Negative Energy and Mass Symmetry

[jbcool]

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?

 

[azntoon]

Noether's Theorem does not apply to systems which are invariant under a switch from E→-E and m→-m. This type of system is known as an "anti-symmetric system", and it does not possess any conserved quantities. However, if the Hamiltonian and Reversal operator commute in such a system, then the potential must be odd under mass negation. This means that the potential must satisfy the condition V(-m)= -V(m). Examples of potentials which satisfy this condition include uniform gravitational potentials, free potentials, and harmonic oscillators. In quantum mechanics, the mass is not an observable, so it does not have an associated operator. Therefore, it is not necessary to consult QFT to answer this question; the same principles apply in non-relativistic quantum mechanics.