Functional integration
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- This article's topic is not that of functional integration (neurobiology).
In mathematical physics, functional integration is integration over certain infinite-dimensional spaces. See functional analysis. Physicists often refer informally to functional integrals over spaces of paths (or field configurations) as path integrals, which are different from path integrals in the usual sense.
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The Feynman integral
Functional integration techniques in physics were pioneered by Richard Feynman, who successfully applied his "path integral formulation" to problems in quantum mechanics and quantum field theory, as well as classical and quantum statistical mechanics.
As of August 2003, no rigorous definition of functional integration has been given which is applicable to all instances where it arises heuristically. Another way to say this is that important problems whose solutions are obtained by heuristic methods involving functional integrals have eluded formulation in terms of any of the existing rigorous definitions of functional integration.
The problem of functional integration is to make sense of expressions such as
- <math>\int \mathcal{D}\phi F[\phi]\geq 0<math>
where <math>\mathcal{D}\phi<math> is "Lebesgue measure" on an infinite dimensional space where φ is valued. The problem with this definition is that there is no translation-invariant measure on an infinite-dimensional vector space. In fact, on an infinite-dimensional space one can really only integrate with respect to probability measures. Infinite-dimensional compact groups admit a translation-invariant Haar measure.
Irving Segal observed that the orthogonal group is much larger than the translation group, and that there are large numbers of probability distributions on infinite-dimensional Hilbert spaces which are invariant under the orthogonal group. However, the rigorous theory of functional integration that he developed is not far-reaching enough to accommodate all the physical applications of the Feynman path integral.
Old material follows that needs to be elaborated
When <math>\phi\geq 0<math>, a functional measure might be possible and we have a Wiener integral. Otherwise, we might have something which looks very fishy, like the use of summing of nonconvergent infinite series and the use of infinitesimals before the introduction of concepts like ε-δ, uniform convergence, etc..
Functional integrals over manifolds are sometimes approximated by a lattice, but there is no guarantee this will give a good approximation or even converge. This is related to statistical field theory. The so-called renormalization group methods allow a rigorous continuum limit if the lattice theory has an ultraviolet fixed point. In fact, the direct naïve approximation by a lattice can have its pitfalls, because, for example, the fermion doubling problem, among other things.
Even simple Gaussian integrals like
- <math>\int D\phi e^{i\int dt\frac{dx}{dt}(t)^2},<math>
where <math>\phi:\mathbb{R}\rightarrow \mathbb{R}<math>, need renormalization to make sense, and only ratios of such integrals can be defined in an invariant manner.
For the use of functional integrals in quantum field theory, see path integral formulation.
Translational invariance
The "functional measure" <math>\mathcal{D}\phi<math> (it's not really a measure in the measure theoretic sense of the word) is said to be translationally invariant if for any differentiable functional A which falls off to zero for large φ,
- <math>\int \mathcal{D}\phi {\delta \over \delta \phi(x)} A[\phi]=0<math>
for any point x where we have taken the functional derivative.
Gaussian integral approximation
Suppose we wish to "evaluate" a functional integral of the form
- <math>\int \mathcal{D}\phi A[\phi]e^{-B[\phi]}<math>
(the case <math>\int \mathcal{D}\phi A[\phi] e^{iB[\phi]}<math> is similar provided it is possible to "Wick rotate" slightly in the imaginary direction to give a "convergent" integral)
where A and B are polynomial functionals in φ. We can write B nonuniquely as the sum of a nonsingular quadratic polynomial functional G and the "remainder" C. Let's assume G's Hessian matrix is positive definite and its minimum is at φ=φ0. If we define a new variable <math>\bar{\phi}<math> as φ-φ0, then <math>B[\bar{\phi}+\phi_0]<math> is a homogeneous quadratic polynomial in <math>\bar{\phi}<math>. Assuming translational invariance, the functional integral is equal to
- <math>\int \mathcal{D}\bar{\phi} A[\bar{\phi}+\phi_0] e^{-G[\bar{\phi}+\phi_0]-C[\bar{\phi}+\phi_0]}<math>
The integrand is still the product of a polynomial with the exponential of a polynomial. Expand
- <math>C[\bar{\phi}+\phi_0]=\sum_{n=0}^N c_{i_1 \dots i_n} \bar{\phi}^{i_1} \cdots \bar{\phi}^{i_n}<math>
and
- <math>A[\bar{\phi}+\phi_0]=\sum_{m=0}^M a_{j_1 \dots j_m} \bar{\phi}^{j_1} \cdots \bar{\phi}^{j_m}<math>
where we are using the deWitt notation and c are the coefficients which happens to be totally symmetric (antisymmetric for odd (fermionic) indices).
Then, we can rewrite the functional integral as
- <math>\sum_{p_0=0}^\infty \cdots \sum_{p_N=0}^\infty \sum_{m=0}^M \int \mathcal{D}\bar{\phi} {(-1)^{p_0+ \dots +p_N} \over {p_0}! \cdots {p_N}!} a_{j_1 \dots j_m} \bar{\phi}^{j_1} \cdots \bar{\phi}^{j_m} \left ( \prod_{n=0}^N \prod_{k=1}^{p_n} c_{i_1 \dots i_n} \bar{\phi}^{i_1} \cdots \bar{\phi}^{i_n} \right ) e^{-G[\bar{\phi}+\phi_0]}<math>
where we have tacitly assumed we can interchange the infinite sum with the functional integral.
Now, each term in the sum can be evaluated according to the Gaussian integral article. It turns out we can keep track of the computations of each term using a graphical notation known as Feynman diagrams. Each occurrence of a c is represented by an internal vertex, each ji by an external vertex and each occurrence of G−1 is represented by an edge. A vertex is incident to an edge if the indices contract. The result is often an infinite series which does not converge. This is only one of the problems. Another is the answer depends upon the choice of the splitting of B into G and C which is unnatural. As such, some people simply think of functional integrals as a mnemonic for deriving the Feynman rules.