Spectral Derivatives
In Finite Dimensional Complex Hilbert Spaces
and C*-algebras


Abstractly write the dynamical equation in Quantum Mechanics (QM),

        H |f(t)>  =  i h (d/dt) |f(t)>

in the Schroedinger picture. The Planck constant 'h' will always represent the normalized constant 'h-bar' where h-bar := (h/2pi). Here, H is the Hamiltonian (energy) operator from the canonical formalism of mechanics, |f(t)> is the state vector that may also depend on variables other than time, and t is a classical variable not attached to any operator and that instead turns out to be the parameter of the one parameter unitary group of dynamical transformations for which H is the generator if H is an operator not explicitly dependent on t. If H is time independent the formal solution to the dynamical equation can be expressed by:

       |f(t)>  =   U(t) |f(0)>
       U(t)  :=  exp( itH/h )

If H is time dependent, the formal solution can be expressed by:

       |f(t)>  =   U(t) |f(0)>


       U(t)  :=  exp( (i/h) INTEGRAL(s=0,s=t) H(s) ds )

   provided, of course, that the time ordered integral exists.
   See notes on the  Lax Equation .

In the Heisenberg picture, the dynamical equation is

        HA - AH - ipart-dA/part-dt  =  -ih DA/Dt

   where "part-d" is the sign of partial differential.
where A is the Hermitean operator representing some alleged "observable" property of the system defined by the Hamiltonian. The operator A may depend both explicitly and implicitly on t. The partial derivative is the derivative of A with respect to its explicit dependence, while the operator D/Dt expresses the total time derivative:
   Say A  =  A( x(t), t ), then

        DA/Dt  =  (part-dA/part-dx)(dx/dt) + (part-dA/part-dt)

In the Heisenberg picture then, the time dependence is attached to the operators and not to the states, so that when A is not explicitly time dependent, the formal solution when H is not explicitly time dependent can be similarly expressed by:

       A(t)  =  U(t) A(0) U^(-1)(t)

       U(t)  :=  exp( itH/h )

   If H is time dependent,
   the formal solution can be expressed by:

       A(t)  =  U(t) A(0) U^(-1)(t)

   where again

       U(t)  :=  exp( (i/h) INTEGRAL(s=0,s=t) H(s) ds )

   and again provided that the integral exists.

The structure of time t in QM is one of a universal Newtonian variable that is algebraically "central", i.e., considered as a legitimate operator, which it is not, it commutes with all operators, and can therefore be said conceptually to be in the center of the algebra of observables. Taking the viewpoint of central time seriously, one can picture the algebra as actually a direct sum (actually a direct integral if time is continuous) of algebras each one of which is an invariant subspace of an alleged time operator. Usually it is more convenient to consider time as simply being the parameter of a unitary group of automorphisms of any one of these algebras, all being unitarily equivalent, in the direct sum. Conceptually we will distinguish the apparently continuous time parameter of QM which is presumed operationally referred to a large laboratory clock from the time told by the clock that is the system itself.

In what follows, the Dirac notation for Hilbert space vectors and their dual linear functionals is used. Standard mathematical notation is altered to fit within the the printing ascii character set. The symbolic expresions, unfortunately, sometimes take on a particularly cumbersome aspect but with some luck, their content will be clear.

Consider the set of formal relations in QM:

        [Q, P]  =  ih I

        <q'|q>  =  delta(q' - q)

        Q|q>  =  q|q>

        P|q>  =  ih (part-d/part-dq)|q>

                           |q'> - |q>
              =  i h  lim  ----------
                     q'->q   q' - q

In this last equation, the action of the operator P is expressed as the limit of a difference of eigenvectors of the operator Q divided by a difference of eigenvalues of the operator Q. It is this relationship that I wish to generalize in defining a spectral derivative associated to operators other than Q. This generalization might then also provide a generalization of the notion of canonically conjugate pairs of observables, as well a generalization of the concept of "dynamical equation", where the time of dynamical evolution is specifically noncentral as an operator which is not a multiple of the identity in the relevant algebra of operators,

The notion of spectral derivative is to be defined using a similar ratio of differences. The limit cannot actually be taken in a finite n-dimensional space with n fixed, but the differences (q'-q) can be taken optimally small throughout the spectrum when the spectrum can be given a suitable ordering.

We want to formulate a "dynamical law" that avoids the assumption of a central time, while maintaining the idea that the values of a time parameter are to be the spectral values of some operator. The physical motivation for such a notion is a theoretical elaboration of dynamics in the context of noncentral time.

