Ground state and functional integral representations of the CCR algebra with free evolution
Abstract
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 functionals. 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 Gaussian state is surprisingly faithful and its GNS representation is characterized in terms of a KMS operator. In the corresponding Euclidean formulation, one has a generalization of the OsterwalderSchrader 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2002
 DOI:
 10.48550/arXiv.mathph/0212037
 arXiv:
 arXiv:mathph/0212037
 Bibcode:
 2002math.ph..12037L
 Keywords:

 Mathematical Physics;
 Mathematics  Mathematical Physics;
 81S40;
 60G10;
 60G15;
 46C20;
 47B50
 EPrint:
 LateX, with correction of misprints and extension of results in sect.2