Use of negative frequency for the sake of simplifying mathematics?

Question

How can we use the idea of negative frequency for the sake of simplifying mathematics if negative frequency does not exist (to my knowledge) in nature ? For example, when plotting the spectra of a Fourier series.

Answer 1

negative frequency does not exist

Depends on how you define frequency. If defining such a thing as negative frequency makes the math easier (it does), why not do it? It's probably less objectionable than defining an imaginary anything, and we do that all the time.

Answer 2

As other answers, if it makes the mathematics smoother or more tractable than equivalent mathematics without the use of negative frequency, why wouldn't you embrace the technique?

However, this is a physics site, so a more satisfying answer is going to give a meaning for negative frequency. Already you have the beginnings of a good answer in user safesphere's comment:

Oscillation is equivalent to rotation where frequency is equivalent to rpm (revolutions per minute). A negative frequency is simply equivalent to rotation in tbe opposite direction.

and his/ her linked DSP SE answer here that fleshes this comment out.

An application of this idea in physics is the diagonalization of the Maxwell curl equations through the use of Riemann-Silberstein vector $\vec{F} = \vec{E} + i\,c\,\vec{B}$, which I discuss in more detail in my answer here. Both Maxwell curl equations are replaced by one:

$$i\, \partial_t \vec{F} = c\,\nabla \times \vec{F}$$

and recover electric and magnetic fields through the real and imaginary parts. The positive frequency parts of this solution represent the left-handed circularly polarized field, the negative frequency parts are the right-handed circularly polarized field.

In the equivalent notation of the exterior calculus, one can build self and anti-self dual parts of the electromagnetic field $\tilde{F} = F + i\,\star F$, but you may not have come across this yet. Its positive and negative frequency parts have the same interpretation in terms of oppositely handed circularly polarized parts of the field.

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However, note that, as discussed in the other answer, the modern use of the Riemann-Silberstein / Self+Anti-Self Dual notation is to use two separate Riemann Silberstein vectors $\vec{F}_\pm = \vec{E} \pm i\,c\,\vec{B}$ fulfilling the separate equations $$i\, \partial_t \vec{F} = \pm c\,\nabla \times \vec{F}_\pm$$ and then to keep only the positive frequency parts. In this usage, the left and right hand circularly polarized field components are separated and given by $\vec{F}_\pm$, respectively.

Answer 3

The only time I’ve seen negative frequency used is in Quantum field theory where it is proportional to the energy of a particle (i.e. $E=\hbar\omega$) In QFT, Feynman interpreted the negative frequency/energy results of the Klein-Gordon equation (used to find the field for a relativistic particle) as corresponding to an antiparticle with positive energy and frequency moving backwards in time rather than a normal particle with negative energy and frequency moving forward in time. This makes computation easier as it allows for the particle and antiparticle parts of the field to be incorporated into one field. I have never seen it used elsewhere, although it may have. I hope this answers your question