Does there exist a sensible quantum theory of an ''algebra-valued'' scalar field

Journal Article · Sat Apr 15 00:00:00 EDT 1989 · Phys. Rev. D; (United States)

DOI:https://doi.org/10.1103/PhysRevD.39.2297· OSTI ID:6188279

Anco, S C; Wald, R M

Consider a scalar field phi in Minkowski spacetime, but let phi be valued in an associative, commutative algebra openA rather than openR. One may view the resulting theory as describing a collection of coupled real scalar fields. At the classical level, theories of this type are completely well behaved and have a global symmetry group which is a nontrivial enlargement of the Poincare group. (They are analogs of the new class of gauge theories for massless spin-2 fields found recently by one of us, whose gauge group is a nontrivial enlargement of the usual diffeomorphism group.) We investigate the quantization of such scalar field theories here by studying the case of a lambdaphi/sup 4/ field, with phi valued in the two-dimensional algebra generated by an identity element e and a nilpotent element v satisfying v/sup 2/ = 0. The Coleman-Mandula theorem, which states that the symmetry group of a nontrivial quantum field theory cannot be a nontrivial enlargement of the Poincare group, is evaded here because the finite ''extra'' symmetries of the classical theory fail to be implemented in the quantum theory by unitary operators and the infinitesimal symmetries (which can be represented in the quantum theory by quadratic forms) connect the one-particle Hilbert space to multiparticle states. Nevertheless, we find that the conventional Feynman rules for this theory lead to vacuum decay at the tree level and fail to yield a well-defined S matrix. Some alternative approaches are investigated, but these also appear to fail.

• Research Organization: Enrico Fermi Institute and Deparment of Physics, University of Chicago, Chicago, Illinois 60637
• OSTI ID: 6188279
• Journal Information: Phys. Rev. D; (United States), Vol. 39:8
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
SCALAR FIELDS
SPACE-TIME
CLASSICAL MECHANICS
GAUGE INVARIANCE
HILBERT SPACE
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MINKOWSKI SPACE
POINCARE GROUPS
QUANTUM FIELD THEORY
S MATRIX
SCATTERING
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ANGULAR MOMENTUM
BANACH SPACE
DIFFERENTIAL EQUATIONS
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FIELD THEORIES
INVARIANCE PRINCIPLES
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PARTIAL DIFFERENTIAL EQUATIONS
PARTICLE PROPERTIES
SPACE
SYMMETRY GROUPS
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645400* - High Energy Physics- Field Theory