It is useful for various purposes to evaluate certain matrix elements and to prove some relations between them. Basic results and some messy details have been relegated to this section and stated in terse mathematical style. It should be used primarily as a reference.

The eigenvectors |q(n, k)> of Q(n) and the eigenvectors |p(n, k)> of P(n) and are null with respect to G(n): <q(n, k)| G(n) |q(n, k)> = 0 <p(n, k)| G(n) |p(n, k)> = 0 and <q(n, k)| P(n) |q(n, k)> = 0 <p(n, k)| Q(n) |p(n, k)> = 0 Proof: The conclusion results from the commutation relation (2.8) <q(n, k)| [Q(n), P(n)] |q(n, k)> = i <q(n, k)| G(n) |q(n, k)> and similarly for |p(n, k)> QED

The result depends only on the fact that Q(n) and P(n) are Hermitean, hence have real eigenvalues and only one dimensional invariant subspaces.

Note that in the present context, the "classical" QM result for the harmonic oscillator, of vanishing of the expectation values of Q and P, is differently interpreted. The Q(n) and P(n) eigenstates are more like "light cone coordinates". See the discussion in [Section IX].

<Lemma 8.2>: The eigenvalues of Q(n) and P(n) are nondegenerate. For k not= j, q(n, k) - q(n, j) not= 0. p(n, k) - p(n, j) not= 0. Proof: Follows directly from the non-degeneracy of the roots of the Hermite polynomials. The roots of any orthogonal polynomial are real and distinct. See also [Section IX]. QED

<Lemma 8.3>: <q(n, k)|n, n-1> = (1/ sqrt(n)) exp(i pi k) <q(n, k)| G(n) |q(n, j)> = delta_(kj) - exp(i pi (k-j)) |<q(n, k)| G(n) |q(n, j)>|^2 = 1 - delta_(kj) and <q(n, k)| G^2(n) |q(n, j)> = delta_(kj) + (n-2)exp(i pi (k-j)) | n(n-2), k = j |<q(n, k)| G^2(n) |q(n, j)>|^2 = | | (n-2)2, k not= j and more generally for m > 0, <q(n, k)| G^m(n) |q(n, j)> = delta_(kj) + g_m exp(i pi (k-j)) where recursively: g_0 = 0 g_1 = -1 g_(m+1) = -g_m (n-1) - 1 which recursion can be solved to yield [-(n-1)]^m - 1 g_m = -------------- n Furthermore, <q(n, k)| G^(-1)(n) |q(n, j)> = delta_(kj) - exp(i pi (k-j))/(n-1) and <q(n, k)| G^(+1/2)(n) |q(n, j)> = delta_(kj) - (1 +|- i sqrt(n-1)) exp(i pi (k-j))/n <q(n, k)| G^(-1/2)(n) |q(n, j)> = delta_(kj) - (1 +|- i/ sqrt(n-1))exp(i pi(k-j))/n Proof: For k not= j and XI(n) defined by (9.15) so, XI(n)|n, k> = |q(n, k)>: <q(n, k)| G(n) |q(n, j)> = <n, k| XI!(n) G(n) XI(n) |n, j> = <n, k| XI!(n) [I(n) - n|n, n-1><n, n-1| XI(n)] |n, j> if j should equal k, then we must have For all j, n<q(n, k)|n, n-1><n, n-1|q(n, k)> = 1 so |<q(n, j)|n, n-1>|^2 = (1/n) therefore <q(n, j)|n, n-1> = (1/sqrt(n)) exp(i theta(n, j) ) but we know that this is a matrix element of XI(n) and that XI(n) is real. So we can write <q(n, j)|n, n-1> = (1/sqrt(n)) exp(i pi theta(n, j)) where now theta(n, j) is an integer valued function. So generally, <q(n, k)| G(n) |q(n, j)> = delta_(kj) - n <n, k|XI!(n)|n, n-1><n, n-1|XI(n)|n, j> = delta_(kj) - n <q(n, k)|n, n-1><n, n-1|q(n, j)> = delta_(kj) - exp(i pi (theta(n, k) - theta(n, j)))

By explicit evaluations, (there should be a more general and rigorous way) it can be seen that theta(n, j) does not really have an n dependency, and that one can take theta(n, j) = j. This choice may be a consequence of the particular ordering of the roots by j in q(n, j) implicit in the particular diagonalizing transformation XI(n) that we have chosen. Cf. [Section IX] and [Section X].

Therefore,

<q(n, k)| G(n) |q(n, j)> = delta_(kj) - exp(i pi (k-j))The matrix elements of G^2(n) can be calculated by simple

matrix multiplication of <q(n, k)| G(n) |q(n, j)> by itself.Prove the general form for the matrix elements of positive powers of G(n) by induction, the recursive formula for g_m, then follows by substitution. Solving the recursion for the form of g_m as a function of its index can be done by reducing g_(m+k) to a function of g_m and letting m=0. collecting the resulting power series gets the result; straight forward, but slightly tedious.

The matrix elements of G^(-1)(n) can be obtained by noting from the known spectrum of G(n), that the inverse exists and that

G^2(n) + (n-2) G(n) - (n-1) I(n-1) = 0is the minimal polynomial for G(n). Multiplying by G^(-1)(n), and solving for G^(-1)(n), expressing it in terms of G(n) and a unit matrix demonstrates the result when previous results of this lemma are used.

For the matrix elements of G^(1/2)(n), guess a form similar to the form of the expressions for the elements of G^m(n), so that for some as yet undetermined a, that

(n-1) SIGMA (delta_(mk) + i a exp(i pi (m-k))) (delta_(kj) + i a exp(i pi (k-j))) = k=0 delta_(mj) - exp(i pi (m-j)) Performing the indicated matrix multiplication delta_(mj) - (n a^2 - 2 i a) exp(i pi (m-j)) = delta_(mj) - exp(i pi (m-j)) Then solving the quadratic equation n a^2 - 2ia - 1 = 0 Determines a and yields the result. Similarly for the matrix elements of G^(-1/2)(n), guess the same form and use the established matrix elements of G^(-1)(n) to determine the value of a. QED

<Lemma 8.4>: <q(n, k)| [Q(n), G(n)] |q(n, j)> = (q(n, k) - q(n, j)) exp(i pi (k-j)) <p(n, k)| [P(n), G(n)] |p(n, j)> = (p(n, k) - p(n, j)) exp(i pi (k-j)) Proof: Both follow directly from Lemma 8.3. QED

<Lemma 8.5>: <q(n, k)| N(n) |q(n, j)> = | (1/2)q^2(n, k) + (1/2)<q(n, k)| P^2(n) |q(n, j)>, k = j | | (1/2)(<q(n, k)| P^2(n) |q(n, j)> - (-1)^((k-j))), k not= j Proof: Follows from Lemma 8.3 and the relation (1/2)(Q^2(n) + P^2(n)) = N(n) + (1/2)G(n)To continue the evaluation to the matrix elements of P^(2)(n) see Theorem 8.3.

QED <Theorem 8.1>: <q(n, k)|P(n)|q(n, j)> = | exp(i pi (k-j)) | -i -------------------- , k not= j | q(n, k) - q(n, j) | | 0 , k = j and <p(n, k)|Q(n)|p(n, j)> = - <q(n, k)|P(n)|q(n, j)> Proof: Again from the commutation relation (2.8) <q(n, k)|[Q(n), P(n)]|q(n, j)> = i <q(n, k)|G(n)|q(n, j)> (q(n, k) - q(n, j))<q(n, k)|P(n)|q(n, j)> = i <q(n, k)|G(n)|q(n, j)> (q(n, k) - q(n, j))<q(n, k)|P(n)|q(n, j)> = i <q(n, k)|G(n)|q(n, j)>lemma 8.1 and lemma 8.2

exp(i pi (k-j)) <q(n, k)|P(n)|q(n, j)> = -i ------------------- , k not= j q(n, k) - q(n, j)For the evaluation of the matrix elements <p(n, k)|Q(n)|p(n, j)> in terms of <q(n, k)|P(n)|q(n, j)>, use equations (7.7) and equation (7.15).

QEDThat <q(n, k)|P(n)|q(n, j)> is not zero when k not= j depends on the non-degeneracy of the eigenvalues q(n, k).

<Corollary 8.1.1>: <q(n, k)|[P(n), G(n)]|q(n, j)> = 1 1 i exp(i pi (k-j)) ( SIGMA ----------------- - SIGMA ----------------- ) (m not=k) q(n, k) - q(n, m) (m not=j) q(n, m) - q(n, j) for k not= j and 0, for k = j Proof:Follows directly from Theorem 8.1 and Lemma 8.3.

