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The Discrete n-Fourier Transforms
Fr(n) and UPSILON(n)

Define an abstract Fourier transform PHI in a C*-algebra as a unitary and therefore diagonalizable element with its spectrum constrained as follows. There exists an ordering by the index k of its orthonormal eigenbasis |k> in the associated Hilbert space so that the spectral resolution can be written:

          PHI  =  SIGMA_k  |k> i^k <k|

   Then, PHI is unitary and idempotent of order 4.

          PHI^4  =  I

   which also expresses PHI as one of the fourth roots of unity.

    <Theorem 7.1>:

   The space of Fourier transforms M_PHI in the unitary group U,
   is a single orbit in U, on which U acts transitively in its
   adjoint action on itself.

   A Fourier transform always has eigenvalues sqrt(1), sqrt(i); 
   and is an element of SU(n) only when

             n  =  4 nu + 1
             for nu  =  1, 2, 3, 4, ...
   i.e., for
             n  =  5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, ...


   Consider the adjoint action of U as a group of automorphisms on itself,

        v  ->  U! v U,  for v in U.

If PHI is a Fourier transform, since the spectrum is invariant under U, then so is (u! PHI u), where u is any element of the unitary group U. Any Fourier transform generates, by the above action of U, an orbit in U, on which U acts transitively. For any faithful, irreducible representation space for U, any orthonormal basis has a Fourier transform. These Fourier transforms are exhaustive. Any orthonormal basis can be mapped to any other orthonormal basis unitarily. So any Fourier transform can be mapped to any other Fourier transform unitarily. Therefore there is only one Fourier orbit in U.

   The Fourier transform being in SU(n) only for the prescribed
   values of n is trivial and seen by inspection.


   The Fourier transform

        Fr(4) = Diag[1, i, -1, -i]

in a four dimensional space is unitarily equivalent to a cyclic operator. If the unitary group U = U(4n), a defining carrier space Hilb(4n), is reduced under the action of PHI into the direct sum of n copies of Hilb(4).

As a vector space, the C*-algebra Alg(Hilb(n)) has a natural decomposition under the main involution of Hermitean conjugation into symmetric and antisymmetric subspaces that are left invariant by the involution.

        Alg( Hilb(n) )  =  H( Hilb(n) ) + S( Hilb(n) )

   where under the main involution of Hermitean conjugation

        Alg( Hilb(n) )!  =  H( Hilb(n) ) - S( Hilb(n) )

Every Fourier transform, being unitary, also leaves these subspaces invariant.

The adjoint action of a Fourier transform PHI on Alg(Hilb(n)) breaks Alg(Hilb(n)) as a vector space into four subspaces which are cyclically permuted under the action of PHI.

Note that Fr(n) acting on the Hermitean generators Q(n) and P(n), permutes only two subspaces. There is a subspace spanned by the operators Q^k(n), k = 0, 1, ..., n-1 and a subspace spanned by P^k(n), k = 0, 1, ..., n-1. They share the one dimensional subspace spanned by I(n).

[Weyl 1946], pp. 178, 194 The isotropy group at PHI, U^PHI is defined as the subgroup of U which leaves PHI fixed. Leaving PHI fixed is equivalent to commuting with PHI. If two elements of U commute with PHI, then they commute with each other, since both must be diagonal when PHI is diagonal. So the isotropy group, U^PHI is Abelian. If U = U(n), then U^PHI is an n-parameter Abelian subgroup of U(n). The isotropy group at any element of U is in fact conjugate to U^PHI, so there is essentially only one isotropy group.

Any u is unitarily equivalent to an element of U^PHI. In the basis |k> for PHI, all elements of the form

        Diag[..., lambda_k, ...],

   where lambda_k, complex unimodular,

        lambda_k lambda_k*  =  1,

are in the isotropy group. Any element of U is reducible to such a form by a similarity transformation. Each element of the isotropy group is a representative member of an equivalence class of elements of U, where the equivalence is the equality of ordered spectra. Let pi_n be the group of permutations on n objects. If U = U(n), then pi_n is isomorphic to a discrete subgroup U_ pi of U, that acts in U considered as a group of automorphisms of itself, as the group of "basis" permutations at an element of u say PHI. It is clear that U_ pi leaves U^PHI invariant.

