On an arbitrary orthonormal basis of an n-dimensional complex Hilbert space, define the truncated Boson annihilation operator
| 0 √1 0 0 0 0 ... 0 |
| 0 0 √2 0 0 0 ... 0 |
| 0 0 0 √3 0 0 ... 0 |
B(n) = | ... | (2.1)
| 0 0 ... √(n-1)|
| 0 0 ... 0 |
with '√' indicating a square root,
and the truncated Boson creation operator
B!(n)
as the Hermitean conjugate (complex conjugate transpose).
Then, the customary number operator
N(n) := B!(n) B(n) (2.2)
can be defined in its spectral representation on its eigenbasis
taken as canonical:
N(n) |n, k> = k |n, k> (2.3)
For k = 0, 1, ..., n-1, explicit matrix elements are,
<n, k|B(n)|n, j> = sqrt(j) delta_k_(j-1) = sqrt(k+1) delta_k_(j-1)
<n, k|B!(n)|n, j> = sqrt(k) delta_k_(j+1) = sqrt(j+1) delta_k_(j+1)
<n, k|N(n)|n, j> = k delta_k_j
Also then.
[B(n), B!(n)] = G(n) (2.4)
where
| I(n-1) | 0 |
| | |
G(n) = | | | (2.5)
|---------------| |
| 0 -(n-1) |
and I(n-1) is the (n-1) x (n-1) identity. The trace of G(n)
Tr( G(n) ) = 0 (2.6)
as it must since the trace of the commutator of any two finite
matrices must vanish
[Halmos 1952]
[Halmos 1954].
The finite dimensional analogs of the position and momentum operators are then defined by
Q(n) = (1/sqrt(2)) ( B!(n) + B(n) ) (2.7a)
P(n) = (i/sqrt(2)) ( B!(n) - B(n) ) (2.7b)
where i := sqrt(-1), so that
[Q(n), P(n)] = i G(n) (2.8)
and the inverse transformation
B(n) = (1/sqrt(2)) ( Q(n) + iP(n) ) (2.9a)
B!(n) = (1/sqrt(2)) ( Q(n) - iP(n) ) (2.9b)
Q(n) and P(n) are Hermitean, i.e. equal to their Hermitean conjugates. The word "Hermitean" is always used in this sense unless further qualified, and denotes Hermiticity with respect to the standard Euclidean inner product of an (here implicitly assumed) underlying Hilbert space.
We also then have the familiar relations as formulae:
B(n) |n, k> = sqrt(k) |n, k-1> (2.10a)
B!(n) |n, k> = sqrt(k+1) |n, k+1> (2.10b)
B^2(n) |n, k> = sqrt(k(k-1)) |n, k-2> (2.10c)
B!^2(n) |n, k> = sqrt((k+1)(k+2)) |n, k+2> (2.10d)
and more generally:
B^j(n) |n, k> = sqrt(k(k-1)...(k-j+1)) |n, k-j> (2.11a)
B!^j(n) |n, k> = sqrt((k+1)(k+2)...(k+j)) |n, k+j> (2.11b)
where it is understood that
|n, k> = 0, if k = 0 (mod n)
Consider the subspace Z(n) of Hilb(n) spanned by vectors
|n, k>, for k = 0, 1, ..., n-3
On Z(n) , G(n) acts as I(n). Symbolically,
G(n) |Z(n)> = I(n) |Z(n)>
Moreover, The subspace is invariant under the action of both Q(n)
and P(n).
Q(n) |Z(n)> = |Z(n)>
P(n) |Z(n)> = |Z(n)>
so
( [Q(n), P(n)] - i G(n) )|Z(n)> = 0
and
[Q(n), G(n)] |Z(n)> = 0
[P(n), G(n)] |Z(n)> = 0
Furthermore, an uncertainty relation of exactly the same form as that of QM holds [Section XIII]. So that there exist algebraic realizations of weak CCR that are finite dimensional. The technical device that is met in the usual realizations by unbounded operators on an infinite dimensional Hilbert space, but not met in the finite dimensional cases is that the domain of definition Z(n) is clearly not dense in Hilb(n). Moreover, Z(n) is also not invariant under the action of quadratics in Q(n) and P(n), and then certainly not invariant under the action of general polynomials in Q(n) and P(n), which is to say, it is not invariant under the obvious representation of the enveloping algebra of a Lie algebra that Q(n), P(n) and G(n) may generate.
The easy observation that G(n) is very much like I(n), and that simply suppressing the one or two errant 1-dimensional subspace makes things look very much like QM kinematics, is the source of this investigation. The questions that are addressed here are basically: How does such a structure connect with QM? What structures are essentially the same? What is lost in passing from the finite dimensional cases to the infinite dimensional case, and similarly for the reverse passage? What is gained in passing from the finite dimensional cases to the infinite dimensional case, and similarly for the reverse passage? Can such a finite dimensional commutator structure define a "local" quantum theory?
