We claim that it does not make sense to calculate the sums of path integrals at an intermediate spacetime point. Only the path integral to the measuring apparatus matters.
In our blog post of Bell's theorem, we observed that we must not assume a wave function collapse at an intermediate stage, before the measurements are analyzed in the head of the scientist. If the collapse happens, for example, in the measuring apparatuses, then wave function information is lost, and the result is not compatible with a local version of quantum mechanics.
It is the interference pattern of wave functions which finally collapses in the head of the scientist.
John Stewart Bell proved that if we assume a collapse at two measuring apparatuses far apart from each other, then quantum mechanics is not compatible with local realism. Superluminal communication would be needed.
Albert Einstein was not satisfied with this "spooky action at a distance".
Conjecture 1. The correct quantum mechanics is a local version where the wave function only collapses inside the head of exactly one scientist.
Assuming the collapse in the heads of two scientists would require the spooky action at a distance. The collapse in just one head is a rather local thing. This may solve the problem of spooky action. Einstein's second problem "God does not play dice" remains. We need to think about the David Bohm model again.
Let us repeat our earlier Conjecture 5 in a different form here:
Conjecture 2. Quantum mechanics is defined by path integrals. It only makes sense to sum the amplitudes in the head of the one scientist. Any sum at an intermediate stage of an experiment is nonsensical.
In most cases we can calculate sums at intermediate stages, and that does help us in the mathematical computation. But there is no physical meaning in these sums.
According to Conjecture 2, the different paths do not exist in the same spacetime. There is no sense in summing them at a certain spacetime point. If we have two different planets, there is no sense in summing the temperatures at 30 degrees east, 60 degrees north, at 12 noon on the two planets.
Intermediate sums may diverge. It is no problem, according to Conjecture 2. A diverging sum of amplitudes happens in the middle of a virtual loop. One cannot meaningfully sum all waves of arbitrarily high momenta.
Claiming that Schrödinger's cat is both alive and dead in the box is talking about an intermediate stage in an experiment. Conjecture 2 says such talk is nonsense.
According to Feynman, a potential can cause an electron to scatter "back in time". That would be backward causation.
Having a loop in the path imposes an extra condition on the path: in the spirit of the vacuum atom, there must be an integer number of wavelengths in the path. It is like the future constraining the past. But another way to interpret this is that a particle and an antiparticle are emitted, and they are only allowed to annihilate when the wavelengths match. Then there would be no backward causation.
http://meta-phys-thoughts.blogspot.com/2018/10/how-to-renormalize-gravity-hawking.html
What about our claim in an earlier post (Conjecture 1 there) that all fermions will eventually scatter back in time? Surely a real electron cannot scatter back in time if we have in the experiment made sure that no positrons can enter. But can we make that sure? We need to perform measurements on the test system. Those measurements might produce positrons.
What if we have a universe where there are no positrons? Then electrons cannot scatter back in time. The wave equation of the electron would be linear then? This reminds us of the free electron. If there are no other particles around, we believe that the electron moves under a totally linear equation and cannot scatter back.
Is there a chicken and egg problem here? If we have a scientist at a location in spacetime, he may measure whether electrons tend to scatter back in time. If they do, he concludes there are a lot of positrons flying around. But if they do not? Then he concludes there are few positrons. Our own universe seems to lack antimatter.
Our conjecture that all fermions will reflect back was probably wrong. It was based on the claim that QED is nonlinear for the electron. But QED seems to be linear for the electron if there are no other particles around. This is consistent with the path integral approach. Only real particles which enter the experiment have an influence. There are no virtual particles popping out from space.
If the wave equation of the electron is linear in the absence of other particles, how can we reconcile this with the fact that the wave of the electron is coupled to the electromagnetic field, and the electromagnetic field is coupled to the electron field, making the wave equation of the electromagnetic field nonlinear? The solution is that an electron is not coupled to the electromagnetic field in the absence of other particles. This is consistent with our claim that the particles "create their own environment". There are no fields or vacuum fluctuations in spacetime which is devoid of particles.
Conjecture 3. The linearity of the wave equation of a particle depends on the existence of other particles in the neighborhood. If there are no other particles, the equation is linear, which means that the particle is not coupled to any other field.