Although this obvious abstraction of the Fourier transform *must* exist someplace in the mathematical literature, I have no references, simply because I have not seen nor heard of any.

Getting on with the specifics of Fourier transforms in FCCR context: It is easy to see by direct calculation from the definitions in section II, that the relations

             [N(n), B(n)]   =  -B(n)                          (7.1a)

             [N(n), B!(n)]  =  +B!(n)                         (7.1b)

   hold, and that consequently

             [N(n), Q(n)]  =  -i P(n)                         (7.2a)

             [N(n), P(n)]  =  +i Q(n)                         (7.2b)

   Without proof, the well known (BCH) Baker-Campbell-Hausdorff
   formula is:

         exp( a B ) A exp( -a B )

         =  SIGMA  (a^k/k!) C(A: B, k)                         (7.3)


     C(A: B, k)  :=  [B, C(A: B, k-1)]                         (7.4a)

               =  [B, [B, ... [B, A] ...] ]

               =  SIGMA (-1)^j (k j) B^(k-j) A B^j

     C(A:B,0)  :=  A                                           (7.4b)

   defines the k-fold commutator of A by B, and

     [A, B] := AB - BA                                         (7.4c)

   Using equations (7.4) above, equations  (2.7),
   and defining a unitary exponential operator of N(n)
   we have:

     exp( i alpha N(n) ) Q(n) exp( -i alpha N(n) )
                 =   Q(n) cos( alpha ) + P(n) sin( alpha )     (7.5a)

     exp( i alpha N(n) ) P(n) exp( -i alpha N(n) )
                 =  -Q(n) sin( alpha ) + P(n) cos( alpha )     (7.5b)

     exp( i alpha N(n) ) B(n) exp( -i alpha N(n) )
                 =   exp( -i alpha B(n) )                      (7.5c)

     exp( i alpha N(n) ) B!(n) exp( -i alpha N(n) )
                 =   exp( -i alpha B!(n) )                     (7.5d)

   If alpha is taken as  pi/2, define n-Fourier transform as

             Fr(n)  =  exp( i (pi/2) N(n) )                    (7.6)

   then it follows that:

             Fr(n) Q(n) Fr!(n)   = +P(n)                       (7.7a)

             Fr(n) P(n) Fr!(n)   = -Q(n)                       (7.7b)


             Fr(n) B(n) Fr!(n)   = -i B(n)                     (7.8a)

             Fr(n) B!(n) Fr!(n)  = +i B!(n)                    (7.8b)

             Fr^2(n) Q(n) Fr!^2(n)   = -Q(n)                   (7.9a)

             Fr^2(n) P(n) Fr!^2(n)   = -P(n)                   (7.9b)

             Fr^4(n)  =  I(n)                                  (7.10)

These are the usual properties of the Fourier transform in QM, between position and momentum representations. Note that (7.9) above is usually interpreted in QM to mean that Fr^2(n) has the property of spatial inversion. Also, since N(n) commutes with G(n), Fr(n) leaves G(n) invariant.

Since Fr(n) is unitary, and P(n) & Q(n) are Hermitean, we have

             Sp( Q(n) )  =  Sp( P(n) )                         (7.11)

that is P(n) and Q(n) have the same spectral set, and the spectral set is real. Furthermore, since Fr^2(n) is unitary, if q in Sp( Q(n) ), then so is -q, and iff n is odd then 0 is in Sp( Q(n) ). Similarly, then for P(n). (This duplicates results of an analysis of Sp( Q(n) ) in [Section IX].


    <Proposition 7.1 >:

   Q(n) and P(n) are both invertible if and only if n is even.

   This is also clear from the detailed analysis of eigenvalues
   in [Section IX], and the fact that zero is a root of any odd
   Hermite polynomial.

The n-Fourier transform provides the map between the spectral representations of Q(n) and P(n). The common spectrum of Q(n) and P(n) is non-degenerate. See Lemma 8.2. To establish notation we write the eigenvalue problem as:

        Q(n) |q(n, k)>  =  q(n, k) |q(n, k)>                   (7.12a)

        P(n) |p(n, k)>  =  p(n, k) |p(n, k)>                   (7.12b)

   and the spectral representations as:

        Q(n)  = SIGMA    |q(n, k)> q(n, k) <q(n, k)|            (7.13a)

        P(n)  = SIGMA    |p(n, k)> p(n, k) <p(n, k)|            (7.13b)

   Now, since

        Fr(n) Q(n)  =  P(n) Fr(n),

   it follows that

         P(n) Fr(n) |q(n, k)>  =  q(n, k) Fr(n) |q(n, k)>

   establishing a fixed correlation for the indices that label
   the eigenvectors of Q(n) and P(n) so that:

                 q(n, k)  =  p(n, k)                           (7.14)


                 Fr(n) |q(n, k)>  =  |p(n, k)>                 (7.15)

   in exact analogy to the standard integral Fourier transform of QM.

As in the harmonic oscillator problem of QM, the canonical basis |n, k> can be generated by repeated application of the n-creation operator:

              |n, k>  =  (1/sqrt(k!)) B!^k(n)|n, 0>             (7.16)


                 B^n(n)  =  0.                                 (7.17)

   For this basis

              Fr(n) |n, k>  =  i^k |n, k>                      (7.18)

   with invariant determinant

              Det( Fr(n) )  =  exp( i (pi/4)n(n-1) )           (7.19)

   and trace

              Tr( Fr(n) )  =  1 - i^n cos( n(pi/4) )             (7.20)

   Equation (7.20) follows from the identity

         SIGMA exp( i k theta )  =

         1 + exp( i n theta/2 ) sin( (n-1) theta/2 )/sin( theta/2 )

Equations (7.19) and (7.20) will, of course, hold for any finite Fourier transform.

It is clear that for any orthonormal basis of an Hilb(n), the self-representation of the basis is canonical, and so the construction of section II will go through and that therefore an n-Fourier transform can be constructed, which has all the present properties. We discuss the concepts of transformations and equivalence further in [Section IX].

There is another Finite Fourier transform that arises in the present context from considering the cyclic transformation C(n) used in the polar decomposition of B!(n) and B(n), introduced in [Section II]. As remarked there, it is unitary and has as its eigenvalues, the n-th roots of unity. Now consider the diagonalization problem. It is important to keep in mind that diagonalization concept is invariant under the discrete group of basis permutations; any diagonalizing transformation may include, an arbitrary permutation of basis. Thus there are n! possible diagonalizing transformations. Out of these, we can pick one by specifying the one to one basis correspondence. We will choose a basis ordering that corresponds to that of |n, k> by the index k. Other than cyclic permutations of this order seem to mix up the concept of "time ordering" as determined by phase. Choosing our particular ordering, amounts to choosing a zero time with discrete time values equidistributed on the unit circle in the complex plane. Since we already know the eigenvalues, let UPSILON(n) be a diagonalizing transformation for C(n), so that

       UPSILON(n) C(n) UPSILON!(n)  =  exp( i 2 (pi/n) X(n) )      (7.21)

   where X(n) has the the same spectrum as N(n),

   i.e., {0, 1, 2, ..., n-1} on some basis |xi(n, k)>, so that,

       UPSILON(n) |n, k>  =  |xi(n, k)>                              (7.22)


       X(n) |xi(n, k)>  =  xi_k |xi(n, k)>                           (7.23)

The basis |xi(n, k)> can be constructed most easily as <n, j|xi(n, k)>. The diagonalizing transformation is then constructed by observing the permutation action of C(n).

        |xi(n, k)>  =

        (1/sqrt(n)) SIGMA exp( i(2 pi/n)kj ) |n, j>                  (7.24)


        C(n) |xi(n, k)>  =  exp( i 2 pi xi_k/n ) |xi(n, k)>          (7.25)

   The columns of the diagonalizing transformation are
   constructed from the eigenvectors, so,

        <n, k| UPSILON(n) |n, j>  =
                   (1/sqrt(n)) exp( i(2 pi/n) kj )        (7.26)

   One has immediately that

     UPSILON^(-1)(n)  =  UPSILON!(n)  =  UPSILON*(n)       (7.27)

   The unitarity of UPSILON(n) is expressed by the identity

      SIGMA exp( i(2 pi/n)(k - j)q ) =  n delta_(kj)       (7.28)

   Equation (7.28) can also be verified by considering the invariant trace

                       | 0, k not= mn
     Tr( C^k(n) )  =   |                                   (7.29)
                       | n, k = mn

   for any integer m.  But, -n < (k-j) < +n, and the result follows.

