# Representation theory of the Galilean group

In

nonrelativistic quantum mechanics , an account can be given of the existence ofmass and spin as follows:The spacetime

symmetry group of nonrelativistic quantum mechanics is theGalilean group . In 3+1 dimensions, this is the subgroup of theaffine group on (t,x,y,z) whose linear part leaves invariant both the metric$(g\_\{mu\; u\})\; =\; diag(1,0,0,0)$ and the (independent) dual metric$(g^\{mu\; u\})\; =\; diag(0,1,1,1)$. A similar definition applies for n+1 dimensions.We are interested in

projective representation s of this group, which are equivalent tounitary representation s of the nontrivial central extension of theuniversal covering group of theGalilean group by the one dimensional Lie group**R**, refer to the articleGalilean group for the central extension of itsLie algebra . We will focus upon the Lie algebra here because it is simpler to analyze and we can always extend the results to the full Lie group thanks to the Frobenius theorem.:$[E,P\_i]\; =0$:$[P\_i,P\_j]\; =0$:$[L\_\{ij\},E]\; =0$:$[C\_i,C\_j]\; =0$:$[L\_\{ij\},L\_\{kl\}]\; =ihbar\; [delta\_\{ik\}L\_\{jl\}-delta\_\{il\}L\_\{jk\}-delta\_\{jk\}L\_\{il\}+delta\_\{jl\}L\_\{ik\}]$:$[L\_\{ij\},P\_k]\; =ihbar\; [delta\_\{ik\}P\_j-delta\_\{jk\}P\_i]$:$[L\_\{ij\},C\_k]\; =ihbar\; [delta\_\{ik\}C\_j-delta\_\{jk\}C\_i]$:$[C\_i,E]\; =ihbar\; P\_i$:$[C\_i,P\_j]\; =ihbar\; Mdelta\_\{ij\}$

If you think about how spatial and time translations, rotations and boosts work, these relations are intuitive (except for the central extension).

The

central charge "M" is of course aCasimir invariant . The mass shell invariant:$ME-\{P^2over\; 2\}$

is a second

Casimir invariant . In 3+1 dimensions, a thirdCasimir invariant is $W^2$ where:$vec\{W\}\; =\; M\; vec\{L\}\; +\; vec\{P\}\; imesvec\{C\}$.More generally, in n+1 dimensions, invariants will be a function of $W\_\{ij\}\; =\; M\; L\_\{ij\}\; +\; P\_i\; C\_j\; -\; P\_j\; C\_i$ and $W\_\{ijk\}\; =\; P\_i\; L\_\{jk\}\; +\; P\_j\; L\_\{ki\}\; +\; P\_k\; L\_\{ij\}$, as well as of the mass shell invariant and central charge.

Using

Schur's lemma , in an irreducible unitary representation, each of these Casimir invariants are multiples of the identity. Let's call these coefficients "m" and "mE"_{0}and (in the case of 3+1 dimensions) "w" respectively. Remember we are talking about unitary representations here, which means these values have to be real. So, "m" > 0, "m" = 0 and "m" < 0. The last case is similar to the first.In 3+1 dimensions, when "m">0, for the third invariant, we can write, $w\; =\; ms$, where $s$ is the represents the spin, or intrinsic angular momentum. More generally, in n+1 dimensions, the generators L and C will be related, respectively, to the total angular momentum and center of mass moment by:$W\_\{ij\}\; =\; M\; S\_\{ij\}$:$L\_\{ij\}\; =\; S\_\{ij\}\; +\; X\_i\; P\_j\; -\; X\_j\; P\_i$:$C\_i\; =\; M\; X\_i\; -\; P\_i\; t$where:$P^2\; t\; =\; vec\{C\}.vec\{P\}$.

From a purely representation theoretic point of view, we'd have to study all of the representations, but we are interested in applications to quantum mechanics here. There, "E" represents the

energy , which has to be bounded from below if we require thermodynamic stability. Consider first the case where m is nonzero. If we look at the $(E,vec\{P\})$ space with the constraint:$mE=mE\_0+\{P^2\; over\; 2\},$

we find the boosts act transitively on this hypersurface. In fact, treating the energy E as the Hamiltonian, differentiating with respect to P, and applying Hamilton's equations, we obtain the mass-velocity relation $mvec\{v\}\; =\; vec\{P\}$. The hypersurface is parametrized by the velocity $vec\{v\}$.

Look at the stabilizer of a point on the orbit, ("E"

_{0}, 0), corresponding to where the velocity is 0. Because of transitivity, we know the unitary irrep contains a nontrivialsubspace with these energy-momentum eigenvalues. (This subspace only exists in arigged Hilbert space because the momentum spectrum is continuous.) It is spanned by "E", $vec\{P\}$, "M" and "L"_{"ij"}. We already know how the subspace of the irrep transforms under all but theangular momentum . Note that the rotation subgroup is Spin(3). We have to look at itsdouble cover because we're considering projective representations. This is called thelittle group , a name given byEugene Wigner . Themethod of induced representation s tells us the irrep is given by thedirect sum of all the fibers in avector bundle over the "mE" = "mE"_{0}+ "P"^{2}/2 hypersurface whose fibers are a unitary irrep of Spin(3). Spin(3) is none other than SU(2). Seerepresentation theory of SU(2) . There, it is shown the unitary irreps of SU(2) are labeled by "s", a nonnegative integer multiple of one half. This is called the spin, due to historical reasons. So, we have shown for "m" not equal to zero, the unitary irreps are classified by "m", "E"_{0}and a spin "s". Looking at the spectrum of "E", we find that if "m", the mass, is negative, the spectrum of "E" is not bounded from below. So, only the case with a positive mass is physical.Now, let's look at the case where "m" = 0. Because of unitarity,

:$mE-\{P^2\; over\; 2\}=\{-P^2\; over\; 2\}$

is nonpositive. Suppose it is zero. Here, the boosts and the rotations form the little group. So, any unitary irrep of this little group also gives rise to a projective irrep of the Galilean group. As far as we can tell, only the case which transforms trivially under the little group has any physical interpretation and it corresponds to the no particle state (vacuum).

The case where the invariant is negative requires additional comment. This corresponds to the representation class for "m" = 0 and non-zero $vec\{P\}$. Extending the

bradyon ,luxon ,tachyon classification from the representation theory of the Poincaré' group to an analogous classification, here, one may term these states as "synchrons". They represent an instantaneous transfer of non-zero momentum across a (possibly large) distance. Associated with them is a 'time' operator:$t=-\{vec\{P\}.vec\{C\}\; over\; P^2\}$which may be identified the time of transfer. These states are naturally interpreted as the carriers of instantaneous action-at-a-distance forces.In the 3+1-dimensional Galilei group, the boost generator may be decomposed into:$vec\{C\}\; =\; \{vec\{W\}\; imesvec\{P\}\; over\; P^2\}\; -\; vec\{P\}t$with $vec\{W\}$ playing a role analogous to

helicity .**ee also***

Representation theory of the Poincaré group

*Wigner's classification

*Representation theory of the diffeomorphism group

*Rotation operator

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