Uniqueness of the Kontsevich-Vishik trace

Maniccia, Lidia; Schrohe, Elmar; Seiler, Jörg

doi:10.1090/s0002-9939-07-09168-x

Cited by 13 publications

(24 citation statements)

References 12 publications

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“…This trace is called the generalized Kontsevich-Vishik trace (the original Kontsevich-Vishik trace is the special case of A 0 being a classical pseudo-differential operator) and given by By construction, the generalized Kontsevich-Vishik trace coincides with tr on trace-class operators. Furthermore, it was shown that the Kontsevich-Vishik trace is the only trace on the algebra of classical pseudo-differential operators that restricts to tr in L(L 2 (X)) [21]. These properties make the generalized Kontsevich-Vishik trace a prime candidate for path integral regularization as such a path integral regularization is consistent with respect to discretization (turning operators into matrices and, thus, trace-class) and Wick rotations (turning the path integral into pseudo-differential operator traces).…”

Section: Resultsmentioning

confidence: 99%

Regularizing Feynman path integrals using the generalized Kontsevich-Vishik trace

Hartung

2017

Journal of Mathematical Physics

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A fully regulated definition of Feynman's path integral is presented here. The proposed re-formulation of the path integral coincides with the familiar formulation whenever the path integral is well-defined. In particular, it is consistent with respect to lattice formulations and Wick rotations, i.e., it can be used in Euclidean and Minkowskian space-time. The path integral regularization is introduced through the generalized Kontsevich-Vishik trace, that is, the extension of the classical trace to Fourier Integral Operators. Physically, we are replacing the time-evolution semi-group by a holomorphic family of operator families such that the corresponding path integrals are welldefined in some half space of C. The regularized path integral is, thus, defined through analytic continuation. This regularization can be performed by means of stationary phase approximation or computed analytically depending only on the Hamiltonian and the observable (i.e., known a priori). In either case, the computational effort to evaluate path integrals or expectations of observables reduces to the evaluation of integrals over spheres. Furthermore, computations can be performed directly in the continuum and applications (analytic computations and their implementations) to a number of models including the non-trivial cases of the massive Schwinger model and a ϕ 4 theory.

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Section: Resultsmentioning

confidence: 99%

Regularizing Feynman path integrals using the generalized Kontsevich-Vishik trace

Hartung

2017

Journal of Mathematical Physics

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“…When combined with Proposition 4.2, Theorem 3.3 gives the existence and the uniqueness of (Res k ) k∈N on L. [11] Uniqueness of traces 181 APPLICATION 4.4. When we combine Proposition 4.2 with the result of [9], the uniqueness of TR on odd-class classical PDOs for odd-dimensional manifolds and Theorem 3.5, we obtain a proof of the uniqueness of TR on L odd in odd dimensions.…”

Section: Then the Operatormentioning

confidence: 93%

“…2 [9] Uniqueness of traces 179 REMARK 3.6. If τ 0 = 0, then any graded trace or ordinary trace extending τ 0 on A vanishes.…”

Section: Proof We Proceed By Induction On Kmentioning

confidence: 99%

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Uniqueness of Traces on Log-Polyhomogeneous Pseudodifferential Operators

Ducourtioux¹,

Ouédraogo²

2011

J. Aust. Math. Soc.

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“…2.1.4 An odd-class symbol as a sum of derivatives Proposition 1 (see Lemma 1.3 in [4], and [14]). Let n ∈ Z be odd.…”

Section: The Noncommutative Residue On Symbolsmentioning

confidence: 99%

Classification of Traces and Associated Determinants on Odd-Class Operators in Odd Dimensions

Jiménez

Ouédraogo

2012

SIGMA

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Abstract. To supplement the already known classification of traces on classical pseudodifferential operators, we present a classification of traces on the algebras of odd-class pseudodifferential operators of non-positive order acting on smooth functions on a closed odd-dimensional manifold. By means of the one to one correspondence between continuous traces on Lie algebras and determinants on the associated regular Lie groups, we give a classification of determinants on the group associated to the algebra of odd-class pseudodifferential operators with fixed non-positive order. At the end we discuss two possible ways to extend the definition of a determinant outside a neighborhood of the identity on the Lie group associated to the algebra of odd-class pseudodifferential operators of order zero.

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Uniqueness of the Kontsevich-Vishik trace

Cited by 13 publications

References 12 publications

Regularizing Feynman path integrals using the generalized Kontsevich-Vishik trace

Regularizing Feynman path integrals using the generalized Kontsevich-Vishik trace

Uniqueness of Traces on Log-Polyhomogeneous Pseudodifferential Operators

Classification of Traces and Associated Determinants on Odd-Class Operators in Odd Dimensions

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