Vol. 137, No. 2, 1989

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Algebraic characterization of the vacuum for quantized fields transforming nonunitarily

Irving E. Segal

Vol. 137 (1989), No. 2, 387–403

The vacuum as an expectation value form on the Clifford or Weyl algebra over an orthogonal or symplectic real linear space, invariant under a given group of automorphisms of such, is treated without assumptions as to self-adjointness or positivity. This is necessary for the quantization of fields that transform non-unitarily, in particular indecomposably, such as the full section spaces of typical conformally invariant bundles over space-times. A stability condition in the nature of positivity of the energy is shown to be sufficient to characterize the vacuum for the basic case of a one-parameter group. In application e.g. to spannor fields transforming under SU(2,2), this results in a vacuum invariant under the maximal subgroup K, giving rise to a natural broken symmetry.

Mathematical Subject Classification 2000
Primary: 81E05, 81E05
Secondary: 46L60, 46N05
Received: 29 March 1988
Published: 1 April 1989
Irving E. Segal