Non-canonical quantum optics (II): Poincar\'e covariant formalism and   thermodynamic limit
  The paper contains further development of the idea of field quantization introduced in M. Czachor, J. Phys. A: Math. Gen. {\bf 33} (2000) 8081-8103. The formalism is extended to the relativistic domain. The link to the standard theory is obtained via a thermodynamic limit. Unitary representations of the Poincar\'e group at the level of fields and states are explicitly given. Non-canonical multi-photon and coherent states are introduced. In the thermodynamic limit the statistics of photons in a coherent state is Poissonian. The $S$ matrix of radiation fields produced by a classical current is given by a non-canonical coherent-state displacement operator, a fact automatically eliminating the infrared catastrophe. Field operators are shown to be operators and not operator-valued distributions, and can be multiplied at the same point in configuration space. An exactly solvable example is used to compare predictions of the standard theory with those of non-canonical quantum optics, and explicitly shows the mechanism of automatic ultraviolet regularization occuring in the non-canonical theory. Similar conclusions are obtained in perturbation theory, where one finds the standard Feynman diagrams, but the Feynman rules are modified. A comparison with the Dicke-Hepp-Lieb model allows to identify the physical structure behind the non-canonical algebra as corresponding to an ensemble of indefinite-frequency oscillators with constant density $N/V$. 
Marek Czachor and Monika Syty