Automatic Regularization by Quantization in Reducible Representations of CCR: Point-Form Quantum Optics with Classical Sources |
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Authors: | Marek Czachor and Klaudia Wrzask |
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Institution: | (1) Katedra Fizyki Teoretycznej i Informatyki Kwantowej, Politechnika Gdańska, 80-952 Gdańsk, Poland;(2) Centrum Leo Apostel (CLEA), Vrije Universiteit Brussel, 1050 Brussels, Belgium |
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Abstract: | Electromagnetic fields are quantized in a manifestly covariant way by means of a class of reducible “center-of-mass N-representations” of the algebra of canonical commutation relations (CCR). The four-potential A
a
(x) transforms in these representations as a Hermitian four-vector field in Minkowski four-position space (without change of
gauge), but in momentum space it splits into spin-1 massless photons and two massless scalars. What we call quantum optics
is the spin-1 sector of the theory. The scalar fields have physical status similar to that of dark matter (spin-1 and spin-0
particle numbers are separately conserved). There are no negative-norm or zero-norm states. Unitary dynamics is given by the
point-form interaction picture, with minimal-coupling Hamiltonian constructed from fields that are free on the null-cone boundary
of the Milne universe. SL(2,C) transformations as well as the dynamics are represented unitarily in the Hilbert space corresponding
to N four-dimensional oscillators. Vacuum is a Bose-Einstein condensate of the N-oscillator gas and is given by any N-oscillator product state annihilated by all annihilation operators. The form of A
a
(x) is determined by an analogue of the twistor equation. The same equation guarantees that the set of vacuum states is Poincaré
invariant. The formalism is tested on quantum fields produced by pointlike classical sources. Photon statistics is well defined
even for pointlike charges, with ultraviolet and infrared regularizations occurring automatically as a consequence of the
formalism. The probabilities are not Poissonian but of a Rényi type with α=1−1/N; the Shannon limit N→∞ is an ultraviolet/infrared-regularized Poisson distribution. The average number of photons occurring in Bremsstrahlung
splits into two parts: The one due to acceleration, and the one that remains nonvanishing even for inertially moving charges.
Classical Maxwell electrodynamics is reconstructed from coherent-state averaged solutions of Heisenberg equations. We show
in particular that static pointlike charges polarize vacuum and produce effective charge densities and fields whose form is
sensitive to both the choice of representation of CCR and the corresponding vacuum state. |
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Keywords: | Field quantization Point-form dynamics Oscillator algebras Renyi entropies Infrared regularization |
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