Quantum theories constructed on the noncommutative spacetime called the Groenewold–Moyal plane exhibit many interesting properties such as Lorentz and CPT noninvariance, causality violation and twisted statistics. We show that such violations lead to many striking features that may be tested experimentally. These theories predict Pauli forbidden transitions due to twisted statistics, anisotropies in the cosmic microwave background radiation due to correlations of observables in spacelike regions and Lorentz and CPT violations in scattering amplitudes.

A systematic analysis is made of the relations between the symmetries of a classical field and the symmetries of the one-particle quantum system that results from quantizing that field in regimes where interactions are weak. The results are applied to gain a greater insight into the phenomenon of antimatter.

Noncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale—and ultimately the concept of a point—makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes’ spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the (...) scalar field Moyal–Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry. (shrink)

Anti-elementarity is a strong way of ensuring that a class of structures, in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the...

Since Bohmian mechanics is explicitly nonlocal, it is widely believed that it is very hard, if not impossible, to make Bohmian mechanics compatible with relativistic quantum field theory. I explain, in simple terms, that it is not hard at all to construct a Bohmian theory that lacks Lorentz covariance, but makes the same measurable predictions as relativistic QFT. All one has to do is to construct a Bohmian theory that makes the same measurable predictions as QFT in one Lorentz frame, (...) because then standard relativistic QFT itself guarantees that those predictions are Lorentz invariant. I first explain this in general terms, then I describe a simple Bohmian model that makes the same measurable predictions as the Standard Model of elementary particles, after which I give some hints towards a more fundamental theory beyond standard model. Finally, I present a short story telling how my views of fundamental physics in general, and of Bohmian mechanics in particular, evolved over time. (shrink)

We show that in the presence of the torsion tensor \, the quantum commutation relation for the four-momentum, traced over spinor indices, is given by \. In the Einstein–Cartan theory of gravity, in which torsion is coupled to spin of fermions, this relation in a coordinate frame reduces to a commutation relation of noncommutative momentum space, \, where U is a constant on the order of the squared inverse of the Planck mass. We propose that this relation replaces the (...) integration in the momentum space in Feynman diagrams with the summation over the discrete momentum eigenvalues. We derive a prescription for this summation that agrees with convergent integrals: \^s}\rightarrow 4\pi U^{s-2}\sum _{l=1}^\infty \int _0^{\pi /2} d\phi \frac{\sin ^4\phi \,n^{s-3}}{[\sin \phi +U\varDelta n]^s}\), where \}\) and \ does not depend on p. We show that this prescription regularizes ultraviolet-divergent integrals in loop diagrams. We extend this prescription to tensor integrals. We derive a finite, gauge-invariant vacuum polarization tensor and a finite running coupling. Including loops from all charged fermions, we find a finite value for the bare electric charge of an electron: \. This torsional regularization may therefore provide a realistic, physical mechanism for eliminating infinities in quantum field theory and making renormalization finite. (shrink)

In my commentary, I will argue that the conclusions drawn in the paper Noncommutative causality in algebraic quantum field theory by Gábor Hofer-Szaboó are incorrect. As proven by J.S. Bell, a local common causal explanation of correlations violating the Bell inequality is impossible.

In this review article we discuss some of the applications of noncommutative geometry in physics that are of recent interest, such as noncommutative many-body systems, noncommutative extension of Special Theory of Relativity kinematics, twisted gauge theories and noncommutative gravity.

We give an almost explicit presentation of exotic functions corresponding to some exotic smooth structure on topologically trivial R4. The construction relies on the model-theoretic tools from the previous paper. We can formulate unexpected, yet direct connection between ‘‘localized’’ exotic small R4’s and some noncommutative spaces. The formalism of QM can be interpreted in terms of exotic smooth R4’s localized in spacetime. A new way of looking at the problem of decoherence is suggested. The 4-dimensional spacetime itself has built-in (...) means which may enforce a kind of decoherence. (shrink)

It is shown how to identify potential signatures of noncommutative geometry within the decay spectrum of a muon in orbit near the event horizon of a microscopic Schwarzschild black hole. This possibility follows from a re-interpretation of Moffat’s nonsymmetric theory of gravity, first published in Phys. Rev. D 19:3554, 1979, where the antisymmetric part of the metric tensor manifests the hypothesized noncommutative geometric structure throughout the manifold. It is further shown that for a given sign convention, the predicted (...) signatures counteract the effects of curvature-induced muon stabilization predicted by Singh and Mobed in Phys. Rev. D 79:024026, 2009. While it is unclear whether evidence for noncommutative geometry may become observable anytime soon, this approach at least provides a useful direction for future quantum gravity research based on the ideas presented here. (shrink)

I consider to what extent the phenomenon of interference precludes the possibility of attributing simultaneously determinate values to noncommuting observables, and I show that, while all observables can in principle be taken as simultaneously determinate, it suffices to take a suitable privileged observable as determinate to solve the measurement problem.

