My reply to Matt Visser tries to once again explain my suspicion that theoretical physics might use complex calculus not as proper as required.
Dear Matt,
You got me and perhaps may essay repeatedly wrong. I am not David Garfinkle, and I hope you will not go on getting me wrong concerning complex calculus. While complex calculus is an application of complex numbers, theorists like you tend to be not aware of a few trivial trifles:
You are guessing that what you called an algebraic field extends the real numbers in the sense it offers additional freedom. Actually, the description of a physical quantity in complex domain is subject to Hermitean symmetry which means, it doesn't convey more essential information as compared with the unilateral and real-valued original function; let's say of either elapsed or future time. Future processes cannot be measured, past processes cannot be prepared.
Certainly you preferred what I consider the risky habit of introducing a physical quantity within complex domain.
In other words, when writing exp(i theta) instead of cos(theta) you are omitting i sin(theta).
A simple objection against this habit is that it implies an arbitrary and therefore unwarranted and often ambiguous choice of the sign of rotation.
When someone like Schrödinger applied complex calculus on a real-valued function of elapsed time (extending from -oo to 0), he applied as do I Heaviside's trick in order to prepare it for the integration (from -oo to +oo) required by Fourier transformation. In a first step, zeros are attributed to the not yet existing future. The now zero-valued future part is then split into a positive and a negative mirror picture of the originally unilateral function. This creates multiple but necessary redundancies: an even and an odd component, both extending from -oo to +oo. The third step, Fourier transformation into complex domain leads to the mentioned symmetries of real part and imaginary part. An original function of time mutates into complex functions of positive and negative frequency (from -oo to +oo). A function of frequency in original domain gets the so called analytic signal, i.e. two complex symmetrical functions of past and future. So far nothing is wrong. We may benefit from complex calculus in either complex frequency or complex time domain. If no seemingly redundant component was omitted and not just the inverse Fourier transformation but also the inversion of preparing operations were properly performed, we safely arrive back in the original unilateral domain.
Schrödinger's original communications, in particular his 4th reveal the way he speculatively introduced the complex wave function as a trick to reduce the degree of equation from four to two.
Perhaps not just Dirac ignored that elapsed as well as future time are always positive too when he argued that frequency (and the Hamiltonian) must always be positive. Weyl confessed: So far there is no explanation in sight for the [mirror] symmetry of time in quantum physics. The envisioned symmetry of almost all fundamental particles with their antiparticles was not found.
There was something that puzzled me for a while when I compared cosine transformation in IR+ with Fourier transformation in IR and questioned the unavoidability of ih. Heisenberg's matrices seem to confirm the necessity of complex calculus in Schrödinger's picture. Meanwhile I understand that Heisenberg's square matrices also correspond to Hermitean symmetry in IR. A real-valued alternative corresponds to (triangular) half-matrices with elements only above or only below the diagonal which may reflect the border between past and future.
Accordingly I felt not just forced to criticize Einstein's imprecise wording "past, present, and future" but also to reintroduce Euclid's notion number as a measure, not a pebble.
Concerning ict, cf. the essay by Phipps.
Matt, I agree with your statement: "If a well-developed branch of pure mathematics turns out to have some use in the natural sciences, then the natural scientists will quickly appropriate that strain of pure mathematics and turn it into applied mathematics..." Yes, G. Cantor's well-developed cardinalities in excess of aleph_1 didn't turn out to have some use in the natural sciences.
Regards, Eckard Blumschein
P.S.: Michael Studencki, I apologize for hurting you. Read my essay(s), and you will find why I reinstate Euclid's notion of number based on the measure one, not on a pebble a la Hausdorff. So far, I did not yet read your bio and also not your essay. I consider Relativity and the belonging held for real spacetime in contrast to reasonable relativity not as harmless as you seems to describe it.
In Germany we have a proverb "sitting between the chairs". Maybe, the truth concerning Relativity/relativity sits likewise between academic pros and cons.