Classical topology and quantum states

Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac’s ‘quantum mechanics’, then a quantum physicist would not be able to tell a torus from a hole in the ground. We argue that there are indeed such axioms involving observables with smooth time evolution: they contain commutative subalgebras from which the spatial slice of spacetime with its topology (and with further refinements of the axiom, its C ^K - and C ^--structures) can be reconstructed using Gel’fand-Naimark theory and its extensions. Classical topology is an attribute of only certain quantum observables for these axioms, the spatial slice emergent from quantum physics getting progressively less differentiable with increasingly higher excitations of energy and eventually altogether ceasing to exist. After formulating these axioms, we apply them to show the possibility of topology change and to discuss quantized fuzzy topologies. Fundamental issues concerning the role of time in quantum physics are also addressed.     

Higgs physics at future colliders: Recent theoretical developments

I review the physics of the Higgs sector in the standard model and its minimal supersymmetric extension, the MSSM. I will discuss the prospects for discovering the Higgs particles at the upgraded Tevatron, at the large hadron collider, and at a future high-energye ⁺ e ⁻ linear collider with centre-of-mass energy in the 350–800 GeV range, as well as the possibilities for studying their fundamental properties. Some emphasis will be put on the theoretical developments which occurred in the last two years.     

Model for Topological Fermions

 We introduce a model designed to describe charged particles as stable topological solitons of a field with values on the internal space S ³. These solitons behave like particles with relativistic properties like Lorentz contraction and velocity dependence of mass. This mass is defined by the energy of the soliton. In this sense this model is a generalization of the Sine-Gordon model¹(We do not chase the aim to give a four-dimensional generalization of Coleman’s isomorphism between the Sine-Gordon model and the Thirring model which was shown in 2-dimensional space-time) from 1 + 1-dimensions to 3 + 1-dimensions, from S ¹ to S ³. For large distances from the centre of solitons this model tends to a dual U(1)-theory with freely propagating electromagnetic waves. Already at the classical level it describes important effects, which usually have to be explained by quantum field theory, like particle-antiparticle annihilation and the running of the coupling. Received November 30, 1999; revised June 20, 2000; accepted for publication October 2, 2000     

Fuzzy Space-Time Geometry and Particle’s Dynamics

The quantum space-time with Dodson-Zeeman topological structure is studied. In its framework, the states of massive particle m correspond to the elements of fuzzy ordered set (Foset), i.e. the fuzzy points. Due to their partial ordering, m space coordinate x acquires principal uncertainty σ _x . Schroedinger formalism of Quantum Mechanics is derived from consideration of m evolution in fuzzy phase space with minimal number of additional axioms. The possible particle’s interactions on fuzzy manifold are studied and shown to be gauge invariant.     

Braid Group Statistics Implies Scattering in Three-Dimensional Local Quantum Physics

It is shown that quantum fields for massive particles with braid group statistics (Plektons) in three-dimensional space-time cannot be free, in a quite elementary sense: They must exhibit elastic two-particle scattering into every solid angle, and at every energy. This also implies that for such particles there cannot be any operators localized in wedge regions which create only single particle states from the vacuum and which are well-behaved under the space-time translations (so-called temperate polarization- free generators). These results considerably strengthen an earlier “NoGo-theorem for ’free’ relativistic Anyons”.     

A priori probability and localized observers

A physical and mathematical framework for the analysis of probabilities in quantum theory is proposed and developed. One purpose is to surmount the problem, crucial to any reconciliation between quantum theory and space-time physics, of requiring instantaneous wave-packet collapse across the entire universe. The physical starting point is the idea of an observer as an entity, localized in space-time, for whom any physical system can be described at any moment, by a set of (not necessarily pure) quantum states compatible with his observations of the system at that moment. The mathematical starting point is the theory of local algebras from axiomatic relativistic quantum field theory. A function defining thea priori probability of mistaking one local state for another is analysed. This function is shown to possess a broad range of appropriate properties and to be uniquely defined by a selection of them. Through a general model for observations, it is argued that the probabilities defined here are as compatible with experiment as the probabilities of conventional interpretations of quantum mechanics but are more likely to be compatible, not only with modern developments in mathematical physics, but also with a complete and consistent theory of measurement.     

Gravitational field of global monopole with semi-classical effects

We find an exact solution for the space-time of a global monopole by using the vacuum expectation value of the stress energy tensor due to an arbitrary collection of conformal mass less free quantum fields as a source. In a particular situation, the solution is shown to possess an interesting feature like ‘wormholes’ space-time. The monopole exerts no gravitational force on the surrounding matter.     

Identical Quantum Particles and Weak Discernibility

Saunders has recently claimed that “identical quantum particles” with an anti-symmetric state (fermions) are weakly discernible objects, just like irreflexively related ordinary objects in situations with perfect symmetry (Black’s spheres, for example). Weakly discernible objects have all their qualitative properties in common but nevertheless differ from each other by virtue of (a generalized version of) Leibniz’s principle, since they stand in relations an entity cannot have to itself. This notion of weak discernibility has been criticized as question begging, but we defend and accept it for classical cases likes Black’s spheres. We argue, however, that the quantum mechanical case is different. Here the application of the notion of weak discernibility indeed is question begging and in conflict with standard interpretational ideas. We conclude that the introduction of the conceptual resource of weak discernibility does not change the interpretational status quo in quantum mechanics.     

