General Principles of Quantum Field Theory pp 417-449 | Cite as

# Fields in an Indefinite Metric

Chapter

## Abstract

We need to have recourse to a picture of fields in a space with an indefinite metric when attempting to formulate the idea of a virtual (or “potential”) state (not unlike the way one uses representations to organize the physical states of quantum systems). The notion of a virtual state can be illustrated by the following “classical” example. As is well known, the state of a free classical electromagnetic field in space-time

**M**is defined by the stress tensor field*F*_{λμ}(*x*) satisfying Maxwell’s equations. This state can also be defined by a vector potential*A*_{ μ }(*x*); however, such a characterization has a certain redundancy since two configurations, say,*A*_{ μ }(*x*) and*A*′_{ μ }(*x*)of the vector potential define the same physical state (that is, they are equivalent) if their stress tensor fields are the same:$$
{F_{\lambda \mu }}\left( { \equiv {\partial _\lambda }A{}_\mu - {\partial _\mu }{A_\lambda }} \right) = {F'_{\lambda \mu }}\left( { \equiv {\partial _\lambda }{{A'}_\mu } - {\partial _\mu }{{A'}_\lambda }} \right).
$$

## Keywords

Gauge Group Abelian Gauge Theory Vacuum Vector Wightman Function Poincare Group
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© Kluwer Academic Publishers 1990