REVIEW
On the unitarity problem in space/time noncommutative theories
High Energy Physics - Theory · Submitted 28 Jan 2002 · 12 references
Pith's verdict: The diagnosis is probably right and the second-order algebra is clean, but the abstract's 'It is shown' outruns a seven-page second-order sketch.
T0 · AI review
Why Pith says this
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Referee report
The paper
The paper claims
The unitarity violation reported for field theories on spacetime with a noncommuting time is argued to be an artifact of one calculational recipe — the modified Feynman rules — rather than a property of the theory, since Hamiltonian andYang
The reading
What carries the argument
The Moyal star product $f *_sigma g(x) = e^{i(2) partial_mu sigma^{mu nu} partial_nu} f(x) g(x)$, which replaces pointwise multiplication and is Hermitean. The load-bearing object is the star product of the retarded and advanced propagators, $i(2-2)$. In ordinary field theory the analogous product of Heaviside functions vanishes, $i(2-2)$, and that vanishing is what unitarity rests on; with a generic $i(2-2)$ the star product does not vanish, and the modified Feynman rules have no term to compensate. The Yang–Feldman construction supplies that term because the time ordering is applied to the propagators before the nonlocal product, and a second structural fact is used repeatedly: under a spacetime integral, one star product may be replaced by a pointwise product, which lets the authors move $i(2-2)$ past the free field and read off the correction to the scattering amplitude.
Extended reading notes
Core claim
The authors recompute the fish graph of a $gvarphi^3$ theory in the modified Feynman rules and split it into a planar piece, identical to ordinary field theory, and a nonplanar piece. The nonplanar piece evaluates to $i(2-2)$, where the first two terms are the ones that appear in the product of two tree graphs and the last two are extra. Those extra terms vanish only when the noncommutativity matrix $i(2-2)$ has a time- or lightlike null vector, so for generic space/time noncommutativity they do not vanish — and that, the authors say, is the entire source of the reported unitarity violation. They then show that the same graph computed in the Hamiltonian approach of Doplicher–Fredenhagen–Roberts, where time ordering is applied to the $t$ variables of each $H_I(t)$ before the star products are taken, satisfies the unitarity identity at second order. The same conclusion is reached in the Yang–Feldman approach, where the interacting field is expanded in powers of the coupling and solved with the retarded propagator: the nonplanar part there equals $i(2-2)$, and the term $i(2-2)$ that was missing in the modified Feynman rules is exactly what makes the second-order identity hold.
Load-bearing premise
The step from 'the interacting field is Hermitean' to 'the S-matrix is unitary' assumes the Yang–Feldman field has well-defined asymptotic free limits, and the paper itself defers the question of asymptotic completeness to later work.
Editorial extensions
If this is right
- The no-go result for space/time noncommutative theories becomes a statement about the Filk-type Feynman rules rather than about the theories themselves, so the class of spacetimes with noncommuting time is reopened for unitary model building.
- For noncommutativity with a time- or lightlike null vector, all three formulations agree and the ordinary unitarity identity $i(2-2)$ is recovered, matching the earlier lightlike case.
- In the Yang–Feldman scheme the scattering amplitude of ordinary field theory is modified by exactly the term $i(2-2)$, which is the term that enforces unitarity rather than breaking it.
- The Hamiltonian formulation gives a formally self-adjoint $H_I(t)$ and hence a formally unitary perturbative expansion, at the price of explicit breaking of Lorentz invariance.
- Any calculation that treats the time-ordering step functions as if they commuted with the star product will misreport the unitarity properties of these theories.
Reading between the lines
- If the diagnosis is right, the same bookkeeping error should be visible in other nonlocal settings — theories with a fundamental length, or with a nonlocal kinetic term — wherever time ordering is folded into a nonlocal product instead of being applied outside it.
- The paper's two alternative formulations are not shown to agree with each other beyond second order; checking the third-order $gvarphi^3$ graph in both would be a sharp internal consistency test.
- Because the extra term is a commutator-like object built from retarded and advanced propagators, it may have a natural interpretation as a causal-dissipation or noise term, which would connect the unitarity question to the fluctuation–dissipation structure of nonlocal theories.
- The authors' remark that the value of $i(2-2)$ is not fixed but should be integrated over the set of compatible eigenvalues suggests that unitarity here may hold only after that averaging, a point the present note explicitly sets aside.
Original abstract
It is shown that the violation of unitarity observed in space/time noncommutative field theories is due to an improper definition of quantum field theory on noncommutative spacetime.
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.
The abstract, sentence by sentence
It is shown that the violation of unitarity observed in space/time noncommutative field theories is due to an improper definition of quantum field theory on noncommutative spacetime.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | It is shown that the violation of unitarity observed in space/time noncommutative field theories is due to an improper definition of quantum field theory on noncommutative spacetime. | result 0.83 | 0.77 | 0.92 | asserted 1.00 |
What was examined, and what was not
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References
How the literature uses it
35 arXiv papers cite this in 54 sentences
Papers citing it · 41 sentences without a confident label
Uses its method or tool 4 papers
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As an approach that does not share this problem, quantization in the Yang-Feldman formalism was proposed in [ref] , and this is the way we proceed here.
Divergences in QFT on the Noncommutative Minkowski Space with Grosse-Wulkenhaar potential · Introduction · 0.96
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Using [ref] [equation] where $Δ_-(x) = Δ_+(-x)$, and the well-known expression for the square of the Feynman propagator (see, e.g., [ref] ), we obtain, after renormalization, in the limit $k^2 →0$, [equation] Here $μ$ is some mass scale that depends on the renormalization condition, and $ε$ is the sign function.
Noncommutative (supersymmetric) electrodynamics in the Yang-Feldman formalism · The planar part · 0.90
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For the implementation of interactions the Yang-Feldman approach [ref] may prove useful.
QFT on homothetic Killing twist deformed curved spacetimes · Conclusions and outlook · 0.88
Cites it as support 3 papers
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From a technical point of view, it is noteworthy that the $T$ time-ordering prescription of [ref] , while preserving unitarity, violates the positive-energy prescription: “negative-energy” particle in the sense of Feynman can propagate in the forward time direction [ref] , thus forbidding the Wick rotation to Euclidean theory (upon the rotation, the contour of integration cuts through the poles).
Noncommutative Time in Quantum Field Theory · Comments · 0.98
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Every attempt to define interactions on this model entailed the breakdown of covariance under Lorentz boosts at some level [ref] .
Twisted Covariance as a Non Invariant Restriction of the Fully Covariant DFR Model · Conclusions · 0.97
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Indeed, it was shown in [ref] that, with the proper treatment of time ordering, the optical theorem for the “fish” graph holds true, and there is no violation of unitarity even if time and space do not commute.
Quantum Spacetime: a Disambiguation · Formal unitarity is not violated! · 0.98
Cites it as background 5 papers
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Different attempts are performed to cure space-time NC-QED [ref] , that in two dimensions are by themselves interesting to study not only because they are the noncommutative counterpart of the well-known Schwinger model [ref] .
Perturbative quantization of two-dimensional space-time noncommutative QED · Concluding remarks · 0.84
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We remark that noncommutative field theories typically contain infinitely many time derivatives in their actions, which may lead to apparent violations of unitarity [ref] and causality [ref] .
Exponentially Long Evaporation of Noncommutative Black Hole · Introduction · 0.84
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In the Minkowskian situation, quite a number of works have investigated how Feynman rules and operator ordering prescriptions are to be modified in order to render a perturbatively constructed interacting theory unitary, starting out from a Moyal-deformed version of a free quantum field on Minkowski spacetime [ref] .
Wick rotation for quantum field theories on degenerate Moyal space(-time) · Introduction · 0.99
Discussion
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