In this paper, the mechanics of E. Cartan is used to formulate the quantum electrodynamics of the meson field. The dynamics of quantized fields are written in the form of Cartan mechanics. One of the Cartan equations - the Schrödinger equation is solved by the perturbation theory method. As a result, the processes of photon boson emission and photon boson absorption are studied. The modern use of the tools of Cartan mechanics for the formulation of all branches of theoretical physics: mechanics, electrodynamics, quantum mechanics also involves the spread of Cartan mechanics in problems of quantum electrodynamics. Along with the well-known mechanics of Lagrange and Hamilton, the use of Cartan mechanics tools has become very promising This paper answers this question. To quantize the meson field, the Lagrangian and Hamiltonian formalism is used. And for quantization of the electromagnetic field, Maxwell's equations and the energy formula of the electromagnetic field are used. The type of electromagnetic current is derived from the Lagrangian invariance concerning the phase Ψ-operator of the meson field. And the form of electromagnetic interaction of the electromagnetic field is. with a meson current from electrodynamics
Along with the well-known mechanics of Lagrange and Hamilton, the use of Cartan mechanics tools has become very promising [5].
Its application first to the problems of mechanics, then, in general relativity, Einstein and, finally, to field theory) showed the universality and convenience of its application to other problems of physics. *
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One of these problems is quantum field theory and, in particular, quantum electrodynamics.
In this paper, the mechanics of E. Cartan is used to formulate the quantum electrodynamics of the meson field.
Mesons are bound states of a quark and an antiquark. Mesons have a baryon number B = 0 and an integer (including zero) spin, i.e., they are bosons. The masses and quantum numbers of mesons are determined by the types of quark and antiquark that make up the meson, their radial quantum numbers, the relative orientation of their spins, and the values of isospins and orbital moments. The interaction caused by the meson field of nuclear forces is carried out using virtual particles.
The quark model allows one to qualitatively describe the structure of mesons and to obtain their quantum numbers.
The dynamics of quantized fields are written in the form of Cartan mechanics. One of the Cartan equations - the Schrödinger equation is solved by the perturbation theory method. As a result, the processes of photon boson emission and photon boson absorption are studied. 2
Main content. E. Cartan mechanics in quantum electrodynamics of a meson field The Lagrangian of the meson field has the form [10]: The Lagrange equation has the form: = ∫ { 12 ∗
− ( ⃗ ∗)∙ ( ⃗ )− ћ22 2 ∗ }
Equation (
− √−ћ2∆ + 2 2 },
really:
(
The Hamiltonian corresponding to the Lagrangian 1 has the form:
= ∫
+ ∗
For the second quantization of mesons, the corresponding ideas of Haken [10] were
used:
= ∫
ћ ∗
− ℒ = ∫ ∗ √−ћ2∆ + 2 2
(
To pass to the representation of secondary quantization, we expand Ψ by the annihilation operators: and + = ∑ − ⃗⃗ ∙⃗ ⃗ = ∑ − ⃗⃗ ∙⃗ ⃗
3 ⃗ , (2 )2 3 + ⃗ concerning the creation operators.
Hamiltonian (
The electric field strength ⃗ and the magnetic field induction ⃗ can be represented
in the form:
= ∑ ⃗ √ћ2 2 + 2 2 +
⃗ ⃗
(
⃗ √4 , ⃗ ⃗ = ∑ ⃗ ⃗ ℎ ⃗ √4 .
Here ћ ⃗ = ⃗ – is the photon momentum, α – indices of 2 directions perpendicular to
it. We substitute the expressions (
leads to: what gives Similarly: what gives
∇⃗ ∙ ⃗ = ∇⃗ ∙ ℎ⃗⃗ = 0, ∇⃗ × ⃗ = − 1 ⃗ = ∑⃗ ⃗ ⃗ ( )⃗∇ × ⃗ = − ∑⃗ ⃗⃗ ( )1 ⃗ ℎ⃗ , gives ⃗⃗ ( ) = − 2⃗ ⃗ и ∇⃗ × ⃗ = ⃗⃗ ℎ⃗⃗
∇⃗ × ⃗ = 1 ⃗
= ∑⃗
⃗
⃗ ∇⃗ × ℎ⃗⃗ = ∑⃗
1
⃗
⃗⃗ ⃗ ,
Calculating the rotor from the expressions (
⃗ .
