′ ′ = ∑ ⃗⃗ + ⃗⃗ ⃗⃗
The Hamiltonian of fermions H in the second quantization representation has the form: = ∫ +{ ⃗ ∙ ⃗ + 2} = ∑ ⃗⃗, ⃗⃗′ , , ′ ∫ + ⃗⃗ ⃗⃗′ ′ ̃ ∗( ⃗⃗ , , ⃗ ){ ⃗ ∙ ⃗ + + 2 } ( ⃗⃗ ′, ′, ⃗ ) = ∑ ⃗⃗, ⃗⃗′ , , ′ ∫ + ⃗⃗
⃗⃗′ ′ ̃ ∗( ⃗⃗ , , ⃗ ) ( ⃗⃗ ′, ′, ⃗ ) ⃗⃗′ ′ = = ∑ ⃗⃗, ⃗⃗′ , , ′ + ⃗⃗ ⃗⃗′ ′ ⃗⃗′ ′ ⃗⃗ ⃗⃗′ ′ = ∑ ⃗⃗ here we have used the explicit form ( ⃗⃗ ′, ⃗ , ′):
3 ( ⃗⃗ ′, ⃗ , ′) = −2 ( ⃗⃗ , ) − ⃗⃗′∙ ⃗ .
The momentum of the Dirac field is equal to the sum of the momenta of its quanta.
To determine the type of current of charged fermions, we use the Hermitian conjugation of the Dirac equation: (29) (30) (31) (32) (33) (34)
Replacing the ordinary derivative with the covariant derivative in equation (28), we obtain a fermion interacting with the electromagnetic field:
Taking the density of the Lagrange function (Bethe G., 1964) in the form: ℒ = ̅ [ ( − ) − ħ ] = ̅ [ [ ( − ) − ħ ] = 0 = 0 + ̅ 1 = ∫ [− ̅ ] = The equations of E. Cartan for 2 forms (40) have the form: (35) (36) (37) (38) tential with an electromagnetic field:
= − ̅ . 0- is the standard Lagrangian of the Dirac equation and ̅ interaction poThe Hamiltonian of the interaction of a fermion with an electromagnetic field is: = ∑ ⃗ , ⃗⃗, ( ⃗ ⃗⃗ +⃗−ħ ⃗⃗,
The obtained results of the second quantization of fermions can be combined by writing in form 2 the forms of E. Cartan for the Cartan equation: = 1 ∧ ɳ1[ , ]+ + 2 ∧ ɳ2[ +, +]+ + 3 ∧ ɳ3 ([ , +]+ − ) + + 1 ∧ ɳ 1 [ , ]− + 2 ∧ 2 [
+, +]− + 3 ∧ 3 ([ , +]− − ) + + ∧ { ћ ∣ 1⟩ − (∑ ⃗⃗,
= [ , ]+ = [ +, +]+ = [ , +]+ − = [ , ]− = = [ +, +]− = [ , +]− − = ћ ∣ 1⟩ − (∑ ⃗⃗,
All equations (41), except [
We rewrite it in the form: where:
ћ ∣ 1⟩ = ( 0 + 1)∣ 1⟩, Representing ∣ 1⟩ = −
ћ ∣ ⟩, we obtain: 0 ћ − 0 − ћ 0 ћ ∣ ⟩ + − 0 ћ ћ ∣ ⟩
0 = 0 − ћ ∣ ⟩ + 1 − 0
ћ ∣ ⟩ Or ћ ∣ ⟩
= 0 ћ 1 − 0 ћ = ∑ ⃗ , ⃗⃗, , ⃗ ⃗⃗ ћ 0 ћ 1 − 0
ћ ∣ ⟩
= 1 ∣ ⟩, In the first order of perturbation theory, its solution has the form: ∣ ⟩ = ∣ 0⟩ + 1 ћ 0
∫ 1( ) ∣ ⟩ ∣ ⟩ ≈ ∣ 0⟩ + 1 ћ 0 ∫ 1( ) ∣ 0⟩ (41) (42) (43) (44) (45) (46)
= +⃗0
∣ to:
At t≫1, this value is equal to: ∣ ⟩ ≈ +⃗0
∣ + neous emission of a photon by an electron. To do this, take: ∣ 0⟩ = ⃗0 theory. Based on the use of these mechanics, additional conditions can be introduced directly into the Cartan equations. Such conditions include, for example, switching conditions between pulses and coordinates, as well as Lorentz calibration conditions. | 2 2 (49) (50) + 1ћ ∫0 ∑ ⃗ , ⃗⃗, , ⃗ ⃗⃗ ћ Thus, the probability for an electron to emit a photon with momentum ħ ⃗⃗ is equal | ⃗0,ћ ⃗⃗, 0|2 2 2( ⃗0−ħ ⃗⃗, − ⃗0, + ћ ⃗⃗ ) =