The following are a few well known facts: A vector |psi> in a Hilbert space Hilb is called a cyclic vector for an operator A iff finite linear combinations of A^k|psi> are dense in Hilb. If Hilb = Hilb(n) is finite dimensional, for k <= n, the A^k|psi> span Hilb and are orthogonal. Again in a finite dimensional Hilb, an operator possesses a cyclic vector iff all its eigenvalues are distinct, i.e., its spectrum is multiplicity free. An operator is diagonalizable if it is normal, i. e.,
[A, A!] = 0
Generally, the following three conditions are equivalent
i) A is multiplicity free
ii) A has a cyclic vector
iii) {X: [A, X] = 0} is an Abelian algebra
For two arbitrary operators A and B
(k-1)
[A^k, B] = SIGMA A^m [A, B] A^(k-1-m)
m=0
and then,
(k-1) (j-1)
[A^k, B^j] = SIGMA SIGMA A^m B^(j-1-n) [A, B] B^n A^(k-1-m)
m=0 n=0
or equivalently,
(k-1) (j-1)
[A^k, B^j] = SIGMA SIGMA A^m B^n [A, B] B^(j-1-n) A^(k-1-m)
m=0 n=0
<Definition 5.1>:
Let Hilb(n) be a complex Hilbert space topologically
isomorphic to C^n, n-dimensional complex space and let Alg(Hilb(n))
be the algebra of complex linear endomorphisms of Hilb(n).
Alg(Hilb(n)) is a finite dimensional C*-algebra
with involution
A -> A!, for all A in Alg(Hilb(n)).
Take A such that A not= A!, i.e., A is non-Hermitean.
Call A a generator for Alg(Hilb(n)),
if any element of Alg(Hilb(n)) can be expressed as a
polynomial in A and A! with complex coefficients xi_(kj):
for any X in Alg( Hilb(n) ),
X = xi_kj A!^k A^j
with summation over k and j.
<Theorem 5.1>:
The annihilation operator B(n) is a generator for Alg( Hilb(n) ).
Proof:
The canonical algebra basis E_(kj) = |n, k><n, j|
spans Alg( Hilb(n) ).
Then, from equation
(2.11),
E_kj = (B!^k(n)/sqrt(k!) |n, 0><n, 0| (B^j(n)/sqrt(j!)
with
0 <= k, j <= n-1
showing that any operator is expressible by a polynomial in B!(n) and B(n).
QED
Consider groups of linear transformations on the two dimensional space of Alg( Hilb(n) ) that is spanned by the generating operators B(n) and B!(n). Let A be in SL(2, C) and let tau_mu be the raw unnormalized spin matrices for mu = 1,2,3, and for mu = 0, let tau_0 = I(2), the two dimensional unit matrix. For a real four-vector x^mu define the associated Cartan matrix with summation convention,
X = x^mu tau_mu
so that
Det( X ) = x^0^2 - x^1^2 - x^2^2 - x^3^2
and
x^mu = (1/2) Tr( X tau_mu )
then let
A! X A = Y
where
Y = y^mu tau_mu
Since SL(2, C) preserves determinants in this action,
Det( Y ) = Det( X )
and so the Minkowskian pseudonorm of the associated four-vector
is preserved.
A! X A = L_mu^nu(A^(-1)) x^mu tau_nu
where L_mu^nu(A^(-1)) is a matrix of the proper orthochronous Lorentz group SO(1, 3) acting on the four-dimensional carrier space of its defining representation.
The tau_mu constitute a basis for the space of 2x2 Hermitean matrices that is isomorphic as an indefinite inner product space to Minkowski space. Thus,
tau_0 is a unit timelike vector
tau_k for k = 1, 2, 3 are unit spacelike vectors
and
tau_0 - tau_k are lightlike vectors.
The tau_mu unit vectors transform as
A! tau_mu A = L_mu^nu(A) tau_nu
The subgroups of SL(2, C), that preserve the representative vectors
tau_0, tau_3 and (tau_0 - tau_3)
under the action
X -> A! X A
are seen to be respectively, SU(2), SU(1, 1) and IU(1),
the local homomorphs of SO(3), SO(2, 1) and ISO(2).
Let
(zeta| = (z_1, z_2) and |zeta) = (z_1, z_2)!
be elements of the dual carrier space and carrier space of the
SL(2, C) representation. Then, the bilinears (zeta|tau_mu|zeta)
transform as the components of a 4-vector.