What is wanted then is the definition of a spectral derivative, or a differentiation of a vector, and then also of an operator, with respect to the spectrum of some given operator. Furthermore we want to define this idea algebraically in the context of finite dimensional complex Hilbert spaces and the finite dimensional C*-algebras of operators that act thereon so that the spectral derivative of an operator is also defined. The reason for considering finite dimensional spaces is that a proper quantum mechanical time operator is known not to exist in a truly reasonable way in the context of infinite dimensional Hilbert spaces with the usual formulation of QM.
Cf. e. g., [Garrison 1970]
A time operator does exist, however, in a certain finite dimensional variant of QM. It is this operator to which the interpretation of time told by the system clock can be attached. Accommodating the structure of the Heisenberg picture where the operators are time dependent and not the Hilbert space vectors means that one should also be able to extend the notion of spectral derivative to express the spectral derivative of an operator with respect to another operator. This is done after first discussing the spectral derivatives of Hilbert space vectors.

Irrespective of the physical motivation, it would seem that the mathematical machinery developed has interest in its own right and there exists the possibility that this machinery may have applications in other areas.

The desired properties of the spectral derivative operator SPD(A)[], with respect to an operator A, in its action on Alg(n), the C*-algebra of linear operators acting on Hilb(n), the complex Hilbert space, should be those, when appropriate, of a standard derivative:

        P1) Linearity of action
            For alpha a scalar and X and Y in the domain of its action
            SPD(A)[ alpha(X + Y) ]  =  alpha SPD(A)[ X ] + alpha SPD(A)[ Y ]

        P2) Skewness with respect to algebra involution '!',
            which concretely becomes Hermitean conjugation, i.e.,
            transpose + complex conjugation.
            If A! = A,
                 SPD!(A)[ X ]  =  - SPD(A)[ X ]
            so then the Hermitean operator will be
                 i SPD!(A)[ X ]

        P3) The Leibnitz properties
            For A, X, Y in the operator algebra acting on a complex
            finite dimensional Hilbert space.
            SPD(A)[ A ]  =  I
            SPD(A)[ XY ]  =  SPD(A)[ X ] Y + X SPD(A)[ Y ]
            i.e., SPD(A)[ . ] is a derivation.

Relevant toplogical and algebraic definitions, and results preliminary to the following material are contained in [Appendix A].

Let Alg(n) be the algebra of linear operators (complex matrices) acting on Hilb(n), a complex Hilbert space with complex dimension n. With the operation of complex conjugate transpose Alg(n) has the involution associated with a C*-algebra, and it inherits the Hilbert space norm to become a C*-algebra. A full matric algebra over a division ring is simple: its only ideals are the trivial ones, the algebra itself and (0).

   Let A be a normal operator in Alg(n):

        [A, A!]  =  0

The operator A is then always diagonalizable by a similarity transformation and has, in general, complex eigenvalues and orthogonal eigenvectors which can be normalized in the usual way [Kato 1966], p. 59. Then we can write,

        A |a_k>  =  a_k |a_k>,     A! |a_k>  =  a_k |a_k>
   and have
        <a_k|a_j>  =  delta_kj

The class of normal operators subsumes those of Hermitean and unitary operators. Normal operators have the properties [Kato 1966], p. 55.

        Norm( A^k )  =  Norm^(k)( A ), where Norm(.) is the C*-algebra norm

        Norm( A )   =  rho( A ), the spectral radius

        A is normal implies p(A) is normal for any polynomial
             p() with complex coefficients.

        If A is normal and nonsingular,
             then its inverse, A^(-1) is normal and nonsingular.

        A^k = 0 for some integer k implies A = 0.

        Similarity equivalence for two normal operators implies
        unitary equivalence. [Putnam 1967], p. 12 

        Let arbitrary A in a complex algebra be represented
             A = H + iJ,
        where H and J are invariant under Hermitean conjugation, then
             [A, A!]  =  -2i[H, J]
        Therefore, if A is normal,
             [H, J]  =  0
        Normal operators are then Abelian complex extensions
        of Hermitean operators.

        Then, assuming A normal, since
             A! = H - iJ,
        define the non-negative Hermitean operator
             R  =  (H^2 + J^2)^(1/2)
                =  A! A  =  A A!
        to obtain a polar decomposition of the normal A
             A  =  U R
        with U unitary, and
             [U, R]  =  0
        If A is invertible, then U is unique.

Label the one dimensional invariant subspaces and their attached eigenvalues by the integers k = 0, 1, 2, ..., n-1 and consider the index labels to be taken mod-n, so the indicies are cyclic. .sp Also let A have nondegenerate spectrum Sp( A ) so that all the a_k are distinct.

If we have in mind that A is to model a kind of time operator, the nondegeneracy assumption can be read descriptively as: "Discrete eigenpoints of time are algebraically separated."

