48 2. THEORIES, LAGRANGIANS AND COUNTERTERMS Recall that we can write the propagator of a massless scalar field theory on a Riemannian manifold M as an integral of the heat kernel. If x, y M are distinct points, then P (x, y) = 0 Kt(x, y)dt where Kt C∞(M 2 ) is the heat kernel. This expression of the propagator is sometimes known as the Schwinger representation, and the parameter t as the Schwinger parameter. One can also interpret the parameter t as proper time, as we will see shortly. The heat kernel Kt(x, y) is the probability density that a particle in Brownian motion on M, which starts at x at time zero, lands at y at time t. Thus, we can rewrite the heat kernel as Kt(x, y) = f:[0,t]→M f(0)=x,f(t)=y DWienerf where DWienerf is the Wiener measure on the path space. We can think of the Wiener measure as the measure for a quantum field theory of maps f : [0,t] M with action given by E(f) = t 0 df, df . Thus, we will somewhat loosely write Kt(x, y) = f:[0,t]→M f(0)=x,f(t)=y e−E(f) where we understand that the integral can be given rigorous meaning using the Wiener measure. Combining these expressions, we find the desired expression for the prop- agator as a one-dimensional functional integral: P (x, y) = φ∈C∞(M) eS(φ)φ(x)φ(y) = t=0 f:[0,t]→M f(0)=x,f(t)=y e−E(f). This expression is the core of the world-line formulation of quantum field theory. This expression tells us that the correlation between the values of the fields at points x and y can be expressed in terms of an integral over paths in M which start at x and end at y. If we work in Lorentzian signature, we find the (formal) identity φ∈C∞(M) eS(φ)iφ(x)φ(y) = t=0 f:[0,t]→M f(0)=x,f(t)=y eiE(f). This expression is difficult to make rigorous sense of I don’t know of a rigorous treatment of the Wiener measure when the target manifold has Lorentzian signature.
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