     <n, k| UPSILON^2(n) |n, j>  =

               =  (1/n) SIGMA exp( i(2 pi/n)(k + j)q )     (7.30)

   In the |n, k> basis,

                       |1 0 0 0      ...      0 0 0|
                       |0 0 0 0      ...      0 0 1|
                       |  ...        ...      0 1 0|
                       |  ...        ...      1 0 0|
     UPSILON^2(n)  =   |0 0 0 0                    |        (7.31)
                       |  ...                     .|
                       |0 0 0 1      ...          0|
                       |0 0 1 0      ...          0|
                       |0 1 0 0      ...          0|


     UPSILON^4(n)  =  I(n)                                  (7.32)

   which holds in any basis.

     UPSILON^3(n)  =  UPSILON!(n)  =  UPSILON*(n)           (7.33)

   So UPSILON(n) satisfies the criteria for an abstract Fourier

We now develop the relation between X(n) and N(n). By explicit calculation of the diagonalized C(n), given by UPSILON(n) C(n) UPSILON!(n), with UPSILON(n) considered as an active transformation we find the matrix on the basis |n, k> to be:

        UPSILON(n) C(n) UPSILON!(n) =
             exp(i(2 pi/n)Diag[0, n-1, n-2, ..., 2, 1])  (7.34)

   So in equation  (7.25)  we can take

        xi_k  =  -k                                      (7.35)

   In a short but elegant paper, the authors in [Chaturvedi 1998] show
   that UPSILON(n) will transform N(n) on its eigenbasis to its maximally
   off diagonal form.

From here to the end of this section, unless other bases are specified, all operators will be considered given in the eigenbasis of N(n) and considered to be active transformations.
One can also consider xi_k when not in the exponential as being in domain of J_n of the integers mod n. J_n has all the properties of an integral domain, except that of the law of cancellation. If n is prime p, however, J_p is an integral domain, and with its natural definition of multiplication a finite field of characteristic p. A formulation of FCCR in terms of finite fields is considered separately in [Appendix J]. Along this line [Ahmavaara 1965a], [Ahmavaara 1965b] suggested some large finite field for standard quantum theory.

   If we define as also in equation  (14.4) 

     OMEGA(n)  :=  (n-1)I(n) - N(n)                              (7.36)


     UPSILON^2(n) N(n) UPSILON^2(n)  =  C!(n) OMEGA(n) C(n)      (7.37)

   and with definition  (2.18c),

     UPSILON^2(n) M(n) UPSILON^2(n)  =  C!^2(n) OMEGA(n) C^2(n)


     UPSILON(n) C(n) UPSILON!(n)
      =  exp( i(2 pi/n) C!(n) UPSILON(n) C(n) )
      =  exp( i(2 pi/n) UPSILON^2(n) N(n) UPSILON^2(n) )
      =  UPSILON(n) exp( i(2 pi/n) UPSILON(n) N(n) UPSILON!(n) ) UPSILON!(n)
      =  UPSILON!(n) exp( i(2 pi/n) UPSILON!(n) N(n) UPSILON(n) ) UPSILON(n)

   To diagonalize C(n) is also to diagonalize X(n).  Then,

     UPSILON(n) exp( i(2 pi/n) F(n) ) UPSILON!(n)

     =  UPSILON(n) exp( i(2 pi/n) UPSILON(n) N(n) UPSILON!(n) ) UPSILON!(n)


     exp( i(2 pi/n) F(n) )  =  exp( i(2 pi/n) UPSILON(n) N(n) UPSILON!(n) )

   determining X(n) in terms of UPSILON(n) and N(n) up to an additional
   multiple of the identity.  So, for some integer k, we can then represent
   X(n) as

        X(n)  =  UPSILON(n) N(n) UPSILON!(n) + k I(n)


