Bell nonlocality from twisted statistics

Aleksandra Go\v{c}anin; Dragoljub Go\v{c}anin; Ivana \DJ or\dj evi\'{c}; Jovan Potrebi\'{c}

REVIEW 2 major objections 4 minor 37 references

Bell nonlocality from twisted statistics

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A free scalar field on a noncommutative Moyal plane can violate the CHSH Bell inequality solely through the phase of twisted multiparticle statistics, reaching the Tsirelson bound when the invariant twist phase equals π.

desk verdict A clean, honest proof-of-principle for Bell violation from twisted Fock statistics, whose only real weakness is an explicitly acknowledged and still unjustified asymmetry between source and detector couplings. read the letter →

arxiv 2608.06359 v1 pith:TKYBOUFT submitted 2026-08-06 quant-ph

Ivana DJ orj̣ ević , Jovan Potrebić , Aleksandra Gočanin , Dragoljub Gočanin This is my paper · ORCID

classification quant-ph

keywords noncommutativespacetimeMoyalplanetwistedstatisticsBellnonlocalityCHSHinequalitydressedFock-spaceoperatorsquantumscalarfieldmomentum-modeentanglement

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that Bell-nonlocal correlations can be generated purely by the twisted exchange structure of multiparticle states, without interactions or modified dynamics. On the Moyal plane, the free scalar field's one-particle sector is unchanged, but the Drinfel'd twist dresses the Fock-space creation operators so two excitations carry a momentum-dependent phase. A weak classical source coupled to the dressed field, acting in two time-ordered pulses, coherently prepares one excitation in each outgoing channel; after conditioning on that two-particle sector, the phase becomes a relative controlled phase between the two wave-packet qubits. With suitable local mode measurements the CHSH parameter reaches $S_\theta^{\max}=2\sqrt{1+\sin^2(\chi_\theta/2)}$, which exceeds the local bound of 2 whenever $\chi_\theta\not\equiv0 \pmod{2\pi}$. This gives an operational, Bell-test signature of noncommutative spacetime structure that lives entirely in the multiparticle sector.

What carries the argument

The machinery is the Drinfel'd twist $\mathcal F_\theta = \exp(-\tfrac i2 \theta^{\mu\nu}\partial_\mu\otimes\partial_\nu)$ and its Fock-space realization, the dressing transformation $d^\dagger_p = b^\dagger_p \exp(\tfrac i2 p\wedge P)$ with $p\wedge q=p_\mu\theta^{\mu\nu}q_\nu$. Since the dressing contains the total momentum operator $P$, creating a second excitation on a state carrying momentum $p$ multiplies the two-particle amplitude by $e^{ip\wedge q/2}$. The key invariant is the combination $\chi=\tfrac12(p_0-p_1)\wedge(q_0-q_1)$, the part of the four pairwise twist phases that survives independent local rephasings; it acts as a controlled phase between the two momentum-mode qubits, and its magnitude sets the concurrence $C_\theta=|\sin(\chi/2)|$.

What would settle it

A decisive check is to recompute the second-order preparation amplitude with the interaction $J(x)\hat\phi_0(x)$ using the bare field: the factor $e^{ip\wedge q/2}$ in Eq. (37) would not appear, the state (50) would reduce to the product state (48), and the maximal CHSH value would be exactly 2 for every momentum configuration. Thus any derivation producing $S>2$ from the bare coupling would falsify the mechanism; equivalently, an experiment with collinear momentum splittings ($\Delta p\parallel\Delta q$) should show no violation at all.

Watch

Extended reading notes

Core claim

The central claim is that the state prepared by a linear source coupled to the twist-dressed field, projected onto one excitation in each outgoing channel, is locally equivalent to $|\Psi_\chi\rangle = \frac12(|00\rangle+|01\rangle+|10\rangle+e^{i\chi}|11\rangle)$, with $\chi = \tfrac12(p_0-p_1)_\mu\theta^{\mu\nu}(q_0-q_1)_\nu$. Because local rephasings of Alice's and Bob's one-particle bases cannot remove this phase, it acts as a genuine nonlocal controlled phase between two momentum-mode qubits. The maximal CHSH value for this state is $S_\theta^{\max}=2\sqrt{1+\sin^2(\chi/2)}$, so any $\chi$ that is not an integer multiple of $2\pi$ violates the local-realistic bound, and $\chi=\pi$ reaches the Tsirelson value $2\sqrt2$. The entanglement is created at the preparation stage by the ordered action $d^\dagger_{p}d^\dagger_{q}|0\rangle = e^{ip\wedge q/2}b^\dagger_{p}b^\dagger_{q}|0\rangle$, not by the free-field Hamiltonian.

Load-bearing premise

The load-bearing premise is that the source couples to the twist-dressed field $\hat\phi_\theta$ while Alice's and Bob's local mode bases stay undeformed; if the source coupled to the bare field instead, or if the measurement basis were twist-deformed, the phase $\chi$ would become a removable global phase and the state would be a product state.

Editorial extensions

If this is right

A free, non-interacting scalar field on the Moyal plane can still produce Bell-nonlocal correlations, because the source-preparation sector, not the dynamics, supplies the entanglement.
The maximum $S_\theta^{\max}=2\sqrt2$ is attained for $\chi=\pi$ mod $2\pi$, so twisted statistics alone can saturate the Tsirelson bound.
Weak noncommutativity gives an excess above the local bound that is quadratic in $\chi_\theta$: $S_\theta^{\max}=2+\chi_\theta^2/4+O(\theta^4)$, while the concurrence grows only linearly.
No violation occurs if Alice's and Bob's momentum splittings are collinear or if all modes lie in one momentum-space dimension, because then $\chi_\theta=0$; the effect is governed by the oriented area $\Delta p\times\Delta q$ in the noncommutative plane.
The Bell test requires only standard which-mode readouts and balanced mode mixers on each side; no twist-deformed detector calibration is needed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If twisted statistics operate in the same way for fermions, an analogous one-excitation-per-channel source would generate a similar invariant phase; since $\tau_\theta^2=1$, this is still ordinary braided statistics, not anyonic, so the violation would remain a two-particle exchange effect.
One could invert the formula $S=2\sqrt{1+\sin^2(\theta k\ell)}$ and use a measured CHSH value to bound the noncommutativity scale $\ell_{NC}$, provided the source-dressing asymmetry is realized and the momentum geometry is known.
The paper's own conclusion flags the decisive open question: whether the preparation/measurement asymmetry is dynamically realizable; if actual detectors operate in a co-deformed basis, $\chi$ may become a removable global phase and the predicted violation could disappear.
Because the relevant quantity is an oriented momentum-space area, matter-wave or interferometric setups that use two non-collinear momentum modes per side would be natural candidates for a laboratory test.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this is a careful, well-written paper that does exactly what it claims, with the central caveat stated openly in the text. If you take the twisted Fock dressing seriously and assume the source couples to the dressed field while Alice and Bob keep their wave-packet basis undeformed, then the state they prepare is the controlled-phase state (50), the CHSH maximum is 2*sqrt(1+sin^2(chi/2)), and the momentum-space area criterion follows directly. The algebra checks out - I went through Eqs. (14)-(20), (37), (47)-(50), (63), and Appendix B; the two-qubit CHSH optimization is standard Horodecki and correctly applied. The specific construction - a source-induced entangled state from the twist dressing, with a closed-form CHSH expression and the geometric interpretation in Sec. IV.C - is new in the literature, even though the ingredients (twisted Fock exchange, Horodecki bound) are known. No data fitting, no self-citation, no hidden circularity: the Bell violation is a genuine prediction conditional on the model.

The soft spot is exactly the one the stress-test note identifies, and I agree it is load-bearing. The phase chi appears because the source sees phi_theta while the detectors see the bare modes. If the source coupled to the bare field, the state would be a product. If the measurement basis were twist-deformed as well, the state would again factor. The authors say this in Sec. V: the preparation/measurement asymmetry is 'the central modeling assumption that remains to be investigated microscopically.' They do not pretend to have solved it. That is honest, but it means the headline claim is not yet a complete physical prediction; it is a proof-of-principle conditional on a noncovariant split. A referee should push on this, but the authors have already flagged the right question.

Two minor points: the postselection onto the one-excitation-per-channel sector is discussed but the detection loophole is not analyzed, and the narrow-wave-packet approximation (45) is reasonable but could use error estimates. Neither undermines the main derivation.

Who gets value: anyone working on noncommutative field theory phenomenology, twisted statistics, or Bell inequalities in QFT. This is a solid contribution that deserves a serious referee, even though my own verdict is conditional rather than unconditional. Recommend peer review with the clear request that the authors either strengthen the microscopic justification for the asymmetry or state more prominently that the effect disappears if the measurement basis is also dressed.

Referee Report

2 major / 4 minor

Summary. The paper studies a free real scalar QFT on the Moyal plane with twisted statistics, implemented by dressed creation and annihilation operators. A classical source is assumed to couple to the dressed field, and two time-ordered pulses prepare, after conditioning on one excitation in each outgoing channel, a two-qubit state with a momentum-dependent phase. Assuming the local measurement basis is undeformed, the state is claimed to be locally equivalent to an entangled two-qubit state, and explicit CHSH measurements are shown to yield S_max = 2 sqrt(1 + sin^2(chi/2)). The paper concludes that a Bell violation can serve as an operational probe of noncommutative spacetime structure encoded in the multiparticle sector.

Significance. If the central preparation/measurement asymmetry can be microscopically justified, the result is conceptually interesting: it shows that the multiparticle-sector deformation, rather than modified dispersion or one-particle physics, can produce Bell-nonlocal correlations, with an explicit formula and a clean geometric condition on momentum separations. The algebraic steps are transparent and the CHSH optimization in Appendix B is correct for the representative state. The authors are also commendably explicit about the status of their main hypothesis and about the postselection issue. The significance is, however, conditional: the advertised violation is fully carried by an unproven asymmetry between the source and detector couplings, and the quantitative formula (63) is derived only for balanced wave-packet amplitudes.

major comments (2)

[Eqs. (47)-(50), (63)] The reduction |Psi_theta> ~ |Psi_chi> is not valid for arbitrary alpha_i, beta_j. After removing local phases, the coefficient matrix is M_ij = |alpha_i| |beta_j| times the controlled-phase matrix, so the concurrence of the normalized state is C = 4 |alpha_0 alpha_1 beta_0 beta_1 sin(chi/2)|, not |sin(chi/2)|. Equation (63) therefore gives the maximal CHSH value only if the mode amplitudes are balanced (|alpha_0|=|alpha_1|=|beta_0|=|beta_1|=1/sqrt(2)), which is not stated anywhere in Section III. Please either impose this condition explicitly when introducing f_A and f_B, or give the general formula S_max = 2 sqrt(1 + C^2) with the correct C. The qualitative existence of a violation for chi != 0 is not affected unless one of the amplitudes vanishes, but the quantitative prediction and the weak-field expansion in Eqs. (64)-(65) are not correct for the general state written in Eq. (47).
[Secs. III and V] The central claim is conditional on the asymmetry expressed in Eq. (24) (source couples to the dressed field phi_theta) and the fixed, undeformed measurement basis in Sec. IV.A. The authors themselves state in Sec. V that this asymmetry 'remains to be investigated microscopically.' The assumption is genuinely load-bearing: if the source coupled to the bare field, Eq. (37) would contain no twist phase and the state would be the product state (48); if the detector basis were also twist-deformed, the relative phase would become a global rephasing and S_max would return to 2. The manuscript therefore does not yet establish that the proposed experiment is the operational probe announced in the title and abstract. A microscopic model of both the source-field and detector-field couplings, or a clearly stated restriction of the physical claim to the postulated split, is needed before the headline conclusion can be accepted.

minor comments (4)

[Fig. 1 caption and Sec. III] The caption below Fig. 1 appears to repeat 'O(B)_S' twice where one occurrence should be 'O(A)_S'.
[Sec. IV.A and Appendix B] The observable family in Eqs. (57)-(58) is restricted to the XZ plane of the local Bloch sphere, while the optimal observables in Appendix B are expressed in the Schmidt basis and require local unitaries U_A, U_B. Please state explicitly that arbitrary local SU(2) mode transformations are available (e.g., through passive mode mixers and phase shifters), so the reader can see that the optimal settings are reachable.
[Sec. III and Sec. IV] The Bell parameter is evaluated on the conditional two-particle sector. The authors correctly note that this should not be a setting-dependent postselection and suggest either an event-ready identification or retaining vacuum/no-detection events. It would be useful to show explicitly how the full coherent state in Eq. (33) yields a CHSH value above 2 when all sectors are retained, since the argument that the excess survives is only sketched in words.
[Sec. IV.C] The phase chi_theta is invariant under local rephasings, but its numerical value depends on the chosen frame through the constant matrix theta^{mu nu}. A sentence clarifying that the protocol is defined with respect to the fixed Moyal twist in the chosen inertial frame would help avoid confusion about Lorentz transformations.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is a self-contained conditional calculation from an explicitly stated source-coupling ansatz, with no fitted parameters and no load-bearing self-citation.

full rationale

The central result S_max^theta = 2 sqrt(1 + sin^2(chi/2)) follows from the prepared state (Eqs. 47 and 50) by the standard two-qubit CHSH maximization reproduced in Appendix B and checked against the known Horodecki result. The phase chi_theta is not fitted to any Bell data; it is computed from the twist parameter and the chosen momentum separations (Eqs. 66-70), so the calculation is predictive within the model. The dressing transformation (Eqs. 14-15) and the dressed-field source coupling (Eq. 24) are explicitly stated modeling assumptions. Section V discloses this: 'The central modeling assumption that remains to be investigated microscopically is the asymmetry between preparation and measurement...' A conditional result whose premise contains the relevant mechanism is a modeling limitation, not a circular derivation. The paper nowhere obtains the effect from a premise that already presupposes the effect, and the external theorems it invokes are standard nontrivial results (CHSH bound, Tsirelson bound, Horodecki maximal violation). References [19,24,25] provide the noncommutative twisted-statistics formalism but are not self-citations by the present authors, and the calculation does not rely on their conclusions beyond defining the dressing representation. No fitted input is relabeled as a prediction, and no uniqueness theorem is used to forbid alternatives. The skeptic's counterfactual, that coupling the source to the bare field or dressing the measurement basis would remove the phase, shows the result is assumption-dependent; that limitation is correctly acknowledged in the paper and is a correctness risk rather than circularity. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data. The only free parameter is the hypothetical theta, whose value the protocol aims to measure. The derivation leans on prior twisted-statistics formalism and on two explicit postulates about source and detector; the CHSH formula itself is standard. No new particles or forces are introduced.

free parameters (1)

theta^{mu nu} = undetermined (target of the probe)
The CHSH excess depends on chi = 1/2 (p0-p1)_mu theta^{mu nu}(q0-q1)_nu. Its value is not fitted; the protocol is intended to probe it, but it is an input hypothesis.

assumptions (6)

domain assumption Twisted statistics and the Fock dressing representation of the Moyal twist (d_p = b_p exp(-i/2 p wedge P)).
Adopted from Refs. [19,24,25] and Appendix A; used in Eqs. (14)-(20) to obtain the two-particle phase.
ad hoc to paper The source couples linearly and locally to the dressed field phi_theta, not to the bare field.
Eq. (24); explicitly called a postulate in Sec. V. The entire Bell effect depends on it.
ad hoc to paper The local measurement basis (mode mixers and detectors) is fixed, undeformed basis.
Sec. IV.A and Sec. V; if the basis is twist-deformed, chi can be removed.
domain assumption Wave packets are narrow enough to replace momentum-dependent phases by central-momentum phases.
Eq. (45); broad wave packets would smear chi.
domain assumption Postselection on the one-excitation-per-channel sector is fair or accompanied by retaining all other outcomes.
Eqs. (34)-(36); CHSH is computed on the conditional state, so the detection/fair-sampling loophole must be controlled.
standard math Horodecki's formula for the maximal CHSH value of a pure two-qubit state.
Appendix B; cited as Ref. [27].

