Born's rule for the QED electron violates causality

[A. Neumaier]

TL;DR Summary

Born's rule for the QED electron violates causality.

Summary: Born's rule for the QED electron violates causality.

[Since the thread where some of this material was presented was closed for further discussion, I summarize here the main content relevant for the above topic.]

The free QED electron can be described in terms of a non-local single-parthicle Hamiltonian, as follows.

The single electron sector of renormalized QED including infrared dressing is invariant under Poincare transformations, since in this sector there is no scattering. Its Hilbert space carries a reducible unitary representation of the Poincare group. The generator of time defines the Hamiltonian $H$. The resolvent $(E-H)^{-1}$ equals the renormalized electron propagator, and is given by the Kallen-Lehmann formula associated with some continuous mass density (due to infrared dressing effects) whose support extends from
the nominal electron mass to infinity.

The mass spectrum is nondegenerate and has a branch point at the nominal electron mass, where the continuous mass spectrum has a sharp peak. This means that the free QED electron has an additional mass degree of freedom, which formally behaves like an additional momentum degree of freedom. This mass degree of freedom generates the
continuous mass spectrum. Therefore the free QED electron is not an elementary particle in the sense of Wigner but a stable infraparticle.

• B. Schroer, Infrateilchen in der Quantenfeldtheorie, Fortschritte der Physik 11 (1963), 1-31.

(In the QM treatment of multielectronic systems, the mass degree of freedom is generally suppressed, a sensible approximation given the peaked mass spectrum. Indeed, infrared problems are not much addressed in the literature.) The details of the mass density are not completely known but the basic structure, including the branch point, of the electron propagator, is discussed in Section II of

• T. Appelquist and J. Carazzone, Infrared singularities and massive fields, Phys. Rev. D 11 (1975), 2856--2861.

See also pp.65-66 in

• D. Buchholz, The physical state space of quantum electrodynamics, Comm. Math. Phys. 85 (1982), 49--71.

Since the QED electron has a positive mass spectrum, its Hamiltonian is bounded from below. Therefore, the results of

• G.C. Hegerfeldt, Instantaneous spreading and Einstein causality in quantum theory, Annalen der Physik 7 (1998), 716-725.

apply. He showed that the positive mass spectrum together with Born's rule
''ensures either instantaneous spreading or confinement in a fixed bounded region for all times'' (quote from p.7).

In other words, a single QED electron prepared locally in an arbitrary state has - according to the Born rule, taken at face value - a nonzero probability of being immediately detected arbitrarily far away.

This violates causality.

 

[A. Neumaier]

Summary: Born's rule for the QED electron violates causality.

Note that the theoretical apparatus of relativistic QFT is about q-expectations of products of field operators (n-point functions), and causality is valid for these.

The above argument only shows that the claim for probabilities of measurement results, governed by Born's rule for the dynamically propagated wave function, violates causality.

Since expectation values say nothing at all about low probability events, the causal commutation relations have no implications for the very low probability of being immediately detected arbitrarily far away.

Thus there is no contradiction between the claim that QED is causal with respect to q-expectations and the claim in the summary. It is a defect of Born's rule and not of QED.

 

[charters]

Others have interpreted this sort of thing as a problem with the proposition that apparta can make measurements within arbitrary and strictly local spacetime volumes, strictly localizing the electron inside (or outside) that volume.

 

[A. Neumaier]

Others have interpreted this sort of thing as a problem with the proposition that apparata can make measurements within arbitrary and strictly local spacetime volumes, strictly localizing the electron inside (or outside) that volume.

Isn't this just another way of saying that causality is violated? That a local apparatus produces immediate nonlocal consequences...

 

[charters]

Isn't this just another way of saying that causality is violated? That a local apparatus producers immediate nonlocal consequences...

The idea is the apparata are nonlocal too. They are also described by an effectively/Compton localized Gaussian around some point x.

So, when an electron is emitted, spacelike separated from the emission event (which occurs at x), the local field still contains the tails of the excited-source field and the electron field vacuum. Inside the light cone of x, the excited-source field is "unzipped" into the ground-source state and the exicted state of the electron field. But this only becomes the exact, proper |1> state of the electron field in the asymptotic limit.