        F(n)  :=  UPSILON(n) N(n) UPSILON!(n)

   Simple observation or numerical calculation for various n,
   (this is no proof) shows that k = 0, so finally,

        X(n)  =  UPSILON(n) N(n) UPSILON!(n)

   For convenience further define:

     F_+(n)  :=  UPSILON(n) N(n) UPSILON!(n)                     (7.38a)

     F_-(n)  :=  UPSILON!(n) N(n) UPSILON(n)                     (7.38b)

     N_+(n)  :=  N(n)                                            (7.38c)

     N_-(n)  :=  C!(n) OMEGA(n) C(n)                             (7.38d)

     M_+(n)  :=  M(n)                                            (7.38e)

     M_-(n)  :=  C!^2(n) OMEGA(n) C^2(n)                         (7.38f)

     E_+(n)  :=  exp(+i(2 pi/n)N(n)) F_+(n) exp(-i(2 pi/n)N(n))      (7.38g)

     E_-(n)  :=  exp(-i(2 pi/n)N(n)) F_+(n) exp(+i(2 pi/n)N(n))      (7.38h)


     UPSILON(n) F_+(n) UPSILON!(n)  =  UPSILON^2(n) N(n) UPSILON!^2(n)
                       =  C!(n) OMEGA(n) C(n)

     UPSILON^2(n) F_+(n) UPSILON!^2(n)  =  UPSILON^3(n) N(n) UPSILON!^3(n)
                         =  UPSILON(n) C!(n) OMEGA(n) C(n) UPSILON!(n)
                         =  F_-(n)

     UPSILON^3(n) C(n) UPSILON!^3(n)   =
     UPSILON^2(n) exp( i2 pi/n M(n) ) UPSILON!^2(n)

   Also by simple computation

     M(n)  =  C(n) N(n) C!(n)                           (7.39)

   With the exponential representations:

     C(n)  =  exp( +i(2 pi/n) F_+(n) )                     (7.40a)
           =  exp( -i(2 pi/n) F_-(n) )

     C!(n)  =  exp( +i(2 pi/n) F_-(n) )                    (7.40b)
            =  exp( -i(2 pi/n) F_+(n) )

   where F_(+|-)(n) is Hermitean, then UPSILON(n) cycles sets of operators:

                | C(n)                |
                |                     |
                | exp(-i(2 pi/n)N(n)) |
     UPSILON(n) |                     | UPSILON!(n)  =
                | C!(n)               |
                |                     |
                | exp(+i(2 pi/n)N(n)) |

                          | exp(-i(2 pi/n)N(n)) |
                          |                     |
                          | C!(n)               |
                          |                     |
                          | exp(+i(2 pi/n)N(n)) |
                          |                     |
                          | C(n)                |


                | F_+(n)            |                 | C!(n)OMEGA(n)C(n) |
                |                   |                 |                   |
                | C!(n)OMEGA(n)C(n) |                 | F_-(n)            |
     UPSILON(n) |                   | UPSILON!(n)  =  |                   |
                | F_-(n)            |                 | N(n)              |
                |                   |                 |                   |
                | N(n)              |                 | F_+(n)            |


                | M(n)                  |
                |                       |
                | E_+(n)                |
     UPSILON(n) |                       | UPSILON!(n)  =
                | C!^2(n)OMEGA(n)C^2(n) |
                |                       |
                | E_-(n)                |

             | E_+(n)                |
             |                       |
             | C!^2(n)OMEGA(n)C^2(n) |
             |                       |
             | E_-(n)                |
             |                       |
             | M(n)                  |


             | G(n)                      |
             |                           |
             | UPSILON(n)G(n)UPSILON!(n) |
  UPSILON(n) |                           | UPSILON!(n)  =
             | C!^2(n)G(n)C^2(n)         |
             |                           |
             | UPSILON!(n)G(n)UPSILON(n) |

                  | UPSILON(n)G(n)UPSILON!(n) |
                  |                           |
                  | C!^2(n)G(n)C^2(n)         |
                  |                           |
                  | UPSILON!(n)G(n)UPSILON(n) |
                  |                           |
                  | G(n)                      |


   Also, as identities,

     C^2(n) UPSILON!^2(n) G(n) UPSILON^2(n) C!^2(n)  =  G(n)       (7.45)

     exp( i(2 pi/n) OMEGA(n) )  =  C(n) exp( -i(2 pi/n) N(n) ) C!(n)

                          =  exp( -i(2 pi/n) M(n) )

forming one cycle of unitary operators and three cycles of Hermitean operators under UPSILON(n).

