Representation theory of the Galilean group

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In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin as follows:

The spacetime symmetry group of nonrelativistic quantum mechanics is the Galilean group. In 3+1 dimensions, this is the subgroup of the affine group on (t,x,y,z) whose linear part leaves invariant both the metric (g_{\mu\nu}) = \mathrm{diag}(1,0,0,0) and the (independent) dual metric (g^{\mu\nu}) = \mathrm{diag}(0,1,1,1). A similar definition applies for n+1 dimensions.

We are interested in projective representations of this group, which are equivalent to unitary representations of the nontrivial central extension of the universal covering group of the Galilean group by the one-dimensional Lie group R, refer to the article Galilean group for the central extension of its Lie 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}]=i\hbar [\delta_{ik}L_{jl}-\delta_{il}L_{jk}-\delta_{jk}L_{il}+\delta_{jl}L_{ik}]
[L_{ij},P_k]=i\hbar[\delta_{ik}P_j-\delta_{jk}P_i]
[L_{ij},C_k]=i\hbar[\delta_{ik}C_j-\delta_{jk}C_i]
[C_i,E]=i\hbar P_i
[C_i,P_j]=i\hbar M\delta_{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 a Casimir invariant. The mass shell invariant

ME-{P^2\over 2}

is a second Casimir invariant. In 3+1 dimensions, a third Casimir invariant is W^2 where

\vec{W} = M \vec{L} + \vec{P}\times\vec{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 mE0 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 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 m\vec{v} = \vec{P}. The hypersurface is parametrized by the velocity \vec{v}.

Look at the stabilizer of a point on the orbit, (E0, 0), corresponding to where the velocity is 0. Because of transitivity, we know the unitary irrep contains a nontrivial linear subspace with these energy-momentum eigenvalues. (This subspace only exists in a rigged Hilbert space because the momentum spectrum is continuous.) It is spanned by E, \vec{P}, M and Lij. We already know how the subspace of the irrep transforms under all but the angular momentum. Note that the rotation subgroup is Spin(3). We have to look at its double cover because we're considering projective representations. This is called the little group, a name given by Eugene Wigner. The method of induced representations tells us the irrep is given by the direct sum of all the fibers in a vector bundle over the mE = mE0 + P2/2 hypersurface whose fibers are a unitary irrep of Spin(3). Spin(3) is none other than SU(2). See representation 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, E0 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}\times\vec{P} \over P^2} - \vec{P}t

with \vec{W} playing a role analogous to helicity.

See also[edit]

References[edit]

  • Lévy-Leblond, Jean-Marc (1967), "Nonrelativistic Particles and Wave Equations", Communications in Mathematical Physics (Springer) 6 (4): 286–311 .
  • Ballentine, Leslie E. (1998). Quantum Mechanics, A Modern Development. World Scientific Publishing Co Pte Ltd. ISBN 981-02-4105-4.