3.1 QFT on a manifold and symmetry
As emphasized, the locality plays the essential role in (topological properties of) QFT. One concequence of the locality is that, given a \(d+1\)-dimensional QFT, we can quantize the theoy on a spacial closed manifold \(M_d\) to get \(\mathcal{H}_{M_d}\).13 In other words a QFT is not just the Hilbert space and the Hamiltonian, but a rule to assign them when a closed manifold \(M_d\) is given. It is more precise to express the same thing in terms of the algebra of operators. For a QFT, given a region \(R \in \mathbb{R}^{d,1}\), there is the algebra of local operators contained in \(R\): \(\mathcal{A}(R)\). Then we “glue” the algebra \(\mathcal{A}(B^{d+1})\) on a ball to get \(\mathcal{A}(R)\) for an arbitrary manifold \(R\). In particular we can get \(\mathcal{A}(M_d\times \mathbb{R})\).14
A symmetry also is expected to be local: that if there is a symmetry \(T\) in a QFT, that means, for arbitrary \(\mathcal{M}_d\), there should exists a symmetry transformation \(T_{M_d}\) on \(\mathbb{P}\mathcal{H}_{M_d}\). This is manifest in the case of continuous symmetry. For example, the unitary operator \(U_\lambda\) for a \(U(1)\) rotation by angle \(\lambda\) is, from the Noether’s theorem, \[\begin{equation} \tag{3.1} U_\alpha = e^{\mathrm{i}\lambda \int_{M_d}j^0 \mathrm{d}^dx}, \end{equation}\] where \(j^\mu\) is the Noether current. In other words the locality of the unitary operator is manifest in the integral in the expression. For a discrete symmetry, the Noether current is not available. However, we can define that a symmetry transformation is a automorphism of \(\mathcal{A}(R)\): \[\begin{equation} \tag{3.2} T \curvearrowright \mathcal{A}(R) \to \mathcal{A}(R), \end{equation}\] and then \(T\) also naturally (projectively) acts on \(\mathcal{H}_{M_d}\) which is a representation of \(\mathcal{A}(R)\). In a more explict term, if the QFT is written in terms of fundamental fields and symmetry acts on these fields, the Hilbert space \(\mathcal{H}_{M_d}\) is the Fock space constructed from creation/annhilation operators for each modes of the fields on \(M_d\), and the action of a symmetry on the fields naturally defines the action on the annihilation operators, in turn defining the action on \(\mathcal{H}_{M_d}\).
For a symmetry transformation \(T\) respecting the locality15, defined in the above sense, to be the symmetry of the theory, the corresponding unitary has to commute with the Hamiltonian. We also assume that the Hamiltonian is local: \[\begin{equation} \tag{3.3} H = \int T_{00} \mathrm{d}^dx, \end{equation}\] where \(T_{\mu\nu}\) is the energy-momentum tensor. For a symmetry in QFT, we demand that \([U,T_{\mu\nu}] = 0\). Otherwise \(U\) does not commute with the spacetime symmetry.
References
We often need an additional structure to \(M_d\). For example, if the theory includes fermionic fields, the manifold \(M_d\) has to be equipped with the spin structure. In this lecture we do not care about thie subtle but interesting and important subtlety and assume that the theory is bosonic, i.e. does not contain fermionic fields. This amounts to assume only the orientation of \(M_d\). (To put a QFT on an unoriented manifold, one needs an anti-unitary symmetry which also be ignored in thie lecture.) The references mentioned in 1.1.4 have contents about fermionic theories.↩︎
This explanation is awfully rough. For a better discussion the reader might refer to [14]↩︎
We cannot call this a “local symmetry” since it means something else.↩︎