(n-1)
S_3(n) = ----- I(n) - N(n) (14.1)
2
= (1/2)( OMEGA(n) - N(n) )
(+|-)S_1(n)
= (1/2)(B!(n) OMEGA_(+|-)^(1/2)(n) + OMEGA_(+|-)^(1/2)(n) B(n))
(14.2)
(+|-)S_2(n)
= (i/2)(B!(n) OMEGA_(+|-)^(1/2)(n) - OMEGA_(+|-)^(1/2)(n) B(n))
(14.3)
where as in [Section VII],
OMEGA(n) := (n-1) I(n) - N(n) (14.4a)
= C(n) UPSILON!(n) F_-(n) UPSILON(n) C!(n)
= C(n) UPSILON(n) F_+(n) UPSILON!(n) C!(n)
The choice of square root sign is a chirality choice.
We use the signless notation for the positive sign:
OMEGA^(1/2)(n) := OMEGA_+^(1/2)(n) (14.4b)
OMEGA(n) is diagonal and equal to N(n) turned upside down; as N(n),
it is not invertible for any n >= 2, since one of its eigenvalues
is always zero. Numerically, on |n, k>, with '/' indicating a
a square root,
| 0 /(n-1)/1 0 0 ... 0 |
|/(n-1)/1 0 /(n-2)/2 0 ... 0 |
| 0 /(n-2)/2 0 /(n-3)/3 ... 0 |
S_1(n) = (1/2)| 0 ...................................... 0 |
| 0 ... 0 /2/(n-2) 0 |
| 0 ... /2/(n-2) 0 /1/(n-1)|
| 0 ... 0 /1/(n-1) 0 |
| 0 -i/(n-1)/1 0 0 ... 0 |
|i/(n-1)/1 0 -i/(n-2)/2 0 ... 0 |
| 0 i/(n-2)/2 0 -i/(n-3)/3 ... 0 |
S_2(n) = (1/2)| 0 ..................................... 0 |
| 0 ... 0 -i/2/(n-2) 0 |
| 0 ... i/2/(n-2) 0 -i/1/(n-1)|
| 0 ... 0 i/1/(n-1) 0 |
| (n-1)/2 - 0 0 0 |
| 0 (n-1)/2 - 1 0 ... 0 |
| 0 0 (n-1)/2 - 3 0 ... 0 |
S_3(n) = | 0 ... 0 |
| 0 ... (n-1)/2 - (n-3) 0 0 0 |
| 0 ... (n-1)/2 - (n-2) 0 |
| 0 ... (n-1)/2 - (n-1) |
So that on the |n, k> basis
S_3(n) |n, k> = [(n-1)/2 - k] |n, k>
Then the commutation relation for the algebra su(2) is obeyed:
[S_a(n), S_b(n)] = i epsilon_(abc) S_c(n) (14.5)
for every n, determining the n-dimensional UIRREP of SU(2) by the
exponential map. The nonhermitean n-dimensional su(1, 1) algebra with
J_1(n) := S_1(n)
i J_2(n) := S_2(n) (14.6)
J_3(n) := S_3(n)
has commutation relations
[J_1(n), J_2(n)] = +J_3(n), [J_2(n), J_3(n)] = +J_1(n)
[J_3(n), J_1(n)] = -J_2(n)
(14.7)
The single Casimir invariants are
S_1^2(n) + S_2^2(n) + S_3^2(n) = (n^2 - 1)/4 I(n) (14.8a)
J_1^2(n) - J_2^2(n) + J_3^2(n) = (n^2 - 1)/4 I(n) (14.8b)
Define also the standard raising and lowering operators
S_(+|-)(n) := S_1(n) (+|-) i S_2(n) (14.9a)
S_1(n) = (1/2)(S_+(n) + S_-(n)) (14.9b)
iS_2(n) = (1/2)(S_+(n) - S_-(n)) (14.9c)
S_+(n) = OMEGA^(1/2)(n) B(n) (14.10a)
S_-(n) = B!(n) OMEGA^(1/2)(n) (14.10b)
implying
S_+(n) S_-(n) = OMEGA(n) M(n) (14.10c)
S_-(n) S_+(n) = OMEGA_1(n) N(n) (14.10d)
where OMEGA_1(n) is defined in equation (14.13).
So on the |n, k> basis
S_+(n) |n, k> = sqrt(n-k)sqrt(k) |n, k+1> (14.11a)
S_-(n) |n, k> = sqrt(n-1-k)sqrt(k+1) |n, k-1> (14.11b)
Interestingly, one cannot reconstruct B(n) and B!(n) and then Q(n) and
P(n) directly from the S_a(n) since OMEGA(n) is not invertible.
Yet since
[N(n), B(n)] = -B(n)
[N(n), B!(n)] = +B!(n)
so
[OMEGA(n), B(n)] = +B(n) (14.12a)
[OMEGA(n), B!(n)] = -B!(n) (14.12b)
[S_+(n), S_-(n)] = OMEGA(n) - N(n) (14.12c)
we can then express the S_a(n) in terms of the operator OMEGA_1(n),
which is invertible on Hilb(n)
OMEGA_1(n) := OMEGA(n) + I(n) (14.13)
Then,
S_1(n) =
(1/2) ( OMEGA_1^(1/2)(n) B!(n) + B(n) OMEGA_1^(1/2)(n) ) (14.14a)
S_2(n) =
(i/2) ( OMEGA_1^(1/2)(n) B!(n) - B(n) OMEGA_1^(1/2)(n) ) (14.14b)
S_3(n) = (1/2) ( OMEGA_1(n) - h(n) ) (14.14c)
where h(n) is defined by equation (2.14),,
h(n) := N(n) + I(n)
which can be inverted to express
B(n) = S_+(n) OMEGA_1^(-1/2)(n) (14.15a)
B!(n) = OMEGA_1^(-1/2)(n) S_-(n) (14.15b)
with
S_+(n) = B(n) OMEGA_1^(1/2)(n) (14.16a)
S_-(n) = OMEGA_1^(1/2)(n) B!(n) (14.16b)
Then also expressing Q(n) and P(n) by:
Q(n) = (1/sqrt(2)) {OMEGA_1^(-1/2)(n), S_1(n)}
+ (i/sqrt(2)) [S_2(n), OMEGA_1^(-1/2)(n)] (14.16c)
P(n) = (i/sqrt(2) [OMEGA_1^(-1/2)(n), S_1(n)]
+ (1/sqrt(2){S_2(n), OMEGA_1^(-1/2)(n)} (14.16d)
Similarly,
S_1(n) = (2)^(-3/2) ( {OMEGA_1^(1/2)(n), Q(n)}
+ i [P(n), OMEGA_1^(1/2)(n)] )
S_2(n) = i(2)^(-3/2) ( [OMEGA_1^(1/2)(n), Q(n)]
- i {P(n), OMEGA_1^(1/2)(n)} )
S_3(n) = (n-1)I(n-1) - (1/2)( Q^2(n) - P^2(n) )
With the nesting of Hilb(n-1) in Hilb(n)
OMEGA_1(n-1) = OMEGA(n) (14.17)
Passage to the complexified Lie algebra for the IRREP of either su(2) or su(1, 1), gives a finite dimensional nonhermitean representation of sl(2, C). For sl(2, C), there is a unique, up to equivalence, IRREP of dimension n, where the algebra is considered as a three dimensional algebra over the complex field. It can be obtained as the differentiated version of the group representation on holomorphic polynomials in two complex, variables of homogeneous degree n. Thus obtaining the IRREP from the defining rep of U(n). The IRREPS of U(n) can be obtained using holomorphic polynomials in n complex variables of homogeneous degree k. [Knapp 1988], p. 120.
