In 1807, Joseph Fourier announced his mathematical equivalence of finite functions with an INFINITE series of simple harmonic functions; this meant any complex tone could be summed (analyzed) into an infinite series of frequencies (base plus overtones). De Broglie's suggestion of a moving point particle being mathematically equivalent to an infinite series of plane wave (2D harmonics).
Schrodinger generalized this analysis of the finite atom (spatially & temporally limited) into its mathematical equivalent: 3D spherical harmonics.
So, what's the big deal? Why do mathematicians reify their representations?
Particles are NOT waves.
I'm not a physicist. However I dare to expose the following: It is true that a particle is not a wave, but perhaps its movement can describe a path that can be described as a wave. In such cases physics needs mathematical tools to model the observed or propose some theory. This enriches the knowledge of both physics and mathematics.
This is a very legitimized question. However, reification is performed not by mathematicians but basically by those physicists who consider an incomplete or even inconsistent "physical model" to represent an almost 1:1-correspondence of "physical reality", without being aware how and to to what exent "physical reality" can be recognized by intelligent human beings at all. In contemporary physics textbooks and publications physical facts are often mixed up with physical phantasies which result from physical hypotheses or mathematical models or both. Although the language of mathematical theories is useful or even necessary to formulate physical facts unambiguously, a pinpoint accuracy in physical statements is often missed which causes lots of nonsense over decades of science. Prime example in traditional QM: ... the probability of "finding" an electron...
For stationary quantum states (my understanding is that) wave-mechanics and Fourier techniques are very similar (e.g. resonances and standing waves). For non-stationary states, when things are no longer isolated and start interacting with each other (i.e. in the "real world") they are not so similar. I would also argue that in the latter case, our simple atomistic view of what is going on (based on ancient Greek philosophy and adapted by Bohr et al) is not very useful and that we should get used to the idea of complexity at all scales, not just at the scales that we can perceive directly with our biological senses.
In the classical theory of EM the field is ubiquitous in space. In self-field theory the photon moves along a path and is a quantized form of field. So this is not like the Fourier transform which I take to be between a continuous field or particle and a discrete field or particle. Of course there is the discrete Fourier transform.
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