In QM, the Hilbert space Hilb on which the algebra of observables acts is endowed with its customary Euclidean inner product. Preservation of inner products is then obtained under an infinite dimensional unitary pseudogroup of transformations, "rotations in Hilb". Passage to the ray or projective space, normalizing the Hilbert vectors, allowing them to be interpreted in the usual way as probability distributions. In the Schroedinger picture, time evolution of the system is described by the time evolution of the state vectors. If in the Schroedinger equation,
H |psi(t)> = -i h-bar (PARTIAL/PARTIAL t) |psi(t)>
the Hamiltonian a generator of time translations is Hermitean,
the one-parameter dynamical group U(t) with time as its parameter
is unitary. The normalization of |psi(t)> is then invariant
with respect to time translations and this invariance
is associated with the conservation of probability.
In the sense of a change of basis, probability conservation expresses the abstract invariance of the physical system in question under the basis change, since the fundamental CCR kinematical postulate is understood at an arbitrary single instant of time. In the dynamical sense, temporal evolution by the one-parameter unitary subgroup of the full pseudounitary group, preserves inner products in the Schroedinger picture or Hermiticity of the observables in the Heisenberg picture. In the Heisenberg picture, CCR as an expression in the observables is left form invariant under the transformations of dynamical evolution. In either picture, the algebraic form of CCR is left invariant under transformations of the full unitary pseudogroup. CCR is also form invariant under translations of the Q and P and under scaling transformations. Generally, we are dealing with the invariance of a commutator and Theorem 5.2 is applicable.
It is readily apparent that the Euclidean Hermiticity of Q(n) and P(n) will not be preserved under the full group of transformations that preserve G(n), but only under some unitary subgroup. As it stands then, FCCR is not strictly form invariant under the full group of automorphisms of the algebra generated by Q(n) and P(n), that preserve G(n). In this section I want to look at ways of reconciling the Euclidean Hermiticity of Q(n) and P(n) with the G(n) invariance group so that a FCCR is form invariant.
In order for FCCR to have CCR as a physical limit, most generally, as n->infinity, and that this limit not depend on a basis of Hilb, it is necessary for the kinematical invariance group to preserve G(n). The largest continuous group of linear transformations U(G, n) on each Hilb(n) that does this is conjugate in the general linear group GL(n, C) to the non-compact Lie group U(n-1, 1). In [Appendix B] the explicit conjugacy to the Group U(n-1, 1) is shown. This group is adopted as the group of canonical transformations.
In this section we examine transformations of FCCR, and seek to uncover the transformation laws for the fundamental Q(n) and P(n) under the invariance group. The fundamental pair is transformed in order to provide FCCR form invariance. Two pairs of operators determine a form invariant FCCR while a linear combination of them correspond to the actual observables. The significance of this is discussed.
The one-parameter class of theories
Consider FCCR given most generally by
[Q(n), P(n)] = i alpha(n) G(n) (11.1)
or
[B(n), B!(n)] = alpha(n) G(n) (11.2)
where Q(n) and P(n) are defined in terms of B(n) and B!(n)
according to equations
(2.7).
No similarity transformation of the operator algebra
generated by B(n) and B!(n) can alter the
value of alpha(n). Its value defines a theory,
to change the value, is to change the theory.
It is assumed then, again for the limit n->infinity
to be QM, that
lim_n->infinity alpha(n) = h-bar . (11.3)
And so, a limit
as alpha(n) -> 0, if it exists presumably defines
a classical limit, both of these limits being understood as a
limit of a sequence of Lie algebras associated with groups;
or more precisely a limit of the sequence of defining representations
of the Lie algebras.
If the non-canonical transformation
B(n) -> a B(n)
B!(n) -> b B!(n)
is performed, then
[B(n), B!(n)] = a b alpha(n) G(n)
but image operators of the transformation are generally no longer
Hermitean conjugates of one another. If b = 1/a
[B(n), B!(n)] = alpha(n) G(n)
If a = b, then B(n) and B(n) are still Hermitean conjugates but the
parameter of the theory is altered.