∇⃗ × (∇⃗ × ⃗ )= ∇⃗∇⃗ ∙ ⃗ − ∆ ⃗ =
⃗⃗ ∇⃗ × ℎ⃗⃗ ,
(
∇⃗ × (∇⃗ × ℎ⃗⃗ )= ∇⃗∇⃗ ∙ ℎ⃗⃗ − ∆ℎ⃗⃗ = ⃗⃗ ∇⃗ × ⃗ , ∆ℎ⃗⃗ = − ( ⃗⃗ )2 ⃗ ℎ⃗ .
Expressions (
Thus, the energy of the electromagnetic field Н2 is equal to: 2 = 1 8 ∫ ( 2 + 2)=
∙ 4 ∫ 1 8 + ∑⃗ ⃗ ´ ´ ⃗ ⃗ ´ ⃗ ⃗ ´ ´ ⃗ ⃗ ´ ´ ⃗ ∙ ⃗ ´ ´)= (∑⃗ ⃗ ´ ´ ⃗
⃗ ´ ´ ∙ ℎ⃗⃗ ∙ ℎ⃗⃗ ´ ´ +
1
2 ∑⃗ ( ⃗
(
So, a consequence of the Lagrangian invariance is the existence of the 4th vector :
ℒ ℒ ̃ = 0 = ̃ = − ∗ ∗.
ℒ ̃ = ( ∗ −
) . Thus, the Hamiltonian of the interaction of an electromagnetic field with a field having an electric charge [16] From the expression for and it can be seen that вз has the form: вз = 1 ∫ . вз = ∑ ⃗ ⃗ ( ⃗ + Using the results of the second quantization of the meson and electromagnetic fields and their interaction, one can formulate quantum electrodynamics in the form of the mechanics of E. Cartan [5].
To do this, take the 2 Cartan form Ω in the form:
= { ћ ∣ 1 ⟩ − [ ∑ √ћ2 2 + 2 + + ∑ ⃗ ћ ⃗ ( ⃗ + ∑ ⃗ ⃗ ( +⃗−⃗ + ∑ ⃗ ⃗ ´[ ⃗ +, +⃗´] ⃗ ∧ ⃗ ´ + ∑ ⃗ ⃗ ´([ ⃗ , +⃗´]− ⃗ ⃗ ´) ⃗ ∧ ⃗ ´ + + ⃗ + 1)+ 2 + ∑ ⃗ ⃗ ´[ ⃗ , ⃗ ´] ⃗ ∧ ⃗ ´ + ∑ ⃗ ⃗ ´[
+⃗, +⃗´] ⃗ ∧ ƺ ⃗ ´. [ , ] = − The equations of E. Cartan [5] give: ⃗⃗ = − { ћ ∣ 1 ⟩ − [ ∑ √ћ2 2 + 2 + + ∑ ⃗ ћ ⃗ ( ⃗ ⃗ ⃗ −⃗
⃗ )] ∣ 1 ⟩ } = [ ⃗ , +⃗´]− ⃗ ⃗ ´ = [ ⃗ , ⃗ ´] == [ ⃗
+, +⃗´] = [ ⃗ , +⃗´]− ⃗ ⃗ ´ = [ ⃗ , ⃗ ´] = [ +⃗, +⃗´].
