Geometric Quantization of Dirac spinor in QFT

I have been using resources such as, Geometric quantization, Baykara Uchicago, to get a deeper insight into geometric quantization. However, it seems to me that this theory is only valid for quantum mechanics instead of also applying to quantum field theory.

To me, the problem seems to be that geometric quantization attempts to quantize functions of the symplectic manifold, where as quantum fields are really sections of the hermitian line bundle. See for example, Andrej Bona, Geometric quantization of the Dirac spinor. The Dirac spinor is naturally a section. How would one address this problem?

My question is, does it make sense to quantize such a section of the polarized manifold, and how does one do that?

• I would not agree that geometric quantization "is only valid for QM instead of also applying to QFT". For example, the geometric quantization of 3d Chern-Simons theory (which is a classical field theory) is a quite famous example on the field theoretic side... Dec 21, 2021 at 12:34
• @G.Blaickner, I wasn't aware of that, thank you. Would you be able to link some papers, or sources from which I could get a deeper understanding? Dec 21, 2021 at 14:47
• See the answer below :-) Dec 21, 2021 at 16:41

Since you asked in the comments, let me provide some references to the geometric quantization of Chern-Simons theory. This does of course not answer your second qustion about how to deal with spinors in geometric quantization and I would be interested in an answer to this myself. However, it goes in the direction of applying geometric quantization in field theory, although the field theory in question might be rather exotic.

So, the field theory in question is 3d-Chern Simons theory, which is a topological gauge theory, defined as follows: Consider the following data:

1. A compact, orientable smooth manifold $$\mathcal{M}$$.
2. A principal $$G$$-bundle $$\pi:P\to\mathcal{M}$$ with Lie group $$G$$, whose Lie algebra $$\mathfrak{g}$$ is equipped with a non-degenerate $$\mathrm{Ad}$$-invariant symmetric bilinear form $$\langle\cdot,\cdot\rangle$$.

Now, lets take a connection 1-form $$A\in\Omega^{1}(P,\mathfrak{g})$$, i.e. a "gauge field" in physics terminology. Then, we define the "Chern-Simons form" by

$$\mathrm{CS}[A]:=\operatorname{tr}(A\wedge\mathrm{d}A)+\frac{2}{3}\mathrm{tr}(A\wedge [A\wedge A])\in\Omega^{3}(P)$$

where "$$\operatorname{tr}(\cdot\wedge\cdot)$$" denotes the induced wedge-product defined using the inner product $$\langle\cdot,\cdot\rangle$$ on $$\mathfrak{g}$$, i.e. $$\mathrm{tr}(\omega\wedge\eta):=\sum_{a,b=1}^{\mathrm{dim}(G)}(\omega^{a}\wedge\eta^{b})\langle T_{a},T_{b}\rangle\in\Omega^{k+l}(P)$$ for all $$\omega\in\Omega^{k}(P,\mathfrak{g})$$ and for all $$\eta\in\Omega^{l}(P,\mathfrak{g})$$, where $$\{T_{a}\}_{a=1}^{\mathrm{dim}(G)}$$ denotes a basis of $$\mathfrak{g}$$ and where $$\omega^{a}\in\Omega^{k}(P)$$ and $$\eta^{b}\in\Omega^{l}(P)$$ denote the coordinate forms with respect to this basis. One can easily verify that this definition is independent of the choice of basis. Note that if $$G$$ is compact and simple and $$\langle\cdot,\cdot\rangle$$ positive definite, then $$\langle\cdot,\cdot\rangle$$ is necessarily a negative multiple of the Killing form of $$\mathfrak{g}$$, which is also the reason why we use the notation "$$\mathrm{tr}$$" above.

