Algebras of creation and destruction operators

A general analysis of bilinear algebras of creation and destruction operators is performed. Generalizing the earlier work on the single-parameterq-deformation of the Heisenberg algebra, we study two-parameter and four-parameter algebras. Two new forms of quantum statistics called orthofermi and orthobose statistics and aq-deformation interpolating between them have been found. In the Fock representation, quadratic relations among destruction operators, wherever they are allowed, are shown to follow from the bilinear algebra of creation and destruction operators. Postitivity of the Hilbert space for the four-parameter algebra has been studied in the two-particle sector, but for the two-parameter algebra, results are presented up to the four-particle sector.     

Algebraical comparison of classical and quantum polynomial observables

The symplectic vector spaceE of theq andp's of classical mechanics allows a basis free definition of the Poisson bracket in the symmetric algebra overE. Thus the symmetric algebra overE becomes a Lie algebra, which can be compared with the quantum mechanical Weyl algebra with its commutator Lie structure. The universality of the Weyl algebra is used to study the well-known ‘classical’ Moyal realisation of the Weyl algebra in the symmetric algebra. Quantisations are defined as linear mappings of the underlying vector spaces of the two algebras. It is shown that the classical Lie algebra is −2 graded, whereas the quantum Lie algebra is not. This proves that they are not isomorphic, and hence there is no Dirac quantisation.     

An Introduction to Many Worlds in Quantum Computation

The interpretation of quantum mechanics is an area of increasing interest to many working physicists. In particular, interest has come from those involved in quantum computing and information theory, as there has always been a strong foundational element in this field. This paper introduces one interpretation of quantum mechanics, a modern ‘many-worlds’ theory, from the perspective of quantum computation. Reasons for seeking to interpret quantum mechanics are discussed, then the specific ‘neo-Everettian’ theory is introduced and its claim as the best available interpretation defended. The main objections to the interpretation, including the so-called “problem of probability” are shown to fail. The local nature of the interpretation is demonstrated, and the implications of this both for the interpretation and for quantum mechanics more generally are discussed. Finally, the consequences of the theory for quantum computation are investigated, and common objections to using many worlds to describe quantum computing are answered. We find that using this particular many-worlds theory as a physical foundation for quantum computation gives several distinct advantages over other interpretations, and over not interpreting quantum theory at all.     

Homotopy Approach to Quantum Gravity

Gravity may be a quantum-space-time effect. General relativity is quantized by small generic changes in its commutation relations that make its Lie algebras simple on all levels, positing extra variables frozen by self-organization as needed. This quantizes space-time coordinates as well as fields and eliminates physical singularities. Fermi statistics and sl (nℝ) Lie algebras are assumed for all levels. Spin 1/2 is taken to be anomalous, arising from vacuum organization; the spin-statistics relation is incorporated. The gravitational field is quartic in Fermi variables. Einstein’s non-commutativity of parallel transport emerges as a vestige of Heisenberg’s quantum non-commutativity near the classical limit.     

Computational Power of Infinite Quantum Parallelism

Recent works have independently suggested that quantum mechanics might permit procedures that fundamentally transcend the power of Turing Machines as well as of ‘standard’ Quantum Computers. These approaches rely on and indicate that quantum mechanics seems to support some infinite variant of classical parallel computing. We compare this new one with other attempts towards hypercomputation by separating (1) its %principal computing capabilities from (2) realizability issues. The first are shown to coincide with recursive enumerability; the second are considered in analogy to ‘existence’ in mathematical logic. PACS (2003): 03.67. Supported by DFG project Zi1009/1-1.     

Glafka–2004 ‘Iconoclasm’: The Soul and Spirit of the Meeting

The following is a brief talk that opened and attempted to set the atmosphere for the first ‘Glafka–2004: Iconoclastic Approaches to Quantum Gravity’ international theoretical physics conference. It aimed to capture the general spirit of the meeting, as well as to inspire and unite its participants under the following envisioned ‘cause’: to bring together and scrutinize certain important current quantum gravity research approaches in a fresh, unconventional, almost unorthodox, way.Introductory remarks to the 1st Glafka–2004: Iconoclastic Approaches to Quantum Gravity international theoretical physics conference, held in Athens, Greece (summer 2004).     