Q(n) -> a Q(n), and P(n) -> a P(n)
[B(n), B!(n)] = a^2 alpha(n) G(n)
[Q(n), P(n)] = i a^2 alpha(n) G(n)
These are the so called pseudo-canonical transformations.
Canonical Transformations U(G, n)
U is in U(G, n) if
U G(n) U! = G(n)
and then since G(n) and U are always invertible.
U^(-1) = G(n) U! G^(-1)(n)
If u is in u(G, n) the Lie algebra of U(G, n),
u G(n) = - G(n) u!
That is, u is skewhermitean with respect to G(n).
The properties of G(n)-hermiticity and G(n)-skewhermiticity are
invariant under U(G, n). The exponential map
u -> exp( u )
maps, as usual, the algebra u(G, n) to the group U(G, n).
Hermitean "time" operators
Cf. Origins of Newtonian Time
For any Q and P representation that satisfy either CCR or FCCR, we can construct the associated creation and annihilation operators. In FCCR (2.8), applying any U(G, n) transformation, is to produce another n-dimensional representation from the oscillator representation. Are there n-dimensional FCCR representations that are not obtainable by applying elements of U(G, n) to the oscillator representation? Equivalently, are there disjoint FCCR orbits in Alg(Hilb(n)) under the action of U(G, n)?
Also, the creation and annihilation operators are linear combinations of Q(n) and P(n), so we transform them by a U(G, n) transformation exactly as the Q(n) and P(n). In the oscillator representation, the B(n) and B!(n) are naturally represented in a polar decomposition, equations (2.15).
B(n) = C(n) N^(1/2)(n), B!(n) = N^(1/2)(n) C!(n)
So FCCR is written
C(n) N(n) C!(n) - N(n) = alpha(n) G(n)
where C(n) is unitary and therefore expressible as
C(n) = exp( i 2 (pi/n) F(n) )
If G(n) were really the identity, then the rewritten FCCR would
imply by way of the BCH formula (7.3), that
i (2 pi/n) [F(n), N(n)] = alpha(n) I(n)
[F(n), N(n)] = -i n 2 pi alpha(n) I(n)
If we define
t(n) = (2 pi/(omega n) ) F(n)
then
[t(n), N(n)] = -i alpha(n) I(n)
and so the Hermitean operator t(n) would essentially be shown to be
canonically conjugate, in the usual sense, to N(n).
For an oscillator, t(n) would be canonically
conjugate to the energy, and therefore an appropriate time operator.
But G(n) is not I(n) just "close to it". We have instead,
exp( i omega t(n) ) N(n) exp( -i omega t(n)) = N(n) + alpha(n) G(n)
More to the point, however, we have shown in
[Section VII] that
there are two time operators t_(+|-)(n)
one forward, one backward, and in
Theorem 8.12 that
that for the oscillator Hamiltonian, and these time operators that,
For for all finite n and for k = j,
<t_k| [H_osc(n), t_(+|-)(n)] |t_j> = 0
and for k not= j, and large n
<t_k| [H_osc(n), t_(+|-)(n)] |t_j> asympt=
(+|-)i h-bar omega_H tau_0 ( n/(2 pi) + i (k-j) )
Q(n) and P(n) are defined to be Hermitean with respect to the Euclidean inner product on Hilb(n). Unitary transformations will preserve the Hermiticty. With a linear homogeneous transformation law, general U(G, n) transformations of FCCR will not preserve Euclidean Hermiticity. They will not preserve unitarity. In order for U(G, n) to be a proper kinematical group, the fundamental Q(n) and P(n) should be Hermitean with respect to G(n).
There is a large subgroup, the maximal compact subgroup of U(G, n) isomorphic to U(n-1) DIRECT-PRODUCT U(1) that preserves G(n) and the Hermiticity of Q(n) and P(n). These are the unitary transformations that leave the |n, n-1> invariant up to a phase. Let SU(G, n) be the "special" subgroup of U(G, n) consisting of those elements of U(G, n) with determinant 1. Then, concomitantly, there is a large subgroup of SU(G, n) conjugate to
S(U(n-1) DIRECT-PRODUCT U(1)) that preserves G(n) and theHermiticity of Q(n) and P(n). S(U(n-1) DIRECT-PRODUCT U(1)) is, in fact, the maximal compact subgroup of SU(n-1, 1) and by conjugacy defines a maximal compact subgroup of SU(G, n).
