**Time-slicing Definition**

For a particle in a smooth potential, the path integral is approximated by zig-zag paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position *x _{a}* at time

*t*to

_{a}*x*at time

_{b}*t*, the time sequence

_{b}can be divided up into *n* + 1 little segments *t _{j}* −

*t*, where

_{j − 1}*j*= 1,...,

*n*+ 1, of fixed duration

This process is called *time-slicing*.

An approximation for the path integral can be computed as proportional to

where is the Lagrangian of the 1d-system with position variable *x*(*t*) and velocity *v* = *ẋ*(*t*) considered (see below), and *dx _{j}* corresponds to the position at the

*j*-th time step, if the time integral is approximated by a sum of

*n*terms.

In the limit *n* → ∞, this becomes a functional integral, which - apart from a nonessential factor - is directly the product of the probability amplitudes - more precisely, since one must work with a continuous spectrum, the respective densities - to find the quantum mechanical particle at *t _{a}* in the initial state

*x*and at

_{a}*t*in the final state

_{b}*x*.

_{b}Actually is the classical Lagrangian of the one-dimensional system considered, also

where is the Hamiltonian,

- , and the above-mentioned "zigzagging" corresponds to the appearance of the terms:

In the Riemannian sum approximating the time integral, which are finally integrated over *x _{1}* to

*x*with the integration measure

_{n}*dx*is an arbitrary value of the interval corresponding to

_{1}...dx_{n}x̃_{j}*j*, e.g. its center, (

*x*x

_{j}+_{j − 1}

*)/2.*

Thus, in contrast to classical mechanics, not only does the stationary path contribute, but actually all virtual paths between the initial and the final point also contribute.

Feynman's time-sliced approximation does not, however, exist for the most important quantum-mechanical path integrals of atoms, due to the singularity of the Coulomb potential *e*2/*r* at the origin. Only after replacing the time *t* by another path-dependent pseudo-time parameter

the singularity is removed and a time-sliced approximation exists, that is exactly integrable, since it can be made harmonic by a simple coordinate transformation, as discovered in 1979 by İsmail Hakkı Duru and Hagen Kleinert. The combination of a path-dependent time transformation and a coordinate transformation is an important tool to solve many path integrals and is called generically the Duru-Kleinert transformation.

Read more about this topic: Path Integral Formulation, Concrete Formulation

### Famous quotes containing the word definition:

“Was man made stupid to see his own stupidity?

Is God by *definition* indifferent, beyond us all?

Is the eternal truth man’s fighting soul

Wherein the Beast ravens in its own avidity?”

—Richard Eberhart (b. 1904)