Avramidi, Ivan G. Heat semigroups on Weyl algebra. (English) Zbl 1456.58020 J. Geom. Phys. 161, Article ID 104044, 25 p. (2021). Summary: We study the algebra of semigroups of Laplacians on the Weyl algebra. We consider first-order partial differential operators \(\nabla_i^\pm\) forming the Lie algebra \([\nabla_j^\pm,\nabla_k^\pm]=i\mathscr{R}_{jk}^\pm\) and \([\nabla_j^+,\nabla_k^-]=i\frac{1}{2}(\mathscr{R}_{j k}^++\mathscr{R}_{j k}^-)\) with some anti-symmetric matrices \(\mathscr{R}_{ij}^\pm\) and define the corresponding Laplacians \(\Delta_\pm=g_\pm^{ij}\nabla_i^\pm\nabla_j^\pm\) with some positive matrices \(g_\pm^{i j} \). We show that the heat semigroups \(\exp(t\varDelta_\pm)\) can be represented as a Gaussian average of the operators \(\exp\left< \xi , \nabla^\pm\right>\) and use these representations to compute the product of the semigroups, \(\exp(t\varDelta_+) \exp(s\varDelta_-)\) and the corresponding heat kernel. MSC: 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 58J35 Heat and other parabolic equation methods for PDEs on manifolds 58J37 Perturbations of PDEs on manifolds; asymptotics 58J53 Isospectrality 58J90 Applications of PDEs on manifolds Keywords:heat semigroup; heat kernel; relative spectral invariants; combined heat traces; spectral asymptotics; Laplace type operators PDFBibTeX XMLCite \textit{I. G. Avramidi}, J. Geom. Phys. 161, Article ID 104044, 25 p. (2021; Zbl 1456.58020) Full Text: DOI arXiv References: [1] Avramidi, I. G., A covariant technique for the calculation of the one-loop effective action, Nuclear Phys. B, B355, 712-754 (1991) [2] Avramidi, I. G., A new algebraic approach for calculating the heat kernel in gauge theories, Phys. Lett. B, B305, 27-34 (1993) [3] Avramidi, I. G., Covariant algebraic method for calculation of the low-energy heat kernel, J. Math. Phys.. J. Math. Phys., J. Math. Phys., 39, 1720-5070 (1998), Erratum · Zbl 1056.58501 [4] Avramidi, I. G., Heat kernel on homogeneous bundles over symmetric spaces, Comm. Math. Phys., 288, 963-1006 (2009) · Zbl 1180.58013 [5] Avramidi, I. G., Heat Kernel Method and Its Applications (2015), Birkhäuser: Birkhäuser Basel · Zbl 1342.35001 [6] Avramidi, I. G., Quantum heat traces, J. Geom. Phys., 112, 271-288 (2017) · Zbl 1354.58021 [7] Avramidi, I. G., Bogolyubov invariant via relative spectral invariants on manifolds, J. Math. Phys., 61, Article 032303 pp. (2020) · Zbl 1439.81079 [8] Avramidi, I. G., Relative spectral invariants of elliptic operators on manifolds, J. Geom. Phys., 150, Article 103599 pp. (2020) · Zbl 1507.58011 [9] Avramidi, I. G.; Buckman, B. J., Heat determinant on manifolds, J. Geom. Phys., 104, 64-88 (2016) · Zbl 1344.53033 [10] Berger, M., A Panoramic View of Riemannian Geometry (1992), Springer: Springer Berlin [11] Bonfiglioli, A.; Fulci, R., Topics in Noncommutative Algebra: The Theorem of Campbell, Baker, Hausdorff and Dynkin (2012), Springer: Springer Berlin · Zbl 1231.17001 [12] Dewitt, B. S., The Global Approach to Quantum Field Theory, Vol. 1 and 2 (2003), Oxford University Press: Oxford University Press Oxford [13] Gilkey, P. B., Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem (1995), CRC: CRC Boca Raton · Zbl 0856.58001 [14] Gordon, C.; Webb, D. L.; Wolpert, S., One cannot hear the shape of a drum, Bull. Amer. Math. Soc. (N.S.), 27, 134-138 (1992) · Zbl 0756.58049 [15] Grib, A. A.; Mamaev, S. G.; Mostepanenko, V. M., Vacuum Quantum Effects in Strong Fields (1994), Friedmann Laboratory: Friedmann Laboratory St. Petersburg [16] L. Parker, D. Toms, Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity, Cambridge: Cambridge Monographs on Mathematical Physics, 2009. · Zbl 1180.81001 [17] Prudnikov, A. P.; Brychkov, Yu. A.; Marychev, O. I., Integrals and Series, Vol. I (1998), CRC: CRC Boca Raton [18] Schueth, D., Continuous families of isospectral metrics on simply connected manifolds, Ann. of Math., 149, 287-308 (1999) · Zbl 0964.53027 [19] Wald, R. M., Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics (1994), University of Chicago Press: University of Chicago Press Chicago · Zbl 0842.53052 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.