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- TL;DR Summary
- I try to clarify some misunderstandings about the general structure of relativistic QFT. Particularly the important defining property of "locality"

In a recent thread about Bell tests etc. it has been claimed my point of view of "locality" where "non-mainstream physics". Of course, I cannot give a complete summary of the foundations in a forum posting, as demanded by @DrChinese there. The most clear treatment, particularly emphasizing the pivotal role the principle of "microcausility" and thus "locality" plays in the QFT formulation of relativistic QT, can be found in

S. Weinberg, The Quantum Theory of Fields, vol. 1, Cam. Uni. Press (1995)

The first step in the construction of a relativistic QT is the analysis of the unitary (ray) representations of the underlying spacetime symmetry, i.e., the proper orthochronous Poincare group, which is a semidirect product of space-time translations and proper orthochronous Lorentz transformations. Instead of this group, its covering group is relevant in QT, i.e., the proper orthochronous Lorentz group is substituted by its covering group, ##\text{SL}(2,\mathbb{C})##. There are non non-trivial central extensions (in contradistinction to the analog case of the Galiei group in non-relativistic QT), i.e., one can concentrate on the unitary irreducible representations.

As it turns out the corresponding analysis (Wigner 1939) leads to two big classes of representations, i.e., the "massive and massless representations", that admit a causal description of the quantum dynamics. The single-particle states (i.e., free-particle states) are characterized by the mass Casimir operators ##m^2 \geq 0## and "spin" ##s##. For massive representations ##s \in \{0,1/2,1,\ldots \}## characterizes the irreducible representation of the rotation group for the "particles at rest", i.e., ##\vec{p}=0## (and since ##p \cdot p=m^2##, ##p^0 \in \{\pm m \}##). For massless representations there is only helicity, i.e., except for ##s=0##, where there is no polarization degree of freedom, there are only two polarization states, ##h=\pm s##, where the helicity is the component of the total angular momentum of the particle in direction of its momentum, and ##p^0 \in \{\pm |\vec{p}| \}##.

As it further turns out, so far the only successful realization of relativistic QT is in terms of local (sic!) relativistic quantum field theories (QFTs), i.e., one realizes the unitary irreps. of the Poincare group, described above with local field operators ##\hat{\psi}_{\sigma}(x)##, which transform according to a finite-dimensional representation of the proper orthochronous Lorentz goup characterized by two numbers ##(k,k')## with ##k,k' \in \{0,1/2,1,\ldots \}##. This representation then contains all the representations of the rotations with ##j \in \{|k-k'|,|k-k'|+1,\ldots k+k' \}##, and the field equations of motion thus must "project out" all the "unwanted" representations except the representation ##j## of the rotation group one likes to describe a particle with this spin, ##j##. This leads to the usual well-known field equations like Klein-Gordon, Dirac (where one puts together ##(1/2,0) \oplus (0,1/2)## to also represent space reflections in addition to the proper orthochronous Lorentz transformations), and massless spin-1 particles (which are necessarily "gauge fields" in order to have only two helicity polarization degrees of freedom rather than some continuos polarization degrees of freedom corresponding to more general representations of the socalled "little group" of the massless representations, which is ISO(2) rather than SO(3) in the massive case).

Then to implement interactions one realizes that to get a well-defined unitary scattering matrix, that is Poincare covariant and fulfilles the cluster-decomposition principle one has to implement the microcausality constraint, i.e., the Hamilton density ##\hat{\mathcal{H}}(x)## must commute with any local observable ##\hat{mathcal{O}}(y)## at space-like distances of their arguments:

$$[\hat{\mathcal{H}}(x),\hat{\mathcal{O}}(y)]=0 \quad \text{for} \quad (x-y)^2<0.$$

It turns out that the local fields corresponding to spin ##s## must be quantized as fermions (bosons) if ##s## is half-integer (integer), which is the famous spin-statistics theorem. In addition for these "locality" properties to hold each field is to be composed of the two possible representations for given ##m^2>0## with ##p^0>0## and ##p^0<0##, i.e., for each "particle state" there's also a correspondin "anti-particle state", both with positive energy ##E=\sqrt{\vec{p}^2+m^2}>0## (by using the "Feynman Stueckelberg trick" to have the ##p^0>0## modes with annihilation and the ##p^0<0## modes with creation operators in the mode decomposition of the free-field operators, which also is necessary to have fields transforming locally under Poincare transformations). Then one can show that also the CPT symmetry holds for any such (interacting) QFT, and the original goal is also achieved: One has a unitary Poincare-covariant S-matrix, fulfilling the cluster-decomposition principle, the latter being the very property demanded as "locality" in this standard sense of relativistic quantum physics.

