On the Heat Kernel and Weyl Anomaly of Schrödinger invariant theory

Abstract

We propose a method inspired from discrete light cone quantization (DLCQ) 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 the Schrödinger field theory of a complex scalar field cannot have any Weyl anomalies. To be precise, we show that the Weyl anomaly ${\mathcal{A}}_{d+1}^G$ for Schrödinger theory is related to the Weyl anomaly of a free relativistic scalar CFT ${\mathcal{A}}_{d+2}^R$ via ${\mathcal{A}}_{d+1}^G=2\pi \delta (m){\mathcal{A}}_{d+2}^R$ where $m$ is the charge of the scalar field under particle number symmetry. We provide further evidence of 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 generalise our method to show that a similar result holds for one time derivative theories with even $z>2$.

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

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• Gravity duals for non-relativistic CFTs
• Nonrelativistic conformal field theories
• Spacetime Symmetries of the Quantum Hall Effect
• A challenge to the $a$-theorem in six dimensions
• Heat kernel methods for Lifshitz theories
• Lifshitz Anomalies, Ward Identities and Split Dimensional Regularization
• Weyl Consistency Conditions in Non-Relativistic Quantum Field Theory
• Constraints on Perturbative RG Flows in Six Dimensions
• On Newton-Cartan local renormalization group and anomalies
• Heat kernel for Newton-Cartan trace anomalies
• Anisotropic Weyl invariance

Cited by in corpus (14)

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• Towards a unitary, renormalizable and ultraviolet-complete quantum theory of gravity
• Unitarity and Universality in non relativistic Conformal Field theory
• Nonrelativistic trace and diffeomorphism anomalies in particle number background
• Triviality of Entanglement Entropy in the Galilean Vacuum
• Non-Relativistic Supersymmetry on Curved Three-Manifolds
• Entanglement entropies of an interval in the free Schrödinger field theory on the half line
• Entanglement entropies of an interval in the free Schrödinger field theory at finite density
• A (1+1)-dimensional Lifshitz Weyl Anomaly From a Schr$\ddot{\mathrm{o}}$dinger-invariant Non-relativistic Chern-Simons Action
• Developments in non-relativistic field theory and complexity