# QED functions E(a to b) and P(a to b)

#### Cemre

Hello all...

In his famous video lectures about quantum electro dynamics,
Richard Feynman is talking about electron and photon functions
E and P. And whole story is about path-integrating those functions.

In wikipedia it is as follows:
If a photon moves from one place and time – in shorthand, A – to another place and time – shorthand, B – the associated quantity is written in Feynman's shorthand as P(A to B). The similar quantity for an electron moving from C to D is written E(C to D).
In Feynman's lectures, he says P is related to ( 1 / ( (t^2) - (x^2) ) )
( minus; reminding of relativity )

What exactly are those E and P functions? ( formula? )

Thanks.

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#### _PJ_

What is referred to is the amplitude of probability waves calculated from the difference in location of the 'particles'

To give a formula,. you would need a summation of the squares of all possible path trajectories between the 'particle' at point A and the particle at point B.

#### Cemre

What is referred to is the amplitude of probability waves calculated from the difference in location of the 'particles'

To give a formula,. you would need a summation of the squares of all possible path trajectories between the 'particle' at point A and the particle at point B.
Hi,

yes, yes, I know that E and P are functions that take a couple of positions and time, one for location 1 , one for location 2, and give out a complex number which is the probability amplitute. I know about probability amplitutes.

path-Integrating complex functions E and P gives the total probability, I understand that.

But :) what exactly are E and P ( like F is F=ma )

Thanks.

#### _PJ_

Hi,

yes, yes, I know that E and P are functions that take a couple of positions and time, one for location 1 , one for location 2, and give out a complex number which is the probability amplitute. I know about probability amplitutes.

path-Integrating complex functions E and P gives the total probability, I understand that.

But :) what exactly are E and P ( like F is F=ma )

Thanks.
They are just to distinguish between the Electron (E) and Photon (P)

#### Cemre

Feynman says in his lecture that

P(e1,e2) is almost equivalent to E(e1,e2)... where e1 and e2 are event1 and event2
in fact P(e1,e2) is E(e1,e2) where rest mass is set to zero.

But he doesn't give the open formula for E(e1,e2)

I would appreciate very much if someone could give me the open formula for E(e1,e2)

They are just to distinguish between the Electron (E) and Photon (P)
_PJ_ , you still don't understand what I am asking...

#### asheg

Are they green's functions and if it's so how they have been calculated?

#### Cemre

Hi again,

I think I found something...

at wikipedia : http://en.wikipedia.org/wiki/Path_integral_formulation

Feynman's interpretation :
...
3. The contribution of a history to the amplitude is proportional to e^{i S/hbar}, where hbar is reduced Planck's constant, and can be set equal to 1 by choice of units, while S is the action of that history, given by the time integral of the "Lagrangian" along the corresponding path.
--> given by the time integral of the "Lagrangian" along the corresponding path

also another wiki article has information about QED Lagrangian.

"Quantum electrodynamic Lagrangian" :
at wikipedia : http://en.wikipedia.org/wiki/Lagrangian

Need verification...
Are those correct ?

Thanks.

#### Avodyne

You might try looking in Feynman's book QED: the Strange Theory of Light and Matter. It's been many years since I read it, but as I recall Feynman used very sloppy and imprecise language to describe the math, incorrectly referring to a propagator as a quantum amplitude. So, if you're interested in learning the real theory, Feynman is not the best source.

#### RedX

You might try looking in Feynman's book QED: the Strange Theory of Light and Matter. It's been many years since I read it, but as I recall Feynman used very sloppy and imprecise language to describe the math, incorrectly referring to a propagator as a quantum amplitude. So, if you're interested in learning the real theory, Feynman is not the best source.
I've never read that particular book, but I'm a big fan on his lectures in physics, volume I.

I vaguely recall Peskin and Schroeder saying that the propagator is the amplitude also, although I've never read their book.

It kind of makes sense on a conversational level. Propagators are joined at vertices, and at each vertex propagators can split into other propagators, sort of like a double slit experiment where at each screen the possible paths split into two. The location of the screens or vertices are integrated over all space, like a path integral. Real particles (say at position x) correspond to sources that inject their physical location at one end of a propagator J(x)delta(x,z), and that equation kind of says that the source at x propagates to z and the amplitude is delta(x,z), and then z meets a vertex (double slit), and finally z leads to another real particle at y via delta(z,y)J(y). So you have the amplitude to go from x to y of a real particle as a path integral through all paths z.

#### RedX

Here's the full propagator for the real Klein-Gordan field:

http://en.wikipedia.org/wiki/Propagator#Feynman_propagator

$$\ = \left \{ \begin{matrix} -\frac{1}{4 \pi} \delta(s) + \frac{m}{8 \pi \sqrt{s}} H_1^{(1)}(m \sqrt{s}) & \textrm{ if }\, s \geq 0 \\ -\frac{i m}{ 4 \pi^2 \sqrt{-s}} K_1(m \sqrt{-s}) & \textrm{if }\, s < 0. \end{matrix} \right.$$

It's pretty nasty looking, but simplifies considerably if the particle is massless to something like one over the (spacetime) distance squared (for both time-like and space-like intervals if I recall).

#### Cemre

You might try looking in Feynman's book QED: the Strange Theory of Light and Matter. It's been many years since I read it, but as I recall Feynman used very sloppy and imprecise language to describe the math, incorrectly referring to a propagator as a quantum amplitude. So, if you're interested in learning the real theory, Feynman is not the best source.
The video lectures are not very clear but I found it in the book... thanks... it is as follows.

P(A to B) is ;
A photon has an amplitude to go from a point A in space-time to another point B.
This amplitude, which I will call P(A to B), is calculated from a formula that depends only on the difference in location (X2-X1) and the difference of the time (T2-T1).
In fact, it's a simple function that is the inverse of the difference of their squares, an "interval," I, that can be written as (X2 -X1)^2 - (T2-T1)^2.
E(A to B) is ;
The formula for E(A to B) is complicated, but there is an interesting
way to explain what it amounts to. E(A to B) can be represented as a giant
sum of a lot of different ways an electron could go from point A to point
B in space-time

the electron could take a "one-hop flight," going directly from A to B;
it could take a "two-hop flight," stopping at an intermediate point C;
it could take a "three-hop flight," stopping at points D and E, and so on.

In such an analysis, the amplitude for each "hop"--from one point F to
another point C, is P(F to G), the same as the amplitude for a photon
to go from a point F to a point G.
The amplitude for each "stop" is represented by n, n being the same number
I mentioned before which we used to make our calculations come out right.
"n" is related to the "mass" of the electron.

The formula for E(A to B) is thus a series of terms:
P(A to B) [the "one-hop" flight]
+ P(A to C)*n2*P(C to B) ["two-hop" flights, stopping at C]
+ P(A to D)*n2*P(D to E)*n2*P(E to B) ["three-hop" flights, stopping at D and E]
+ . . . for all possible intermediate points C, D, E, and so on.

Note that when n increases, the nondirect paths make a greater contribution
to the final arrow. When n is zero (as for the photon), all terms
with an n drop out (because they are also equal to zero), leaving only the
first term, which is P(A to B). Thus E(A to B) and P(A to B) are closely
related.