Partial integration of equivariant half-density

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Partial integration of equivariant half-density

Now we will consider partial integration when the half-density satisfies the equivariant Master Equation. We show that the resulting effective theory still satisfies the equivariant Master Equation.

Suppose that a Lie group

acts on

by canonical transformations, and moreover that every Hamiltonian

is constant on the fibers of

Suppose that we are given an equivariant half-density

, i.e.:

(7)

Then the result of the partial integration is an equivariant half-density on

(8)

Indeed, Eq. (7) is equivalent to the statement that for any two functions

and

(9)

Eq. (8) follows from Eqs. (3) and (9) considering the case when both

and

are constant along the fiber of

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