Now look at things from the viewpoint of the irreducible finite dimensional representations of SL(2, C). That group has two Casimir invariants. So the Irreducible representations are labeled by two integers. Refer to such a matrix representation as D^(k/2, j/2). If an IRREP of SL(2, C) is restricted to unitary matrices, a representation of SU(2) results. The IRREPS of SU(2) are equivalent to D^(k/2, 0). But the representation of SU(2) obtained by a unitary restriction of D^(k/2, j/2) is not irreducible; it is equivalent to a product of two IRREPS of SU(2), D^(k/2) DIRECT-PRODUCT D^(k/2). The usual Clebsch-Gordon reduction then shows that the direct product is unitarily equivalent to a direct sum of SU(2) UIRREPS:
(j+k)/2
D^(j/2) DIRECT-PRODUCT D^((k/2) = DIRECT-SUM D^(J)
J = |j-k|/2
with J integral valued.
In the direct product space (Hilb(n) DIRECT-PRODUCT Hilb(m))
construct the direct product of the n and the m dimensional FCCR, so:
[B(n) DIRECT-PRODUCT I(m), B!(n) DIRECT-PRODUCT I(m)]
= G(n) DIRECT-PRODUCT I(m)
[I(n) DIRECT-PRODUCT B(m), I(n) DIRECT-PRODUCT B!(m)]
= I(n) DIRECT-PRODUCT G(m)
From the these operators form the corresponding IRREPS of su(2).
Call these S_a(n), and K_a(m) respectively. Then,
[S_a(n), S_b(n)] = i epsilon^(abc) S_c(n)
[K_a(m), K_b(m)] = i epsilon^(abc) K_c(m)
[S_i(n), K_j(m)] = 0
Define the skew Hermitean operators M_k,
and the Hermitean N_k by E.g. [Schweber 1961], p. 41,
i M_a = K_a + S_a
N_a = K_a - S_a
so the inverse transformation is
K_a = (i/2) (M_a - i N_a)
S_a = (i/2) (M_a + i N_a)
Then
[M_a, M_b] = +epsilon^(abc) M_c
[N_a, N_b] = -epsilon^(abc) M_c
[M_a, N_b] = +epsilon^(abc) N_c
exhibiting the form of a Cartan canonical decomposition.
[appendix B]
These operators constitute an (nm)-dimensional representation of the Lie algebra sl(2, C) which is the Lie algebra of SL(2, C) the simply connected covering group of the proper orthochronous Lorentz group. The operators, therefore, yield a representation also of the proper orthochronous Lorentz group. The M_a are the generators of pure spatial rotations and the N_a are the generators of the hyperbolic rotations in the 2-planes formed with the time axis. With the index assignments determined by:
A_21 = M_1, A_02 = M_2, A_10 = M_3,
A_30 = N_1, A_31 = N_2, A_32 = N_3
and
A_ab := - A_ba
so 0, 1, 2 are the spatial indicies and 4 is the time index,
the preceding commutation relations can be written as:
[A_ab, A_st] =
eta_as A_bt + eta_bt A_as - eta_bs A_at - eta_at A_bs
where eta_ab is the Minkowski metric tensor Diag[+1, +1, +1, -1].
(1/2) A_ab A^ab = M^2 - N^2
and using the standard four dimensional epsilon density
(1/8) epsilon_abst A_ab A^st = M DOT N
where M and N are the 3-vectors of operators defined by
M := (M_1, M_2, M_3)
N := (N_1, N_2, N_3)
In terms of the S_a(n), and K_a(m) we can write out the usual specifics
of the sl(2, C) representation.
Let
|n, k; m, j> := |n, k> DIRECT-PRODUCT |m, j>
These span the direct product representation space. Then with
S_(+|-)(n), defined as before in (14.9) and K_(+|-)(m), defined similarly
S_+(n)|n, k; m, j> = sqrt(n-k) sqrt(k) |n, k-1; m, j>
S_-(n)|n, k; m, j> = sqrt(n-1-k) sqrt(k+1) |n, k+1; m, j>
S_3(n)|n, k; m, j> = [(n-1)/2 - k] |n, k; m, j>
K_+(n)|n, k; m, j> = sqrt(m-j) sqrt(j) |n, k; m, j-1>
K_-(n)|n, k; m, j> = sqrt(m-1-j) sqrt(j+1) |n, k; m, j+1>
K_3(n)|n, k; m, j> = [(m-1)/2 - j] |n, k; m, j>
In Hilb(n), the S_a(n) represent su(2) associated with a spin j=(n-1)/2.
The usual spin state label m goes from +j to -j in integer decrements.
Our k goes from 0 to n-1, so k = j - m = (n-1)/2 - m.
For the usual spin basis with "origin at the center":
S_(+|-)(n)|n, k; m, j> =
sqrt[(k -|+ n)(k (+|-) n + 1)] |n, k(+|-)1; m, j>
S_3(n)|n, k; m, j> = k |n, k; m, j>
K_(+|-)(n)|n, k; m, j> =
sqrt([(j -|+ m)(j (+|-) m + 1)] |n, k; m, j(+|-)1>
K_3(n)|n, k; m, j> = j |n, k; m, j>
In an irreducible representation, the values of the Casimir
operators are:
M^2 - N^2 = (1 - (n^2 + m^2)/2) I(mn)
M DOT N = (n^2 - m^2)/4 I(mn)
If we simply take the S_a and K_a to be
S_a = S_a(n)
K_a = -iS_a(n)
they satisfy the commutation relations
[S_a(n), S_b(n)] = +i epsilon^(abc) S_c(n)
[K_a(m), K_b(m)] = +1 epsilon^(abc) K_c(m)
[S_a(n), K_b(m)] = 0
where
i M_a = K_a + S_a
N_a = K_a - S_a
M_a = -(i + 1) S_a
N_a = -(i - 1) S_a
and the Casimir operators have the values
M^2 - N^2 = 0
M DOT N = -i(n^2 - 1)/2 I(n)
To complete the picture of all the irreducible representations
of the restricted Lorentz group see [Naimark 1957].
It is not unreasonable to claim that the essential content of the principle of relativistic invariance is contained in the representations of Lorentz group. Representations of the covering group SL(2, C) is necessary if known half-odd integral spin is to be possible. The requirement of full Poincare invariance in order for a theory to dubbed relativistic may be too much to impose: in the large, Poincare invariance is violated in the context of spacetime manifold curvature of General Relativity, and in the small, by expected curvature fluctuations that are the result of locally high energy densities. There is an extensive regime within which Poincare invariance is a good approximation, but it certainly can not be expected to be good in a theoretical context where the space and time quantities themselves may already be discrete. In view of the above, it is of interest to carry out the calculation of the commutators of the S_a(n), with Q(n) and P(n), then similarly for the direct product.
