Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1199A very practical way to ensure (6.308) is to use a Lagrangian whichitself is Q−exact, L = {Q, V }, for some operator V . This choice has anextra virtue, which we can see if we explicitly include Planck’s constant inour description: the quantum measure then readsexp i {Q,Then, we can use exactly the same argument as before to show that∫MV}.dd 〈O i 1 · · · O in 〉 = 0.That is, the correlators we are interested in are independent of , and wecan therefore calculate them exactly in the classical limit.Descent equationsAn important property of cohomological field theories is that, given a scalarphysical operator on M – where by ‘scalar’ we mean an operator that doesnot transform under coordinate transformations of M, so in particular ithas no α−indices – we can construct further operators which behave likep−forms on M. The basic observation is that we can integrate (6.308) overa spatial hypersurface to get a similar relation for the momentum operators:P α = {Q, G α },where G α is a fermionic operator. Now consider the operatorO (1)α = i{G α , O (0) },where O (0) (x) is a scalar physical operator: {Q, O (0) (x)} = 0.calculateddx α O(0) = i[P α , O (0) ] = [{Q, G α }, O (0) ]= ±i{{G α , O (0) }, Q} − i{{O (0) , Q}, G α } = {Q, O (1)α }.Let usIn going from the second to the third line, we have used the Jacobi identity.The first sign in the third line depends on whether O (0) is bosonic orfermionic, but there is no sign ambiguity in the last line. By defining the1–form operatorO (1) = O α(1) dx α