207 Description of the Interaction of Fermions with an Electromagnetic Field Based on Cartan Mechanics* Anatoliy N. Kazak 1 [0000-0001-7678-921], Nikolay N. Oleinikov 1 [0000-0002-9348-9153], Yuri D. Mendygulov 1 [0000-0003-3490-9143], Dmitry V. Samokhvalov 2 [0000-0002-5127-2511] 1 V.I. Vernadsky Crimean Federal University, Simferopol, Russia 2 Saint Petersburg Electrotechnical University "LETI", St. Petersburg, Russia kazak_a@mail.ru Abstract. In this paper, the mechanics of Eli Cartan is used, which is an alterna- tive to the Lagrange-Hamiltonian formalism, has certain advantages in the for- mulation 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 quan- tum field theory. Based on the use of these mechanics, additional conditions can be introduced directly into the Cartan equations. Such conditions include, for ex- ample, switching conditions between pulses and coordinates, as well as Lorentz calibration conditions Keywords: Cartan mechanics; quantum electrodynamics, fermions, electro- magnetic interaction. 1 Introduction The mechanics of Eli Cartan, which is an alternative to the Lagrange-Hamiltonian for- malism, 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 Uh- lenbeck 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 [1-4]. 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). 208 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 [5-7]. 2 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 En- rico 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 [8-10]. 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 statis- tics, Dirac called bosons [11-13]. The carriers of all interactions — the photon in the electromagnetic, the gluons in the strong, and the W and Z particles in the weak — are 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 likeli- hood 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 [9, 10]. 