Zeta-Regularized Traces Versus the Wodzicki Residue as Tools in Quantum Field Theory and Infinite Dimensional Geometry

Paycha, Sylvie

doi:10.1007/978-1-4020-2468-9_5

Cited by 3 publications

(4 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…which is a generalization of [17]*equation (2.21) and [22]*equation (9). Furthermore, we obtain for A(z) = [B, CQ z ] with invertible Q, that ζ(A) = 0, i.e.…”

Section: 1mentioning

confidence: 63%

“…Using the Laurent expansion, we can reproduce many well-known facts about ζ-functions of pseudo-differential operators and Fourier Integral Operators like [17]*equation (2.21), [22]*equation (9), [23]*equations (0.12), (0.14), (0.17), (0.18), and (2.20).…”

Section: Introductionmentioning

confidence: 99%

“…In section 3, we will see that Corollary 2.14 applied to pseudo-differential operators implies many well-known formulae, e.g. [17]*equation (2.21), [22]*equation (9), and [23]*equation (2.20).…”

mentioning

confidence: 99%

See 2 more Smart Citations

A generalized Kontsevich-Vishik trace for Fourier Integral Operators and the Laurent expansion of $ζ$-functions

Hartung,

Scott

2015

Preprint

View full text Add to dashboard Cite

Based on Guillemin's work on gauged Lagrangian distributions, we will introduce the notion of a poly-log-homogeneous distribution as an approach to ζ-functions for a class of Fourier Integral Operators which includes cases of amplitudes with asymptotic expansion ∑ k∈N am k where each am k is log-homogeneous with degree of homogeneity m k but violating Re(m k ) → −∞. We will compute the Laurent expansion for the ζ-function and give formulae for the coefficients in terms of the phase function and amplitude as well as investigate generalizations to the Kontsevich-Vishik quasi-trace. Using stationary phase approximation, series representations for the Laurent coefficients and values of ζ-functions will be stated explicitly. Additionally, we will introduce an approximation method (mollification) for ζ-functions of Fourier Integral Operators whose symbols have singularities at zero by ζ-functions of Fourier Integral Operators with regular symbols.

show abstract

“…which is a generalization of [17]*equation (2.21) and [22]*equation (9). Furthermore, we obtain for A(z) = [B, CQ z ] with invertible Q, that ζ(A) = 0, i.e.…”

Section: 1mentioning

confidence: 63%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A generalized Kontsevich-Vishik trace for Fourier Integral Operators and the Laurent expansion of $ζ$-functions

Hartung,

Scott

2015

Preprint

View full text Add to dashboard Cite

show abstract

“…Non-discretized path integrals in Euclidean space-time can be studied within the framework of classical pseudo-differential operators and their traces and determinants [23]. These traces and determinants are defined using ζ-regularization which also gives rise to the Kontsevich-Vishik trace [16,17].…”

Section: Introductionmentioning

confidence: 99%

Regularizing Feynman path integrals using the generalized Kontsevich-Vishik trace

Hartung

2017

Journal of Mathematical Physics

View full text Add to dashboard Cite

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.

show abstract

Dixmier traces and extrapolation description of noncommutative Lorentz spaces

Gayral

Sukochev

2014

Journal of Functional Analysis

View full text Add to dashboard Cite

Abstract. We study the connections between Dixmier traces, ζ-functions and traces of heat semigroups beyond the dual of the Macaev ideal and in the general context of semifinite von Neumann algebras. We show that the correct framework for this investigation is that of operator Lorentz spaces possessing an extrapolation description. We demonstrate the applicability of our results to Hörmander-Weyl pseudo-differential calculus on R n . In this context, we prove that the Dixmier trace of a pseudodifferential operator coincides with the 'Dixmier integral' of its symbol.

show abstract

Zeta-Regularized Traces Versus the Wodzicki Residue as Tools in Quantum Field Theory and Infinite Dimensional Geometry

Cited by 3 publications

References 13 publications

A generalized Kontsevich-Vishik trace for Fourier Integral Operators and the Laurent expansion of $ζ$-functions

A generalized Kontsevich-Vishik trace for Fourier Integral Operators and the Laurent expansion of $ζ$-functions

Regularizing Feynman path integrals using the generalized Kontsevich-Vishik trace

Dixmier traces and extrapolation description of noncommutative Lorentz spaces

Contact Info

Product

Resources

About