# How fundamental are complex numbers in quantum theory?

The initial development of QM inherited the use of complex numbers from Fourier analysis. Had Hartley analysis been invented first, is it possible that QM might have been formulated in terms of real-valued quantities instead, or are complex numbers in some sense natural or necessary when describing quantum phenomena?

The initial development of QM inherited the use of complex numbers from Fourier analysis. Had Hartley analysis been invented first, is it possible that QM might have been formulated in terms of real-valued quantities instead, or are complex numbers in some sense natural or necessary when describing quantum phenomena?
Try this:

http://arxiv.org/abs/0907.0909

Interesting, but it seems to assume a priori the use of 2-dimensional quantities of some sort, though not complex numbers specifically. Has anyone devised a similar argument that rules out the reals?

jtbell
Mentor
Had Hartley analysis been invented first
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.

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

I see nothing fundamental about complex numbers; quantum mechanics can be expressed just as well in the language of 2-d vectors (where vector addition corresponds to complex addition, and complex multiplication corresponds to adding up angles and multiplying lengths).

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.

The initial development of QM inherited the use of complex numbers from Fourier analysis. Had Hartley analysis been invented first, is it possible that QM might have been formulated in terms of real-valued quantities instead, or are complex numbers in some sense natural or necessary when describing quantum phenomena?
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.).

I found very interesting and illuminating the paper posted by "Friend". I have one doubt. What are your interpretations of the following, and perhaps more important, statement of that paper:

"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.

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.
Hartley and Fourier both give phase spectra.

Guillemet said:
Hartley and Fourier both give phase spectra.
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.

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.