QED <Corollary 8.1.4>:The following results are obtained from Theorem 8.1 immediately:

Define for a real parameter a, f(a) := exp( i a Q(n) ) P(n) exp( -i a Q(n) ) then <q(n, k)| f(a) |q(n.j)> = <q(n, k)| P(a) |q(n.j)> exp( ia( q(n, k) - q(n, j) ) ) = exp(i pi (k-j)) -i ------------------- exp( ia( q(n, k) - q(n, j) ) ), k not= j q(n, k) - q(n, j) 0 , k = j Similarly define for a real parameter a, g(a) := exp( i a P(n) ) Q(n) exp( -i a P(n) ) then <p(n, k)| g(a) |p(n.j)> = <p(n, k)| Q(a) |p(n.j)> exp( ia( q(n, k) - q(n, j) ) ) = exp(i pi (k-j)) +i ------------------- exp( ia( q(n, k) - q(n, j) ) ), k not= j q(n, k) - q(n, j) and 0, k = j Differentiating the definitions of f(a) and g(a) with respect to a, f'(a) = i[Q(n), f(a)] g'(a) = i[P(n), g(a)] and generally for the mth derivative f^(m)(a) = i^m C( f(a), Q(n), m ) g^(m)(a) = i^m C( g(a), P(n), m ) where C(...) is the mth commutator of its first argument by its second. Differentiating the above result with respect to a, <q(n, k)| (-i)^m C( f(a), Q(n), m )|q(n, j)> = <q(n, k)| f^(m)(a) |q(n, j)> = (d/da)^m<q(n, k)| f(a) |q(n, j)> = <q(n, k)| f(a) |q(n, j)> i^m (q(n, k) - q(n, j))^(m-1) = <q(n, k)|P(a)|q(n, j)> X exp(ia(q(n, k) - q(n, j))) i^m (q(n, k) - q(n, j))^(m-1)

<p(n, k)| (-i)^m C( g(a), P(n), m )|p(n, j)> = <p(n, k)| g^(m)(a) |p(n, j)> = (d/da)^m<p(n, k)| g(a) |p(n, j)> = <p(n, k)| g(a) |p(n, j)> i^m ( p(n, k) - p(n, j) )^(m-1) = <p(n, k)|P(a)|p(n, j)> X exp( ia(p(n, k) - p(n, j)) ) i^m (p(n, k) - p(n, j))^(m-1)

<Theorem 8.2>: For alpha_k all complex, let (n-1) |psi> = SIGMA alpha_k |q(n, k)> k=0 then, <psi| {Q(n), P(n)} |psi> = q(n, k) + q(n, j) = 2 SIGMA Im( alpha_j alpha_k*) ----------------- (-1)^((k-j)) j < k q(n, k) - q(n, j)and

<psi| [Q(n), P(n)] |psi> = = -2i SIGMA Re( alpha_j alpha_k*) ) (-1)^((k-j)) j < k where Re(.) and Im(.) are the real and imaginary parts. Proof: <psi|Q(n)P(n) +|- P(n)Q(n)|psi> = = SIGMA (<psi|Q(n)|q(n, k)><q(n, k)|P(n)|psi> k +|- <psi|Q(n)|q(n, k)><q(n, k)|P(n)|psi>)

= SIGMA q(n, k) ( <psi|q(n, k)><q(n, k)|P(n)|psi> k +|- <psi|q(n, k)><q(n, k)|P(n)|psi> )

= SIGMA q(n, k) (alpha_i* alpha_j) <q(n, i)|q(n, k)><q(n, k)|P(n)|q(n, j)> i,j,k +|- alpha_i* alpha_j <q(n, j)|q(n, k)><q(n, k)|P(n)|q(n, i)> )

= SIGMA q(n, k) (alpha_i* alpha_j) delta_ik) <q(n, k)|P(n)|q(n, j)> i,j,k +|- alpha_i* alpha_j delta_jk <q(n, k)|P(n)|q(n, i)> )

= SIGMA q(n, k) alpha_k* alpha_j <q(n, k)|P(n)|q(n, j)> j,k +|- SIGMA q(n, k) alpha_i* alpha_k <q(n, k)|P(n)|q(n, i)> i,k

= SIGMA q(n, k) ( alpha_k* alpha_j <q(n, k)|P(n)|q(n, j)> j,k +|- alpha_j* alpha_k <q(n, k)|P(n)|q(n, j)> )

= SIGMA alpha_j alpha_k* ( q(n, k) <q(n, k)|P(n)|q(n, j)> j,k +|- q(n, j) <q(n, k)|P(n)|q(n, j)> ) = SIGMA alpha_j alpha_k* ( q(n, k) +|- q(n, j) ) <q(n, k)|P(n)|q(n, j)> j,k

exp(i pi (k-j)) = -i SIGMA alpha_j alpha_k* ( q(n, k) +|- q(n, j) ) --------------------- j not=k ( q(n, k) - q(n, j) ) Combining upper triangular and lower triangular terms

i <psi|G(n)|psi> = <psi|Q(n)P(n) - P(n)Q(n)|psi> = = -i SIGMA (alpha_j alpha_k* exp(i pi (k-j)) j < k + alpha_k alpha_j* exp(i pi (j-k))) = -i SIGMA (alpha_j alpha_k* exp(i pi (k-j)) j < k + alpha_k alpha_j* exp(-i pi (k-j))) = -i SIGMA ( alpha_j alpha_k* exp(i pi (k-j)) j < k + (alpha_j alpha_k*) exp(i pi (j-k)))* ) = -2i SIGMA Re( alpha_j alpha_k* ) (-1)^(k-j) j < k and similarly derive <psi|Q(n)P(n) + P(n)Q(n)|psi> = q(n, k) + q(n, j) = 2 SIGMA Im( alpha_j alpha_k* ) ----------------- (-1)^(k-j) j < k q(n, k) - q(n, j) QED <Corollary 8.2.1>: If for all k, alpha_k = alpha_k* in (n-1) |psi> = SIGMA alpha_k |q(n, k)> k=0 or (n-1) |psi> = SIGMA alpha_k |n, k> k=0 then <psi|{Q(n), P(n)}|psi> = 0. Proof:Since the transformation XI(n) defined by (9.15) connecting the basis |n, k> with |q(n, k)> is an element of the Lie group SO(n), it is real and so the expansion coefficients of |psi> expanded on |n, k> are real if and only if the expansion coefficients of |psi> expanded on |q(n, k)> are real. In either case then <psi|{Q(n), P(n)}|psi> = 0.

QED <Corollary 8.2.2>: There exist |psi>, <psi|psi> not= 0, and <psi|G(n)|psi> not= 0, such that <psi|{Q(n), P(n)}|psi> = 0. Proof: Follows from Corollary 8.2.1. QED <Theorem 8.3>: 1 <q(n, k)|P^2(n)|q(n, k)> = SIGMA --------------------- k not=j ( q(n, k) - q(n, j) )^2 and for k not= j <q(n, k)|P^(2)(n)|q(n, j)> = 2 exp(i pi (k-j)) ---------------------- ( q(n, k) - q(n, j) )^2 R_k H_(n-1)( q(n, k) ) - R_j H_(n-1)( q(n, j) ) - 2 n exp(i pi (k-j)) ----------------------------------------------- ( q(n, k) - q(n, j) ) where R_k is the residue in the partial fraction expansion of 1/H_n( z ). See [Appendix E]. Proof: For k = j, expand as a product of two matricies using Theorem 8.2, <q(n, k)|P^(2)(n)|q(n, k)> = SIGMA |<q(n, j)|P(n)|q(n, k)>|^2 j The result follows from Theorem 8.2. For k not= j, again expand as a product of two matrices <q(n, k)|P^(2)(n)|q(n, j)> = exp(i pi (k-j)) -------------------- X ( q(n, k) - q(n, j) ) 1 1 SIGMA [ ----------------- - ----------------- ] i not= k,j q(n, i) - q(n, k) q(n, i) - q(n, j) but, see [Appendix E]. 1 SIGMA ----------------- = i not= k,j q(n, i) - q(n, k) 1 lim[(z - q(n, k)) (d/dz)log( H_n(z) )] - ----------------- z->q(n,k) q(n, j) - q(n, k) QED

<Theorem 8.4>: <q(n, k)|P(n)|q(n, j)> = <p(n, k)|Q(n)|p(n, j)> Proof: From the Fourier transform (7.6), Fr(n)|p(n, k)> = |p(n, k)> and Fr(n)Q(n)Fr!(n) = P(n). The result follows immediately. QED

<Theorem 8.5>: For the operator F_+(n) = UPSILON(n) N(n) UPSILON!(n) defined by (7.38a) <n, k|F_+(n)|n, j> = <n, k| UPSILON(n) N(n) UPSILON!(n) |n, j> = -(1/2)(1 + i cot( pi (k-j)/n ) ) when k not= j, and <n, k|F_+(n)|n, k> = (n-1)/2 when k = j. Proof: [New and Improved: December 3, 2001] From equation (7.38a) (n-1) <n, k|F_+(n)|n, j> = (1/n) SIGMA m exp{ i 2 pi m/n (k - j) } m=0 Define a function analytic in z, n-1 f_n(z) = SIGMA exp( i m z ) m=0 Understanding this as a geometric series, f_n(z) = ( exp( i n z ) - 1 ) / ( exp( i z ) - 1 ) In its original form, the first derivative n-1 -i f'_n(z) = SIGMA m exp( i m z ) m=0 which is what we wish to compute in closed form. Using the second closed form of the geometric series. f_n(z) = ( exp( i n z ) - 1 ) / ( exp( i z ) - 1 ) sin( n z/2 ) = ------------ exp( i (n-1) z ) sin( z/2 ) Differentiating w.r.t. z, (n/2) cos( n z/2 ) f'_n(z) = ------------------ exp( i (n-1) z ) sin( z/2 ) sin( n z/2 ) - ------------- (1/2) cos( z/2 ) exp( i (n-1) z ) sin^2( z/2 ) sin( n z/2 ) + ------------ i (n-1) exp( i (n-1) z ) sin( z/2 ) Now substitute z = (2 pi)/n (k-j); a little reduction and trigonometry gets the answer, (n-1) SIGMA m exp( i m 2 pi (k-j)/n ) = -(n/2)(1 + i cot( pi (k-j)/n ) ) m=0 If theta = 0, the LHS summation yields n(n-1)/2 directly, and the result follows. QED Addendum: Useful in the calculation of more general sums (n-1) SIGMA m^a exp( i m 2 pi (k-j)/n ) m=0 is an easily obtained recursion relation -i d/dz f_n(z) = n f_n(z) + (n+1) / (exp(iz) - 1)