From the orthonormal eigenbasis |a_k>, using Dirac notation, we can construct a cyclic operator C_A! defined on A's eigenbasis by

        C_A! |a_k>  =  |a_(k+1)>

which extends to full action on Hilb(n) by linearity. Again the basis label is understood mod-n so that

        C_A! |a_(n-1)>  =  |a_n>  =  |a_0>

   C_A! is, of course unitary and further,

        C_A! |a_k>  =  |a_(k-1)>

        <a_k| C_A!   =  <a_(k-1)|

        <a_k| C_A  =  <a_(k+1)|

Ordering of the eigenvalues of the normal,
multiplicity free operator A

For a_k given in polar form, there is a natural ordering and then also a natural ordering of |a_k>:

   Let the eigenvalues be given in polar form
        a_k  =  r_k exp( i omega_k )
        0 <=  r_k,    0  <=  omega_k  <  2pi

            r_k   >   r_j
   then let
            a_k   >   a_j
   else if
            r_k   <   r_j
   then let
            a_k   <   a_j
   else if
            r_k   =   r_j
        we cannot have omega_k  =  omega_j,

        since this violates the zero multiplicity of A.

             omega_k  >  omega_j,
        then let
                 a_k  >  a_j
        else let
                 a_k  <  a_j

   Adopt this ordering so that for k = 0, 1, 2, ..., n-2

             a_k < a_(k+1)
             a_(n-1) > a_n = a_0

   Then, where rho( A ) is the spectral radius of A,

     sup |a_k|  =  rho( A )  =  r_(n-1)  =  Norm( A )

   For the cyclic operator C_A, for example,
   the eigenvalues are the n-th roots of unity

        {exp( i(2pi/n)k ): k = 0, 1, ..., n-1},

   since C_A is idempotent of order n, and n is the minimal order
   of idempotency.  The ordering of these eigenvalues by the above
   definition becomes

        exp( i(2pi/n)k )  <  exp( i(2pi/n)j )

when k < j, so that k = 0, starts the ordering at the minimal eigenvalue which is 1, and proceeds around the unit circle in the complex plane, in a counterclockwise direction. This ordering statement for an extended cyclic operator obviously extends to j and k outside of the prescribed range when both j and k are considered mod-n. On an infinitely sheeted Riemann surface associated with k and j extended to full range of the integers, project the points to the central sheet, to determine the ordering.

Diagonalizing C_A

Let |ca_k> be the orthonormal eigenbasis in Hilb(n) of C_A.

   For k = 0, 1, ..., n-1,

        C_A |ca_k>  =  exp( i(2pi/n)k ) |ca_k> 

   Let UPSILON_A be defined by its components in |ca_k>

        <a_k| UPSILON_A |a_j>  =  (1/sqrt(n)) exp( +i(2pi/n)kj )

        <a_k| UPSILON_A! |a_j>  =  (1/sqrt(n)) exp( -i(2pi/n)kj )

   and let N_A be the number operator defined on |a_k>

         N_A |a_k>  :=  k |a_k>

   Then it is easily verified that:

        UPSILON_A  |a_k>   =  |ca_k>
        UPSILON_A! |ca_k>  =  |a_k>

        UPSILON_A C_A UPSILON_A!  =  exp( i(2pi/n)N_A )

        UPSILON_A^2 |a_0>  =  |a_0>
        UPSILON_A^2 |a_k>  =  |a_(n-k)>,  k not 0

        UPSILON_A^3  =  UPSILON_A!
        UPSILON_A^4  =  I(n)

Coincidentally, the diagonalizing transformation UPSILON_A then has the properties of a finite (discrete) Fourier transform f.

             i) f not= I
            ii) f is unitary: f^(-1)  =  f!

           iii) f is idempotent of order 4: f^4  =  I;
                and therefore has as eigenvalues, the fourth roots
                of unity.

The Difference operators

   Define the difference operators

        (DELTA^+A)  :=  C_A A C_A!  -  A
        (DELTA^-A)  :=  C_A! A C_A  -  A

   and the operators that extract the Hermitean and
   Skewhermitean parts of A,

        He( A )  :=  (1/2)(A + A!)
        Sk( A )  :=  (1/2)(A - A!)