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 be- tween particles with an electric charge [14-17]. According to the generally accepted view, such an interaction between charged particles does not occur directly, but only using an electromagnetic field. In the framework of quantum field theory [11, 16], such an interaction is carried by a massless boson — a photon. 209 Fermions are among the fundamental particles that have an electric charge and par- ticipate in electromagnetic interaction. Along with electromagnetic, there are also weak [3, 7, 9] and strong interactions. The electromagnetic interaction is distinguished by its long-range nature. According to Coulomb's law, the force of interaction between charges decreases only as of the second power of the distance. Gravitational interaction also complies with this law, but it is much weaker than electromagnetic [17-19]. According to the classical (non-quantum) approach, electromagnetic interaction is described by classical electrodynamics [17-20]. First, we quantize the electromagnetic field. Consider a 2-form Ω of the form: 1 𝜕𝐴𝜈 𝜕𝐴𝜈 𝛼𝛽 𝛺 = ∫ 𝑑𝑉 ′ {𝑑 ( 𝑔 ) ∧ 𝑑𝑡} = 2 𝜕𝑥 ′𝛼 𝜕𝑥 ′𝛽 2 𝜈 𝜕 𝐴 ∫ 𝑑𝑉 ′ 𝜕𝑥 ′𝛼𝜕𝑥 ′𝛽 𝑑 𝐴𝜈 𝑔𝛼𝛽 ∧ 𝑑𝑡, 𝜈, 𝛼, 𝛽 = ̅̅̅̅ 0,3 (1) The equation of E. Cartan for her has the form: 𝛿𝛺 𝜕 2 𝐴𝜈 𝛾 𝜕 2 𝐴𝛾 (𝑥) 0= = ∫ 𝑑𝑉 ′ 𝛿 (𝑥⃗ ′ − 𝑥⃗ )𝛿𝜈 𝑔𝛼𝛽 𝑑𝑡 = 𝑔𝛼𝛽 𝑑𝑡 = 0 𝛿𝑑𝐴𝜈 𝜕𝑥 ′𝛼𝜕𝑥 ′𝛽 𝜕𝑥 𝛼𝜕𝑥 𝛽 𝜕 2 𝐴𝛾(𝑥) The equation: 𝜕𝑥 𝛼𝜕𝑥 𝛽 = 0 – is the equation of the dynamics of the electromagnetic field vector potential. We introduce the vectors 𝐸⃗⃗ and 𝐻 ⃗⃗, which also describe the electromagnetic field: ⃗⃗𝐴0 − 1 𝜕𝐴⃗ 𝐸⃗⃗ = −∇ (2) 𝑐 𝜕𝑡 ⃗⃗ = ∇ 𝐻 ⃗⃗ × 𝐴⃗. (3) ⃗⃗ and 𝐸⃗⃗ we obtain the equations: Then for 𝐻 ⃗⃗ ∙ 𝐻 ∇ ⃗⃗ = ∇ ⃗⃗ ∙ ∇ ⃗⃗ × 𝐴⃗ = 0 2 0 2 0 ⃗⃗ ∙ 𝐸⃗⃗ = −∆𝐴0 − 1 𝜕 ∇ ∇ ⃗⃗ ∙ 𝐴⃗ = −∆𝐴0 + 12 𝜕 𝐴2 = 𝑔 𝛼𝛽 𝜕𝛼𝐴 𝛽 = 0 (4) 𝑐 𝜕𝑡 𝑐 𝜕𝑡 𝜕𝑥 𝜕𝑥 Lorentz calibration used here: 𝜕𝐴0 𝜕𝐴𝛼 ⃗⃗ ∙ 𝐴⃗ = 0 = 𝛼 +∇ (5) 𝑐𝜕𝑡 𝜕𝑥 ⃗⃗ × 𝐸⃗⃗ = − 1 𝜕 ∇ ∇ ⃗⃗ ∙ 𝐴⃗ = − 1 𝜕𝐻⃗⃗ (6) 𝑐 𝜕𝑡 𝑐 𝜕𝑡 ⃗⃗ ∙ 𝐻 ∇ ⃗⃗ = ∇ ⃗⃗ × (∇ ⃗⃗ × 𝐴⃗) = ∇ ⃗⃗ ∇ ⃗⃗ 1 𝜕𝐴0 − ∆𝐴⃗ = ⃗⃗ ∙ 𝐴⃗ − ∆𝐴⃗ = −∇ 𝑐 𝜕𝑡 210 2 ⃗⃗ 1 𝜕𝐴0 − 12 𝜕 𝐴2⃗ = 1 𝜕 (−∇ = −∇ ⃗⃗𝐴0 − 1 𝜕𝐴⃗) = 1 𝜕𝐸⃗⃗. (7) 𝑐 𝜕𝑡 𝑐 𝜕𝑡 𝑐 𝜕𝑡 𝑐 𝜕𝑡 𝑐 𝜕𝑡 This is the continuity equation for the energy of an electromagnetic field [21-23]. To record the energy of the electromagnetic field in the secondary quantization rep- resentation, we express 𝐸⃗⃗ and 𝐻 ⃗⃗ through the generalized coordinates and momenta of the electromagnetic field: The energy of the electromagnetic field is: 1 ℋ = 8𝜋 ∫ 𝑑𝑉 (𝐸⃗⃗ 2 + 𝐻 ⃗⃗2 ). (8) Really: 𝜕ℋ 1 𝜕𝐸⃗⃗ ⃗⃗ 𝜕𝐻 1 = 4𝜋 ∫ 𝑑𝑉 (𝐸⃗⃗ ∙ 𝜕𝑡 + 𝐻 ⃗⃗ ∙ ) = (𝑐E ⃗⃗ ∙ (∇ ⃗⃗ × 𝐻 ⃗⃗) − 𝑐H ⃗⃗⃗ ∙ (∇ ⃗⃗ × 𝐸⃗⃗ )) = 𝜕𝑡 𝜕𝑡 4𝜋 ⃗⃗ ∙ 𝑐 𝐸⃗⃗ × 𝐻 −∇ ⃗⃗ 4𝜋 𝜕 𝑐 ∫ 𝑑𝑉 ℋ = ∫𝑑𝑉 − ⃗⃗ × 𝐻 𝑉(E ⃗⃗, ⋯ , ⋯ ) 𝜕𝑡 𝑉 4𝜋 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 𝐸⃗⃗ and 𝐻 ⃗⃗ through the generalized coordinates and momenta of the electromag- netic field: 𝐻 ⃗⃗ ∑𝛼 𝑃𝐾⃗⃗𝛼 (𝑡)ℎ⃗⃗𝐾⃗⃗𝛼 (𝑟⃗)√4𝜋; 𝐸⃗⃗ = ∫ 𝑑𝐾 ⃗⃗ = ∫ 𝑑𝐾 ⃗⃗ ∑𝛼 𝜔𝐾⃗⃗ 𝑞⃗𝐾⃗⃗𝛼 (𝑡)𝑒⃗𝐾⃗⃗𝛼 (𝑟⃗)√4𝜋 And 𝑃 ̇_ (𝐾 ⃗𝛼) = − 〖𝜔 ^ 2〗 _𝐾 ⃗ 𝑞_ (𝐾 ⃗𝛼) Using Maxwell's equations ⃗⃗ ∙ 𝐸⃗⃗ = ∇ ∇ ⃗⃗ ∙ 𝐻 ⃗⃗ = 0 ⃗⃗ × 𝐸⃗⃗ = − 1 𝜕𝐻⃗⃗ ∇ ⃗⃗ = 1 𝜕𝐸⃗⃗, ⃗⃗ × 𝐻 ∇ 𝑐 𝜕𝑡 𝑐 𝜕𝑡 ⃗⃗ ∙ ℎ⃗⃗𝐾⃗⃗𝛼 (𝑟⃗) = ∇ we get ∇ ⃗⃗ ∙ 𝑒⃗𝐾⃗⃗𝛼 (𝑟⃗) = 0. (9) ⃗⃗ × 𝐸⃗⃗ = ∫ 𝑑𝐾 ∇ ⃗⃗ × 𝑒⃗𝐾⃗⃗𝛼 (𝑟⃗) = − 1 𝜕𝐻⃗⃗ = ∫ 𝑑𝐾 ⃗⃗ ∑𝛼 𝜔𝐾⃗⃗ 𝑞𝐾⃗⃗𝛼 (𝑡)∇ ⃗⃗ ∑𝛼 − 1 𝑃̇𝐾⃗⃗𝛼 (𝑡)ℎ⃗⃗𝐾⃗⃗𝛼 (𝑟⃗) 𝑐 𝜕𝑡 𝑐 𝜔 ⃗⃗ × 𝑒⃗𝐾⃗⃗𝛼 (𝑟⃗) = 𝐾⃗⃗⃗⃗ ℎ⃗⃗𝐾⃗⃗𝛼 (𝑟⃗) Hence: ∇ (10) 𝑐 and 𝑃̇𝐾⃗⃗𝛼 = −𝜔2 𝐾⃗⃗ 𝑞𝐾⃗⃗𝛼 (11) ⃗⃗ × 𝐻 ∇ ⃗⃗ = ∫ 𝑑𝐾 ⃗⃗ ∑𝛼 𝑃𝐾⃗⃗𝛼 (𝑡)∇ ⃗⃗ ∑𝛼 𝜔𝐾⃗⃗⃗⃗ 𝑞̇ 𝐾⃗⃗𝛼 (𝑡)𝑒⃗𝐾⃗⃗𝛼 (𝑟⃗). ⃗⃗ × ℎ⃗⃗𝐾⃗⃗𝛼 (𝑟⃗) = 1 𝜕𝐸⃗⃗=∫ 𝑑𝐾 𝑐 𝜕𝑡 𝑐 ⃗⃗ × ℎ⃗⃗𝐾⃗⃗𝛼 (𝑟⃗) = 𝜔⃗𝐾⃗⃗ 𝑒⃗𝐾⃗⃗𝛼 Hence: ∇ (12) 𝑐 and 𝑃𝐾⃗⃗𝛼 (𝑡) = 𝑞̇ 𝐾⃗⃗𝛼 (𝑡). (13) 211 Thus, 𝑃𝐾⃗⃗𝛼 (𝑡) и 𝑞𝐾⃗⃗𝛼 (𝑡) are canonical Hamilton variables, their equation: 𝑞̇ 𝐾⃗⃗𝛼 = 𝑃𝐾⃗⃗𝛼 { ̇ (14) 𝑃𝐾⃗⃗𝛼 = −𝜔2 𝐾⃗⃗ 𝑞𝐾⃗⃗𝛼 It describes a mathematical pendulum for each momentum ћ𝐾 ⃗⃗ and polarization 𝛼. Let us prove