What exactly is "sum over path?"

[Obnoxious]

I read about Richard Feynman's theory of sum over path in a book early this year, but I still don't exactly understand it - just how does he assign numbers to each different path an object could take and cancel out all but one path?

[Severian]

Each possible trajectory for a particle between two fixed points is a 'path'. To make an analogy, if you fly from New York to Tokyo you could fly to LA and the to Tokyo, or fly to Europe and then on to Tokyo. Each of these is a possible path.

 

Now, of course, for a single particle with no interaction, the path which will be taken is obviously a straight line between the two points, which is pretty trivial. But it becomes more interesting when you have mutiple particles which scatter off one another. Then the 'path' is really the particular movement of all the particles from their initial to final points.

 

So, what is this sum over paths? Feynman's theory assigns a complex number to each path. Technically, it is the exponential of [math]i=\sqrt{-1}[/math] times the "action". In turn, the action is the integral of the Lagraingian over the path (or if you prefer to remain covariant, the Lagrange density integrated over space-time):

 

i.e. [math]e^{i \int d^4x {\cal L}}[/math]

 

The Lagrainagian is defined by the theory you are looking at - for example, for a non-interating electron of mass m, it would be:

 

[math] {\cal L} = \bar \psi \left( i \gamma^\mu \partial_\mu -m \right) \psi [/math]

 

(Notice the similarity to the Dirac Equation - the Dirac Equation is the Euler-Lagrange equation for this Lagrangian.)

 

Then you add up all the paths, weighted by this exponential. Most of the contributions will cancel against each other, and the one which is dominant will be the one for which the action is minimised. This is known as "the principle of least action".

[□h=-16πT]

Have a look at Feynman's path or functional integral formulation of quantum field theory.

 

The path integral gives the amplitude for a particle, described by a quantised field operator, to move between two given locations. The amplitude consists of contributions from every possible path a particle can take in a time T, a linear superposition of contributions from each path weighted by a pure phase term which involves the action.

 

The formula is

 

[math]\langle x_b|e^{iHT}|x_a\rangle =\int {\cal D}xe^{i \int d^4x {\cal L}/\hbar}[/math]

 

Where [math]e^{-iHT/\hbar}[/math] is a "time evolution" operator that takes the position [math]x_a[/math] into [math]x_b[/math] in a time T, the functional integral [math]\int {\cal D}x[/math] runs over each path and [math]{\cal L}[/math] is the Lagrangian density of the theory. The Lagrangian is a fundamental scalar quantity describing the whole of the theory. Have a look at Severian's post for an example of a Lagrangian (the free Dirac field Lagrangian).

 

The usefulness of such a formulation is it offers a simpler mechanism for calculating propogation amplitudes and such like in perturbation theory.

 

Certain paths will contribute more to the integral above than others, thus tending towards the classical path assumed by the particle, ie. that which satisfies the principle of least action.

[□h=-16πT]

Have a look at Feynman's path or functional integral formulation of quantum field theory.

 

The path integral gives the amplitude for a particle, described by a quantised field operator, to move between two given locations. The amplitude consists of contributions from every possible path a particle can take in a time T, a linear superposition of contributions from each path weighted by a pure phase term which involves the action.

 

The formula is

 

[math]\langle x_b|e^{iHT}|x_a\rangle =\int {\cal D}xe^{i \int d^4x {\cal L}/\hbar}[/math]

 

Where [math]e^{-iHT/\hbar}[/math] is a "time evolution" operator that takes the position [math]x_a[/math] into [math]x_b[/math] in a time T, the functional integral [math]\int {\cal D}x[/math] runs over each path and [math]{\cal L}[/math] is the Lagrangian density of the theory. The Lagrangian is a fundamental scalar quantity describing the whole of the theory. Have a look at Severian's post for an example of a Lagrangian (the free Dirac field Lagrangian).

 

The usefulness of such a formulation is it offers a simpler mechanism for calculating propogation amplitudes and such like in perturbation theory.

 

Certain paths will contribute more to the integral above than others, thus tending towards the classical path assumed by the particle, ie. that which satisfies the principle of least action.

[□h=-16πT]

Have a look at Feynman's path or functional integral formulation of quantum field theory.

 

The path integral gives the amplitude for a particle, described by a quantised field operator, to move between two given locations. The amplitude consists of contributions from every possible path a particle can take in a time T, a linear superposition of contributions from each path weighted by a pure phase term which involves the action.

 

The formula is

 

[math]\langle x_b|e^{iHT}|x_a\rangle =\int {\cal D}xe^{i \int d^4x {\cal L}/\hbar}[/math]

 

Where [math]e^{-iHT/\hbar}[/math] is a "time evolution" operator that takes the position [math]x_a[/math] into [math]x_b[/math] in a time T, the functional integral [math]\int {\cal D}x[/math] runs over each path and [math]{\cal L}[/math] is the Lagrangian density of the theory. The Lagrangian is a fundamental scalar quantity describing the whole of the theory. Have a look at Severian's post for an example of a Lagrangian (the free Dirac field Lagrangian).

 

The usefulness of such a formulation is it offers a simpler mechanism for calculating propogation amplitudes and such like in perturbation theory.

 

Certain paths will contribute more to the integral above than others, thus tending towards the classical path assumed by the particle, ie. that which satisfies the principle of least action.

[□h=-16πT]

Shit, sorry about that.

[□h=-16πT]

The functional integral formalism allows for simple computation of amplitudes/correlation functions by use of

 

[math]\int {\cal D}\psi e^{\lbrack i\int d^4x{\cal L}\rbrack}[/math]

 

Where the functional integral [math]\int {\cal D}\psi [/math] runs over each configuration of the field [math]\psi [/math] (with Lagrangian density [math]{\cal L}[/math]) between the end points.