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From scale invariance to Lorentz symmetry

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I will discuss the enhancement of space-time symmetries to Lorentz (rotation) invariance at the renormalization group fixed points of non-relativistic (anisotropic) field theories. Upon describing examples from the condensed matter physics, I will review the general argument for the stability of the infrared fixed points with the enhanced symmetry. Then I will focus on unitary field theories in (1+1) space-time dimensions which are invariant under translations, isotropic scale transformations and satisfy the requirement that the velocity of signal propagation is bounded from above. No a priori Lorentz invariance will be assumed. Still, I will prove that above properties are sufficient to ensure the existence of an infinite dimensional symmetry given by one or a product of several copies of conformal algebra. In particular, this implies presence of one or several Lorentz groups acting on the operator algebra of the theory. I will conclude by discussing the challenges in extending this result to higher space-time dimensions.