Supersymmetric Extension of Quantum Scalar Field Theories

Arai, Asao

doi:10.1007/978-94-017-2823-2_6

Cited by 10 publications

(7 citation statements)

References 23 publications

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“…The de Rham complex on infinite product manifolds with Gibbs measures (which appear in connection with problems of classical statistical mechanics) was constructed in [1], [2] (see also [19] for the case of the infinite-dimensional torus). We should also mention the papers [49], [15], [16], [17], [7], where the case of a flat Hilbert state space is considered (the L 2 -cohomological structure turns out to be nontrivial even in this case due to the existence of interesting measures on such a space).…”

mentioning

confidence: 99%

$L^2$-Betti Numbers of Inﬁnite Conﬁguration Spaces

Albeverio¹,

Daletskii²

2006

Publ. Res. Inst. Math. Sci.

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The space Γ X of all locally finite configurations in a infinite covering X of a compact Riemannian manifold is considered. The de Rham complex of square-integrable differential forms over Γ X , equipped with the Poisson measure, and the corresponding de Rham cohomology and the spaces of harmonic forms are studied. A natural von Neumann algebra containing the projection onto the space of harmonic forms is constructed. Explicit formulae for the corresponding trace are obtained. A regularized index of the Dirac operator associated with the de Rham differential on the configuration space of an infinite covering is considered.

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confidence: 99%

$L^2$-Betti Numbers of Inﬁnite Conﬁguration Spaces

Albeverio¹,

Daletskii²

2006

Publ. Res. Inst. Math. Sci.

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“…Dirichlet forms and processes in connection with non-commutative C*-algebras were considered in, e.g., [Gr, AH-K, DavLin]. In an infinite dimensional situation, such questions were discussed in the flat case in [Ar1,Ar2,ArM,AK]. A regularized heat semigroup on differential forms over the infinite dimensional torus was studied in [BeLe].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Analysis on Product Manifolds: Dirichlet Operators on Differential Forms

Albeverio

Daletskii

Kondratiev

2000

Journal of Functional Analysis

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We define a de Rham complex over a product manifold (infinite product of compact manifolds), and Dirichlet operators on differential forms, associated with differentiable measures (in particular, Gibbs measures), which generalize the notions of Bochner and de Rham Laplacians. We give probabilistic representations for corresponding semigroups and study properties of the corresponding stochastic dynamics.

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“…[2]), the author introduced a general class of infinite dimensional Dirac type operators on the abstract boson-fermion Fock space F(H, K) associated with the pair (H, K) of complex Hilbert spaces (for the definition of F(H, K) and , see Section 2), where is a densely defined closed linear operator from H to K. The operator gives an infinite dimensional and abstract version of finite dimensional Dirac type operators. In applications to physics, unifies self-adjoint supercharges (generators of supersymmetry) of some supersymmetric quantum field models (e.g., [3][4][5][6]).…”

Section: Introductionmentioning

confidence: 99%

A Special Class of Infinite Dimensional Dirac Operators on the Abstract Boson-Fermion Fock Space

Arai

2014

Journal of Mathematics

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Spectral properties of a special class of infinite dimensional Dirac operators ( ) on the abstract boson-fermion Fock space F(H, K) associated with the pair (H, K) of complex Hilbert spaces are investigated, where ∈ C is a perturbation parameter (a coupling constant in the context of physics) and the unperturbed operator (0) is taken to be a free infinite dimensional Dirac operator. A variety of the kernel of ( ) is shown. It is proved that there are cases where, for all sufficiently large | | with < 0, ( ) has infinitely many nonzero eigenvalues even if (0) has no nonzero eigenvalues. Also Fredholm property of ( ) restricted to a subspace of F(H, K) is discussed. ,V ( ) is the main object of our analysis in the present paper. We prove the self-adjointness of ( ) and the essential selfadjointness of it on a suitable dense subspace of F(H, K) with some other properties (Theorem 14). Also the spectra of ( ) are identified (Theorem 16). Moreover, it is shown that the domain of ( ) is equal to that of (0) = for all sufficiently small | | (Theorem 17). Section 5 is devoted to analysis of the kernel of ( ). We see that the kernel property of ( ) may be sensitive to conditions for ( , * , ). In Section 6 we consider Fredholm property of ( ) + , the restriction of ( ) to a subspace of F(H, K), in comparison with that of ( ) + = (0) + . We obtain some classification on Fredholm property of ( ) + (Theorem 26). An interesting phenomenon occurs in the following sense: for a constant 0 ̸ = 0 (resp., 0 ̸ = 0), ( 0 ) + is not semi-Fredholm even if ( ) + is Fredholm (resp., semi-Fredholm). In the last

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Supersymmetric Extension of Quantum Scalar Field Theories

Cited by 10 publications

References 23 publications

$L^2$-Betti Numbers of Inﬁnite Conﬁguration Spaces

$L^2$-Betti Numbers of Inﬁnite Conﬁguration Spaces

Stochastic Analysis on Product Manifolds: Dirichlet Operators on Differential Forms

A Special Class of Infinite Dimensional Dirac Operators on the Abstract Boson-Fermion Fock Space

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