        A   =  He( A ) + Sk( A )
        A!  =  He( A ) - Sk( A )

        C_A! (DELTA^+A) C_A  =  -(DELTA^-A)
        C_A! (DELTA!^+A) C_A  =  -(DELTA^-A)!
        C_A! (DELTA^+A)^(-1) C_A  =  -(DELTA^-A)^(-1)

        [(DELTA^+A), (DELTA^+A)!]  =  C_A [A, A!] C_A! - [A, A!]
                            + 2 Sk( [C_A A! C_A!, A] )

   Since [C_A A! C_A!, A!] = 0
   identically and [A, A!] = 0 for a normal operator, we have
   A normal implies

        [(DELTA^+A), (DELTA^+A)!]  =  0

   and similarly

        [(DELTA^-A), (DELTA^-A)!]  =  0

        (DELTA^+A) |a_k>  =  (a_(k+1) - a_k) |a_k>
        <a_k| (DELTA^+A)  =   <a_k| (a_(k+1) - a_k)

        (DELTA^-A) |a_k>  =  (a_(k-1) - a_k) |a_k>
        <a_k| (DELTA^-A)  =   <a_k| (a_(k-1) - a_k)
        (DELTA^+A)! |a_k>  =  (a_(k+1)^* - a_k) |a_k>
        <a_k| (DELTA^+A)!  =   <a_k| (a_(k+1)^* - a_k)

        (DELTA^-A)! |a_k>  =  (a_(k-1)^* - a_k^*) |a_k>
        <a_k| (DELTA^-A)!  =   <a_k| (a_(k-1)^* - a_k^*)

   Although A has not been assumed to be invertible
   (0 may be in Sp( A ), but only once), the operators

        (DELTA^+A), (DELTA^-A), (DELTA^+A)! and (DELTA^-A)!

   are nevertheless all invertible.
   This is an obvious consequence of the assumption of zero
   multiplicity for A.


        (DELTA^+A)^(-1) |a_k>  =  ------------- |a_k>
                                  a_(k+1) - a_k

        (DELTA^-A)^(-1) |a_k>  =  ------------- |a_k>
                                  a_(k-1) - a_k

        (DELTA^+A!)^(-1) |a_k>  =  ----------------- |a_k>
                                   a_(k+1)^* - a_k^*

        (DELTA^-A!)^(-1) |a_k>  =  ----------------- |a_k>
                                   a_(k-1)^* - a_k^*

   It is clear that the operator set

    { A, DELTA^+A, DELTA^-A, DELTA^+A!, DELTA^-A!,

     (DELTA^+A)^(-1), (DELTA^-A)^(-1), (DELTA^+A!)^(-1), (DELTA^-A!)^(-1) }

   is Abelian since the operators share an eigenbasis spanning Hilb(n).

   In particular then reproduce the result

        [(DELTA^+A), (DELTA^+A!)]  =  0
        [(DELTA^-A), (DELTA^-A!)]  =  0

   so that (DELTA^+A) and (DELTA^-A) are normal operators.

Since the cyclic operators C_A! and C_A have their eigenvalues on the unit circle, the operators

             (C_A! - I) and (C_A - I)

have their eigenvalues on a circle of radius one centered at the complex Cartesian point (-1,0). For any n, +1 is always a root of unity. Then, for any n, zero is in

              Sp( (C_A! - I) ) and Sp( (C_A - I) ).

The operators

              (C_A! - I) and (C_A - I)

are never invertible, never Hermitean and never unitary. They are clearly, however, always normal.

If A is assumed to be Hermitean, clearly (C_A! - I) and
(C_A - I) are not affected, but then the Abelian set

     {A, DELTA^+A, DELTA^-A, DELTA^+A!, DELTA^-A!,

     (DELTA^+A)^(-1), (DELTA^-A)^(-1), (DELTA^+A!)^(-1), (DELTA^-A!)^(-1)}

is also Hermitean.

From the definition of the DELTA_As, assuming only that A is normal, commutation properties of the operators constructed so far are:

        [A, C_A!]  =  C_A! (DELTA^+A)
        [A, C_A]  =  C_A (DELTA^-A)

        [(DELTA^+A), A]  =  0
        [(DELTA^-A), A]  =  0

        [C_A!, (DELTA^+A)]  =  [A, C_A!] - (DELTA^+A)C_A!
        [C_A, (DELTA^+A)]  =  [A, C_A] + C_A(DELTA^+A)
        [C_A!, (DELTA^-A)]  =  [A, C_A!] + C_A!(DELTA^-A)
        [C_A, (DELTA^-A)]  =  [A, C_A] - (DELTA^-A)C_A

        [(DELTA^+A), C_A!] - [(DELTA^-A), C_A!]
                        =  (DELTA^+A)C_A! + C_A!(DELTA^-A)

        {(DELTA^+A), C_A!}  =  (DELTA^+A - DELTA^-A) C_A!

   where [A, B] := AB - BA, and {A, B} := AB + BA,
   define the commutator and anticommutator respectively.

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Created: 1997
Last Updated: May 28, 2000
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