Justin Bond writes about the argument by Freeman Dyson.
Suppose that the perturbative Dyson series of Feynman diagrams converges for a weak repulsion between electrons. It produces nice, well behaved results.
Let the coupling constant be a small positive number
e² > 0.
We assume that we can calculate a physical quantity F(e²) with a perturbative series:
Let the series be analytic in e². We can then analytically continue the series into small negative numbers
e² < 0,
and the perturbative series still converges.
A negative value of e² means that electrons attract electrons and positrons attract positrons.
e- ------------------- e-
|
------------ e+
|
------------ e-
|
e- ------------------- e-
--> t
But then the system can tunnel into a lower energy state where we have a large number of electrons close to each other, and elsewhere in space, an equal number of positrons close to each other. The attractive potential of these gatherings is hugely negative, more than the mass-energy of the electrons and the positrons. The system has tunneled into a lower energy state where a huge number of electron-positron pairs came into existence.
In the diagram above, we have a pair born spontaneously.
Such tunneling should be visible in the perturbative series, and should cause divergences in the series?
What is the problem? Is it so that we cannot determine physical quantities with a nice analytic series?
Collapse into a black hole spoils quantum mechanics?
Let us model any physical system with quantum mechanics. There is always a chance that the system will tunnel into a black hole. Is quantum mechanics useless then?
No. The probability of tunneling is negligible in, say, 1 hour.
Feynman cannot handle time-dependent phenomena?
In the Feynman approach, we work in "momentum space", and ignore the time and position coordinates. If the tunneling is time-dependent, then the Feynman approach cannot handle it. What about regularized and renormalized Feynman integrals? We know that they cannot handle "bound" states.
Freeman Dyson argued that the Feynman perturbation series cannot converge. Is this a valid conclusion? The large groups of electrons and positrons form bound states. Can the Feynman series recognize this?
What if it does converge for e² < 0, but simply does not calculate right the collapse of the state to a huge number of electrons and positrons?
John Baez (2016) presents a simple model where a Taylor series E(β) which is supposed to find the ground state energy of a particle in this potential, does not converge:
V = x² + β x⁴,
Barry Simon proved in 1969 that the series does not converge.
Finding the ground state energy is not what a Feynman diagram does. It is not clear if the β model is relevant for Feynman diagrams.
How does a Feynman diagram handle attractive forces in QED? Bound states cannot happen
In quantum electrodynamics, the electron and the positron have an attractive force. If they come very close to each other, they are annihilated.
e+ --------------- ~~~~~~~ γ
|
e- --------------- ~~~~~~~ γ'
--> t
Suppose that an annihilation is not possible. What happens?
γ
/
/
/
e+ ----------------------------
|
e- ----------------------------
--> t
Can the pair emit a large real photon γ as bremsstrahlung? The electron and the positron move at a very slow speed, come close to each other, and emit a lot of energy in a large photon.
But this is prohibited by the Feynman rules. The rules assume that all real particles coming out from the diagram are free. They cannot be in a potential pit. The energy coming into the diagram must equal the energy coming out.
It looks like the Feynman rules prohibit the collapse suggested by Freeman Dyson. The collapse is no problem for Feynman diagrams because the collapse cannot happen in them.
The large populations of electrons and positrons suggested by Dyson cannot be created because energy would not be conserved.
Dyson's argument is not about Feynman diagrams – what does the argument then prove?
The argument seems to prove this:
- If we have an analytic, converging series which calculates correctly a physical quantity for a repulsive force, then the same series calculates something for the attractive force, but it is not the "collapse of the universe". We can say that the series miscalculates for the attractive force.
Is this a fundamental flaw? No. For the attractive force, we need another formula, if the problem is well defined at all. If the problem is not well defined, then we do not need any formula.
Specifically, it is possible that the series of Feynman diagrams does converge. Freeman Dyson did not prove that that is impossible.
Why do people care so much about the behavior of Feynman diagrams? They should not
Feynman diagrams assume that free real particles meet, and the output is another set of free particles.
1. This only covers a very limited set of physical phenomena.
2. The Feynman formulae are a crude mathematical approximation about what might happen.
3. There is no a priori reason why the crude mathematical approximation would work at all. It is surprising that it does work well in many cases.
Freeman Dyson's argument against convergence forgets item 1 above.
People claiming that "new physics" must exist at the Planck scale forget item 3.
Also people claiming that we must modify quantum mechanics, to remove divergences in Feynman integrals (e.g., string model) forget item 3.
Gravity does play a role at the Planck scale. But if we study just QED, there is no need to claim that there is new physics at the Planck scale.
"Supersymmetry" was developed in order to get certain Feynman integrals to converge. Since the divergence problem is a mathematical error, there is no need to assume that supersymmetry is true. The LHC proved that if supersymmetric particles do exist, they are hard to find.
Some people hoped that the LHC will reveal extra dimensions. Extra dimensions were added to get string models to work. And string models were supposed to solve diverging Feynman integrals. Again, we see that new physics were speculated about to fix a mathematical error. That is an almost hopeless strategy. The LHC did not find any extra dimensions.
Conclusions
Freeman Dyson did not prove that series of Feynman diagrams diverge.
Divergence in Feynman diagrams is a mathematical error. This explains why the LHC did not find supersymmetric particles or extra dimensions. It is a bad strategy to fix mathematical errors by modifying the physics.
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