In the remainder of this section, some useful operators
are defined that relate to the structure of FCCR.
Let
| 0 1 0 0 0 ... 0 |
| 0 0 1 0 0 ... 0 |
SHA(n) := | 0 0 0 1 0 ... 0 | (2.12)
| ... |
| 0 0 0 ... 0 1 |
| 0 0 0 ... 0 0 |
be the annihilating shift operator, so that
SHA(n) |n, k> = |n, k-1> (2.13)
where,
|n, k> = 0, if k < 0 or if k > (n - 1)
and define
h(n) := (I(n) + N(n)) (2.14)
then
B(n) = h^(1/2)(n) SHA(n) (2.15a)
B!(n) = SHA!(n) h^(1/2)(n) (2.15b)
and alternatively
B(n) = SHA(n) N^(1/2)(n) (2.16a)
B!(n) = N^(1/2)(n) SHA!(n) (2.16b)
This is the analog of a proper pseudopolar decomposition for the
creation annihilation pair. h(n) has no zero eigenvalue and is
therefore strictly invertible.
Neither SHA(n) nor SHA!(n)
is unitary, as they should be, however, for a true polar decomposition.
This is a straightforward mimic of the "correct" procedure in QM.
Although the existence of a true polar decomposition for the creation
and annihilation operators was at one time assumed, to prove an
uncertainty relation between the standard number operator N and
and a reputed phase operator, a contradiction can be seen to arise,
which leads to the nonexistence of a such a polar decomposition in QM.
Cf.
[Carruthers 1968].
In a finite dimensional space, however, the polar decomposition
does exist and is constructed below:
Define the cyclic shift operators C(n) and C!(n) so that,
| 0 1 0 0 0 ... 0 |
| 0 0 1 0 0 ... 0 |
C(n) := | 0 0 0 1 0 ... 0 | (2.17)
| ... |
| 0 0 0 ... 0 1 |
| 1 0 0 ... 0 0 |
then the following hold:
C!(n) C(n) = C(n) C!(n) = I(n), i.e., C(n) is unitary.
C^n(n) = I(n), i.e., C(n) is idempotent of order n and
therefore has as its eigenvalues,
the n-th roots of unity.
SHA^n(n) = 0, i.e., SHA(n) is nilpotent of order n.
SHA(n) SHA!(n) = Diag[ I(n-1) 0 ]
SHA!(n) SHA(n) = Diag[ 0 I(n-1) ]
C(n) = SHA(n) + SHA!^(n-1)(n)
C!(n) = SHA!(n) + SHA^(n-1)(n)
The operators SHA(n) and C(n) are close in appearance but quite different in properties. Intuitively they can both converge, in some sense, to the same operator in some n -> infinity limit. But there is something interesting about their closeness. It is a point of consternation mentioned above, that in the creation and annihilation operator formalism of QM, that these operators cannot have a polar decomposition, into Hermitean and unitary operators. Equations (2.15) reflect the compromise of custom. For the mathematics of this point see the discussion in [Carruthers 1968]. In FCCR, however, a polar decomposition is possible:
B(n) = C(n) N^(1/2)(n) (2.18a)
B!(n) = N^(1/2)(n) C!(n) (2.18b)
Note that what looks to be an alternative polar decomposition
given by (2.19) below, does NOT give B(n) and B!(n).
A(n) = h^(1/2)(n) C(n) (2.19a)
A!(n) = C!(n) h^(1/2)(n) (2.19b)
[A(n), A!(n)] = C!(n)G(n)C(n) (2.20)
Yet there is an alternative decomposition
B(n) = M^(1/2)(n) C(n) (2.21a)
B!(n) = C!(n) M^(1/2)(n) (2.21b)
where
M(n) := B(n) B!(n) (2.21c)
An important consequence of this decomposition is that a true phase operator exists for every finite n, corresponding, apparently, to the classical phase of Hamilton-Jacobi theory. Moreover, then a "time" operator suitable for an harmonic oscillator as clock is also definable. We develop some of the consequences of this later on, noting here that this leads to a notion of a "world event state" to generalize the concept of "position eigenstate", and more generally the notion that QM "state" should here be replaced by those of "event" and "process". [Section XV] The time of the system clock will be seen to be distinct from the central time that should be read from some laboratory reference clock. But, a discussion of any possible interpretations here is decidely premature.
