The two recognized forms of the standard CCR are that of Heisenberg (1.5a) and that of Weyl (1.4). Related by a formal calculation, they are not equivalent. More precisely: Modulo unitary equivalence, the Weyl relations have only one irreducible solution. The standard Schroedinger representation exponentiates correctly to the Weyl relation. Von Neumann's theorem [Neumann 1931] states that the solution of the Weyl relations is the Schroedinger representation of the Heisenberg relations exponentiated. A corollary of von Neumann's theorem is that any generator for a Weyl pair has a continuous spectrum that is the real line. This rather strong statement can be avoided by considering the Heisenberg relations rather than the Weyl relations, as there are solutions to the former that are not solutions to the latter. See [Garrison 1970] for an example. The conditions on a Heisenberg pair for it to exponentiate to the Weyl relations are roughly that the dense domain on which the Heisenberg CCR is defined and valid be invariant under the action of each of the selfadjoint P and Q, and also that the operator P^2 + Q^2 be essentially selfadjoint on the dense invariant domain. E.g. [Reed 1972], p. 274.

Uncertainty relations are always valid for the states in the domain of the selfadjoint generators for the Weyl relations. For the Heisenberg relations, however, the uncertainty relations can fail. Examples of this can be found in [Garrison 1970], [Lerner 1970], and [Carruthers 1968]. Garrison gives an interesting construction of just such a Heisenberg pair for a number operator and a phase operator implying an energy-time pair for an harmonic oscillator.

The FCCR corresponds to Heisenberg relation, and yet for finite n is always exponentiable. It is clear therefore that the limit of FCCR as n->infinity is not unique and depends on how the limit is taken. It also seems apparent that while we exhibit FCCR as a truncation of the harmonic oscillator type representation, there is no similar procedure of truncation that can be used for a Schroedinger representation. Without making a complete analysis of the situation, the following is a discussion of some aspects of the limit.

First set up the usual situation for an irreducible representation of CCR on a separable Hilbert space expressed in terms of the creation and annihilation operators. So that we know where we are going [Putnam 1967]:

Let Hilb be a separable complex Hilbert space, and let {|k>}, k = 0, 1, 2, ... be a complete orthonormal basis for Hilb.

   For |f> in Hilb given by

     |f> = SIGMA alpha_k |k>                                     (3.1)
             k 

   |f> iis in Hilb iff,

     SIGMA |alpha_k|^2 < infinity                               (3.2)
       k

     B |k> = /k |k-1>,   B! |k> = /(k+1) |k+1>          (3.3)

     Z  :=  { |f> = SIGMA alpha_k |k>
                      k
              : SIGMA (k+1) |alpha_k |^2 < infinity }     (3.4)
                  k

   Furthermore, B is closed on Z, and

        ([B, B!] - I)|f> = 0                              (3.5)

   for |f> in Z, and the operator,

        N := B! B                                 (3.6)

   is selfadjoint on Z having spectrum 0, 1, 2, ...
   with no multiplicity, and satisfies

     N|k> = k |k>                                (3.7)

   A Nesting of the Hilb(n) in Hilb

   Let the map

               |n, k>  ->  |k>          (3.8)

   for |n, k> is in Hilb(n), |k> is in Hilb,
   define the nesting as subspaces

       Hilb(2)  <  ... <  Hilb(n-1)  <  Hilb(n)  <  ...  <  Hilb

               B(n)     ->    B,
               B!(n)    ->    B!,
               G(n)     ->    I,

               Tr( G(n) ) = 0.                           (3.9)

               SIGMA <k|I|k>                       (3.10)
                 k

The last diagonal element of G(n) is

     <n, n-1|G(n)|n, n-1>  =  -(n-1).                (3.11)

This element is essentially responsible for the commutator substructure that appears for any finite n (See [Section IV] and [Section V]) and that disappears when the limit n->infinity is taken. A picture of the convergence is that the effect of this element is damped in the limit to combat the nascent singularity, and that the contributing edge effects eventually "fall off the edge" in a limit weaker than the norm topology.