<Corollary 8.5.1>: For F_-(n) as defined by equation (7.38b) <n, k|F_-(n)|n, j> = <n, j|F_+(n)|n, k> = <n, k|F_+(n)|n, j>* = <n, j| UPSILON!(n) N(n) UPSILON(n) |n, k> = -(1/2)(1 - i cot( pi (k-j)/n ) ) when k not= j, and <n, k|F_-(n)|n, k> = (n-1)/2 when k = j. Proof: A trivial exercise using equations (7.38a) and (7.38b). QED It is important to note that F_-(n) is defined by F_-(n) = UPSILON^2(n) F(n) UPSILON!^2(n) That it is the complex conjugate of F_+(n) = F(n) in the |n, k> basis, is specifically, however, a basis dependent statement.

<Corollary 8.5.2>: <n, k|[N(n), F_(+|-)(n)]|n, j> = | (k-j)(n-1)/2, k = j | | -((k-j)/2)(1 +|- i cot( pi (k-j)/n ) ), k not= j Proof: Since M(n) - N(n) = G(n), using (7.21) we have UPSILON(n) C(n) UPSILON!(n) = UPSILON(n) exp( i 2 pi/n F(n)) UPSILON!(n) = exp( i 2 pi/n M(n) ) = exp( i 2 pi/n ( N(n) + G(n) ) ) Now consider the effective matrix elements <n, k| exp( i 2 pi/n G(n) ) |n, j> = | 0, k not= j = | exp(i2 pi/n), k = j not= n-1 | exp(i(2 pi/n)(1-n)), k = j = n-1 Thus, exp( i 2 pi/n G(n) ) acts in the |n, k> basis as the operator exp(i 2 pi/n)I(n), a multiple of the identity, and therefore it acts so in in every basis. From the above we can conclude: UPSILON(n) F(n) UPSILON!(n) = N(n) + I(n) Then [N(n), F(n)] = [N(n), UPSILON!(n) N(n) UPSILON(n)] so <n, k| [N(n), F(n)] |n, j> = (k-j) <n, k| UPSILON!(n) N(n) UPSILON(n) |n, j> and <n, k| UPSILON!(n) N(n) UPSILON(n) |n, j> (n-1) = (1/n) SIGMA m exp( i(2 pi/n)(j-k)m ) m=0 Therefore, combining the results and using Corollary 8.5.1 the desired results are obtained. QED

<Corollary 8.5.3>: [F_+(n), F_-(n)] = 0 <n, k|F_(+|-)^2(n)|n, j> | (1/2) csc^2( ( pi/n)(k - j) ) | - (n/2)(1 +|- i cot( pi (k-j)/n ) ), k not= j = | | (n-1)(2n-1)/6, k = j Also <n, k|F_+(n) F_-(n)|n, j> | -(1/2) csc^2( (pi/n)(k - j) ), k not= j = | | (n^(2)-1)/6, k = j Proof: For the first result abstractly, from equation (7.35), F_+(n) F_-(n) = UPSILON(n) N_+(n) N_-(n) UPSILON!(n) = UPSILON(n) N_-(n) N_+(n) UPSILON!(n) = F_-(n) F_+(n) The result is also easily shown from the form of the matrix elements of the products above.For the second result, expand the individual products on the eigenbasis of N(n), using the matrix element representation for F_(+|-)(n) as a sum of exponentials given at the start of the proof of Theorem 8.5. Use formula (7.28) expressing the unitarity of UPSILON(n), to reduce the resulting expression. The finite summation,

(n-1) SIGMA m^2 exp( i m theta ) m=0will be needed. This can be gotten by differentiating both sides with respect to z of the formula in theorem 8.5 where the cofactor of the exponential in the sum is linear. Making the substitution for z simplifies the expression. The vanishing of z is again a special case that is the sum of squares of integers, which has a standard formula #19, [Jolley 1961]. Thus showing that

(n-1) SIGMA m^2 exp( i m theta ) m=0 | (n/2) csc^2( (pi/n)(k - j) ) | - (n^2/2)(1 +|- i cot( pi(k-j)/n ) ), k not= j = | | n(n-1)(2n-1)/6, k = j

For the third result, similarly obtain <n, k|F_(+|-)(n) F_(-|+)(n)|n, j> = SIGMA SIGMA SIGMA ( m p exp( (+|-)i(2 pi/n)(km + jp) ) s m p exp( (-|+)i(2 pi/n)(m + p)s ) )The second exponential factor after the independent summation over s is the matrix element of UPSILON^2(n). Cf. equation (7.31), which is nonzero only when p = n-m and p not= 0, or when p=m=0. The latter case has no contribution to the sum. Then also summing over p:

<n, k|F_(+|-)(n) F_(-|+)(n)|n, j> = SIGMA m exp( (+|-)i(2 pi/n)(k-j) ) m - (1/n) SIGMA m^2 exp( (-|+)i(2 pi/n)(k-j)m ) m These sums have already been evaluated, and the result follows with a little algebra. QED

<Theorem 8.6>: UPSILON(n) G(n) UPSILON!(n) = (n-1) (n-1) I(n) - SIGMA SIGMA exp( -i 2 pi (k-j)/n ) |n, k><n, j| k=0 j=0 UPSILON(n) G(n) UPSILON!(n) |n, k> = SIGMA exp(i 2 pi (k-j)/n) |n, j> j not= k <n, k| UPSILON(n) G(n) UPSILON!(n) |n, j> = delta_(kj) - exp( i (2 pi/n) (k+j+2) ) [UPSILON(n), G(n)] = (n-1) sqrt(n) SIGMA ( exp( 2 pik/n)|n, n-1><n, k| - exp(-i 2 pi k/n)|n, k><n, n-1| ) k=0 <n, k| [F_(+|-)(n), G(n)] |n, j> = -(n/2)( delta_(k(n-1)) (1 +|- i cot( (pi/n)(n - j - 1) )) - delta_(j(n-1)) (1 -|+ i cot( (pi/n)(n - k - 1) )) ) Proof: All of the formulas are directly calculable from previously stated results. The details are left to the reader.

<Theorem 8.7>: C(n) N(n) C!(n) = N(n) + G(n) C^k(n) exp(i2 pi/n N(n)) C!^k(n) = exp(i2 pi/n( N(n) + k I(n) )) Proof: The formulas are directly calculable from previously stated results. The details are left to the reader.

<Theorem 8.8>: The matrix elements of the Fourier transform Fr(n) on the |q(n, j)> basis are given by: <q(n, k)|p(n, j)> = <q(n, k)|Fr(n)|q(n, j)> = <n, k| XI!(n) Fr(n) XI(n) |n, j> (n-1) = SIGMA exp(i pi m/2) <n, k|XI!(n)|n, m><n, m|XI(n)|n, j> m=0 (n-1) = SIGMA exp(i pi m/2) (d_m)^(-1) <n, k|XI!(n)|n, m><n, m|XI(n)|n, j> m=0 (n-1) = SIGMA i^m (d_m)^(-1) ~H_m( q(n, k) ) ~H_m( q(n, j) ) m=0 where by equation (9.14), polynomial (d_m)^2 := SIGMA ~H_m^2( q(n, i) ) i and ~H_m(z) is the normalized m-th Hermite polynomial. Similarly <p(n, j)|q(n, k)> = <p(n, k)|q(n, j)> = (n-1) = SIGMA (-i)^m (d_m)(-1) ~H_m( q(n, k) ) ~H_m( q(n, j) ) m=0 Proof:The statement of the results virtually constitutes the proof. Refer to [Sections VII] and [Section IX]. In the n->infinity limit, the matrix elements go over into those of the usual Fourier kernel. Compare the expansion of the Fourier kernel in Hermite polynomials in [Appendix E].

<Corollary 8.8.1>: (n-1) SIGMA (-1)^m (d_m)(-1) ~H_m( q(n, k) ) ~H_m( q(n, j) ) m=0 = delta_(k(n-j)) Proof: From Theorem 8.8 the expression for the matrix elements <q(n, k)| Fr^2(n) |q(n, j)> can also be written down by simply replacing the imaginary i with -1. Equation (10.6) then proves the result. QED

<Theorem 8.9>: The matrix elements of exp( i alpha G(n) ) are given by <q(n, k)| exp( i alpha G(n) ) |q(n j)> = (-1)^(k-j) exp( i alpha ) [ delta_(kj) + ---------- ( exp(-i alpha n) - 1 ) ] n Then also if alpha = (2 pi/n)m for any integer m, <q(n, k)| exp( i alpha G(n) ) |q(n j)> = exp( i alpha ) delta_(kj) Proof:Expand the exponential about 0, and recollect the terms after using Lemma 8.3 to evaluate the matrix elements of powers of G(n).

