A Linear Solution of the Four-Dimensionality Problem

“…Theorem 5.1 [19] Every home paradigm of the R-observer is isomorphic to the quaternion algebra H with a family of Minkowski sensory forms.…”

Section: Definition 57mentioning

confidence: 99%

“…Due to the results of [19] and [20], in the remainder of the paper we shall deal exclusively with the R-observer, henceforth referred to simply as the observer. If not mentioned explicitly, it is assumed in the following that the constructs under consideration always refer to the observer.…”

Section: Observersmentioning

confidence: 99%

GR-Friendly Description of Quantum Systems

Trifonov

2007

“…It has a natural Hermitian form which induces a Euclidean scalar product on its additive vector space S H . There is also a family of natural indefinite scalar products of signature 2 on S H (Trifonov, 1995), induced by the structure tensor H of the quaternion algebra. This result came out of a study of relationship between natural metric properties of unital algebras and internal logic of topoi they generate.…”

Section: Introductionmentioning

confidence: 99%

“…This result came out of a study of relationship between natural metric properties of unital algebras and internal logic of topoi they generate. It was shown in Trifonov (1995) that if the logic of a topos is bivalent Boolean then the generating algebra is isomorphic to the quaternion algebra with a family of natural scalar product of signature 2. Such scalar products can be defined on any linear algebra over a field F. In this note we show that for a unital algebra these scalar products can be naturally extended over the Lie group of its invertible elements, producing a family of principal metrics.…”

Section: Introductionmentioning

confidence: 99%

Natural Geometry of Nonzero Quaternions

Trifonov

2007

“…marriage of QT with GR. Here too, category, (pre)sheaf and topos theory has been anticipated to play a central role for many different reasons, due to various different motivations, and with different aims in mind, depending on the approach to QG that one favors [9,98,6,67,29,79,30,31,32,33, 36,37,77,79, 80].Akin to the present work is the recent paper of Christensen and Crane on so-called 'causal sites' (causites) [8]. Like the Novosibirsk endeavors in classical GR mentioned above, this is an axiomatic looking scheme based on Grothendieck-type of 2-categories (:2-sites) in which the topological and causal structure of spacetime are intimately entwined and, when endowed with some suitable finiteness conditions, appear to be well prepared for quantization using combinatory-topological…”

mentioning

confidence: 99%

Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity

Raptis

2007

The pentalogy [60,61,62,76,63] is brought to its categorical climax by organizing the curved finitary spacetime sheaves of quantum causal sets involved therein, on which a finitary (:locally finite), singularity-free, background manifold independent and geometrically prequantized version of the gravitational vacuum Einstein field equations were seen to hold, into a topos structure DT f cq . We show that the category of finitary differential triads DT f cq is a finitary instance of an elementary topos proper in the original sense due to Lawvere and Tierney. We present in the light of Abstract Differential Geometry (ADG) a Grothendieck-type of generalization of Sorkin's finitary substitutes of continuous spacetime manifold topologies, the latter's topological refinement inverse systems of locally finite coverings and their associated coarse graining sieves, the upshot being that DT f cq is also a finitary example of a Grothendieck topos. In the process, we discover that the subobject classifier Ω f cq of DT f cq is a Heyting algebra type of object, thus we infer that the internal logic of our finitary topos is intuitionistic, as expected. We also introduce the new notion of 'finitary differential geometric morphism' which, as befits ADG, gives a differential geometric slant to Sorkin's purely topological acts of refinement (:coarse graining). Based on finitary differential geometric morphisms regarded as natural transformations of the relevant sheaf categories, we observe that the functorial ADG-theoretic version of the principle of general covariance of General Relativity is preserved under topological refinement. The paper closes with a thorough discussion of four future routes we could take in order to further develop our topos-theoretic perspective on ADG-gravity along certain categorical trends in current quantum gravity research. With respect to QT proper, topos theory appears to be a suitable and elegant framework in which to express the non-objective, non-classical (ie, non-Boolean), so-called 'neo-realist' (ie, intuitionistic), and contextual underpinnings of the logic of (non-relativistic) Quantum Mechanics (QM), as manifested for example by the Kochen-Specker theorem in standard quantum logic [4,5,3,6,83]. Recently, Isham et al.'s topos perspective on the Kochen-Specker theorem and the Boolean algebra-localized (:contextualized) logic of QT has triggered research on applying categorytheoretic ideas to the 'problem' of non-trivial localization properties of quantum observables [103]. Topos theory has also been used to reveal the intuitionistic colors of the logic underlying the 'noninstrumentalist', non-Copenhagean, 'quantum state collapse-free' consistent histories approach to QM [28].At the same time, topos theory has also been applied to General Relativity (GR), especially by the Siberian school of 'toposophers' [21,22,23,24,25,20]. Emphasis here is placed on using the intuitionistic-type of internal logic of a so-called 'formal smooth topos', which is assumed to replace the (category of finit...