that ℎ⃗⃗𝐾⃗⃗𝛼 are orthogonal to ℎ⃗⃗𝐾⃗⃗′𝛼′ for 𝐾 ⃗⃗ ≠ 𝐾 ⃗⃗ , α≠α', as well as 𝑒⃗𝐾⃗⃗𝛼 : Using ′ (10) we obtain [22-25]: ⃗⃗ × 𝑒⃗𝐾⃗⃗𝛼 (𝑟⃗)) = −∆𝑒⃗𝐾⃗⃗𝛼 (𝑟⃗) = 𝜔𝐾⃗⃗⃗ ∇ ⃗⃗ × (∇ ∇ ⃗⃗ × ℎ⃗⃗𝐾⃗⃗𝛼 = (𝜔𝐾⃗⃗⃗ )2 𝑒⃗𝐾⃗⃗𝛼 (𝑟⃗). (15) 𝑐 𝑐 Using (13) we obtain: ⃗⃗ × ℎ⃗⃗𝐾⃗⃗𝛼 (𝑟⃗)) = −∆ℎ⃗⃗𝐾⃗⃗𝛼 (𝑟⃗) = 𝜔⃗⃗⃗⃗𝐾 ∇ ⃗⃗ × (∇ ∇ ⃗⃗ × 𝑒⃗𝐾⃗⃗𝛼 (𝑟⃗) = (𝜔⃗⃗⃗⃗𝐾)2 ℎ⃗⃗𝐾⃗⃗𝛼 (𝑟⃗). (16) 𝑐 𝑐 Equations (15) and (16) show that 𝑒⃗𝐾⃗⃗𝛼 (𝑟⃗) and ℎ⃗⃗𝐾⃗⃗𝛼 (𝑟⃗) are eigenfunctions of the op- 𝜔 erator - ∆ (self-adjoint) with eigenvalues( ⃗𝐾⃗⃗⃗)2 . 𝑐 Hence: ∫ 𝑑𝑉 𝑒⃗𝐾⃗⃗𝛼 ∙ 𝑒⃗𝐾⃗⃗′𝛼′ = ∫ 𝑑𝑉 ℎ⃗⃗𝐾⃗⃗𝛼 ∙ ℎ⃗⃗𝐾⃗⃗′𝛼′ = 𝛿𝐾⃗⃗𝐾⃗⃗′ 𝛿𝛼𝛼′ . Therefore: 1 ℋ = 2 ∫ 𝑑𝑉 (∫ 𝑑𝐾 ⃗⃗ 𝑑𝐾 ⃗⃗ ′ ∑𝛼𝛼′ 𝜔𝐾⃗⃗ 𝜔𝐾⃗⃗′ 𝑞𝐾⃗⃗𝛼 (𝑡)𝑞𝐾⃗⃗′𝛼′ (𝑡) × 𝑒⃗𝐾⃗⃗𝛼 (𝑟⃗) ∙ 𝑒⃗𝐾⃗⃗′𝛼′ (𝑟⃗) + ⃗⃗ ′ ∑𝛼𝛼′ 𝑃𝐾⃗⃗𝛼 (𝑡)𝑃𝐾⃗⃗′𝛼′ (𝑡) × ℎ⃗⃗𝐾⃗⃗𝛼 (𝑟⃗) ∙ ℎ⃗⃗𝐾⃗⃗′𝛼′ (𝑟⃗)) = ⃗⃗ 𝑑𝐾 ∫ 𝑑𝐾 1 ⃗⃗ 𝑑𝐾 = 2 (∫ 𝑑𝐾 ⃗⃗ ′ ∑𝛼𝛼′ 𝜔𝐾⃗⃗ 𝜔𝐾⃗⃗′ 𝑞𝐾⃗⃗𝛼 (𝑡)𝑞𝐾⃗⃗′𝛼′ (𝑡) ∫ 𝑑𝑉 𝑒⃗𝐾⃗⃗𝛼 (𝑟⃗) ∙ 𝑒⃗𝐾⃗⃗′𝛼′ (𝑟⃗) + + ∫ 𝑑𝐾 ⃗⃗ ′ ∑𝛼𝛼′ 𝑃𝐾⃗⃗𝛼 (𝑡)𝑃𝐾⃗⃗′𝛼′ (𝑡) ∫ 𝑑𝑉 ℎ⃗⃗𝐾⃗⃗𝛼 (𝑟⃗) ∙ ℎ⃗⃗𝐾⃗⃗′𝛼′ (𝑟⃗)) = 1 ∫ 𝑑𝐾 ⃗⃗ 𝑑𝐾 ⃗⃗ ∑𝛼 (𝑃2 𝐾⃗⃗𝛼 (𝑡) + 2 +𝜔2 𝐾⃗⃗ 𝑞 2 𝐾⃗⃗𝛼 (𝑡)). (17) The energy of the electromagnetic field ℋ is the Hamiltonian of the Hamilton equa- tion for the electromagnetic field (14). And can be represented in a second quantized form [26]: 1 𝜔 1 ⃗⃗ ∑𝛼(𝜔2 𝐾⃗⃗ 𝑞 2 ⃗⃗ + 𝑃2 𝐾⃗⃗𝛼 ) = ∫ 𝑑𝐾 𝐻 = 2 ∫ 𝑑𝐾 ⃗⃗ ∑𝛼 ħ𝜔𝐾⃗⃗ ( 𝐾⃗⃗⃗⃗ 𝑞 2 ⃗⃗ + 𝑃2 𝐾⃗⃗𝛼 ) = 𝐾𝛼 2ħ 𝐾𝛼 2ħ𝜔 ⃗⃗⃗⃗ 𝐾 1 𝜔 1 𝜔 1 ⃗⃗ ħ𝜔𝐾⃗⃗ [(√ = ∑𝛼 ∫ 𝑑𝐾 𝑃𝐾⃗⃗𝛼 + 𝑖√ 2ħ𝐾⃗⃗⃗⃗ 𝑞𝐾⃗⃗𝛼 ) × (√2ħ𝜔 𝑃𝐾⃗⃗𝛼 − 𝑖√ 2ħ𝐾⃗⃗⃗ 𝑞𝐾⃗⃗𝛼 ) + 2] = 2ħ𝜔 ⃗⃗⃗⃗ 𝐾 ⃗⃗⃗⃗ 𝐾 ⃗⃗ ħ𝜔𝐾⃗⃗ (𝑎+ 𝐾⃗⃗𝛼 𝑎𝐾⃗⃗𝛼 + 1) = ∑𝛼 ∫ 𝑑𝐾 (18) 2 The following notation is introduced here: 1 𝜔 𝑎+ 𝐾⃗⃗𝛼 = √2ħ𝜔 𝑃𝐾⃗⃗𝛼 + 𝑖√ 2ħ⃗𝐾⃗⃗⃗ 𝑞𝐾⃗⃗𝛼 (19) ⃗𝐾 ⃗⃗⃗ 212 1 𝜔 𝑎𝐾⃗⃗𝛼 = √ 𝑃𝐾⃗⃗𝛼 − 𝑖√ ⃗⃗⃗⃗𝐾 𝑞𝐾⃗⃗𝛼 (20) 2ħ𝜔⃗⃗⃗⃗ 𝐾 2ħ As is well known, fermions obey the Dirac equation. To obtain it, it is enough to extract the square root from the Klein-Gordon equation. Klein-Gordon equation: 𝜕2 𝑚2𝑐 2 𝑔𝛼𝛽 𝜕𝑥 𝛼𝜕𝑥 𝛽 𝜑 + 𝜑=0 (21) ħ2 1 Using equality [24-26]: 2 (𝛾 𝛼 𝛾 𝛽 + 𝛾 𝛽 𝛾 𝛼 ) = 𝑔𝛼𝛽 , we obtain from (21): 1 𝜕2 𝑚2𝑐 2 𝜕2 𝑚2𝑐 2 (𝛾 𝛼 𝛾 𝛽 + 𝛾 𝛽 𝛾 𝛼 ) 𝜑+ 𝜑 = 0 = (𝛾 𝛼 𝛾 𝛽 𝜕𝑥 𝛼𝜕𝑥 𝛽 𝜑 + 𝜑) = 2 𝜕𝑥 𝛼𝜕𝑥 𝛽 ħ2 ħ2 𝜕 𝑚𝑐 𝜕 𝑚𝑐 𝜕 𝑚𝑐 = (𝑖𝛾 𝛼 + ) (𝑖𝛾 𝛽 − ) 𝜑, from here: (𝑖𝛾 𝛼 − ) 𝛹 = 0. (22) 𝜕𝑥 𝛼 ħ 𝜕𝑥 𝛽 ħ 𝜕𝑥 𝛼 ħ This is the Dirac equation of the 𝛾 𝛼 - matrix 4х4, and 𝛹- is a 4-component column vector. This equation can also be obtained as the Cartan equation by taking the 2-form Ω in the form: ̅ ∧ 𝑑𝛹𝑖ħ − 𝑑𝐻 ∧ 𝑑𝑡} 𝛺 = ∫ 𝑑𝑥⃗ {𝑑𝛹 (23) taking H in the form: ̅ {𝑐𝛼⃗ ∙ 𝑝⃗ + 𝑚𝑐 2 𝛽 }𝛹𝑑𝑉 𝐻 = ∫𝛹 (24) Then the equations of E. Cartan give: 𝛿𝛺 𝛿𝛺 0 = 𝛿𝑑𝛹̅ = 𝛿𝑑𝛹 = 𝑖ħ𝑑𝛹 − {𝑐𝛼⃗ ∙ 𝑝⃗ + 𝑚𝑐 2 𝛽 }𝛹𝑑𝑡 = ̅ −{𝑐𝛼⃗ ∙ 𝑝⃗ + 𝑚𝑐 2 𝛽 }+ 𝛹 = −𝑖ħ𝑑𝛹 ̅ 𝑑𝑡 (25) Hence: 𝜕𝛹 𝑖ħ 𝜕𝑡 = {𝑐𝛼⃗ ∙ 𝑝⃗ + 𝑚𝑐 2 𝛽 }𝛹 (26) Using 𝛼⃗ + = 𝛼⃗, 𝛽 + = 𝛽 , we obtain: ̅ 𝜕𝛹 𝑖ħ ̅ = −{𝑐𝛼⃗ ∙ 𝑝⃗ + 𝑚𝑐 2 𝛽 }𝛹 (27) 𝜕𝑡 We introduce new matrices to write these equations in covariant form: 𝛽 = 𝛾0 𝛽 2 = (𝛼 𝑘 )2 = 1 𝛽𝛼⃗ = 𝛾⃗ 𝑥 𝜇 = (𝑐𝑡, 𝑟⃗) Multiplying equation (26) by𝛾 0 = 𝛽 and dividing by ħ𝑐, we obtain: 𝜕 𝑚𝑐 (𝑖𝛾 𝛼 𝜕𝑥 𝛼 − ħ ) 𝛹 = 0 (28) 213 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 eigen- values of the operator H of equation (26): ⃗⃗ ′ , 𝑟⃗, 𝑖 ′ ) = 𝐸𝐾⃗⃗′𝑖 ′ 𝒱(𝐾 {𝑐𝛼⃗ ∙ 𝑝⃗ + 𝑚𝑐 2 𝛽 }𝒱(𝐾 ⃗⃗ ′ , 𝑟⃗, 𝑖 ′ ) (29) We expand the solutions of equation (26) in these vectors: ⃗⃗ , 𝑟⃗, 𝑖)𝑎𝐾⃗⃗𝑖 (t) 𝛹 (𝑟⃗, 𝑡) = ∑𝐾⃗⃗,𝑖 𝒱(𝐾 𝑖 = ̅̅̅̅ 1,4 (30) 𝛹 + (𝑟⃗, 𝑡) = ∑𝐾⃗⃗,𝑖 𝒱̃ ∗ (𝐾 ⃗⃗ , 𝑟⃗, 𝑖)𝑎+ 𝐾⃗⃗𝑖 (t) ̅̅̅̅ 𝑖 = 1,4 (31) 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: 𝒱 – 4 component spinor, 𝒱̃ ∗ - complex conjugated transposed spinor. The operator of the number of fermions N has the form: 𝑁 = ∫ 𝑑𝑉 𝛹 + 𝛹 = ∫ 𝑑𝑉 ∑⃗⃗⃗⃗ 𝐾,𝐾 + ⃗⃗ ′ ,𝑖,𝑖 ′ 𝑎 𝐾 ̃ ∗ ⃗⃗ ⃗)𝒱(𝐾 ⃗⃗𝑖 𝒱 (𝐾 , 𝑖, 𝑟 ⃗⃗𝑖 𝑎𝐾 ⃗⃗ ′ , 𝑖 ′, 𝑟⃗) = = ∑𝐾⃗⃗,𝐾⃗⃗′ ,𝑖,𝑖 ′ 𝑎+ 𝐾⃗⃗𝑖 𝑎𝐾⃗⃗′𝑖 ′ 𝛿𝐾⃗⃗𝐾⃗⃗′ 𝛿𝑖𝑖 ′ = ∑𝐾⃗⃗𝑖 𝑎+ 𝐾⃗⃗𝑖 𝑎𝐾⃗⃗𝑖 (32) The Hamiltonian of fermions H in the second quantization representation has the form: 𝐻 = ∫ 𝑑𝑉𝛹 + {𝑐𝛼⃗ ∙ 𝑝⃗ + 𝑚𝑐 2 }𝛹 = ∑𝐾⃗⃗,𝐾⃗⃗′,𝑖,𝑖 ′ ∫ 𝑑𝑉 𝑎+ 𝐾⃗⃗𝑖 𝑎𝐾⃗⃗′𝑖 ′ 𝒱̃ ∗ (𝐾 ⃗⃗ , 𝑖, 𝑟⃗){𝑐𝛼⃗ ∙ 𝑝⃗ + 2 } +𝑚𝑐 𝛽 𝒱(𝐾⃗⃗ , 𝑖 , 𝑟⃗) = ∑𝐾⃗⃗,𝐾⃗⃗′,𝑖,𝑖 ′ ∫ 𝑑𝑉 𝑎 𝐾⃗⃗𝑖 𝑎𝐾⃗⃗′ 𝑖 ′ 𝒱̃ (𝐾 ′ ′ + ∗ ⃗⃗ , 𝑖, 𝑟⃗)𝒱(𝐾 ⃗⃗ , 𝑖 ′ , 𝑟⃗)𝐸𝐾⃗⃗′𝑖 ′ = ′ = ∑𝐾⃗⃗,𝐾⃗⃗′ ,𝑖,𝑖 ′ 𝑎+ 𝐾⃗⃗𝑖 𝑎𝐾⃗⃗′𝑖 ′ 𝐸𝐾⃗⃗′𝑖 ′ 𝛿𝐾⃗⃗𝐾⃗⃗′ 𝛿𝑖𝑖 ′ = ∑𝐾⃗⃗𝑖 𝐸𝐾⃗⃗𝑖 𝑎+ 𝐾⃗⃗𝑖 𝑎𝐾⃗⃗𝑖 . 