Both U(n-1) DIRECT-PRODUCT U(n) and S(U(n-1) DIRECT-PRODUCT U(n)) therefore preserve the form of FCCR.The complication with respect to form invariance arises from the "boost" transformations or hyperbolic rotations of U(G, n).
Q(n) and P(n) are Hermitean with respect to the Euclidean inner product of Hilb(n), which property is not preserved under U(G, n). We seek therefore, transformed Q(n) and P(n), that obey FCCR and have an Hermitean property with respect to G(n). Tentatively consider the operators
q(n) = G^(1/2)(n) Q(n) G^(-1/2)(n)
p(n) = G^(1/2)(n) P(n) G^(-1/2)(n)
Formally, these arise from a similarity transformation
applied to both sides of FCCR, so that then
[q(n), p(n)] = i G(n)
The problem is that G(n) although Euclidean Hermitean,
is not a positive operator, so that its
square root is not real, i.e, not Euclidean Hermitean, and
furthermore it is not uniquely defined.
Consider first the idea of taking a square root of I(n), that is, finding all A, such that A^2 = I(n). If A^2 = I, then A!^2 = I, and A is invertible with A = A^(-1). Therefore, A = A!. Further then, A! = A^(-1). So A is Hermitean and unitary.Any such A is therefore diagonalizable by a unitary transformation. Consider it diagonalized by a basis transformation; on the diagonal we must find a distribution of +1 and -1; there are 2^n possible distributions, hence 2^n possible square roots obtained in this manner; only one of these will be strictly positive. Any unitary transformation of any of these will also satisfy the criteria for being a square root of I(n). Clearly there are many square roots of unity. Similarly there are m^n mth roots of I(n).
Now consider square roots of G(n). The previous considerations
apply to the I(n-1) suboperator of G(n), to which one then
appends the square roots (+|-)i sqrt(n-1).
Of all the possibilities, we can select a specific conjugate
pair of square roots of G(n). That the other possible square
roots of G(n) may be useful to consider we leave open for the moment.
Specific square root operators can be defined which obey:
On the canonical basis,
G_(+|-)^(1/2)(n) := Diag[ I(n-1), (+|-)i sqrt(n-1) ]
G_(+|-)^(1/2)(n) G_(-|+)^(-1/2)(n) = I(n)
G_(+|-)^(1/2)(n) G_(+|-)^(-1/2)(n) = gamma(n)
G_(+|-)^(1/2)(n) G_(+|-)^(1/2)(n) = G(n)
G_(+|-)^(1/2)(n) G_(-|+)^(1/2)(n) = gamma(n) G(n)
G_(+|-)^(1/2)!(n) gamma(n) G_(+|-)^(1/2)(n) = G(n)
G_(+|-)^(1/2)!(n) = G_(-|+)^(1/2)(n)
G_(+|-)^(-1/2)!(n) = G_(-|+)^(-1/2)(n)
[G_(+|-)^(1/2)(n), G_(-|+)^(1/2)(n)] = 0
G(n) G_(+|-)^(-1/2)(n) = G_(-|+)^(1/2)(n)
G^(-1(n) G_(+|-)^(1/2)(n) = G_(-|+)^(-1/2)(n)
gamma(n) G_(+|-)^(1/2)(n) = G_(-|+)^(1/2)(n)
gamma(n) G_(+|-)^(-1/2)(n) = G_(-|+)^(-1/2)(n)
where gamma(n) is
gamma(n) = Diag[ I(n-1), -1]
(11.4)
as also defined in
[Appendix D].
gamma(n) is also the involutive automorphism of the the complexified
algebra u(n, C) isomorphic to gl(n, C) that expresses
the duality of the symmetric spaces U(n)/U(n-1)
and U(n-1, 1)/U(n-1). See
[Appendix D]
and
[Section XII].
Note that only in the single case n=2, is G_(+|-)^(1/2)(n) unitary.
