We suspect that the sign of the probability amplitude in the Feynman vacuum polarization diagram is flipped. We have lots of open questions about the phase of the probability amplitude which various Feynman diagrams calculate.
In the link are listed the Feynman rules from Bjorken and Drell.
Let us look at the derivation of the vacuum polarization correction in the book by Hagen Kleinert (2016). It is in Section 12.16.
e+ q - p
_____
/ \
~~~~~ ~~~~~
\______/
virtual e- q + p virtual photon q
photon q
We assume that q is a pure spatial momentum exchange, for example, in the direction of the x axis.
In the diagram, p is arbitrary 4-momentum. We must integrate over all possible values of p up to some cutoff
|p| < Λ,
where the norm |p| is the euclidean norm, not the Minkowski space norm. That is,
|(t, x, y, z)| = sqrt( t² + x² + y² + z² ).
Let us then look at how the 99 page document in the link determines the phase of the correction. Is it 1, i, -1, or -i?
The photon propagator (12.92) is
-g^μν * i / q²,
where g is the Minkowski metric. In our blog we use the "east coast" sign convention, where g is
-1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1.
In the matrix, the first coordinate is the time coordinate. Note that the square of the 4-momentum q² is the square in the Minkowski metric, not in the euclidean metric. In Feynman diagrams, one always uses the Minkowski metric, unless explicitly mentioned.
The square in the Minkowski metric, with our east coast sign convention, is
(t, x, y, z)² = -t² + x² + y² + z².
The propagator of the second photon line in the diagram brings the coefficient
-i
to the Feynman formula for the vacuum polarization diagram. In literature, the calculated correction is real. Thus, the loop has to get an imaginary value. Let us find out from where does the coefficient i come to the loop.
The imaginary value of the vacuum loop integral
Formula (12.446) contains no gamma matrices and is easy to grasp. We believe that M there is the real electron mass, and
M²
is a positive real number.
In Formulae (12.452) and (12.453) we see that the imaginary factor i comes from a contour integral around the poles of
1 / ( p² - M² )².
Does this make sense? A real-valued integral got an imaginary value.
We have suggested to put the cutoff at
|p| < |q|.
If |q| is small, the pole is not even in the integration 4-volume. The real-valued integral should stay real.
Could we define that the integral in the Feynman calculus is always made imaginary, either through a contour integral or through explicitly multiplying it by i?
The pole corresponds to a real electron. Its propagator in the Feynman calculus is infinite. If q is small, there is not enough energy to create a real electron. The role of almost real electrons needs to be studied. Are they important in vacuum polarization?
The sign flip: negative or positive contribution?
This is a harder problem than the extra i in the value of the integral.
For example, the sign of the contour integral depends on the way we travel around the poles: clockwise or counter-clockwise.
Wikipedia lists different contours: retarded, advanced, and Feynman propagators.
It is suspicious that the sign of a physical entity, the wave function, depends on an ad hoc choice of the integration contour. We need to find a more robust way to calculate the limits of the integrals close to the poles.
Finding correct phase factors for Feynman diagrams
In the Schrödinger equation, with various potentials V, the phase of a wave may change any amount. Reflection from a hard wall makes a 180 degree shift to the phase. Reflection from a soft wall may cause any phase shift.
Feynman rules determine the phase in a way which is probably too crude to be true. Multiplication by -1 or i does not bring all the possibilities.
We need to develop new rules for calculating the phase in Feynman diagrams. We can already suggest one rule:
The phase does not change over a virtual pair loop or any diagram which does not have any external input. The phase cannot change if no external factor affects the system. There is no Baron Munchausen trick.
We may think of the system as isolated if no external lines come in. In quantum mechanics we assume that the phase of a whole "system" does not change if there is no external interaction. The system may be a molecule, for example. We can perform the double slit experiment with molecules. Their phase behaves in a predictable way.
We have conservation of momentum and conservation of the speed of the center of mass. Conservation of the phase might be a similar law of nature.
Question. Can we derive conservation of the phase from other conservation laws of nature?
Why did Richard Feynman and many others think that the phase could change over a virtual pair loop? Maybe they had in mind the Dirac sea of negative energy electrons. Those electrons, if they exist, would be an external factor which would make the phase to change.
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