Definition 1. In Proposition 1 of the previous blog post we introduced an exponentially decaying factor which attenuates the Feynman formulas for a virtual pair. Let us name the modified formulas as Feynman tunneling formulas.
The decaying factor removes both the ultraviolet and the infrared divergence of the Feynman vacuum polarization loop integral. Why Feynman himself did not introduce the decaying factor? Maybe he was worried about the breach of Lorentz invariance and time symmetry, which such a factor involves. The decay rate is calculated in the preferred coordinate system of the center of mass of the collision. A preferred coordinate system is not compatible with Lorentz invariance.
But, actually, original Feynman formulas do have a decaying wave function! Feynman removes all solutions where an off-shell particle flies away. That means that such a wave function decays with time!
The iε trick in quantum field theory textbooks is probably the way in which the decaying wave function is smuggled into the machinery. It does break Lorentz invariance, but the problem has been swept under the carpet.
The iε trick prunes away off-shell particles which fly "away" from the collision area. But it depends on the inertial coordinate system who flies "away". There is a preferred inertial coordinate system. That breaks Lorentz invariance.
If the wave function decays very slowly, then approximate Lorentz invariance is preserved in small local patches of timespace.
Theorem 2. The original machinery of Feynman, as well as the one used in all textbooks, does break Lorentz invariance. QED.
What about time symmetry? Does the original Feynman machinery break also that? Only on-shell particles enter the collision. The Feynman machinery prunes away solutions with off-shell particles flying away. If the pruning is gradual, then there are "tails" of off-shell solutions that leave the collision area, and could be detected by some instrument which is very close to the collision.
But only on-shell particles enter the collision area. The setup is time asymmetric in the Feynman machinery, as well as in our machinery of Definition 1.
Since the virtual photon represents the energy and momentum flow in the electric field of the colliding particles, there certainly are very good grounds for using a preferred coordinate system. We could claim that the virtual particles in Feynman diagrams are only interactions of the real particles. There is no reason to assume that the virtual particles would obey Lorentz invariance at "long" distances.
When a particle moves away from the collision, the interactions quickly grow very small. It is logical that the wave function of an off-shell particle very soon decays to infinitesimal. An off-shell particle cannot exist without interactions. The iε trick of textbooks delays the decay in an unrealistic fashion.
If we have a real photon which flies as a part of a planar wave that fills the whole spacetime, there is lots of symmetry. There are grounds to require Lorentz invariance from it.
We need a more detailed analysis of what the virtual photon and the virtual pair are in a collision.
The last section of the previous blog post introduced the concept of a tachyon traveling with the pair. The wave function of the tachyon decays exponentially with time.
No comments:
Post a Comment