By direct calculation:
note that
G(n) OMEGA(n) = OMEGA(n),
and consequently that
G(n) OMEGA_1(n) = OMEGA_1(n) + G(n) - I(n),
and then further:
[OMEGA^(1/2)(n), B(n)] = + (OMEGA^(1/2)(n) - OMEGA^(1/2)(n-1) ) B(n)
= + (OMEGA^(1/2)(n) - C(n) OMEGA^(1/2)(n) C!(n)) B(n)
[OMEGA^(1/2)(n), B!(n)] = + B!(n) (OMEGA^(1/2)(n) - OMEGA^(1/2)(n-1) )
= - B!(n) (OMEGA^(1/2)(n) - C!(n) OMEGA^(1/2)(n) C(n))
[OMEGA^(1/2)(n-k), B(n)] = + (OMEGA^(1/2)(n-k) - OMEGA^(1/2)(n-k-1) ) B(n)
= + (OMEGA^(1/2)(n-k) - OMEGA^(1/2)(n-k-1) ) B(n-k-1)
[OMEGA^(1/2)(n-k), B!(n)] =
+ B!(n) (OMEGA^(1/2)(n-k) - OMEGA^(1/2)(n-k-1) )
= + B!(n-k-1) (OMEGA^(1/2)(n-k) - OMEGA^(1/2)(n-k-1) )
[OMEGA_1^(1/2)(n), B(n)] = + (OMEGA_1^(1/2)(n) - OMEGA^(1/2)(n) ) B(n)
[OMEGA_1^(1/2)(n), B!(n)] = - B!(n) (OMEGA_1^(1/2)(n) - OMEGA^(1/2)(n) )
[OMEGA_1^(1/2)(n), B^2(n)] =
+ (3 OMEGA_1^(1/2)(n) - 2 OMEGA^(1/2(n) - OMEGA^(1/2)(n-1) ) B^2(n) =
+ (3 OMEGA_1^(1/2)(n) - 2 OMEGA_1^(1/2)(n-1) - OMEGA_1^(1/2)(n-2) ) B^2(n)
[OMEGA_1^(1/2)(n), B!^2(n)] =
- B!^2(n) (3 OMEGA_1^(1/2)(n) - 2 OMEGA^(1/2)(n) - OMEGA^(1/2)(n-1) ) =
- B!^2(n) (3 OMEGA_1^(1/2)(n) - 2 OMEGA_1^(1/2)(n-1) - OMEGA_1^(1/2)(n-2) )
[S_3(n), B!(n)] = -B!(n)
[S_+(n), B!(n)] = -( N(n) OMEGA_1^(1/2) - M(n) OMEGA^(1/2) )
[S_-(n), B!(n)] = OMEGA_1^(1/2) B!^2(n) - B!^2(n) OMEGA^(1/2)
[S_3(n), B(n)] = +B(n)
[S_+(n), B(n)] = OMEGA^(1/2) B^2(n) - B^2(n) OMEGA_1^(1/2)
[S_-(n), B(n)] = +( N(n) OMEGA_1^(1/2) - M(n) OMEGA^(1/2) )
[S_1(n), B(n)] = +(1/2)OMEGA^(1/2) ( B^2(n) - M(n) )
-(1/2)( B^2(n) - N(n) ) OMEGA_1^(1/2)
=
[S_2(n), B(n)] = -(i/2)OMEGA^(1/2) ( B^2(n) + M(n) )
+(i/2)( B^2(n) + N(n) ) OMEGA_1^(1/2)
=
[S_1(n), B!(n)] = -(1/2) ( B!^2(n) - M(n) )OMEGA^(1/2)
+(1/2) OMEGA_1^(1/2)( B!^2(n) - N(n) )
=
[S_2(n), B!(n)] = +(i/2) ( B!^2(n) + M(n) ) OMEGA^(1/2)
-(i/2) OMEGA_1^(1/2) ( B!^2(n) + N(n) )
=
and
[S_+(n), Q(n)] =
+ (1/sqrt(2)) OMEGA^(1/2) (B^2(n) + M(n))
- (1/sqrt(2)) (B^2(n) + N(n)) OMEGA_1^(1/2)
[S_-(n), Q(n)] =
+ (1/sqrt(2)) OMEGA_1^(1/2) (B!^2(n) + N(n))
- (1/sqrt(2)) (B!^2(n) + M(n)) OMEGA^(1/2)
[S_+(n), P(n)] =
+ (i/sqrt(2)) (B^2(n) - N(n)) OMEGA_1^(1/2)
- (i/sqrt(2)) OMEGA^(1/2) (B^2(n) - M(n))
[S_-(n), P(n)] =
+ (i/sqrt(2)) OMEGA_1^(1/2) (B!^2(n) - N(n))
- (1/sqrt(2)) (B!^2(n) - M(n)) OMEGA^(1/2)
[S_1(n), Q(n)] =
+ (1/sqrt(2)) ( OMEGA^(1/2) Y_1(n) - Y_1(n) OMEGA_1^(1/2) )
- (1/sqrt(2)) Y_3(n) ( OMEGA^(1/2) - OMEGA_1^(1/2) )
[S_2(n), Q(n)] = ???
[S_3(n), Q(n)] = +i P(n)
[S_1(n), P(n)] = ???
[S_2(n), P(n)] = ???
[S_3(n), P(n)] = -i Q(n)
[S_1(n), G(n)] = +i 2n sigma_2(n)
= +i 2sqrt(2) n^2 sqrt(n-1) X_2(n)
[S_2(n), G(n)] = +i 2n sigma_1(n)
= +i 2sqrt(2) n^2 sqrt(n-1) X_1(n)
[S_3(n), G(n)] = 0
where we have defined the bilinears Cf. [Section V]:
Y_1(n) := (1/2) (Q^2(n) - P^2(n))
= (1/2) (B!^2(n) + B^2(n))
= Q^2(n) - N(n) - (1/2)G(n)
= Q^2(n) - Y_3(n)
Y_2(n) := (1/2) {Q(n), P(n)}
= (i/2) (B!^2(n) - B^2(n))
= Q(n)P(n) - (i/2)G(n)
= P(n)Q(n) + (i/2)G(n)
Y_3(n) := (1/2) (Q^2(n) + P^2(n))
= (1/2) {B!(n), B(n)}
= (1/2) (N(n) + M(n))
= N(n) + (1/2)G(n)
= M(n) - (1/2)G(n)
[S_+(n), Y_1(n)] = ???
[S_+(n), Y_2(n)] = ???
[S_+(n), Y_3(n)] = ???
[S_-(n), Y_1(n)] = ???
[S_-(n), Y_2(n)] = ???
[S_-(n), Y_3(n)] = ???
[S_1(n), Y_1(n)] = ???
[S_1(n), Y_2(n)] = ???
[S_1(n), Y_3(n)] = ???
[S_2(n), Y_1(n)] = ???
[S_2(n), Y_2(n)] = ???
[S_2(n), Y_3(n)] = ???
[S_3(n), Y_1(n)] = +2i Y_2(n)
[S_3(n), Y_2(n)] = -2i Y_1(n)
[S_3(n), Y_3(n)] = 0
[Y_1(n), B(n)] = +(1/2) (I(n) + G(n)) B!(n)
[Y_1(n), B!(n)] = +(1/2) B(n) (I(n) + G(n))
[Y_1(n), Q(n)] = +(i/2) {P(n), G(n)}
[Y_1(n), P(n)] = +(i/2) {Q(n), G(n)}
[Y_1(n), G(n)] =
+(i/2) n^2(n-1) ( {Q(n), X_2(n)} + {P(n), X_1(n)} )
[Y_2(n), B(n)] = -(i/2) (I(n) + G(n)) B!(n)
[Y_2(n), B!(n)] = -(i/2) B(n) (I(n) + G(n))
[Y_2(n), Q(n)] = -(i/2) {Q(n), G(n)}
[Y_2(n), P(n)] = +(i/2) {P(n), G(n)}
[Y_2(n), G(n)] =
-(i/2) n^(2(n-1) ( {Q(n), X_1(n)} - {P(n), X_2(n)} )
Analog commutators for the dynamical equations of an oscillator
[Y_3(n), B(n)] = -(1/2) {B(n), G(n)}
[Y_3(n), B!(n)] = +(1/2) {B!(n), G(n)}
[Y_3(n), Q(n)] = -i P(n) -i (1/2) n^(2(n-1) X_2(n)
= -i P(n) -i (n/2) (P(n) - P(n-1))
[Y_3(n), P(n)] = +i Q(n) -i (1/2) n^(2(n-1) X_1(n)
= +i Q(n) -i (n/2) (Q(n) - Q(n-1))
[Y_3(n), G(n)] = 0
See equations (4.11).
For all n, the su(2) algebra representation is not real; in particular,
the representations for n odd are not real.
These are the representations of the real Lie algebra so(3).
The generated rotations cannot then act directly on the Cartesian components
of Euclidean tensors; they must act instead on certain complex linear
combinations of them.
For n=3, the vector representation,
| 0 sqrt(2) 0 | |0 1 0|
S_1(3) = (1/2) |sqrt(2) 0 sqrt(2) | = (1/sqrt(2)) |1 0 1|
| 0 sqrt(2) 0 | |0 1 0|
| 0 -sqrt(2) 0 | | 0 -i 0|
S_2(3) = (i/2) |+sqrt(2) 0 -sqrt(2) | = (1/sqrt(2)) |+i 0 -i|
| 0 +sqrt(2) 0 | | 0 +i 0|
|+1 0 0 |
S_3(3) = | 0 0 0 |
| 0 0 -1 |
while the representation that generates the rotations of
the Cartesian components of a vector is written
| 0 0 0 |
l_1(3) = | 0 0 1 |
| 0 -1 0 |
| 0 0 1 |
l_2(3) = | 0 0 0 |
|-1 0 0 |
| 0 +1 0 |
l_3(3) = |-1 0 0 |
| 0 0 0 |
The l_a(3) are skewsymmetric matrices generating
real orthogonal transformations
exp( theta_a l_a(3) )
for real parameters theta_a.
