A paradox about superselection from algebraic QFT

Question

In N.P. Landsman's review of Haag's Local Quantum Physics (p. 523), he notes the following paradox. Haag shows that in infinite volume, the velocity of an electron obeys a superselection rule, which makes quantum interference impossible. But this contradicts experimental observations! The full quote from Haag (p. 282) is:

As a consequence of this superselection rule no coherent superposition of weights $\sigma_p$ with different values of $p$ are possible, no wave packets corresponding to normalizable strict 1-election states can be formed. Yet electron interference is a salient fact which is explained in quantum mechanics by applying the superposition principle to 1-electron wave packets. Obviously this quantum mechanical idealization is good enough for the discussion of electron interference in spite of the fact that QED tells us that, strictly speaking, there are no such coherent wave packets. The seeming paradox may serve as a warning against overrating the significance of idealizations in the mathematical description of a physical situation. The reader is encouraged to work out how the quantum mechanical description of an electron interference experiment can be justified within the field theoretic setting. Here we only remark that such an experiment concerns the partial state in a finite space-time region and that the initial information we have about it is only up to some background with energy density below some threshold. The phenomenon studied must be insensitive to this ignorance.

I have two related questions.

First, how exactly do the factors noted by Haag help evade the theorem? He appears to be gesturing at QED being some kind of effective field theory that can't be trusted below certain length-scales. But I am not sure what the precise argument is.

Second, all of our experiments take place in finite space-time regions. Taken literally, Haag's argument suggests that we should never trust the results of infinite volume algebraic/axiomatic quantum field theory. My impression is that such results, like the CPT theorem, are generally accepted as physically relevant by physicists. If so, why should we trust AQFT's claims about CPT symmetry but not electrons?

Answer 1

The quote is about the ''quantum mechanical idealization'' of quantum field theory, not about quantum field theory per se.

Haag's theorem only refers to what happens in unbounded spacetime, where the same (sheaf of) local Fock description(s) gives rise to an uncountable multitude of nonequivalent states, leading to multiple superselection sectors. Only one of them is the (global) Fock sector, and it carries only states of noninteracting theories.

States cannot be distinguished experimentally if they agree on the expectation values of observables whose support is in a bounded region of spacetime, where the local Fock description is valid. This is formally encapsulated in the Haag-Kastler axioms, which give a purely local characterization of relativistic quantum field theories.

The observation of states in a bounded region by a local observer can always be simulated in the Fock sector by adding far away charged particles that have negligible influence in the region of observation but approximate arbitrarily well the global state in any given larger bounded region. The colloquial expression of this is to say that that one can add ''particles behind the moon''.

But this does not imply ''that we should never trust the results of infinite volume algebraic/axiomatic quantum field theory''. Indeed, by your reasoning (''Taken literally ...'') we should never trust any asymptotic approximation or finite truncation anywhere in physics! However, physics very often has to rely on these since exact results are comaratively rare and restricted to simple, idealized situations.

The proper consequence of the matter is that one should regard ''infinity'' and ''very large'' as very useful informal approximations of each other, while for quantitative assessments one has to quantify the errors made by such an approximation.

In particular, this is the standard way how statistical mechanics treats the thermodynamic limit to get a thermal description in terms of nonrelativistic field theory, where distinct phase also represent superselection sectors.