Timeline for Compact or non-compact boson from bosonization?
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| Oct 8, 2021 at 22:35 | comment | added | Zack | Perhaps it's just terminological, but by "Gaussian" I mean that the path integral and correlation functions can be evaluated by a simple Gaussian integral. This is not the case for the compact boson, where a Gaussian integral amounts to neglecting the compactness of $\phi$ and "misses" vortices. Perhaps the confusion is because Coleman's paper only treats the zero charge sector of the Thirring model, which means no winding of the boson. But I'm not totally positive, I would think even with no winding there could still be vortices so long as the total charge is zero. | |
| Oct 8, 2021 at 19:16 | comment | added | Connor Behan | Also, compact and non-compact free bosons are both Gaussian if you just look at the action. The radius is important for determining which local operators are admissible in the theory. But if someone just hands you an operator built from $\phi$, you can read off its scaling dimension without knowing the radius. | |
| Oct 8, 2021 at 19:12 | comment | added | Connor Behan | I haven't read Coleman's steps in awhile but I thought SG described a compact boson. In $\cos(\beta \phi)$, you can shift $\phi$ by $\frac{2\pi}{\beta}$ meaning the radius should be $\frac{1}{\beta}$. If this is true, (1.9) agrees with what the current-current term is supposed to do. You get an infinite radius at $g = \infty$ but a finite radius at $g = 0$. | |
| Oct 8, 2021 at 17:31 | comment | added | Zack | So, should I regard SG as a theory of a compact or non-compact boson, and does the RG analysis change between the two cases? | |
| Oct 8, 2021 at 17:31 | comment | added | Zack | Thanks for your comments -- I'm still learning CFT, so I don't yet understand the meaning of the partition function you wrote. But your last point does raise another confusion of mine: although the Sine-Gordon action seems to respect the periodicity of $\phi$, my understanding was that the SG theory was a theory of a non-compact boson -- see, ex, the mode expansion in Coleman's Eq. 2.1. This is also important in the traditional discussion of the KT transition in the SG theory, where the scaling dimension of $\cos \phi$ is evaluated using the Gaussian action. | |
| Oct 8, 2021 at 17:07 | history | answered | Connor Behan | CC BY-SA 4.0 |