For linear transformations on the "generating subspace of the adjoint representation", preservation of the commutator is equivalent to preservation of the indefinite form tau_3 (=j_3). First look at the algebra of the invariance group.
If j is left-skew with respect to tau_3,
tau_3 j = -j! tau_3
a linearly independent set of j's can be taken as
|0 i| |0 -1| |1 0| |i 0|
|i 0| |1 0| |0 -1| |0 i|
which form a Lie algebra u(1, 1).
We call these respectively, 2j_1, 2j_2, 2j_3, 2j_4.
The commutation relations for the first three are given in [Appendix C]. Clearly j_4 commutes with that set.
For any j that is a real linear combination of these generators,
exp( j! ) tau_3 exp( j ) = tau_3
and
[exp( j )]! tau_3 [exp( j )] = tau_3
The transformations of the one parameter subgroup generated by
j_4 belong to GL(n, C) but not to SL(2, C).
Moreover, they map
G(n) -> exp( i theta/2 G(n) )
Aside:
Note that uncertainty relations
[Section XIII]
are invariant
under such transformations, and also that the usual map between
a (Q, P) and a (B, B!) pair involve such transformations
and that this transformation is considered physical in the sense
of not destroying the physical structure of the system that is represented.
This is to say that these transformations as well as the transformations
that are real conformal transformations of G(n) may not not be without
interest: electromagnetic interactions can be derived from U(1) gauge
invariance in QM; local metrical structure in GR can be reduced to lightcone
structure and a real conformal factor. We are presumably dealing with
quantum theoretical expression in some bounded spacetime patch.
Nevertheless, for the moment discarding the phase transformation with generator j_4, the transformations of the real parameter Lie group that preserve the commutator then belong to SU(1, 1) locally isomorphic to Sp(1, R).
In the adjoint action of SU(1, 1) on its algebra, with
V in SU(1, 1)
3
V j_k V^(-1) = SIGMA L_k^l(V) j_l
l=1
and the L_k^l(V) are the matrices of the defining representation
of the group SO(2, 1) that is covered by SU(1, 1).
Define the operator 2-spinor and its Hermitean dual
|beta) = (B!(n), B(n))-TRANSPOSE
(beta| = (B(n), B!(n))
then the bilinears
(beta| tau_mu |beta)
transform as the components of an SL(2, C)
vector operator. These evaluate as the forms:
(beta| tau_0 |beta) = {B(n), B!(n)}
= (Q^(2(n) + P^(2(n))
(beta| tau_1 |beta) = (B!^(2(n) + B^(2(n))
= (Q^(2(n) - P^(2(n))
(beta| tau_2 |beta) = i (B!^(2(n) - B^(2(n))
= {Q(n), P(n)}
= M(n) + N(n)
(beta| tau_3 |beta) = [B(n), B!(n)]
= -i [Q(n), P(n)]
= G(n)
The first and the last bilinears are related by
2 N(n) = {B(n), B!(n)} - [B(n), B!(n)]
The component with subscript 0, or "time" component of the vector
operator is the Hamiltonian of an oscillator;
this is left invariant by an SU(2) subgroup.
The component with subscript 1 is the Lagrangian of the oscillator.
The component with subscript 2 is an analog of a correlation coefficient.
The third spacelike component is G(n), left invariant by an SU(1, 1) subgroup.
The cognate lightlike operators corresponding to (tau_0 +|- tau_k)
are:
(beta|(tau_0 - tau_1)|beta) = -(B!(n) - B(n))^2
= 2 P^2(n)
(beta|(tau_0 - tau_1)|beta) = -i (B!(n) + i B(n))^2
= (Q(n) - P(n))^2
(beta|(tau_0 - tau_3)|beta) = 2 N(n)
(beta|(tau_0 + tau_1)|beta) = (B!(n) + B(n))^2
= 2 Q^2(n)
(beta|(tau_0 + tau_1)|beta) = +i (B!(n) - i B(n))^2
= (Q(n) + P(n))^2
(beta|(tau_0 + tau_3)|beta) = 2 N(n) + 2 G(n)
= 2 M(n)
with Q(n), P(n), N(n) and M(n) defined by equations
(2.7),
(2.2),
and
(2.21)
respectively.
These are all left invariant by an IU(1) subgroup of SL(2, C).
