Energy conservation for virtual photon

[neelakash]

In introducing the concept of 'virtual photon',Halzen Martin writes (ch#1,P#7) "An ekectron emits a photon (the quantum of electromagnetic field) and as a result,recoils in order to conserve momentum.it is clearly impossible to conserve energy as well,so the emitted photon is definitely not a real photon"...

Why energy and momentum cannot be simoultaneously satisfied?Is momentum conservation is a bit prefferred over energy conservation?

 

[tiny-tim]

Hi neelakash!

To simplify the calculations, let's do it in the frame in which the initial and final velocities of the electron are equal and opposite …

change in energy-momentum 4-vector = (E,p) - (E,-p) = (0,2p), which is faster than light (infinitely fast, in this case), and so it can't be the energy-momentum 4-vector of a real photon (or a real anything)!

(same in any other frame … the change cemes out as (∆E,∆p) with ∆E² < ∆p², which isn't allowed)

 

[Phiphy]

I think the statement that "energy is not conserved for a virtual particle" is not right. The 4-momentum including energy is always conserved as the requirement of Lorentz covariance. This is manifest in Feynman diagrams: when we go from the space-time representation to momentum representation, each vertex contributes a delta function which exactly results in 4-momentum conservation.

The true difference between a real and a virtual particle is, a real particle is on-shell, ie., satisfying Einstein's energy momentum relationship: E^2=m^2+p^2 (with c=1); while a virtual particle does not satisfy this relationship. This difference manifests the difference between classical mechanics and quantum mechanics. A real particle satisfies the classical equation of motion(equivalent to the on-shell condition), so it follows a classical path of motion. A virtual particle is caused by quantum fluctuation, and follows "other paths" in the path integral, so it's off-shell.

 

[tiny-tim]

Hi Phiphy!

(try using the X² tag just above the Reply box )

I think the statement that "energy is not conserved for a virtual particle" is not right.

ah, but nobody said that "energy is not conserved for a virtual particle" …

Halzen Martin (ch#1,P#7) says …

it is clearly impossible to conserve energy as well,so the emitted photon is definitely not a real photon

and I said …

… so it can't be the energy-momentum 4-vector of a real photon (or a real anything)!

We're all saying that energy-momentum is not conserved for a real particle.

The true difference between a real and a virtual particle is, a real particle is on-shell, ie., satisfying Einstein's energy momentum relationship: E^2=m^2+p^2 (with c=1); while a virtual particle does not satisfy this relationship.

Nah … the true difference between a real and a virtual particle is, a real particle is real and a virtual particle isn't.

(the clue's in the name! )