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

“primary” speed of light, included in the free Hamiltonian. Taking into account this consideration, we split the complete Hamiltonian (17) into the central part and the perturbation differently, namely S C H H H = + , (18) Where we choose the self-consistent (or approximating) Hamiltonian in the form similar to the free Hamiltonian (17), but with the speed of light c  renormalized due to the photon-photon interaction: 0 , S k kj kj k j H a a E ω + = + ∑  , (19) Where k ck ω =   The correlation Hamiltonian describing the interaction of renormalized or “dressed” photons is chosen so that the total Hamiltonian remains unchanged: ( ) 0 , C k k kj kj k I k j k H a a E H ω ω ω + = − + − + ∑ ∑    . (20) This Hamiltonian describes the interaction of photons propagating at a renormalized speed of light, which we will not consider. Formulas (19), (20) include a term that does not contain operators, taking into account which is necessary for the correct formulation of the self-consistent field model. We chose it because the approximating Hamiltonian (19) is as close as possible to the exact Hamiltonian. It means that it is necessary to require that the value S C I H H H ≡ − = be minimal, i.e., equal to zero. From here, we obtain the conditions natural for the theory of a self-consistent field: , 0 S C H H H = = (21) Averaging is performed using the statistical operator ( ) exp S F H ρ β = − , (22) Where F is the free energy and is the reciprocal temperature value 1 T β = . Condition (21) allows us to determine the non-operator part of Hamiltonian (19): ( ) 0 2 k I k k E c c k f ck H = − + + ∑ ∑    (23) where the distribution function of renormalized photons has the Planck form ( ) 1 exp 1 k kj kj k f a a β ω + = = −  (24) and does not depend on the polarization index. From the normalization condition of the statistical operator (22) Sp =1 ρ follows the expression for the free energy of radiation ( ) ( ) 2 2 ln 1 k k I k k k F c c k f ck H T e β ω − = − + + + − ∑ ∑ ∑     . (25) If we neglect the photon-photon interaction and zero-point fluctuations, formula (25), follows the usual formulas of the thermodynamics of black radiation [22]. It is natural to require that in the approximation used with Hamiltonian (19) and free energy (25), as in the case of a gas of non-interacting photons, thermodynamic relations must be satisfied. Since the introduced renormalized speed itself, in principle, can depend on thermodynamic variables to satisfy the thermodynamic relations, the following condition must be met: 0 F c ∂ = ∂  . (26) From this condition and formula (25) follows the relationship that determines the renormalized speed: 2 I k k H c c c k f c ∂ ∂ − = ∂ ∂ ∑     (27) Since (27) includes the temperature-dependent distribution function (24), then, naturally, the speed of light is a function of temperature. Thus, the average of the interaction Hamiltonian should be calculated. In this case, as in the theory of phonons in solids [19, 20], divergent integrals appear. When describing phonons within the continuum model, it is natural to cut off such integrals at a wave number equal to the inverse of the average distance between particles or, when integrating over frequency, at the Debye frequency. In the case of photons, the divergent integrals will be cut off at the wave number, the choice of which will be discussed a little later. Taking this into account, calculating the average of the interaction Hamiltonian (13) gives 4 2 2 2 1312 15 4 m I V k H D c J J π   = ⋅ ⋅ +      (28) where, ( ) 4 6 4 T J c ζ   =      , ( ) 4 4 90 1, 0823 ζ π = ≈ is the zeta function. Let be the ratio of the temperature- dependent speed of light to the “base” speed of light. Considering that 2 , 2 k k V k f J π = ∑   from (27), we find the equation for: © 2023 Global Journals 1 Year 2023 38 Frontier Research Volume XXIII Issue ersion I VXI ( A ) Science Global Journal of Fine Structure Constants Across Cosmic Realms: Exploring