The Photon Gas in the Quantized Friedmann Universe

Rainer Collier

doi:10.37532. /puljmap.2023 .6(3); 01-03

Rainer Collier^*

Institute of Theoretical Physics, Friedrich Schiller University of Jena, Germany, Email: rainer@dr-collier.de

^*Correspondence: Rainer Collier, Institute of Theoretical Physics, Friedrich Schiller University of Jena, Germany, Email: rainer@dr-collier.de

Received: 21-Jul-2023, Manuscript No. puljmap-23-6608; Editor assigned: 23-Jul-2023, Pre QC No. puljmap-23-6608(PQ); Accepted Date: Aug 10, 2023; Reviewed: 28-Jul-2023 QC No. puljmap-23-6608(Q); Revised: 01-Aug-2023, Manuscript No. puljmap-23-6608(R); Published: 15-Aug-2023, DOI: 10.37532. /puljmap.2023 .6(3); 01-03

Citation: Collier R. The Photon gas in the quantized Friedmann universe. J Mod Appl Phys. 2023, 6(3):1-3

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Abstract

A semi classical fundamental equation of quantum gravitation is applied to a flat Friedmann universe containing a photon gas. The mean energy of a photon is determined by the method of canonical partition and the classical one-particle state function for the ideal photon gas. For very high temperatures, there is a deviation from the classical equipartition theorem

Key Words

Semi classical quantum gravitation; Friedmann universe; Planck´s radiation law; statistical canonical partition

Introduction

If, on the smallest spatial and temporal scales, the universe has a grainy structure, Planck’s electromagnetic radiation law ought to have a modified structure, too. Radiation propagating over a spacetime range that has a lattice structure will not be able to fall below a bottom limit of the particle wavelength and, thus, will also have to maintain an upper limit of the particle energy.

For the theoretical implementation of this idea there are several approaches. The most difficult problem is encountered in supporting these investigations by a generalized special theory of relativity that would permit an invariant limit for a minimum limit unit of length or a maximum of particle energy. We refer to Ref and, especially for the thermodynamics of the photon gas, to [1-10]. To extricate ourselves us from these principal problems, let us first look for a first correction of Planck`s radiation formula as it would appear, e.g., in a quantized Friedmann universe

The hamiltonian operator

Let our investigations start from the Hamiltonian operator of matter in a flat Friedmann universe. In (Ref) C. Kiefer and T. P. Singh have developed a semi classical quantum gravitation from WheelerDeWitt‘s equation of quantum gravitation (ref) and applied this to the Friedmann universe [9-12]. For the motion of matter in this universe, a corrected Schrödinger equation of the following type is given:

equation

From this fundamental equation, therefore, we derive a non-linear semi classical Hamiltonian of the following structure for the motion of matter in the flat Friedmann universe.

equation

or also, with the scale factor a equation

equation

wherein is equation the Hamiltonian of the matter in the Minkowski space and

equation

are the Planck-length, Planck-energy and the Planck-mass

Note: In Dirac’s theory, an analogue procedure leads to a semi classical Hamiltonian that is similar to the above gravitative case [13]:

equation

The correction term in (2.6) has the same structure as that in (2.3), but with a negative sign.

Application to the photon gas

Let us now consider an ensemble of harmonic oscillators in the heating bath having a temperature T (Einstein‘s photon model), and determine the mean energy ε of a photon therein. Let us use for that the canonical partition with the one-particle partition sum Z and the Hamiltonian Hˆ in the flat Friedmann universe from (2.3)

equation

As the λ term is a sufficiently small correction term only, we can readily note

equation

The mean one-particle energy equation then, as we know, results from

equation

Z₀, however, is the one-particle partition sum of the Planck partition in the flat Minkowski space

equation

so that, from (3.5), (3.6) and (3.7), it follows that

equation

Our interest is focused on the correction of Planck´s radiation law at very high temperatures kT» ħω or βε« 1. For that case we obtain, with (3.7) and (3.8),

equation

With (3.1) and (3.2), therefore, we obtain

The occurrence of the parameter equation points to the influence of the Friedmann universe. We recognize that the number of classical degrees of freedom per particle deviates from 2, although this correction, due to the magnitude of the Planck energy is extremely little.

A value for α

Can we in (2.3) set a value forα? In Ref we established an approach to a modified Planck’s radiation law, which we developed in an imitation of Einstein’s well-known laser model [5, 14]. The modified Planck formula, which was based also on statistical considerations, reads as follows:

equation

For temperaturesT →∞ this converts to

equation

If, however, we remain at a temperature of we obtain from (4.2) for this case:

equation

If we now compare (4.3) with our result (3.13) for the Friedmann universe, we can readily conclude that α =1 and also, therefore, from (2.3), that equation This is well in agreement with the ground state of a quantized universe. A quantization of the Friedmann cosmos provides namely a discretization of the cosmic scale factor and the energy content of the cosmos,

how we can find in [1,15,16]. With n =1 we read off, for ground state length and ground state energy, respectively, the values equation 1 and such that α =1 seems to be a reasonable value. According to formula (4.3), however, corrections to the classical Rayleigh-Jeans behavior and, thus, to the classical equipartition theorem pro photon, will not appear before the temperatures come close to the Planck temperature Such temperatures will at best appear at a cosmologically short time after the big bang.

Conclusion

We applied the semiclassical fundamental equation of quantum gravitation, derived from the Wheeler-DeWitt equation by C. Kiefer, to the electromagnetic radiation in a flat Friedmann universe. We used the method of canonical partition and the state function for the ideal photon gas to determine the mean energy a photon. For very high temperatures (but still kT« E*), there is a deviation from the classical behavior, which now took the form equation being influenced by the quantity of the Planck energy