Aside:
The form that is being preserved is a "spinor metric", or
from a different point of view what is being preserved is the
symplectic structure of the bivector space associated with
an abstract Lie algebra as vector space. The abstract Lie
algebra itself as the carrier space for representation is
either the adjoint or coadjoint representation.
Under the complex translations
B(n) -> B(n) + zeta I(n)
B!(n) -> B(n) + zeta* I(n)
for zeta complex,
(beta| tau_0 |beta)
-> {B(n), B!(n)} + 2( zeta B!(n) + B(n) zeta* + 2 |zeta|^2
(beta| tau_1 |beta)
-> (B!^2(n) + B^2(n))
+ 2( zeta B!(n) + B(n) zeta* + (zeta*^2 + zeta^2)
(beta| tau_2 |beta)
-> i (B!^2(n) - B^2(n))
+ i 2( zeta B!(n) - B(n) zeta* + (zeta*^2 - zeta^2)
(beta| tau_3 |beta)
-> [B(n), B!(n)]
Under the complex translations of the B pair that are consistent with
their relation by Hermitean conjugation, of the four components of the
SL(2, C) vector operator, only the component that is the commutator
[B(n), B!(n)] is invariant. Equivalently,
[Q(n), P(n)] = i G(n)
is form invariant under the real translations,
Q(n) -> Q(n) + a I(n)
P(n) -> P(n) + b I(n)
for a, b real.
We want to note someplace that the I(n) in the translation transformations
belongs to gl(2, C)
Still further,
[B(n), B!(n)] = G(n)
is form invariant under the scaling transformations
B(n) -> xi B(n)
B!(n) -> xi*^(-1) B(n)
for xi complex, or equivalently,
[Q(n), P(n)] = i G(n)
is form invariant under the scaling transformations,
Q(n) -> xi Q(n)
P(n) -> xi^(-1) P(n)
for complex xi.
These have the generators j_3 and (i j_3).
The preceding considerations show:
<Theorem 5.2>:
The SL(2, C) action (in fact GL(2, C) on the two dimensional generating
subspace spanned by B(n) and B!(n)
X -> A! X A
for X in GL(2, C) and for A in SL(2, C)
extends to an automorphism of the algebra of complex linear
operators Alg( Hilb(n) ).
Construction of main vector operator.
An invariance subgroup of the fundamental commutator is contained
in each of the customarily singled out isotropy subgroups of
SL(2, C)?
Larger invariance group contained in GL(2, C)?
Largest invariance group is an affine conformal group?
|0 i 0| |0 -1 0|
j_1 = |i 0 0| j_2 = |1 0 0|
|0 0 0| |0 0 0|
|1 0 0| |i 0 0|
j_3 = |0 -1 0| j_3 = |0 -i 0|
|0 0 0| |0 0 0|
|0 0 1| |0 0 i|
x_1 = |0 0 1| x_2 = |0 0 -i|
|0 0 0| |0 0 0|
The generators j_1, j_2, j_3 form an su(1, 1) subalgebra,
with commutation relations:
[j_1, j_2] = +i j_3, [j_2, j_3], = +i j_1, [j_3, j_1] = -i j_2
while x_1 and x_2 form a translation subalgebra.
The addition of j_3 = i j_3, causes closure under commutation,
on the algebra of sl(2, C), since
[i j_1, i j_2] = -i j_3,
[i j_2, i j_3], = -i j_1,
[i j_3, i j_1] = +i j_2
and
[i j_1, j_2] = - j_3, [i j_2, j_3], = - j_1, [i j_3, i j] = + j_2
which is not, however, an invariance algebra of the commutator.
For the commutation relations with the translation generators:
[x_1, x_2] = 0
[x_1, j_1] = -i x_1, [x_1, j_2] = +i x_2, [x_1, j_3] = -i x_2
[x_2, j_1] = +i x_2, [x_2, j_2] = +i x_1, [x_2, j_3] = +i x_1
[x_1, ij_1] = + x_1, [x_1, ij_2] = - x_2, [x_1, ij_3] = + x_2
[x_2, ij_1] = - x_2, [x_2, ij_2] = - x_1, [x_2, ij_3] = - x_1
thus introducing the additional translation generators ix_1 and ix_2,
so that the algebra now closes on the inhomogeneous isl(2, C).