QED <Theorem 8.10>: Let H_osc(n) = (h-bar omega_H/2)( Q^2(n) + P^2(n) ) = (h-bar omega_H)( N(n) + (1/2)G(n) ) be the analog of the QM harmonic oscillator Hamiltonian, and let t_(+|-)(n) = (2 pi/(n omega_T)) F_(+|-)(n) be the time operators [Section VII] so FCCR can be written exp( i omega_T t_+(n) ) N(n) exp( -i omega_T t_+(n) ) - N(n) = G(n). The eigenbasis of t(n) = t_+(n) is denoted by |t_k>, with eigenvalues t_k. We also have UPSILON(n) |n, k> = |t_k> omega_T = (2 pi)/(tau_0 n) t_k = -(2 pi k)/(omega_T n) = -k tau_0 which is equivalent to t_k = (n-k) tau_0 where tau_0 is a fundamental unit of time, perhaps but not necessarily equal to the Planck time. omega_T t_k = 2 pi k/n omega_H = 2 pi/tau_0 = n omega_T Then <n, k| t_(+|-)(n) |n, j> = | | -(tau_0/2)(1 +|- i cot( pi (k-j)/n) ), k not= j | | tau_0 (n-1)/2, k = j | and [NB also compute the matrix elements of the energy squared for computing uncertainty of energy in time eigenstate.] <t_k| H_osc(n) |t_j> = | | i(h-bar omega_H/2) exp(i 2 pi (k-j)/n) cot( pi (k-j)/n), k not= j | | | h-bar omega_H (n-1)/2, k = j | Proof: From exp( +i omega_T t_+(n) ) N(n) exp( +i omega_T t_+(n) ) - N(n) = G(n) show that <t_k| H_osc(n) |t_j> = (h-bar omega_H/2)(1 + exp( i omega_T (t_k - t_j))) <t_k| N(n) |t_j> = (h-bar omega_H/2)(1 + exp( i omega_T (t_k -t_j))) <n, k|F_-(n)|n, j>A straightforward evaluation using Corollary 8.5.1 yields the results. Fine details are left for the reader's amusement. There are no calculational surprises or subtleties. The future projected expectation values of obvervables at a future time are given by suitable algorithms, even if they do not progress differentiably.

QED <Corollary 8.10.1>: <n, k| t_(+|-)^2(n) |n, j> = | | (tau_0^2)/2) csc^2( pi (k-j)/n) ) + n<n, k| t_(+|-)(n) |n, j>, k not= j | | (tau_0 (2n-1)/3) <n, k| t_(+|-)(n) |n, j>, k = j | <n, k| t_(+|-)(n) |n, j>^2 = | | -(tau_0^2)/4) csc^2( pi (k-j)/n ) - | tau_0^2 <n, k| t_(+|-)(n) |n, j>, k not= j | | (tau_0 (n-1)/2)^2, k = j | |<n, k| t_(+|-)(n) |n, j>|^2 = | | (tau_0^2)/4) csc^2( pi (k-j)/n ), k not= j | | (tau_0)(n-1)/2)^(2), k = j | and from the first two of these relations, the uncertainty of t_(+|-)(n) in an energy eigenstate can be calculated as (DELTA t_(+|-)(n)) = (< t_(+|-)^2(n)> - <t_(+|-)(n)>^2)^(1/2) = (tau_0 /2) ((n^2 - 1)/3)^(1/2) For n = 2 (DELTA t_(+|-)(2)) = <t_(+|-)(n)> = (tau_0 /2) while for large n (DELTA t_(+|-)(n)) = (tau_0 / 2) (n / sqrt(3))Note:

Considering the oscillator as a clock, the expected value of the time in an energy state is about midway in the cycle and for large n, plus or minus about 15% of a full cycle. Thus the time in an energy eigenstate is not totally indeterminate, but clearly these do not make for good clock states.

Proof:The results follow directly from Theorem 8.10 and Corollary 8.5.3.

QED <Theorem 8.11>: For the eigenbasis of the oscillator time operator t(n) as defined in Theorem 8.10, with Q(n) and P(n) defined by equation (2.7), <t_k| Q(n) |t_j> = = (1/(n sqrt(2))) ( + exp( +i 2 pi k/n ) + exp( -i 2 pi j/n ) ) X (n-1) SIGMA sqrt(m) exp( -i 2 pi (k-j)m/n ) m=0 <t_k| P(n) |t_j> = = (i/(n sqrt(2))) ( - exp( +i 2 pi k/n ) + exp( -i 2 pi j/n) ) X (n-1) SIGMA sqrt(m) exp( -i 2 pi (k-j)m/n ) m=0 <t_k| G(n) |t_j> = delta_(kj) - exp( i(2 pi / n) (k - j) ) <t_k| G^2(n) |t_j> = delta_(kj) + (n-2) exp( i(2 pi / n)(k - j) ) i^n - 1 <t_k| Fr(n) |t_j> = (1/n) ------------------------------ i exp( +i(2 pi/n)(k - j) ) - 1Note special case in Fourier transform for n a multiple of 4. The Numerator of fraction vanishes and the denominator can vanish when k-j = n/4. Cf. [Theorem 7.3],

Then specializing to the "expectation values": NB: Compare the QM & classical SHO, [Messiah 1965] vol. I, p. 444 <t_k| Q(n) |t_k> (n-1) = [(sqrt(2)/n) SIGMA sqrt(m)] cos(2 pi k/n) m=0 <t_k| P(n) |t_k> (n-1) = [(sqrt(2)/n) SIGMA sqrt(m)] sin(2 pi k/n) m=0 <t_k| G(n) |t_k> = 0 <t_k| G^2(n) |t_k> = (n-1) <t_k| Fr(n) |t_k> = (1/n) (i^n - 1)/(i-1) = (n-1) = (1/n) SIGMA i^m = (1/n) Tr( Fr(n) ) m=)and the amplitude of the sine and cosine is always less than the largest eigenvalue of Q(n), in fact less than sqrt(n-1). Note that the square of the amplitude is equal to a "square mean root" of the energy on a pure state of equally distributed |n, k>:

(n-1) [(sqrt(2)/n) SIGMA sqrt(m)]^2 = 2 [(1/n) Tr( N^(1/2)(n) )]^(2) m=0 Proof: Again the results are straightforwardly arrived at using |t_k> = UPSILON(n) |n, j> <n, k| Q(n) |n, j> = (1/sqrt(2))( sqrt(k) delta_(j(k-1)) + sqrt(j) delta_(k(j-1)) ) <n, k| P(n) |n, j> = (i/ sqrt(2))( sqrt(k) delta_(k(j-1)) - sqrt(j) delta_(k(j-1)) ) and a small amount of index fiddling.The matrix elements of G^2(n) are calculated in the manner of Lemma 8.3.

QED <Corollary 8.11.1>: <t_k| exp( i alpha G(n) ) |t_j> = exp( i(2 pi/n)(k-j) ) exp( i alpha ) [ delta_(kj) + --------------------- ( exp( -i alpha n ) - 1 ) ] n and just as in Theorem 8.9, if alpha = (2 pi/n)m for any integer m, <t_k| exp( i alpha G(n) ) |t_j> = exp( i alpha ) delta_(kj) Proof:See Theorem 8.9, Theorem 8.11 and Lemma 8.3.

QED <Corollary 8.11.2>: If alpha_m(n) := (2 pi/n)m forThe Kronecker delta in corollary 8.11.1 is invariant under arbitrary changes of bases instituted by similarity transformationsanyinteger m, exp( i alpha_m(n) G(n) ) = exp( i alpha_m(n) ) I(n) Proof:

O -> S O S^(-1)with S in GL(n, C). Therefore, although the evaluation was performed in a convenient basis, it holds for all bases.

QEDNote that exp( i alpha_m(n) ) I(n) are the elements of the discrete center of SU(n), and that while I(n) is an element of U(n) and a generator for u(n), it is neither an element of SU(n) nor a generator for su(n). Cf. [Appendix B].

Because of the cyclic operator C(n) in the statement of FCCR, the "time" appears already integrated (Weyl); the position does not. We unintegrate by the BCH formula (7.3) to derive the time-energy commutation relation for the harmonic oscillator.

[B(n), B!(n)] = G(n) M(n) - N(n) = G(n) C(n) N(n) C!(n) - N(n) = G(n)In the following form, FCCR shows an inherent ambiguity with regard to t(n) eigenvalues. Negative eigenvalues will do, but they can also be considered as being understood mod n. Then finite fields of prime characteristic or finite commutative rings may become important. Cf. [Appendix J].

