Does there exist a sensible quantum theory of an ‘‘algebra-valued’’ scalar field?

Stephen C. Anco; Robert M. Wald

doi:10.1103/PhysRevD.39.2297

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Does there exist a sensible quantum theory of an ‘‘algebra-valued’’ scalar field?

Stephen C. Anco and Robert M. Wald

Enrico Fermi Institute and Deparment of Physics, University of Chicago, Chicago, Illinois 60637

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Phys. Rev. D 39, 2297 – Published 15 April, 1989

DOI: https://doi.org/10.1103/PhysRevD.39.2297

Abstract

Consider a scalar field φ in Minkowski spacetime, but let φ 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 Poincaré 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 λ $φ^{4}$ field, with φ valued in the two-dimensional algebra generated by an identity element e and a nilpotent element v satisfying $v^{2}$ =0. The Coleman-Mandula theorem, which states that the symmetry group of a nontrivial quantum field theory cannot be a nontrivial enlargement of the Poincaré 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. Thus, although the classical theory is perfectly well behaved, it seems that there does not exist a sensible quantum theory of an algebra-valued scalar field.

References (10)

C. Cutler and R. M. Wald, Class. Quantum Gravit. 4, 1267 (1987).
R. M. Wald, Class. Quantum Gravit. 4, 1279 (1987).
These notions are made precise in Ref. 2. Roughly, an ``algebra manifold'' is based on an associative, commutative algebra, openA, with identity element, and is defined by replacing openR with openA everywhere in the definition of an ordinary manifold. (The ``everywhere'' includes the differentiability of overlap charts, and for this a notion of ``algebra differentiability'' is defined.)
S. Coleman and J. Mandula, Phys. Rev. 159, 1251 (1967).
For complex algebras (i.e., which possess an element j satisfying $j^{2}$ =-e), our assertions remain true if the ``most nilpotent subspace'' is two-real dimensional, i.e., one-complex dimensions, where a complex subspace is one where v cdot j is in the subspace whenever v is.
R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980).
Note that this rule assumes the vacuum is stable, and comes from normalizing all S-matrix amplitudes by the vacuum-to-vacuum amplitude.
G. 't Hooft and M. Veltman, in Particle Interactions at Very High Energies, edited by D. Speiser, F. Halzen, and J. Weyers (Plenum, New York, 1974), Part B, p. 177.
It is easy to see that the ``representation'' on ${scrW}_{0}$ cannot be equivalent to the ``representation'' on any other ${scrW}_{n}$ on account of the uniqueness of the Poincaré-invariant vacuum, |0 > member ${scrW}_{0}$ .

[c1] C. Cutler and R. M. Wald, Class. Quantum Gravit. 4, 1267 (1987).

[c2] R. M. Wald, Class. Quantum Gravit. 4, 1279 (1987).

[c3] These notions are made precise in Ref. 2. Roughly, an ``algebra manifold'' is based on an associative, commutative algebra, openA, with identity element, and is defined by replacing openR with openA everywhere in the definition of an ordinary manifold. (The ``everywhere'' includes the differentiability of overlap charts, and for this a notion of ``algebra differentiability'' is defined.)

[c4] S. Coleman and J. Mandula, Phys. Rev. 159, 1251 (1967).

[c5] For complex algebras (i.e., which possess an element j satisfying $j^{2}$ =-e), our assertions remain true if the ``most nilpotent subspace'' is two-real dimensional, i.e., one-complex dimensions, where a complex subspace is one where v cdot j is in the subspace whenever v is.

[c6] R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).

[c7] C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980).

[c8] Note that this rule assumes the vacuum is stable, and comes from normalizing all S-matrix amplitudes by the vacuum-to-vacuum amplitude.

[c9] G. 't Hooft and M. Veltman, in Particle Interactions at Very High Energies, edited by D. Speiser, F. Halzen, and J. Weyers (Plenum, New York, 1974), Part B, p. 177.

[c10] It is easy to see that the ``representation'' on ${scrW}_{0}$ cannot be equivalent to the ``representation'' on any other ${scrW}_{n}$ on account of the uniqueness of the Poincaré-invariant vacuum, |0 > member ${scrW}_{0}$ .

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Does there exist a sensible quantum theory of an ‘‘algebra-valued’’ scalar field?

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