The analysis of the behavioral trend of particle using Klein-Gordon and Dirac equation that was minimally coupled to electromagnetic wave four-vector potential has been carried out. In the analysis, it was clearly observed that each of them has non-relativistic limit at one stage or the other and based on this limitation, there is a great challenge posed on the idea of single particle interpretation since in each case there is a particle and an antiparticle. It therefore reveals the fact that there is no conceptually real existence of single particle in isolation when it comes to relativistic quantum mechanics for any of the equations being used to study the interaction of particle with electromagnetic field.
We consider the stochastic limit of the standard non relativistic QED (but our results also hold for the polaron interaction of a particle with a Boson field).Extending the Fock case results of
[AcLu92] , we take the initial state of the field to be a Gibbs state at a given temperature.We show that a new algebra, with commutation relations depending on the temperature and acting on a Hilbert module, emerges.This algebra, that we call the Hot Ree Algebra, generalizes the QED Hilbert module algebra in the sense of
[AcLuv097C] and [Ske96] and therefore also the Ree (or Boltzmannian) algebra.It is interesting to notice that, when the module structure is neglected, the algebra we find is precisely the algebra that was found in the singleton independence central limit theorem of
[Ah098a], [Ah098b] .So the present result also gives a natural physical interpretation for that algebra.