Global Journal of Science Frontier Research, A: Physics and Space Science, Volume 23 Issue 11

determines the energy of a non-interacting electromagnetic field before critical values electric strengths Schwinger’s field Es = 1.32 × 10¹ ⁶ [V × cm ˉ ¹] and magnetic field strength H = 10¹ ⁶ [Gs]. The second term, w ᵉͬ ( w ₁ ), describes the interaction of photons due to the production of electron-positron pairs [10]: ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 3 7 I w D D     = − − + + +     E E H H E H H E EH HE (6) The constant D can be calculated by the methods of quantum electrodynamics [10] and in Gaussian units ≡ ћ ³ m⁴c⁵ (7) Where η is the dimensionless coefficient η ≡ α ² 45×(4 π ) ² ≈ 7.5 × 10ˉ⁹ (8) α is the fine structure constant (1), m is the mass of the electron, c is the speed of light. It is convenient D to write the coefficient through the Compton wavelength of the electron mc =   in the form [10]: 3 2 D mc η =  (9) It is of interest to estimate the value of the energy 0 I w w ratio. This quantity is equal to the ratio of the energy of the field contained in the volume 3  to the rest energy of the electron. In addition, this ratio must also be multiplied by a small dimensionless coefficient η . For a magnetic field strength of the order of 6 10 H Gs  , we obtain 20 0 10 I w w −  , so the contribution of the interaction to the total field energy is indeed minimal. Let us move on to the description of the electromagnetic field in terms of the Fourier components of fields, using the expansion of fields in plane waves, ( ) ( ) ( ) ( ) , , , i i k k k k t t e t t e = = ∑ ∑ kr kr E r E H r H (10) Then the total Hamiltonian of the field in the volume, following (2), is the sum of the free Hamiltonian and the interaction Hamiltonian 0 I H H H = + , (11) ( ) 0 8 k k k k k V H π + + = + ∑ E E H H , (12) ( ) ( ) ( ) ( ) { { } ( ) ( ) ( ) ( ) } ( ) ( ) ( ) ( ) ( ) { } ( ) { } 1 2 3 4 1 2 3 4 1 2 3 4 4 3 2 1 1 2 3 4 4 3 2 1 1 2 3 4 1 2 3 4 2 3 7 . i i I k k k k k k k k k k k k k k k k k k k k k k k k k k H VD VD + + + + + + + + + + + + = − − − − ∆ − + − + + + ∆ − + − ∑ ∑ E E E E H H H H E E H H H H E E k k k k E H E H H E H E k k k k (13) Here ( ) 1 ∆ = k , if 0 = k and ( ) 0 ∆ = k if 0 ≠ k . In (12) and (13), we can move on to the operators of the creation kj a + and destruction kj a of photons, using the representations of the operators of the Fourier components of fields: ( ) ( ) ( ) ( ) 2 , 2 , k k kj kj j j k kj kj j j k i a a V ic a a V π ω π ω + − + − = − −   = + ×  ∑ ∑ E e k H k e k   (14) Where, k ck ω = , and for the polarization vectors ( ) j e k , the orthonormality and completeness conditions are valid: ( ) ( ) ( ) ( ) 1 2 1 2 2 , j j j j j j j k k e e k α α α α αα δ δ ′ ∗ ∗ ′ ′ = = − ∑ e k e k k k , (15) as well as conditions ( ) ( ) ( ) 0, j j ∗ = − = ke k e k e k (16) The free Hamiltonian of the field (12) is reduced to the sum of the Hamiltonians of harmonic oscillators, 0 1 2 k kj kj kj H a a ω +   = +     ∑  (17) The electromagnetic field, taking into account nonlinear effects, is characterized by the complete Hamiltonian (11). To take into account the interaction in a many-particle system, as a rule, the Hamiltonian of non-interacting particles is chosen as the leading approximation, in our case it is (17), and the interaction Hamiltonian (13) is considered as a perturbation. This choice, as noted above, is not the best, since in the leading approximation the effects caused by the interaction are entirely ignored, which in the case under consideration, although small, can, as we will see, lead to qualitatively new effects. From the self-consistent approach to the description of many-particle systems it is known that taking into account interaction effects in the central approximation leads to a change in the dispersion law of the original particles and, thus, we move from the representation of free particles to the language of collective excitations - quasiparticles. It is natural to assume that in the case considered here, interaction effects will lead to a renormalization of the 1 Year 2023 37 Frontier Research Volume XXIII Issue ersion I VXI ( A ) Science © 2023 Global Journals Global Journal of Fine Structure Constants Across Cosmic Realms: Exploring