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A Holographic Description of Negative Energy States

Using the AdS/CFT duality, we study the expectation value of stress tensor in $2+1$-dimensional quantum critical theories with a general dynamical scaling $z$, and explore various constrains on negative energy density for strongly coupled field theories. The holographic dual theory is the theory of gravity in 3+1-dimensional Lifshitz backgrounds. We adopt a consistent approach to obtain the boundary stress tensor from bulk construction, which satisfies the trace Ward identity associated with Lifshitz scaling symmetry. In particular, the boundary stress tensor, constructed from the gravitational wave deformed Lifshitz geometry, is found up to second order in gravitational wave perturbations. {The result} is compared to its counterpart in free {scalar} field theory at the same order in an expansion of small squeezing parameters. This allows us to relate the boundary values of gravitational waves to the squeezing parameters of squeezed vacuum states. We find that, in both cases with $z=1$, the stress tensor satisfies the averaged null energy condition, and is consistent with the quantum interest conjecture. Moreover, the negative lower bound on null-contracted stress tensor, which is averaged over time-like trajectories along nearly null directions, is obtained. We find a weaker constraint on the magnitude and duration of negative null energy density in strongly coupled field theory as compared with the constraint in free relativistic field theory. The implications are discussed.