Loop-and-allied QG bibliography


Science Advisor
Simplicial Palatini action
V.M. Khatsymovsky
(Submitted on 18 May 2017)
We consider the simplicial complex (piecewise flat spacetime) and a simplicial analog of the Palatini form of the general relativity action where the discrete Christoffel symbols are given on the tetrahedra (3-dimensional simplexes) as variables that are independent of the metric. Excluding these variables with the help of the equations of motion gives exactly the Hilbert-Einstein action or, in the present context, Regge action. The present paper continues our previous work. Now we include the parity violation term and the analogue of the Barbero-Immirzi parameter introduced in the Cartan-Weyl representation of the Einstein action with orthogonal connection. The path integral is considered and elementary areas are shown to be fixed at some Planck scale values.

Local Lorentz covariance in finite-dimensional Local Quantum Physics
Matti Raasakka
(Submitted on 18 May 2017)
We show that local Lorentz covariance arises canonically as the group of transformations between local thermal states in the framework of Local Quantum Physics, given the following three postulates: (i) Local observable algebras are finite-dimensional. (ii) Minimal local observable algebras are isomorphic to M2(C), the observable algebra of a single qubit. (iii) The vacuum restricted to any minimal local observable algebra is thermal. The derivation reveals a new and surprising relation between spacetime structure and local quantum states. In particular, we show how local restrictions of the vacuum can determine the connection between different local inertial reference frames.

Spacetime has a `thickness'
Samir D. Mathur
(Submitted on 18 May 2017)
Suppose we assume that (a) information about a black hole is encoded in its Hawking radiation and (b) causality is not violated to leading order in gently curved spacetime. Then we argue that spacetime cannot just be described as a manifold with a shape; it must be given an additional attribute which we call `thickness'. This thickness characterizes the spread of the quantum gravity wavefunctional in superspace -- the space of all 3-geometries. Low energy particles travel on spacetime without noticing the thickness parameter, so they just see an effective manifold. Objects with energy large enough to create a horizon do notice the finite thickness; this modifies the semiclassical evolution in such a way that we avoid horizon formation and the consequent violation of causality.
Signatures of extra dimensions in gravitational waves
David Andriot, Gustavo Lucena Gómez
(Submitted on 24 Apr 2017)
Considering gravitational waves propagating on the most general 4+N-dimensional space-time, we investigate the effects due to the N extra dimensions on the four-dimensional waves. All wave equations are derived in general and discussed. On Minkowski4 times an arbitrary Ricci-flat compact manifold, we find: a massless wave with an additional polarization, the breathing mode, and extra waves with high frequencies fixed by Kaluza-Klein masses. We discuss whether these two effects could be observed.
Comments: 21 pages + appendices, comments welcome!
Subjects: High Energy Physics - Theory (hep-th); Cosmology and Nongalactic Astrophysics (astro-ph.CO); High Energy Astrophysical Phenomena (astro-ph.HE); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Phenomenology (hep-ph)
Cite as: arXiv:1704.07392 [hep-th]
(or arXiv:1704.07392v1 [hep-th] for this version)
Area Operator in Loop Quantum Gravity
Adrian P. C. Lim
(Submitted on 10 May 2017)
A hyperlink is a finite set of non-intersecting simple closed curves in R×R3. Let S be an orientable surface in R3. The dynamical variables in General Relativity are the vierbein e and a su(2)×su(2)-valued connection ω. Together with Minkowski metric, e will define a metric g on the manifold. Denote AS(e) as the area of S, for a given choice of e.
The Einstein-Hilbert action S(e,ω) is defined on e and ω. We will quantize the area of the surface S by integrating AS(e) against a holonomy operator of a hyperlink L, disjoint from S, and the exponential of the Einstein-Hilbert action, over the space of vierbeins e and su(2)×su(2)-valued connections ω. Using our earlier work done on Chern-Simons path integrals in R3, we will write this infinite dimensional path integral as the limit of a sequence of Chern-Simons integrals. Our main result shows that the area operator can be computed from a link-surface diagram between L and S. By assigning an irreducible representation of su(2)×su(2) to each component of L, the area operator gives the total net momentum impact on the surface S.
Comments: arXiv admin note: text overlap with arXiv:1701.04397, arXiv:1705.00396
Subjects: Mathematical Physics (math-ph); General Relativity and Quantum Cosmology (gr-qc); Differential Geometry (math.DG)
MSC classes: 83C45, 81S40, 81T45, 57R56
Cite as: arXiv:1705.06577 [math-ph]
Renormalizability, fundamentality and a final theory: The role of UV-completion in the search for quantum gravity
Karen Crowther, Niels Linnemann
(Submitted on 18 May 2017)
Principles are central to physical reasoning, particularly in the search for a theory of quantum gravity (QG), where novel empirical data is lacking. One principle widely adopted in the search for QG is UV completion: the idea that a theory should (formally) hold up to all possible high energies. We argue---\textit{contra} standard scientific practice---that UV-completion is poorly-motivated as a guiding principle in theory-construction, and cannot be used as a criterion of theory-justification in the search for QG. For this, we explore the reasons for expecting, or desiring, a UV-complete theory, as well as analyse how UV completion is used, and how it should be used, in various specific approaches to QG.
Subjects: History and Philosophy of Physics (physics.hist-ph); General Relativity and Quantum Cosmology (gr-qc)
Cite as: arXiv:1705.06777 [physics.hist-ph]
(or arXiv:1705.06777v1 [physics.hist-ph] for this version)
mprint of quantum gravity in the dimension and fabric of spacetime
Giovanni Amelino-Camelia, Gianluca Calcagni, Michele Ronco
(Submitted on 13 May 2017)
We here conjecture that two much-studied aspects of quantum gravity, dimensional flow and spacetime fuzziness, might be deeply connected. We illustrate the mechanism, providing first evidence in support of our conjecture, by working within the framework of multifractional theories, whose key assumption is an anomalous scaling of the spacetime dimension in the ultraviolet and a slow change of the dimension in the infrared. This sole ingredient is enough to produce a scale-dependent deformation of the integration measure with also a fuzzy spacetime structure. We also compare the multifractional correction to lengths with the types of Planckian uncertainty for distance and time measurements that was reported in studies combining quantum mechanics and general relativity heuristically. This allows us to fix two free parameters of the theory and leads, in one of the scenarios we contemplate, to a value of the ultraviolet dimension which had already found support in other quantum-gravity analyses. We also formalize a picture such that fuzziness originates from a fundamental discrete scale invariance at short scales and corresponds to a stochastic spacetime geometry, recovering the structure of Nottale scale relativity.
Comments: 5 pages
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc)
Cite as: arXiv:1705.04876 [hep-th]
(or arXiv:1705.04876v1 [hep-th] for this version
The Thiemann Complexifier in Loop Cosmology
Jibril Ben Achour, Etera R. Livine
(Submitted on 10 May 2017)
In the context of Loop Quantum Gravity (LQG), we study the fate of the complexifier, that is the generator of the canonical transformations shifting the Immirzi parameter, for the homogeneous and isotropic FRW cosmology. We focus on the closed CVH algebra for canonical general relativity consisting in the complexifier, the 3d volume and the Hamiltonian constraint. In standard cosmology, for gravity coupled to a scalar field, the CVH algebra is identified as a su(1,1) Lie algebra, with the Hamiltonian as a null generator and the complexifier as a boost. The su(1,1) Casimir is given by the matter density. In the loop gravity cosmology approach, the gravitational Hamiltonian is regularized in terms of SU(2) holonomies. In order to keep a closed CVH algebra, we show that the complexifier and inverse volume factor needs to be similarly regularized. Then the su(1,1) Casimir is given by the matter density and the volume gap. The action of the Hamiltonian constraints and the complexifier can be exactly integrated. This is straightforward to extend to the quantum level: the cosmological evolution is described in terms of SU(1,1) coherent states and the regularized complexifier generates unitary transformations. This means that, in the physical Hilbert space, the Immirzi ambiguity is to be distinguished from the volume gap, it can be rescaled unitarily and ultimately disappears from physical predictions of the theory. Finally, we show that the complexifier becomes the effective Hamiltonian when deparametrizing the dynamics using the scalar field as a clock, thus underlining the deep relation between cosmological evolution and scale transformations.
Comments: 37 pages
Subjects: General Relativity and Quantum Cosmology (gr-qc)
Cite as: arXiv:1705.03772 [gr-qc]
(or arXiv:1705.03772v1 [gr-qc] for this version)
Pre-inflationary universe in loop quantum cosmology
Tao Zhu, Anzhong Wang, Gerald Cleaver, Klaus Kirsten, Qin Sheng
(Submitted on 22 May 2017)
The evolutions of the flat FLRW universe and its linear perturbations are studied systematically in {\em the dressed metric approach} of LQC. When the evolution of the background at the quantum bounce is dominated by the kinetic energy of the inflaton, it can be divided into three different phases prior to the preheating, {\em bouncing, transition and slow-roll inflation}. During the bouncing phase, the evolution is independent of not only the initial conditions, but also the inflationary potentials. In particular, the expansion factor can be well described by the same exact solution in all the cases considered. In contrast, in the potential dominated case such a universality is lost. It is also because of this universality that the linear perturbations are independent of the inflationary models, too, and are obtained exactly. During the transition phase, the evolution of the background is first matched to that given in other two phases, whereby the e-folds of the expansion are obtained. In this phase the perturbation modes are all oscillating, and are matched to the ones given in other phases. Considering two different sets of initial conditions, one is imposed during the contracting phase and the other is at the bounce, we calculate the Bogoliubov coefficients and find that the two sets yield the same results and all lead to particle creations at the onset of the inflation. Due to the pre-inflationary dynamics, the scalar and tensor power spectra become scale-dependent. Comparing with the Planck 2015 data, we find constraints on the total e-folds that the universe must have expanded since the bounce, in order to be consistent with current observations.
Comments: revtex4, 24 figures, and 5 tables
Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO); High Energy Physics - Phenomenology (hep-ph); High Energy Physics - Theory (hep-th)
Cite as: arXiv:1705.07544 [gr-qc]
(or arXiv:1705.07544v1 [gr-qc] for this version)


Science Advisor
The ıε prescription in the SYK model
Razvan Gurau
(Submitted on 24 May 2017)
We introduce an ıϵ prescription for the SYK model both at finite and at zero temperature. This prescription regularizes all the naive ultraviolet divergences of the model. As expected, the prescription breaks the conformal invariance, but the latter is restored in the ϵ→0 limit. We prove rigorously that in the limit ϵ→0 the Schwinger Dyson equation of the resummed two point at low momentum is recovered.


Gold Member

Einstein Equation from Covariant Loop Quantum Gravity and Semiclassical Continuum Limit
Muxin Han
(Submitted on 25 May 2017)
In this paper we explain how 4-dimensional general relativity and in particular, the Einstein equation, emerge from the spinfoam amplitude in loop quantum gravity. We propose a new limit which couples both the semiclassical limit and continuum limit of spinfoam amplitudes. The continuum Einstein equation emerges in this limit. Solutions of Einstein equation can be approached by dominant configurations in spinfoam amplitudes. A running scale is naturally associated to the sequence of refined triangulations. The continuum limit corresponds to the infrared limit of the running scale. An important ingredient in the derivation is a regularization for the sum over spins, which is necessary for the semiclassical continuum limit. We also explain in this paper the role played by the so-called flatness in spinfoam formulation, and how to take advantage of it.


Science Advisor
Immirzi parameter without Immirzi ambiguity: Conformal loop quantization of scalar-tensor gravity
Olivier J. Veraguth, Charles H.-T. Wang
(Submitted on 25 May 2017)
Conformal loop quantum gravity provides an approach to loop quantization through an underlying conformal structure i.e. conformally equivalent class of metrics. The property that general relativity itself has no conformal invariance is reinstated with a constrained scalar field providing the physical scale. Conformally equivalent metrics have recently been shown to be amenable to loop quantization including matter coupling. It has been suggested that conformal geometry may provide an extended symmetry to allow a reformulated Immirzi parameter necessary for loop quantization to behave like an arbitrary group parameter that requires no further fixing as its present standard form does. In this work, we find that this can be naturally realized via conformal frame transformations in scalar-tensor gravity. Such a theory generally incorporates a dynamical scalar gravitational field and reduces to general relativity when the scalar field becomes a pure gauge. Specifically, we introduce a "conformal Einstein frame" in which loop quantization is implemented and demonstrate that different Immirzi parameters under this description are associated with different conformal frames related by a global conformal transformation. Nevertheless, they share the same quantization having, for example, the same area gaps, modulated by the scalar gravitational field.

What are we missing in our search for quantum gravity?
Lee Smolin
(Submitted on 25 May 2017)
Some reflections are presented on the state of the search for a quantum theory of gravity. I discuss diverse regimes of possible quantum gravitational phenomenon, some well explored, some novel.

Gravity from Quantum Spacetime by Twisted Deformation of the Quantum Poincaré Group
Cesar A. Aguillón, Albert Much, Marcos Rosenbaum, J. David Vergara
(Submitted on 24 May 2017)
We investigate a quantum geometric space in the context of what could be considered an emerging effective theory from Quantum Gravity. Specifically we consider a two-parameter class of twisted Poincar\'e algebras, from which Lie-algebraic noncommutativities of the translations are derived as well as associative star-products, deformed Riemannian geometries, Lie-algebraic twisted Minkowski spaces and quantum effects that arise as noncommutativities. Starting from a universal differential algebra of forms based on the above mentioned Lie-algebraic noncommutativities of the translations, we construct the noncommutative differential forms and Inner and Outer derivations, which are the noncommutative equivalents of the vector fields in the case of commutative differential geometry. Having established the essentials of this formalism we construct a bimodule, required to be central under the action of the Inner derivations in order to have well defined contractions and from where the algebraic dependence of its coefficients is derived. This again then defines the noncommutative equivalent of the geometrical line-element in commutative differential geometry. We stress, however, that even though the components of the twisted metric are by construction symmetric in their algebra valuation, this is not so for their inverse and thus to construct it we made use of Gel'fand's theory of quasi-determinants, which is conceptually straightforward but computationally becoming quite complicate beyond an algebra of 3 generators. The consequences of the noncommutativity of the Lie-algebra twisted geometry are further discussed.
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Loop Quantum Gravity in the Momentum Representation
W. F. Chagas-Filho
(Submitted on 26 May 2017)
We present a generalization of the first-order formalism used to describe the dynamics of a classical system. The generalization is then applied to the first-order action that describes General Relativity. As a result we obtain equations that can be interpreted as describing quantum gravity in the momentum representation.
Subjects: General Relativity and Quantum Cosmology (gr-qc)
Cite as: arXiv:1705.09471 [gr-qc]
No alternative to proliferation
Daniele Oriti
(Submitted on 27 May 2017)
We reflect on the nature, role and limits of non-empirical theory assessment in fundamental physics, focusing in particular on quantum gravity. We argue for the usefulness and, to some extent, necessity of non-empirical theory assessment, but also examine critically its dangers. We conclude that the principle of proliferation of theories is not only at the very root of theory assessment but all the more necessary when experimental tests are scarce, and also that, in the same situation, it represents the only medicine against the degeneration of scientific research programmes.
Comments: 15 pages; contribution to the volume "Why trust a theory?", edited by: R. Dardashti, R. Dawid, K. Thebault, to be published by Cambridge University Press
Subjects: History and Philosophy of Physics (physics.hist-ph); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th)
Cite as: arXiv:1705.09858 [physics.hist-ph]
Renormalization and Coarse-graining of Loop Quantum Gravity
Christoph Charles
(Submitted on 31 May 2017)
The continuum limit of loop quantum gravity is still an open problem. Indeed, no proper dynamics in known to start with and we still lack the mathematical tools to study its would-be continuum limit. In the present PhD dissertation, we will investigate some coarse-graining methods that should become helpful in this enterprise. We concentrate on two aspects of the theory's coarse-graining: finding natural large scale observables on one hand and studying how the dynamics of varying graphs could be cast onto fixed graphs on the other hand.
To determine large scale observables, we study the case of hyperbolic tetrahedra and their natural description in a language close to loop quantum gravity. The surface holonomies in particular play an important role. This highlights the structure of double spin networks, which consist in a graph and its dual, which seems to also appear in works from Freidel et al. To solve the problem of varying graphs, we consider and define loopy spin networks. They encode the local curvature with loops around an effective vertex and allow to describe different graphs by hidding them in a coarse-graining process. Moreover, their definition gives a natural procedure for coarse-graining allowing to relate different scales.
Together, these two results constitute the foundation of a coarse-graining programme for diffeomorphism invariant theories.
Comments: PhD Thesis, Ecole Normale Superieure de Lyon, 303pages, many figures
Subjects: General Relativity and Quantum Cosmology (gr-qc)
Cite as: arXiv:1705.10984 [gr-qc]
(or arXiv:1705.10984v1 [gr-qc] for this version)


Science Advisor
The continuum approach to the BF vacuum: the U(1) case
Patryk Drobiński, Jerzy Lewandowski
(Submitted on 27 May 2017)
A quantum representation of holonomies and exponentiated fluxes of a U(1) gauge theory that contains the Pullin-Dittrich-Geiller (DG) vacuum is presented and discussed. Our quantization is performed manifestly in a continuum theory, without any discretization. The discretness emerges on the quantum level as a property of the spectrum of the quantum holonomy operators. The new type of a cylindrical consistency present in the DG approach, now follows easily and naturally. A generalization to the non--Abelian case seems not difficult.
Comments: 12 pages
Subjects: General Relativity and Quantum Cosmology (gr-qc)
Cite as: arXiv:1705.09836 [gr-qc]
Diffeomorphism invariant cosmological sector in loop quantum gravity
Christopher Beetle, Jonathan Steven Engle, Matthew Ernest Hogan, Phillip Mendonca
(Submitted on 8 Jun 2017)
In this paper we work out in detail a new proposal to define rigorously a sector of loop quantum gravity at the diffeomorphism invariant level corresponding to homogeneous and isotropic cosmologies, and propose how to compare in detail the physics of this sector with that of loop quantum cosmology. The key technical steps we have completed are (a) to formulate conditions for homogeneity and isotropy in a diffeomorphism covariant way on the classical phase space of general relativity, and (b) to translate these conditions consistently using well-understood techniques to loop quantum gravity. To impose the symmetry at the quantum level, on both the connection and its conjugate momentum, the method used necessarily has similiarities to the Gupta-Bleuler method of quantizing the electromagnetic field. Lastly, a strategy for embedding states of loop quantum cosmology into this new homogeneous isotropic sector, and using this embedding to compare the physics, is presented.
Comments: 25 pages
Subjects: General Relativity and Quantum Cosmology (gr-qc)
Cite as: arXiv:1706.02424 [gr-qc]
(or arXiv:1706.02424v1 [gr-qc] for this version)
Pulsations of a black hole in loop quantum gravity
Changjun Gao, Youjun Lu, You-Gen Shen, Valerio Faraoni
(Submitted on 24 Jun 2017)
The Hawking-Penrose singularity theorem states that a singularity forms inside a black hole in general relativity. To remove this singularity one must resort to a more fundamental theory. Using the corrected dynamical equation of loop quantum cosmology and braneworld models, we study the gravitational collapse of a perfect fluid sphere with a rather general equation of state. In the frame of an observer comoving with this fluid, the sphere pulsates between a maximum and a minimum size, avoiding the singularity. The exterior geometry is also constructed. There are usually {an outer and an inner apparent horizon}, resembling the Reissner-Nordstr\"om situation. For a distant observer the {horizon} crossing occurs in an infinite time and the pulsations of the black hole quantum "beating heart" are completely unobservable. However, it may be observable if the black hole is not spherical symmetric and radiates gravitational wave due to the quadrupole moment, if any.
Comments: 24 pages, 2 figures
Subjects: General Relativity and Quantum Cosmology (gr-qc); Cosmology and Nongalactic Astrophysics (astro-ph.CO); High Energy Physics - Theory (hep-th)
Cite as: arXiv:1706.08009 [gr-qc]


Science Advisor
Cosmological Effective Hamiltonian from full Loop Quantum Gravity Dynamics
Andrea Dapor, Klaus Liegener
(Submitted on 29 Jun 2017)
The concept of effective dynamics has proven successful in LQC, the cosmological sector of LQG. We apply the same idea in the full theory, by computing the expectation value of the scalar constraint with respect to some coherent states peaked on the phase-space variables of flat Robertson-Walker spacetime. We comment on the relation with effective LQC and find a deviation stemming from the Lorentzian part of the Hamiltonian.
Quantum Reduced Loop Gravity with matter: eigenvectors of the Hamiltonian operator in isotropic cosmology
Jakub Bilski, Suddhasattwa Brahma, Antonino Marciano
(Submitted on 30 Jun 2017)
Introducing a new method, we demonstrate how the action of reduced operators can be derived without resorting to a recoupling theory and how they exactly reproduce the results obtained in the standard approach of Quantum Reduced Loop Gravity (QRLG). This is particularly relevant while dealing with volume operator when dealing with the coupling of matter fields to gravity. Apart from reinforcing the close link between QRLG and loop quantum cosmology (LQC), this procedure also sheds new light on the issue of how to extract the continuum limit, without resorting to the large-jexpansion, thereby pointing towards a new approach to tackle this problem.
Comments: 14 pages
Subjects: General Relativity and Quantum Cosmology (gr-qc)
Cite as: arXiv:1707.00065 [gr-qc]
(or arXiv:1707.00065v1 [gr-qc] for this version)
2+1 homogeneous Loop Quantum Gravity with a scalar field clock
Jakub Bilski, Antonino Marciano
(Submitted on 3 Jul 2017 (v1), last revised 5 Jul 2017 (this version, v2))
We focus on three-dimensional QRLG with the purpose of shedding light on the link between reduced LQG and LQC in four space-time dimensions. Considering homogeneous three-dimensional LQG, the theory simplifies to QRLG. We then implement Thiemann's Quantum Spin Dynamics for Euclidean three-dimensional space-time in presence of a real scalar matter field. We deploy a polymer quantization of the scalar field while using methods of quantum reduced loop gravity. We compute the scalar Hamiltonian operator on the states of the kinematical Hilbert space of the theory, and exhibit its matrix elements that are derived using a new simplified method. The coupling to matter, which plays the role of a carrier of dynamics, opens the pathway to the study of phenomenological implications. We finally comment on the relations between three-dimensional QRLG and LQC, as well as on the appearance of the correspondence principle for the scalar field.
Comments: 15 pages, typos corrected, cross-citation added
Subjects: General Relativity and Quantum Cosmology (gr-qc)
Cite as: arXiv:1707.00723 [gr-qc]
A new duality between Topological M-theory and Loop Quantum Gravity
Andrea Addazi, Antonino Marciano
(Submitted on 17 Jul 2017)
Inspired by the long wave-length limit of topological M-theory, which re-constructs the theory of 3+1D gravity in the self-dual variables' formulation, we conjecture the existence of a duality between Hilbert spaces, the H-duality, to unify topological M-theory and loop quantum gravity (LQG). By H-duality non-trivial gravitational holonomies of the kinematical Hilbert space of LQG correspond to space-like M-branes. The spinfoam approach captures the non-perturbative dynamics of space-like M-branes, and can be claimed to be dual to the S-branes foam. The Hamiltonian constraint dealt with in LQG is reinterpreted as a quantum superposition of SM-brane nucleations and decays.
Subjects: High Energy Physics - Theory (hep-th)
Cite as: arXiv:1707.05347 [hep-th]
Induced loop quantum cosmology on a brane via holography
C. A. S. Silva
(Submitted on 20 Jul 2017)
Based on the holographic principle, it is demonstrated that loop quantum Friedmann equations can be induced on a brane, corresponding to a strongly coupled string regime in the bulk, and have braneworld cosmology equations as its low energy limit. Such result can establish a possible connection between loop quantum gravity and string theory.
Comments: 6 pages, 1 figure
Subjects: General Relativity and Quantum Cosmology (gr-qc)
Cite as: arXiv:1707.07586 [gr-qc]


Science Advisor
Kochen-Specker theorem revisited
Del Rajan (Victoria University of Wellington), Matt Visser (Victoria University of Wellington)
(Submitted on 4 Aug 2017)
The Kochen-Specker theorem is a basic and fundamental 50 year old non-existence result affecting the foundations of quantum mechanix, strongly implying the lack of any meaningful notion of "quantum realism", and typically leading to discussions of "contextuality" in quantum physics. Original proofs of the Kochen-Specker theorem proceeded via brute force counter-examples; often quite complicated and subtle (albeit mathematically "elementary") counter-examples. Only more recently have somewhat more "geometrical" proofs been developed. We present herein yet another simplified geometrical proof of the Kochen-Specker theorem, one that is valid for any number of dimensions, that minimizes the technical machinery involved, and makes the seriousness of the issues raised manifest.

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