Undecidability of the spectral gap

Cubitt, Toby S.; Pérez-Garcı́a, David; Wolf, Michael M.

doi:10.1038/nature16059

Cited by 249 publications

(308 citation statements)

References 44 publications

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“…34 The adage Weber reports in Kronecker's name is known not to appear in any of Kronecker's writings; see Ewald [44, p. 942, note a].…”

Section: ] (Emphasis Added) Haim Gaifman Writesmentioning

confidence: 99%

“…It is worth pondering the fact that nonconstructive mathematics is routinely used in physics (see e.g., the discussions of the Hawking-Penrose singularity theorem and the CalabiYau manifolds in [95], undecidability of the spectral gap [34]), without scholars jumping to the conclusion that physical reality is somehow non-constructive. The non-standard proofs require a fair amount of general machinery to set up, but conversely, once all the machinery is up and running, the proofs become slightly shorter, and can exploit tools from (standard) In this section we will find the exact solutions for the modes of a uniform beaded string having N beads and with fixed ends.…”

Section: 104mentioning

confidence: 99%

See 1 more Smart Citation

Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others

Fletcher¹,

Hrbacek²,

Kanovei³

et al. 2017

Real Analysis Exchange

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Abstract. This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis. We highlight some applications including (1) Loeb's approach to the Lebesgue measure, (2) a radically elementary approach to the vibrating string, (3) true infinitesimal differential geometry. We explore the relation of Robinson's and related frameworks to the multiverse view as developed by Hamkins.Keywords: axiomatisations, infinitesimal, nonstandard analysis, ultraproducts, superstructure, set-theoretic foundations, multiverse, naive integers, intuitionism, soritical properties, ideal elements, protozoa. The last third of the 19th century saw (at least) two dynamic innovations: set theory and the automobile. Consider the following parable.A silvery family sedan is speeding down the highway. It enters heavy traffic. Every epsilon of the road requires a new delta of patience on APPROACHES TO ANALYSIS WITH INFINITESIMALS 3 the part of the passengers. The driver's inquisitive daughter Sarah, 1 sitting in the front passenger seat, discovers a mysterious switch in the glove compartment. With a click, the sedan spreads wings and lifts off above the highway congestion in an infinitesimal instant. Soon it is a mere silvery speck in an infinite expanse of the sky. A short while later it lands safely on the front lawn of the family's home.Sarah's cousin Georg 2 refuses to believe the story: true Sarah's father is an NSA man, but everybody knows that Karl Benz's 1886 PatentMotorwagen had no wing option! At a less parabolic level, some mathematicians feel that, on the one hand, "It is quite easy to make mistakes with infinitesimals, etc." (Quinn [137, p. 31]) while, as if by contrast, "Modern definitions are completely selfcontained, and the only properties that can be ascribed to an object are those that can be rigorously deduced from the definition. . . Definitions that are modern in this sense were developed in the late 1800s." (ibid., p. 32).We will have opportunity to return to Sarah, cousin Georg, switches, and the heroic "late 1800s" in the sequel; see in particular Section 7.4. IntroductionThe 2.1. Audience. The text presupposes some curiosity about infinitesimals in general and Robinson's framework in particular. While the text is addressed to a somewhat informed audience not limited to specialists in the field, an elementary introduction is provided in Section 3.Professional mathematicians and logicians curious about Robinson's framework, as well as mathematically informed philosophers are one possible (proper) subset of the intended audience, as are physicists, engineers, and economists who seem to have few inhibitions about using terms like infinitesimal and infinite number. , and (2) the Intended Interpretation hypothesis (II), entailing an identification of a standard N in its set-theoretic context, on the one hand, with ord...

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“…34 The adage Weber reports in Kronecker's name is known not to appear in any of Kronecker's writings; see Ewald [44, p. 942, note a].…”

Section: ] (Emphasis Added) Haim Gaifman Writesmentioning

confidence: 99%

Section: 104mentioning

confidence: 99%

Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others

Fletcher¹,

Hrbacek²,

Kanovei³

et al. 2017

Real Analysis Exchange

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show abstract

“…In [106,107], the authors explicitly claim that "for any consistent, recursive axiomatisation of mathematics, there exist specific Hamiltonians for which the presence or absence of a spectral gap is independent of the axioms". The same philosophy is developed in [108,109].…”

Section: Gödel-cohen Incompletenessmentioning

confidence: 99%

Quantum Mechanics, vacuum, particles, Gödel-Cohen incompleteness and the Universe

Gonzalez‐Mestres

2017

EPJ Web Conf.

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Abstract. Are the standard laws of Physics really fundamental principles? Does the physical vacuum have a more primordial internal structure? Are quarks, leptons, gauge bosons... ultimate elementary objects? These three basic questions are actually closely related. If the deep vacuum structure and dynamics turn out to be less trivial than usually depicted, the conventional "elementary" particles will most likely be excitations of such a vacuum dynamics that remains by now unknown. We then expect relativity and quantum mechanics to be low-energy limits of a more fundamental dynamical pattern that generates them at a deeper level. It may even happen that vacuum drives the expansion of the Universe from its own inner dynamics. Inside such a vacuum structure, the speed of light would not be the critical speed for vacuum constituents and propagating signals. The natural scenario would be the superbradyon (superluminal preon) pattern we postulated in 1995, with a new critical speed c s much larger than the speed of light c just as c is much larger than the speed of sound. Superbradyons are assumed to be the bradyons of a super-relativity associated to c s (a Lorentz invariance with c s as the critical speed). Similarly, the standard relativistic space-time with four real coordinates would not necessarily hold beyond low-energy and comparatively local distance scales. Instead, the spinorial space-time (SST) with two complex coordinates we introduced in 1996-97 may be the suitable one to describe the internal structure of vacuum and standard "elementary" particles and, simultaneously, Cosmology at very large distance scales. If the constituents of the preonic vacuum are superluminal, quantum entanglement appears as a natural property provided c s c . The value of c s can even be possibly found experimentally by studying entanglement at large distances. It is not excluded that preonic constituents of vacuum can exist in our Universe as free particles ("free" superbradyons), in which case we expect them to be weakly coupled to standard matter. If a preonic vacuum is actually leading the basic dynamics of Particle Physics and Cosmology, and standard particles are vacuum excitations, the Gödel-Cohen incompleteness will apply to vacuum dynamics whereas the conventional laws of physics will actually be approximate and have error bars. We discuss here the possible role of the superbradyonic vacuum and of the SST in generating Quantum Mechanics, as well as the implications of such a dynamical origin of the conventional laws of Physics and possible evidences in experiments and observations. Black holes, gravitational waves, possible "free" superbradyons or preonic waves, unconventional vacuum radiation... are considered from this point of view paying particular attention to LIGO, VIRGO and CERN experiments. This lecture is dedicated to the memory of John Bell

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“…Specifically, if is the error allowed, we will focus on the scaling of error with rather than n. In other settings, such as infinite translationally invariant Hamiltonians, it is possible for the complexity to grow rapidly with 1/ even for fixed local dimension [6]. Another example closer to the current work is [7], which showed that approximating quantum interactive proofs to high accuracy (specifically with the bits of precision polynomial in the message dimension) corresponds to the complexity class EXP rather than PSPACE.…”

Section: Introductionmentioning

confidence: 99%

An Improved Semidefinite Programming Hierarchy for Testing Entanglement

Harrow

Natarajan

2017

Commun. Math. Phys.

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We present a stronger version of the Doherty-Parrilo-Spedalieri (DPS) hierarchy of approximations for the set of separable states. Unlike DPS, our hierarchy converges exactly at a finite number of rounds for any fixed input dimension. This yields an algorithm for separability testing which is singly exponential in dimension and polylogarithmic in accuracy. Our analysis makes use of tools from algebraic geometry, but our algorithm is elementary and differs from DPS only by one simple additional collection of constraints.

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Undecidability of the spectral gap

Cited by 249 publications

References 44 publications

Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others

Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others

Quantum Mechanics, vacuum, particles, Gödel-Cohen incompleteness and the Universe

An Improved Semidefinite Programming Hierarchy for Testing Entanglement

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