For conformal invariance of the commutator, add the two generators
d1 = i d2,
|1 0 0| |i 0 0|
d_1 = |0 1 0| d_2 = |0 i 0|
|0 0 0| |0 0 0|
with commutators:
[d_1, d_2] = 0
[d_1, j_k] = 0, [d_2, j_k] = 0
[x_1, d_1] = -i x_1, [x_1, d_2] = -1 x_1 = +i (ix_1)
[x_2, d_1] = -i x_2, [x_2, d_2] = -1 x_2 = +i (ix_2)
The additional commutators close on the inhomogeneous algebra igl(2, C),
the most general coordinate transformation of an affine Euclidean space C^2.
These invariance transformations of the FCCR commutator form do not,
however, all preserve the Hermiticity of Q(n) and P(n), or equivalently
the relation between B(n) and B!(n) of being Hermitean conjugates?
The form invariance group of commutators is the inhomogeneous Lie group
ISU(1, 1).
The linear transformation that connects the (B(n), B!(n)) pair with the
(Q(n), P(n)) pair is recognized as a physical isomorphism in QM, however,
and is obviously a transformation that does not preserve Hermiticty.
Preservation of Hermiticty is then not a requirement of physical isomorphism.
Hermiticity v. Analyticity.
Group of transformations that preserve the anticommutator {B, B!} ?
This covariance of commutators has no sense of course, in the context of abstract Lie algebras but rather in the context of a representation where the Lie product is represented by commutators of an associated algebra of linear operators acting on the carrier space of the representation. We consider the pairs of (Q(n), P(n)) for different values of n to be physically different, and concentrate on the primacy of fundamental or defining representations, although other representations of the abstracted structure may also be of use. The cognates in QM of these other representations would seem to appear in direct products of CCR representations which then describe systems of many particles.
Note that the linear transformation taking the pair (B(n), B!(n)) to
the pair (Q(n), P(n)) is:
| Q(n) | | 1 1 | | B!(n) |
| | = (1/sqrt(2)) | | | |
| P(n) | | i -i | | B(n) |
| B!(n) |
:= u | |
| B(n) |
and the determinant of the transformation is -i, so
it is unitary, but not special unitary.
See equations
(10.1)
and the subsection "Spin 1/2 Revisited" in
[Section XVI].
The inverse transformation is obviously
u! =
| 1 -i |
u^(-1) = (1/sqrt(2)) | |
| 1 i |
Furthermore, u may be decomposed as
| 1 0 | | 1 1 |
u = | | (1/sqrt(2)) | |
| 0 -i | | -1 1 |
| 1 0 | | cos( pi/4) sin( pi/4) |
= | | | |
| 0 -i | | -sin( pi/4) cos( pi/4) |
| exp(+i pi/4) 0 | | cos( pi/4) sin( pi/4) |
= exp(-i pi/4) | | | |
| 0 exp(-i pi/4) | | -sin( pi/4) cos( pi/4) |
= exp( -i( pi/4)tau_0 ) exp( -i( pi/4)tau_3 ) exp( -i( pi/4)tau_2 )
= exp( -i( pi/2)sigma_0 ) exp( -i( pi/2)sigma_3 ) exp( -i( pi/2)sigma_2 )
Under the mapping by u, the matrices of the commutator invariance
algebra map:
j_1 -> i j_3
j_2 -> - j_1
j_3 -> i j_2
j_4 -> j_4
or
| j_1' | | 0 0 i 0 | | j_1 |
| j_2' | | -1 0 0 0 | | j_2 |
| j_3' | = | 0 i 0 0 | | j_3 |
| j_4' | | 0 0 0 1 | | j_4 |
The question raised by theorem 5.2 is whether the pseudocanonical
transformations can be implemented also by similarity transformations.
The similarity transformations cannot be unitary because they are
finite dimensional representations of a noncompact group.
<Lemma 5.3.1>:
For B(n), B!(n) and N(n) defined by equations (2.1) and (2.2),
[N(n), B^k(n)] = - k B^k(n)
[N(n), B!^k(n)] = + k B!^k(n)
Therefore, for an arbitrary polynomial function f(),
[N(n), f(B(n))] = - B(n) (d/dB(n)) f(B(n))
[N(n), f(B!(n))] = + B!(n) (d/dB(n)) f(B!(n))
and
[N^k(n), B(n)] = - B(n) ( N^k(n) - (N(n) - I(n))^k )
[N^k(n), B!(n)] = + ( N^k(n) - (N(n) - I(n))^k ) B!(n)
Proof:
By unsurprising induction. The details are left as amusement.
