Comments on microcausality, chaos, and gravitational observables

Marolf, Donald

doi:10.1088/0264-9381/32/24/245003

Cited by 21 publications

(47 citation statements)

References 68 publications

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“…While this calculation closely follows [10], there is a subtly for theories coupled to matter [52]. Therefore, we will present the calculation in complete generality before specializing to the hyperscaling violating geometries (17).…”

Section: Appendix B: Holographic Calculation Of the Butterfly Velocitymentioning

confidence: 99%

Lieb-Robinson Bound and the Butterfly Effect in Quantum Field Theories

2016

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As experiments are increasingly able to probe the quantum dynamics of systems with many degrees of freedom, it is interesting to probe fundamental bounds on the dynamics of quantum information. We elaborate on the relationship between one such bound-the Lieb-Robinson bound-and the butterfly effect in strongly coupled quantum systems. The butterfly effect implies the ballistic growth of local operators in time, which can be quantified with the "butterfly" velocity v B . Similarly, the Lieb-Robinson velocity places a state-independent ballistic upper bound on the size of time evolved operators in nonrelativistic lattice models. Here, we argue that v B is a state-dependent effective Lieb-Robinson velocity. We study the butterfly velocity in a wide variety of quantum field theories using holography and compare with freeparticle computations to understand the role of strong coupling. We find that v B remains constant or decreases with decreasing temperature. We also comment on experimental prospects and on the relationship between the butterfly velocity and signaling. DOI: 10.1103/PhysRevLett.117.091602 In relativistic systems with exact Lorentz symmetry, causality requires that spacelike-separated operators commute. In nonrelativistic systems, there is no analogous notion: a local operator Vð0Þ at the origin need not commute with another local operator Wðx; tÞ at position x at a later time t, even if the separation is much larger than the elapsed time jxj ≫ t. This can be understood by considering the Baker-Campbell-Hausdorff formula for the expansion of Wðx; tÞ ¼ e iHt WðxÞe −iHt ,where H is the Hamiltonian which is assumed to consist of bounded local terms. As long as there is some sequence of terms in H that connect the origin and point x (and absent any special cancellations), the operator Wðx; tÞ will generically fail to commute with Vð0Þ. This does not necessarily imply that the magnitude commutator ½Wðx; tÞ; Vð0Þ between distance operators must be large. A bound of Lieb and Robinson [1], along with many subsequent improvements [2][3][4], limits the size of commutators of local operators separated in space and time, even in nonrelativistic systems. In terms of the Heisenberg operator Wðx; tÞ at position x and time t and an operator V at the origin of space and time, the bound reads ∥½Wðx; tÞ; Vð0Þ∥where K 0 and ξ 0 are constants, ∥ · ∥ indicates the operator norm, and v LR is the Lieb-Robinson velocity. The growth of the commutator is controlled by v LR , which is a function of the parameters of the Hamiltonian. Hence, although operators separated by a distance x may cease to exactly commute for any t > 0, the Lieb-Robinson bound implies that their commutator cannot be Oð1Þ until t ≳ x=v LR . Thus, the Lieb-Robinson velocity provides a natural notion of a "light" cone for nonrelativistic systems. Even for relativistic systems, where causality implies that the commutator of local operators must be exactly zero for t < x (in this Letter we have set the speed of light to unity, c ¼ 1), the Lieb-Robinson cone, ...

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Section: Appendix B: Holographic Calculation Of the Butterfly Velocitymentioning

confidence: 99%

Lieb-Robinson Bound and the Butterfly Effect in Quantum Field Theories

2016

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“…Ref. [22] argues that this classical localization can be extended to the quantum context to define localized subalgebras. While this is an interesting topic for future investigation, there are some significant obstacles.…”

Section: A Relational Field Theory Observablesmentioning

confidence: 99%

Observables, gravitational dressing, and obstructions to locality and subsystems

2016

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Quantum field theory -our basic framework for describing all non-gravitational physics -conflicts with general relativity: the latter precludes the standard definition of the former's essential principle of locality, in terms of commuting local observables. We examine this conflict more carefully, by investigating implications of gauge (diffeomorphism) invariance for observables in gravity. We prove a dressing theorem, showing that any operator with nonzero Poincaré charges, and in particular any compactly-supported operator, in flat-spacetime quantum field theory must be gravitationally dressed once coupled to gravity, i.e. it must depend on the metric at arbitrarily long distances, and we put lower bounds on this nonlocal dependence. This departure from standard locality occurs in the most severe way possible: in perturbation theory about flat spacetime, at leading order in Newton's constant. The physical observables in a gravitational theory therefore do not organize themselves into local commuting subalgebras: the principle of locality must apparently be reformulated or abandoned, and in fact we lack a clear definition of the coarser and more basic notion of a quantum subsystem of the Universe. We discuss relational approaches to locality based on diffeomorphism-invariant nonlocal operators, and reinforce arguments that any such locality is state-dependent and approximate. We also find limitations to the utility of bilocal diffeomorphisminvariant operators that are considered in cosmological contexts. An appendix provides a concise review of the canonical covariant formalism for gravity, instrumental in the discussion of Poincaré charges and their associated long-range fields.

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“…), with P a polynomial, do not exist in diffeomorphism-invariant theories. Instead, one may consider so-called relational observables -see, e.g., [44][45][46][47] and [48] for a recent review containing further references -which characterise the state of one dynamical field with respect to another dynamical field, and are mildly non-local. These have been used in the cosmological context in [49,50] and have a clear geometric interpretation.…”

Section: How Do Gauge Invariant Variables Help With Quantization?mentioning

confidence: 99%

Approaches to linear local gauge-invariant observables in inflationary cosmologies

Fröb¹,

Hack²,

Khavkine³

2018

Class. Quantum Grav.

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We review and relate two recent complementary constructions of linear local gauge-invariant observables for cosmological perturbations in generic spatially flat single-field inflationary cosmologies. After briefly discussing their physical significance, we give explicit, covariant and mutually invertible transformations between the two sets of observables, thus resolving any doubts about their equivalence. In this way, we get a geometric interpretation and show the completeness of both sets of observables, while previously each of these properties was available only for one of them.

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Comments on microcausality, chaos, and gravitational observables

Cited by 21 publications

References 68 publications

Lieb-Robinson Bound and the Butterfly Effect in Quantum Field Theories

Lieb-Robinson Bound and the Butterfly Effect in Quantum Field Theories

Observables, gravitational dressing, and obstructions to locality and subsystems

Approaches to linear local gauge-invariant observables in inflationary cosmologies

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