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Annales De L Institut Henri Poincare-physique Theorique

On the classical limit and the infrared problem for non-relativistic fermions interacting with the electromagnetic field

F. Ruiz Ruiz, R. F. Álvarez-Estrada

Published Jan 1, 1984• 0 citations

Abstract

The classical limit and the infrared divergence problem for a non-relativistic charged quantum particle interacting with the quantized electromagnetic field are analized. Overall three momentum conservation is taken into account. A unitary transformation associated to the coherent state corresponding to a particle surrounded by a cloud of soft photons is performed upon the hamiltonian and the particle-field states. The transformed state represnting a moving dressed quantum particle and its energy are given by the Brillouin-Wigner perturbation theory. It is shown formally that the quantum energy approaches the classical electromagnetic field at the Plank constant h goes to zero. Moreover,all Feynman diagrams contributing to this quantum energy are infrared finite, without needing add diagrams of the same order in the electric charge to obtain the infrared finiteness. Those properties justify the usefulness of the unitary transformation. The Compton effect in the forward direction is studied using dressed charged particle states after the unitary transformation has been performed. The quantum cross section approaches the classical limit (Thomson´s formula) as h ͢͢͢͢͢ 0, and the Feynman diagrams are free of infrared divergences.

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The classical limit and the infrared divergence problem for a non-relativistic charged quantum particle interacting with the quantized electromagnetic field are analized. Overall three momentum conservation is taken into account. A unitary transformation associated to the coherent state corresponding to a particle surrounded by a cloud of soft photons is performed upon the hamiltonian and the particle-field states. The transformed state represnting a moving dressed quantum particle and its energy are given by the Brillouin-Wigner perturbation theory. It is shown formally that the quantum energy approaches the classical electromagnetic field at the Plank constant h goes to zero. Moreover,all Feynman diagrams contributing to this quantum energy are infrared finite, without needing add diagrams of the same order in the electric charge to obtain the infrared finiteness. Those properties justify the usefulness of the unitary transformation. The Compton effect in the forward direction is studied using dressed charged particle states after the unitary transformation has been performed. The quantum cross section approaches the classical limit (Thomson´s formula) as h ͢͢͢͢͢ 0, and the Feynman diagrams are free of infrared divergences.

The quantum theory of radiation, formulated in wave-mechanical language but in terms of running waves, is shown to lead under suitable conditions to the classical theory of emission by a particle as a first approximation. Non-relativistic mechanics is used, and the limitations of the method are brought out in terms of classical theory. The characteristic quantum phenomenon of the Compton effect requires that the Compton recoil shall exceed the spread in momentum of the particle, and so that the wave-packet shall greatly exceed in diameter the wave-length of the incident radiation; this condition is realizable on any scale of magnitude. For quasi-classical motion just the reverse is required; furthermore, if the motion is to last for a number of waveperiods, the wave-length must greatly exceed

\frac{h}{\mathrm{mc}}

(

m=\mathrm{mass}\mathrm{of}\mathrm{the}\mathrm{particle}

). The classical packet-motion arises from combinations between quantum-states of the particle-field system; this fact suggests that the photon is not a fundamental constituent of the field but only one possible aspect of its action. The infinite "zero-energy" associated apparently even with the lowest energy-state of the field implies no electromagnetic field intensities at any point. Whereas for problems on energy or momentum it is customary to represent an incident wave-train by a single typical quantum state, to obtain motional phenomena of classical type a more complete expression such as that given in the paper must be used. The physically empty field likewise corresponds to a mixture of quantum states but without correlation among their amplitudes.

Abstract Ehrenfest theorem is extended to the relativistic quantum theory of charged particles, moving under the influence of a classical electromagnetic field. In order to extend the original Ehrenfest result to the relativistic domain we bypassed the problems with the relativistic position operator by deriving directly Newton’s second law. Our approach is characterized by its universality. The detailed form of the wave equation is not needed. All that is required is the existence of the conserved electric four-current built from the particle wave function. The derivation is based on the conservation laws for the energy and momentum.

We give here a field-theoretical derivation of the Hamiltonian of the non-relativistic quantum electrodynamics in the Coulomb gauge using the Lagrange formalism. It leads to the same result as the usual derivation, where one just replaces the classical vector potential in the minimal coupling of the second quantized electron Hamiltonian by the quantized one and adds the photon energy. This approach illustrates however the proper use of the Euler-Lagrange equations and the canonical formalism that fail if one tries to build up the many-body quantum theory starting from the classical theory of point-like particles interacting with electromagnetic fields.

We call attention to the well-known fact that the electromagnetic (EM) field consists of bound and radiation components, and only their sum provides for the implementation of energy-momentum conservation for interacting classical charges.In this context, we then focus our attention on quantum systems of electrically bound charges, which do not radiate in the stationary energy states and thus their EM field comprises only the bound component.The non-applicability of Maxwell's equations to quantum mechanics does not permit, in general, ignoring the problem of recovering the energy-momentum conservation for such pure bound EM field systems, and we explore this problem within the Schrödinger-Dirac quantization scheme.

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 Poincaré 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

.