I was wondering about a system, specifically quantum, though classical solutions are still welcome, which was resisting all applications of Noether's Theorem, and related techniques. If a system is invariant under a switch from E→-E AND m→-m, then what are the conserved quantities (in analogy to a system invariant under time or space parity). I found that if the Hamiltonian and Reversal operator (the one that negates mass and energy) commute, then the potential is odd under mass negation (such as a uniform gravitational potential, free potential, or a harmonic oscillator). The problem however (I think), is that in quantum mechanics, the mass is not an observable, and therefore does not correlate with an operator. Is it necessary to consult QFT for this question to have any meaning (with relativistic mass and energy), or is it still of merit in NRQM?