Divine Neutrality

In the October 2007 issue of Physics Today beginning on page 78 there is an article on Wavelets by Ivan Selesnick, son of my friend Stephen Selesnick. He must be a very proud father.

I read the article. I wish I understood what I read. What does sparse mean?

Here is my picture of the story: A fourier transform tells us to what extant the signal is periodic; it displays the frequencies embedded in the signal. Signal polution ‘has no frequency’ - white noise - yielding small amplitude at all frequencies. Thus it can, in the transform, be distinguished from what we are looking for - periodicity. A spectrum shows us what frequencies are present.

But signals may have distinguishing features other than frequency for which we want to search. The idea is to create a transform to display the distinguishing feature we wish to favor and to blunt what we wish to devalue. Wavelets are the transforms that do this: enhance what we favor, blunt what we devalue.

Is this a valid overview? Have I got it right?

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A commonplace computational practice in quantum mechanics generates the most profound conceptual challenge to the theory. The challenge is called the measurement problem. Here are some quotes summarizing the problem.

“The quantum measurement parodox.. stated succinctly… In quantum mechanics all possibilities… are left open whereas in … experience a definite outcome always (occurs).”
A. J. Leggett in Foundations of Physics. 18, 939 (1988)

“How is the measuring instrument proded into making up its mind which value it has observed?”
Bryce S. Dewitt, Physics Today 23, 30 (1970)

“Some explanation must be provided for the fact that the Hilbert—space vector… collapses onto a certain eigenvector during a measurement process…”
J. Bub, Nuovo Cimento v. 57, Nr.2, 503 (1968)

The probability amplitudes evolve deterministically until a measurement is made: the measurement stops the evolution. What is the essential element that changes the evolution of the system from being in a state
|S> = (superposition sum of many states |n>),
into being in a state, say, |n=3>, one from among the many in the superposition?
Marvin Chester, never published

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