<Theorem 8.12>: For the oscillator Hamiltonian and associated time operators t_(+|-)(n) defined by equation (7.49): for k = j and every finite n, <t_k| [t_(+|-)(n), H_osc(n)] |t_j> = 0 for k not= j, <t_k| [t_(+|-)(n), H_osc(n)] |t_j> = (-|+) i n h-bar tau_0 omega_H/(2 pi) X [ pi (k-j)/n ] X ----------------- exp( i (2 pi/n) (k-j) ) cos[ pi (k-j)/n ] sin[ pi(k-j)/n ] Then for k not= j, the real and imaginary parts are, Re( <t_k| [t_(+|-)(n), H_osc(n)] |t_j> ) = sin( theta ) (2 theta) (+|-) h-bar omega_H (k-j) ------------- -------------- cos( theta ) theta sin( 2 theta ) Im( <t_k| [t_(+|-)(n), H_osc(n)] |t_j> ) = sin( theta ) (-|+) h-bar omega_H n tau_0/(2 pi) ------------ cos( 2 theta ) cos( theta ) theta where theta := pi (k - j)/n and for very large n and 0 < |theta| << pi, asymptotically Re( <t_k| [t_(+|-)(n), H_osc(n)] |t_j> ) = (+|-) h-bar omega_H (k-j) Im( <t_k| [t_(+|-)(n), H_osc(n)] |t_j> ) = (-|+) h-bar omega_H (n tau_0/(2 pi)) Re( <t_k| [t_(+|-)(n), H_osc(n)] |t_j> ) is asymptotically independent of n, while Im( <t_k| [t_(+|-)(n), H_osc(n)] |t_j> ) is asymptotically independent of (k-j). <t_k| [t_(+|-)(n), H_osc(n)] |t_j> -gt; (-|+) i h-bar omega_H tau_0 ( n/(2 pi) + i(k - j) ) Furthermore, for k = j <t_k| [t_(+|-)(n), H_osc(n)] |t_j> = 0 = <t_k| G(n) |t_j> and for k not= j <t_k| [t_(+|-)(n), H_osc(n)] |t_j> = (-|+) i n h-bar tau_0 omega_H/(2 pi) theta cot( theta ) exp( i 2 theta ) = (+|-) i n h-bar tau_0 omega_H/(2 pi) theta cot( theta ) <t_k| G(n) |t_j> For any finite k not= j, lim theta cot( theta ) = 1 n->infinity Therefore, asymptotically for all k, j < n <t_k| [t_(+|-)(n), H_osc(n)] |t_j> = = (+|-) i n h-bar tau_0 omega_H/(2 pi) <t_k| G(n) |t_j> Since this is true for all matrix elements on both sides of the equation, and further by remembering that omega_H = 2 pi/tau_0, conclude that asymptotically [t_(+|-)(n), H_osc(n)] -> (+|-)i h-bar n G(n) If P(n) and Q(n) are canonically conjugate, then alpha t_(+|-) and beta H_osc(n) where (alpha beta) = n, are canonically conjugate in asymptopia in the same sense. Also for m > 1, C(H_osc(n): t_(+|-)(n), m) = -i (n h-bar/2) m C(G(n): t_(+|-)(n), m-1) Proof #1:The results for the single commutator follow easily from the results of Theorem 8.10. For higher commutators, write FCCR as,

exp( +i omega_T t(n) ) N(n) exp( -i omega_T t(n) ) - N(n) = G(n) where omega_T = 2 pi/(n tau_0). With H_osc(n) = (h-bar omega_H/2)(N(n) + (1/2)G(n)) this becomes exp( +i omega_T t_(+|-)(n) ) H_osc(n) exp( -i omega_T t_(+|-)(n) ) - H_osc(n) = (+|-) (h-bar omega_H)/2) X [exp( +i omega_T t_(+|-)(n) ) G(n) exp( -i omega_T t_(+|-)(n) ) + G(n)] Now expand both sides with the BCH formula (7.3). infinity SIGMA (i omega_T)^m/m! C(H_osc(n): t_(+|-)(n), m) - H_osc(n) = m=0 infinity (h-bar omega_H/2) (SIGMA (i omega_T)^m/m! C(G(n): t_(+|-)(n), m) + G(n) ) m=0With a cancellation of H_osc(n) terms and collecting terms in powers of omega_T, also using

omega_H = n omega_T this becomes infinity SIGMA (i^m/m!) C(H_osc(n): t_(+|-)(n), m) omega_T^m m=1 infinity - SIGMA (i^m/(m-1)!) (n h-bar/2) C(G(n): t_(+|-)(n), m-1) omega_T^m m=2 = (+|-) (n h-bar omega_T) G(n) Equating term by term in powers of omega_T: for the first power, [t_(+|-)(n), H_osc(n)] = (-|+) i n h-bar G(n) and for m > 1, C(H_osc(n): t_(+|-)(n), m) = -i (n h-bar/2) m C(G(n): t_(+|-)(n), m-1) QED Proof #2: In the known relation C(n) N(n) C!(n) - N(n) = G(n) C(n) is the cyclic operator in the eigenbasis of N(n), which is therefore unitary and can be represented by C(n) = exp( i (2 pi/n) t(n) ) Consider the associated one-parameter unitary group C(n, theta) = exp( i theta t(n) ) so C!(n, theta) = exp( -i theta t(n) ) since t!(n) = t(n) and the expansion by the BCH theorem C(n, theta) N(n) C!(n, theta) = N(n) + i theta [t(n), N(n)] + (i theta)^2/2 [t(n), [t(n), N(n)]] + ... and for small theta we can write C(n, theta) N(n) C!(n, theta) - N(n) = i theta [t(n), N(n)]By letting theta = (2 pi/n) and taking n large enough we can make theta as small as desired, therefore, for n large enough

G(n) = i(2 pi/n) [t(n), N(n)] Now C^2(n, theta) N(n) C^2!(n, theta) - C(n, theta) N(n) C!(n, theta) = C(n, theta) G(n) C!(n, theta) Using BCH expansions as before and approximating for small theta, C^2(n, theta) N(n) C^2!(n, theta) = G(n) + i 2 theta [t(n), N(n)] C(n, theta) G(n) C!(n, theta) = G(n) + i theta [t(n), G(n)] we have N(n) + i 2 theta [t(n), N(n)] - N(n) = G(n) + G(n) + i theta [t(n), G(n)] N(n) + i 2 theta ( -i theta^(-1) G(n) ) - N(n) = G(n) + G(n) + i theta [t(n), G(n)] Therefore, for large n [t(n), G(n)] = 0 and therefore for H(n) := N(n) + (1/2)G(n) [t(n), H(n)] = -i(n/(2 pi)) G(n) QED <Theorem 8.13>: exp(i alpha G(n))|t_j> = exp(i alpha) |t_j> + exp(i alpha) (exp(-i alpha n) - 1)exp(-i(2 pi/n)j) sqrt(n) |n, n-1> Proof: exp(i alpha G(n))|t_j> = SIGMA |t_m> <t_m| exp( i alpha G(n) ) |t_j> = m using Corollary 8.11.1 = exp(ialpha) |t_j)> + [SIGMA exp(i(2 pi/n)m) |t_m)>] X m exp(i alpha) (exp(-i alpha n) - 1) exp(-i(2 pi/n)j) But SIGMA exp(+i(2 pi/n)m) = SIGMA exp(-i(2 pi/n)(n-1)m) and |n, n-1> = (1/ sqrtn) SIGMA exp(-i(2 pi/n)(n-1)m) |t_m)> The result follows. QED Corollary 8.13.1: exp( (2 pim/n) (N(n) + (1/2)G(n)) ) |t_k)> = exp( ( pim/n) G(n) ) |t_k+m)> = | exp(i( pim/n)) |t_k+m)> for m even | | exp(i( pim/n)) [|t_k+m)> + 2 sqrt(n) exp(-i(2 pi/n)(k+m)) |n, n-1>] for m odd In particular then, for integral m, exp( (4 pim/n) (N(n) + (1/2)G(n)) ) |t_k)> = exp(i(2 pim/n)) |t_k+2m)> Proof: The first reduction is a result of Theorem 7.3, the second from Theorem 8.13. QED

For n = 2, corollary 8.13.1 is trivial, See also from [Section XVI]. For n > 2, as m takes on integral values m = 1, 2, ..., n, the operator in the special case of corollary 8.13.1 cycles through the eigenvectors |t_k)> in one double cycle if n is odd and in two disconnected cycles if n is even.