At this point, I wish to pursue another train of thought. Heisenberg's original motivation for CCR in the form of (1.1a) was said to be that one could derive the uncertainty relation, which he considered fundamental. There are minor variants of CCR, that also lead to the same uncertainty relation, that are derived in [Section XIII]. Consider the polar decomposition (2.18). In both, a square root is taken. There is then a natural choice of sign, which could be absorbed by the unitary operator, essentially rotating its eigenvalues by pi or equivalently flipping about the imaginary axis. I prefer to keep the sign choice with Hermitean factor. In (2.18), both roots are implicitly chosen positive. If both roots for B(n) and B!(n) are chosen negative, this amounts to nothing more than a canonical transformation, producing no real difference in FCCR. If, however, opposite root signs are taken and we form an alternative creation annihilation pair by defining the skew-Hermitean representations of FCCR:
B_-(n) = i C(n) N^(1/2)(n) (2.22a)
B_-!(n) = -i N^(1/2)(n) C!(n) (2.22b)
so
[B_-(n), B_-!(n)] = G(n) (2.22c)
Then,
B_-(n) |n, k> = i (k)^(1/2) |n, k-1> (2.23a)
B_-!(n) |n, k> = -i (k+1)^(1/2) |n, k+1> (2.23b)
N_-(n) := B_-!(n)B_-(n) (2.24)
= N(n)
N_-(n)|n, k> = k|n, k> (2.25)
Hermitean Q_-(n) and P_-(n) which will also satisfy equation
(2.8)
can be defined as before by taking:
Q_-(n) = (i/sqrt(2)) ( B_-!(n) + B_-(n) ) (2.26a)
P_-(n) = (1/sqrt(2)) ( B_-!(n) - B_-(n) ) (2.26b)
Physically, there is no difference between,
these two representations, when they are isolated.
Putting them together on the same space is another matter.
Define
N_-(n) := B_-!(n) B(n) (2.27)
= B!(n) B_-(n)
= -N(n)
which has eigenvalues that are the negative of N(n).
Also, then
[B_-(n), B!(n)] = i G(n) (2.28)
Extending the energy spectrum in a palindromatic way
to negative values is consistent with and indeed
demanded by relativistic principles.
Using both representations together effects such
an extension in the readily available oscillator
Hamiltonian analog
h-bar omega ( N(n) + (1/2)G(n) ).
Consider the direct sum Hilb_+(n) + Hilb_-(n),
with Hilb_+(n) and Hilb_-(n)
being copies of Hilb(n).
Construct a standard Hermitean representation and the
skew representation of FCCR on both
Hilb_+(n) and Hilb_-(n).
Now define the operator
N_R(n) := N(n) + N_-(n) (2.29)
and let |n, k> designate the canonical basis of Hilb_+(n)
as before and let |n, -k> designate the
matching basis in Hilb_-(n).
The existence of a doubly degenerate vacuum should be noted.
There is an alternative way of looking at this construction as a direct product of Hilb(2) X Hilb_-(n) (See [Section XIV].) So that
N_R(n) := sigma_3 + N_-(n) (2.30)
where sigma_3 is the third Pauli spin matrix
numerically given by Diag[1, -1].
The a basis for the direct product space space is given by
|n, k, s> = |s> X |n, k>
where again k = 0, 1, ..., n-1; and now s = +1, -1.
An Aside On Operator Convergence
Consider now the limiting situation n -> infinity. SHA(n) passes naturally to a generalized nilpotent operator SHA on the one-sided sequentially infinite Hilbert space Hilb:
SHA = lim SHA(n) (2.31)
n->infinity
where
lim || SHA^n ||^(1/n) = 0. (2.32)
n->infinity
Then SHA as well as SHA(n), has zero spectral radius
[Rickart 1960],
[Putnam 1951],
therefore Sp( SHA(n) ) and Sp( SHA )
contain only the value zero.
If the sequence Hilb(n) is considered as a sequential nesting of
subspaces of Hilb,
then SHA(n) is the projection of SHA onto Hilb(n).
SHA on Hilb is, of course, not unitary, and not isometrical;
it is a norm-decreasing map.
Although SHA! is also not unitary, it is
isometrical while SHA!(n) is clearly not.
SHA is not normal (neither is SHA(n)):
[SHA, SHA!] = |0><0| (2.33)
yet it is hyponormal in that the commutator as an Hermitean
operator satisfies
[SHA!, SHA] < 0 (2.34)
while SHA(n) is not hyponormal. SHA! as an isometrical yet not normal operator, has Sp( SHA! ) equal to the closed unit disc; and as a hyponormal operator possesses a unitary extension to the two-sided shift operator on the two sided sequentially infinite Hilbert space (Hilb X Hilb) [Putnam 1967],
Further, neither SHA nor SHA! are completely continuous on Hilb; and therefore, the limit in which (2.8) is to be understood is decidedly not in the sense of the norm topology. We return to the question of the sense of the limit (2.8) in [Section III].
