The Feynman path integral derived from the definition of the Dirac delta function.

The Dirac delta is defined as

[1]

And for any function f(x), we get

[2]

But if f(x₁) is the Dirac delta function itself, then

[3]

This is a transitive property that can be applied as many time as desired. Applying it again gives

[4]

And applying it an infinite number of times gives

[5]

And then if we let

[6]

with

[7]

then rearranging terms as before

[8]

which equals

[9]

we see that

[10]

is equal to

[11]

And by gathering terms, this is equal to

[12]

which is the Feynman path integral formulation of quantum mechanics.

This is usually shortened to the notation:

[13]

This derivation is part of a larger effort to derive physics from logic. There it is seen that the Dirac delta function fulfills the role of a mathematical expression for the material implication of propositional logic.