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The question is in the title. Can a Poincare invariant Lagrangian lead to a path integral that is not Lorentz or Poincare invariant? If so, can I have an example?

A related confusion: on page 426 of Weinberg I, he seems to show that this is not the case, at least for the translation subgroup. However, on the next page, he uses the same argument to show that electric charge is always conserved, even non-perturbatively. Now this is not always true. Is it the case that he is actually showing it implicitly only to all orders in perturbation theory (but not at finite coupling), and that the caveat is that the free states do not necessarily span the Hilbert space?

gmarocco
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1 Answers1

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Yes, e.g. the conformal/Weyl anomaly in string theory can be tied to the breaking of:

  • Lorentz symmetry of the target space in the light-cone gauge quantization$^1$, cf. e.g. Refs. 1 & 2;

  • BRST/gauge symmetry in the covariant quantization.

References:

  1. M.B. Green, J.H. Schwarz and E. Witten, Superstring theory, Vol. 1, 1986; eqs. (2.3.24-35).

  2. B. Zwiebach, A first course in String Theory, 2nd edition, 2009; section 12.5.

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$^1$ The breaking of manifest Lorentz covariance in intermediate steps is a well-known feature of canonical quantization. The point here is that there might not be any possible way at all to restore the Lorentz covariance of the classical model at quantum level.

Qmechanic
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