In QM one would have CCR: [Q, P] = i I, (h-bar=1)
with
[P, I] = [Q, I] = 0,
expressing the nilpotency of the Heisenberg Lie algebra;
with FCCR we have equation (2.8) with:
[G(n), Q(n)] := i 2 n^2 (n-1) X_2(n) (4.1)
= i alpha^(-2)(n) X_2(n)
[P(n), G(n)] := i 2 n^2 (n-1) X_1(n) (4.2)
= i alpha^(-2)(n) X_1(n)
[X_1(n), X_2(n)] := i alpha(n) X_3(n) (4.3)
with the definition
alpha(n) := (n sqrt(2(n-1)))^(-1) (4.4)
(so alpha(n) goes to zero as n^(-3/2) when n->infinity)
and with Einstein summation convention, permutation symbol,
and Kronecker delta,
1
[X_i(n), X_j(n)] = i -------------- epsilon_(ijk) X_k(n) (4.5)
n sqrt(2(n-1))
= i alpha(n) epsilon_(ijk) X_k(n)
{X_i(n), X_j(n)} = (1/4) n^(-2) (n-1)^(-1) delta_i_j (4.6)
= (1/2) alpha^2(n) delta_i_j
3
X_1^2(n) + X_2^2(n) + X_3^2(n) = ----------- (O(n-2) + I(2)) (4.7)
8n^(2)(n-1)
= (3/4) alpha^(2)(n) (O(n-2) + I(2))
where i, j, k = 1, 2, 3.
If on this two dimensional subspace we define the normalized
Pauli spin matrices and the unit matrix:
|0 1|
sigma_1(n) = (1/2) | |
|1 0|
|0 -i|
sigma_2(n) = (1/2) | |
|i 0|
|1 0|
sigma_3(n) = (1/2) | |
|0 -1|
|1 0|
sigma_0(n) = | |
|0 1|
(4.8)
which obey
[sigma_i(n), sigma_j(n)] = i epsilon_(ijk) sigma_k(n) (4.9)
for i,j,k = 1,2,3
where epsilon_(ijk) is the standard completely antisymmetric
epsilon-density, giving the 2x2 IRREP of the algebra su(2),
which acts on the subspace of Hilb(n) spanned by |n, n-1>
and |n, n-2>, then
X_k(n) = alpha(n) sigma_k(n) (4.10)
for k = 1,2,3
and
Q(n) = Q(n-1) + (n alpha(n))^(-1) sigma_1(n)
= Q(n-1) + (n alpha(n))^(-2) X_1(n)
P(n) = P(n-1) + (n alpha(n))^(-1) sigma_2(n)
= P(n-1) + (n alpha(n))^(-2) X_2(n)
G(n) = G(n-1) + (n alpha(n))^(-2) sigma_3(n)
= G(n-1) + (n alpha(n))^(-3) X_3(n)
(4.11)
The three X_k(n) obviously close on themselves
under commutation forming an su(2) algebra, equation
(4.5).
The scale factor alpha(n) of the structure constants
goes to zero as n -> infinity,
as alpha(n) is of the order of n^(-3/2).
If they are further commuted with Q(n), P(n)
and G(n), the commutators close with the X_k(n)
to span an su(3) algebra in its defing representation.
For the su(3) algebra, explicitly take algebra generators:
|0 1 0| |0 0 1|
lambda_1(n) = |1 0 0| lambda_4(n) = |0 0 0|
|0 0 0| |1 0 0|
|0 -i 0| |0 0 -i|
lambda_2(n) = |i 0 0| lambda_5(n) = |0 0 0|
|0 0 0| |i 0 0|
|1 0 0| |0 0 0|
lambda_3(n) = |0 -1 0| lambda_6(n) = |0 0 1|
|0 0 0| |0 1 0|
|0 0 0| |0 0 0|
lambda_7(n) = |0 0 -i| lambda_8(n) = |0 1 0|
|0 i 0| |0 0 -1|
then
sqrt(n-2)
[Q(n), X_1(n)] = +i ------------ lambda_5(n)
2n sqrt(n-1)
1 sqrt(n-2)
[Q(n), X_2(n)] = +i --- lambda_8(n) - i ------------ lambda_4(n)
n 2n sqrt(n-1)
1 sqrt(n-2)
[Q(n), X_3(n)] = -i --- lambda_7(n) + i ------------ lambda_2(n)
n 2n sqrt(n-1)
1 sqrt(n-2)
[P(n), X_1(n)] = -i --- lambda_8(n) - i ------------ lambda_4(n)
n 2n sqrt(n-1)
sqrt(n-2)
[P(n), X_2(n)] = -i ------------ lambda_5(n)
2n sqrt(n-1)
1 sqrt(n-2)
[P(n), X_3(n)] = +i --- lambda_6(n) - i ------------ lambda_1(n)
n 2n sqrt(n-1)
1
[G(n), X_1(n)] = -i ------------- lambda_7(n) = +i n X_2(n)
sqrt(2 (n-1))
1
[G(n), X_2(n)] = -i ------------- lambda_6(n) = -i n X_1(n)
sqrt(2 (n-1))
[G(n), X_3(n)] = 0
Repeating the procedure generates su(4). An extrapolation suggests that the ultimate closure is on the algebra su(n) in its defining representation. This "Aufbau" of su(m) terminating in su(n) by successive commutation produces a subalgebra chain, with containment:
su(2) < su(3) < ... < su(n-1) < su(n)
It can be seen that FCCR (2.8) can be represented in terms of the generators of the su(n) algebra. The commutator of either Q(n) or P(n) with any of these basis generators does not not vanish. For n > 1, su(n) is simple, that is, does not contain any invariant Lie subalgebra.
<Theorem 4.1>:
Q(n) and P(n) generate by commutation, the defining
representation of the Lie algebra su(n), and the operators
for k = 0, 1, 2, ..., n-1,
h_k(n) = -(1/n) C!^(k+1)(n) G(n) C^(k+1)(n)
so that
Tr( h_k)(n) ) = 0
form a Cartan subalgebra Cf.
[Appendix B].
The real polynomials in Q(n) and P(n) are a nonfaithful representation
of the universal enveloping algebra of su(n).
Proof:The h_k(n) are all diagonal, and since their traces vanish, there are n-1 linearly independent h_k(n). For the cyclic operator C(n)
C^n(n) = I(n) = C^0(n)
C^(n-k)(n) = I(n) = C^k(n)
Define,
B_k(n) = (1//n) C!^(k+1)(n) B(n) C^(k+1)(n)
so
B_k!(n) = (1//n) C!^(k+1)(n) B(n) C^(k+1)(n)
then since,
[B!(n), B(n)] = -G(n)
it follows that
[B!_k(n), B_k(n)] = h_k(n)
and of course
[h_k(n), h_j(n)] = 0
and
n n-1
SIGMA h_k(n) = SIGMA h_k(n) = 0
k=1 k=0
Q(n) and P(n) are both Hermitean and of trace zero, so their
commutator, -i [Q(n), P(n)]
and subsequent commutators of commutators formed in this fashion will
also be Hermitean and of trace zero.
From the expansions of Q(n) and P(n) as linear
combinations of the su(n) generators
given in
[Appendix B]
it can be seen that the only
matrices that can commute with both Q(n) and P(n) are multiples of the
identity.
By Schur's lemma, the algebra that Q(n) and P(n) generate
is irreducible.
It is then the full complement of nxn Hermitean
matrices of trace zero, that is, the Lie algebra su(n).
Email me, Bill Hammel at: