Spiros leaked a trivial observation from a talk given by John Preskill: in quantum mechanics, the position and the momentum may be measured totally accurately and simultaneously, but only modulo \(\Delta X\) and \(h / \Delta X\), respectively.

The proof is a simple application of the CBH-like formulae such as\[

\exp(A) \exp(B) = \exp(B) \exp(A) \exp([A,B])

\] whenever \([A,B]\) is a \(c\)-number. And indeed, it is the case if \(A,B\) are linear in \(x,p\), respectively. The precise position \(X\) modulo \(\Delta X\) may be expressed by the (unitary) operator\[

\exp(A)=\exp(2\pi i X / \Delta X)

\] while the analogous momentum \(P\) modulo the inverse spacing \(\Delta P = 2\pi \hbar / \Delta X\) is expressed as\[

\exp(B) = \exp(i P \Delta X/\hbar)

\] Nice. So apply the CBH formula to get\[

\exp(A)\exp(B) = \exp(B)\exp(A) \exp([A,B])

\] to see that the commutator appearing as the last exponent is \([A,B]=-2\pi i\) and, as you know, its exponential is therefore one. If you choose units with \(\hbar=1\) and a dimensionless \(X,P\), you may democratically pick the spacing \(\Delta X =\Delta P = \sqrt{2 \pi}\), as announced in the original tweet.

So when the spacings \(\Delta X,\Delta P\) are such that the corresponding grid is as dense as "one phase space cell per grid site", then the corresponding unitary operators of \(X,P\) that measure \(X,P\) precisely but modulo the grid conspire in such a way that the two operators commute with each other. It means that they can be simultaneously diagonalized.

However, note that the two operators aren't Hermitian, something that you normally expect when you list your eigenvalues and orthogonal eigenstates. On the other hand, both operators are unitary i.e. they are "normal", too. So they could have a continuous basis of orthogonal eigenstates. What are they?

Let us look at them in the position basis. The eigenstates of \(A\), morally the "exponential of \(X\)", look like a sum of multiples of delta-functions that are only supported at values of \(X\) that differ by an integer multiple of \(\Delta X\) from each other. On the other hand, the eigenvalues of \(B\) are those that are periodic with the period \(\Delta X\) up to the multiplication by a phase.

So the simultaneous eigenstates are simply the grids with equally spaced delta-functions decorated by a linearly changing phase – you may choose where the \(\delta\)-functions sit in the interval \((0,\Delta X)\); and you may choose the phase by which the \(\delta\)-functions rotate from one to the neighbor. A subtlety is that if you write the precise wave functions as a function of these two phases (\(X\) and \(P\) modulo the spacing, stretched to the intervals \((0,2\pi)\), let us call the phases \(\alpha,\beta\)), then the phase of the wave function depends not only on \(\alpha\) modulo \(2\pi\) and \(\beta\) modulo \(2\pi\) but also on the (integer part) of the ratios \(\alpha,\beta\) modulo \(2\pi\).

At any rate, it's cute that in quantum mechanics, the infinite two-dimensional fuzzy \(XP\)-plane is equivalent to a two-dimensional non-fuzzy continuous basis of simultaneous eigenstates that only live in one phase space cell. In some sense, quantum mechanics says that you may always assume that \(X,P\) are only allowed to live in the single unit square of the phase space! When I wrote this blog post – directly, without preparations – it looked completely trivial to me. I feel that I must have used this result many times, e.g. when I considered the continuous limits of the "fuzzy membranes" in Matrix theory.

But I seem unable to find a particular place where this point of Preskill's, about the limited ability to measure \(X,P\) at the same moment, is articulated (not even in our textbook We Grow Linear Algebra that dedicates a few pages to the CBH formulae). I feel that there could have been places of my research where I was unaware of this rudimentary fact – when I was trying to describe some relationships between the grids and the noncommutative geometry. Was it in my considerations involving the Dirac quantization rule for monopoles?

The allowed electric and magnetic charges live in a grid, as explained by Dirac. The observation in this blog post probably allow you to express a general two-dimensional wave function \(\psi(x_E,x_M)\) as some twisted product of a general function on the phase space cell (the elementary square, see above) and a function that is only defined on the two-dimensional grid of allowed electric and magnetic charges. Note that the two spaces are "equally large" and this splitting into a tensor product is therefore analogous to the splitting of QM with coordinates \(x,y\) to two copies of one-dimensional QM; but the grid-cell splitting is "asymmetric". I think that this statement must have been relevant for something I did at some point but I didn't quite appreciate it. Some research direction could have ended with a "dead end" answer although it shouldn't have.

Too bad that I can't Google search in all ideas that I have ever thought about LOL. Well, maybe I could if I were recording everything that goes through my consciousness. It could actually be a good idea to capture one's thoughts and write them down into some mind-reading diary. The relevant data could be just a few kilobytes per day, when textualized and digitized, but it could make a difference.

P.S.: I must emphasize that this measurement of \(X,P\) modulo the tiny spacing is totally incompatible with any classical limit. Classical physics only works when \(\Delta X \cdot \Delta P \gg \hbar/2\), when the product is much greater than the minimum size dictated by quantum mechanics. The precise measurement of \(X,P\) modulo something has nothing to do with the "approximate measurement" of \(X,P\) without any "modulo". In fact, as indicated in the tensor decomposition discussion above, these two parts of the information may be considered exactly disjoint.

## No comments:

## Post a Comment