Now define the operators,
Q_(+|-)(n) := G_(+|-)^(1/2)(n) Q(n) G_(-|+)^(-1/2)(n) (11.5a)
P_(+|-)(n) := G_(+|-)^(1/2)(n) P(n) G_(-|+)^(-1/2)(n) (11.5b)
so the complex conjugate transposes are
Q_(+|-)!(n) = G_(+|-)^(-1/2)(n) Q(n) G_(-|+)^(1/2)(n) (11.6a)
P_(+|-)!(n) = G_(+|-)^(-1/2)(n) P(n) G_(-|+)^(1/2)(n) (11.6b)
Each of these satisfy FCCR with G(n) as defined by:
[Q_(+|-)(n), P_(+|-)(n)] = i G(n) (11.7a)
[Q_(+|-)(n), P_(-|+)(n)] = i G(n) (11.7b)
[Q_(+|-)!(n), P_(+|-)!(n)] = i G(n) (11.7c)
[Q_(+|-)(n), Q_(-|+)(n)] =
(+|-)(i/2) sqrt(n-2) (n lambda_4(n) - i (n-2) lambda_5(n)) (11.7d)
[P_(+|-)(n), P_(-|+)(n)] = [Q_(+|-)(n), Q_(-|+)(n)] (11.7e)
Note that for n = 2, the generators of SU(3), lambda_k(2) vanish.
Cf.
[section IV].
Then also the operators Q_(+|-)(n) and P_(+|-)(n) are
right G(n)-Hermitean in that,
Q_(+|-)(n) G(n) = G(n) Q_(+|-)!(n) (11.8a)
and
P_(+|-)(n) G(n) = G(n) P_(+|-)!(n) (11.8b)
So now we have an FCCR expression, expressed in terms of right
G(n)-hermitean operators, which is therefore form invariant under
the canonical group U(G, n).
In the canonical basis, we have
Q_(+|-)(n) =
Q(n-1) (+|-) (1/sqrt(2))(n/2 sigma_2(n) + i(n/2 - 1)sigma_1(n)) (11.9a)
P_(+|-)(n) =
P(n-1) (+|-) (1/sqrt(2))(n/2 sigma_1(n) - i(n/2 - 1)sigma_2(n)) (11.9b)
Then defining, Euclidean Hermitean and skew parts. This separation is not U(G, n) invariant, but is invariant under the action of the maximal compact subgroup of U(G, n): in either the adjoint or coadjoint case, the algebra u(G, n) has the invariant Cartan canonical decomposition with respect to the adjoint/coadjoint action of its maximal compact subgroup into the direct sum of the algebra of its maximal compact subgroup and an invariant subspace. Q(n-1) belongs to the algebra of the maximal compact subgroup and the other term belongs to the invariant subspace.
_hQ(n) = Q(n-1) (11.10a)
_hP(n) = P(n-1) (11.10b)
_sQ(n) =
(1/sqrt(2))(n/2 sigma_2(n) + i(n/2 - 1)sigma_1(n)) (11.11a)
_sP(n) =
(1/sqrt(2))(n/2 sigma_1(n) - i(n/2 - 1)sigma_2(n)) (11.11b)
we have commutation relations:
[Q_(+|-)(n), G(n)] = (-|+)i _sP(n) (11.12a)
[P_(+|-)(n), G(n)] = (+|-)i _sQ(n) (11.12b)
[_hQ(n), _hP(n)] = iG(n-1) (11.12c)
[_sQ(n), _sP(n)] = -i (n-1) sigma_3(n)
= -i sqrt(2) n (n-1)^(3/2) X_3(n)
= -i ( G(n) - G(n-1) )
:= -i (DELTA G(n))/(DELTA n)
(11.12d)
[_sQ(n), _hP(n)] =
+ sqrt(n-2)/4 (i n lambda_5(n) - (n-2) lambda_4(n)) (11.12e)
[_hQ(n), _sP(n)] = [_sQ(n), _hP(n)] (11.12f)
[_hQ(n), _sQ(n)] =
- sqrt(n-2)/4 (i n lambda_4(n) + (n-2) lambda_5(n)) (11.12g)
[_hP(n), _sP(n)] = [_hQ(n), _sQ(n)] (11.12h)
Transformation Laws
Consider now the requirement that the new FCCR transform correctly (form invariantly) under the canonical group. Let Q_(+|-)(n) transform according to the covariant group:
Q_(+|-)(n) -> U Q_(+|-)(n) U!