Local cohomology and superselection structure

IfA ^μ is a vector field satisfying ?^μ A ^v ?? ^v A ^μ=0 can one find a scalar field φ such thatA ^μ=?^μφ? A novel quantum analogue of this classical problem incorporating locality is introduced and is shown to generate those super-selection sectors whose charge can be strictly localized. In a 2-dimensional space-time there are further possibilities; in particular, soliton sectors can be generated by this procedure.     

Coherence vs. decoherence in (some) problems of condensed matter physics

We present an ‘overview’ of coherence-to-decoherence transition in certain selected problems of condensed matter physics. Our treatment is based on a subsystem-plus-environment approach. All the examples chosen in this paper have one thing in common — the environmental degrees of freedom are taken to be bosonic and their spectral density of excitations is assumed to be ‘ohmic’. The examples are drawn from a variety of phenomena in condensed matter physics involving, for instance, quantum diffusion of hydrogen in metals, Landau diamagnetism and c-axis transport in high T _c superconductors.     

Demokrit – Planck

Demokrit – Planck A branch of physics exists closely linked to the constant h and associated with atomism. It is this h-physics that Planck originated. But atomism like existence of localized, charged particles with different masses does not follow from this physics, especially the charge quant. Hence Demokrit asserted more then quantum physics is competent to answer.     

Quantum Brownian Motion on Non-Commutative Manifolds: Construction,Deformation and Exit Times

We begin with a review and analytical construction of quantum Gaussian process (and quantum Brownian motions) in the sense of Franz (The Theory of Quantum Levy Processes, [math.PR], 2009), Schürmann (White noise on bioalgebras. Volume 1544 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1993) and others, and then formulate and study in details (with a number of interesting examples) a definition of quantum Brownian motions on those non-commutative manifolds (a la Connes) which are quantum homogeneous spaces of their quantum isometry groups in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009). We prove that bi-invariant quantum Brownian motion can be ‘deformed’ in a suitable sense. Moreover, we propose a non-commutative analogue of the well-known asymptotics of the exit time of classical Brownian motion. We explicitly analyze such asymptotics for a specific example on non-commutative two-torus A_q{\mathcal{A}_\theta} , which seems to behave like a one-dimensional manifold, perhaps reminiscent of the fact that A_q{\mathcal{A}_\theta} is a non-commutative model of the (locally one-dimensional) ‘leaf-space’ of the Kronecker foliation.     

Entanglement and Quantum Superposition of a Macroscopic-Macroscopic system

Two quantum Macro-states and their Macroscopic Quantum Superpositions (MQS) localized in two far apart, space-like separated sites can be non-locally correlated by any entangled couple of single-particles having interacted in the past. This novel “Macro-Macro” paradigm is investigated on the basis of a recent study on an entangled Micro-Macro system involving N≈10⁵ particles. Crucial experimental issues as the violation of Bell’s inequalities by the Macro-Macro system are considered.     

Quantum Instantons with Classical Moduli Spaces

 We introduce a quantum Minkowski space-time based on the quantum group SU(2) _q extended by a degree operator and formulate a quantum version of the anti-self-dual Yang-Mills equation. We construct solutions of the quantum equations using the classical ADHM linear data, and conjecture that, up to gauge transformations, our construction yields all the solutions. We also find a deformation of Penrose's twistor diagram, giving a correspondence between the quantum Minkowski space-time and the classical projective space ℙ³. Received: 10 May 2002 / Accepted: 10 January 2003 Published online: 5 May 2003 Communicated by L. Takhtajan     

Warped Convolutions,Rieffel Deformations and the Construction of Quantum Field Theories

Warped convolutions of operators were recently introduced in the algebraic framework of quantum physics as a new constructive tool. It is shown here that these convolutions provide isometric representations of Rieffel’s strict deformations of C ^*–dynamical systems with automorphic actions of \mathbb Rⁿ{\mathbb R^n} , whenever the latter are presented in a covariant representation. Moreover, the device can be used for the deformation of relativistic quantum field theories by adjusting the convolutions to the geometry of Minkowski space. The resulting deformed theories still comply with pertinent physical principles and their Tomita–Takesaki modular data coincide with those of the undeformed theory; but they are in general inequivalent to the undeformed theory and exhibit different physical interpretations.     

All basic condensed matter physics phenomena and notions mirror in biology — A hypothesis,two examples and a novel prediction

A few billion years of evolutionary time and the complex process of ‘selection’ has given biology an opportunity to explore a variety of condensed matter phenomena and situations, some of which have been discovered by humans in the laboratory, that too only in extreme non-biological conditions such as low temperatures, high purity, high pressure etc., in the last centuries. Biology, at some level, is a complex and self-regulated condensed matter system compared to the ‘inanimate’ condensed matter systems such as liquid ⁴He, liquid water or a piece of graphite. In this article I propose a hypothesis that ‘all basic condensed matter physics phenomena and notions (already known and ones yet to be discovered) mirror in biology’. I explain this hypothesis by considering the idea of ‘Bose condensation’ or ‘momentum space order’ and discuss two known example of quantum magnetism encountered in biology. I also provide some new and rather speculative possibility, from light harvesting in biological photosynthesis, of mesoscopic excition condensation related phenomena at room temperature.