The announcement the world has been waiting for can now be made: the paper “Morlet wavelets in quantum mechanics” has been updated. The latest version has a much clearer explanation of the point, a number of errors corrected, and some stylistic infelicities eliminated.
This version has been uploaded to the physics archive.
Wavelets offer significant advantages for the analysis of problems in quantum mechanics. Because wavelets are localized in both time and frequency they avoid certain subtle but potentially fatal conceptual errors that can result from the use of plane wave or delta function decomposition. Morlet wavelets are particularly well-suited for this work: as Gaussians, they have a simple analytic form and they work well with Feynman path integrals. To take full advantage of Morlet wavelets we need an explicit form for the inverse Morlet transform and a manifestly covariant form for the four-dimensional Morlet wavelet. We supply both here.
Why quantum time?
A few years ago I was looking for an interpretation/formalism for quantum mechanics which would be manifestly symmetric between time and space. The first question I had was:
Is time already quantized?
Is time treated using the same quantum rules as space is? can quantum mechanics be written in a way which is manifestly covariant?
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Wavelets are like musical notes: they are wave forms limited in both time and frequency. What makes them particularly useful is that any reasonable wave function may be written as a sum over them.
Usually we think of music in terms of pure tones, in terms of its Fourier components. But pure tones can be a bit too pure. For one thing, if a tone is to be completely pure it has to last forever, not a characteristic associated with practical questions. Wavelets are impure tones, and therefore a better match to the real world.
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