Then in order that FCCR be form invariant,
P_(+|-)(n) must transform according to
the contravariant group:
G(n)-conjugate Creation and Annihilation operators
iQ_(+|-)(n) and iP_(+|-)(n) are, of course
invariant under right G(n)-conjugation and belong to the coadjoint
algebra u(G, n). The left G(n)-hermitean pair
Q^(+|-)(n) := G^(-1)(n) Q_(+|-)(n) G(n)
P^(+|-)(n) := G^(-1)(n) P_(+|-)(n) G(n)
can also be defined. If we now define,
B_(+|-)(n) = (1/sqrt(2))( Q_(+|-)(n) + iP_(+|-)(n) )
then the right G(n)-conjugate is
B_(+|-)^G(n) = (1/sqrt(2))( Q_(+|-)(n) - iP_(+|-)(n) )
These satisfy
[B_(+|-)(n), B_(+|-)^G(n)] = G(n)
The operators
B_(+|-)^G(n) B_(+|-)(n) and B_(+|-)(n) B_(+|-)^G(n)
are right G(n)-hermitean.
The operators Q(n) and Q_(+|-)(n) are related by a similarity transformation; under such transformations the spectrum is invariant. So the eigenvalues of Q_(+|-)(n) are exactly those of Q(n) which have been determined in [Section IX]. The eigenvectors and the diagonalizing transformation, on the other hand are changed. From [Section IX] we know that
XI!(n) Q(n) XI(n) = Q_d(n)
is diagonal where XI(n) is in SO(n). Then,
XI!(n) G_(-|+)^(-1/2)(n) Q_(+|-)(n) G_(+|-)^(+1/2)(n) XI(n) = Q_d(n)
and since
G_(+|-)^(+1/2)(n) Q_d(n) G_(-|+)^(-1/2)(n) = Q_d(n)
we have
XI_(+|-)^G(n) Q_(+|-)(n) XI_(+|-)(n) = Q_d(n)
where the diagonalizing transformation for Q_(+|-)(n) is defined as
XI_(+|-)(n) := G_(+|-)^(+1/2)(n) XI(n) G_(-|+)^(-1/2)(n)
so
XI_(+|-)^G(n) = G_(+|-)^(+1/2)(n) XI!(n) G_(-|+)^(-1/2)(n)
= G_(+|-)^(+1/2)(n) XI^(t(n) G_(-|+)^(-1/2)(n)
and
XI_(+|-)^G(n) XI_(+|-)(n) = I(n)
Furthermore the eigenkets
|q_(+|-)(n, k)> = XI_(+|-)^G(n) |n, k>
Clearly diagonalization of Q_(+|-)(n) by XI_(+|-)^G(n)
is not a kinematical transformation, indicating that once more, the
eigenkets of the "Q" operator are not physically realizable from the
canonical basis |n, k>. Instead of the defining condition
for U(G, n) we have
XI_(+|-)(n) G(n) XI_(+|-)^t(n) = G(n)
showing that XI_(+|-)(n) leaves G(n) invariant.
Similar considerations apply in considering the eigenvalue problem for P_(+|-)(n).
The action of U(G, n) on Hilb(n)
The action of U(G, n) on Hilb(n) breaks Hilb(n) into three classes disjoint orbits, exactly as does the Lorentz group acting on Minkowski spacetime. By analogy, Minkowski space would be assumed to have the signature (+ + + -).
The group U(G, n) acts transitively, in Hilb(n) on each of its orbits. These can be divided into three classes:
Hilb_+a(n) := {|psi> in Hilb(n): <psi|G(n)|psi> = +a}
Hilb_0(n) := {|psi> in Hilb(n): <psi|G(n)|psi> = 0}
Hilb_-a(n) := {|psi> in Hilb(n): <psi|G(n)|psi> = -a}
where a is real and non-negative.
