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 suggest 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?

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?

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 superseleciton 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 suggest 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?

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 suggest 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?