Define for any n, rotation operators in the |q(n, k)> basis
by equality of the matrix elements
<n, k| S_a(n) |n, j> := <q(n, k)| S^(XI)_a(n) |q(n, j)>
so
S_a(n) = XI(n) S^(XI)_a(n) XI!(n)
This definition is specifically made so that
[Q(n), S^(XI)_3(n)] = 0
and so S^(XI)_3(n) is then the generator of rotations
about the Q-axis. Using the S^(XI)_a(n) as generators
of spatial rotations, the operators associated with two other axes
can be defined. Alleged position operators defined in this way
clearly behave more like angular momentum operators.
Elementary quantized spacetime can then to be thought of more
in terms of a local,
noncommutative geometry arising from quantized directions
than from a notion of continuous extension.
The idea that this is conceptually
appropriate has been discovered before [Penrose 1972].
From the su(2) algebra commutation relations (14.5)
and the BCH formula
(7.3),
one can prove:
exp[ -i(pi/2) S_3(n) ] S_1(n) exp[ +i(pi/2) S_3(n) ] = +S_2(n)
exp[ -i(pi/2) S_3(n) ] S_2(n) exp[ +i(pi/2) S_3(n) ] = -S_1(n)
exp[ -i(pi/2) S_2(n) ] S_3(n) exp[ +i(pi/2) S_2(n) ] = +S_1(n)
exp[ -i(pi/2) S_2(n) ] S_1(n) exp[ +i(pi/2) S_2(n) ] = -S_3(n)
exp[ -i(pi/2) S_1(n) ] S_2(n) exp[ +i(pi/2) S_1(n) ] = +S_3(n)
exp[ -i(pi/2) S_1(n) ] S_3(n) exp[ +i(pi/2) S_1(n) ] = -S_2(n)
and then by exponentiating,
exp[-i(pi/2) S_3(n)] exp[+ialpha S_1(n)] exp[+i(pi/2) S_3(n)]
= exp[+ialpha S_2(n)]
exp[-i(pi/2) S_3(n)] exp[+ialpha S_2(n)] exp[+i(pi/2) S_3(n)]
= exp[-ialpha S_1(n)]
exp[-i(pi/2) S_2(n)] exp[+ialpha S_3(n)] exp[+i(pi/2) S_2(n)]
= exp[+ialpha S_1(n)]
exp[-i(pi/2) S_2(n)] exp[+ialpha S_1(n)] exp[+i(pi/2) S_2(n)]
= exp[-ialpha S_3(n)]
exp[-i(pi/2) S_1(n)] exp[+ialpha S_2(n)] exp[+i(pi/2) S_1(n)]
= exp[+ialpha S_3(n)]
exp[-i(pi/2) S_1(n)] exp[+ialpha S_3(n)] exp[+i(pi/2) S_1(n)]
= exp[-ialpha S_2(n)]
which relations are true in particular for alpha = (+|-) pi/2.
Note that S_3(n) and OMEGA(n) are closely related
to the Fourier transform Fr(n), (7.6).
exp( -i (pi/2) S_3(n) ) = e^(-i pi(n-1)/4) Fr(n)
= i^(-(n-1)/2) Fr(n)
exp( -i pi/2 OMEGA(n) ) = (-i)^n Fr(n)
Particularly then we have the relations,
Fr(n) S_1(n) Fr!(n) = + S_2(n)
Fr(n) S_2(n) Fr!(n) = - S_1(n)
and recalling equations (7.7),
Fr(n) Q(n) Fr!(n) = + P(n)
Fr(n) P(n) Fr!(n) = - Q(n)
So under rotations generated by S_3(n) P(n) and Q(n)
behave like S_1(n) and S_2(n) respectively.
The group of rotations generated by the three S_a(n),
appears to be more related to Fourier-like rotations than spatial
rotations.
We can exploit this situation to define spatially rotated
operators, nonetheless.
Let the eigenbasis of each of the S_a(n) be notationally
defined by the eigenvalue equations
S_a(n) |s_a(n, k)> = s(n, k) |s_a(n, k)>
where
s_a(n, k) = s(n, k)
for a = 1, 2, 3. Then identically,
|s_3(n, k)> = |n, k>
and
s(n, k) = (1/2)(n-1) - k
Then the matrix elements numerically obey
<n, k| S_3(n) |n, j>
= <s_a(n, k)| S_a(n) |s_a(n, j)>
= delta_kj s(n, k)
In each of the bases labeled by index "a", set up a triple of operators
by numerical equivalence of matrix elements:
<n, k| Q(n) |n, j> =
<s_a(n, k)| Q_a(n) |s_a(n, j)>
<n, k| P(n) |n, j> =
<s_a(n, k)| P_a(n) |s_a(n, j)>
<n, k| G(n) |n, j> =
<s_a(n, k)| G_a(n) |s_a(n, j)>
so that
[Q_a(n), P_a(n)] = i G_a(n)
is obeyed in |s_a(n, k)>, hence in all bases.
Note the distinction between a true Fourier transform which is
idempotent of order 4, and the exponentiated rotation generators below
with angles pi/2.
| + I(n) for n odd
[ exp( -i pi/2 S_a(n) ) ]^4 = |
| - I(n) for n even
[ exp( -i pi/2 S_a(n) ) ]^8 = I(n)
calling to mind the octonian Cayley algebra, and the mod 8
theorem in the structure theory of Clifford algebras.
For n odd, the operators within the faithful representations
of SO(3) are idempotent of order 4; but those within the double
valued spin representations of SO(3), for n even, the
operators are idempotent of order 8.
However, the factor exp(i pi(n-1)/4) which
automatically appears
in exponentiating equation
(14.1) compensates so that that the Fourier
transforms
Fr_a(n) := exp(i pi(n-1)/4) exp( -i pi/2 S_a(n) )
with
Fr_3(n) = Fr(n)
are in fact idempotent of order 4 for both n odd and even.
If the compensating phase factors were omitted from the definitions,
there would not be any effect of the double valuedness in a similarity
transformation, since the two negative factors cancel.
The only difference in effect between n odd and even would be in the
Fourier transform of a vector, which would appear as a phase difference.
Then for a = 1, 2, 3,
Fr_a(n) Q_a(n) Fr_a!(n) = + P_a(n)
Fr_a(n) P_a(n) Fr_a!(n) = - Q_a(n)
Fr_a^2(n) Q_a(n) Fr_a^2!(n) = - Q_a(n)
Fr_a^2(n) P_a(n) Fr_a^2!(n) = - P_a(n)
Fr_a^4(n) = I(n)
Fr_a!(n) Fr_a(n) = I(n)
[Fr_a(n), G_a(n)] = 0
which can be verified with aid of the following formulas, and
their conjugates.
With the Fr_a(n) so defined, the relations
Fr_3(n) Fr_1(n) Fr_3!(n) = Fr_2(n)
Fr_3(n) Fr_2(n) Fr_3!(n) = Fr_1!(n)
Fr_2(n) Fr_3(n) Fr_2!(n) = Fr_1(n)
Fr_2(n) Fr_1(n) Fr_2!(n) = Fr_3!(n)
Fr_1(n) Fr_2(n) Fr_1!(n) = Fr_3(n)
Fr_1(n) Fr_3(n) Fr_1!(n) = Fr_2!(n)
can be read from the above formulae for exponentials of the S_a(n).
The abstract spatial axes are being considered to be
represented by three operators.
The same representational information is contained in the eigenvector
sets for the three operators which determine their relative orientation,
and their common spectrum.