For Hilb_+a(n) and Hilb_-a(n)
one can adopt the representative orbits
Hilb_+(n) and Hilb_-(n) with a=1.
See
[Section XII].
By way of analogy, the connected component of the Inhomogeneous Lorentz group IL_+(+) has a covering group
P_+(+) = T(4, R) SEMI-DIRECTPROD SL(2, C)
where T(4, R) is the translation group in R^4,
SEMI-DIRECTPROD is a semidirect product
[Appendix B]
and SL(2, C) is the
covering group of proper orthochronous homogeneous Lorentz group.
A group element of P_+(+) can be designated by (LAMBDA, a).
Group multiplication is to the left:
(LAMBDA_1, a_1)(LAMBDA_2, a_2) =
(LAMBDA_1 LAMBDA_2, a_1 + LAMBDA_1 a_2)
Unitary irreducible representations of T(4, R) are one-dimensional
and of the form exp( ip_mu a_mu )
where a_mu are the group parameters and
p_mu in R^4 is the character.
If we let P_+(+) act on the a_mu, there is an associated
action on the p_mu. The orbit of p is
defined to be the subset
pg in R^4 such that g is in P_+(+).
Then, R^4
breaks up into disjoint orbits of which there are six kinds:
The isotropy groups:
1) p^2 = -m^2 p_4 > 0 (timelike) SU(2)
2) p^2 = -m^2 p_4 < 0 (timelike) SU(2)
3) p^2 = +m^2 (spacelike) SU(1, 1)
4) p^2 = 0 p_4 < 0 (lightlike) ISO(2)
5) p^2 = 0 p_4 > 0 (lightlike) ISO(2)
6) p = 0 (trivial) SL(2, C)
where m^2 is a fixed positive number.
The isotropy group is the subgroup of P_+(+)
which leaves p in an orbit, fixed.
Similarly the following table lists the isotropy groups for the relevant noncompact groups acting on the homogeneous spaces of their defining representation. <gamma> is the value of the indefinite bilinear form that is preserved by the group action. This subgroup structure has equivalence to that of U(G, n) by the similarity mapping of conjugacy in GL(n, C).
<gamma>: ( > 0 ) ( = 0 ) ( < 0 )
|
Group: |---------------------------------------------------
|
U(n-1, 1) | U(n-2, 1) IU(n-2) U(n-1) DIRECT-PRODUCT U(1)
|
SU(n-1, 1) | SU(n-2, 1) ISU(n-2) S(U(n-1) DIRECT-PRODUCT U(1))
|
SO(n-1, 1) | SO(n-2, 1) ISO(n-2) SO(n-1)
where 'I' as a prefix indicates the associated inhomogeneous group
that is expressed as a semidirect product of the homogeneous group
with a group of translations.
The algebra of a noncompact translation group is isomorphic to the algebra of a compact toroidal group. (The real line is the covering space of one-dimensional torus and R^n is covering space of the topological product of one-dimensional tori.)
Euler decomposition
As with the Lorentz group and others, we can decompose a general U(G, n) transformation into
u = R^(-1) B R
a "spatial rotation" to an axis, followed by a
boost along that axis; then an inverse of the first rotation.
This is a variation on the decomposition by Euler angles
for SO(3): an incomplete Euler decomposition.
The boost transformations BU(G, n)
form a group that is conjugate in GL(2, C) to U(1, 1).
Consult and compare to
[Appendix B].
Define the matrices
|1 0 | |1 0 |
t(n) = | | t^(-1)(n) = | |
|0 1/sqrt(n-1)| |0 sqrt(n-1)|
|1 0 | |1 0|
g(n) = | | gamma(2) = | |
|0 -(n-1)| |0 -1|
Then for U in U(1, 1), U_b in BU(G, n)
t!(n) g(n) t(n) = gamma(2)
U! gamma(2) U = gamma(2), U_b! g(n) U_b = g(n)
and therefore
t^(-1)(n) U t(n) is in BU(G, n)
U(G, n) Covariance and the Harmonic Oscillator
[UNFINISHED]
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