That a pair of a cycle can be expressed as complex conjugates of one another is a basis dependent statement. Such a relation is not preserved under unitary or pseudounitary transformations. Since F_+(n) is directly related to a time operator, as shown with specificity in the later [Derivation of Local Newtonian Time], it would appear that in the |n, k> basis, time reversal is effected by complex conjugation. This is consistent with the fact that, also in this basis,

     Q*(n)  =  +Q(n)                              (7.46a)

     P*(n)  =  -P(n)                              (7.46b)

   when Q(n) and P(n) are to be treated like the position and
   momentum operators of QM.

It must be remarked that while Fr(n) happens to be a unitary transformation that commutes with G(n), the Fourier transform UPSILON(n) does not commute with G(n), and is therefore not a kinematically admissible transform if preservation of G(n) is to be a physical principle [Section XI]. The situation is similar to that of the diagonalizing transformations of Q(n) and P(n). [Section IX] and [Sections X]. Although, their eigenvectors are mathematically available, they are not physically available. For the oscillator analog, the energy as a Hamiltonian depends linearly on N(n) and G(n). The vectors |n, k> are then energy eigenvectors. The operator F(n) is related to a time operator, paired with the Hamiltonian. With the energy diagonalized, the time eigenvectors are not physically available. The picture is then that if the energy eigenvectors are physically available, then also, those of position, momentum and time are not. They all must possess an intrinsic fuzziness if the strict invariance of G(n) is insisted upon.

The invariance group of G(n) is not unitary, and so the alleged energy operator of the oscillator cannot be a real Hamiltonian: it is not an invariant of the motion, which is entirely the point of being a Hamiltonian function.

   Although, in general for any n > 2,

        [Fr(n), UPSILON(n)] not= 0,

   we have:

    <Theorem 7.2>:

   If n is odd, then

        [Fr^2(n), UPSILON^2(n)]  not=  0                          (7.47)

   and if n = 2 nu is even

        [Fr^2(n), UPSILON^2(n)]  =  0                             (7.48)

        C^nu(n)  =  C!(n)


   Calculate the commutator in |n, k> basis, noting equation
    (7.31).  Its zero or non-zero value will then hold in any basis.


   By simple inspection:

    <Lemma 7.1>:

   For any n

     Fr(n) C(n) Fr!(n)   =  (+i)^(n-1) ( C(n) - SHA(n) ) - i SHA(n)
     Fr(n) C!(n) Fr!(n)  =  (-i)^(n-1) ( C!(n) - SHA!(n) ) + i SHA!(n)

     C(n) Fr(n) C!(n)   =  +i Fr(n)
     C(n) Fr!(n) C!(n)  =  -i Fr!(n)

    <Theorem 7.3>:

   Iff n is a multiple of 4,


        {C(n), Fr^2(n)}  =  0

        {C^2(n), Fr(n)}  =  0

        [C^4(n), Fr(n)]  =  0

        Fr(n) C(n) Fr!(n)   =   -i C(n)

        Fr^2(n) C(n) Fr^2!(n)   =   -1 C(n)
        Fr^3(n) C(n) Fr^3!(n)   =   +i C(n)

        Fr(n) C!(n) Fr!(n)  =   +i C!(n)

        C(n) Fr(n) C!(n)    =  +i Fr(n)
        C^2(n) Fr(n) C^2!(n)  =  -1 Fr(n)
        C^3(n) Fr(n) C^3!(n)  =  -i Fr(n)
        C^4(n) Fr(n) C^4!(n)  =  +1 Fr(n)
        C^5(n) Fr(n) C^5!(n)  =  +i Fr(n)
        C^6(n) Fr(n) C^6!(n)  =  -1 Fr(n)
        C^7(n) Fr(n) C^7!(n)  =  -i Fr(n)


        C!(n) Fr(n) C(n)  =  -i Fr(n)

   Further with n = 4 nu

        UPSILON(n) Fr(n) UPSILON!(n)  =   C^nu(n)

        UPSILON(n) Fr!(n) UPSILON!(n)  =   C^nu!(n)

         Tr( Fr(n) )  =  0

         Det( Fr(n) )  =  (-1)^nu

   and the operator


   also now has the properties of a Fourier transform;

   while it is generally true for any n that

        UPSILON(n) Fr!(n) UPSILON!(n), UPSILON(n) Fr(n) UPSILON!(n),

        Fr(n) UPSILON!(n) Fr!(n), Fr(n) UPSILON(n) Fr!(n),

   are all Fourier transforms, it is also true for any n that,

        Fr(n) SHA!(n) Fr!(n)  =  +i SHA!(n)

        Fr!(n) SHA(n) Fr(n)  =  +i SHA(n)

        Fr!(n) SHA!(n) Fr(n)  =  -i SHA!(n)

   Proof: Left as an exercise

   With the time operators defined by

        t_+|-(n)  :=  (2 pi/(n omega_T)) F_+|-(n)                    (7.49)


        UPSILON(n) |n, k>  =  |t_k>                                  (7.50)

   mapping the "energy" vector with value (h-bar omega_H (k + 1/2))
   to the time vector with value (k tau_0) where tau_0 is some
   time quantum.  Then also as a numerical equality,

     <n, k| UPSILON(n) |n, k>  =

                          <t_k| UPSILON(n) |t_k>         (7.51)

   In the t(n) eigenbasis, UPSILON^2(n) inverts the
   direction of the clock sense and is thus a time reversal operator.
   Read  Theorem 7.2  then as saying that for n even the parity operator
   commutes with the time reversal operator.

   The following defines notation for the usual quantities associated
   with the eigenvalues of forward and reversed time operators:

   As a passive diagonalizing transformation of C(n) in the N(n)

     <n, k|UPSILON(n) C(n) UPSILON!(n)|n, j>  =

                           delta_kj exp( -i(2 pi/n) k )     (7.52)

   where k is understood mod n,

        -k  =  n - k  mod n.                                 (7.53)

   We define the forward clock basis vectors

        |t_+k>  =  UPSILON(n) |n, k>                              (7.54)

   and the backward clock basis vectors (merely a relabelling)

        |t_-k>  =  UPSILON!(n)|n, k>                              (7.55)

   so that  

        <t_+k|  =  <n, k|UPSILON!(n)                              (7.56a)


        <t_-k|  =  <n, k|UPSILON(n)                              (7.56b)


        C(n)   =  exp( +i(2 pi/n) F_+(n) )
               =  exp( +iomega_T t_+(n) )                   (7.57a)

        C!(n)  =  exp( +i(2 pi/n) F_-(n) )
               =  exp( +iomega_T t_-(n) )                   (7.57b)


        omega_T  =  (2 pi)/(n tau_0)                        (7.58)

    it is clear from  (7.40)  that

     exp( -iomega_T t_+(n) )  =  exp( +iomega_T t_-(n) )     (7.59)

   We have chosen this particular separation of time operator and
   frequency so that time operator eigenvalues are integral multiples
   of a necessary fundamental time quantum tau_0.

        <n, k|UPSILON(n) C(n) UPSILON!(n)|n, j>  =

     <t_-k| C(n) |t_-k> =  delta_kj exp( -iomega_T t_+k )             (7.60)


        t_+k  =  k tau_0                                   (7.61)

   Also then

     <t_-k| C(n) |t_-k> =  <t_+k| C!(n) |t_+k>                   (7.62a)


     <t_-k| C!(n) |t_-k> =  <t_+k| C(n) |t_+k>                   (7.62b)

   Also define

        t_-k  =  - k tau_0
             =  t_+(n-k)  (mod n)                   (7.63)


     t_+k + t_-k  =  0 (mod n)                   (7.64)


        t_+(n) |t_+k>  =  t_+k |t_+k>                   (7.65a)

        t_+(n) |t_-k>  =  t_-k |t_-k>                   (7.65b)


        t_-(n) |t_-k>  =  t_+k |t_-k>                   (7.66a)

        t_+(n) |t_-k>  =  t_-k |t_-k>                   (7.66b)

   Duration of a clock cycle:  T_n  =  n tau_0  =  2 pi/omega_T   (7.67)

   Clock precision: DELTAt = t_+(k+1) - t_+k  =  tau_0            (7.68)


     UPSILON^2(n) F_+(n) UPSILON!^2(n)  =  F_-(n)                 (7.69)

     t_-(n)  =  UPSILON^2(n) t(n) UPSILON!^2(n)                   (7.70)

defines a time operator of reversed clock time.