Sp( Q_1(n) ) = Sp( Q_2(n) ) = Sp( Q_3(n) )
The information on relative orientation is also expressed by
the unitary transformations that connect the basis sets.
Define the unitary operators rotating the abstract axes,
represented by the eigenbases, into each other:
From the relations above,
exp[ +i(pi/2) S_2(n) ] S_1(n) exp[ -i(pi/2) S_2(n) ] = +S_3(n)
exp[ -i(pi/2) S_1(n) ] S_2(n) exp[ +i(pi/2) S_1(n) ] = +S_3(n)
exp[ +i(pi/2) S_3(n) ] S_2(n) exp[ -i(pi/2) S_3(n) ] = +S_1(n)
Then taking matrix elements in the eigenbasis of the operator on the
right, it is easy to see that,
exp[ -i(pi/2) S_2(n) ] |s_3(n, k)> = |s_1(n, k)>
exp[ +i(pi/2) S_1(n) ] |s_3(n, k)> = |s_2(n, k)>
exp[ -i(pi/2) S_3(n) ] |s_1(n, k)> = |s_2(n, k)>
So from the viewpoint of the |s_3(n, k)> basis
the rotated QPG triples can be written:
Q_1(n) = exp[ -i(pi/2) S_2(n) ] Q(n) exp[ +i(pi/2) S_2(n) ]
= Fr_2(n) Q(n) Fr_2!(n)
Q_2(n) = exp[ +i(pi/2) S_1(n) ] Q(n) exp[ -i(pi/2) S_1(n) ]
= Fr_1!(n) Q(n) Fr_1(n)
Q_3(n) = Q(n)
P_1(n) = exp[ -i(pi/2) S_2(n) ] P(n) exp[ +i(pi/2) S_2(n) ]
= Fr_2(n) P(n) Fr_2!(n)
P_2(n) = exp[ +i(pi/2) S_1(n) ] P(n) exp[ -i(pi/2) S_1(n) ]
= Fr_1!(n) P(n) Fr_1(n)
P_3(n) = P(n)
G_1(n) = exp[ -i(pi/2) S_2(n) ] G(n) exp[ +i(pi/2) S_2(n) ]
= Fr_2(n) G(n) Fr_2!(n)
G_2(n) = exp[ +i(pi/2) S_1(n) ] G(n) exp[ -i(pi/2) S_1(n) ]
= Fr_1!(n) G(n) Fr_1(n)
G_3(n) = G(n)
The unitary transformations that map the operator triple
(Q_3(n), P_3(n), G_3(n))
to
(Q_1(n), P_1(n), G_1(n))
and to
(Q_2(n), P_2(n), G_2(n))
are now considered as active transformations connecting different
sets of operators and not transformations of basis.
Cross-Commutators
The Q-P cross commutators can be expressed in terms of the Q-Q
commutators and the P-P commutators.
We will use use this expression later when the asymptotic forms
for the Q-Q commutators and the P-P commutators can be then be
used to provide asymptotic forms for the cross commutators.
[Q_1(n), P_2(n)] = Fr_2(n) [Q_3(n), Q_2(n)] Fr_2!(n)
[Q_2(n), P_1(n)] = Fr_2(n) [P_3(n), P_2(n)] Fr_2!(n)
[Q_1(n), P_3(n)] = Fr_1(n) [P_2(n), P_1(n)] Fr_1!(n)
[Q_3(n), P_1(n)] = Fr_1(n) [Q_2(n), Q_1(n)] Fr_1!(n)
[Q_2(n), P_3(n)] = Fr_2!(n) [P_2(n), P_1(n)] Fr_2(n)
[Q_3(n), P_2(n)] = Fr_2!(n) [Q_2(n), Q_1(n)] Fr_2(n)
The matrix with elements
<n, k| XI(n) |n, j>
determined in [Section IX] diagonalize Q(n) and determine the map
between the basis |n, k> and the eigenbasis of Q(n), |q(n, k)>,
XI(n) |q(n, k)> = |n, k>
Consider a new set of rotation generators S^(XI)_a(n)
that is designed so that S^(XI)_3(n)
commutes with Q_3(n).
This will clearly be the case if S^(XI)_3(n) is diagonal
in the |q(n, k)> basis.
Therefore define S^(XI)_3(n) by
<q(n, k)| S^(XI)_3(n) |q(n, j)> = <n, k| S_3(n) |n, j>
so
S^(XI)_3(n) = XI!(n) S_3(n) XI(n)
where XI(n) is considered as an active transformation of S_3(n).
If all three S_a(n) are transformed to define
S^(XI)_a(n) := XI!(n) S_a(n) XI(n)
then, clearly, the commutation relations
[S^(XI)_a(n), S^(XI)_b(n)] = i epsilon^(abc) S^(XI)_c(n)
are preserved and,
[S^(XI)_3(n), Q_3(n)] = 0
Now define a new set of operators
Q^(XI)_1(n) :=
exp[ -i(pi/2) S^(XI)_2(n) ] Q^(XI)_3(n) exp[ +i(pi/2) S^(XI)_2(n) ]
Q^(XI)_2(n) :=
exp[ +i(pi/2) S^(XI)_1(n) ] Q^(XI)_3(n) exp[ -i(pi/2) S^(XI)_1(n) ]
Q^(XI)_3(n) := Q_3(n)
These rotate into each other under the action of the group
generated by the S^(XI)_a(n) and,
[S^(XI)_a(n), Q^(XI)_a(n)] = 0
So S^(XI)_a(n) can then be considered to be a
generator of rotations about the axis Q^(XI)_a(n).
Similarly, define an algebra of generators for the rotations of momenta
by
S^(PI)_a(n) := PI!(n) S_a(n) PI(n)
where from section X
PI(n) = Fr(n) XI(n) Fr!(n)
is the diagonalizing transformation for P(n).
Then, again, the commutation relations
[S^(PI)_a(n), S^(PI)_b(n)] = i epsilon^(abc) S^(PI)_c(n)
are preserved and,
[S^(PI)_3(n), P_3(n)] = 0
Again define a new set of operators
P^(PI)_1(n) :=
exp[ -i( pi/2) S^(PI)_2(n) ] P^(PI)_3(n) exp[ +i( pi/2) S^(PI)_2(n) ]
P^(PI)_2(n) :=
exp[ +i( pi/2) S^(PI)_1(n) ] P^(PI)_3(n) exp[ -i( pi/2) S^(PI)_1(n) ]
P^(PI)_3(n) := P_3(n)
These rotate into each other under the action of the group
generated by the S^(PI)_a(n) and,
[S^(PI)_a(n), P^(PI)_a(n)] = 0
So S^(PI)_a(n) can then be considered to be a generator of rotations
about the axis P^(PI)_a(n).
The two sets of rotation generators S^(XI)_a(n), and S^(PI)_a(n) can be
mapped to one another with Fr_3(n)
Fr_3(n) S^(XI)_1(n) Fr_3!(n) = + S^(PI)_2(n)
Fr_3(n) S^(XI)_2(n) Fr_3!(n) = - S^(PI)_1(n)
Fr_3(n) S^(XI)_3(n) Fr_3!(n) = S^(PI)_3(n)
Or more obviously
S^(XI)_a(n) = XI!(n) PI(n) S^(PI)_a(n) PI!(n) XI(n)
Let
R_32(n, +(pi/2) ) := exp[ +i(pi/2) S_1(n) ]
R_31(n, -(pi/2) ) := exp[ -i(pi/2) S_2(n) ]
R_21(n, +(pi/2) ) := exp[ +i(pi/2) S_3(n) ]
where the axes have been assumed to form a right handed system and
rotation senses have been taken to be counterclockwise looking down positive
end of the rotation axis, thus conforming to a right and rule for rotations.
