Thus, all the possible expressions, Equation (6) (and Equation (7), since by Equation (35), it is independent of α), are equivalent, under a similarity transformation, to Equation (8), which is self-adjoint under the measure

$\text{d}W=\text{d}x\left(\frac{\text{d}W}{\text{d}x}\right)$ . The measure is automatically generated by the canonical trans

formation. We also point out that one could have chosen the constant of integration, $g\left(x\right)$ in Equation (6) to ensure that one obtains Equation (8), independent of the choice of $\alpha $ .

7. Generalized Harmonic Oscillators, Generalized Coherent States and Generalized Fourier Transforms

In this Section, we point out that the operators $\stackrel{^}{W},{\stackrel{^}{p}}_{W}$ can also be used to define a generalized harmonic oscillator with Hamiltonian

${\stackrel{^}{H}}_{W}=\frac{1}{2}\left({\stackrel{^}{p}}_{W}^{2}+{\stackrel{^}{W}}^{2}\right),$ (39)

with $\stackrel{^}{W},{\stackrel{^}{p}}_{W}$ given by Equation (11). The eigenstates are given by

${\psi}_{j}\left(W\right)={c}_{j}{H}_{j}\left(W\right)\mathrm{exp}\left[-\frac{{W}^{2}}{2}\right],$ (40)

where the coefficients ${c}_{j}$ are the usual normalization constants and the functions ${H}_{j}\left(W\right)$ are standard Hermite polynomials of the variable W. They are a complete, orthonormal basis, so that

$\underset{j=0}{\overset{\infty}{\sum}}{\psi}_{j}^{\ast}}\left(W\right){\psi}_{j}\left({W}^{\prime}\right)=\delta \left(W-{W}^{\prime}\right)$ (41)

and

${\int}_{-\infty}^{\infty}\text{d}W\text{\hspace{0.05em}}{\psi}_{j}^{\ast}\left({W}^{\prime}\right){\psi}_{{j}^{\prime}}\left({W}^{\prime}\right)}={\delta}_{j{j}^{\prime}}.$ (42)

Just as in the case of the standard harmonic oscillator, where $\Delta x\Delta k=1/2$ for the ground state, the W-harmonic oscillator ground state for a given polynomial W(x) will satisfy $\Delta W\left(x\right)\Delta W\left(k\right)=1/2$ . Next we note that Equation (39) can be factored in terms of the ladder operators

${\stackrel{^}{a}}_{W}=\frac{1}{\sqrt{2}}\left(\frac{\text{d}}{\text{d}W}+W\right)$ (43)

${\stackrel{^}{a}}_{W}^{+}=\frac{1}{\sqrt{2}}\left(-\frac{\text{d}}{\text{d}W}+W\right).$ (44)

Thus,

${\stackrel{^}{H}}_{W}={\stackrel{^}{a}}_{W}^{+}{\stackrel{^}{a}}_{W}+1/2$ (45)

$={\stackrel{^}{a}}_{W}{\stackrel{^}{a}}_{W}^{+}-1/2.$ (46)

It follows that one can construct coherent states using the W-displacement operator or as eigenvectors of the lowering operator. The resulting coherent states are over-complete and can be used as basis-functions for a variety of calculations. They also can be used to explore new types of semiclassical approximations. In addition, one can generate W-generalized Wigner distributions in the $W,{P}_{W}$ phase space. As for the new MUB, these all may be implemented in the coordinate representation with the correct measure guaranteed by the canonical transformation. We note that the Heisenberg-Weyl Lie algebra governs the W-oscillator, in contrast to the work of Williams, et al. [34] .

Of course, the orthonormal eigenstates of the W-harmonic oscillator are eigenstates of the W-momentum eigenstates (interpreted as generalized Fourier transform kernels). That is, the ${\psi}_{j}\left({W}^{\prime}\right)$ satisfy the generalized Fourier transform relation

$\begin{array}{l}\langle {\varphi}_{{p}_{W}}|{\displaystyle {\int}_{-\infty}^{\infty}\text{d}W|W\rangle}\langle W|{\psi}_{j}\rangle \\ =\frac{1}{\sqrt{\text{2\pi}}}{\displaystyle {\int}_{-\infty}^{\infty}\text{d}W\text{\hspace{0.05em}}{\text{e}}^{-i{p}_{W}W}}{\psi}_{j}\left(W\right)={\psi}_{j}\left({p}_{W}\right).\end{array}$ (47)

In the coordinate representation, this is

$\frac{1}{\sqrt{2\text{\pi}}}{\displaystyle {\int}_{-\infty}^{\infty}\text{d}x\text{\hspace{0.17em}}\frac{\text{d}W}{\text{d}x}{\text{e}}^{-iW\left(k\right)W\left(x\right)}}{\psi}_{j}\left(W\left(x\right)\right)={\psi}_{j}\left(W\left(k\right)\right),$ (48)

and it is in this form that the transformation will be used. We also note that time-frequency analyses can be carried out by a windowed generalized Fourier transform, with the window being the ground state of the W-harmonic oscillator, or any other convenient window. Such transformations should be of interest for the analysis of chirps.

Finally, we note that one can also use the W-Gaussian to generate new “minimum uncertainty wavelets” and the closely related “Hermite Distributed Approximating Functionals” that have proved to be extremely useful computational tools in a number of areas, as well as for digital signal processing [35] - [41] .

8. Results and Discussion

For invertible canonical transformations of $x,{p}_{x}$ to Cartesian-like variables

$W,{p}_{W}$ , Dirac quantization results in the unique operators $\stackrel{^}{W}=W,{\stackrel{^}{p}}_{W}=-i\frac{\text{d}}{\text{d}W}$ .

Additionally, simple replacement of the usual position and momentum variables

by $\stackrel{^}{x}=x,\stackrel{^}{p}=-i\frac{\text{d}}{\text{d}x}$ in any of the infinitely many classical expressions for

$W,{p}_{W}$ typically leads to non-Hermitian operators. These are normally rejected as valid operators but they can all be transformed to the W-representation Hermitian operators, $\stackrel{^}{W},{\stackrel{^}{p}}_{W}$ . The non-Hermitian x-representation operators yield biorthogonal, complete basis sets and the Hermitian cases yield a unique orthonormal complete basis set (all with Dirac delta normalization). These non- Hermitian operators all are examples of quasi-Hermitian operators.

We’ve shown that possessing the canonical commutation relation corresponding to a unit Poisson bracket is not sufficient [23] to produce MUB. The standard method of defining a self-adjoint momentum operator as the average of the corresponding classical expressions satisfies the correct commutation relation but does not lead directly to MUBs. Neither does the symmetric or asymmetric quantization of the canonical momentum as defined in Equation (6), directly yield an MUB in the coordinate representation. Rather, it is only in the new, W-representation that one obtains an MUB [23] . In addition, for any given choice of W, this MUB is unique (as seen when expressed in the x-representa- tion) and independent of the particular way in which one arranges the canonical momentum prior to quantization!

In the above study, we restricted ourselves to odd power dominated, non- negative coefficient polynomial choices of the generalized position. We ask now what happens if we choose only the even powered polynomials. In that case, the eigenstates of ${\stackrel{^}{p}}_{W}$ have exactly the same form as before. However, there are now additional, standard, normalizable eigenstates in the ${L}^{2}$ sense for complex eigenvalues ${p}_{W}={p}_{W}^{real}+i{p}_{W}^{imag}={W}^{real}\left(k\right)+i{W}^{imag}\left(k\right)$ , of the form

${\phi}_{W}\left(x\right)=\mathrm{exp}\left[i{W}^{real}\left(k\right)W\left(x\right)-{W}^{imag}\left(k\right)W\left(x\right)\right],$ (49)

$W\left(x\right)\stackrel{\left|x\right|\to \infty}{\to}+\infty ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{W}^{imag}\left(k\right)>0.$ (50)

These tend to zero as $x\to \pm \infty $ and so the eigenstate structure is more complicated. In addition, their modulus is not constant and eigenstates for different eigenvalues are not orthogonal. A third important issue is the fact that the domain of W with even powers of x is $0\le W<\infty $ . Thus, the new variable is not “Cartesian-like”, since one no longer has the full real line as the domain of the new variable but only the half line. The transform is also no longer invertible except on the half line, $0\le x<\infty $ . This is suggestive of a “radial-like” behavior, and is a situation which will be studied further, along with more general choices of W (e.g., fractional powers of $x$ and others).

We also see that it is possible to obtain new, unexpected self-adjoint operators by properly accounting for the Jacobian of the canonical transformation. This avoids forcing self-adjointness by symetrization techniques by taking advantage of the fact that the chain rule automatically ensures that one particular ordering of the classical variables, Equation (8), is manifestly self-adjoint, with the proper measure. There is a sense in which this is an obvious point (it is well known that there are operators that are self-adjoint under one measure and not another). The point is that the change in measure is a natural consequence of our seeking to find more natural coordinates to describe the systems of interest. We stress that this has already lead us in previous studies to develop new coherent states whose convergence properties for excited states are superior to bases that are not defined using information about the ground state [31] . Our strategy here is simply to use the same ideas that make canonical transformations so useful in classical dynamics for the quantum mechanical case. The result is that we now have an infinite number of W choices resulting in sets of operators whose eigenstates form continuous MUBs, as well as continuous, complete biorthogonal and orthogonal basis sets and over-complete coherent states!

