Results 1  10
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28
qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expec ..."
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Cited by 98 (3 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
Convoluted generalized white noise, Schwinger functions and their continuation to Wightman functions
, 2008
"... We construct Euclidean random fields X over IR d, by convoluting generalized white noise F with some integral kernels G, as X = G ∗ F. We study properties of Schwinger (or moment) functions of X. In particular, we give a general equivalent formulation of the cluster property in terms of truncated Sc ..."
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Cited by 24 (14 self)
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We construct Euclidean random fields X over IR d, by convoluting generalized white noise F with some integral kernels G, as X = G ∗ F. We study properties of Schwinger (or moment) functions of X. In particular, we give a general equivalent formulation of the cluster property in terms of truncated Schwinger functions which we then apply to the above fields. We present a partial negative result on the reflection positivity of convoluted generalized white noise. Furthermore, by representing the kernels Gα of the pseudo–differential operators (− ∆ + m 2 0) −α for α ∈ (0,1) and m0> 0 as Laplace transforms we perform the analytic continuation of the (truncated) Schwinger functions of X = Gα ∗ F, obtaining the corresponding (truncated) Wightman distributions on Minkowski space which satisfy the relativistic postulates on invariance, spectral property, locality and cluster property.
Models of local relativistic quantum fields with indefinite metric (in all dimensions
 Commun. Math. Phys
, 1997
"... A condition on a set of truncated Wightman functions is formulated and shown to permit the construction of the Hilbert space structure included in the Morchio–Strocchi modified Wightman axioms. The truncated Wightman functions which are obtained by analytic continuation of the (truncated) Schwinger ..."
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Cited by 18 (11 self)
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A condition on a set of truncated Wightman functions is formulated and shown to permit the construction of the Hilbert space structure included in the Morchio–Strocchi modified Wightman axioms. The truncated Wightman functions which are obtained by analytic continuation of the (truncated) Schwinger functions of Euclidean scalar random fields and covariant vector (quaternionic) random fields constructed via convoluted generalized white noise, are then shown to satisfy this condition. As a consequence such random fields provide relativistic models for indefinite metric quantum field theory, in dimension 4 (vector case), respectively in all dimensions (scalar case).
Unitary representations and OsterwalderSchrader Duality
 PROC. SYMPOS. PURE MATH
, 1998
"... The notions of reflection, symmetry, and positivity from quantum field theory are shown to induce a duality operation for a general class of unitary representations of Lie groups. The semisimple Lie groups which have this cduality are identified and they are placed in the context of HarishChandr ..."
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Cited by 11 (5 self)
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The notions of reflection, symmetry, and positivity from quantum field theory are shown to induce a duality operation for a general class of unitary representations of Lie groups. The semisimple Lie groups which have this cduality are identified and they are placed in the context of HarishChandra’s legacy for the unitary representations program. Our paper begins with a discussion of path space measures, which is the setting where reflection positivity (OsterwalderSchrader) was first identified as a useful tool of analysis.
Minoru W.: Multiple Stochastic Integrals Construction of nonGaussian Reflection Positive Generalized Random Fields
"... Multiple stochastic integrals construction of nonGaussian reflection positive generalized random fields S. Albeverio1),2), M. W. Yoshida3)∗ ..."
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Cited by 9 (8 self)
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Multiple stochastic integrals construction of nonGaussian reflection positive generalized random fields S. Albeverio1),2), M. W. Yoshida3)∗
Markov quantum fields on a manifold
"... We study scalar quantum field theory on a compact manifold. The free theory is defined in terms of functional integrals. For positive mass it is shown to have the Markov property in the sense of Nelson. This property is used to establish a reflection positivity result when the manifold has a reflect ..."
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Cited by 5 (0 self)
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We study scalar quantum field theory on a compact manifold. The free theory is defined in terms of functional integrals. For positive mass it is shown to have the Markov property in the sense of Nelson. This property is used to establish a reflection positivity result when the manifold has a reflection symmetry. In dimension d=2 we use the Markov property to establish a sewing operation for manifolds with boundary circles. Also in d=2 the Markov property is proved for interacting fields.
Construction of relativistic quantum fields in the framework of white noise analysis
 J. Math. Phys
, 1999
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On the path Hölder continuity in models of Euclidean quantum field theory., preprint Inst
, 1996
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COMPLEX CLASSICAL FIELDS: A FRAMEWORK FOR REFLECTION POSITIVITY
, 2012
"... Abstract. We explore a framework for complex classical fields, appropriate for describing quantum field theories. Our fields are linear transformations on a Hilbert space, so they are more general than random variables for a probability measure. Our method generalizes Osterwalder and Schrader’s cons ..."
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Cited by 3 (2 self)
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Abstract. We explore a framework for complex classical fields, appropriate for describing quantum field theories. Our fields are linear transformations on a Hilbert space, so they are more general than random variables for a probability measure. Our method generalizes Osterwalder and Schrader’s construction of Euclidean fields. We allow complexvalued classical fields in the case of quantum field theories that describe neutral particles. From an analytic pointofview, the key to using our method is reflection positivity. We investigate conditions on the Fourier representation of the fields to ensure that reflection positivity holds. We also show how reflection positivity is preserved by various