These obey:
R_32(n, +(pi/2) ) = R_32!(n, -(pi/2) )
R_31(n, -(pi/2) ) = R_31!(n, +(pi/2) )
R_21(n, +(pi/2) ) = R_21!(n, -(pi/2) )
and
R_31(n, +(pi/2) ) R_32(n, +(pi/2) ) R_31(n, -(pi/2) )
= R_21(n, +(pi/2) )
R_21(n, +(pi/2) ) R_31(n, +(pi/2) ) R_21(n, -(pi/2) )
= R_32(n, +(pi/2) )
R_32(n, +(pi/2) ) R_21(n, +(pi/2) ) R_32(n, -(pi/2) )
= R_31(n, +(pi/2) )
For any n >= 2 define in the |q(n, k)> basis, the rotated
Q-operators:
Q_1(n) := R^(XI)_31(n, -(pi/2) ) Q(n) R^(XI)_31(n, +(pi/2) )
Q_2(n) := R^(XI)_32(n, +(pi/2) ) Q(n) R^(XI)_32(n, -(pi/2) )
Q_3(n) := Q(n)
so for the generators of the Q-rotations,
[Q_1(n) S^(XI)_1(n)] = 0
[Q_2(n) S^(XI)_2(n)] = 0
[Q_3(n) S^(XI)_3(n)] = 0
Further defining the rotated P-operators:
P_1(n) := R^(XI)_31(n, -(pi/2) ) P(n) R^(XI)_32(n, +(pi/2) )
P_2(n) := R^(XI)_32(n, +(pi/2) ) P(n) R^(XI)_32(n, -(pi/2) )
P_3(n) := P(n)
and rotated G-operators
G_1(n) := R^(XI)_31(n, -(pi/2) ) G(n) R^(XI)_32(n, +(pi/2) )
G_2(n) := R^(XI)_32(n, +(pi/2) ) G(n) R^(XI)_32(n, -(pi/2) )
G_3(n) := G(n)
then applying the rotation operators to equation (2.8),
[Q_a(n), P_a(n)] = i G_a(n)
for a = 1, 2, 3.
Note that
[G_a(n) S^(XI)_a(n)] not= 0
for a = 1, 2, 3.
The eigenvalue equations can then be written:
Q_a(n) |q_a(n, k)> = q_a(n, k) |q_a(n, k)>
P_a(n) |p_a(n, k)> = p_a(n, k) |p_a(n, k)>
G_a(n) |g_a(n, k)> = g_a(n, k) |g_a(n, k)>
to establish notation.
The commutators [Q_a(n), Q_b(n)] can be
messy to calculate for general finite n, but asymptotically for large n,
a relatively simple structure arises. Looking at the commuting pair Q_3(n)
and S^(XI)_3(n) we see that S^(XI)_3(n) has a
linear symmetric spectrum and from
Theorem 9.4 that Q_3(n) also
has a linear symmetric
spectrum in asymptopia so that Q_3(n)
and S^(XI)_3(n) are asymptotically linearly related.
More specifically it is easy to see using equations
(9.22) and from
Theorem 9.4 that
asymptotically
Q_3(n) |q(n, k)> apprx= (DELTA q(n)) [(n-1)/2 - k] |q(n, k)>
and from the definition of S^(XI)_3(n),
S^(XI)_3(n) |q(n, k)> = [(n-1)/2 - k] |q(n, k)>
for k = 0, 1, 2, ..., n-1,
that Q_3(n) and S^(XI)_3(n) are asymptotically conformally related:
Q_3(n) apprx= (DELTA q(n)) S^(XI)_3(n)
with DELTAq(n) given by equation (9.22c):
DELTAq(n) = (pi)/(2 sqrt(n))
Similarly then for the three pairs Q_a(n) and S^(XI)_a(n).
Q_a(n) apprx= (DELTA q(n)) S^(XI)_a(n)
Using (14.5), we have the result stated as
<Theorem 14.1>:
For large n, asymptotically,
[S^(XI)_a(n), Q_b(n)] apprx= i (DELTA q(n)) epsilon_abc S^(XI)_c(n)
(This is the relation that is satisfied when the Q_a(n) are
components of a vector operator under rotations generated by
S^(XI)_a(n).)
or
[S^(XI)_a(n), Q_b(n)] apprx= i epsilon_(abc) Q_c(n)
and,
[Q_a(n), Q_b(n)] apprx= i (DELTA q(n)) epsilon_(abc) Q_c(n)
<Corollary 14.1.1>:
Asymptotically for large n,
[S^(PI)_a(n), P_b(n)] apprx= i (DELTA q(n)) epsilon_(abc) S^(PI)_c(n)
or
[S^(PI)_a(n), P_b(n)] apprx= i epsilon_(abc) P_c(n)
and,
[P_a(n), P_b(n)] apprx= i (DELTA p(n)) epsilon_(abc) P_c(n)
where (DELTA p(n)) is the asymptotic eigenvalue spacing for P(n),
hence for all P_a(n); and numerically,
(DELTA p(n)) = (DELTA q(n)) = (pi)/(2 sqrt(n))
<Corollary 14.1.2>:
When
[Q_a(n), Q_b(n)] apprx= i (DELTA q(n)) epsilon_(abc) Q_c(n)
and
[P_a(n), P_b(n)] apprx= i (DELTA p(n)) epsilon_(abc) P_c(n)
Q_a(n) and P_a(n) are each conformal
to distinct IRREPS of su(2) both acting on Hilb(n).
Since then the IRREPS are unitarily related and since
the the conformal factors (DELTA q(n)) and (DELTA p(n)) are
equal, there exists a unitary V(n), such that
V(n) Q_a(n) V!(n) = P_a(n)
In fact,
V(n) = XI!(n) PI(n)
= XI!(n) Fr(n) XI(n) Fr!(n)
<Theorem 14.2>:
Write the full set of commutation relations for Q_b(n),
and P_b(n) as
[Q_a(n), P_b(n)] = i G_ab(n)
so
G_aa(n) := G_a(n)
and
Tr( G_ab(n) ) = 0
for all a,b = 1,2,3, and further, asymptotically
[S^(XI)_a(n), S^(PI)_b(n)] apprx= i(4n/(pi)) G_ab(n)
For large n, using the commutator results of
Theorem 14.1,
and the previously calculated cross-commutators,
the components of G_ab(n) are asymptotically:
_______________________________________________________________________
G_a1(n) b=1 b=2 b=3
_______________________________________________________________________
|
a=1 | Fr_2(n) G(n) Fr_2!(n)
|
a=2 | -(DELTA p(n)) Fr_2(n) P_1(n) Fr_2!(n)
|
a=3 | -(DELTA q(n)) Fr_1!(n) Q_3(n) Fr_1(n)
|
_______________________________________________________________________
G_a2(n) b=1 b=2 b=3
_______________________________________________________________________
|
a=1 | -(DELTA q(n)) Fr_2(n) Q_1(n) Fr_2!(n)
|
a=2 | Fr_1!(n) G(n) Fr_1(n)
|
a=3 | -(DELTA q(n)) Fr_2!(n)Q_3(n)Fr_2(n)
|
_______________________________________________________________________
G_a3(n) b=1 b=2 b=3
_______________________________________________________________________
|
a=1 | -(DELTA p(n)) Fr_1(n) P_3(n) Fr_1!(n)
|
a=2 | -(DELTA p(n)) Fr_2!(n) P_3(n) Fr_2(n)
|
a=3 | G(n)
|
_______________________________________________________________________
Theorem 14.3:
Further asymptotics of G_ab(n)
It is not unreasonable to base a model of an isotropic
(i.e., rotationally selfsimilar) chunk
of quantized three-space on the operator set Q_a(n),
interpreting them to be the operators corresponding to the quantized
coordinate operators associated with quantized space, perhaps
the quantized interior of a black hole. In the black hole case
one must categorize and distinguish "irreducible black holes"
from "reducible black holes"; the context is now, of course,
a small part of the elusive theory of quantum gravity.
To the general end, consider the radial vector operator components and
the quadratic operator of it Euclidean magnitude,
R_Q(n) := (Q_1(n), Q_2(n), Q_3(n))
3
R_Q^2(n) = SIGMA Q_a^2(n)
a=1
The Euclidean nature of the magnitude is a consequence of defining
an SU(2)-invariant operator by using the Cartan metric of the Lie group SU(2).
