Classical and Quantum Theory with a New Symmetry (original) (raw)

A Gauge-theoretical Treatment of the Gravitational Field: Classical

2008

In the geometrodynamical setting of general relativity one is concerned mainly with Riemannian metrics over a manifold M . We show that for the space M := Riem(M), we have a natural principal fiber bundle (PFB) structure Diff(M) →֒ M π → M/Diff(M), first hinted at in [1]. This construction makes the gravitational field amenable to exactly the same gauge-theoretic treatment given in [2], where it is used to separate rotational and vibrational degrees of freedom of n-particle systems, both classically and quantum mechanically. Furthermore, we show how the gauge connection in this PFB setting can be seen as a realization of Mach’s Principle of Relative Motion, in accordance with Barbour’s et al work on timeless gravitational theories [3] using best-matching. We show Barbour’s reconstruction of GR is obtained by requiring the connection to be the one induced by the deWitt metric in M. As a simple application of the gauge theory, we put the ADM lagrangian in a Kaluza-Klein context, in wh...

The General-Covariant and Gauge-Invariant Theory of Quantum Particles in Classical Backgrounds

International Journal of Modern Physics D, 2003

A new approach to the concept of particles and their production in quantum field theory is developed. A local operator describing the current of particle density is constructed for scalar and spinor fields in arbitrary gravitational and electromagnetic backgrounds. This enables one to describe particles in a local, general-covariant and gauge-invariant way. However, the current depends on the choice of a 2-point function. There is a choice that leads to the local non-conservation of the current in a gravitational or an electromagnetic background, which describes local particle production consistent with the usual global description based on the Bogoliubov transformation. The most natural choice based on the Green function calculated using the Schwinger–DeWitt method leads to the local conservation of the current, provided that interactions with quantum fields are absent. Interactions with quantum fields lead to the local non-conservation of the current which describes local particle...

Covariant quantum mechanics and quantum symmetries

2002

We sketch the basic ideas and results on the covariant formulation of quantum mechanics on a curved spacetime with absolute time equipped with given gravitational and electromagnetic fields. Moreover, we analyse the classical and quantum symmetries and show their relations.

The Quantum Gauge Principle

Time, Temporality, Now, 1997

We consider the evolution of quantum fields on a classical background space-time, formulated in the language of differential geometry. Time evolution along the worldlines of observers is described by parallel transport operators in an infinite-dimensional vector bundle over the space-time manifold. The time evolution equation and the dynamical equations for the matter fields are invariant under an arbitrary local change of frames along the restriction of the bundle to the worldline of an observer, thus implementing a "quantum gauge principle". We derive dynamical equations for the connection and a complex scalar quantum field based on a gauge field action. In the limit of vanishing curvature of the vector bundle, we recover the standard equation of motion of a scalar field in a curved background space-time.

A general theory of quantum relativity

Physics Letters B, 2004

The geometric form of standard quantum mechanics is compatible with the two postulates: 1) The laws of physics are invariant under the choice of experimental setup and 2) Every quantum observation or event is intrinsically statistical. These postulates remain compatible within a background independent extension of quantum theory with a local intrinsic time implying the relativity of the concept of a quantum event. In this extension the space of quantum events becomes dynamical and only individual quantum events make sense observationally. At the core of such a general theory of quantum relativity is the three-way interplay between the symplectic form, the dynamical metric and non-integrable almost complex structure of the space of quantum events. Such a formulation provides a missing conceptual ingredient in the search for a background independent quantum theory of gravity and matter. The crucial new technical element in our scheme derives from a set of recent mathematical results on certain infinite dimensional almost Kahler manifolds which replace the complex projective spaces of standard quantum mechanics. 1

Symmetries in covariant quantum mechanics

2001

The thesis is divided in two parts. In the first part of the thesis, I present an updated version of the theory of covariant quantum mechanics and of the associated classical background. This theory has originally been proposed by Jadczyk and Modugno and further developed in cooperation with other people. It provides a geometric model for quantum mechanics of a scalar and spin particle in a curved spacetime with absolute time equipped with a given classical gravitational and electromagnetic field. Starting from minimal axioms, this model yields in a covariant way the Schrodinger equation by means of a global Lagrangian formalism and the quantum operators associated with classical quantisable functions by means of classification of distinguished quantum vector fields. In the second part of the thesis, I investigate systematically the infinitesimal symmetries of the classical and quantum geometric structures. This part deals with the main original contributions of the thesis. Some res...

Classical and quantum mechanics of non-abelian gauge fields

Nuclear Physics B, 1984

We i n v estigate the classical and quantum properties of a system of SU(N) non-Abelian Chern-Simons (NACS) particles. After a brief introduction to the subject of NACS particles, we rst discuss about the most general phase space of SU(N) i n ternal degrees of freedom or isospins which can be identied as one of the coadjoint orbits of SU(N) group by the method of symplectic reduction. A complete Dirac's constraint analysis is carried out on each orbit and the Dirac bracket relations among the isospin variables are calculated. Then, the spatial degrees of freedom and interaction with background gauge eld are introduced by considering the phase space of associated bundle which has one of the coadjoint orbit as the ber. Finally, the theory is quantized by using the coherent state method and various quantum mechanical properties are discussed in this approach. In particular, a coherent state representation of the Knizhnik-Zamolodchikov equation is given and possible solutions in this representation are discussed.

On the Integration of General Relativity with Quantum Theory and the Standard Model

2018

We propose (1) that the flat space-time metric that defines the traditional covariant Heisenberg algebra commutation rules of quantum theory between the four-vector position and momentum, be generalized to be the space-time dependent Riemann metric satisfying Einstein’s equations for general relativity (GR), which determine the metric from the energy-momentum tensor. The metric is then a function of the four-vector position operators which are to be expressed in the position representation. This then allows one (2) to recast the Christoffel symbols, and the Riemann and Ricci tensors in Einstein’s GR differential equations for the metric, as an algebra of commutation relations among the four-vector position and momentum operators (a generalized Lie algebra). This then (3) defines the structure constants of the rest of the Poincare algebra with the space-time dependent metric of general relativity tightly integrating it with quantum theory. (4) We propose that the four momentumoperat...

The Project of the Quantum Relativity

The intrinsic unification of the quantum theory and relativity has been discussed here in the light of the last developments. Such development is possible only on the way of the serious deviation from traditional assumptions about a priori spacetime structure and the Yang-Mills generalization of the well known U (1) Abelian gauge symmetry of the classical electrodynamics. In fact, more general gauge theory should be constructed. Formally we deal with the quantum version of the gauge theory of the deformable bodies-the gauge theory of the deformable quantum state. More physically this means that the distance between quantum states is strictly defined value whereas the distance between bodies (particle) is an approximate value, at best. Thereby, all well known solid frames and clocks even with corrections of special relativity should be replaced by the flexible and anholo-nomic quantum setup. Then Yang-Mills arguments about the spacetime coordinate dependence of the gauge unitary rotations should be reversed on the dependence of the spacetime structure on the gauge transformations of the flexible quantum setup. One needs to build " inverse representation " of the unitary transformations by the intrinsic dynamical spacetime transformations. In order to achieve such generalization one needs the general footing for gauge fields and for " matter fields ". Only fundamental pure quantum degrees of freedom like spin, charge, hyper-charges, etc., obey this requirement. One may assume that they correspond some fundamental quantum motions in the manifold of the unlocated quantum states (UQS's). Then " elementary particles " will be represented as a dynamical process keeping non-linear coherent superposition of these fundamental quantum motions.