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16 1 Introduction to path integrals Mode regularization with ΔLM R ❅ ❅ ❅ Dimensional regularization with ΔLDR ❅ ❅ ❅ Other reg. schemes with ΔL (ΔL fixed by renormalization conditions) Formal configuration space path integral with L + ΔL ✻ΔL unknown Loops Solution of Schr¨ odinger equation (heat kernel) ❅ ❘ ❅ β ˆ z| exp(− h ¯ H)|y ❅ ❅ Time slicing Trotter Weyl Berezin ✒ ✻ Direct operator method ❄ ✲ Discrete phase-space path integral ❄ ❄ Final result Continuous phase-space path integral with ΔLT S Discrete configuration-space path integral Products of distributions ✻ Loops ❄ Matthews ✲ Continuous configuration-space path integral with ΔLT S No products of distributions Fig.
34) is independent of gn . This implies that at the one- and two-loop level one can construct an infinity of divergent graphs from a given divergent graph by inserting gn vertices. The following diagrams illustrate this fact . As we shall see, the ghost cancels the leading divergences, but we repeat that finite ambiguities remain which must be fixed by renormalization conditions. Of course, general coordinate invariance must also be imposed, but this symmetry requirement is not enough to fix all of the renormalization conditions.
The notation x2 , t2 |x1 , t1 is due to Dirac who called this element a transformation function. ) Dirac knew that in classical mechanics the time evolution of a system could be written as a canonical transformation, with Hamilton’s principal function S(x2 , t2 |x1 , t1 ) as the generating functional [53]. This function S(x2 , t2 |x1 , t1 ) is the classical action evaluated along the classical path that begins at the point x1 at time t1 and ends at the point x2 at time t2 . In his 1932 article Dirac wrote that x2 , t2 |x1 , t1 corresponds to exp ¯hi S(x2 , t2 |x1 , t1 ).