Asymptotically,
R_Q^2(n) apprx= [DELTA q(n)]^2 (n^2 - 1)/4 I(n)
= ( pi^2 (4n)^(-1) (n^2 - 1)/4 ) I(n)
= ( pi^2 / 16) (n - 1/n) I(n)
apprx= ((pi^2 / 16) n ) I(n)
Consider a spatial point (necessarily fuzzed by uncertainty)
being associated with the expectation values of a radial operator,
<R_Q(n)> = (<Q_1(n)>, <Q_2(n)>, <Q_3(n)>)
Conceptually, this set of expectation values is no different from the
set of p and q expectation values given the CCR of QM, except, of course,
that the expectation values cannot be considered as taken simultaneously
in the standard sense; neither is it taken is the standard sense
in the standard construction of a "hashed phase space" For quantum
statistical mechanics.
Theorem 14.1,
the expected values of the operator squares for fixed n,
always obey asymptotically,
<Q_1^2(n)> + <Q_2^2(n)> + <Q_3^2(n)> apprx= ( pi^2 /16) (n - n^(-1))
apprx= (( pi^2 / 16) n )
and similarly,
<P_1^2(n)> + <P_2^2(n)> + <P_3^2(n)> apprx= pi^2 / 16) (n - n^(-1))
From the definition of the uncertainty
SIGMA_a (DELTA Q_a^2(n))^2
= SIGMA_a <Q_a^2(n)> - SIGMA_a <Q_a(n)>^2
apprx= (n pi^2 / 16) - SIGMA_a <Q_a(n)>^2
From the CRs of the Q_a(n),
SIGMA_a (DELTA Q_a^2(n))^2 > 0
since the uncertainty in all three Q_a(n) cannot vanish
simultaneously. Therefore,
<R_Q(n)>^2 := SIGMA_a <Q_a(n)>^2 < <R_Q^2(n)>
or
<R_Q(n)>^2 < (n pi^2 / 16)
A point on the surface of the sphere of radius (pi / 4) sqrt(n)
is not a possible expectation value for any Q_a(n) or P_a(n);
this sphere then acts like a Q-thoretical impassable membrane.
The expectation values of R_Q(n) are on a 3-dimensional ball of
asymptotic radius (pi/4)sqrt(n), then of asymptotic surface area and
volume,
S = (4 pi) (n pi^2 / 16)
= ( pi^3 n/4 )
V = ( 4/3 pi (n pi^2 / 16)^(3/2)
= ( 4/3 pi^4 / 4^3 ) n^(3/2)
= ( pi^4 / (3 * 16) ) n^(3/2)
= ( pi^4 / 48 ) n^(3/2)
respectively.
The points of the bounding sphere are not realizable for finite n.
The vector |q_a(n, 0)> is the one
associated with the maximal eigenvalue of Q_a(n).
From
<q_3(n, 0)| Q_1^2(n) |q_3(n, 0)> + <q_3(n, 0)| Q_2^2(n) |q_3(n, 0)>
= <q_3(n, 0)| R_Q^2(n) |q_3(n, 0)> - q_3^2(n, 0)
<q_3(n, 0)| Q_1^2(n) |q_3(n, 0)> + <q_3(n, 0)| Q_2^2(n) |q_3(n, 0)> =
= [DELTAq(n)]^2 [(n^2 - 1)/4 - (n-1)^2 / 4 ]
= [ pi^2 / (4 n)] [1 - 1/n]
= [ pi^2 / 4] [n - 1]
apprx= [ (pi^2 / 4) (n - 1) ], for large n
The maximal value of any <Q_1^2(n)> never quite reaches the
the defined asymptotic coordinate sphere.
If each Hilb(n) is associated with a spatial sphere of radius
(pi/4) sqrt(n),
and we want to consider a spherical subset of quantized space of radius
(pi/4) sqrt(n),
it would appear that this must correspond to considering also any smaller
spheres that can be nested within the sphere associated with Hilb(n).
With the assumption that these nested spheres can be considered independent
and noninteractive (perhaps not always valid), the full physical spatial
subset should then be represented by considering the direct sum
n
DIRECT-SUM Hilb(k)
k=2
Since
<Q_1^2(n)> + <Q_2^2(n)> + <Q_3^2(n)> - ( (pi/4) sqrt(n) )^2 apprx=10:%00 0
and the eigenvectors of the Q_a^2(n) are G(n)-null,
it is not unreasonable to associate each Hilb(n) and its sphere
with a light sphere of a quantized spacetime.
The consequent time operator then has each of the Hilb(k) Hilbert
subspaces in the above direct sum as an invariant subspace associated
with an eigenvalue.
From the asymptotic commutation relations among
the Q_a(n) the uncertainty relations
(DELTA Q_1(n)) (DELTA Q_2(n)) <= |<Q_3(n)>| (DELTA q(n))
and cyclically, hold.
The maximal value of |<Q_3(n)>| by the
asymptotic approximation of
[Section IX] is (DELTA q(n))(1/2)(n-1);
the minimal value is zero.
Then in asymptopia, express the uncertainty relations for the worst or
greatest lower bound by,
(DELTA Q_1(n)) (DELTA Q_2(n)) >= (1/2)[DELTA q(n)]^2 (n-1)
apprx= ( pi^2 / 8)(1 - n^(-1) )
So, as n becomes very large the greatest
lower bound on the products of
uncertainties in two coordinates decreases to an ultimate lower
bound of (pi^2/8) = 1.23370, multiplied by
the square of some appropriate fundamental spatial quantum.
Physically, this is, of course, not unreasonable.
Taking the product of the three uncertainty relations, and then
the positive square root,
(DELTA Q_1(n)) (DELTA Q_2(n)) (DELTA Q_3(n)) >=
[|<Q_1(n)><Q_2(n)><Q_3(n)>|]^(1/2) [DELTA q(n)]^(3/2)
which expresses the spatial volumetric uncertainty as a function
of n and the expectation values in the coordinates.
It is not difficult to show that in terms of
the operator trace norm, that
Tr( Q_a^2(n) ) = n(n-1)/2
Tr( S_a^2(n) ) = (n/12)(n^2 - 1)
so
Tr( Q_a^2(n) ) = [6/(n+1)] Tr( S_a^2(n) )
where [6/(n+1)]^(1/2) is the value of the refined eigenvalue
spacing (DELTA q(n)) calculated in
[Appendix E].
Topological Differences in the Sphere for n odd and even
UNFINISHED
Limits of the P-Q Commutator System
From the first expression for the commutator,
<q_c(n, k)|[Q_a(n), Q_b(n)]|q_c(n, j)>
-> i (DELTA q(n)) epsilon_(abc) <q_c(n, k)|Q_b(n)|q_c(n, j)>
= i epsilon_(abc) delta_(kj) [DELTA q(n)]^2 [(n-1)/2 - k]
= i epsilon_(abc) delta_(kj) [pi/2]^2 [1/2 - 1/(2n) - k/n]
and so for any finite k and j,
lim <q_c(n, k)|[Q_a(n), Q_b(n)]|q_c(n, j)> =
n->infinity
i epsilon_(abc) delta_(kj) [ pi^2/8]
But we know that, in the limit, eigenvectors <q_c(infinity, k)|,
should be physically forbidden since they violate the uncertainty
relation for Q_c(infinity) and P_c(infinity).
Theorem 14.4:
Noncommutative quantized geometry, and limits.
QM and the quantized geometry of spaces of constant curvature.
In the context of the Lorentz group acting on Minkowski space of absolute
spacetime events,
the subgroup of spatial rotations is the maximal compact subgroup.
The elements of Hilb(n) would appear to replace the concept of
points of Minkowski space and so represent a local patch of Minkowski
space in a quantized context. From the natural appearance of su(2)
IRREPS and from the natural nesting Hilb(n-1) SUBSPACE OF Hilb(n),
a reasonable approach to defining the idea of "local spatial rotations in
a three dimensional subspace of a quantized spacetime patch" seems to
be by using a subgroup of the maximal compact subgroup of the invariance
group of FCCR. The generators of this su(2) subgroup are easily seen
to be S_a(n-1) where the matrices are actually nxn, with last
rows and columns being zero. Similarly, a subdiagonalization of Q(n)
can be performed using XI(n-1) as constructed in section IX.