As Weinberg stresses, it might not be the only way to build a QFT with these properties, but so far there is no example for an alternative construction, and so far these standard formulation with local QFTs describe all the known particles and their interactions with high precision.

Last but not least it also describes all the Bell tests with photons correctly, i.e., it is a way to have "locality" on the one hand, i.e., a theory, within which it is impossible to have causal effects between space-like separated events due to the microcausality constraint, but on the other hand as any QT also relativistic QFT implies the existence of entangled states and the corresponding long-range correlations described by them, which can be used for all the successful Bell tests.

Partiucularly there is no contradiction between the "locality" of the QFT in the above described sense and the possibility to prepare entangled photon pairs of before uncorrelated photon pairs via "entanglement swapping" as described in the other thread based on the work by Zeilinger et al,

Jian-Wei Pan, Dik Bouwmeester, Harald Weinfurter, and Anton Zeilinger, Experimental Entanglement Swapping: Entangling Photons That Never Interacted, PRL

The entanglement of photons 1 and 4 for the subensemble selected by a projective Bell measurement of photon 2 and 3 is due to the original preparation of the entangled pair of photons 1 and 2 as well as photons 3 and 4 (where the pairs (12) and (34) themselves are uncorrelated). Both the creation of these original entangled pairs as well as the selection process through coincidence measurements on photons (23) are explained by local interactions of the em. field with the various "devices" used to manipulate them (parametric down conversion in BBO crystals, beam splitters, mirrors, lenses, etc. etc.), i.e., by standard QED. The selection through the measurement on the pair (23) and the measurements on photons 1 and 4 to establish their entanglement for the corresponding subensemble can be space-like seperated, excluding any causal influence of the measurement event with the pair (23) on photon 1 and/or photon 4. This is also what is meant in the paper, when they say the photons 1 and 4 "never interacted with one another or which have never been dynamically coupled by any other means":

So the results of this and other experiments with entangled states is in full accordance with both "locality" of interactions and the long-ranged correlations described by the "inseparability" of far distantly observed parts of an entangled system, leading to the specific correlations contradicting the predictions of all "local realistic" theories in Bell's sense, and afaik also Bell uses "local" in this standard sense of relativistic physics, i.e., that space-like separated events cannot be causally connected.

S. Weinberg, The Quantum Theory of Fields, vol. 1, Cam. Uni. Press (1995)

The first step in the construction of a relativistic QT is the analysis of the unitary (ray) representations of the underlying spacetime symmetry, i.e., the proper orthochronous Poincare group, which is a semidirect product of space-time translations and proper orthochronous Lorentz transformations. Instead of this group, its covering group is relevant in QT, i.e., the proper orthochronous Lorentz group is substituted by its covering group, ##\text{SL}(2,\mathbb{C})##. There are non non-trivial central extensions (in contradistinction to the analog case of the Galiei group in non-relativistic QT), i.e., one can concentrate on the unitary irreducible representations.

As it turns out the corresponding analysis (Wigner 1939) leads to two big classes of representations, i.e., the "massive and massless representations", that admit a causal description of the quantum dynamics. The single-particle states (i.e., free-particle states) are characterized by the mass Casimir operators ##m^2 \geq 0## and "spin" ##s##. For massive representations ##s \in \{0,1/2,1,\ldots \}## characterizes the irreducible representation of the rotation group for the "particles at rest", i.e., ##\vec{p}=0## (and since ##p \cdot p=m^2##, ##p^0 \in \{\pm m \}##). For massless representations there is only helicity, i.e., except for ##s=0##, where there is no polarization degree of freedom, there are only two polarization states, ##h=\pm s##, where the helicity is the component of the total angular momentum of the particle in direction of its momentum, and ##p^0 \in \{\pm |\vec{p}| \}##.

As it further turns out, so far the only successful realization of relativistic QT is in terms of local (sic!) relativistic quantum field theories (QFTs), i.e., one realizes the unitary irreps. of the Poincare group, described above with local field operators ##\hat{\psi}_{\sigma}(x)##, which transform according to a finite-dimensional representation of the proper orthochronous Lorentz goup characterized by two numbers ##(k,k')## with ##k,k' \in \{0,1/2,1,\ldots \}##. This representation then contains all the representations of the rotations with ##j \in \{|k-k'|,|k-k'|+1,\ldots k+k' \}##, and the field equations of motion thus must "project out" all the "unwanted" representations except the representation ##j## of the rotation group one likes to describe a particle with this spin, ##j##. This leads to the usual well-known field equations like Klein-Gordon, Dirac (where one puts together ##(1/2,0) \oplus (0,1/2)## to also represent space reflections in addition to the proper orthochronous Lorentz transformations), and massless spin-1 particles (which are necessarily "gauge fields" in order to have only two helicity polarization degrees of freedom rather than some continuos polarization degrees of freedom corresponding to more general representations of the socalled "little group" of the massless representations, which is ISO(2) rather than SO(3) in the massive case).

Then to implement interactions one realizes that to get a well-defined unitary scattering matrix, that is Poincare covariant and fulfilles the cluster-decomposition principle one has to implement the microcausality constraint, i.e., the Hamilton density ##\hat{\mathcal{H}}(x)## must commute with any local observable ##\hat{mathcal{O}}(y)## at space-like distances of their arguments:

$$[\hat{\mathcal{H}}(x),\hat{\mathcal{O}}(y)]=0 \quad \text{for} \quad (x-y)^2<0.$$

It turns out that the local fields corresponding to spin ##s## must be quantized as fermions (bosons) if ##s## is half-integer (integer), which is the famous spin-statistics theorem. In addition for these "locality" properties to hold each field is to be composed of the two possible representations for given ##m^2>0## with ##p^0>0## and ##p^0<0##, i.e., for each "particle state" there's also a correspondin "anti-particle state", both with positive energy ##E=\sqrt{\vec{p}^2+m^2}>0## (by using the "Feynman Stueckelberg trick" to have the ##p^0>0## modes with annihilation and the ##p^0<0## modes with creation operators in the mode decomposition of the free-field operators, which also is necessary to have fields transforming locally under Poincare transformations). Then one can show that also the CPT symmetry holds for any such (interacting) QFT, and the original goal is also achieved: One has a unitary Poincare-covariant S-matrix, fulfilling the cluster-decomposition principle, the latter being the very property demanded as "locality" in this standard sense of relativistic quantum physics.

As Weinberg stresses, it might not be the only way to build a QFT with these properties, but so far there is no example for an alternative construction, and so far these standard formulation with local QFTs describe all the known particles and their interactions with high precision.

Last but not least it also describes all the Bell tests with photons correctly, i.e., it is a way to have "locality" on the one hand, i.e., a theory, within which it is impossible to have causal effects between space-like separated events due to the microcausality constraint, but on the other hand as any QT also relativistic QFT implies the existence of entangled states and the corresponding long-range correlations described by them, which can be used for all the successful Bell tests.

Partiucularly there is no contradiction between the "locality" of the QFT in the above described sense and the possibility to prepare entangled photon pairs of before uncorrelated photon pairs via "entanglement swapping" as described in the other thread based on the work by Zeilinger et al,

Jian-Wei Pan, Dik Bouwmeester, Harald Weinfurter, and Anton Zeilinger, Experimental Entanglement Swapping: Entangling Photons That Never Interacted, PRL

**80**, 3891 (1998)The entanglement of photons 1 and 4 for the subensemble selected by a projective Bell measurement of photon 2 and 3 is due to the original preparation of the entangled pair of photons 1 and 2 as well as photons 3 and 4 (where the pairs (12) and (34) themselves are uncorrelated). Both the creation of these original entangled pairs as well as the selection process through coincidence measurements on photons (23) are explained by local interactions of the em. field with the various "devices" used to manipulate them (parametric down conversion in BBO crystals, beam splitters, mirrors, lenses, etc. etc.), i.e., by standard QED. The selection through the measurement on the pair (23) and the measurements on photons 1 and 4 to establish their entanglement for the corresponding subensemble can be space-like seperated, excluding any causal influence of the measurement event with the pair (23) on photon 1 and/or photon 4. This is also what is meant in the paper, when they say the photons 1 and 4 "never interacted with one another or which have never been dynamically coupled by any other means":

We experimentally entangle freely propagating particles that never physically interacted with one

another or which have never been dynamically coupled by any other means. This demonstrates that

quantum entanglement requires the entangled particles neither to come from a common source nor to

have interacted in the past. In our experiment we take two pairs of polarization entangled photons and

subject one photon from each pair to a Bell-state measurement. This results in projecting the other two

outgoing photons into an entangled state. [S0031-9007(98)05913-4]

So the results of this and other experiments with entangled states is in full accordance with both "locality" of interactions and the long-ranged correlations described by the "inseparability" of far distantly observed parts of an entangled system, leading to the specific correlations contradicting the predictions of all "local realistic" theories in Bell's sense, and afaik also Bell uses "local" in this standard sense of relativistic physics, i.e., that space-like separated events cannot be causally connected.

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