𝑖 = ̅̅̅̅ 1,4 (33) The energy of the fermion field is equal to the sum of the energies of its quanta. The momentum of the fermion field 𝑃⃗⃗ is equal to: 𝑃⃗⃗ = ∫ 𝑑𝑉𝛹 + − 𝑖ħ∇ ⃗⃗ 𝛹 = ∑𝐾⃗⃗,𝐾⃗⃗′,𝑖,𝑖 ′ ∫ 𝑑𝑉 𝑎+ 𝐾⃗⃗𝑖 𝑎𝐾⃗⃗′𝑖 ′ 𝒱̃ ∗ (𝐾 ⃗⃗ , 𝑟⃗, 𝑖) − 𝑖ħ∇ ⃗⃗𝒱(𝐾 ⃗⃗ ′ , 𝑟⃗, 𝑖 ′ ) = ⃗⃗ ′ ∫ 𝑑𝑉 𝒱̃ ∗ (𝐾 = ∑𝐾⃗⃗,𝐾⃗⃗′ ,𝑖,𝑖 ′ 𝑎+ 𝐾⃗⃗𝑖 𝑎𝐾⃗⃗′𝑖 ′ ħ𝐾 ⃗⃗ , 𝑟⃗, 𝑖)𝒱(𝐾 ⃗⃗ ′ , 𝑟⃗, 𝑖 ′ ) = ⃗⃗ ′ 𝛿𝐾⃗⃗𝐾⃗⃗′ 𝛿𝑖𝑖 ′ = ∑𝐾⃗⃗𝑖 ħ𝐾 = ∑𝐾⃗⃗,𝐾⃗⃗′ ,𝑖,𝑖 ′ 𝑎+ 𝐾⃗⃗𝑖 𝑎𝐾⃗⃗′𝑖 ′ ħ𝐾 ⃗⃗ ′ 𝑎+ 𝐾⃗⃗𝑖 𝑎𝐾⃗⃗𝑖 ⃗⃗ ′ , 𝑟⃗, 𝑖 ′ ): here we have used the explicit form𝒱(𝐾 3 𝒱(𝐾 ⃗⃗ , 𝑖)𝑒 −𝑖𝐾⃗⃗′∙𝑟⃗ . ⃗⃗ ′ , 𝑟⃗, 𝑖 ′) = 𝐿−2 𝑈(𝐾 (34) 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 conju- gation of the Dirac equation: 214 𝜕 𝑚𝑐 𝜕𝛹 𝜕𝛹 𝑚𝑐 (𝑖𝛾 𝜇 𝜕𝑥 𝛼 − ħ ) 𝛹 = 0 = 𝑖𝛾 0 𝜕𝑐𝑡 + 𝑖𝛾 𝐾 𝜕𝑥 𝐾 − ħ 𝛹 Hermitian mating: 𝑖 𝜕𝛹 + 𝜕𝛹 + 𝑚𝑐 − 𝑐 𝜕𝑡 𝛾 0 − 𝑖 𝜕𝑥 𝐾 (−𝛾 𝐾 ) − ħ 𝛹 + = 0 (35) Multiplying (35) on the right by 𝛾 0 and using: 𝛾 𝐾 𝛾 0 = −𝛾 0 𝛾 𝐾 , we obtain: 𝜕𝛹 + 𝜕𝛹 + 𝑚𝑐 ̅ 𝜕𝛹 ̅ 𝜕𝛹 𝑚𝑐 ̅ =0= 𝑖 𝜕𝑥 0 𝛾 0 𝛾 0 + 𝑖 𝜕𝑥 𝐾 𝛾 0 𝛾 𝐾 − ħ 𝛹 + 𝛾 0 = 0 = 𝑖 𝜕𝑥 0 𝛾 0 + 𝑖 𝜕𝑥 𝐾 𝛾 𝐾 + ħ 𝛹 ̅ 𝜕𝛹 𝑚𝑐 ̅=0 = 𝑖 𝜕𝑥 𝜇 𝛾 𝜇 + ħ 𝛹 (36) Here 𝛹 ̅ ≡ 𝛹 +𝛾 0. Replacing the ordinary derivative with the covariant derivative in equation (28), we obtain a fermion interacting with the electromagnetic field: 𝜕 𝑚𝑐 [𝑖𝛾 𝛼 ( − 𝑖𝑒𝒜𝛼 ) − ħ ] 𝛹 = 0 (37) 𝜕𝑥 𝛼 Taking the density of the Lagrange function (Bethe G., 1964) in the form: 𝜕 𝑚𝑐 𝜕 𝑚𝑐 ̅ [𝑖𝛾 𝛼 ( 𝛼 − 𝑖𝑒𝒜𝛼 ) − ] 𝛹 = 𝛹 ℒ=𝛹 ̅ [𝑖𝛾 𝛼 𝛼 − ] 𝛹 + 𝑖 − 𝑖𝑒𝛹 ̅ 𝛾 𝛼 𝛹𝒜𝛼 = 𝜕𝑥 ħ 𝜕𝑥 ħ ̅ 𝛾𝛼𝛹 = 𝐿0 + 𝑒𝛹 (38) ̅ 𝛾 𝛼 𝛹𝒜𝛼 interaction po- 𝐿0 - is the standard Lagrangian of the Dirac equation and 𝑒𝛹 ̅ 𝛼 tential with an electromagnetic field: 𝑈 = −𝑒𝛹 𝛾 𝛹𝒜𝛼 . The Hamiltonian of the interaction of a fermion