Sneaking √-1 in
When you see a paper that disingenuously sneaks the imaginary unit into an equation — drink!
I’ve mentioned in a couple of posts now (here, here) that a lot of work “deriving” quantum mechanics from other classical physics is a game of “spot where they're sneaking i in”. The imaginary unit i = √−1 brings in complex numbers and turns real functions into waves — the foundational mathematics of quantum mechanics.
Wavefunctions reside in a complex function space, Hilbert space, and their absolute square determines the probability of observations — the Born rule. These are unnecessary in classical physics (classical mechanics, thermodynamics, special & general relativity). Therefore, finding i magically appearing in, say, a thermodynamic model would be a link to quantum physics. However, the complex numbers in quantum physics are also the results of postulates. This leaves some people asking “why?”
In addition, there are three major ways General Relativity (GR) and Quantum Mechanics / Quantum Field Theory1 (QM/QFT) are different from each other. Those complex numbers in the postulates are one. Another is the fact that time is a parameter in QM / QFT but a coordinate in GR (spacetime) — the idea of clocks are very different in the two theories. The third is more technical, but naive quantum gravity should have gravitons as spin-2 excitations of the “gravity field” but those would not be renormalizable in QFT (i.e. they would lead to infinities that can't be subtracted at all orders of perturbation theory).
There are a lot of papers out there trying to “explain” the postulates — my two posts on the “elsewhere” interpretation of quantum mechanics are working at trying to explain the postulates in terms of special relativity. I even use one of the tricks for sneaking i in (Wick rotation — see below), but I admit up front that it’s a kludge. It’s for intuition, not rigor. However other papers can be a bit more ... sales-oriented.
So I’d like to give a short list of the “tricks” and a little bit of explanation of what is wrong with them. This list might grow during my literature review ...
Wick Rotation
This is basically time t → i t. That is to say: just put i in. No sneaking necessary. This transforms 3+1 D Minkowski space into 4 D Euclidean space (which is very much like 3 D Euclidean space we are familiar with). It turns diffusion equations into quantum wave equations. (“The Schrodinger equations is a diffusion equation in imaginary time.”) Wipe your hands and you’re done.
But you can’t just do that. The structure of Minkowski space is very different from Euclidean space. The asymptotic boundary of Minkowski space has the symmetries of the BMS group. The asymptotic boundary of Euclidean space ... does not.
There are specific problems in quantum field theory where you can’t just do that either. It’s called Wick rotation because it rotates the contour integrals involved in Feynman diagrams — and doing that may move “poles” (singularities in the complex integral) inside or outside of the contour. That’s a problem because those poles are generally what contribute to the contour integral — no poles can mean the integral is zero.
Heuristically, since we don’t know the pole structure of the fundamental theory of the universe, we don’t really know if we can just Wick rotate.
This particular trick is widespread and not specific to any kind of approach.
Make up something else that behaves just like i
Discovering this particular rabbit hole was the reason for writing this post. This trick has been championed by David Hestenes and B. J. Hiley (the latter worked a lot with David Bohm, another prominent questioner of QM postulates). The idea is that we replace complex numbers (and other stuff) with ... Clifford Algebras. This creates a set of mathematical objects that work just like Dirac matrices, Pauli matrices, and i. Basically — a rose by any other name.
They even call the thing that behaves like i ... i. “But it’s not i, you see. It’s a pseudoscalar in an exterior algebra.”
I personally don’t think this is any different from just using i. I imagine Hestenes and Hiley might beg to differ, but it’s essentially just a notation change. And sometimes not even a notation change. This is not to say this is entirely contentless. Rephrasing the mathematics in terms of Clifford Algebras could lead to new insights. Hestenes’ idea that the appearance of iℏ in QM is basically spin despite the Schrodinger equation not having spin is neat. However, all we’ve done is change the QM postulates from “Hilbert space” to “Clifford Algebra”.
Take it out of the determinant in GR
This was a funny one. The Einstein-Hilbert action (density) is often written R √−g. Well, can’t we just factor the √−1 out? Well, yes but that just makes √g complex as if spacetime was a Kahler manifold. You sneak in i by saying 1 = √−(−1) = −√−1 * √−1 = −i * √−1
Note: you can just write √−g = √|det g| which ruins this particular bit of fun.
BTW, QFT is basically just the consistent combination of QM + Special Relativity