Dipartimento di Fisica , Universita ’ di Pisa and INFN , Pisa F . Strocchi Scuola Normale Superiore and INFN , Pisa

The problem of existence of ground state representations on the CCR algebra with free evolution are discussed and all the solutions are classified in terms of non regular or indefinite invariant function-als. In both cases one meets unusual mathematical structures which appear as prototypes of phenomena typical of gauge quantum field theory, in particular of the temporal gauge. The functional integral representation in the positive non regular case is discussed in terms of a generalized stochastic process satisfying the Markov property. In the indefinite case the unique time translation and scale invariant Gaus-sian state is suprisingly faithful and its GNS representation is characterized in terms of a KMS operator. In the corresponding Euclidean formulation, one has a generalization of the Osterwalder-Schrader reconstruction and the indefinite Nelson space, defined by the Schwinger functions, has a unique Krein structure, allowing for the construction of Nelson projections, which satisfy the Markov property. Even if Nelson positivity is lost, a functional integral representation of the Schwinger functions exists in terms of a Wiener random variable and a Gaussian complex variable.