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Cite this review

Pith. "Pith review of Bell nonlocality from twisted statistics." pith.science (2026). https://pith.science/paper/TKYBOUFT

@misc{pith2026260806359,
  author       = {Pith},
  title        = {Pith review of: Bell nonlocality from twisted statistics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TKYBOUFT}},
  note         = {Machine review of arXiv:2608.06359}
}

read the original abstract

We investigate Bell correlations for a free real quantum scalar field on the noncommutative Moyal plane. Although the free field dynamics and the one-particle sector remain unchanged, the deformation enters through twisted multiparticle statistics and its Fock-space dressing representation. A classical external source coupled locally to the twist-dressed quantum field prepares coherent superpositions of momentum-pair configurations propagating toward two spacelike-separated laboratories. The momentum-dependent twist phases are generally nonfactorizable and generate entanglement between the corresponding wave-packet modes. We show that suitable local mode measurements lead to a violation of the CHSH Bell inequality. The resulting correlations provide an operational probe of the noncommutative structure encoded in the multiparticle sector of the quantum field.

Figures

Figures reproduced from arXiv: 2608.06359 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Moyal twist and Poincar´ e covariance For the remainder of this Appendix, we specialize in Minkowski spacetime and adapt the twist vector fields to some globally inertial frame,X I =∂ µ, which gives us the standard Moyal twist Fθ = exp − i 2 θµν∂µ ⊗∂ ν .(A7) The associated Moyal⋆-product is f ⋆ g=fexp i 2 ← −∂ µθµν− →∂ ν g.(A8) For plane waves ep(x) =e −i...

work page
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This momentum-space interpretation also clarifies the role of the common preparation stage

The NC contribution vanishes whenever θkℓ=nπ,(75) in which caseS θ = 2. This momentum-space interpretation also clarifies the role of the common preparation stage. The source must coherently populate at least two distinct momentum modes in each outgoing sector, and the corresponding local mode separations must probe a nonvanishing com- ponent ofθ µν. The ...

work page 2025
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This is no longer true after twisting the coproduct

Twisted flip and twisted statistics For two identical particles in the commutative theory, particle exchange is implemented by the ordinary flip op- erator τ0(ϕ⊗ψ) =ψ⊗ϕ.(A28) The ordinary bosonic and fermionic projectors are Π± = 1 2 (1±τ 0).(A29) Because the primitive coproduct is co-commutative, [τ0,∆ 0(g)] = 0,(A30) ordinary symmetrization and antisymm...

work page
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Twisted oscillator algebra We now pass to the second-quantized description. Let bp andb † p denote the ordinary (bare) bosonic annihilation and creation operators, satisfying [bp, b† q] = (2π)3 2Ep δ(3)(p−q),(A37) together with [bp, bq] = 0,[b † p, b† q] = 0.(A38) The twisted (dressed) bosonic creation and annihila- tion operators satisfy d† pd† q =e ip∧q...

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(A39)–(A41)

Dressing transformation The twisted oscillator algebra can be represented on the ordinary bosonic Fock space through the dressing transformation dp =b p exp − i 2 p∧P ,(A42) d† p =b † p exp i 2 p∧P ,(A43) where p∧P:=p µθµνPν,(A44) and Pµ = Z dµ(k)k µ b† kbk (A45) is the total momentum operator, with dµ(k) = d3k (2π)3 2Ek , E k = p k2 +m 2.(A46) Using [Pµ,...

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The dressed scalar field The free real scalar field with twisted statistics is ex- panded as ˆϕθ(x) = Z dµ(p) dpe−ip·x +d † peip·x .(A54) The corresponding ordinary free scalar field is ˆϕ0(x) = Z dµ(p) bpe−ip·x +b † peip·x .(A55) Substituting the dressing transformation into Eq. (A54) gives ˆϕθ(x) = ˆϕ0(x) exp 1 2 ← −∂ µθµνPν .(A56) The derivative in thi...