Suppose we normalize the elements |f> is in Hilb(n),

                       <f|f>  =  1                         (3.12)

                <G(n)>  =  <f|f> - n |<f|n, n-1>|^2
                        =  1 - n |<f|n, n-1>|^2           (3.13)

   We would like

                lim <G(n)>  =  1                              (3.14)
                n->infinity

   Therefore require that for every n:

             n-1
     |f> = SIGMA alpha_k |k>                       (3.15)
             k=0


      n-1
     SIGMA (k + 1) |alpha_k|^2 < infinity          (3.16)
      k=0

   So, as n->infinity,
   |<f|n, n-1>|^2 must approach zero faster than n^(-2).

Let |k> in Hilb be the eigenbasis of the number operator N in the standard representation of CCR associated with creation and annihilation operators and the simple harmonic oscillator. Let SHA be the shift operator on Hilb defined by

               SHA |k>  =  |k-1>

       B(k_max) |f>  =  B(k_max + 1) |f>  =  B |f>

       lim  B(n > k_max) |f>
       n->infinity 

       lim  B!(n > k_max) |f>
       n->infinity

       G(k_max) |f>   =  G(k_max + 1) |f>  =  I |f>

       lim  G(n > k_max) |f>  =  I |f>
       n->infinity

A point to be made with respect to the formalistic sensibility of FCCR is the following: In the usual QM formalism we postulate the strong form of CCR, and then as one might with any Lie algebra, seek its representations within an algebra of operators acting on some Hilbert space. It then becomes clear that not only must the space in question be infinite dimensional, but that at least one of the operators p and q must be unbounded. The only known way to deal concretely with the unbounded operator situation is to restrict the domain of definition, and try to find one such, that is dense in the Hilbert space and invariant under the action of B and B!. It turns out that this is usually possible. If, however, we begin by allowing that CCR should be postulated in its projective form, so that it is to hold only when there is a restriction of states to some proper, but "sufficiently large" subspace of the projective Hilbert space, then the search for representations will automatically yield the finite dimensional cases presented here.

The simple act of suppressing the subspace spanned by |n, n-1>, is premature and not the only way of enacting the restriction as can be seen in the sections that deal with invariance and symmetry groups and with uncertainty relations.

That the Heisenberg algebra is an Inönü-Wigner contraction of su(2) is no news; it has been known for about 40 years. [Hioe 1974a]. This contraction is really a matter of metricity on the category of Lie algberas; Robert Hermann suggested on several occasions that there there were more general limiting procedures of interest for physics. Consider this as one of them.

A second point to be made for the sensibility of FCCR has to do with the notion of "canonically conjugate pair". The idea stemming from the Hamiltonian or Canonical formulation of classical mechanics, expresses a formal symplectic symmetry in pairs of physical variables. The physical point of this symmetry, however, is the specification of generators of groups of motions. One can demonstrate a generalized definition for canonically conjugate pairs in finite dimensional linear spaces that also fulfills this desire for such specification and provides the formal symmetry.

In classical Hamiltonian mechanics and in quantum mechanics as well as quantum field theory and special relativity, we define "momentum", conceptually, as the generator of position translations, and in a conjugate way say that position is the generator of momentum translations. To say this in the language of quantum mechanical operators on a projective Hilbert space,

          exp( i alpha Q ) |f(q)>  =  |f(q +  alpha)>
   and
          exp( i beta P ) |f(p)>  =  |f(p +  beta)>

   or in terms of an operator algebra,

          exp( +i alpha Q ) P exp( -i alpha Q ) =  P +  alpha I
   and
          exp( +i beta P ) Q exp( -i beta P ) =  Q +  beta I