As a next step, note that the bundle $$P$$ is trivial if and only if it admits a smooth global section. In particular, this is the case if $$G$$ is compact and simply-connected and if $$\mathcal{M}$$ has dimension $$\leq 3$$. The "Chern-Simons action'' is then defined to be the functional $$S_{\mathrm{CS}}[s,A]:=\int_{\mathcal{M}}\,s^{\ast}\mathrm{CS}[A],$$ where $$s:\mathcal{M}\to P$$ is a global gauge. At this point, the definition clearly depends on the choice of chosen gauge. Let $$f\in\mathcal{G}(P)$$ be a gauge transformation (i.e. a bundle automorphism of $$P$$). Then, after a straight-forward calculation, one finds that $$S_{\mathrm{CS}}[f\circ s,A]-S_{\mathrm{CS}}[s,A]=S_{\mathrm{CS}}[s,f^{\ast}A]-S_{\mathrm{CS}}[s,A]=-\frac{1}{6}\int_{\mathcal{M}}\,s^{\ast}\mathrm{tr}(\theta\wedge [\theta\wedge\theta]),$$ with $$\theta:=\sigma_{f}^{\ast}\mu_{G}$$, where $$\mu_{G}\in\Omega^{1}(G,\mathfrak{g})$$ denotes the Maurer-Cartan form on $$G$$ and where $$\sigma_{f}\in C^{\infty}(P,G)$$ is the map defined by $$f(p)=p\cdot\sigma_{f}(p)$$ for all $$p\in P$$. One can show that the integral on the right-hand side is always an element in $$6\cdot\mathbb{Z}$$. To sum up, the Chern-Simons action $$S_{\mathrm{CS}}[s,A]$$ is independent of the choice of global gauge modulo $$\mathbb{Z}$$. Hence, we have a well-defined action of the type $$S_{\mathrm{CS}}[A]:=S_{\mathrm{CS}}[s,A]\in\mathbb{C}/\mathbb{Z}.$$

As a quantum theory, we hence get a well-defined theory, since for example in the formal path integral, we consider the complex exponential of the action and, after choosing a convenient normalization, different choices of gauges leave the exponential invariant.

More details about Chern-Simons theory can be found in the papers by D. S. Freed:

1. D. S. Freed: Classical Chern-Simons Theory, Part 1. Advances in Mathematics, 113(2):237–303, 1995. Preprint.
2. D. S. Freed: Classical Chern-Simons theory, Part 2. Houston Journal of Mathematics, 28(2):293–310, 2002. See here.

The first part covers simply connected gauge groups and the second part covers arbitrary compact Lie groups.

Now, lets turn to the question of quantization: The geometric quantization of 3d-Chern Simons theory goes back to the seminal paper by Edward Witten from 1989:

• E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121(3):351–399, 1989. See here.

This is basically one of the Fields-Medaille winning works of E. Witten establishing a link between QFT and knot theory. A summary of the geometric quantization of Chern-Simons theory with a lot of references can be found in this nlab article. I also remember some very good (hand-written) lecture notes on the Geometric Quantization of Chern-Simons theory, namely at the university of Munich (Germany) by M. Schottenloher. See here (last chapter).

Note that there is also an alternative way of quantizing 3d-Chern Simons theory, namely the 3-dimensional topological field theory defined by the Reshetikhin–Turaev model. However, it was shown for example in

• J. Roberts: Skein theory and Turaev-Viro invariants, Topology 34(4):771-787, 1995. See here.

that the two approaches are equivalent.

As a last remark, althout Chern-Simons theory might seem to be a little bit exotic, let me just say the general relativity in 3-dimensions is in fact a Chern-Simons theory with gauge group $$\mathrm{SU}(2)$$ (Euclidean case) and $$\mathrm{SO}(1,2)$$ (Lorentzian case) (without a cosmological constants; otherwise you have to add a volume term). Hence, 3d-Quantum Chern-Simons Theory can be taken as a theory of 3-dimensionl quantum gravity. In fact, it has been shown that this theory is related to other quantum gravity theories in 3d, like the Ponzano-Regge spin foam model, the Boulatov group field theory model, LQG and quantization of 3d-BF theory.