Results 1  10
of
14
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 expectation ..."
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Cited by 64 (2 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].
Noncommutative martingale inequalities
, 1997
"... We prove the analogue of the classical BurkholderGundy inequalites for noncommutative martingales. As applications we give a characterization for an ItoClifford integral to be an Lpmartingale via its integrand, and then extend the ItoClifford integral theory in L2, developed by Barnett, Streater ..."
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Cited by 41 (9 self)
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We prove the analogue of the classical BurkholderGundy inequalites for noncommutative martingales. As applications we give a characterization for an ItoClifford integral to be an Lpmartingale via its integrand, and then extend the ItoClifford integral theory in L2, developed by Barnett, Streater and Wilde, to Lp for all 1 < p < ∞. We include an appendix on the noncommutative analogue of the classical Fefferman duality between H¹ and BMO.
Optimal Hypercontractivity for Fermi Fields and Related NonCommutative Integration Inequalities
, 1992
"... : Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of noncommutativ ..."
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Cited by 15 (1 self)
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: Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of noncommutative integration are established. On leave from School of Math., Georgia Institute of Technology, Atlanta, GA 30332 Work supported by U.S. National Science Foundation grant no. PHY9019433A01. 1 I. INTRODUCTION Observables pertaining to the configuration of a quantum system with n degrees of freedom are operators Q 1 ; Q 2 ; : : : ; Q n which, depending on the system, may or may not commute. Our main concern is with the case in which the configuration variables are amplitudes of certain field modes. For boson fields, these configuration observables do commute, and the state space H can be taken as the space of all complex square integrable functions on their joint spectrum. This is t...
Multiplicity of ground states in quantum field . . .
, 2008
"... The ground states of an abstract model in quantum field theory are investigated. By means of the asymptotic field theory, we give a necessary and sufficient condition for that the expectation value of the number operator of ground states is finite, from which we obtain a wideusable method to estima ..."
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Cited by 14 (3 self)
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The ground states of an abstract model in quantum field theory are investigated. By means of the asymptotic field theory, we give a necessary and sufficient condition for that the expectation value of the number operator of ground states is finite, from which we obtain a wideusable method to estimate an upper bound of the multiplicity of ground states. Ground states of massless GSB models and the PauliFierz model with spin 1/2 are investigated as examples.
Lowest energy states in nonrelativistic QED: atoms and ions in motion
, 2006
"... Within the framework of nonrelativisitic quantum electrodynamics we consider a single nucleus and N electrons coupled to the radiation field. Since the total momentum P is conserved, the Hamiltonian H admits a fiber decomposition with respect to P with fiber Hamiltonian H(P). A stable atom, resp. io ..."
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Cited by 11 (3 self)
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Within the framework of nonrelativisitic quantum electrodynamics we consider a single nucleus and N electrons coupled to the radiation field. Since the total momentum P is conserved, the Hamiltonian H admits a fiber decomposition with respect to P with fiber Hamiltonian H(P). A stable atom, resp. ion, means that the fiber Hamiltonian H(P) has an eigenvalue at the bottom of its spectrum. We establish the existence of a ground state for H(P) under (i) an explicit bound on P, (ii) a binding condition, and (iii) an energy inequality. The binding condition is proven to hold for a heavy nucleus and the energy inequality for spinless electrons.
Holomorphic functions and the heat kernel measure on an infinite dimensional complex orthogonal group
 Potential Analysis
"... Abstract. The heat kernel measure µt is constructed on an infinite dimensional complex group using a diffusion in a Hilbert space. Then it is proved that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure. The closure of these polynomials, HL2 (SOHS, µ ..."
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Cited by 11 (7 self)
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Abstract. The heat kernel measure µt is constructed on an infinite dimensional complex group using a diffusion in a Hilbert space. Then it is proved that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure. The closure of these polynomials, HL2 (SOHS, µt), is one of two spaces of holomorphic functions we consider. The second space, HL2 (SO(∞)), consists of functions which are holomorphic on an analog of the CameronMartin subspace for the group. It is proved that there is an isometry from the first space to the second one. The main theorem is that an infinite dimensional nonlinear analog of the Taylor expansion defines an isometry from HL2 (SO(∞)) into the Hilbert space associated with a Lie algebra of the infinite dimensional group. This is an extension to infinite dimensions of an isometry of B. Driver and L. Gross for complex Lie groups. All the results of this paper are formulated for one concrete group, the HilbertSchmidt complex orthogonal group, though our methods can be applied in more general situations. 1.
Hypercontractivity in noncommutative holomorphic spaces
 Commun. Math. Phys
, 2005
"... ABSTRACT. We prove an analog of Janson’s strong hypercontractivity inequality in a class of noncommutative “holomorphic ” algebras. Our setting is the qGaussian algebras Γq associated to the qFock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq, a qSeg ..."
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Cited by 8 (6 self)
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ABSTRACT. We prove an analog of Janson’s strong hypercontractivity inequality in a class of noncommutative “holomorphic ” algebras. Our setting is the qGaussian algebras Γq associated to the qFock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq, a qSegalBargmann transform, and prove Janson’s strong hypercontractivity L 2 (Hq) → L r (Hq) for r an even integer. 1.
Fiber Hamiltonians in the nonrelativistic quantum electrodynamics
 J. Funct. Anal
, 2007
"... A translation invariant Hamiltonian H in the nonrelativistic quantum electrodynamics is studied. This Hamiltonian is decomposed with respect to the total momentum PT: H = H(P)dP, R d where the selfadjoint fiber Hamiltonian H(P) is defined for arbitrary values of coupling constants. It is discussed ..."
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Cited by 8 (4 self)
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A translation invariant Hamiltonian H in the nonrelativistic quantum electrodynamics is studied. This Hamiltonian is decomposed with respect to the total momentum PT: H = H(P)dP, R d where the selfadjoint fiber Hamiltonian H(P) is defined for arbitrary values of coupling constants. It is discussed a relationship between rotation invariance of H(P) and polarization vectors, and functional integral representations of n point Euclidean Green functions of H(P) is given. From these, some applications concerning with degeneracy of ground states, ground state energy and expectation
Functional Integral Representation of the PauliFierz Model with Spin 1/2
, 706
"... A FeynmanKactype formula for a Lévy and an infinite dimensional Gaussian random process associated with a quantized radiation field is derived. In particular, a functional integral representation of e−tHPF generated by the PauliFierz Hamiltonian with spin 1/2 in nonrelativistic quantum electrody ..."
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Cited by 6 (6 self)
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A FeynmanKactype formula for a Lévy and an infinite dimensional Gaussian random process associated with a quantized radiation field is derived. In particular, a functional integral representation of e−tHPF generated by the PauliFierz Hamiltonian with spin 1/2 in nonrelativistic quantum electrodynamics is constructed. When no external potential is applied HPF turns translation invariant and it is decomposed as a direct integral HPF = ∫ ⊕ R3 HPF(P)dP. The functional integral representation of e−tHPF(P) is also given. Although all these Hamiltonians include spin, nevertheless the kernels obtained for the path measures are scalar rather than matrix expressions. As an application of the functional integral representations energy comparison inequalities are derived. 1 2 The PauliFierz model with spin 1
On a certain semiclassical problem on Wiener spaces
 PUBL.RES.INST.MATH.SCI.
, 2003
"... We study asymptotic behavior of the spectrum of a Schrödinger type operator L V on the Wiener space as # ##. Here L ..."
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Cited by 3 (3 self)
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We study asymptotic behavior of the spectrum of a Schrödinger type operator L V on the Wiener space as # ##. Here L