QED
<Lemma 5.3.2>:
From lemma 5.3.1 derive easily, the following:
N^k(n) B(n) = B(n) ( N(n) - I(n) )^k
B!(n) N^k(n) = ( N(n) - I(n) )^k B!(n)
[N(n), B!^k(n) B^j(n)] = -(k-j) B!^k(n) B^j(n)
[N(n), B^k(n) B!^j(n)] = -(k-j) B^k(n) B!^j(n)
<Definition 5.2>:
Let f(z) be an analytic function of its argument,
so that it is representable by a convergent power series
within a circle of convergence in the complex plane.
Consider the power series expansion for f( A(n) ),
where A(n) is in Alg( Hilb(n) ).
It is well defined and convergent if Sp( A(n) ) lies within the
circle of convergence.
In analogy to the Frechet derivative,
define a finite Frechet difference, by
delta f( A(n) ) := f( A(n) ) - f( A(n) - I(n) )
The second Frechet difference similarly
delta^(2) f(A(n)) := f(A(n)) - 2f(A(n) - I(n)) + f(A(n) - 2I(n))
and more generally the m-th Frechet difference
m
delta^(m) f( A(n) ) := SIGMA (-1)^j B(m j) f( A(n) - j I(n) )
j=0
where B(m j) are the binomial coefficients, and
delta^(m) f( A(n) ) = 0 if m < 0, while
delta^(m) f( A(n) ) = A(n) if m = 0.
Consequently, (without proof)
delta^(m) [alpha f(A(n))] = alpha [delta^(m) f(A(n))]
delta^(m) [f(A(n)) + g(A(n))] = delta^(m) f(A(n)) + delta^(m) g(A(n))
delta^(m) [f(A(n)) g(A(n))]
= [delta^(m) f(A(n))]g(A(n)) + f(A(n))[delta^(m) g(A(n))]
delta^(m) delta^(k) f(A(n)) = delta^(m+k) f(A(n))
<Lemma 5.3.3>:
The k-fold commutator can be expressed by,
C(A: B, k) = k! SIGMA (-1)^m (B^j A B^m/(j! m!))
j+m=k
Proof:
From the Baker-Campbell-Hausdorff formula (7.3),
C(A: B, k) = (d/d alpha)^k ( exp(+alpha B) A exp(-alpha B) |
| alpha = 0
and expanding the exponentials
exp(+alpha B) A exp(-alpha B) =
SIGMA (-1)^m alpha^(j+m)/(j! m!) B^j A B^m
j,m
To obtain the formula,
perform the k-th derivative of this expression and set alpha = 0.
QED
<Lemma 5.3.4>:
From Lemma 5.3.1, and assuming a convergent power series expansion for
f( N(n) ), we can write
[f( N(n) ), B(n)] = - B(n) delta f( N(n )
[delta f( N(n) ), B(n)] = - B(n) delta^(2) f( N(n )
[delta^(m) f( N(n) ), B(n)] = - B(n) delta^(m+1) f( N(n )
[delta^(m) f( N(n) ), B^2(n)]
= - B^2(n) (2 delta^(m+1) f( N(n ) ) - delta^(m+2) f( N(n) ) )
[delta^(m) f( N(n) ), B^k(n)]
k
= - B^k(n) SIGMA (-1)^(j-1) B(m j) delta^(m+j) f( N(n ) )
j=1
[f( N(n) ), B!(n)] = + delta f( N(n ) B!(n)
[delta f( N(n) ), B!(n)] = + delta^(2) f( N(n ) B!(n)
[delta^(m) f( N(n) ), B!(n)] = + delta^(m+1) f( N(n ) B!(n)
[delta^(m) f( N(n) ), B!^2(n)]
= + (2 delta^(m+1) f( N(n ) ) - delta^(m+2) f( N(n) ) ) B!^2(n)
[delta^(m) f( N(n) ), B!^k(n)]
k
= + SIGMA (-1)^(j-1) B(m j) delta^(m+j) f( N(n ) ) B!^k(n)
j=1
C(f( N(n) ): B(n), m)
= (-1)^m B^m(n) (2 delta^(m-1) f( N(n) ) - delta^(m) f( N(n) ) )
C(f( N(n) ): B!(n), m)
= (2 delta^(m-1) f( N(n) ) - delta^(m) f( N(n) ) ) B!^m(n)
where C( . : . , m) is the m-fold commutator defined by
equation (7.4a).
Proof:
Email me, Bill Hammel at: