Periodic functions in 1-dimension

Assume for a complex function of a dimensionless real variable , the boundary condition

(1.1)

This is a transformation from the real line to the complex plane.

(1.2)

The Fourier series is

(1.3)

(1.4)

Orthogonality of the basis set of functions

(1.5)

Mother pre-wavelet of Fourier series is obviously

(1.6)

The basis is generated from it by integer dilation of the phase x.

Parseval identity of signal conservation of energy under basis transformations

(1.7)

The squeezed-shifted wavelet: Zoom in and zoom out!

We now allow non-periodic functions over the whole real line without the periodic boundary condition (1.1). Replace the sinusoidal wave mother pre-wavelet by a localized mother wavelet packet or window function. We must shift it to cover whole real line. One obvious way is to use integer shifts

(1.8)

We also need to dilate the mother wavelet. Communication engineers like to partition the frequencies into bands of consecutive “octaves” using integer powers of 2. The convention is

(1.9)

This is a combined operation of squeezing by a factor of with a translation not of , but of rescaled . That is

(1.10)

Therefore, squeezes the domain of support (where ) into a smaller region as well as decreasing the shift. This is a zoom-out transformation like decreasing the magnification of a microscope showing less detail, less resolution of the image. On the contrary, does the opposite zoom-in increasing the resolution of the image or increasing the magnification of the microscope. Note that ordinary Fourier analysis in physics does not have this capability that is particularly suitable for nonstationary statistical processes with fast changes especially for open non-equilibrium systems.

A wavelet series can be of the form

(1.11)

provided that we have a complete orthonormal basis[3] or “frame of reference”

(1.12)

In terms of quantum measurement theory, the squeezed-shifted wavelet base functions are formally “filters” or eigenfunctions of some “wavelet observable” if there is also a directly measurable eigenvalue by some constructable physical detector. This formal filter structure may or may not translate to something objectively real depending on more specific physical information.

Just like in Fourier analysis we can generalize a discrete integer wavelet series to a continuous integral wavelet transform where squeeze-shift integers j,k are replaced by real parameters a, b

(1.13)

The notation is that is the integral wavelet transform, whilst is the orthodox integral Fourier transform. We can, of course, still keep as a special case

(1.14)

consistent with (1.10). Note that the wavelet literature calls the binary dilation, and the dyadic position when the octave algorithm (1.14) is used.

“while the two components of Fourier analysis… the Fourier series and the Fourier transform are basically unrelated; the two corresponding components of wavelet analysis, namely the wavelet series (1.11) and the integral wavelet transform (1.13) have an intimate relationship as described in (1.14)” p. 6 Chui

(1.15)

Think of as an analog signal. This would be some kind of stochastic time series coming out of a detector. You can generalize this to a classical gauge force field configuration with spatial frequencies or wave numbers as well as temporal frequencies. We want to do better than Fourier analysis, which is only really best, suited for stationary uniform stochastic processes. The familiar Fourier transform is the spectrum of the temporal signal. Parseval’s theorem (1.7) is a special form of what is called a unitary transformation in quantum physics in which inner products are invariant under change of frame of reference (basis). The integral of the squared modulus of the spectrum is a measure of the total signal power.

“to extract the spectral information from the analog signal … takes an infinite amount of time, using both past and future information just to evaluate the spectrum at a single frequency .. the [Fourier] formula does not even reflect frequencies that change with time. What is really needed is … to be able to determine the time intervals what yield the spectral information on any desirable range of frequencies (or frequency band) … for high-frequency spectral information, the time interval should be relatively small to give better accuracy, and for low-frequency spectral information, the time interval should be relatively wide to give complete information… it is important to have a flexible time-frequency window that automatically narrows at high “center-frequency” and widens at low-center frequency … the integral wavelet transform [(1.13)] relative to some basic wavelet … has this … zoom-in and zoom-out capability.” Pp. 6-7 Chui

Windows on the world above and below.

Both basic “Mother” wavelet window function and its classical Fourier transform must decay to zero sufficiently fast in order to define both a center and a width in both time and frequency. The product of the complementary widths of course still obey the Heisenberg uncertainty noise of a quantized area in phase space. However, we will see later how W. Zurek shows substructure inside this quantized area under special conditions. Therefore, we can, it appears, in principle, pierce the veil and see inside h per conjugate pair degree of freedom, where h is Planck’s quantum of action.[4] Extending Zurek’s idea to a Bose-Einstein condensate shows why the latter is protected against decoherence of the quantum phasing. The argument is essentially, a single particle state whose Wigner phase space density extends over a large phase space “area” shows “spotty” sub-quantum pilot qubit wave structure on the scale . In which is the sensitivity of the large single-particle nonlocal “Schrodinger Cat” state to environmental quantum decoherence. This is why it is impossible to see single particles in many places at once, so to speak, on a large scale. The quantum phase coherence is rapidly destroyed the larger . My new idea not found in Zurek’s Nature paper is that Bose-Einstein condensates work exactly the dual opposite way. Now the effective phase space volume per particle inside the superfluid is , where is the number of Bose-Einstein condensed identical quanta of spin 0 or spin 1, real or virtual. Therefore, is large which means London rigidity. The macro-quantum order parameter

(1.16)

is robust, rigid, insensitive to decoherence. Zurek’s argument is turned topsy-turvy upside-down.

Let be a window function. Its center is defined as

(1.17)

The window function’s width is defined as twice where

(1.18)

All windows functions must obey

(1.19)

Let Mother wavelet and its Fourier transform both be kosher windows functions[5] in their time and frequency domains respectively with centers and widths . Heisenberg’s uncertainty principle, even in the classical domain, sans of course, is that

(1.20)