Quantum Theory and Statistical Thermodynamics by Peter Hertel

Author:Peter Hertel
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

5.5.3 Black-Body Radiation

The photon behaves as a boson. Although it has only two polarization states, right hand or left hand circular polarization, it is described by a vector-vector field with an auxiliary condition. For massive particles there is always a frame of reference where the particle is at rest. Galilei transformations which respect this are rotations. The unitary irreducible representations of this so-called little group SU are characterized by a spin quantum number which is either (bosons) or (fermions). The photon has no mass and there is no frame of reference where the particle is at rest. It has a different little group, namely E, the unitary irreducible representations of which are either one- or infinite-dimensional. The photon transforms according to a one-dimensional representation which is characterize by the projection of angular momentum onto the direction of motion, namely . Because space reflexion is a symmetry, there is an equivalent state with .

would also be possible, as for the anti-neutrino, or for the neutrino. However, these particles are not the same. Neutrinos and anti-neutrinos are fermions. The photon, however, which coincides with its anti-particle, is a boson with two polarization states.

The Photon Gas

Recall how we deduced the grand-canonical Gibbs state. Entropy should be maximal with the constraints that close-by operators must be states and that the average energy and particle number are prescribed. This is why we encounter three Lagrange multiplier, namely free energy, temperature and chemical potential. For massless particles, however, their average number cannot be prescribed. We need no corresponding Lagrange multiplier. This boils down to choosing .

We may apply our result for bosons. The one-particle energy is

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