with an electromagnetic field is: ̅ 𝛾 𝜇 𝛹𝒜𝜇 ] = 𝐻1 = ∫ 𝑑𝑉[−𝑒𝛹 = ∑𝑃⃗⃗,𝐾⃗⃗,𝑛𝑖 (𝑔𝑃⃗⃗𝐾⃗⃗𝑖 𝑎𝑃+⃗⃗−ħ𝐾⃗⃗,𝑖 𝑎𝑃⃗⃗𝑖 𝑏𝑛𝐾 + ∗ + ⃗⃗ + 𝑔 𝑃⃗⃗ 𝐾𝑖 𝑎𝑃⃗⃗ 𝑖 𝑎𝑃⃗⃗ −ħ𝐾 ⃗⃗,𝑖 𝑏𝑛𝐾 ⃗⃗ ) (39) 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: 𝑗 𝑗 𝑗 𝛺 = 𝑑𝜉1𝑖 ∧ 𝑑ɳ1 [𝑎𝑖 , 𝑎𝑗 ] + 𝑑𝜉2𝑖 ∧ 𝑑ɳ2 [𝑎𝑖+ , 𝑎𝑗+ ] + 𝑑𝜉3𝑖 ∧ 𝑑ɳ3 ([𝑎𝑖 , 𝑎𝑗+ ] − 𝛿𝑖𝑗 ) + + + + 𝑗 𝑗 𝑗 +𝑑𝜁1𝑖 ∧ 𝑑ɳ𝛳1 [𝑏𝑖 , 𝑏𝑗 ]− + 𝑑𝜁2𝑖 ∧ 𝑑𝛳2 [𝑏𝑖+ , 𝑏𝑗+ ] + 𝑑𝜁3𝑖 ∧ 𝑑𝛳3 ([𝑏𝑖 , 𝑏𝑗+ ] − 𝛿𝑖𝑗 ) + − − ⃗⃗ ћ𝜔𝐾⃗⃗ (𝑏+ 𝐾⃗⃗𝛼 𝑏𝐾⃗⃗𝛼 + 1) + +𝑑𝜇 ∧ {𝑖ћ𝑑∣ 𝑡1⟩ − (∑𝐾⃗⃗,𝑖 𝐸𝐾⃗⃗𝑖 𝑎+ 𝐾⃗⃗𝑖 𝑎𝐾⃗⃗𝑖 + ∑𝛼 ∫ 𝑑𝐾 2 + ∑𝑃⃗⃗,𝐾⃗⃗,𝛼,𝑖 𝑔𝑃⃗⃗𝐾⃗⃗𝑖 𝑎𝑃+⃗⃗−ħ𝐾⃗⃗,𝑖 𝑎𝑃⃗⃗𝑖 𝑏𝐾+⃗⃗𝛼 + 𝑔∗ 𝑃⃗⃗𝐾𝑖 𝑎𝑃+⃗⃗𝑖 𝑎𝑃⃗⃗−ħ𝐾⃗⃗,𝑖 𝑏𝐾⃗⃗𝛼 ) × ∣ 𝑡1⟩𝑑𝑡} (40) The equations of E. Cartan for 2 forms (40) have the form: 𝜕2𝛺 𝜕2𝛺 𝜕2𝛺 𝜕2𝛺 𝜕2𝛺 𝜕2𝛺 0 = 𝜕𝑑𝜉 𝑖 𝜕𝑑ɳ1𝑗 = 𝜕𝑑𝜉 𝑖 𝜕𝑑ɳ2𝑗 = 𝑗 = 𝑗 = 𝑗 = 𝑗 = 1 2 𝜕𝑑𝜉3𝑖 𝜕𝑑ɳ3 𝜕𝑑𝜉3𝑖 𝜕𝑑ɳ3 𝜕𝑑𝜁1𝑖 𝜕𝑑𝛳1 𝜕𝑑𝜁2𝑖 𝜕𝑑𝛳2 215 𝜕2𝛺 𝜕𝛺 = 𝑗 = = [𝑎𝑖 , 𝑎𝑗 ] = [𝑎𝑖+ , 𝑎𝑗+ ] = [𝑎𝑖 , 𝑎𝑗+ ] − 𝛿𝑖𝑗 = [𝑏𝑖 , 𝑏𝑗 ] = 𝜕𝑑𝜁3𝑖 𝜕𝑑𝛳3 𝜕𝑑𝜇 + + + − = [𝑏𝑖+ , 𝑏𝑗+ ] = [𝑏𝑖 , 𝑏𝑗+ ] − 𝛿𝑖𝑗 = 𝑖ћ𝑑∣ 𝑡1⟩ − (∑𝐾⃗⃗,𝑖 𝐸𝐾⃗⃗𝑖 𝑎+ 𝐾⃗⃗𝑖 𝑎𝐾⃗⃗𝑖 + − − ⃗⃗ ћ𝜔𝐾⃗⃗ (𝑏+ 𝐾⃗⃗𝛼 𝑏𝐾⃗⃗𝛼 + 1) + ∑𝑃⃗⃗,𝐾⃗⃗,𝛼,𝑖 𝑔𝑃⃗⃗𝐾⃗⃗𝑖 𝑎+⃗⃗ ⃗⃗ 𝑎𝑃⃗⃗𝑖 𝑏+⃗⃗ + + ∑𝛼 ∫ 𝑑𝐾 2 𝑃 −ħ𝐾,𝑖 𝐾𝛼 +𝑔∗ 𝑃⃗⃗𝐾𝑖 𝑎𝑃+⃗⃗𝑖 𝑎𝑃⃗⃗−ħ𝐾⃗⃗,𝑖 𝑏𝐾⃗⃗𝛼 ) ∣ 𝑡1⟩𝑑𝑡 (41) All equations (41), except [25, 26] the last, are permutation relations between the operators, indicating that the operators 𝑎𝑖 , 𝑎𝑗+ - are fermionic, and 𝑏𝑖 , 𝑏𝑗+ - are bosonic operators. The last equation is the Schrödinger equation for interacting fields: fermionic with electromagnetic (bosonic) fields. We rewrite it in the form: 𝑑∣𝑡1⟩ 𝑖ћ 𝑑𝑡 = (𝐻0 + 𝐻1 )∣ 𝑡1⟩, (42) where: 1 ⃗⃗ ћ𝜔𝐾⃗⃗ (𝑏+ 𝐾⃗⃗𝛼 𝑏𝐾⃗⃗𝛼 + ) 𝐻0 = ∑𝐾⃗⃗,𝑖 𝐸𝐾⃗⃗𝑖 𝑎+ 𝐾⃗⃗𝑖 𝑎𝐾⃗⃗𝑖 + ∑𝛼 ∫ 𝑑𝐾 2 𝐻1 = ∑𝑃⃗⃗,𝐾⃗⃗,𝛼,𝑖 𝑔𝑃⃗⃗𝐾⃗⃗𝑖 𝑎𝑃+⃗⃗−ħ𝐾⃗⃗,𝑖 𝑎𝑃⃗⃗𝑖 𝑏𝐾+⃗⃗𝛼 +𝑔∗ 𝑃⃗⃗𝐾𝑖 𝑎𝑃+⃗⃗𝑖 𝑎𝑃⃗⃗−ħ𝐾⃗⃗,𝑖 𝑏𝐾⃗⃗𝛼 . 