Heat kernel and Weyl anomaly of Schrödinger invariant theory

We propose a method inspired by discrete light cone quantization to determine the heat kernel for a Schrödinger field theory (Galilean boost invariant with z=2 anisotropic scaling symmetry) living in d+1 dimensions, coupled to a curved Newton-Cartan background, starting from a heat kernel of a relativistic conformal field theory (z=1) living in d+2 dimensions. We use this method to show that the Schrödinger field theory of a complex scalar field cannot have any Weyl anomalies. To be precise, we show that the Weyl anomaly Ad+1G for Schrödinger theory is related to the Weyl anomaly of a free relativistic scalar CFT Ad+2R via Ad+1G=2πδ(m)Ad+2R, where m is the charge of the scalar field under particle number symmetry. We provide further evidence of the vanishing anomaly by evaluating Feynman diagrams in all orders of perturbation theory. We present an explicit calculation of the anomaly using a regulated Schrödinger operator, without using the null cone reduction technique. We generalize our method to show that a similar result holds for theories with a single time-derivative and with even z>2.

Note:

27 pages, v2: 31 pages, clarifications regarding Heat Kernel added, reference updated, comments regarding anti-commuting fields added; v3: 40pages, 2 appendices are added, one of which includes an independent calculation verifying the earlier result, one line is added to abstract, references are updated, matches the version accepted to Journal

heat kernel
anomaly: Weyl
discrete light cone quantization
scaling: anisotropy
symmetry: scaling
invariance: Galilei
field theory: conformal
field theory: scalar
field theory: complex
perturbation theory

References(52)

Figures(0)

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