Our previous blog entry contains a crucial observation: if we try to estimate what portion of a flux of particles passes through a high potential wall, and we use the Feynman propagator method where we represent the wave function ψ as a sum of "spike" wave functions and let the spike wave functions develop in time, then we easily get huge or even infinite estimation errors.
Definition 1. A spike wave function is a wave function that originally at time t is zero everywhere except at a space point x. A more formal way to define a spike wave function is to set it to a Dirac delta function at the spacetime point (x, t). We use the word spike because it is shorter and more intuitive than a Dirac delta.
A spike wave function describes a particle with an arbitrarily high momentum and kinetic energy, and such a particle can easily travel over a high potential wall. The time development of an individual spike wave function would suggest that a sizeable portion of the flux travels over the wall, which is a completely wrong conclusion.
In reality, only a small portion of the flux passes through the wall because there is an almost complete destructive interference of wave functions on the other side of the wall. But to prove the destructive interference we would need to calculate with extreme precision. Since we are using a relatively rough perturbative approximation method, we do not have extreme precision available. Thus, if we use spike wave functions, we easily get huge or even infinite estimation errors.
Definition 2. A path C is the full time development of a wave function ψ over the full spacetime, or a relevant portion of spacetime. Note that a path typically is not a linear curve in 4D spacetime but rather the full development of a wave function over a spacetime zone.
Definition 3. An integral over a path C is the integral of the lagrangian density of a physical theory over a single path C.
Definition 4. A (Feynman) path integral is the "sum" of the integrals over all allowed paths C, weighed by a "functional measure" of the path. We may restrict paths C to be analytic functions. We will define the functional measure in each case that we will treat.
In the perturbative method, we typically want to solve a problem of the following type:
We want to estimate the effect of an electromagnetic field on a path integral by switching on the field in some small patch of spacetime, say, 10^-15 m in diameter and 10^-23 s in time.
Within an individual path C of the full path integral, the spacetime patch may contain virtual or real electrons or positrons.
spacetime patch
__________
/ \
| e- |
| e+ |
\ ___________/
electric field E -->
The interaction term in the QED lagrangian
Dirac adjoint(ψ) i e B_μ ψ
inside the patch will affect the path integral over the path C. Now, the important observation is that the effect depends quite smoothly on the positions of electrons and positrons that are contained in the small spacetime patch.
If we in some way "merge" several paths together to form a "bundle" of paths, then we can sum the various spike wave functions that describe the various configurations of electrons and positrons in the patch, and we obtain a relatively smooth wave function which is concentrated in the 10^-15 m patch.
Since the sum wave function is smooth, it does not contain arbitrarily high momenta p, and we can estimate its time development with a better precision than individual spike wave functions.
Using a smooth wave function might be equivalent to restricting the momentum |p| in the original Feynman integral by some relatively high cutoff, or "regularizing" it.
Conjecture 5. The Feynman integral formula is an approximation which is highly unstable under small estimation errors, while the regularization method is much more stable and is the "correct" way to estimate the path integral.
There is no need to speculate about "new physics" at high momenta or energies. The divergence of Feynman integrals is a weakness of the approximation method and does not tell anything about the QED lagrangian itself.
Our analysis above concerns the vacuum polarization divergence. We will analyze the self-energy and vertex correction divergences later. Do they have the same origin in using spike wave functions?
We need to calculate some kind of error margins for the Feynman integral method and the regularization method to prove that the regularization method really is the "correct" approximation method for the path integral.
Also, we need to check the effect of these ideas on quantum gravity. Feynman diagrams for quantum gravity are "non-renormalizable". Why is that?
There are lots of questions remaining about the QED lagrangian and Feynman diagrams:
The "virtual photon" in a Feynman diagram describes the electric field of an electron or positron and its interaction with another charged particle. What is this "virtual photon" exactly in a path C of the wave function?
How do we define the field A of an electron when the electron itself is described by the Dirac wave equation and we do not have a location for a point-like electron?
Is the field A, in a sense, generated by an "interaction" of the Dirac field to the electromagnetic field or should we take A as a "built-in feature" of the Dirac field?
Does A really have an effect on the electron itself, as Feynman claims?
Can we really model the interaction electromagnetic field -> Dirac field with the QED lagrangian?
Can we interpret a virtual electron and a virtual positron as a zero energy positronium atom which gets excited?
No comments:
Post a Comment