Quantum geons and noncommutative spacetimes

Physical considerations strongly indicate that spacetime at Planck scales is noncommutative. A popular model for such a spacetime is the Moyal plane. The Poincaré group algebra acts on it with a Drinfel’d-twisted coproduct, however the latter is not appropriate for more complicated spacetimes such as those containing Friedman-Sorkin (topological) geons. They have rich diffeomorphisms and mapping class groups, so that the statistics groups for N identical geons is strikingly different from the permutation group S _N . We generalise the Drinfel’d twist to (essentially all) generic groups including finite and discrete ones, and use it to deform the commutative spacetime algebras of geons to noncommutative algebras. The latter support twisted actions of diffeomorphisms of geon spacetimes and their associated twisted statistics. The notion of covariant quantum fields for geons is formulated and their twisted versions are constructed from their untwisted counterparts. Non-associative spacetime algebras arise naturally in our analysis. Physical consequences, such as the violation of Pauli’s principle, seem to be one of the outcomes of such nonassociativity. The richness of the statistics groups of identical geons comes from the nontrivial fundamental groups of their spatial slices. As discussed long ago, extended objects like rings and D-branes also have similar rich fundamental groups. This work is recalled and its relevance to the present quantum geon context is pointed out.     

de Broglie-Bohm formulation of quantum mechanics, quantum chaos and breaking of time-reversal invariance

A recently developed unified theory of classical and quantum chaos, based on the de Broglie-Bohm (Hamilton-Jacobi) formulation of quantum mechanics is presented and its consequences are discussed. The quantum dynamics is rigorously defined to be chaotic if the Lyapunov number, associated with the quantum trajectories in de Broglie-Bohm phase space, is positive definite. This definition of quantum chaos which under classical conditions goes over to the well-known definition of classical chaos in terms of positivity of Lyapunov numbers, provides a rigorous unified definition of chaos on the same footing for both the dynamics. A demonstration of the existence of positive Lyapunov numbers in a simple quantum system is given analytically, proving the existence of quantum chaos. Breaking of the time-reversal symmetry in the corresponding quantum dynamics under chaotic evolution is demonstrated. It is shown that the rigorous deterministic quantum chaos provides an intrinsic mechanism towards irreversibility of the Schrodinger evolution of the wave function, without invoking ‘wave function collapse’ or ‘measurements’     

Can Spacetime be a Condensate?

We explore further the proposal [Hu, B. L. (1996). General relativity as geometro-hydrodynamics. (Invited talk at the Second Sakharov Conference, Moscow, May 1996); gr-qc/9607070.] that general relativity is the hydrodynamic limit of some fundamental theories of the microscopic structure of spacetime and matter, i.e., spacetime described by a differentiable manifold is an emergent entity and the metric or connection forms are collective variables valid only at the low-energy, long-wavelength limit of such micro-theories. In this view it is more relevant to find ways to deduce the microscopic ingredients of spacetime and matter from their macroscopic attributes than to find ways to quantize general relativity because it would only give us the equivalent of phonon physics, not the equivalents of atoms or quantum electrodynamics.It may turn out that spacetime is merely a representation of certain collective state of matter in some limiting regime of interactions, which is the view expressed by Sakharov [Sakharov, A. D. (1968). Soviet Physics-Doklady 12, 1040–1041; Sakharov, A. D. (1967). Vacuum quantum fluctuations in curved space and the theory of gravitation. Doklady Akad. Nauk S.S.R. 177, 70; Adler, S. L. (1982). Reviews of Modern Physics 54, 729]. In this talk, working within the conceptual framework of geometro-hydrodynamics, we suggest a new way to look at the nature of spacetime inspired by Bose–Einstein condensate (BEC) physics. We ask the question whether spacetime could be a condensate, even without the knowledge of what the‘atom of spacetime’ is. We begin with a summary of the main themes for this new interpretation of cosmology and spacetime physics, and the ‘bottom-up’ approach to quantum gravity. We then describe the ‘Bosenova’ experiment of controlled collapse of a BEC and our cosmology-inspired interpretation of its results. We discuss the meaning of a condensate in different context. We explore how far this idea can sustain, its advantages and pitfalls, and its implications on the basic tenets of physics and existing programs of quantum gravity.     

