Does quantum mechanics predict all the effects of relativity except gravity?

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If you're only looking at non-relativistic quantum mechanics, no, it does not. The proper way to add special relativity to quantum mechanics leads to Quantum Field Theory, and that is the theory that properly takes into account all the effects of relativity. There were attempts to modify the Hamiltonian of the non-relativistic theory to make it relativistic, but it turns out that all such theories don't work. To add relativity you need to change the dynamical objects of the theory from particles, all the way to fields, and this is a much larger change than just adding relativity to classical mechanics.

You can sort of do relativistic quantum mechanics (RQM), and that's what Dirac came up with. The problem is that the Lorentz group is non-compact, and so there is no natural ground state in RQM. Dirac worked around it by inventing anti-particles. It gets you a fair ways toward describing phenomena where relativistic effects are important, mostly for free non-interacting cases. Eventually, however, you are forced to quantize the electromagnetic field, similarly to the way you quantize particles in non-relativistic QM. It gets very messy in a hurry, and it took decades to make the whole foundation look more than just ad hoc kludges to remove infinities from the calculations, but it holds together well for the most part.

Sean Carroll's program is to come at the whole thing from a different direction: assume the universe is a vector in a very large Hilbert space, together with some Hamiltonian that is basically a collection of energy eigenvalues in its own basis, and derive the rest from it (including QFT, gravity, many worlds, the Born rule and the classical world). It is a bit ambitious, but all the low-hanging fruit in foundations of physics has been picked clean.

(My reading of physics books is progressing very slowly, so I'll be done in a year at best, so I'm asking this question here.) Does quantum mechanics describe the slowing down of time and the contraction of objects when approaching the speed of light, as well as other effects (special?) theory of relativity?