Unitarity and Universality in non relativistic Conformal Field theory

Abstract

We relate the notion of unitarity of a $SL(2,\mathbf{R})$ invariant field theory with that of a Schrodinger field theory using the fact that $SL(2,\mathbf{R})$ is a subgroup of Schrodinger group. Exploiting $SL(2,\mathbf{R})$ unitarity, we derive the unitarity bounds and null conditions for a Schrödinger field theory (for the neutral as well as the charged sector). In non integer dimensions the theory is shown to be non-unitary. Furthermore, the use of $SL(2,\mathbf{R})$ subgroup opens up the possibility of borrowing results from 1D $SL(2,\mathbf{R})$ invariant field theory to explore Schrodinger field theory, in particular, the sector with zero charge. We explore the consequences of $SL(2,\mathbf{R})$ symmetry e.g. the convergence of operator product expansion in the kinematic limit, where all the operators (neutral and/or charged) are on same temporal slice ($x=constant$), the universal behavior of weighted spectral density function, existence of infinite number of $SL(2,\mathbf{R})$ primaries, the analytic behavior of three point function as a function of spatial separation. We discuss the implication of imposing parity invariance ($\tau \to -\tau$) in addition to Schrodinger invariance and emphasize its difference from time reversal ($\tau \to -\tau$ with charge conjugation).

arXiv comment12 Pages (two column), 5 Figures v2: added section on subtleties with crossing symmetry v3: ref. added, appendix C removed, app. B extended with discussion of time ordering, typos fixed, discussion of parity mended, v4: Reformatted paper, matches with the published version (except appendix E)

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• Correlators in phase-ordering from Schrödinger-invariance
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