TransWikia.com

Integral eigenvalues in compact rank-2 symmetric $U(1)$ gauge theory

Physics Asked by Waterfall on January 11, 2021

I am reading a paper related to rank-2 symmetric $U(1)$ gauge theory:
Fracton topological order from the Higgs and partial-confinement mechanisms of rank-two gauge theory (or arXiv:1802.10108).

My question concerns a skipped calculation in Sec. II. There, the authors write that $A_{munu}$ being compact (mod $2pi$) and the canonical commutator $[A_{munu},E_{munu}]=-i$ implies that the eigenvalues of $E_{munu}$ are integers. I don’t follow the authors’ reasoning here and I don’t see why the eigenvalues should be integers. Would someone please enlighten me?

Note: cross-posted on Physics Overflow

One Answer

The range of $A_{munu}$ and its commutation relation with $E_{munu}$ are essentially the same as the those of the azimuthal angle $phi$ and the angular momentum component $L_{z}$. $phi$ covers the compact range from $0$ to $2pi$, then wraps around on itself, just like $A_{munu}$. The corresponding conjugate momentum operator is $L_{z}=frac{hbar}{i}frac{partial}{partialphi}$, and the commutator of the two is $[phi,L_{z}]=ihbar$. This is the same as the commutation relation $[A_{munu},E_{munu}]$, apart from an overall factor of $-hbar$. It is well known that the operator $L_{z}$ has eigenvalues $pm mhbar$, for integer values of $m$; this is required for the system to have a wave function that is single valued under the coordinate redefinition $phirightarrowphi+2pi$. For exactly the same reason, in order that the (physically meaningless) affine shift $A_{munu}rightarrow A_{munu}+2pi$ not change the wave function, $E_{munu}$ must be quantized the same way as $L_{z}$ was. Accounting for the difference of $-hbar$, this means that the eigenvalues of $E_{munu}$ are integers.

Correct answer by Buzz on January 11, 2021

Add your own answers!

Ask a Question

Get help from others!

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP