Suppose that we have a real electron which moves with momentum p to some direction and it encounters a perpendicular gently sloping potential wall for which the electron does not have enough energy to penetrate.
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e ---> /
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The diagram above is misleading: the slope should be gentle, not steep.
While the electron is moving in empty space, we can describe it roughly with a planar wave that moves to the direction of p.
In quantum mechanics, a particle can penetrate to a zone where the its potential energy is bigger than the original kinetic energy of the particle. We may say that the kinetic energy of the particle is negative in that zone. The absolute value |ψ| of its wave function falls exponentially in that zone.
Let us describe a single possible path of a particle with an integral exp(i L(x)) over a single path C. If the kinetic energy of the particle stays positive along the path, the path integral absolute value stays the same but the phase of the integral rotates. If the kinetic energy is negative, the phase of the integral is "frozen" and the absolute value falls exponentially, as we progress along the path.
We may imagine that when the phase of the integral turns, then the particle progresses in real time, but when the phase is frozen, the particle progresses in "imaginary time". The Wick rotation is based on this idea.
Definition 1. If the kinetic energy of a particle is negative, we may call it a virtual particle. A particle with positive kinetic energy is real. If we have real particles hitting a potential wall, we call the zone where the particle kinetic energy is negative the virtual zone, and similarly for the real zone.
Question 2. How do virtual particles "move"? Do they obey "trajectories" similar to real particles? To be more precise, what does the wave function ψ look like in the virtual particle zone and how does it phase and absolute value there behave?
If we have waves in water and they encounter a gently sloping shore, the trajectory of single water molecule bears some resemblance to the electron wave function above. As long as water is deep enough, the molecule will move up and down: its "phase" will rotate. But if the molecule comes close to the waterline, it will be washed on land in a crest of a wave: its "phase" is frozen to the "up" position. It is very unlikely that the molecule will be able to climb high on land. That corresponds to the exponentially falling absolute value of the wave function.
Suppose that we have an eternal, unchanging, coherent flux ψ of particles hitting a high potential wall. Let us calculate the phase and absolute value of ψ at a spacetime point (x, t) inside the virtual zone. We may use a path integral, for example, in the calculation. A symmetry argument then shows that the phase of ψ will rotate with time in the same fashion inside the virtual zone as in the real zone.
What about the phase of ψ at two points z1 and z2 in the virtual zone, where z1 is closer to the real zone than z2?
Is the phase at z1 "later" than at z2? Or is the phase the same at z1 and z2?
The path integral over C which we described above is very different from the path integral Feynman uses in his diagrams. Feynman's idea is to represent the wave function as an uncountable sums of sharp "spikes" for each spacetime point x, and calculate how the spike evolves over time. The propagator at spacetime point will tell the value of the wave function at that (later in time) point.
Suppose that we have a particle that approaches a high potential wall which it has very meager changes of penetrating. Let us represent the particle as a wave function. If we try to calculate the penetration probability with the spike method, we are in trouble. Each spike represents a particle with a potentially infinite kinetic energy, because the Fourier decomposition contains every momentum p. Particles with high kinetic energy will pass the potential wall without problems. We should show that destructive interference of various probability amplitudes causes the wave function to have very meager values on the other side of the wall.
Any small error in the calculation of destructive interference may make the result infinite.
A better way to calculate is to make a Fourier decomposition of the incoming flux of particles. We cannot produce particles with infinite energy. The Fourier decomposition will have only moderate values of momentum p. We calculate the wave function on the other side of the wall for each of these moderate p.
In Feynman diagrams, incoming particles have moderate values of p. Each vertex must conserve p. We get moderate values of p for each line, except in the case of a loop. The vacuum polarization loop is an example.
The momentum p is not restricted inside a loop? Why not? Because we assume that the virtual electron and the positron are born as a spike wave function at a spacetime point. If we could show that they are born in a bigger zone and have a smoother wave function, then we would have a natural cutoff for the momentum p, and the Feynman integral would converge.
Why does Feynman assume that the particles are born at a point?
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