A Renyi Quantum Null Energy Condition: Proof for Free Field Theories

Abstract

The Quantum Null Energy Condition (QNEC) is a lower bound on the stress-energy tensor in quantum field theory that has been proved quite generally. It can equivalently be phrased as a positivity condition on the second null shape derivative of the relative entropy $S_{\text{rel}}(\rho ||\sigma )$ of an arbitrary state $\rho$ with respect to the vacuum $\sigma$. The relative entropy has a natural one-parameter family generalization, the Sandwiched Renyi divergence $S_n(\rho ||\sigma )$, which also measures the distinguishability of two states for arbitrary $n\in [1/2,\infty )$. A Renyi QNEC, a positivity condition on the second null shape derivative of $S_n(\rho ||\sigma )$, was conjectured in previous work. In this work, we study the Renyi QNEC for free and superrenormalizable field theories in spacetime dimension $d>2$ using the technique of null quantization. In the above setting, we prove the Renyi QNEC in the case $n>1$ for arbitrary states. We also provide counterexamples to the Renyi QNEC for $n<1$.

arXiv comment39 pages, 6 figures

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