Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1201boundary conditions in the path–integral formalism. In its simplest form,this relation looks like∫ BC2BC1Dφ · · · e iS[φ] = 〈BC1| T (· · · ) |BC2〉. (6.309)Here, we included a time–ordering operator for completeness, but as wehave stated before, for the theories we are interested in this ordering isirrelevant. The notation |BCi〉 indicates the state corresponding to theincoming or outgoing boundary condition. For example, if on the pathintegral side we prescribe all fields at a certain initial time, φ(t = t 1 ) = f(t 1 ), on the operator side this corresponds to the incoming state satisfyingφ(t 1 )|BC1〉 = f(t 1 )|BC1〉. (6.310)where on the l.h.s. we have an operator acting, but on the r.h.s. there isa simple scalar multiplication. More generally, in the operator formalismwe can have linear combinations of states of the type (6.310). Therefore,we should allow for linear combinations on the l.h.s. of (6.309) as well. Inother words, in the path–integral formalism, a state is an operator whichadds a number (a weight) to each possible boundary condition on the fields.From this point of view, the states in (6.310) are like ‘delta–functionals’:they assign weight 1 to the boundary condition φ(t 1 ) = f(t 1 ), and weight0 to all other boundary conditions.Now, let us specialize to 2D field theories. Here, the boundary of acompact manifold is a set of circles. Let us for simplicity assume that the‘incoming’ boundary consists of a single circle. We can now define a statein the above sense by doing a path integral over a second surface withthe topology of a hemisphere. This path integral gives a number for everyboundary condition of the fields on the circle, and this is exactly whata state in the path–integral formalism should do. In particular, one canuse this procedure to define a state corresponding to every operator O aby inserting O a on the hemisphere and then stretching this hemisphere toinfinite size. An expectation value in the operator formalism, such as〈O a |O b (x 2 )O c (x 3 )|O d 〉 cyl , (6.311)on a cylinder of finite length, can then schematically be drawn as in Figure6.24. Here, instead of first doing the path integrals over the semi–infinitehemispheres and inserting the result in the path integral over the cylinder,one can just as well integrate over the whole surface at once. However, notethat in topological field theories, there is no need to do the stretching, since