The famous problem of the infinite energy of vacuum fluctuations would be solved with the following conjecture:
Conjecture 4. All fluctuations of fields are causally created by the real particles in the neighborhood, and the total energy of those fluctuations is the energy of the particles. There is no "vacuum energy" without particles.
A first step in proving Conjecture 4 is to show that Feynman diagrams do not require one to assume any vacuum fluctuations. But there might be "non-perturbative" effects of vacuum fluctuations. We will analyze the Casimir effect later. Robert Jaffe in his 2005 paper claimed that the effect does not have anything to do with vacuum fluctuations.
https://en.wikipedia.org/wiki/Casimir_effect
Our discussion above suggests that Feynman was wrong in speaking about electrons scattered back in time. There is no backward causation unless positrons are around, and in that case it is really not scattering back in time but annihilation.
Classical reflection from an accelerating mirror
Our blog post about a time mirror a few days back was based on a backward causation model. It seems to be wrong. If we let a laser beam reflect from an accelerating mirror, how do we explain the negative frequencies which appear in the reflected beam? It is a classical process. We should not need quantum mechanics in explaining this.
wave ---->
-----------------------|------------------
fixed point
Suppose that we have a string attached to a fixed point. We rotate the string with our hand to generate a circular right-hand wave. Since the end of a string is attached to a fixed point, we need to imagine a right-hand wave approaching from the opposite direction (reflection wave), to cancel out the circular wave at the fixed point, and keep it fixed. There is also a phase shift of 180 degrees in the reflection wave.
What if the fixed point is accelerating to the left? Then the reflection wave must have a complex waveform, to keep the fixed point fixed. It has to be a chirp, and must contain left-handed component waves. The left-handed waves are the "negative frequencies" that we have blogged about.
From where does the energy come to the negative frequencies? It has to come from the incoming wave or from the fixed point movement.
^ string tension
\
\
| ------------> string tension
fixed point
The diagram shows the situation, greatly exaggerated. The sum of forces on the fixed point pulls to the right. In the reflection, there is a small impulse on the fixed point towards the right.
If the fixed point is accelerating to the left, it will feed more and more energy into the reflected wave. It is a blueshift. What is the role of the left-hand waves that the fixed point feeds into the reflection?
One way of modeling the reflection is to assume that a pair of waves is born at the fixed point, like a pair of particles. A left-hand wave travels to the right and "annihilates" the incoming wave. The right-hand wave that is born travels to the left.
If the fixed point is accelerating, then there also are very weak left-hand waves traveling to the left, besides the righ-hand main reflection. If these left-hand waves were born with a right-hand partner, what happened to the partner? Was it reflected from the fixed point? But then it would annihilate the left-hand wave. Strange.
The spider model of pair production; is charge conserved?
wave
-----> <---- //\O/\\ spider
------------------------------------------string
Let us replace the fixed point with a spider on a string. It will rotate the string with its one leg.
Pair production by a spider. When the spider rotates the string, a right-handed wave is produced in one direction and a left-handed one to the other. This corresponds to pair production in quantum field theory.
The spider may simulate a fixed point in the string by producing a left-handed wave which exactly cancels (annihilates) the incoming right-handed wave. The right-handed wave which the spider produces is the reflected wave.
Reflection by a spider. The spider rotates the string in a way which exactly cancels the incoming wave. The partner wave which the spider produces is the reflection.
What about a reflection from an accelerating fixed point? The spider runs on the string with an accelerating speed. The spider sees the incoming wave as a chirp. A chirp contains both positive and negative frequencies, that is, both right-hand and left-hand waves. To annihilate the incoming wave, the spider must produce both right-hand and left-hand waves.
Reflection from an accelerating mirror by a spider. The spider runs and produces a chirp to both directions. One of the chirps exactly annihilates the incoming wave. The other chirp is the reflection of the incoming wave.
Question 5. Is electric charge conserved when an electron bounces from an accelerating charge?
In the accelerating spider model a purely right-hand wave is converted to a mixture of right-hand and left-hand waves.
A right-hand wave corresponds to a positive frequency, that is, an electron. The spider model suggests that we can create positrons without the accompanied electron. The flux of such positrons is extremely weak. It is like the (nonexistent) Unruh radiation.
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