<Lemma 8.6>: For k, j = 0, 1, ..., n-1 (n-1) SIGMA cos( (pi/2)(k - j) ) k not= j | n, if n = 2 nu and nu is even | = | (n-1), if n = 2 nu + 1 | | (n-2), if n = 2 nu and nu is odd for integral nu. For k, j = 0, 1, ..., n-1 (n-1) SIGMA (-1)^(k-j) k not= j | if n = 2 nu = | -2 nu | if n = 2 nu + 1 for integral nu. (n-1) SIGMA (-1)^k k k=0 | -(n/2) if n is even = | | +(n-1)/2 if n is odd (n-1) SIGMA (-1)^(k) k=0 | 0 if n is even | | 1 if n is odd Proof: The proofs are not difficult, but are either obvious or tedious. They are omitted and left for amusement. <Theorem 8.14>: For a real uniform distribution of |q(n, k> and also therefore for a real uniform distribution of |p(n, k)>, (n-1) |psi> := SIGMA (1/sqrt(n)) |q(n, k)> k=0 | +1 for n = 2nu <psi|G(n)|psi> = | | +(2 nu)/(2 nu + 1) for n = 2 nu + 1 = 2 nu/n Proof: Use the second formula of Lemma 8.6, and Lemma 8.3. QED <Theorem 8.15>: With m-fold commutator defined by equation (7.4): <q(n, k)| C(Q(n): G(n), m) |q(n, j)> = (q(n, k) - q(n, j))^(m) (delta_kj) - exp(i pi(k-j))) <q(n, k)| exp( +i alpha Q(n) ) G(n) exp( -i alpha Q(n) )|q(n, j)> = exp[ i alpha (q(n, k) - q(n, j)) ] (delta_kj) - exp(i pi(k-j))) <q(n, k)| C(G(n): Q(n), m) |q(n, j)> = (n-1) - exp( i pi k ) SIGMA exp(i pi l) <q(n, l)| C(G(n): Q(n), m-1) |q(n, j)> l=0 (n-1) + exp( i pi j ) SIGMA exp(i pi l) <q(n, k)| C(G(n): Q(n), m-1) |q(n, l)> l=0 <q(n, k)| C(G(n): Q(n), 2) |q(n, j)> = n [(-1)^j q(n, k) - (-1)^k q(n, j)] <q(n, k)| C(G(n): Q(n), 3) |q(n, j)> = n^2 [(-1)^j q(n, k) + (-1)^k q(n, j)] - 2n exp(i pi(k-j)) SIGMA_l) exp(i pil) q(n, l) which for large n becomes, | n^2[(-1)^j q(n, k) + (-1)^k q(n, j)] | - (pi/2) n^(3/2) exp(i pi(k-j)), for n even | | n^2[(-1)^j q(n, k) + (-1)^k q(n, j)], for n odd Proof:

The first formula follows easily from Lemma 8.3, the second from the first formula. The third formula follows from Lemma 8.3 and the recursion relation for the m-fold commutator. The fourth and fifth formulas follow directly from Lemma 8.3 and the sixth as a special case of formula three using formulas four and five. The last approximation for large n result from the linear approximation of q(n, k) given by equation (9.24). The two summations that result are evaluated using the last two formulas of Lemma 8.6. This merely sketches the elementary proofs.

QED <Lemma 8.7>: For large n, large j and large n-j, asymptotically, with the singular term for k=j omitted, (n-1) SIGMA 1/(q_k - q_j) -> (2/pi) sqrt(n) ln( j/(n-j) ) k=0 Proof: Consider the summation for integral m, j and l, +l SIGMA 1/(m+k) k = -l If m = 0, the summation is symmetric about zero and vanishes; m > 0 shifts the values in the denominators to the right, so we may write +l l+m l-m SIGMA 1/(m+k) = SIGMA 1/k - SIGMA 1/k k = -l k=1 k=1 Asymptotically, [Jolley 1961], #70 n SIGMA 1/k -> gamma + ln n + 1/(2n) - 1/(12n(n+1)) - ... k=1 with additional terms that decrease more rapidly with increasing n, and where gamma is the Euler-Mascheroni constant. So, +l SIGMA 1/(m+k) -> gamma + ln(l+m) - g - ln(l-m) = ln((l+m)/(l-m)) k = -l which can be verified to hold for either sign of m. Further from the symmetry of the summation range, then l SIGMA 1/(m-k) -> gamma + ln(l+m) - g - ln(l-m) = ln((l+m)/(l-m)) k = -l Asymptotically by equation (9.22a) q_k -> (pi/(2 sqrt(n))) ( (n-1)/2 - k ) so 1/(q_k - q_j) -> (2 sqrt(n)/ pi) (1/(j-k)) The result follows then easily from the previous results where a 1 is dropped, being small compared to n. QED <Theorem 8.16>: Asymptotically, the matrix elements <p(n, k)| [G(n), Q(n)] |p(n, j)> = <q(n, k)| [P(n), G(n)] |q(n, j)> -> i (2 sqrt(n)/pi) exp( i pi (k-j) ) ln(k/j) Proof:

The first equality follows from application of the Fourier transform Fr(n) defined by equation (7.6). For the asymptotic behavior, expand the products in the commutator by inserting the identity operator expanded in q-eigenbasis. Use Lemma 8.3 to evaluate the matrix elements of G(n) in the q-eigenbasis and Theorem 8.1 to evaluate the matrix elements of P(n) in the q-eigenbasis. After evaluating the Kronecker deltas that will appear and collecting terms, find that,

<q(n, k)| [P(n), G(n)] |q(n, j)> = = i exp( i pi (k-j) ) SIGMA ( 1/(q_k -q_l) + 1/(q_j - q_l) ) (l not= k, l not= j) Now using Lemma 8.7 <q(n, k)| [P(n), G(n)] |q(n, j)> -> i exp( i pi (k-j) ) 2 sqrt(n)/pi ln( (k(n-j))/(j(n-k)) ) For very large n (k(n-j))/(j(n-k)) -> k/j and the result follows. QED <Theorem 8.17>: For the eigenvectors of Q(n) and t(n) associated with the oscillator: <t_k|q(n, j)> = (n d_j)^(-1/2) SIGMA exp( -i (2 pi/n) kl ) ~H_j( q(n, l) ) l and |<t_k|q(n, j)>|^2 = (n d_j)^(-1) SIGMA exp( i(2 pi/n)k(s-l) ) ~H_j( q(n, s) ) ~H_j( q(n, l) ) (s,l) = (n d_j)^(-1) SIGMA cos( (2 pi/n)k(s-l) ) ~H_j( q(n, s) ) ~H_j( q(n, l) ) (s,l) with ~H() defined by equation (9.13), and d_j defined by equation (9.14). Proof: From <t_k|q(n, j)> = <n, k| UPSILON!(n) XI(n) |n, j>

expand as the product of two matrices in |n, k> basis whose elements are given by equations (7.26) and (9.15). The first result follows immediately. The second follows from an additional computation of a matrix product.

QED Can the q-t conditional probability expression of theorem 8.17, be expressed in asymptopia as an evaluable integral? <Theorem 8.18>:Let B be an arbitrary operator in Alg(n) acting on Hilb(n), |a_j> an arbitrary orthonormal basis of Hilb(n), and |phi> an arbitrary element of Hilb(n).

For the amplitudes <phi|B|a_j> and <a_j|B|phi>interpreted as transition amplitudes that distinguish the "before-bra" from the "after-ket", one can construct the normalized transition or conditional probabilities