A relation ≺ϕ between noncommuting 1-0 quantum observables (i.e., projections) is introduced, ϕ being the state vector of the system. This relation extends the empirical implication between commuting projections. An operational interpretation of the new relation is given, which can be expressed also in counterfactual terms. It is shown that a relation proposed some years ago by Hardegree, namely the Sasaki arrow ↪ϕ, can be interpreted in terms of the relation ≺ϕ; furthermore, this new relation turns out to be successful (...) also in cases in which the Sasaki arrow fails. (shrink)

A simple noncommutative probability theory is presented, and two examples for the difference between that theory and the classical theory are shown. The first example is the well-known formulation of the Heisenberg uncertainty principle in terms of a variance inequality and the second example is an interpretatio of the Bell paradox in terms of noncommuntative probability.

Abstract.A keystone of the theory of noncommutative noetherian rings is the theorem that establishes a necessary and sufficient condition for a given ring to have a quotient ring. We trace the development of this theorem, and its applications, from its first version for noncommutative domains in the 1930s to Goldie’s theorems in the late 1950s.

We use joint probabilities to analyze the EPR argument in the Bohm's example of spins.(1) The properties of distribution functions for two, three, or more noncommuting spin components are explicitly studied and their limitations are pointed out. Within the statistical ensemble interpretation of quantum theory (where only statements about repeated events can be made), the incompleteness of quantum theory does not follow, as the consistent use of joint probabilities shows. This does not exclude a completion of quantum mechanics, going beyond (...) it, by a more general theory of single events, using hidden variables, for example. (shrink)

We show that the Bohmian approach in terms of persisting particles that move on continuous trajectories following a deterministic law can be literally applied to QFT. By means of the Dirac sea model – exemplified in the electron sector of the standard model neglecting radiation – we explain how starting from persisting particles, one is led to standard QFT employing creation and annihilation operators when tracking the dynamics with respect to a reference state, the so-called vacuum. Since on the level (...) of wave functions, both formalisms are mathematically equivalent, this proposal provides for an ontology of QFT that includes a dynamics of individual processes, solves the measurement problem and explains the appearance of creation and annihilation events. (shrink)

A (to our knowledge) novel Generalized Nonlinear Schrödinger equation based on the modifications of Nottale-Cresson’s fractal-scale calculus and resulting from the noncommutativity of the phase space coordinates is explicitly derived. The modifications to the ground state energy of a harmonic oscillator yields the observed value of the vacuum energy density. In the concluding remarks we discuss how nonlinear and nonlocal QM wave equations arise naturally from this fractal-scale calculus formalism which may have a key role in the final formulation of (...) Quantum Gravity. (shrink)

Despite quantum theory’s remarkable success at predicting the statistical results of experiments, many philosophers worry that it nonetheless lacks some crucial connection between theory and experiment. Such worries constitute the Quantum Measurement Problems. One can broadly identify two kinds of worries: (1) pragmatic: it is unclear how to model our measurement processes in order to extract experimental predictions, and (2) realist: we lack a satisfying metaphysical account of measurement processes. While both issues deserve attention, the pragmatic worries have worse consequences (...) if left unanswered: If our pragmatic theory-to-experiment linkage is unsatisfactory, then quantum theory is at risk of losing both its evidential support and its physical salience. Avoiding these risks is at the core of what I will call the Pragmatic Measurement Problem. Fortunately, the pragmatic measurement problem is not too difficult to solve. For non-relativistic quantum theory, the story goes roughly as follows: One can model each of quantum theory’s key experimental successes on a case-by-case basis by using a measurement chain. In modeling this measurement chain, it is pragmatically necessary to switch from using a quantum model to a classical model at some point. That is, it is pragmatically necessary to invoke a Heisenberg cut at some point along the measurement chain. Past this case-by-case measurement framework, one can then strive for a wide-scoping measurement theory capable of modeling all (or nearly all) possible measurement processes. For non-relativistic quantum theory, this leads us to our usual projective measurement theory. As a bonus, proceeding this way also gives us an empirically meaningful characterization of the theory’s observables as (positive) self-adjoint operators. But how does this story have to change when we move into the context of quantum field theory (QFT)? It is well known that in QFT almost all localized projective measurements violate causality, allowing for faster-than-light signaling; These are Sorkin’s impossible measurements. Thus, the story of measurement in QFT cannot end as it did before with a projective measurement theory. But does this then mean that we need to radically rethink the way we model measurement processes in QFT? Are our current experimental practices somehow misguided? Fortunately not. I will argue that (once properly understood) our old approach to modeling quantum measurements is still applicable in QFT contexts. We ought to first use measurement chains to build up a case-by-case measurement framework for QFT. Modeling these measurement chains will require us to invoke what I will call a QFT-cut. That is, at some point along the measurement chain we must switch from using a QFT model to a non-QFT model. Past this case-by-case measurement framework, we can then strive for both a new wide-scoping measurement theory for QFT and an empirically meaningful characterization of its observables. It is at this point that significantly more theoretical work is needed. This paper ends by briefly reviewing the state of the art in the physics literature regarding the modeling of measurement processes involving quantum fields. (shrink)

In this essay I examine a recent argument by Steven Weinberg that seeks to establish local quantum field theory as the only type of quantum theory in accord with the relevent evidence and satisfying two basic physical principles. I reconstruct the argument as a demonstrative induction and indicate it's role as a foil to the underdetermination argument in the debate over scientific realism.

In this paper we explore the possibility of giving a justification of the “semantic information” content and measure, in the framework of the recent coalgebraic approach to quantum systems and quantum computation, extended to QFT systems. In QFT, indeed, any quantum system has to be considered as an “open” system, because it is always interacting with the background fluctuations of the quantum vacuum. Namely, the Hamiltonian in QFT always includes the quantum system and its inseparable thermal bath, formally “entangled” like (...) an algebra with its coalgebra, according to the principle of the “doubling” of the degrees of freedom between them. This is the core of the representation theory of cognitive neuroscience based on QFT. Moreover, in QFT, the probabilities of the quantum states follow a Wigner distribution, based on the notion and measure of quasiprobability, where regions integrated under given expectation values do not represent mutually exclusive states. This means that a computing agent, either natural or artificial, in QFT, against the quantum Turing machine paradigm, is able to change dynamically the representation space of its computations. This depends on the possibility of interpreting QFT system computations within the framework of category theory logic and its principle of duality between opposed categories, such as the algebra and coalgebra categories of QFT. This allows us to justify and not only to suppose, like in the “theory of strong semantic information” of L. Floridi, the definition of modal “local truth” and the notion of semantic information as a measure of it, despite both measures being defined on quasiprobability distributions. (shrink)

Philosophical analysis of spontaneous symmetry breaking in particle physics has been hindered by the unavailability of rigorous formulations of models in quantum field theory. A strategy for addressing this problem is to use the rigorous models that have been constructed for SSB in quantum statistical mechanics systems as a basis for drawing analogous conclusions about SSB in QFT. Based on an analysis of this strategy as an instance of the application of the same mathematical formalism to different domains and as (...) an instance of drawing analogies between domains, I conclude that certain structural explanations can be exported from QSM to QFT, but that causal explanations cannot. (shrink)

We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic. The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffie algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs in (...) CyLL + MIX. This can be viewed as a fully faithful representation of a free *-autonomous category, canonically enriched over vector spaces. This paper is a natural extension of the authors' previous work, "Linear Lauchli Semantics", where a similar theorem is obtained for the commutative logic MLL + MIX. In that paper, we interpret proofs as dinaturals which are invariant under certain actions of the additive group of integers. Here we also present a simplification of that work by showing that the invariance criterion is actually a consequence of dinaturality. The passage from groups to Hopf algebras in this paper corresponds to the passage from commutative to noncommutative logic. However, in our noncommutative setting, one must still keep the invariance condition on dinaturals. (shrink)

Proposed quantum experiments in deep space will be able to explore quantum information issues in regimes where relativistic effects are important. In this essay, we argue that a proper extension of quantum information theory into the relativistic domain requires the expression of all informational notions in terms of quantum field theoretic (QFT) concepts. This task requires a working and practicable theory of QFT measurements. We present the foundational problems in constructing such a theory, especially in relation to longstanding causality and (...) locality issues in the foundations of QFT. Finally, we present the ongoing Quantum Temporal Probabilities program for constructing a measurement theory that (i) works, in principle, for any QFT, (ii) allows for a first- principles investigation of all relevant issues of causality and locality, and (iii) it can be directly applied to experiments of current interest. (shrink)

This work is devoted to the relations between Lambek’s Syntactic Calculus and noncommutative variants of Girard’s Linear Logic; in particular the paper will consider: the geometrical representation of the laws of LC by means of proof-nets; the discovery - due to such a geometrical representation - of some laws of LC not yet considered; the discussion of possible linguistic uses of these new laws.

