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Suppose $\mathbb{U}$ is an unitary operator acting on the Hilbert space of states representing a symmetry transformation such as rotation, translation etc. $\mathbb{U}$ is said to be a symmetry of non-relativistic quantum mechanics (NRQM) if leaves the Hamiltonian $H$ invariant i.e., $$\mathbb{U}^\dagger H\mathbb{U}=H.$$ This is the statement of symmetry in NRQM in its Hamiltonian formulation.

What is the statement of symmetry in NRQM in its Lagrangian or path-integral formulation? In this case, one uses Lagrangian instead of Hamiltonians and the operators don't make explicit appearance.

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In the path integral formulation of NRQM a symmetry of the theory is one which leaves the measure of the path integral invariant (i.e. the jacobian of the transformation is 1). If the measure remains invariant, then a simple change of variables inside the path integral ensures any transition amplitudes will be the same.

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  • $\begingroup$ I know this true in quantum field theory. It would be great if you can give an example from NRQM where the path-integral measure is really not invariant under some transformation. @CsarAlgebra $\endgroup$ – SRS Jul 1 '17 at 15:31

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