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In somes references of Relativistic Quantum Mechanics, the one-particle states are given by representation theory of Poincaré algebra.

Could I mimics this for the non-relativistic case? States in non-relativistic quantum mechanics could be seen as representation of Galilei algebra (or Galilei group)?

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I deleted my answer, which I think missed the point of your question, and told you things you probably already know. Sorry about that... – user566 Jan 26 '11 at 19:56
Well, in the galilean algebra, the boost commutator has a central element which is identified as the mass. Besides, it tells that the center of mass moves with constant velocity (for systems without external potentials, of course). It's more or less non-trivial, so to say. – Rafael Jan 26 '11 at 20:00
Hum... I guess my comment lost sense after the others deleted theirs. Anyway, look at Schwinger's book Particles, Fields and Sources. He has this discussion there right in the beginning. – Rafael Jan 26 '11 at 20:01
Yes. There is a well-known procedure for this called the Inonu-Wigner contraction. – user346 Jan 27 '11 at 12:52
up vote 2 down vote accepted

Dear Leandro, in the relativistic case, the full Hilbert space is a unitary representation of the full Poincaré group. However, one-particle states are representations of the translations simply because they carry momenta $p$ and under translations by $\Delta x^\mu$, they get multiplied by the phase $\exp(ip\cdot\Delta x)$.

The translation generators of the Poincaré groups commute with each other, so you may diagonalize the Hilbert space and divide it into subspaces or "sectors" with fixed values of $d$-momenta - imagine $d=4$ in the most well-known dimensionality. These subspaces have well-defined transformation laws under the translations and under the boosts.

Now, if you look at the subspace corresponding to the momentum $k^\mu$, it contains states that are eigenstates of translations. They map to completely different states - in different sectors - under the boosts. However, we want to classify particles with momentum $k^\mu$ according to the rotations. Why is it so?

It's because the little group is defined as the subgroup of the Lorentz group that preserves $k^\mu$ - so it's exactly the group that maps the sector with a given value of $k^\mu$ onto itself. For massive particles, the little group is $SO(d-1)$; for massless particles, it is a semidirect product of $SO(d-2)$ with some $d-2$ semi-light-like mutually commuting generators of the Lorentz group, $J_{i-}$. For tachyons, if they could exist, the little group would be $SO(d-2,1)$ - a signature flip from the massive particles.

So the particles are then classified according to representations of the little group. For the massless particles, only the compact part $SO(d-2)$ is really important.

Now, the Galilean group is a contraction (infinite $c$ limit) of the Poincaré group; it has the same number of generators. In fact, the search for the representations would follow the very same prescription. You may also define energy and momenta - except that in non-relativistic theories, these two types of a quantity are not related by any speed of light.

Then you could also look at a given sector with a fixed energy and fixed 3-momentum, and find out that the sector is only preserved by the little group. Again, the boosts - in the non-relativistic case, they're simply $\vec x'=\vec x+\vec v t$ - would map the sector onto other sectors. The little group would always be $SO(d-1)$, like for massive relativistic particles, because $c$ is sent to infinity so no particles can move by the speed of light or faster. All of them resemble the relativistic, slowly moving massive particles. Well, you could have "instantaneous particles" (a new type of representation) that move by the infinite speed but they would cause lots of problems to the theory - and you could integrate their fast effects out immediately, anyway.

Obviously, you will find out that the non-relativistic particles simply carry some energy, some momentum, and transform under the little group - the usual $(2j+1)$-tuplet we know from non-relativistic quantum mechanics in the case $d=4$.

It's a somewhat different question how one could construct and classify off-shell fields and field theories (and their Lagrangians if they have one). Obviously, the time coordinate may be treated as a scalar and off-shell fields only have to form representations under $SO(d-1)$. The non-relativistic theories are much less constrained because one may break a spacetime tensor from relativity into time-like, space-like, and perhaps various mixed components and treat them separately, erase some of them while keeping others, give them different interactions etc.

The theories may be nonlocal - because there is no speed-of-light limit on the velocity - which makes the theories even less constrained. The invariance under the non-relativistic boosts is also easier to satisfy.

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Much better, thanks. Sometimes I type faster than I think. – user566 Jan 26 '11 at 21:13

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