We present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffie algebra. Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs (...) in CyLL + MIX. This can be viewed as a fully faithful representation of a free *-autonomous category, canonically enriched over vector spaces. This paper is a natural extension of the authors' previous work, "Linear Lauchli Semantics", where a similar theorem is obtained for the commutative logic MLL + MIX. In that paper, we interpret proofs as dinaturals which are invariant under certain actions of the additive group of integers. Here we also present a simplification of that work by showing that the invariance criterion is actually a consequence of dinaturality. The passage from groups to Hopf algebras in this paper corresponds to the passage from commutative to noncommutative logic. However, in our noncommutative setting, one must still keep the invariance condition on dinaturals. (shrink)

The present paper is the first part of a work which follows up on H. Kummer: “A constructive approach to the foundations of quantum mechanics,”Found. Phys. 17, 1–63 (1987). In that paper we deduced the JB-algebra structure of the space of observables (=detector space) of quantum mechanics within an axiomatic theory which uses the concept of a filter as primitive under the restrictive assumption that the detector space is finite-dimensional. This additional hypothesis will be dropped in the present paper.It turns (...) out that the relevant mathematics for our approach to a quantum mechanical system with infinite-dimensional detector space is the noncommutative spectral theory of Alfsen and Shultz.We start off with the same situation as in the previous paper (cf. Sects. 1 and 2 of the present paper). By postulating four axioms (Axioms S, DP, R, and SP of Sec. 3), we arrive in a natural way at the mathematical setting of Alfsen and Shultz, which consists of a dual pair of real ordered linear spaces 〈Y, M〉: A base norm space, called the strong source space (which, however, in slight contrast to the setting of Alfsen and Shultz, is not 1-additive) and an order unit space, called the weak detector space, which is the norm and order dual space of Y. The last section of part I contains the guiding example suggested by orthodox quantum mechanics. We observe that our axioms are satisfied in this example. In the second part of this work (which will appear in the next issue of this journal) we shall postulate three further axioms and derive the JB-algebra structure of quantum mechanics. (shrink)

This essay touches on a number of topics in philosophy of quantum field theory from the point of view of the LSZ asymptotic approach to scattering theory. First, particle/field duality is seen to be a property of free field theory and not of interacting QFT. Second, it is demonstrated how LSZ side-steps the implications of Haag's theorem. Finally, a recent argument due to Redhead, Malament and Arageorgis against the concept of localized particle states is addressed. Briefly, the argument observes that (...) the Reeh-Schlieder theorem entails that correlations between spacelike separated vacuum expectation values of local field operators are always present, and this, according to the above authors, dictates against the notion of a localized particle state. I claim that this moral is excessive and that a coherent notion of localized particles is given by the LSZ approach. The underlying moral to be drawn from this analysis is that questions concerning the ontology of interacting QFT cannot be appropriately addressed if one restricts oneself to the free theory. (shrink)

We consider several subtle aspects of the theory of neutrino oscillations which have been under discussion recently. We show that the S-matrix formalism of quantum field theory can adequately describe neutrino oscillations if correct physics conditions are imposed. This includes space-time localization of the neutrino production and detection processes. Space-time diagrams are introduced, which characterize this localization and illustrate the coherence issues of neutrino oscillations. We discuss two approaches to calculations of the transition amplitudes, which allow different physics interpretations: (i) (...) using configuration-space wave packets for the involved particles, which leads to approximate conservation laws for their mean energies and momenta; (ii) calculating first a plane-wave amplitude of the process, which exhibits exact energy-momentum conservation, and then convoluting it with the momentum-space wave packets of the involved particles. We show that these two approaches are equivalent. Kinematic entanglement (which is invoked to ensure exact energy-momentum conservation in neutrino oscillations) and subsequent disentanglement of the neutrinos and recoiling states are in fact irrelevant when the wave packets are considered. We demonstrate that the contribution of the recoil particle to the oscillation phase is negligible provided that the coherence conditions for neutrino production and detection are satisfied. Unlike in the previous situation, the phases of both neutrinos from Z 0 decay are important, leading to a realization of the Einstein-Podolsky-Rosen paradox. (shrink)

The idea that spacetime has to be replaced by Clifford space (C-space) is explored. Quantum field theory (QFT) and string theory are generalized to C-space. It is shown how one can solve the cosmological constant problem and formulate string theory without central terms in the Virasoro algebra by exploiting the peculiar pseudo-Euclidean signature of C-space and the Jackiw definition of the vacuum state. As an introduction into the subject, a toy model of the harmonic oscillator in pseudo-Euclidean space is studied.