Tensor product composition of algebras in Bose and Fermi canonical formalism

We consider a graded algebra with two products (σ, α) over anε-factor commutation. One of the products (σ) isε-commutative, but, in general non-associative; and the other (α) is a graded Lieε-product and a gradedε-derivative with respect to the first (σ). Using the obvious mathematical condition, namely—the tensor product of two graded algebras with the sameε-factors is another with the sameε-factor, we determine the complete structure of a two-product (σ, α) graded algebra. When theε-factors are taken to be unity and the gradation structure is ignored, we recover the algebras of the physical variables of classical and quantum systems, considered by Grgin and Petersen. With the retention of the gradation structure and the possible choice of two ε-factors we recover the algebras of the canonical formalism of boson and fermion systems for the above classical and quantum theories. We also recover in this case the algebra of anticommutative classical systems considered by Martin along with its quantum analogue.     

$${\phi}$$ -Coordinated Quasi-Modules for Quantum Vertex Algebras

We develop a theory of f{\phi} -coordinated (quasi-) modules for a general nonlocal vertex algebra where f{\phi} is what we call an associate of the one-dimensional additive formal group. By specializing f{\phi} to a particular associate, we obtain a new construction of weak quantum vertex algebras in the sense of Li (Selecta Mathematica (New Series) 11:349–397, 2005). As an application, we associate weak quantum vertex algebras to quantum affine algebras, and we also associate quantum vertex algebras and f{\phi} -coordinated modules to a certain quantum βγ-system explicitly.     

q-Deformation of algebraic structures of quantum and classical mechanics

There exists a coassociative and cocommutative coproduct in the linear space spanned by the two algebraic products of a classical Hamilton algebra (the algebraic structure underlying classical mechanics [1]). The transition from classical to quantum Hamilton algebra (the algebraic structure underlying quantum mechanics) is anħ-deformation which preserves not only the Lie property of the classical Hamilton algebra but also the coassociativity and cocommutativity of the above coproduct. By explicit construction we obtain the algebraic structures of theq-deformed Hamilton algebras which preserve the said properties of the coproduct. Some algorithms of these structures are obtained and their implications discussed. The problem of consistency of time evolution with theq-deformed kinematical structure is discussed. A characteristic distinction between the parametersħ andq is brought out to stress the fact thatq cannot be regarded as a fundamental constant.     

The algebra and geometry ofSU(3) matrices

We give an elementary treatment of the defining representation and Lie algebra of the three-dimensional unitary unimodular groupSU(3). The geometrical properties of the Lie algebra, which is an eight dimensional real linear vector space, are developed in anSU(3) covariant manner. Thef andd symbols ofSU(3) lead to two ways of ‘multiplying’ two vectors to produce a third, and several useful geometric and algebraic identities are derived. The axis-angle parametrization ofSU(3) is developed as a generalization of that forSU(2), and the specifically new features are brought out. Application to the dynamics of three-level systems is outlined.     

Semiclassical statistical mechanics ofv-dimensional fluid mixture of hardv-spheres

The problem of calculating the equilibrium properties ofv-dimensional fluid mixture of hardv-spheres is studied. High temperature expansion for the density independent radial distribution function is derived for a hardv-sphere mixture. The ‘excess’ quantum corrections to the second virial coefficient and the excess free energy are also studied. Significant features are the large increase in ‘excess’ quantum correction with increasing dimensionality.     

Quasi-Hopf Deformations of Quantum Groups

The search for elliptic quantum groups leads to a modified quantum Yang–Baxter relation and to a special class of quasi-triangular quasi-Hopf algebras. This Letter calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac–Moody algebras are more rigid than their loop algebra quotients and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with sl(n) are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang–Baxter relation and are calculated more or less explicitly. The modified classical Yang–Baxter relation is obtained and the elliptic solutions are worked out explicitly.The same method is used to construct the Universal R-matrices associated with Felder's quantization of the Knizhnik–Zamolodchikov–Bernard equation, to throw some light on the quasi-Hopf structure of conformal field theory and (perhaps) the Calogero–Moser models.     

Quantum Field Theory on Spacetimes with a Compactly Generated Cauchy Horizon

We prove two theorems which concern difficulties in the formulation of the quantum theory of a linear scalar field on a spacetime, , with a compactly generated Cauchy horizon. These theorems demonstrate the breakdown of the theory at certain base points of the Cauchy horizon, which are defined as ‘past terminal accumulation points’ of the horizon generators. Thus, the theorems may be interpreted as giving support to Hawking's ‘Chronology Protection Conjecture’, according to which the laws of physics prevent one from manufacturing a ’time machine‘. Specifically, we prove: Theorem 1. There is no extension to of the usual field algebra on the initial globally hyperbolic region which satisfies the condition of F-locality at any base point. In other words, any extension of the field algebra must, in any globally hyperbolic neighbourhood of any base point, differ from the algebra one would define on that neighbourhood according to the rules for globally hyperbolic spacetimes. Theorem 2. The two-point distribution for any Hadamard state defined on the initial globally hyperbolic region must (when extended to a distributional bisolution of the covariant Klein-Gordon equation on the full spacetime) be singular at every base point x in the sense that the difference between this two point distribution and a local Hadamard distribution cannot be given by a bounded function in any neighbourhood (in M × M) of (x,x). In consequence of Theorem 2, quantities such as the renormalized expectation value of φ² or of the stress-energy tensor are necessarily ill-defined or singular at any base point. The proof of these theorems relies on the ‘Propagation of Singularities’ theorems of Duistermaat and H?rmander. Received: 14 March 1996/Accepted: 11 June 1996     

On the fundamental representation of borcherds algebras with one imaginary simple root

Borcherds algebras represent a new class of Lie algebras which have almost all the properties that ordinary Kac-Moody algebras have, but the only major difference is that these generalized Kac-Moody algebras are allowed to have imaginary simple roots. The simplest nontrivial examples one can think of are those where one adds by hand one imaginary simple root to an ordinary Kac-Moody algebra. We study the fundamental representation of this class of examples and prove that an irreducible module is given by the full tensor algebra over some integrable highest weight module of the underlying Kac-Moody algebra. We also comment on possible realizations of these Lie algebras in physics as symmetry algebras in quantum field theory.Supported by Konrad-Adenauer-Stiftung e.V.Supported by Deutsche Forschungsgemeinschaft.     

An invitation to higher gauge theory

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2-group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2-group’, which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2-group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2-group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2-group’. We also touch upon higher structures such as the ‘gravity 3-group’, and the Lie 3-superalgebra that governs 11-dimensional supergravity.     

Division Algebras and Quantum Theory

Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’. It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly ‘complex’ representations), those that are self-dual thanks to a symmetric bilinear pairing (which are ‘real’, in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are ‘quaternionic’, in that they are the underlying complex representations of representations on quaternionic Hilbert spaces). This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. More generally, Hilbert spaces of any one of the three kinds—real, complex and quaternionic—can be seen as Hilbert spaces of the other kinds, equipped with extra structure.     

Quantum Einstein’s equations and constraints algebra

In this paper we shall address this problem: Is quantum gravity constraints algebra closed and what are the quantum Einstein’s equations. We shall investigate this problem in the de-Broglie-Bohm quantum theory framework. It is shown that the constraint algebra is weakly closed and the quantum Einstein’s equations are derived.