Functional Integration - Examples

Examples

Most functional integrals are actually infinite but the quotient of two functional integrals can be finite. The functional integrals that can be solved exactly usually start with the following Gaussian integral:

\frac{ \int{ e^{i \int{ -\frac{1}{2}f(x).K(x,y).f(y) dxdy} + \int{ J(x).f(x) dx} } } } { \int{ e^{i \int{ -\frac{1}{2}f(x).K(x,y).f(y) dxdy} } } } = e^{i \frac{1}{2}\int{ J(x).K^{-1}(x,y).J(y) dxdy } }

By functionally differentiating this with respect to J(x) and then setting J to 0 this becomes an exponential multiplied by a polynomial in f. For example setting we find:

\frac{ \int{ f(a) f(b) e^{i \int{ f(x) \Box f(x) dx^4}} } }{ \int{ e^{i \int{ f(x) \Box f(x) dx^4}} } } = K^{-1}(a,b) = \frac{1}{|a-b|^2}

where a,b and x are 4-dimensional vectors. This comes from the formula for the propagation of a photon in quantum electrodynamics. Another useful integral is the functional delta function:

\int{ e^{i \int{ f(x) g(x) dx}} } = \delta = \prod_x\delta( g(x) )

which is useful to specify constraints. Functional integrals can also be done over Grassmann-valued functions where which is useful in quantum electrodynamics for calculations involving fermions.

Read more about this topic:  Functional Integration

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