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 Schr\"odinger 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).