|<phi|B|a_j>|^2 P( B: phi -> a_j ) = --------------- <phi|B B!|phi> provided the denominator is non zero, and |<a_j|B|phi>|^2 P( B: a_j -> phi ) = --------------- <phi|B! B|phi> provided the denominator is non zero. If B is Hermitean (B = B!), then P( B: a_j -> phi ) = P( B: phi -> a_j ) = |<a_j|B|phi>|^2 = --------------- <phi|B^2|phi> Cf. also Theorem 8.21: Proof: SIGMA_j |<phi|B|a_j>|^2 = SIGMA_j <phi|B|a_j><phi|B|a_j>* = SIGMA_j <phi|B|a_j><a_j|B!|phi> = <phi|B B!|phi> where (B B!) is a positive, Hermitean and therefore normal operator. Similarly, SIGMA_j |<a_j|B|phi>|^2 = <phi|B! B|phi> which provides the normalizing factors for the probability distribution. QED The following generalizes a result of Theorem 8.12.Let H(n), C_H(n), C_H!(n), |E_k>, |t_k>, and UPSILON_H(n) be as they are defined in Theorem 8.19. As Ansatz, let t_H(n, a) be defined by<Theorem 8.19>:Let H(n) be an arbitrary Hamiltonian operator and C_H(n) the cyclic operator in the H(n) eigenbasis |E_k> having the same numerical elements as C(n) in the eigenbasis of N(n).C_H(n) |E_k> = |E_(k-1)> C_H(n) is unitary and can be expressed C_H(n) = exp( i(2 pi/n) t_H(n) ) which defines the time operator associated with the Hamiltonian.The time operator is Hermitean, has an integer valued additive spectrum. Call its orthonormal eigenbasis |t_k> It has a unitary diagonalizing transformation UPSILON_H(n) which connects |E_k> to |t_k>UPSILON_H(n) |E_k> = |t_k>UPSILON_H(n) has the properties of a finite Fourier transform (Cf. [Section VII] ) and has representation<E_k| UPSILON_H(n) |E_j> = (1/sqrt(n)) exp( i(2 pi/n) kj )(The Fourier transform relates the orthonormal eigenbases of H(n) and t_H(n). It does not relate the operators. (We don't even know what the spectrum of H(n) is.) Then for asymptotically large n,[t_H(n), H(n)] = -i (n/(2 pi)) DELTA( H(n) ) where DELTA( H(n) ) = C_H(n) H(n) C_H!(n) - H(n) and [t_H(n), DELTA( H(n) )] = 0 Proof: Define the continuous one-parameter group of unitary transformations C(theta, n) = exp( +itheta t_H(n)) ) C!(theta, n) = exp( -itheta t_H(n)) ) and further define DELTA_theta( H(n) ) = C(theta, n) H(n) C!(theta, n) - H(n) From the BCH expansion (7.3) DELTA_theta( H(n) ) = i theta [t_H(n), H(n)] + (i theta)^2/2! [t_H(n), [t_H(n), H(n)]] + ... For very small theta we can approximate the expansion as simply DELTA_theta( H(n) ) = i theta [t_H(n), H(n)] By letting theta = (2 pi/n) and taking n large enough, theta can be made as small as desired for the approximation to be valid, and so for large n [t_H(n), H(n)] = -i (2 pi/n)^(-1) DELTA_(2 pi/n)( H(n) ) where in the basis |E_k>, DELTA_(2 pi/n)( H(n) ) has the form Diag[ (E_1) - E_0)), (E_2) - E_1)), ..., (E_n-1) - E_n-2)), (E_0) - E_n-1))] For the commutator [t_H(n), DELTA( H(n) )]: C(theta, n) H(n) C!(theta, n) - H(n) = DELTA_theta( H(n) ) C^2(theta, n) H(n) C!^2(theta, n) - C(theta, n) H(n) C!(theta, n) = C(theta, n) DELTA_theta( H(n) ) C!(theta, n) C^2(theta, n) H(n) C!^2(theta, n) - H(n) - DELAT_theta( H(n) ) = C(theta, n) DELTA_theta( H(n) ) C!(theta, n) C^2(theta, n) H(n) C!^2(theta, n) - H(n) = DELTA_theta( H(n) ) + C(theta, n) DELTA_theta( H(n) ) C!(theta, n) But for small theta C^2(theta, n) H(n) C!^2(theta, n) = H(n) + i 2 theta [t_H(n), H(n)] and C(theta, n) DELTA_theta( H(n) ) C!(theta, n) = DELTA_theta( H(n) ) + i theta [t_H(n), DELTA_theta( H(n) ) Substituting back i 2 theta [t_H(n), H(n)] = 2 DELTA_theta( H(n) ) + i theta [t_H(n), DELTA_theta( H(n) ) i 2 theta (-i theta^(-1) DELTA_theta( H(n) )) = 2 DELTA_theta( H(n) ) + i theta [t_H(n), DELTA_theta( H(n) ) 2 DELTA_theta( H(n) ) = 2 DELTA_theta( H(n) ) + i theta [t_H(n), DELTA_theta( H(n) ) Therefore, asymptotically [t_H(n), DELTA_theta( H(n) ) = 0 QED <Theorem 8.20>:Let H(n), C_H(n), C_H!(n), |E_k>, |t_k>, UPSILON_H(n) and t_H(n) be as they are defined in Theorem 8.19. Define C_T(n) and C_T!(n) as the cyclic operators defined in eigenbasis |t_k>, and UPSILON_T(n) as the diagonalizing transformation of C_T(n).Then UPSILON_T(n) = UPSILON_H(n) [H(n), C_T ] = 0, [H(n), C_T!] = 0 and, [t(n), C_H ] = 0, [t(n), C_H!] = 0 Proof: From the definition of t(n) we know that UPSILON_H(n) |E_k> = |t_k> <t_k| UPSILON_H!(n) = <E_k| UPSILON_H!(n) |t_k> = |E_k> <E_k| UPSILON_H(n) = <t_k| and that <E_k| UPSILON_H(n) |E_j> = <E_k|t_j> = (1/sqrt(n)) exp( i(2 pi/n) kj ) <t_k| UPSILON_H(n) UPSILON_H(n) UPSILON_H!(n) |t_j> = <t_k| UPSILON_H(n) |t_j> = (1/sqrt(n)) exp( i(2 pi/n) kj ) Similarly <t_k| UPSILON_T(n) |t_j> = <E_k|t_j> = (1/sqrt(n)) exp( i(2 pi/n) kj ) Therefore UPSILON_T(n) = UPSILON_H(n)The diagonalizing transformation of C_H(n) has the same components in the |E_j> basis as it does in the |t_j> basis. It will therefore also diagonalize the cyclic operator C_T(n) and in doing so maps C_T(n) to its diagonalized form in the |E_j> basis. That is, Both H and C_T(n) are diagonal in the same basis, and therefore commute:[H(n), C_T(n)] = 0 [H(n), C_T!(n)] = 0 and of course it was already obvious that [t(n), C_H(n)] = 0 [t(n), C_H!(n)] = 0 QED <Theorem 8.21>:Let H(n), C_H(n), C_H!(n), |E_k>, |t_k>, UPSILON_H(n) and t_H(n) be as they are defined in Theorem 8.19. The transition probabilities for H(n) between the |t_k> are thenP_(kj)(n) = (Tr( H^2(n) )/n)^(-1) |<t_k|H(n)|t_j>|^2 Proof: <t_k|H(n)|t_j> = SIGMA_l <t_k|E_l>E_l<E_l|t_j> = SIGMA_l (1/n) E_l exp( i(2 pi/n)l(k-j) ) |<t_k|H(n)|t_j>|^2 = SIGMA_(lm) (1/n)^2 E_l E_m exp( i(2 pi/n)(l-m)(k-j) ) SIGMA_j |<t_k|H(n)|t_j>|^2 = SIGMA_j (SIGMA_(l=m) + SIGMA_(lnot=m)) (1/n)^2 E_l E_m exp( i(2 pi/n)(l-m)(k-j) ) = SIGMA_j SIGMA_l (1/n)^2 E_l^2 + SIGMA_j SIGMA_(lnot=m) (1/n)^2 E_l E_m exp( i(2 pi/n)(l-m)(k-j) ) = (1/n) SIGMA_l E_l^2 + SIGMA_j SIGMA_(lnot=m) (1/n)^2 E_l E_m exp( +i(2 pi/n)k(l-m) ) X exp( -i(2 pi/n)j(l-m) ) = (1/n) SIGMA_l E_l^2 + SIGMA_(lnot=m) (1/n)^2 E_l E_m exp( +i(2 pi/n)k(l-m) ) X SIGMA_j exp( -i(2 pi/n)j(l-m) ) = (1/n) SIGMA_l E_l^2 + SIGMA_(lnot=m) (1/n)^2 E_l E_m exp( +i(2 pi/n)k(l-m) ) n delta_(lm) = (1/n) SIGMA_l E_l^(2) Therefore, SIGMA_j |<t_k|H(n)|t_j>|^2 = (1/n) Tr( H^2(n) ) since the normalizing factor has been computed, the result follows. QEDNotice that the normalization factor in theorem 8.21 is essentially an invariant trace, so the total probability of this distribution over transitions is conserved under any general linear transformation H(n) -> S^(-1) H(n) S.

<Lemma 8.20>:

C_H(n) = exp( i omega_t(n, a) t_H(n, a) ) with omega_t(n, a) := (2 pi/tau_0 n^(1-a)) So, Sp( t_H(n, a) ) = { tau_0 k/n^a }, k = 0, 1, ..., n-1 Further define the uniform spectral spacing tau_0(n, a) = tau_0/n^a and the period for a system cycle T_0(n, a) = n tau_0(n, a) So, omega_t(n, a) tau_0(n, a) = (2 pi/n) and omega_t(n, a) T_0(n, a) = 2 piThen, if UPSILON_H(n) is identified as the abstract precursor of the QM Fourier kernel

exp( i E_k t/h-bar ) dt,H(n) must have an additive spectrum with spacing say epsilon(n) and

epsilon(n) tau_0(n, a) = (2 pi/n) h-bara form of an almost classical uncertainty relation. Furthermore, H(n) is then a linear function of N(n). More precisely, for the eigenvalues E_k(n),

E_k(n) = (h/tau_0(n)) (k/n + r(n, k))

where r(n, k) is an arbitrary integer that can depend on n and k. Then H(n) = ( h/tau_0(n) ) ( (1/n)N(n) + R(n) ) where [N(n), R(n)] = 0 and Sp( R(n) ) = {r(n, k)} Proof: Identifying UPSILON_H(n) with the precursor of the QM Fourier kernel, we have exp( i (2 pi/n)kj ) = exp( i E_k t_j / h-bar ) so E_k t_j/h-bar = (2 pi/n) kj + 2 pi m'(n) with m'(n) and arbitrary integer. Rewriting t_j = (h/E_k)( kj/n + m'(n) ) From equation (7.61), we know that Sp( t_H(n, a) ) is additive and that t_j = tau_0(n, a) j then m'(n) = 0 and so tau_0(n, a) j = (h/E_k)( kj/n ) thus tau_0(n, a) will be forced to be also a function of k unless E_k = epsilon(n) k Note: It may not, however, be totally lunatic to consider a multiplicity of operators t_(+|-)(n, k), one for each E_k. then tau_0(n, a) j = ( h/(n epsilon(n)) ) j and so tau_0(n, a) epsilon(n, a) = (2 pi/n) h-bar QED With the Energy-time Fourier Ansatz, the dynamical Schroedinger equation is implied: <Theorem 8.22>:Let H(n), C_H(n), C_H!(n), |E_k>, |t_k>, and UPSILON_H(n) be as they are defined in Theorem 8.19, and let t_H(n, a) be defined as it is in Lemma 8.20. Then, the existence of UPSILON_H(n) and the kernel identification of the previous lemma implies a formal precursor to the Schroedinger equation of motion.