Useful relations with SHA and SHA! on Hilb are
for any n:
SHA^n (SHA!)^n = I (2.35a)
while
(SHA!)^n SHA^n = I - I(n) (2.35b)
so
[SHA^n, (SHA!)^n] = I(n) (2.35c)
and
[SHA^n, (SHA!)^n] SHA = SHA(n) (2.35d)
Summary and Discussion of II:
Begin with truncation of Bosonic creation and annihilation operators. Commutation brings in an indefinite form G(n). For n=2, we have the Fermionic creation and annihilation operators [Section XVI], spin-(1/2) revisited. If we stay finite, "normalizing states" doesn't seem like a good idea. General unitary transformations at finite n, will disturb the form of G(n). After the transformation is made, a limiting procedure, will probably not give QM. What does it give? I have not investigated this as a classification problem. The appearance of an indefinite form and the desirability of its preservation under some group of canonical transformations acting on Hilb(n) leads immediately to a noncompact group, conjugate to U(n-1, 1) in GL(n, C) acting on Hilb(n), as the kinematical invariance group [Section XI]. For n>3, U(n-1, 1) directly contains the Lorentz group as a subgroup, which will then leave invariant in its action on Hilb(n), the quantity <x|G(n)|x>. Even though we cannot consistently normalize the elements of Hilb(n), passing effectively to the projective ray space, expectation values can still be computed by dividing by the Euclidean norm of a state. The existence of a genuine "phase operator", the lost (in QM) conjugate to the number operator, that would be used to construct a time operator for the harmonic oscillator is recaptured.
The fact that the various decompositions (2.15), (2.17), (2.18) and (2.21) exist may be of physical significance in distinguishing a transient process passing through a spacetime patch from a captured or cyclic process that is confined to the patch. If so, a superposition of these two kinds of process is also possible since,
B(n) = (a C(n) + b SHA(n)) N^(1/2)(n) (2.36a)
B!(n) = N^(1/2)(n) (a C!(n) + b SHA!(n)) (2.36b)
where a and b are real and a + b = 1.
Consider eliminating 'i' as an artifice for understanding how it
must be put back in? Cf.
[Mackey 1968], p. 107.
Let P'(n) = -iP(n), then
[Q(n), P'(n)] = G(n)
where Q is real symmetric and P' is real antisymmetric.
Q is diagonalizable by the orthogonal transformation XI(n)
developed later,
[Section IX]
and
[Section X].
P' is now, however, real antisymmetric,
hence the generator of a rotation in a real n-dimensional
space. The operator exp( aP'(n) ) is the rotation.
If n is even, Det( P'(n) ) is nonzero.
Also when n is even any rotation can be decomposed into
a product of mutually commuting rotations in 2-planes.
Similarly P' may be expressed as
P' = A B A^(-1)
where A is orthogonal (A^(-1) = A-transpose)
and B^2 is diagonal
A Second reason for this being interesting:
The eigenvectors of Q(n) are real and G-null, "on the light cone".
If real positive l.c.s are taken of these eigenvectors, the
result will always be within the forward lightcone - "causal process".
<f|G(n)|f> > 0
As expected,
if real uniformly phased l.c.s are taken of these eigenvectors, the
result will also always be within the forward lightcone.
<f|G(n)|f> > 0
Each nonzero value of <f|G(n)|f> is associated with (n-1)-dim hyperboloid and an analog "proper time" of the "process" |f>. Presumably, position on the hyperbolid has something to do with the relative velocity of some observer. If the coefficients of a l.c. forming |f> have to do with probabilities, then these change relative to the observer; this is not what would be normally desireable. But, then we don't know yet what these changes might be, nor how they would be effected for very large n. Also, what happens to the notion of l.c. itself of various |f> and what hyperbolid does a|f> + b|f'> belong to, is a question that needs an answer. Is there another defining state composition rule like the SR rule for adding velocities? Expectation values of Q(n) on such states --- Normalization allows that the expectation values be bounded by the maximal eigenvalue of Q(n). Without the 'i', Q & P' are NOT connected by a Fourier transform. The forward and backward cones are disconnected in a real space but become "connected by phase" in a complex space. Note that our Complex extension of a real n-dim pseudoeuclidean Hilbert space is not like the "complexified Minkowski space" used in QM & QFT where the functions (operator valued in QFT) are continued analytically, and distributions are defined by jump functions between functions analytically defined in the upper half plane and those defined analytically in the lower half plane.
Email me, Bill Hammel at: