Time-energy uncertainty and relativistic canonical commutation relations in quantum spacetime

It is shown that the time operatorQ ⁰ appearing in the realization of the RCCR's [Q,P^v]=–jhg^v, on Minkowski quantum spacetime is a self adjoint operator on Hilbert space of square integrable functions over _m =×v _m , where is a timelike hyperplane. This result leads to time-energy uncertainty relations that match their space-momentum counterparts. The operators Q appearing in Born's metric operator in quantum spacetime emerge as internal spacetime operators for exciton states, and the condition that the metric operator should possess a ground exciton state assumes the significance of achieving minimal spacetime4-momentum uncertainty in fundamental standards for spacetime measurements.Supported in part by NSERC research grant No. A5206.     

Flux vacua in DBI type Einstein–Maxwell theory

We study compactification of extra dimensions in a theory of Dirac–Born–Infeld type gravity. We investigate the solution for Minkowski spacetime with an S ² extra space as well as that for de Sitter spacetime (S ⁴) with an S ² extra space. They are derived by the effective potential method in the presence of the magnetic flux on the extra sphere. We also consider the higher-dimensional generalization of the solutions. We find that, in a certain model, the radius of the extra space has a minimum value independent of the higher-dimensional Newton constant.     

Noncommutative parameters of quantum symmetries and star products

The star product technique translates the framework of local fields on noncommutative spacetime into nonlocal fields on standard spacetime. We consider the example of fields on κ-deformed Minkowski space, transforming under κ-deformed Poincaré group, with noncommutative parameters. By extending the star product to the tensor product of functions on κ-deformed Minkowski space and κ-deformed Poincaré group we represent the algebra of noncommutative parameters of deformed relativistic symmetries by functions on classical Poincaré group.     

Amplitude phase-space model for quantum mechanics

We show that there is a close relationship between quantum mechanics and ordinary probability theory. The main difference is that in quantum mechanics the probability is computed in terms of an amplitude function, while in probability theory a probability distribution is used. Applying this idea, we then construct an amplitude model for quantum mechanics on phase space. In this model, states are represented by amplitude functions and observables are represented by functions on phase space. If we now postulate a conjugation condition, the model provides the same predictions as conventional quantum mechanics. In particular, we obtain the usual quantum marginal probabilities, conditional probabilities and expectations. The commutation relations and uncertainty principle also follow. Moreover Schrödinger's equation is shown to be an averaged version of Hamilton's equation in classical mechanics.     

Doubled conformal compactification

We use Weyl transformations between the Minkowski spacetime and dS/AdS spacetime to show that one cannot well define the electrodynamics globally on the ordinary conformal compactification of the Minkowski spacetime (or dS/AdS spacetime), where the electromagnetic field has a sign factor (and thus is discountinuous) at the light cone. This problem is intuitively and clearly shown by the Penrose diagrams, from which one may find the remedy without too much difficulty. We use the Minkowski and dS spacetimes together to cover the compactified space, which in fact leads to the doubled conformal compactification. On this doubled conformal compactification, we obtain the globally well-defined electrodynamics.     

κ-Minkowski spacetime and the star product realizations

We investigate a Lie algebra-type κ-deformed Minkowski spacetime with undeformed Lorentz algebra and mutually commutative vector-like Dirac derivatives. There are infinitely many realizations of κ-Minkowski space. The coproduct and the star product corresponding to each of them are found. An explicit connection between realizations and orderings is established and the relation between the coproduct and the star product, provided through an exponential map, is proved. Utilizing the properties of the natural realization, we construct a scalar field theory on κ-deformed Minkowski space and show that it is equivalent to the scalar, nonlocal, relativistically invariant field theory on the ordinary Minkowski space. This result is universal and does not depend on the realizations, i.e. the orderings, used.     

The unpredictability of quantum gravity

Quantum gravity seems to introduce a new level of unpredictability into physics over and above that normally associated with the uncertainty principle. This is because the metric of spacetime can fluctuate from being globally hyperbolic. In other words, the evolution is not completely determined by Cauchy data at past or future infinity. I present a number of axioms that the asymptotic Green functions should obey in any reasonable theory of quantum gravity. These axioms are the same as for ordinary quantum field theory in flat spacetime, except that one axiom, that of asymptotic completeness, is omitted. This allows pure quantum states to decay into mixed states. Calculations with simple models of topologically non-trivial spacetime indicate that such loss of quantum coherence will occur but that the effect will be very small except for fundamental scalar particles, if any such exist.     

Lattice Wess-Zumino-Witten model and quantum groups

Quantum groups play the role of symmetries of integrable theories in two dimensions. They may be detected on the classical level as Poisson-Lie symmetries of the corresponding phase spaces. We discuss specifically the Wess-Zumino-Witten conformally invariant quantum field model combining two chiral parts which describe the left- and right-moving degrees of freedom. On one hand, the quantum group plays the role of the symmetry of the chiral components of the theory. On the other hand, the model admits a lattice regularization (in Minkowski space) in which the current algebra symmetry of the theory also becomes quantum, providing the simplest example of a quantum group symmetry coupling space-time and internal degrees of freedom. We develop a free field approach to the representation theory of the lattice sl (2)-based current algebra and show how to use it to rigorously construct an exact solution of the quantum SL (2) WZW model on lattice.     

How Does an Electron Tell the Time?

We examine a model of a digital clock to clarify the origin of the spacetime approach in special relativity. Specifically, we consider a two photon clock and assemble a statistical mechanics of such clocks to see how Minkowski space relates to local finite frequency clock behaviour. The result suggests that finite frequency clocks measure spacetime area and it is this feature that provides a simple mechanism behind Minkowski space on large scales. The same feature appears to implicate quantum mechanics on small scales.     

Topological Geometrodynamics

The identification of spacetime as a 4-surface in the space H =M⁴×CP₂ (product of Minkowski space and complex projective space of complex dimension two) as means of obtaining Poincare invariant theory of gravitation was the triggering idea of topological geometrodynamics (TGD), which can be regarded as an attempt to unify basic interactions in terms of submanifold geometry instead of abstract manifold geometry as in case of General Relativity. One can however regard TGD also as a generalization of string model: instead of strings free particles are regarded as 3-surfaces. In this article I want to describe these two approaches and to show how they merge into a single coherent scheme provided macroscopic 3-space with matter is identified as a 3-surface containing particles as topological inhomogenities. Also the quantization program of TGD based on the idea that interacting field theory can be regarded as a classical, free field theory for Grassmann algebra valued Schrödinger amplitude in the space of all possible 3-surfaces of H, is described.     

Gravitational Collapse without a Remnant

We investigate the gravitational collapse of a spherically symmetric, inhomogeneous star, which is described by a perfect fluid with heat flow and satisfies the equation of state p=ρ/3 or p=C ρ ^γ at its center. Different from the ordinary process of gravitational collapsing, the energy of the whole star is emitted into space. And the remaining spacetime is a Minkowski one at the end of the process.     

Contrasting Classical and Quantum Vacuum States in Non-inertial Frames

Classical electron theory with classical electromagnetic zero-point radiation (stochastic electrodynamics) is the classical theory which most closely approximates quantum electrodynamics. Indeed, in inertial frames, there is a general connection between classical field theories with classical zero-point radiation and quantum field theories. However, this connection does not extend to noninertial frames where the time parameter is not a geodesic coordinate. Quantum field theory applies the canonical quantization procedure (depending on the local time coordinate) to a mirror-walled box, and, in general, each non-inertial coordinate frame has its own vacuum state. In particular, there is a distinction between the “Minkowski vacuum” for a box at rest in an inertial frame and a “Rindler vacuum” for an accelerating box which has fixed spatial coordinates in an (accelerating) Rindler frame. In complete contrast, the spectrum of random classical zero-point radiation is based upon symmetry principles of relativistic spacetime; in empty space, the correlation functions depend upon only the geodesic separations (and their coordinate derivatives) between the spacetime points. The behavior of classical zero-point radiation in a noninertial frame is found by tensor transformations and still depends only upon the geodesic separations, now expressed in the non-inertial coordinates. It makes no difference whether a box of classical zero-point radiation is gradually or suddenly set into uniform acceleration; the radiation in the interior retains the same correlation function except for small end-point (Casimir) corrections. Thus in classical theory where zero-point radiation is defined in terms of geodesic separations, there is nothing physically comparable to the quantum distinction between the Minkowski and Rindler vacuum states. It is also noted that relativistic classical systems with internal potential energy must be spatially extended and can not be point systems. The classical analysis gives no grounds for the “heating effects of acceleration through the vacuum” which appear in the literature of quantum field theory. Thus this distinction provides (in principle) an experimental test to distinguish the two theories.     

