2. FUNCTIONAL INTEGRALS IN QUANTUM FIELD THEORY 5 The fundamental quantities one wants to compute are the correlation functions of a set of observables, defined by the heuristic formula O1,...,On = φ∈C∞(M) eiS(φ)/ O1(φ) · · · On(φ)Dφ. Here D φ is the (non-existent!) Lebesgue measure on the space C∞(M). The non-existence of a Lebesgue measure (i.e. a non-zero translation invariant measure) on an infinite dimensional vector space is one of the fundamental diﬃculties of quantum field theory. We will refer to the picture described here, where one imagines the existence of a Lebesgue measure on the space of fields, as the naive functional integral picture. Since this measure does not exist, the naive functional integral picture is purely heuristic. 2.2. Throughout this book, I will work in Riemannian signature, in- stead of the more physical Lorentzian signature. Quantum field theory in Riemannian signature can be interpreted as statistical field theory, as I will now explain. Let M be a compact manifold of Riemannian signature. We will take our space of fields, as before, to be the space C∞(M, R) of smooth functions on M. Let S : C∞(M, R) → R be an action functional, which, as before, we assume is the integral of a Lagrangian. Again, a typical example would be the φ4 action S(φ) = − 1 2 x∈M φ(D +m2)φ + 1 4! φ4. Here D denotes the non-negative Laplacian. 2 We should think of this field theory as a statistical system of a random field φ ∈ C∞(M, R). The energy of a configuration φ is S(φ). The behaviour of the statistical system depends on a temperature parameter T : the system can be in any state with probability e−S(φ)/T . The temperature T plays the same role in statistical mechanics as the pa- rameter plays in quantum field theory. I should emphasize that time evolution does not play a role in this pic- ture: quantum field theory on d-dimensional space-time is related to statis- tical field theory on d-dimensional space. We must assume, however, that the statistical system is in equilibrium. As before, the quantities one is interested in are the correlation functions between observables, which one can write (heuristically) as O1,...,On = φ∈C∞(M) e−S(φ)/T O1(φ) · · · On(φ)Dφ. 2 Our conventions are such that the quadratic part of the action is negative-definite.

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