The present paper comprises Sects. 5–8 of a work which proposes an axiomatic approach to quantum mechanics in which the concept of a filter is the central primitive concept. Having layed down the foundations in the first part of this work (which appeared in the last issue of this journal and comprises Sects. 0–4), we arrived at a dual pair 〈Y, M〉 consisting of abase norm space Y and anorder unit space M, being in order and norm duality with respect (...) to each other. This is precisely the setting of noncommutative spectral theory, a theory which has been developed during the late nineteen seventies by Alfsen and Shultz. (2,3) In this part we add to the four axioms (Axioms S, DP, R, SP) of Sect. 3 three further axioms (Axioms E, O, L). These axioms are suggested by the work of Alfsen and Shultz and enable us to derive the JB-algebra structure of quantum mechanics (cf. Theorem 8.9). (shrink)

The essence of the method of physics is inseparably connected with the problem of interplay between local and global properties of the universe. In the present paper we discuss this interplay as it is present in three major departments of contemporary physics: general relativity, quantum mechanics and some attempts at quantizing gravity (especially geometrodynamics and its recent successors in the form of various pregeometry conceptions). It turns out that all big interpretative issues involved in this problem point towards the necessity (...) of changing from the standard space-time geometry to some radically new, most probably non-local, generalization. We argue that the recent noncommutative geometry offers attractive possibilities, and gives us a conceptual insight into its algebraic foundations. Noncommutative spaces are, in general, non-local, and their applications to physics, known at present, seem very promising. One would expect that beneath the Planck threshold there reigns a "noncommutative pregeometry", and only when crossing this threshold does the usual space-time geometry emerges. (shrink)

Compact Bilinear Logic , introduced by Lambek [14], arises from the multiplicative fragment of Noncommutative Linear Logic of Abrusci [1] by identifying times with par and 0 with 1. In this paper, we present two sequent systems for CBL and prove the cut-elimination theorem for them. We also discuss a connection between cut-elimination for CBL and the Switching Lemma from [14].

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

Linear Logic is a versatile framework with diverse applications in computer science and mathematics. One intriguing fragment of Linear Logic is Multiplicative-Additive Linear Logic (MALL), which forms the exponential-free component of the larger framework. Modifying MALL, researchers have explored weaker logics such as Noncommutative MALL (Bilinear Logic, BL) and Cyclic MALL (CyMALL) to investigate variations in commutativity. In this paper, we focus on Cyclic Nonassociative Bilinear Logic (CyNBL), a variant that combines noncommutativity and nonassociativity. We introduce a sequent system (...) for CyNBL, which includes an auxiliary system for incorporating nonlogical axioms. Notably, we establish the cut elimination property for CyNBL. Moreover, we establish the strong conservativeness of CyNBL over Full Nonassociative Lambek Calculus (FNL) without additive constants. The paper highlights that all proofs are constructed using syntactic methods, ensuring their constructive nature. We provide insights into constructing cut-free proofs and establishing a logical relationship between CyNBL and FNL. (shrink)

A suitable unified statistical formulation of quantum and classical mechanics in a *-algebraic setting leads us to conclude that information itself is noncommutative in quantum mechanics. Specifically we refer here to an observer's information regarding a physical system. This is seen as the main difference from classical mechanics, where an observer's information regarding a physical system obeys classical probability theory. Quantum mechanics is then viewed purely as a mathematical framework for the probabilistic description of noncommutative information, with the (...) projection postulate being a noncommutative generalization of conditional probability. This view clarifies many problems surrounding the interpretation of quantum mechanics, particularly problems relating to the measuring process. (shrink)

We consider the Lambek calculus, or noncommutative multiplicative intuitionistic linear logic, extended with iteration, or Kleene star, axiomatised by means of an$\omega $-rule, and prove that the derivability problem in this calculus is$\Pi _1^0$-hard. This solves a problem left open by Buszkowski (2007), who obtained the same complexity bound for infinitary action logic, which additionally includes additive conjunction and disjunction. As a by-product, we prove that any context-free language without the empty word can be generated by a Lambek grammar (...) with unique type assignment, without Lambek’s nonemptiness restriction imposed (cf. Safiullin, 2007). (shrink)