𝑖𝐻0 𝑡 Representing ∣ 𝑡1⟩ = 𝑒 − ћ ∣ 𝑡⟩, we obtain: 𝑖𝐻0 𝑡 𝑖𝐻0 𝑡 𝑖𝐻0 𝑡 𝑖𝐻0 𝑡 𝑖𝐻 𝑑∣𝑡⟩ 𝑖ћ − ћ0 𝑒 − ћ ∣ 𝑡⟩ + 𝑒 − ћ 𝑖ћ 𝑑𝑡 = 𝐻0 𝑒 − ћ ∣ 𝑡⟩ + 𝐻1 𝑒 − ћ ∣ 𝑡⟩ 𝑖𝐻0 𝑡 𝑖𝐻0 𝑡 𝑑∣𝑡⟩ Or 𝑖ћ 𝑑𝑡 = 𝑒 ћ 𝐻1 𝑒 − ћ ∣ 𝑡⟩ (43) 𝑖𝐻0 𝑡 𝑖𝐻0 𝑡 𝑖 (𝐸⃗𝑃⃗⃗−ħ𝐾 ⃗⃗⃗,𝑖−𝐸⃗𝑃 ⃗⃗,𝑖 +ћ𝜔⃗𝐾 ⃗⃗ )𝑡 + 𝑒 ћ 𝐻1 𝑒 − ћ = ∑𝑃⃗⃗,𝐾⃗⃗,𝛼,𝑖 𝑔𝑃⃗⃗𝐾⃗⃗𝑖 𝑒 ћ 𝑎𝑃⃗⃗−ħ𝐾⃗⃗,𝑖 𝑎𝑃⃗⃗𝑖 𝑏𝐾+⃗⃗𝛼 + 𝑖 (𝐸𝑃 ⃗⃗⃗,𝑖−𝐸𝑃 ⃗⃗⃗⃗,𝑖−ћ𝜔𝐾 ⃗⃗⃗⃗ )𝑡 + +𝑔∗ 𝑃⃗⃗𝐾𝑖 𝑒 ћ ⃗⃗⃗−ħ𝐾 𝑎𝑃⃗⃗𝑖 𝑎𝑃⃗⃗−ħ𝐾⃗⃗,𝑖 𝑏𝐾⃗⃗𝛼 (44) We represent the differential equation: 𝑑∣𝑡⟩ 𝑖ћ 𝑑𝑡 = 𝐻1 ∣ 𝑡⟩, (45) in the form of an integral equation: 1 𝑡 ∣ 𝑡 ⟩ = ∣ 0⟩ + ∫ 𝐻 (𝛳)𝑑𝛳 ∣ 𝛳⟩ (46) 𝑖ћ 0 1 In the first order of perturbation theory, its solution has the form: 1 𝑡 ∣ 𝑡⟩ ≈ ∣ 0⟩ + 𝑖ћ ∫0 𝐻1 (𝛳)𝑑𝛳 ∣ 0⟩ (47) 216 In the first order of the perturbation theory, we calculate the probability of sponta- neous emission of a photon by an electron. To do this, take: ∣ 0⟩ = 𝑎𝑃+⃗⃗0𝑖 ∣ 𝑣𝑎𝑐𝑢𝑢𝑚⟩. Then: ∣ 𝑡⟩ ≈ 𝑎𝑃+⃗⃗0𝑖 ∣ 𝑣𝑎𝑐𝑢𝑢𝑚⟩ + 𝑖 1 𝑡 (𝐸𝑃 ⃗⃗⃗⃗,𝑖−𝐸𝑃 ⃗⃗⃗,𝑖+ћ𝜔𝐾 ⃗⃗⃗⃗ )𝛳 + + 𝑖ћ ∫0 𝑑𝛳 ∑𝑃⃗⃗,𝐾⃗⃗,𝛼,𝑖 𝑔𝑃⃗⃗𝐾⃗⃗𝑖𝛼 𝑒 ћ ⃗⃗⃗−ħ𝐾 𝑎𝑃⃗⃗−ħ𝐾⃗⃗,𝑖 𝑎𝑃⃗⃗𝑖 𝑏𝐾+⃗⃗𝛼 𝑎𝑃+⃗⃗0𝑖0 ∣ 𝑣𝑎𝑐𝑢𝑢𝑚⟩ = = 𝑎𝑃+⃗⃗0𝑖0 ∣ 𝑣𝑎𝑐𝑢𝑢𝑚⟩ − 𝑖 (𝐸⃗⃗⃗ ⃗⃗⃗⃗ −𝐸⃗⃗⃗ +ћ𝜔⃗⃗⃗⃗ )𝑡 𝑔𝑃 ћ 𝑃−ħ𝐾,𝑖 𝑃,𝑖 𝐾 −1 ⃗⃗⃗𝑖𝛼𝑒 ⃗⃗⃗𝐾 − ∑𝑃⃗⃗,𝐾⃗⃗,𝛼,𝑖 𝑎𝑃+⃗⃗−ħ𝐾⃗⃗,𝑖 𝑏𝐾+⃗⃗𝛼 𝛿𝑃⃗⃗𝑃⃗⃗0 𝛿𝑖𝑖0 ∣ 𝑣𝑎𝑐𝑢𝑢𝑚⟩ = 𝐸𝑃 ⃗⃗⃗⃗,𝑖 −𝐸𝑃 ⃗⃗⃗−𝐾 ⃗⃗⃗,𝑖+ћ𝜔𝐾 ⃗⃗⃗⃗ 𝑖 (𝐸⃗⃗⃗ ⃗⃗⃗⃗,𝑖−𝐸⃗⃗⃗ +ћ𝜔⃗⃗⃗⃗ )𝑡 ћ 𝑃0 −ħ𝐾 𝑃0 ,𝑖 𝐾 −1 𝑔⃗𝑃⃗⃗ 𝐾 ⃗⃗⃗⃗𝑖𝑒 = 𝑎𝑃+⃗⃗0𝑖 ∣ 𝑣𝑎𝑐𝑢𝑢𝑚⟩ − ∑𝐾⃗⃗,𝛼 0 𝐸 𝑎𝑃+⃗⃗0−ħ𝐾⃗⃗,𝑖0 𝑏𝐾+⃗⃗𝛼 ∣ 𝑣𝑎𝑐𝑢𝑢𝑚⟩ (48) ⃗⃗⃗,𝑖0 −𝐸𝑃 ⃗⃗⃗0 −ћ𝐾 𝑃 ⃗⃗⃗0 ,𝑖 +ћ𝜔𝐾 ⃗⃗⃗⃗ ⃗⃗ is equal Thus, the probability for an electron to emit a photon with momentum ħ𝐾 to: 2 2 (𝐸⃗⃗⃗ ⃗⃗⃗,𝑖0 −𝐸𝑃 ⃗⃗⃗0 ,𝑖+ћ𝜔𝐾 ⃗⃗⃗ ) 𝑃0 −ћ𝐾 |𝑔⃗⃗⃗ ⃗⃗⃗⃗,𝑖 𝑃 ,ћ𝐾 | |sin2 𝑡| 0 0 2 𝐸⃗⃗⃗ 2 (49) ⃗⃗⃗,𝑖0 −𝐸⃗𝑃 𝑃0 −ћ𝐾 ⃗⃗0 ,𝑖+ћ𝜔⃗𝐾 ⃗⃗⃗ | | 2 At t≫1, this value is equal to: 2 |𝑔𝑃⃗⃗0,ћ𝐾⃗⃗,𝑖0 | 𝜋 2 𝛿 2 (𝐸𝑃⃗⃗0−ħ𝐾⃗⃗,𝑖 − 𝐸𝑃⃗⃗0,𝑖 + ћ𝜔𝐾⃗⃗ ) = 2 1 𝑇 = |𝑔𝑃⃗⃗0,ћ𝐾⃗⃗,𝑖0 | 𝜋 2 𝛿(𝐸𝑃⃗⃗0−ħ𝐾⃗⃗,𝑖 − 𝐸𝑃⃗⃗0,𝑖 + ћ𝜔𝐾⃗⃗ ) ∫ 𝑒 𝑖∆𝐾 𝑑𝐾 = 2𝜋 −𝑇 2 = |𝑔𝑃⃗⃗0,ћ𝐾⃗⃗,𝑖0 | 𝜋𝛿(𝐸𝑃⃗⃗0−ħ𝐾⃗⃗,𝑖 − 𝐸𝑃⃗⃗0,𝑖 + ћ𝜔𝐾⃗⃗ )𝑇 (50) The probability of radiation per unit time W is equal to: 2 𝑊 = |𝑔𝑃⃗⃗0,ћ𝐾⃗⃗,𝑖0 | 𝜋𝛿(𝐸𝑃⃗⃗0−ħ𝐾⃗⃗,𝑖 − 𝐸𝑃⃗⃗0,𝑖 + ћ𝜔𝐾⃗⃗ ) 3 Conclusions We demonstrated the possibility of using the mechanics of E. 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