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

Journal Article · Tue Jan 01 00:00:00 EST 1980 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.524315· OSTI ID:5727575

Gessner, W; Ernst, V

The indefinite metric space O/sub M/ of the covariant form of the quantized Maxwell field M is analyzed in some detail. S/sub M/ contains not only the pre-Hilbert space X/sup 0/ 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 L/sup q/ disjunct up to the zero element o of S/sub M/. The L/sup q/ are the maximal subspaces of S/sub M/ which allow the usual statistical interpretation. Each L/sup q/ corresponds uniquely to one square integrable, spatial distribution j/sup o/(x) of the total charge Q=0. If M is in any state from L/sup q/, the bare charge j/sup 0/(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 L/sup 0/ of the free Maxwell field. Each L/sup q/ contains a secondary rosette of continuously many, up to o disjunct, isomorphic Hilbert spaces H/sub g//sup q/ related to different electromagnetic gauges. The space H/sub o//sup q/, 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 H/sub g//sup q/, the bare 4-current j/sup 0/(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.

• Research Organization: Sektion Physik der Universitaet Muenchen, Theresienstrasse 37, D 8000 Muenchen 2, Germany
• OSTI ID: 5727575
• Journal Information: J. Math. Phys. (N.Y.); (United States), Vol. 21:1
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
MAXWELL EQUATIONS
METRICS
QUANTUM FIELD THEORY
COMMUTATION RELATIONS
COULOMB FIELD
ELECTROMAGNETIC FIELDS
GAUGE INVARIANCE
HILBERT SPACE
BANACH SPACE
DIFFERENTIAL EQUATIONS
ELECTRIC FIELDS
EQUATIONS
FIELD THEORIES
INVARIANCE PRINCIPLES
MATHEMATICAL SPACE
SPACE
645400* - High Energy Physics- Field Theory