Dark matter from a gas of wormholes

The simplistic model of the classical spacetime foam is considered, which consists of static wormholes embedded in Minkowski spacetime. We explicitly demonstrate that such a foam structure leads to a topological bias of point-like sources which can equally be interpreted as the presence of a dark halo around any point source. It is shown that a non-trivial halo appears on scales where the topological structure possesses local inhomogeneity, while the homogeneous structure reduces to a constant renormalization of the intensity of sources. We also show that in general dark halos possess both (positive and negative) signs depending on scales and specific properties of the topological structure of space.     

Field theories on a lattice

Lattice field theories are described as a way to regularize continuum quantum field theories. They are obtained by replacing ordinary space time by a lattice, space time derivatives by suitable differences and Minkowski by Euclidean space. The connection between a quantum field theory isd space dimension and classical statistical mechanics in (d+1) dimensions is brought outvia elementary examples. The problem of regaining the continuum limit and of handling nonabelian gauge theories are briefly discussed.     

Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion

The perturbative treatment of quantum field theory is formulated within the framework of algebraic quantum field theory. We show that the algebra of interacting fields is additive, i.e. fully determined by its subalgebras associated to arbitrary small subregions of Minkowski space. We also give an algebraic formulation of the loop expansion by introducing a projective system ?^{( n )} of observables “up to n loops”, where ?⁽⁰⁾ is the Poisson algebra of the classical field theory. Finally we give a local algebraic formulation for two cases of the quantum action principle and compare it with the usual formulation in terms of Green's functions. Received: 9 February 2000 / Accepted: 21 March 2000     

Axiomatic Quantum Field Theory in Curved Spacetime

The usual formulations of quantum field theory in Minkowski spacetime make crucial use of features—such as Poincaré invariance and the existence of a preferred vacuum state—that are very special to Minkowski spacetime. In order to generalize the formulation of quantum field theory to arbitrary globally hyperbolic curved spacetimes, it is essential that the theory be formulated in an entirely local and covariant manner, without assuming the presence of a preferred state. We propose a new framework for quantum field theory, in which the existence of an Operator Product Expansion (OPE) is elevated to a fundamental status, and, in essence, all of the properties of the quantum field theory are determined by its OPE. We provide general axioms for the OPE coefficients of a quantum field theory. These include a local and covariance assumption (implying that the quantum field theory is constructed in a local and covariant manner from the spacetime metric and other background structure, such as time and space orientations), a microlocal spectrum condition, an “associativity” condition, and the requirement that the coefficient of the identity in the OPE of the product of a field with its adjoint have positive scaling degree. We prove curved spacetime versions of the spin-statistics theorem and the PCT theorem. Some potentially significant further implications of our new viewpoint on quantum field theory are discussed.     

The differential and cone structures of spacetime

We propose to model spacetime by a differential space rather than by a differential manifold. A differential space is the pair (M, C), where M is any set, and C a family of real functions on M, satisfying certain axioms; C is called a differential structure of a corresponding differential space. This concept suitably generalizes the manifold concept. We show that C can be chosen in such a way that it contains all information about the causal structure of spacetime. This information can be read out of C with the help of only one postulate, namely that physical signals travel along piecewise smooth curves in (M, C). We effectively construct the Minkowski spacetime, with its cone structure, in this way. Some comments are made.     

Glimmers of a Pre-geometric Perspective

Spacetime measurements and gravitational experiments are made by using objects, matter fields or particles and their mutual relationships. As a consequence, any operationally meaningful assertion about spacetime is in fact an assertion about the degrees of freedom of the matter (i.e. non gravitational) fields; those, say for definiteness, of the Standard Model of particle physics. As for any quantum theory, the dynamics of the matter fields can be described in terms of a unitary evolution of a state vector in a Hilbert space. By writing the Hilbert space as a generic tensor product of “subsystems” we analyse the evolution of a state vector on an information theoretical basis and attempt to recover the usual spacetime relations from the information exchanges between these subsystems. We consider generic interacting second quantized models with a finite number of fermionic degrees of freedom and characterize on physical grounds the tensor product structure associated with the class of “localized systems” and therefore with “position”. We find that in the case of free theories no spacetime relation is operationally definable. On the contrary, by applying the same procedure to the simple interacting model of a one-dimensional Heisenberg spin chain we recover the tensor product structure usually associated with “position”. Finally, we discuss the possible role of gravity in this framework.