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What kinds of inconsistencies would one get if one starts with Lorentz noninvariant Lagrangian of QFT? The question is motivated by this preprint arXiv:1203.0609 by Murayama and Watanabe.

Also, what is the basic difference between

  • a theory that is non-relativistic approximation to relativistic theory, and

  • a fundamentally Lorentz non-invariant theory?

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Lorentz invariant theories are desirable because their behaviour does not depend on a specific frame of reference.

A theory non-relativistic approximation to a relativistic theory can usually be acquired by expanding physical quantities which are part of a relativistic around low velocities or momenta. A fundamentally Lorentz non-invariant theory is just some theory which does not possess the property of Lorentz-invariance.

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