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  • 1
    Electronic Resource
    Electronic Resource
    Cohomology of power sets with applications in quantum probability (1989)
    Springer
    Communications in mathematical physics 124 (1989), S. 337-364 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Square integrable Wiener functionals may be represented as sums of multiple Itô integrals. This leads to an identification of such functionals with square integrable functions on the symmetric measure space of the Lebesgue spaceR +. When the pointwise product of Wiener functionals is thus carried over, the product takes a pleasing form (cf. Wick's theorem) and various non-commutative perturbations of this “Wiener product” have been considered. Here we employ cohomological arguments to analyse deformations of an abstract Wiener product. This leads to the construction of Lévy fields which are neither bosonic nor fermionic, and also gives rise to homotopies between quasi-free boson and fermion fields. Finally we unify existence and uniqueness results for quantum stochastic differential equations by treating mixed noise differential equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01219655
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  • 2
    Electronic Resource
    Electronic Resource
    Projective unitary antiunitary representations of locally compact groups (1969)
    Springer
    Communications in mathematical physics 15 (1969), S. 305-328 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We give here a systematic presentation of the theory of projective representations when antiunitary operators are present. In particular the imprimitivity theorem of Mackey is proved in this situation and all the unitary antiunitary representations of the extended Poincaré group are derived.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01645530
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  • 3
    Electronic Resource
    Electronic Resource
    Unification of fermion and Boson stochastic calculus (1986)
    Springer
    Communications in mathematical physics 104 (1986), S. 457-470 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Fermion annihilation and creation processes are explicitly realised in Boson Fock space as functions of the corresponding Boson processes and second quantisations of reflections. Conversely, Boson annihilation and creation processes can be constructed from the Fermion processes. The existence of unitary stochastic evolutions driven by Fermion and gauge noise is thereby reduced to an equivalent Boson problem, which is then solved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01210951
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  • 4
    Electronic Resource
    Electronic Resource
    Infinitely divisible representations and positive definite functions on a compact group (1970)
    Springer
    Communications in mathematical physics 16 (1970), S. 148-156 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract A complete description of the infinitely divisible positive definite functions on a compact group is given. Their relation to infinitely divisible representations is also discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01646621
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  • 5
    Electronic Resource
    Electronic Resource
    A new method for constructing factorisable representations for current groups and current algebras (1976)
    Springer
    Communications in mathematical physics 50 (1976), S. 167-175 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract LetC e ∞ (R n ,G) denote the group of infinitely differentiable maps fromn-dimensional Euclidean space into a simply connected and connected Lie group, which have compact support. This paper introduces a class of factorisable unitary representations ofC e ∞ (R n ,G) with the property that the unitary operatorU f corresponding to a functionf inC e ∞ (R n ,G) depends not only onf, but also on the derivatives off up to a certain order. In particular these representations can not be extended to the group of all continuous functions fromR n toG with compact support.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01617994
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  • 6
    Electronic Resource
    Electronic Resource
    On the derivation of the Schroedinger equation in a Riemannian manifold (1968)
    Springer
    Communications in mathematical physics 9 (1968), S. 303-312 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Under certain conditions it is shown that the kinetic part of the dynamical operator of a quantum mechanical system with a Riemannian manifold as configuration space is the Laplace-Beltrami operator.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01654284
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  • 7
    Electronic Resource
    Electronic Resource
    Quantum Ito's formula and stochastic evolutions (1984)
    Springer
    Communications in mathematical physics 93 (1984), S. 301-323 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Using only the Boson canonical commutation relations and the Riemann-Lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes. This leads to a noncommutative Ito product formula, a realisation of the classical Poisson process in Fock space which gives a noncommutative central limit theorem, the construction of solutions of certain noncommutative stochastic differential equations, and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps. The classical Ito product formula for stochastic differentials with respect to Brownian motion and the Poisson process is a special case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01258530
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  • 8
    Electronic Resource
    Electronic Resource
    Time-orthogonal unitary dilations and noncommutative Feynman-Kac formulae (1982)
    Springer
    Communications in mathematical physics 83 (1982), S. 261-280 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract An analysis of Feynman-Kac formulae reveals that, typically, the unperturbed semigroup is expressed as the expectation of a random unitary evolution and the perturbed semigroup is the expectation of a perturbation of this evolution in which the latter perturbation is effected by a cocycle with certain covariance properties with respect to the group of translations and reflections of the line. We consider generalisations of the classical commutative formalism in which the probabilistic properties are described in terms of non-commutative probability theory based on von Neumann algebras. Examples of this type are generated, by means of second quantisation, from a unitary dilation of a given self-adjoint contraction semigroup, called the time orthogonal unitary dilation, whose key feature is that the dilation operators corresponding to disjoint time intervals act nontrivially only in mutually orthogonal supplementary Hilbert spaces.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01976044
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  • 9
    Electronic Resource
    Electronic Resource
    Stop times in fock space stochastic calculus (1987)
    Springer
    Probability theory and related fields 75 (1987), S. 317-349 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A stop time S in the boson Fock space ℋ over L 2(ℝ)+ is a spectral measure in [0,∞] such that {S([0,t])} is an adapted process. Following the ideas of Hudson [6], to each stop time S a canonical shift operator U Sis constructed in ℋ. When S({∞}) has the vacuum as a null vector U Sbecomes an isometry. When S({∞})=0 it is shown that ℋ admits a factorisation ℋ S]⊗ℋ{S where ℋ{S is the range of U Sand ℋ S] is a suitable subspace of ℋ called the Fock space upto time S. This, in particular, implies the strong Markov property of quantum Brownian motion in the boson as well as fermion sense and the Dynkin-Hunt property that the classical Brownian motion begins afresh at each stop time. The stopped Weyl and fermion processes are defined and their properties studied. A composition operation is introduced in the space of stop time to make it a semigroup. Stop time integrals are introduced and their properties constitute the basic tools for the subject.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00318706
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  • 10
    Electronic Resource
    Electronic Resource
    On the exact bound of the Coulomb potential with respect to the Dirac operator (1978)
    Springer
    Letters in mathematical physics 2 (1978), S. 413-415 
    Details
    ISSN: 1573-0530
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Summary The exact bound of the Coulomb potential with respect to the self adjoint Dirac operator is evaluated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00400168
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