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FCCR Limits as n->infinity
and Quantum Mechanical Analogs

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)

   |f> iis in Hilb iff,

     SIGMA |alpha_k|^2 < infinity                               (3.2)

Define the linear operators B, and B! by their action on the basis |k> as

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

with |-1> := 0. Then, B and B! both closed operators, can be extended to a common invariant domain Z dense in Hilb, where

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

   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)

With regard to the above, stated without proof, see [Putnam 1967] and [Reed 1972].

   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

We would like to have in the limit as n -> infinity, that

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

where I is the identity operator of the C*-algebra of bounded operators, acting on Hilb. It is, of course, a bounded operator. While for any finite n, G(n) is clearly of finite rank, the trace of G(n) exists and

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

Vanishing of the trace of G(n) is necessary since it is the commutator of two finite matrices. The identity, however, is not even of trace class. since

               SIGMA <k|I|k>                       (3.10)

is not bounded. A sufficient condition for a bounded operator not of finite rank to be of trace class, is that the spectral values have only one accumulation point, that it be zero, and that convergence to zero is sufficiently rapid that the sum analogous to (3.10) be bounded. A simple convergence in the norm topology is impossible since B, B!, and N are all unbounded operators.

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)

effectively passing to the projective ray space, where a "state" is associated with the equivalence class {e^i theta |f>}. Then the expectation value <G(n)> defined in equation (8.1) is given by

                <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)

[NB: In the Schoedinger realization 'I' is a continuous idendity, while in the oscillation realization, it is discrete.] The simple convergence of a linear combination (3.1) on an orthonormal set requires that (3.2) be met. This requires that the |alpha_k|^2->0 faster than (1/k). But, we want to make the operator transit, not into the full Hilb, but rather into the domain Z, defined by equation (3.4). Consider the algebra of operators Alg(Hilb(n)) to act not on Hilb(n), but rather on the projective space of equivalence classes of elements of Z such that for every n, |f_1(n)> is equivalent to |f_2(n)> both in Z, iff their projections onto the n-dimensional subspace isomorphic to Hilb(n) are equal. What they are on the complementary subspace is undetermined except by the requirement of membership in Z.

   Therefore require that for every n:

     |f> = SIGMA alpha_k |k>                       (3.15)

     SIGMA (k + 1) |alpha_k|^2 < infinity          (3.16)

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

This is sufficient to insure the existence of the limit (13.4). For further possibilities in considering the limit as n->infinity see [Appendix D] where a limit is taken after a compactification of a hyperbolic complex space that is a submanifold of Hilb(n) of physical vectors is performed.

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>

Let B be the annihilation operator. B, which is unbounded, is defined on Z the subspace of Hilb spanned by finite linear combinations of |k>. The set Z is dense in Hilb, and invariant under the action of B and B! . Let |f> be in Z. Then, in a finite linear combination of |k>, there exists a k_max, so

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

Therefore, for any |f> in Z,

       lim  B(n > k_max) |f>

exists, and is equal to B|f>. Similarly, one can show that

       lim  B!(n > k_max) |f>

exists, and is equal to B!|f>, and that

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


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

Therefore, FCCR converges to CCR in the strong operator topology.

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)>
          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
          exp( +i beta P ) Q exp( -i beta P ) =  Q +  beta I

Passing from kinematics to dynamics we also say that "energy" is generator of time translations. If energy and time were related in quantum mechanics as operators in the same way that position and momentum are, and one could say more meaningfully that time was the generator of energy translations.

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

Since both A and B are multiplicity free, there exist cyclic vectors |f_A> and |f_B> for each operator. That is for k <= n, the vectors

          A^k |f_A>

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

          B^k |f_B>

         |a, k>  =  A^k |f_A>/(/<f_A|A^2k|f_A>)
         |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
          <b, k|b, j>  =  delta_kj

Call each of these spanning sets the orthonormal cyclic basis for A and B, respectively. Clearly any one of the generated vectors

          A^k |f_A>

or any one of the vectors of the orthonormal cyclic basis could be taken as a primary cyclic vector.

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> }

There is a ambiguity regarding the relative orderings of the bases, but there is always a linear permutation transformation U_p(A, B) (which is unitary) that carries the set { |a, k> } to the set { |b_k> } so that

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

Also, the orthonormal eigenbases are connected by a unitary transformation,

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

(Two normal operators equivalent by similarity are unitarily equivalent. [Putnam 1967], p. 12. While the eigenbases are connected by a unitary transformation, the operators A and B may not be. Fr(A, B) maps A to B only if

	   Sp( A )  =  Sp( B )

Since the spectrum of an operator is an invariant under similarity transformations.

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]

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Created: August 1997
Last Updated: August 5, 2000
Last Updated: January 5, 2006