In its action of subdiagonalization it is an element of the SO(n-1)
subgroup of the maximal compact subgroup,
and therefore rotates "spatially".
First perform the subdiagonalization:
| | 0 | | | 0 | | | 0 |
| XI!(n-1) | | | Q(n-1) | 0 | | XI(n-1) | |
|__________|___| |_____________| a_n | |_________|___|
| 0 ... 0 | 1 | | 0...0 a_n | 0 | | 0 ... 0 | 1 |
| | |
| Q_d(n-1) | a_n XI!(n-1)|n-1, n-2> |
= | | |
|_______________________|________________________|
| a_n <n-1, n-2|XI(n-1) | 0 |
| | |
| Q_d(n-1) | a_n |q(n-1, n-2)> |
= | | |
|_______________________|________________________|
| a_n <q(n-1, n-2)| | 0 |
where Q_d(n-1) is diagonal, and a_n := [(n-1)/2]^(1/2).
Also define then, the subspace rotation operators:
| R_32(n-1, +(pi/2))| 0 |
r_32(n, +( pi/2) ) := |___________________|___|
| 0 | 1 |
| R_31(n-1, -(pi/2))| 0 |
r_31(n, -( pi/2) ) := |___________________|___|
| 0 | 1 |
| R_21(n-1, +(pi/2))| 0 |
r_21(n, +( pi/2) ) := |___________________|___|
| 0 | 1 |
which similarly obey
r_32(n, +( pi/2) ) = r_32!(n, -( pi/2) )
r_31(n, -( pi/2) ) = r_31!(n, +( pi/2) )
r_21(n, +( pi/2) ) = r_21!(n, -( pi/2) )
and
r_21(n, +(pi/2)) r_31(n, +(pi/2)) r_21(n, -(pi/2)) = r_32(n, +(pi/2))
r_31(n, +(pi/2)) r_32(n, +(pi/2)) r_31(n, -(pi/2)) = r_21(n, +(pi/2))
r_32(n, +(pi/2)) r_21(n, +(pi/2)) r_32(n, -(pi/2)) = r_31(n, +(pi/2))
From this it is clear that the rotations generated by the S_a(n)
are not like spatial rotations or rotations in momentum space, but are more
like the rotations of Fourier transforms that exchange position and momentum.
As a note in passing, the bilinears do not close under
commutation for finite n, but asymptotically as n->infinity,
they close on a Lie algebra structure of su(1,1). Generally
the commutators have the values:
[Y_1(n), Y_2(n)] =
= iY_3(n) - (i/2){B(n), G(n)B!(n)}
[Y_2(n), Y_3(n)] =
= -2iY_1(n) + [Y_2(n), G(n)]
[Y_3(n), Y_1(n)] =
= -2iY_2(n) - [Y_1(n), G(n)]
Y^(2_1(n) + Y^(2_2(n) - Y^(2_3(n)
= (3/4)G^(2(n) + n(n - 1)/2 [ P(n)X_2(n) + Q(n)X_1(n) ]
In a formal limit n -> infinity, we have the algebra su(1, 1) = so(2, 1)
[Y_1(infinity), Y_2(infinity)] =
= +2i Y_3(infinity)
[Y_2(infinity), Y_3(infinity)] =
= -2i Y_1(infinity)
[Y_3(infinity), Y_1(infinity)] =
= -2i Y_2(infinity)
Y^2_1(infinity) + Y^2_2(infinity) - Y^2_3(infinity)
= (3/4)I(infinity)
Using the above results, the commutators for the direct product
operators can be calculated, althought it is not done here.
Matrix elements of interaxial rotations
To define rotations in a thee space, take the Eulerian
angles: (alpha, beta, gamma).
The following conventions are consistent with conventions of
[Tinkham 1964] p. 112,
and
[Rose 1957].
alpha positive rotation about the z-axis
beta positive rotation about the new y-axis
gamma positive rotation about the new z-axis
Then (alpha, beta, gamma)
( pi/2, 0, 0): x -> y
(0, pi/2, 0): z -> x
( pi/2, - pi/2, 0): z -> y
with the reverse rotations
(- pi/2, 0, 0): y -> x
(0, - pi/2, 0): x -> z
(- pi/2, pi/2, 0): y -> z
The last two are the rotations between the z-axis into the x and y axes
respectively. The matrix elements of a rotation for
j = 0, 1/2, 1, 3/2, 2, ...
that rotates functions (an active rotation) in terms of the Eulerian
angles are:
D^(j)(alpha, beta, gamma)_(ab) =
exp(-i a alpha ) exp(-i b gamma ) SIGMA_k [ (-1)^k/k! c(j, a, b; k) X
(cos beta/2)^(2j-2k-a+b) (-sin beta/2)^(2k+a-b) ]
where the indicies a and b take on the values of the
"magnetic quantum number"
-j <= a, b <= +j
with successive values differing by 1. and where the coefficients
c(j, a, b; k) are defined as:
sqrt[(j+b)! (j-b)! (j+a)! (j-a)!]
c(j, a, b; k) := -----------------------------------
(j+b-k)! (j-a-k)! (k+a-b)!
Note that the general D^(j) matrix does not transform the Cartesian
components of a vector, since then the matrix elements would have to
be real. The D^(j) rather act on the "standard components",
T_-1 := (1/sqrt(2))(V_x + i V_y)
T_0 := V_z
T_+1 := (-1/sqrt(2))(V_x - i V_y)
so,
V_x := (1/(i sqrt(2))) (T_- + T_+)
V_y := (1/sqrt(2)) (T_- - T_+)
V_z := T_0
For the rotation of the z-axis into the x-axis,
D^(j)(0, - pi/2, 0)_ab =
SIGMA_k (-1)^k/k! c(j, a, b; k) X
(cos pi/4)^(2j-2k-a+b) (sin pi/4)^(2k+a-b)
= SIGMA_k (-1)^k/k! c(j, a, b; k) X
(1/sqrt(2))^(2j-2k-a+b) (1/sqrt(2))^(2k+a-b)
= 2^(-j) SIGMA_k (-1)^k/k! c(j, a, b; k)
For the rotation of the z-axis into the y-axis,
D^(j)( pi/2, - pi/2, 0)_ab =
exp(-i a pi/2) SIGMA_k (-1)^k/k! c(j, a, b; k) X
(cos pi/4)^(2j-2k-a+b) (-sin pi/4)^(2k+a-b)
= exp(-i a pi/2) SIGMA_k (-1)^k/k! c(j, a, b; k) X
(1/sqrt(2))^(2j-2k-a+b) (-1/sqrt(2))^(2k+a-b)
= exp(+i a pi/2) exp(-i b pi) 2^(-j) SIGMA_k (-1)^k/k! c(j, a, b; k)
To count the number of terms as a function of j, a, b:
we know that for a = j, hence also for b = j that there is only one term.
Generally, the number of terms in the sum is equal to
1 + Min{ j(+|-)a, j(+|-)b }
E.g., [Messiah 1965] , II p. 1073.
<Theorem 14.5>:
The asymptotic behavior of the matrix elements of the above z->x
and z->y rotation matrices for large n, is given by:
D^(j)(0, - pi/2, 0)_ab ->
= exp(a-b) 2^(-j) J_(a-b)( 2j )
= ( pi j)^(-1/2) 2^(-j) exp(a-b) cos( 2j - pi(a-b)/2 - pi/4 )
D^(j)( pi/2, - pi/2, 0)_ab =
= i^a (-1)^b D^(j)(0, - pi/2, 0)_ab
where J_(a-b)( 2j ) is a Bessel function of integral order.
Proof:
See [Appendix H].
QED
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Created: August 1997
Last Updated: August 25, 2000
Last Updated: November 8, 2011
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