After sufficient wave packet spreading, the electron inside the light come can be approximately understood as having a decomposition along a Newton-Wigner-esque basis, but we avoid the NW pathologies by settling for only effective localization. See David Wallace, pgs 12 and 37 here: https://arxiv.org/abs/quant-ph/0112149

If detection is associated with detecting the centroid of such effectively localized states, there is no causality violation, as the centroids propagate causally. See Barat and Kimball here: https://arxiv.org/abs/quant-ph/0111060

Basically, the causality problem comes from a mistake in the approximation, where we pretend that we can represent the emission of a particle by applying a free theory creation operator at a local point. Since free theory particle states are global states, this is bound to be a problem when scrutinized. But I think this really isn't how particles in a world of interacting, relativistic fields are born and die.

 

[meopemuk]

In other words, a single QED electron prepared locally in an arbitrary state has - according to the Born rule, taken at face value - a nonzero probability of being immediately detected arbitrarily far away.

This violates causality.

The superluminal spreading of the wavefunction does not violate causality by itself. In order to prove the violation of causality you have to show that the moving observer will see that the electron appears "there" before it was emitted from "here."

However, if we apply a boost transformation to the localized electron state at time=0, we'll get a function delocalized over entire world. So, the moving observer will say that the electron was "there" already at time=0, so he cannot claim an unambiguous causality violation.

Eugene.

 

[A. Neumaier]

The superluminal spreading of the wavefunction does not violate causality by itself. In order to prove the violation of causality you have to show that the moving observer will see that the electron appears "there" before it was emitted from "here."

However, if we apply a boost transformation to the localized electron state at time=0, we'll get a function delocalized over entire world. So, the moving observer will say that the electron was "there" already at time=0, so he cannot claim an unambiguous causality violation.

This does not improve the case for Born's rule in the relativistic domain!

 

[A. Neumaier]

Others have interpreted this sort of thing as a problem with the proposition that apparta can make measurements within arbitrary and strictly local spacetime volumes, strictly localizing the electron inside (or outside) that volume.

If there is no local apparatus then there are also no local detection events, and by generalization no local events at all. This would remove the whole empirical basis of relativity theory...

 

[charters]

If there is no local apparatus then there are also no local detection events, and by generalization no local events at all. This would remove the whole empirical basis of relativity theory...

There are local events, but these local events just change the global configurations, describing both the apparatus and target particle, at the event site - then those changes propagate in the causal neighborhood/lightcone of the event.

 

[vanhees71]

In usual QFT one considers the S-matrix, i.e., the transition-rate amplitudes for scattering events described by asymptotic free in states (in QED one has indeed to use the formalism described in #1 to get the correct asymptotic free states to avoid IR problems) going into asymptotic free out states. Nothing is said about transient states and how to interpret them in terms of particles or otherwise.

Then in any decent quantum-optics books you find that what's describes photon detectors are the autocorrelation functions of the electromagnetic field operators, and nothing violates causality.

I don't know, why this should disprove the validity of Born's rule in relativistic QFT since it's based on Born's rule to define the transition probabilities as well as the meaning of the autocorrelation functions of field operators.

Also it's well known that a first-quantization interpretation with wave functions, no matter whether you take the problems with infraparticles into account or not doesn't work in the relativistic realm simply by the acausality such an interpretation would imply. The physical reason is that to localize a relativsitic particle in smaller and smaller regions rather leads to pair creation than to confining the particles further.

With photons as well as with massive particles "localization" means to have a detector defining the position of a particle with a finite spatial resolution. Again trying to resolve the position of a particle more precisely than some given critical resolution of the order $h/m$ should lead to pair creation rather than to gain this better resolution.

I don't know, whether this has been ever discussed in the literature, but shouldn't this resolve all these formal "causality problems"?

 

[charters]

With photons as well as with massive particles "localization" means to have a detector defining the position of a particle with a finite spatial resolution. Again trying to resolve the position of a particle more precisely than some given critical resolution of the order h/mh/mh/m should lead to pair creation rather than to gain this better resolution.

I don't know, whether this has been ever discussed in the literature, but shouldn't this resolve all these formal "causality problems"?

Pair production occurs when the state is effectively or strictly localized below the Compton scale. However, regardless of this no N particle state can be strictly localized (aka Knight localized) with compact support in *any* bounded region regardless of size - this includes strictly localizing an electron in even a 1 km^3 box where there is no meaningful pair production due to "squeezing" the particle. This is due to Hegerfeldt and Reeh Schlieder. So, the issue is if you try to prepare a strictly localized N particle state, it either picks up infinite tails instantly or it can never leave the original region of compact support.

 

[vanhees71]

Well, this just tells you once more that you cannot localize a particle. So what? There are no particles, only quantum fields!