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Journal of Mathematical Physics / Volume 21 / Issue 1

J. Math. Phys. 21, 93 (1980); http://dx.doi.org/10.1063/1.524315 (19 pages)

The rosette of rosettes of Hilbert spaces in the indefinite metric state space of the quantized Maxwell field

W. Gessner and V. Ernst

Sektion Physik der Universität München, Theresienstrasse 37, D 8000 München 2, Germany

The indefinite metric space OM of the covariant form of the quantized Maxwell field M is analyzed in some detail. SM contains not only the pre‐Hilbert space X0 of states of transverse photons which occurs in the Gupta–Bleuler formalism of the free M, but a whole rosette of continuously many, isomorphic, complete, pre‐Hilbert spaces Lq disjunct up to the zero element o of SM. The Lq are the maximal subspaces of SM which allow the usual statistical interpretation. Each Lq corresponds uniquely to one square integrable, spatial distribution jo(x) of the total charge Q=0. If M is in any state from Lq, the bare charge j0(x) appears to be inseparably dressed by the quantum equivalent of its proper, classical Coulomb field E(x). The vacuum occurs only in the state space L0 of the free Maxwell field. Each Lq contains a secondary rosette of continuously many, up to o disjunct, isomorphic Hilbert spaces Hgq related to different electromagnetic gauges. The space Hoq, which corresponds to the Coulomb gauge within the Lorentz gauge, plays a physically distinguished role in that only it leads to the usual concept of energy. If M is in any state from Hgq, the bare 4‐current j0(x), j(x), where j(x) is any square integrable, transverse current density in space, is endowed with its proper 4‐potential which depends on the chosen gauge, and with its proper, gauge independent, Coulomb–Oersted field E(x), B(x). However, these fields exist only in the sense of quantum mechanical expectation values equipped with the corresponding field fluctuations. So they are basically different from classical electromagnetic fields.

KEYWORDS and PACS

Keywords

HILBERT SPACE, METRICS, ELECTROMAGNETIC FIELDS, MAXWELL EQUATIONS, COULOMB FIELD, COMMUTATION RELATIONS, GAUGE INVARIANCE, QUANTUM FIELD THEORY

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.524315

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

For access to fully linked references, you need to log in.
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    J. M. Cook, J. Math. Phys. 2, 33 (1961)JMAPAQ000002000001000033000001.

    R. J. Glauber, Phys. Rev. 130, 2529 (1963)
    131, 2766 (1963).



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