Let A and B be two Hermitean operators acting on an n-dimensional complex Hilbert space Hilb(n). Consequently, the eigenvalues of A and B, a_k and b_k respectively, are real. Further let the spectra Sp( A ) and Sp( B ) be multiplicity free. Then the eigenspaces of A and B are all one dimensional, mutually orthogonal, and can be taken as normalized with respect to the inner product:

          A |a_k>  =  a_k |a_k>

          B |b_k>  =  b_k |b_k>

          <a_k|a_j>  =  delta_k_j

          <b_k|b_j>  =  delta_k_j

          A^k |f_A>

   span Hilb(n) and are mutually orthogonal, and similarly,

          B^k |f_B>

   Let
         |a, k>  =  A^k |f_A>/(/<f_A|A^2k|f_A>)
   and
         |b, k>  =  B^k |f_B>/(/<f_B|B^2k|f_B>)

   where the index k = 0, 1, ..., n-1, so that

          <a, k|a, j>  =  delta_kj
   and
          <b, k|b, j>  =  delta_kj

          A^k |f_A>

For constants alpha_k, beta_k depending on the operator structure, and for k = 0, 1, ..., n-1

          A |a, k>  =  alpha_k+1 |a, k+1>

          B |b, k>  =  beta_k+1 |b, k+1>

   where it is understood that the cyclic bases' indicies are cyclic,

          |a, 0>  =  |a, n>

          |b, 0>  =  |b, n>

A and B are defined to be a generalized canonically conjugate pair iff the following equalities hold for the sets of orthonormal bases:

          { |b, k> }  =  { |a_k> }

          { |a, k> }  =  { |b_k> }

          U_p(A, B) |a, k>  =  |b_k>

          Fr(A, B) |a_k>  =  |b_k>

	   Sp( A )  =  Sp( B )

The operators Q(n) and P(n) are proved to have the same spectrum, (7.11) and are, in fact a generalized canonically conjugate pair [corollary 8.1.2].

Limits are discussed again in [Section XVII] allowing for physical units and scaling functions that are powers of n.

   Addendum Notes and questions:

   Can the definition be meaningfully extended to from
   Hermitean operators to normal operators?

   Deal with modification due to spectral multiplicity.
   The eigenspaces are then not all of dimension one.
   Are such eigenspaces motion attractors?

   Degeneracy of the energy spectrum occurs in atomic
   quantum mechanics, and is broken typically by perturbation
   of the energy by some environmental potential field.
   An m-dimensional oscillator has a degenerate energy spectrum.

   What does one and can one expect with regard to possible
   multiplicity of a time operator's spectrum?

   Can anything be said about the commutator [A, B]?

   <a, k|[A, B]|a, j>  =
        alpha_j+1* b_j delta_j_(k+1) - alpha_j+1 b_k delta_k_(j+1)

   Are both equalities in the definition generalized canonically
   conjugate pair really necessary, i.e. independent?
   I think so.

   When are the alpha_k and beta_k real?

   Reversing the direction of the cycle?

   Suppose, zero is in the spectrum of A.
   Then there is a eigenvector, call it |a_0> such that

                 A |a_0>   =  0

   What is the concept of equivalence.
   How many  generalized canonically conjugate pairs are there?

   Defining a conjugate time operator given a
   multiplicity free Hamiltonian H:
   See [Lemma 8.20]

   0) Find a cyclic vector for the Hamiltonian H

   1) Find the cyclic orthonormal basis for H

   2) On this basis construct a Number operator N_t(n)
      so that there are also creation and annihilation operators
      for time.  For some suitable constant with dimensions of time
      say tau, the time operator is defined as

                              tau N_t(n)

      Clearly then H acts cyclically on its own cyclic basis
      which is the eigenbasis of the time operator.

      Does the time operator necessarily act cyclically on the
      Hamiltonian's eigenbasis?  This is probably where specifying
      the commutator comes in.

      Such a time operator (any time operator) should interfere with
      the equivalence of the Heisenberg and Schroedinger pictures.

      How does the time operator relate relativistically
      to the position operator?
      See [Appendix K]