13. VECTOR-BUNDLE VALUED FIELD THEORIES 73 13.2. The space E2 of non-propagating fields is introduced into this definition with an eye to future applications: none of the examples treated in this book will have non-propagating fields. Thus, the reader will lose nothing by ignoring the space of non-propagating fields. For those who are interested, however, let me briefly explain the reason for considering non-propagating fields. Let us consider a free scalar field theory on a Riemannian manifold M with metric g0. Let us consider per- turbing the metric to g0 + h. The action of the scalar field theory is given by S(φ) = φ g0+h φ, as usual. Notice that the action is not just local as a function of the field φ, but also local as a function of the perturbation h of the metric. However, h is not treated as a quantum field, only as a classical field: we do not consider integration over the space of metrics. In this situation, the tensor h is said to be a background field, or (in the terminology adopted here) a non-propagating field. In this example, the space of fields is E = C∞(M) ⊕ Γ(M, Sym2 TM) where the space E1 of propagating fields is C∞(M), and the space E2 of non-propagating fields is Γ(M, Sym2 TM). The action of the theory is S(φ) = M φ g0+h φ. This action can, as usual, be split into quadratic and interacting parts: S(φ) = M φ g0 φ + M φ ( g 0 +h − g0 ) φ. The quadratic part of the action is the only part relevant to the definition of a free field theory as presented above. As we see, the quadratic part only depends on the propagating field φ, and not on h. However, the interaction term depends both on φ and h. It is a general feature of the interaction terms that they must have some dependence on φ they canot be functions just of the non-propagating field h. We will build this into our definition of interactions in the presence of non-propagating fields shortly. If we take the free theory associated to this example – given by discarding the interacting terms in the action S – we fit it into the general definition as follows. The operator DE 1 is the Laplacian g0 : C∞(M) → C∞(M). The operator D is the identity map C∞(M) → C∞(M). 13.3. Now that we have the general definition of a free field theory, we can start to define the concept of effective interaction in this context. First, we have to define the heat kernel. If the manifold M is compact, there is a unique heat kernel Kt ∈ E ! 1 ⊗ E1 ⊗ C∞(R 0 ) ⊗ A for the operator DE 1 . Composing with the operator D gives an element D Kt ∈ E1 ⊗ E1 ⊗ C∞(R 0 ) ⊗ A . We will view this as an element of E ⊗ E ⊗ C∞(R 0 ).

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