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- Thread starter Guillemet
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Try this:

http://arxiv.org/abs/0907.0909

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jtbell

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Can you provide a reference about this? A Google search for "Hartley analysis" yields only pages about the British writer L. P. Hartley, as far as I can tell (clicking through five pages of search results). Nothing to do with mathematics.Had Hartley analysis been invented first

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Wikipedia: Hartley transformCan you provide a reference about this?

Weird, I get three math-related matches on the first page: http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=272142 http://dx.doi.org/10.1109/CIC.1989.130514 http://www-hsc.usc.edu/~jadvar/CinC-Cepstrum.pdf [Broken]A Google search for "Hartley analysis" yields only pages about the British writer L. P. Hartley, as far as I can tell (clicking through five pages of search results). Nothing to do with mathematics.

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See this thread about writing the Schrodinger equation as two real differential equations.

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Bottom line is, you need a system of encoding 'amplitude' and 'phase'. Both are important. Phase is important because it enables interference to take place.

The Hartley transform does not give you the phase information, so no, I don't see any (obvious) way that quantum mechanics could be adapted to use Hartley analysis instead, though it might be possible through some non-obvious method.

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As I wrote several times in this forum, a lot more can be done in quantum theory using just real numbers (not pairs of real numbers) than people tend to think. For example, for each solution of the Klein-Gordon equation there is a physically equivalent solution (coinciding with the original solution up to a gauge transform) with a real matter field (E. Schroedinger, Nature (London) 169, 538 (1952) ). Furthermore, in a general case, the Dirac equation can be rewritten as an equation for just one real function (http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf , published in Journ. Math. Phys.).

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"each sequence of measurement outcomes obtained in a given experiment is represented

by a pair of real numbers"

I have an obvious intuition but I am searching for a precise (mathematical) definition of what the word "represented" means. Something like "a sequence of measurement outcomes is said to be represented by a mathematical object when ..."

Any thoughts would be very welcome.

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Hartley and Fourier both give phase spectra.Bottom line is, you need a system of encoding 'amplitude' and 'phase'. Both are important. Phase is important because it enables interference to take place.

The Hartley transform does not give you the phase information, so no, I don't see any (obvious) way that quantum mechanics could be adapted to use Hartley analysis instead, though it might be possible through some non-obvious method.

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Nope, Hartley doesn't. Each fourier coefficient encodes the amplitude and phase of the specified frequency, whereas to determine phases in the Hartley transform you need to look at all the coefficients together (using cepstral methods for example). In other words, the transform does not directly give you the phase information.Guillemet said:Hartley and Fourier both give phase spectra.

As an example, you can consider a gabor function (sinusoid multiplied by gaussian). The fourier transform would be a peak at the frequency of the sinusoid which falls off as a gaussian curve. The phase of the peak in the frequency domain tells you the offset of the function in the time domain. The magnitude of the peak tells you the amplitude of the signal. The Hartley transform of this function has a similar structure to the real part of the fourier transform, BUT the height of the peak is no longer independent of phase.

As I said, there might be some convoluted method to get Hartley analysis to work as a substitute for complex numbers. There is no obvious or straightforward way, though.