Saturday, February 26, 2011

τ keeps on slipping...

Apologies: this post is somewhat more specialized than my normal fare, and probably will be boring as hell without a some mathematical knowledge.

Electrons have a negative charge. When you think about it, this really doesn't make much sense; after all, electrons are the charge carriers for electric charge, and so we would hope to assign their charge a positive value. That electrons carry a negative charge isn't a fundamental statement about reality, though, but rather an unfortunate consequence of an arbitrary decision made early on when electricity was being studied, but when electrons were still undiscovered.

Similar cases of unfortunate arbitrary conventions can be seen in other areas of mathematics and science. Recently, for instance, Michael Hartl has argued that $\pi$ is not the right constant to use in the equations governing such things as circles, frequencies and angles. Rather, Hartl argues that $\tau = 2\pi$ is a much more natural choice. Using this convention, the circumference $c$ of a circle is $c=\tau\ r$, eliminating the awkward factor of 2 in $c=2\pi r$. It may seem that we lose something when considering the area $A = \frac12 \tau\ r^2$ of a circle in this notation, but in fact this is much more natural for expressing as an integral, as those familiar with calculus will be happy to note.

Today, I'd like to show you somewhere else in physics where changing notation makes things much more natural. Concretely, I'd like to argue that using a different $\tau$ makes quite a lot of sense, when we using $\tau = it$ as a replacement for the time $t$ in equations. In fact, I take it as a lesson of quantum mechanics that we should consider time to lie along an imaginary axis and not along the real axis. This notational trick, known as Wick rotation, simplifies many physical equations, such as Schrödinger's equation. I find that \(\frac{d}{dt} \left\vert\psi\right\rangle = i \hat{H} \left\vert\psi\right\rangle\)  makes much more sense expressed in imaginary time:
\[\frac{d}{d\tau} \left\vert\psi\right\rangle = \hat{H} \left\vert\psi\right\rangle\] Adopting this convention also makes it manifestly clear why complex conjugation is intimately related to time reversal, since $\tau^* = -\tau$.

It is, in fact, quite rare for $t$ to appear in quantum mechanics without a factor of $i$ attached. Even when describing a classical object interacting with a quantum mechanical system, such as an oscillating field introducing a time-varying term to a system's Hamiltonian (that is, the operator which describes the energy of a system--- if that makes no sense, don't worry), we write something like \[\hat H(t) = \cos(\omega t)\ \hat\sigma_x + \sin(\omega t)\ \hat\sigma_y.\] But wait!, you say! There's no $it$ in that equation! As it turns out, there actually is, but we've hidden it by using trigonometric functions where an exponential function is more natural: \[ \hat{H}(\tau) = e^{-\omega\tau\hat\sigma_z/2}\hat\sigma_x e^{\omega\tau\hat\sigma_z/2} \] This form also has the advantage of making it manifest that the oscillation of the classical field can be thought of as a coordinate rotation of a time-independent field.

Other key results of quantum mechanics become much cleaner with the imaginary-time convention. For instance, this convention along with the natural units convention that $\hbar = 1$ makes Ehrenfest's theorem much less awkward to write: \[\frac{d}{d\tau}\left\langle \hat A\right\rangle = \left\langle \frac{d\hat A}{d\tau}\right\rangle + \left\langle[\hat H, \hat A]\right\rangle\]

At the end of the day, such notational choices as the sign of an electron's charge, the choice of circle constant, or the axis which we use to represent time are all arbitrary. We can do physics quite well even when a choice lacks something in mathematical beauty. My point, then, in exploring the fun of $\tau$ is to show that even though our choice of notation is an arbitrary choice made for the convenience of the humans that work with it, by making our notational choices carefully, we can coax out and make manifest deep truths.

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