Non locality and causal evolution in QFT

Fabrizio Buscemi; Giuseppe Compagno

REVIEW

Non locality and causal evolution in QFT

Fabrizio Buscemi, Giuseppe Compagno

Quantum Physics · Submitted 27 Feb 2006 · 31 references

Pith's verdict: A correct all-orders displacement identity for a solvable scalar-source model, with a real but fixable gap between the theorem's hypotheses and the paper's 'any good observable' claims.

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arXiv IDquant-ph/0602221 (version 1)

Pith IDWBTXEHGJ, Pith's permanent record of this paper and its review

CitationFabrizio Buscemi, Giuseppe Compagno. "Non locality and causal evolution in QFT." arXiv:quant-ph/0602221 (2006). https://pith.science/paper/WBTXEHGJ

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Claim paper

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Editor's note

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I read this paper with the stress-test note in hand. The central result holds up. Eq. (52) says that for the model, the expectation value of an analytic single-point function of the field and its first derivatives on the evolved vacuum equals the vacuum expectation of the same function with Φ shifted by the retarded response g~Δ_R. The derivation in Appendix A is a clean induction using the model's commutation relations; I do not see a gap in the algebra. This is a genuine all-orders extension of the authors' earlier second-order perturbation result and of the specific operator cases treated in refs [2,4]. That is the real contribution, and it is correct as far as it goes.

The free-field discussion in Sec. 2 is largely expository, but it makes a fair point: the nonlocal tails in Φ(x)|0⟩ and in the Feynman propagator are tied to spatially extended states and do not indicate acausal signal propagation. That reading is consistent with Hegerfeldt's theorem and with the earlier literature the authors cite.

The soft spots are real but not fatal. The abstract and Sec. 5 say 'any arbitrary good local observable,' but Eq. (40) restricts the class to formal power series in Φ, ∂_tΦ, and ∇Φ at a single point. Same-point products of these operators are UV divergent without normal ordering or point-splitting, so the identity (52) is a statement about the formal algebra, not about observables in a measurement sense. The authors acknowledge the detector-model gap in the conclusion, and the two-point result (54) covers only one special case. So the main theorem's scope should be stated more carefully, but the causal separation claim is valid for the class actually treated. Minor issues: Eq. (9) is asserted without derivation, and the vacuum-subtraction condition could be stated more prominently.

Who is this for? People working on causality in QFT, the Fermi problem, and detector models. The paper gives an exact solvable benchmark and a clear statement of what a detector model would need to add. It deserves a serious referee: the algebra should be checked once more, and the authors should revise the claims to match the hypotheses. I would not cite it in my own immediate work, but I would keep it in mind as a reference for exactly solvable scalar-source causality.

Referee report

Significance

The algebraic core of the paper is sound and useful. Appendix A gives a clean inductive proof of the displacement identity U⁻¹ΦᵐU = (Φ + gΔ̃_R)ᵐ, the retarded Green function arguments are standard, and the exactness in the coupling constant is a genuine advantage over the earlier perturbative analyses cited in the paper. The model contains no fitted quantities, so the causal structure of the result is a structural feature of the model rather than a curve fit. If the observable class is properly delimited and the ultraviolet issues in same-point products are addressed, the paper provides a clear benchmark illustration of how nonlocal vacuum terms can be separated from causal source-dependent terms in an exactly solvable QFT model. The main limitation is that the advertised scope ('any good local observable') is broader than the class actually proven.

Major objections (2)

[Abstract; Sec. 5; Eq. (40)] The abstract and the conclusions state that expectation values of 'any arbitrary good local observable' evolve causally, but the theorem in Eqs. (40)-(52) is restricted to operators that are analytic functions (formal power series) of Φ(x), ∂ₜΦ(x), and ∇Φ(x) at a single spacetime point. Observables used in realistic detection, such as couplings to local detectors [30,31], are generally not of this form, and non-analytic or extended observables are not covered. Because the causal claim is the paper's central message, the statements should be qualified to the class defined by Eq. (40), with an explicit statement that the extension to detector observables is open.
[Sec. 4, Eqs. (41), (46), (50)-(53)] The same-point products Φᵐ(x), (∂ₜΦ)ᵐ(x), and (∇Φ)ᵐ(x) are not well-defined as operators without a normal-ordering or point-splitting prescription, and their vacuum expectation values ⟨0|Φᵐ(x)|0⟩ for m ≥ 2 are ultraviolet divergent. These divergent vacuum terms also appear inside source-dependent coefficients in Eq. (41): for example, the ν = 1 term of the binomial sum is m g Δ̃_R(x−y) ⟨0|Φᵐ⁻¹(x)|0⟩, which is divergent for m ≥ 3. Thus Eq. (52) is a valid formal algebraic identity for the shifted field, but the displayed expectation values are not defined until a regularization is specified. The manuscript should state a prescription (for instance, normal ordering with respect to the free vacuum or point-splitting) and verify that the causal, source-dependent part of the subtracted expectation value is independent of that prescription.

Minor objections (4)

[Sec. 2, Eq. (9)] Equation (9) is asserted without derivation, although it is used to support the claim that the energy density has a nonlocal expectation value on the states Φ(y)|0⟩; please provide a derivation or an explicit reference. In addition, the term −2∇ₓ(∆₊(x−y)∆₊(y−x)) appears to be vector-valued while the left-hand side is a scalar, so a squared gradient or a divergence is likely intended.
[Sec. 2, Eq. (13)] In Eq. (13) both asymptotic branches are labeled 'for m ≠ 0'; the second branch should read 'for m = 0'.
[Throughout] There are several typographical errors that should be corrected: 'I-Mode na' should be 'Modena', 'attributabed' should be 'attributed', 'cab be' should be 'can be', 'Philisophy' should be 'Philosophy', and 'Princetone' should be 'Princeton'.
[Sec. 3, Figure 1] The caption of Figure 1 is vague; it should define R and R′ explicitly and explain how the separation regions relate to the causal statement made in the text.

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The paper

The paper claims

For a quantum scalar field driven by a classical source, every source-dependent expectation value of a local observable is retarded at all orders in the coupling.

The reading

This paper asks whether the nonlocal features familiar in quantum field theory—the tails of the Feynman propagator, the spreading of one-particle amplitudes—violate relativistic causality. It answers no for a model of a quantum scalar field linearly coupled to a classical source, a model that can be solved exactly. The central result is a shift identity: the expectation value of any analytic local observable built from the field and its derivatives on the source-driven state equals the vacuum expectation value of the same observable with the field shifted by the retarded solution of the classical field equation. Therefore every source-dependent part of the expectation value propagates inside the light cone; the nonlocal pieces that remain are source-independent vacuum terms. The authors conclude that apparent nonlocality in this model is not an acausal signal, while leaving the question of whether realistic detectors can observe it to a separate detector analysis.

What carries the argument

The load-bearing identity is the exact disentangling of the time-evolution operator for a linear coupling, whose Hamiltonian at different times has a c-number commutator. This lets $U(t)$ be written as a product of exponentials of the positive- and negative-frequency parts of the interaction plus c-number phases $\xi(t)$ and $\alpha(t)$. Iterating the resulting commutator $[U^{-1}(t),\Phi^m(x)]$ gives the binomial shift $(\Phi(x)+g\tilde{\Delta}_R(x-y))^m$, and the same argument applies to $\dot{\Phi}$ and $\nabla\Phi$ because their commutation with $U(t)$ yields derivatives of $\tilde{\Delta}_R$. Thus the full dynamics of all one-point observables in the allowed class is carried by the single retarded c-number function $\tilde{\Delta}_R$; no perturbative expansion in $g$ is needed.

Extended reading notes

Core claim

The paper's central claim is that in the exactly solvable model with Hamiltonian $H = H_0 + g\int d^3x\,\Phi(x)j(x)$, with $j$ a classical source localized in a finite spacetime region, the evolved state $|t\rangle=U(t)|0\rangle$ satisfies, for every “good” observable $\hat{O}$ that is an analytic function of $\Phi$, $\dot{\Phi}$, and $\nabla\Phi$ at one spacetime point, $$\langle t|\hat{O}(\Phi,\dot{\Phi},\nabla\Phi)|t\rangle = \langle 0|\hat{O}\big(\Phi+g\tilde{\$\Delta$}_R,\dot{\Phi}+g\partial_t\tilde{\$\Delta$}_R,\nabla\Phi+g\nabla\tilde{\$\Delta$}_R\big)|0\rangle,$$ where $\tilde{\Delta}_R$ is the retarded solution sourced by $j$. Since $\tilde{\Delta}_R$ vanishes outside the forward light cone of the source, all $g$-dependent terms in any such expectation value are causally retarded to all orders. The nonlocal contributions at spacelike separations are exactly the vacuum expectation values, independent of the source. The same structure holds for the two-point function: a source-induced correlation appears only where both points lie in the causal future of the source.

Load-bearing premise

The causal conclusion applies to analytic one-point functions of the field and its derivatives after vacuum subtraction; real detectors, whose couplings are typically extended or non-analytic functionals of the field, fall outside this class, and the paper itself says a detector model is needed to decide whether the residual nonlocality is observable.

Editorial extensions

If this is right

If the shift identity holds, then after vacuum subtraction every “good” local observable has source-dependent support only inside the source's forward light cone, at every order in $g$.
The tails of the Feynman propagator and the spreading of one-particle amplitudes do not by themselves demonstrate acausal signaling, because they are features of the spatially extended states that the field operator creates from the vacuum.
A source switched on in a finite region produces a two-point correlation between two spacelike-separated points only through their common causal connection to the source; no source dependence appears outside that common future.
For a pointlike source the exact expressions reduce to the earlier second-order perturbative formulas, showing those formulas to be exact for this model.
The result offers a nonperturbative benchmark for testing approximate treatments of causality in more complex matter-field models.

Reading between the lines

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

Editorial inference: the same shift argument should apply to observables depending on the field at several spacetime points, but the proof would need to insert $U^{-1}(t)U(t)$ between every pair of fields and track the ordering; the two-point case shown here is the first such step.
Editorial inference: computing the excitation probability of a detector that is locally coupled to the field would test whether the restriction to analytic single-point observables is physically relevant; a source-dependent detector response outside the light cone would mark the boundary of the paper's causal claim.
Editorial inference: the identity suggests a practical picture in which the quantum field behaves as the vacuum plus a classical retarded field, so that nonlocality belongs to the vacuum state rather than to the transmitted signal; whether this picture survives quantum sources or self-interactions is an open question.

Original abstract

Non locality appearing in QFT during the free evolution of localized field states and in the Feynman propagator function is analyzed. It is shown to be connected to the initial non local properties present at the level of quantum states and then it does not imply a violation of Einstein's causality. Then it is investigated a simple QFT system with interaction, consisting of a classical source coupled linearly to a quantum scalar field, that is exactly solved. The expression for the time evolution of the state describing the system is given. The expectation value of any arbitrary ``good'' local observable, expressed as a function of the field operator and its space and time derivatives, is obtained explicitly at all order in the field-matter coupling constant. These expectation values have a source dependent part that is shown to be always causally retarded, while the non local contributions are source independent and related to the non local properties of zero point vacuum fluctuations.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,940 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.21 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Non locality appearing in QFT during the free evolution of localized field states and in the Feynman propagator function is analyzed. It is shown to be connected to the initial non local properties present at the level of quantum states and then it does not imply a violation of Einstein's causality. Then it is investigated a simple QFT system with interaction, consisting of a classical source coupled linearly to a quantum scalar field, that is exactly solved. The expression for the time evolution of the state describing the system is given. The expectation value of any arbitrary ``good'' local observable, expressed as a function of the field operator and its space and time derivatives, is obtained explicitly at all order in the field-matter coupling constant. These expectation values have a source dependent part that is shown to be always causally retarded, while the non local contributions are source independent and related to the non local properties of zero point vacuum fluctuations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Non locality appearing in QFT during the free evolution of localized field states and in the Feynman propagator function is analyzed.	contribution 0.77	0.76	0.07	asserted 0.70
s1	It is shown to be connected to the initial non local properties present at the level of quantum states and then it does not imply a violation of Einstein's causality.	interpretation 0.75	0.86	0.77	asserted 1.00
s2	Then it is investigated a simple QFT system with interaction, consisting of a classical source coupled linearly to a quantum scalar field, that is exactly solved.	method 0.59	0.90	0.03	asserted 0.95
s3	The expression for the time evolution of the state describing the system is given.	contribution 0.46	0.90	0.03	asserted 0.97
s4	The expectation value of any arbitrary ``good'' local observable, expressed as a function of the field operator and its space and time derivatives, is obtained explicitly at all order in the field-matter coupling constant.	result 0.55	0.93	0.06	asserted 1.00
s5	These expectation values have a source dependent part that is shown to be always causally retarded, while the non local contributions are source independent and related to the non local properties of zero point vacuum fluctuations.	result 0.92	0.92	0.42	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Source forward lightcone: the effects present in separation regions between R and R’ may be local for a part of source and non local for another. Thus, taking into account that h0|Φ(x)|0i = 0 and introducing the retarded propagator ∆ret(x) = Θ(t)∆(x), the expectation value (35) takes the form: ht|Φ(x)|ti = g Z t 0 dt ′ Z d 3x ′ j(x ′ , t′ )∆ret(x − x ′ , t − t ′ ) = g∆e R(x − y). (37) In equation (37) it is t > t′ > 0, while (x − y) ≡ (x − y, t − y0) and (y, y0) indicates a spacetime point inside the spacetime region where j(x, t) 6= 0. Therefore the function ∆e R(x−y) defined by…

Figures reproduced from arXiv:quant-ph/0602221 by the authors.

References

Works cited (31)

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Antoniou I, Karpov E, and Pronko G 2001 Found. of Phys. 31 1641

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Ordonez G, Petrosky T, and Prigogine I 2000 Phys. Rev. 62 042106

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Buscemi F and Compagno G 2005 Phys. Lett. A 334 357

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Buchholz D and Yngvason J 1994 Phys. Rev. Lett. 73 613

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Milonni P V, James D F V and Fearn H 1995 Phys. Rev. A 52 1525

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Compagno G, Palma G M, Passante R, and Persico F 1995 Chem. Phys. 198 19

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Hegerfeldt G C 1994 Phys. Rev. Lett. 10 596

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Hegerfeldt G C 1974 Phys. Rev. D 10 3320

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Hegerfeldt G C and Ruijsenaars N M 1980 Phys. Rev. D 22 377

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Perez J F and Wilde I F 1977 Phys. Rev. D 16 315

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Rosenstein B and Usher M 1987 Phys. Rev. D 36 2381

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Stone M 2000 The Physics of Quantum Fields (Springer)

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Veltman M 1995 Diagrammatica (London: Cambridge, University Press)

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[20]

Karpov E, Ordonez G, Petrosky T, Prigogine I, and Pronko G 20 00 Phys. Rev. A 62 012103

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[21]

Compagno G, Passante R and Persico F 1995 Atom-ﬁeld Interaction and Dressed Atoms (London: Cambridge University Press)

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[22]

Peskin M E and Schroeder D V 1995 An introduction to Quantum Field Theory (Westview Press)

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Bjorken J D and Drell S D 1965 Relativistic Quantum Fields (Mcgraw-Hill)

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[24]

Newton T D and Wigner E P 1949 Rev. Mod. Phys. 41 400

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Fleming G and Butterﬁeld J 1998 Physics to Philisophy: Essays in Honor Of Micheal Redhead (London: Cambridge University Press )

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Wheeler J A and Zurek W H 1983 Quantum Theoy and Measurements (Princetone, NI: Princetone University Press)

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[27]

Barat N and Kimball J C 2003 Phys. Lett. A 308 110

View on Pith 2003
[28]

Biswas A K, Compagno G, Palma G M, Passante R, and Persico F 199 0 Phys. Rev. A 42 4291

View on Pith
[29]

Compagno G, Passante R, and Persico F 1990 J. Mod. Opt. 37 1337

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Glauber R J 1963 Phys. Rev. 130 2529 ——1963 Phys. Rev. 131 2766

View on Pith 1963
[31]

Unruh W G 1976 Phys. Rev. D 14 870

View on Pith 1976

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Fermi E 1932 Rev. Mod. Phys. 4 87

View on Pith 1932

[2] [2]

Antoniou I, Karpov E, and Pronko G 2001 Found. of Phys. 31 1641

View on Pith 2001

[3] [3]

Ordonez G, Petrosky T, and Prigogine I 2000 Phys. Rev. 62 042106

View on Pith 2000

[4] [4]

Maiani L and Testa M 1995 Phys. Lett. A 356 319

View on Pith 1995

[5] [5]

Power E and Thirunamachandran T 1997 Phys. Rev. A , 56 3395

View on Pith 1997

[6] [6]

Rubin M H 1987 Phys. Rev. D 35 3836

View on Pith 1987

[7] [7]

Berman P R and Dubetsky B 1997 Phys. Rev. A 55 4060

View on Pith 1997

[8] [8]

Berman P R 2004 Phys. Rev. A 69 022101

View on Pith 2004

[9] [9]

Buscemi F and Compagno G 2005 Phys. Lett. A 334 357

View on Pith 2005

[10] [10]

Buchholz D and Yngvason J 1994 Phys. Rev. Lett. 73 613

View on Pith 1994

[11] [11]

Milonni P V, James D F V and Fearn H 1995 Phys. Rev. A 52 1525

View on Pith 1995

[12] [12]

Compagno G, Palma G M, Passante R, and Persico F 1995 Chem. Phys. 198 19

View on Pith 1995

[13] [13]

Hegerfeldt G C 1994 Phys. Rev. Lett. 10 596

View on Pith 1994

[14] [14]

Hegerfeldt G C 1974 Phys. Rev. D 10 3320

View on Pith 1974

[15] [15]

Hegerfeldt G C and Ruijsenaars N M 1980 Phys. Rev. D 22 377

View on Pith 1980

[16] [16]

Perez J F and Wilde I F 1977 Phys. Rev. D 16 315

View on Pith 1977

[17] [17]

Rosenstein B and Usher M 1987 Phys. Rev. D 36 2381

View on Pith 1987

[18] [18]

Stone M 2000 The Physics of Quantum Fields (Springer)

View on Pith 2000

[19] [19]

Veltman M 1995 Diagrammatica (London: Cambridge, University Press)

View on Pith 1995

[20] [20]

Karpov E, Ordonez G, Petrosky T, Prigogine I, and Pronko G 20 00 Phys. Rev. A 62 012103

View on Pith

[21] [21]

Compagno G, Passante R and Persico F 1995 Atom-ﬁeld Interaction and Dressed Atoms (London: Cambridge University Press)

View on Pith 1995

[22] [22]

Peskin M E and Schroeder D V 1995 An introduction to Quantum Field Theory (Westview Press)

View on Pith 1995

[23] [23]

Bjorken J D and Drell S D 1965 Relativistic Quantum Fields (Mcgraw-Hill)

View on Pith 1965

[24] [24]

Newton T D and Wigner E P 1949 Rev. Mod. Phys. 41 400

View on Pith 1949

[25] [25]

Fleming G and Butterﬁeld J 1998 Physics to Philisophy: Essays in Honor Of Micheal Redhead (London: Cambridge University Press )

View on Pith 1998

[26] [26]

Wheeler J A and Zurek W H 1983 Quantum Theoy and Measurements (Princetone, NI: Princetone University Press)

View on Pith 1983

[27] [27]

Barat N and Kimball J C 2003 Phys. Lett. A 308 110

View on Pith 2003

[28] [28]

Biswas A K, Compagno G, Palma G M, Passante R, and Persico F 199 0 Phys. Rev. A 42 4291

View on Pith

[29] [29]

Compagno G, Passante R, and Persico F 1990 J. Mod. Opt. 37 1337

View on Pith 1990

[30] [30]

Glauber R J 1963 Phys. Rev. 130 2529 ——1963 Phys. Rev. 131 2766

View on Pith 1963

[31] [31]

Unruh W G 1976 Phys. Rev. D 14 870

View on Pith 1976