Editing Path integral formulation (section)

=== Functionals of fields ===

However, the path integral formulation is also extremely important in ''direct'' application to quantum field theory, in which the "paths" or histories being considered are not the motions of a single particle, but the possible time evolutions of a [[field (physics)|field]] over all space.  The action is referred to technically as a [[functional (mathematics)|functional]] of the field: {{math|''S''[''ϕ'']}}, where the field {{math|''ϕ''(''x<sup>μ</sup>'')}} is itself a function of space and time, and the square brackets are a reminder that the action depends on all the field's values everywhere, not just some particular value. ''One'' such given function {{math|''ϕ''(''x<sup>μ</sup>'')}} of [[spacetime]] is called a ''field configuration''. In principle, one integrates Feynman's amplitude over the class of all possible field configurations.

Much of the formal study of QFT is devoted to the properties of the resulting functional integral, and much effort (not yet entirely successful) has been made toward making these [[functional integral]]s mathematically precise.

Such a functional integral is extremely similar to the [[partition function (statistical mechanics)|partition function]] in [[statistical mechanics]].  Indeed, it is sometimes ''called'' a [[partition function (quantum field theory)|partition function]], and the two are essentially mathematically identical except for the factor of {{mvar|i}} in the exponent in Feynman's postulate 3. [[Analytic continuation|Analytically continuing]] the integral to an imaginary time variable (called a [[Wick rotation]]) makes the functional integral even more like a statistical partition function and also tames some of the mathematical difficulties of working with these integrals.