We constructed four distinct types of bases. These include the MUB generated by the three operators, $\left\{\stackrel{^}{W},{\stackrel{^}{p}}_{W},\stackrel{^}{W}+{\stackrel{^}{p}}_{W}\right\}$ . Because these can be implemented in the x-representation (with the new measure automatically taken into account), these new MUB make possible an added layer of security in applications to quantum cryptography. Next we have used the W-representation “position” and “momentum” to define the Hamiltonian of a generalized harmonic oscillator, resulting in the W-representation orthonormal basis functions. Again, these will typically be employed in the x-representation. We have generated new continuous, complete biorthogonal and orthogonal bases using the x-representation of the various $\stackrel{^}{W},{\stackrel{^}{p}}_{W}$ operators. This is despite the fact that the relevant operators are not self adjoint in the x-representation. We also have used the fact that the W-representation harmonic oscillator Hamiltonian can be factored into the corresponding raising and lowering operators to generate new coherent states. Additionally, we also obtain new generalized Wigner distributions based on the W-HO ground states. We note that the eigenstates of ${\stackrel{^}{p}}_{W}$ (in the x-represen- tation) generalize the Fourier transform, so that we have new tools (including new windowed, non-linear transforms) to carry out signal processing of non-li- near, non-stationary time-frequency signals (e.g., chirps) that are not amenable to the standard Fourier transform. The eigenstates of $\stackrel{^}{W}+{\stackrel{^}{p}}_{W}$ (Equation (14)) are clearly linear chirps in the variable W (and therefore highly non-linear chirps when expressed as functions of x).

It is perhaps useful to illustrate the robustness of one realization of the MUB in terms of a sparseness of representation condition. The prime example against which we compare is the continuous MUB arising from the operator set $\left\{\stackrel{^}{x},{\stackrel{^}{p}}_{x},\stackrel{^}{x}+{\stackrel{^}{p}}_{x}\right\}$ . In this case, we deal with the Fourier transform and recognize that the Gaussian, $\mathrm{exp}\left[-{x}^{2}/2\right]$ , is invariant under it. Other functions that are narrower than the Gaussian (e.g., $\mathrm{exp}\left[-{x}^{6}/2\right]$ , corresponding (roughly) to the SUSY definition $W\left(x\right)={x}^{5}$ or precisely to the W-harmonic oscillator choice of $W\left(x\right)={x}^{3}$ ) in the x-representation have much slower decay than the Gaussian after Fourier transforming to the k-representation. This means that their Fourier k-domain representation is not sparse. Thus, the Fourier transform is the optimum basis for representing the Gaussian; it gives the sparsest representation possible for that function but not for others. In like manner, the W-harmonic oscillator choice of $W={x}^{3}$ leads to $\mathrm{exp}\left[-{x}^{6}/2\right]$ , which is invariant under

the generalized Fourier transform kernel ${\varphi}_{{p}_{W}}=\frac{{\text{e}}^{-i{k}^{3}{x}^{3}}}{\sqrt{2\text{\pi}}}$ . Thus, as expected, we

have obtained the optimum (sparsest) basis for describing a system having the ground state $\mathrm{exp}\left[-{x}^{6}/2\right]$ . In addition, we also expect this generalized Fourier transform will be well suited for describing other states that are characterized by the same “coordinate” W(x) [31] .

9. Conclusion

Canonically conjugate transformations lead automatically to quasi-Hermitian operators, and in the form of Equation (8), the new momentum operator is automatically self-adjoint in the new W-representation. Thus, no symmetrization is required to obtain the valid operator observables, $\stackrel{^}{W},{\stackrel{^}{p}}_{W}$ and $\stackrel{^}{W}+{\stackrel{^}{p}}_{W}$ . The Jacobian of the canonical transformation automatically supplies the required measure for self-adjointness. The uniqueness of the self-adjoint $\stackrel{^}{W},{\stackrel{^}{p}}_{W}$ operators follows from the facts that: a) Dirac quantization of the Poisson bracket gives the same result independent of how the classical momentum is defined and b) the form, Equation (8), is “universal” independent of how the classical momentum is defined since all result from a similarity transformation of Equation (8). Finally, canonical quantization is justified because the canonical transformation is constructed to ensure a Cartesian metric for the new position. Additionally, the spectrum of ${\stackrel{^}{p}}_{W}$ is structured identically to that of $\stackrel{^}{W}$ . We call such variables “Cartesian-like”. Finally, canonical transformations lead to a rich structure of new, complete bases, generalized harmonic oscillators and Fourier kernels.

Acknowledgements

The author D. J. K. also gratefully acknowledges partial support of this research from The Fritz Haber Research Center for Molecular Dynamics of the Hebrew University of Jerusalem from 6/16 through 7/16, under the auspices of Professor R. Baer. Extensive discussions about Mutually Unbiased Bases with H. S. Eisenberg are gratefully acknowledged. Several helpful comments on the manuscript by J. Klauder are also gratefully acknowledged.

Fund

This research was supported in part under R. A. Welch Foundation Grant E- 0608.

Conflicts of Interest

The authors declare no conflicts of interest.

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