Proof: <t_k| UPSILON_T(n) |t_j> = <E_k|t_j)> = (1/sqrt(n)) exp( i(2 pi/n) kj ) |t_k> = SIGMA_j (1 sqrt(n)) exp( i(2 pi/n) kj ) |E_j> and with the kernel identification of Lemma 8.20, |t_k> = SIGMA_j (1/sqrt(n)) exp( i (1/h-bar) E_j t_k ) |E_j> Then |t_(k+1)> - |t_k> = SIGMA_j (1/sqrt(n)) exp( i (1/h-bar) E_j t_k ) X (exp( i (tau_0(n)/h-bar E_j ) - 1) |E_j> = ( exp( i (tau_0(n)/h-bar) H(n)) - 1 ) |t_k> or (|t_(k+1)> - |t_k>)/tau_0(n) = ( exp( +i (tau_0(n)/h-bar) H(n)) - 1 )/tau_0(n) |t_k)> and by Hermitean Conjugation (<t_(k+1)| - <t_k|)/tau_0(n) = <t_k| ( exp( -i (tau_0(n)/h-bar) H(n)) - 1 )/tau_0(n) For tau_0(n) small enough (|t_(k+1)> - |t_k>)/tau_0(n) -> (+i/h-bar) H(n)) |t_k)> and (<t_(k+1)| - <t_k|)/tau_0(n) -> (+i/h-bar) <t_k| H(n) If tau_0(n) -> 0 appropriately, write the limiting equations as n -> infinity as differential equations d(|t>)/dt = (+i/h-bar) H(n) |t> and d(<t|)/dt = (-i/h-bar) <t| H(n)This is very good approximation for very large n. The representation of the states |t> can, in a sense, actually survive the limit, but the existence of any time operator with the properties at finite n is exceedingly doubtful.

QED With the Energy-time Fourier Ansatz, the dynamical Heisenberg equation is implied. <Theorem 8.23>:Let H(n), C_H(n), C_H!(n), |E_k>, |t_k>, and UPSILON_H(n) be as they are defined in Theorem 8.18. and let t_H(n, a) be defined as it is in Lemma 8.20. Then, the existence of UPSILON_H(n) and the kernel identification of the previous lemma implies a formal precursor to the Heisenberg equation of motion.

Proof: For an arbitrary Hermitean operator O, write the representation O = SIGMA_(kj) o_(kj) |t_k><t_j)| If DELTA|t_k> := |t_(k+1)> - |t_k> DELTA<t_k| := <t_(k+1)| - <t_k| and DELTA has the Leibnitz product rule, so DELTA (|t_k><t_j|) := (DELTA |t_k>) <t_j| + |t_k> (DELTA <t_j|) we can write DELTA O = SIGMA_(kj) o_(kj) DELTA (|t_k><t_j|) = SIGMA_(kj) o_(kj) (DELTA |t_k>) <t_j| + |t_k> (DELTA <t_j|) Then, since (DELTA|t_k>)/tau_0(n) = ( exp( +i (tau_0(n)/h-bar) H(n)) - 1 )/tau_0(n) |t_k> and (DELTA <t_k|)/tau_0(n) = <t_k| ( exp( -i (tau_0(n)/h-bar) H(n)) - 1 )/tau_0(n) we have (DELTA O)/tau_0(n) = ( exp( +i (tau_0(n)/h-bar) H(n)) - 1 )/tau_0(n) O + O ( exp( -i (tau_0(n)/h-bar) H(n)) - 1 )/tau_0(n) which for small tau_0(n) is well approximated by (DELTA O)/tau_0(n) = (+i/h-bar) [H(n), O] Assuming the limit tau_0(n) -> 0 is well defined, write d O/dt = (+i/h-bar) [H(n), O] QED

The point of Theorem 8.22 and Theorem 8.23 is that the existence of operator t_H(n, a) defined as the generator of the cyclic operator in the H(n) eigenbasis is enough to imply a QM type "equation of motion". This equation of motion is then identically satisfied for finite n and becomes somewhat irrelevant. The important aspect of the indeterminate "dynamics" rests then in the notion of transition probabilities. One might see in this example an instantiation of a seemingly paradoxical catch phrase attributed to J. A. Wheeler: "dynamics without dynamics" spoken in the context of geometrodynamics. When the limit is actually taken, the time operator does not exist and the dynamical equation of motion becomes an important constraint that must be reinstituted "by hand", since a straightforward limit of the equation cannot exist: a limit for the time operator does not exist.

The physical significance of this situation is that quantum theoretical states themselves in the ultrasmall do not evolve smoothly and determinately. Not only is the outcome of a measurement not determinate, but now the evolution of the state of the system is not even determinate. The Old One may play more games of chance that thought previously.

The state evolution itself can only be given probabilistically; this introduces a level of quantization beyond that of QM, while maintaining a certain consistency with QM when n becomes large, i.e., when the ultrasmall becomes from the viewpoint of QM, merely "small".

The indeterminacy of the state, forces attention yet again on the question of it ontological status. If the evolution of the state is indeterminate, can the state itself ever be considered to have been determinate? If a positive ontological status is ascribed to the state, then ontology itself, particularly a temporally persistant ontology, is a suspect concept, at least in the ultrasmall. The process of passing from ontology to ontology, becomes a matter of interest. It is even a better trick than pulling rabbits out of seemingly empty hats, and it is natural enough to be curious about the "how" of that.

If a negative ontological status is ascribed to the state, so that it is considered to be a repository for the statistics of the outcomes of all measurements, merely a convenient mathematical construct, "the hat" so to speak, in and out of which rabbits are to be spirited when no one is looking, it is learned that the concept of "concept" and recursively on up the line is just as suspect as a modeled determinate reality itself. It has been cogently argued in many places and times that the quantum mechanical state function actually tells much more about the measurement and measurer than it does about the system being measured, so perhaps a lesson here is that in the ultrasmall we just can't have very good measurers, nor measurements.

Ontology has not truly been negated, but "the concept" and any implicit epistemological concepts used to model it have indeed been negated. We would seem then to have Everett-Wheeler branching without Everett-Wheeler branching. The local state itself, evolves in a noncomputable, yet probabilistic sequence of the branching of its possibilities. A minimal reflection on the added indefiniteness of state, is that a state as represented by a density matrix is more reasonable than a state represented by an element of a projective Hilbert space. The states of Alg(n) are local and in contact with a larger thermal reservoir: a larger proper superset of states Alg(m), where m >> n. The states of Alg(n) can be expected to be quantum mechanically 'entangled' with those of Alg(m).

In the specific case of the simple harmonic oscillator note that when a limit is taken, the time eigenvalues

t_k(n) = tau_0k/n -> twhere t is a continuous variable defined in the interval [0,\ 1], parametrizing the torus group but can be extended to the covering group, to translations of the real line. An answer to the question as to why one should pass from the infinitely multiply connected toridal group T^(1) to its simply connected covering group REAL, could be no better than an answer as why to include all physical rotations by passing from the obvious SO(3), doubly connected symmetry group of an isotropic space to its simply connected covering group SU(2). The simplest argument is that apparently the physics is in the algebra and not in the group. A small step from such an argument in which correct physical thinking favors the local over the global, just as it seems to favor differential equations, over functional or integral equations in fundamental matters is that physics *is* fundamentally algebraic and that to look at and for fundamental physical models of reality, abstract algebra seems to be the place to start.

A complete and systematic limit for the harmonic oscillator is discussed in the subsubsection on the oscillator with n energy levels in [Section XVI]. In this limit the interval for the limiting time variable is [0, infinity].

<Theorem 8.24>: The FCCR analog of the propagator or Feynman kernel of QM is <q_k, t_l|q_j, t_m> = <q_k|C!^l(n) C^m(n)|q_j> = <q_k| C^(m-l)(n) |q_j> = <q_k| C!^(l-m)(n) |q_j> with SIGMA_l <q_k, t_l|q_j, t_l> = n delta_(kj) and SIGMA_k <q_k, t_l|q_k, t_m> = Tr( C^(m-l)(n) ) = n delta_(ml) where the Kronecker delta in the second expression can be understood modularly so a non zero value obtains when m and l are equal mod n.Proof:

In the QM case where all forces are the consequence of a single time independent potential, the time dependent energy eigenfunctions can be written

PSI_k(x, t) = phi_k(x) exp( -iE_k/h-bar t ) and the propagator can be written <x, t|x', t'> = SIGMA_k PSI_k(x, t) PSI_k*(x', t') = SIGMA_k phi_k(x) phi_k*(x') exp( -iE_k/h-bar (t - t') ) = <x| SIGMA_k |E_k> exp( -iE_k/h-bar (t - t') ) <E_k|x'> = <x| SIGMA_k |E_k> exp( -iE_k/h-bar t ) exp( +iE_k/h-bar t' ) <E_k|x'> = <x| SIGMA_k exp( -iH(n)/h-bar t )|E_k><E_k|exp( +iH(n)/h-bar t' ) |x'> Switching to notation for discrete q and discrete additive t, <q_k, t_l|q_j, t_m> = <q_k|SIGMA_k exp( -iH(n)/h-bar t_l )|E_k><E_k|exp( +iH(n)/h-bar t_m )|q_j> = <q_k| SIGMA_k C_T!^l(n)|E_k><E_k|C_T^m(n) |q_j> = <q_k| C_T!^l(n) SIGMA_k |E_k><E_k|C_T^m(n) |q_j> = <q_k| C_T!^l(n) C_T^m(n) |q_j> and C_T!^l(n) = C_T^(-l)(n) The two summation rules are easy. The propagator folds by a sum over q_j SIGMA <q_k, t_l|q_j, t_m><q_j, t_s|q_i, t_r> = <q_k, t_(l+s)|q_i, t_(m+r)> QED

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