Semiclassical Quantization

Abstract

We want to investigate the semiclassical or one-loop approximation of our Chern–Simons model:

$$\displaystyle{ S = S_{0} + S_{\text{CS}}\;, }$$

(29.1)

where

$$\displaystyle\begin{array}{rcl} & & S_{0}[\eta,A] =\int _{ 0}^{T}dt\left [\frac{1} {2}\eta ^{a}\omega _{ ab}\dot{\eta }^{b} - H(\eta ) -\sum _{ i}A_{i}(t)J_{i}{\bigl (\eta (t)\bigr )}\right ] \\ & & S_{\text{CS}}[A] =\int _{ 0}^{T}dt\,\sum _{ i}k_{i}A_{i}(t) {}\end{array}$$

(29.2)

and k _i is fixed. We shall see that consistency requires k _i to assume (half-) integer values only. In the following, all fields are defined on [0, T] and are assumed to be periodic.