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Relation to the Bell protocol The formal relation directly relevant for the Bell pro- tocol is Eq. (A51). If a source coherently prepares several momentum-pair configurations, the two-particle state has the form |Ψθ⟩ ∝ X i,j Aij d† pi d† qj |0⟩.(A57) In terms of the ordinary creation operators, |Ψθ⟩ ∝ X i,j Aij e i 2 pi∧qj b† pi b† qj |0⟩.(A58) For a sing...

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Moyal twist and Poincar´ e covariance For the remainder of this Appendix, we specialize in Minkowski spacetime and adapt the twist vector fields to some globally inertial frame,X I =∂ µ, which gives us the standard Moyal twist Fθ = exp − i 2 θµν∂µ ⊗∂ ν .(A7) The associated Moyal⋆-product is f ⋆ g=fexp i 2 ← −∂ µθµν− →∂ ν g.(A8) For plane waves ep(x) =e −i...

work page

[2] [2]

This momentum-space interpretation also clarifies the role of the common preparation stage

The NC contribution vanishes whenever θkℓ=nπ,(75) in which caseS θ = 2. This momentum-space interpretation also clarifies the role of the common preparation stage. The source must coherently populate at least two distinct momentum modes in each outgoing sector, and the corresponding local mode separations must probe a nonvanishing com- ponent ofθ µν. The ...

work page 2025

[3] [3]

This is no longer true after twisting the coproduct

Twisted flip and twisted statistics For two identical particles in the commutative theory, particle exchange is implemented by the ordinary flip op- erator τ0(ϕ⊗ψ) =ψ⊗ϕ.(A28) The ordinary bosonic and fermionic projectors are Π± = 1 2 (1±τ 0).(A29) Because the primitive coproduct is co-commutative, [τ0,∆ 0(g)] = 0,(A30) ordinary symmetrization and antisymm...

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[4] [4]

Twisted oscillator algebra We now pass to the second-quantized description. Let bp andb † p denote the ordinary (bare) bosonic annihilation and creation operators, satisfying [bp, b† q] = (2π)3 2Ep δ(3)(p−q),(A37) together with [bp, bq] = 0,[b † p, b† q] = 0.(A38) The twisted (dressed) bosonic creation and annihila- tion operators satisfy d† pd† q =e ip∧q...

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(A39)–(A41)

Dressing transformation The twisted oscillator algebra can be represented on the ordinary bosonic Fock space through the dressing transformation dp =b p exp − i 2 p∧P ,(A42) d† p =b † p exp i 2 p∧P ,(A43) where p∧P:=p µθµνPν,(A44) and Pµ = Z dµ(k)k µ b† kbk (A45) is the total momentum operator, with dµ(k) = d3k (2π)3 2Ek , E k = p k2 +m 2.(A46) Using [Pµ,...

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(A54) gives ˆϕθ(x) = ˆϕ0(x) exp 1 2 ← −∂ µθµνPν .(A56) The derivative in this expression acts only on the field ˆϕ0(x), whileP ν acts on the Fock space

The dressed scalar field The free real scalar field with twisted statistics is ex- panded as ˆϕθ(x) = Z dµ(p) dpe−ip·x +d † peip·x .(A54) The corresponding ordinary free scalar field is ˆϕ0(x) = Z dµ(p) bpe−ip·x +b † peip·x .(A55) Substituting the dressing transformation into Eq. (A54) gives ˆϕθ(x) = ˆϕ0(x) exp 1 2 ← −∂ µθµνPν .(A56) The derivative in thi...

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Relation to the Bell protocol The formal relation directly relevant for the Bell pro- tocol is Eq. (A51). If a source coherently prepares several momentum-pair configurations, the two-particle state has the form |Ψθ⟩ ∝ X i,j Aij d† pi d† qj |0⟩.(A57) In terms of the ordinary creation operators, |Ψθ⟩ ∝ X i,j Aij e i 2 pi∧qj b† pi b† qj |0⟩.(A58) For a sing...

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J. S. Bell, Physics Physique Fizika1, 195 (1964)

1964