     UPSILON^2(n) N_+(n) UPSILON!^2(n)  =  N_-(n)                 (7.71)

so that energy is also reverted.

If the energies h-bar omega_H(k + 1/2) are considered as given with k understood as k (mod n), then energy reversion is equivalent to mapping positive to negative energies. Although energy differences can be understood, energies are not ordered in the strict sense. In both the energy basis and the time basis, the operation of complex conjugation is a consistent time reversal, in a relativistic sense in that time reversal is accompanied by energy reversion.

   There are similar maps

     UPSILON(n) C!^j(n) |n, k>  =  |t_k+j>                    (7.72)

where k+j is understood modulo n. This could also be a time-energy transform, where the reversions are about something other than zero. Either such a distinction has physical significance or it is a kind of gauge freedom. Then also consider this possibility for Fr(n).

Note: if the Fourier transforms Fr(n) and UPSILON(n) commute, then their product is also a Fourier transform.

For any n, UPSILON(n) leaves the subspace in the algebra of Hermitean operators that is spanned by N_+(n), F_+(n), N_-(n) = C(n)OMEGA(n)C!(n), and F_-(n) invariant. The action on these spanning operators as basis is equivalent to a cyclic transformation:

                |N_+(n)|                 | 0  1  0  0 | |N_+(n)|
                |      |                 |            | |      |
                |F_+(n)|                 | 0  0  1  0 | |F_+(n)|
     UPSILON(n) |      | UPSILON!(n)  =  |            | |      |    (7.73)
                |N_-(n)|                 | 0  0  0  1 | |N_-(n)|
                |      |                 |            | |      |
                |F_-(n)|                 | 1  0  0  0 | |F_-(n)|

   which is, in fact, the matrix C(4) = exp( i( pi/2)F(4) ),
   which is diagonalizable by UPSILON(4).

                          |  1   1   1   1 |
                          |                |
                          |  1   i  -1  -i |
     UPSILON(4)  =  (1/2) |                |                   (7.74)
                          |  1  -1  -1  -1 |
                          |                |
                          |  1  -i  -1   i |

   The analogy to the action of Fr(n) on Q(n) and P(n) and
   the inversion of Fr^2(n) is clear.

           |+Q(n)|            | 0  1  0  0 | |+Q(n)|
           |     |            |            | |     |
           |+P(n)|            | 0  0  1  0 | |+P(n)|
     Fr(n) |     | Fr!(n)  =  |            | |     |                   (7.75)
           |-Q(n)|            | 0  0  0  1 | |-Q(n)|
           |     |            |            | |     |
           |-P(n)|            | 1  0  0  0 | |-P(n)|

The time operators appear immediately in an exponential as generators of unitary transformations; that is in a Weyl form rather than a Heisenberg form. Formally then, a group is immediately considered rather than the algebra of the group. Since different Lie groups can have the same Lie algebra, it might be suggested that a further generalization is available by postulating FCCR structure in terms of the algebra instead of the group. This consideration is left open for the present.

   From equations  (7.57b)  and  (7.72)  we have

     exp( +iomega_Tt_+(n) j ) |n, k>  =  |n, k+j>          (7.76a)


     exp( +i(2 pi/n)N(n) j ) |t_k>  =  |t_k+j>             (7.76b)

That is, the operator (omega_T t_+(n)) is the generator of forward N-translations and the operator ((2 pi/n)N(n)) is the generator of forward t-translations. Then we have the following.

    <Theorem 7.4>:

   The operators (omega_T t_+(n))
   and ((2 pi/n)N(n)) are generalized canonically conjugate
   in the sense defined in  [Section III].

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