Equation (
ћ ∣1 ⟩ = [ ∑ √ћ2 2 + 2 +
+ ∑ ⃗ ћ ⃗ ( ⃗
+ ⃗ + 1)+
2
+
+ ∑ ⃗ ⃗ ( ⃗ −⃗
0 = ∑ √ћ2 2 + 2 +
+ ∑ ⃗ ћ ⃗ ( ⃗
and Н1 for the Hamiltonian of the interaction of the meson field with the
electromagnetic field:
we rewrite equation (
And let's move on to the description of the interaction [10]: what gives:
∣ 1 ⟩ = − 0 ћ ћ ∣ ⟩ − 0 ћ ∣ ⟩ 0∣ 1 ⟩ + = 0∣ 1 ⟩ + 1 − 0 ћ ∣ ⟩, ћ ∣ ⟩ = 0 ћ 1 − 0 ћ
∣ ⟩ = 1( )∣ ⟩ 1( )= 0 ћ 1 − 0 ћ ⃗ ( )= 0 ћ ⃗ − 0 ћ = ⃗ + ⃗ ⃗ ⃗ − ⃗ + ⃗ . ⃗ ⃗ ( ) = ⃗ [ + ⃗ , ⃗ ] = − ⃗ ⃗ ( ). ⃗
⃗ ( )= − ⃗ ⃗
and ⃗ ( )= ⃗
+
+⃗ .
⃗⃗ ( )
= − √ ⃗ 2 +
(
In the first order of perturbation theory, integral equation (36) takes the form: + 1 Suppose that in the initial state ∣ 0⟩ there is a charged boson with momentum ћ ⃗ 1:∣ 0⟩ = +⃗1∣ vacuum⟩.
Here: ∣ ⟩ ≈ +⃗1∣ vacuum⟩ + 1 = +⃗1∣ vacuum⟩ +
⃗ 1ћ ∑⃗ ⃗⃗ 1⃗ ( ⃗⃗ 1⃗ − 1) +⃗1−⃗ +⃗ ∣ vacuum⟩ Therefore: ⃗ 1⃗ = √( ⃗ − ) + 2
Thus, we obtained a linear combination of the initial state +⃗1∣ vacuum⟩ of the existence of one charged boson and the state ∑⃗ ⃗ +⃗1−⃗ +⃗ ∣ vacuum⟩, in which this boson emitted one photon with momentum ћ and polarization α.
The probability of emitting a photon with ћ and α is | ⃗ | : 2 2 | ⃗ | = 1 ћ2 | ⃗ |
If in the initial state ∣ 0⟩ there is a charged boson with momentum ћ ⃗ 1 and an electromagnetic field quantum with momentumћ⃗ 2 and polarization 1: Then approximate equality (37) takes the form: ∣ 0⟩ = ⃗ 1 + + Thus, we obtained a linear combination of two initial states: a boson with momentum ћ ⃗ 1 and a photon with momentum ћ ⃗ 2 nd polarization 1 and the final state (after interaction): the state of the boson with momentum ћ( ⃗ 1 + ⃗ 2), which describes the absorption of the photon boson. The probability of this process is ’:
C q ћ
2 4sin2 K1K2,K2 t g* 2 2
K1K2,K2 (42) (43) (44) (45) ond):
K1K2 ,K2
t . (46) t Based on (46), we obtain the probability of photon absorption per unit time (per secW ’q
’ d C q dt 2 g* 2
K1K2 ,K2 (47) So, we have shown that the mechanics of E. Cartan allows us to formulate quantum electrodynamics in a form convenient for calculations. 3
Conclusions
For the second quantization of mesons, ideas were used [10], which lead to equation
(
To quantize the electromagnetic field, we represent E and H in the form of (
The modern use of the tools of Cartan mechanics for the formulation of all branches of theoretical physics: mechanics, electrodynamics, quantum mechanics [3], also involves the spread of Cartan mechanics in quantum electrodynamics asks.
This paper answers this question. To quantize the meson field, the Lagrangian and Hamiltonian formalism is used. Moreover, for quantization of the electromagnetic field, Maxwell's equations and the energy formula of the electromagnetic field are used. The type of electromagnetic current is derived from the Lagrangian invariance concerning the phase Ψ-operator of the meson field.
Also, the form of electromagnetic interaction of the electromagnetic field is with a meson current from electrodynamics.