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arXiv:physics/0611179

Quantum Geometry of the Dynamical Space-time

Peter Leifer

Published Nov 20, 2006• 0 citations
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Abstract

Quantum theory of field (extended) objects without a priori space-time geometry has been represented. Intrinsic coordinates in the tangent fibre bundle over complex projective Hilbert state space $CP(N-1)$ are used instead of space-time coordinates. The fate of quantum system modeled by the generalized coherent states is rooted in this manifold. Dynamical (state-dependent) space-time arises only at the stage of the quantum "yes/no" measurement. The quantum measurement of the gauge ``field shell'' of the generalized coherent state is described in terms of the affine parallel transport of the local dynamical variables in $CP(N-1)$.

DOI
10.48550/arxiv.physics/0611179

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Quantum theory of field (extended) objects without a priori space-time geometry has been represented. Intrinsic coordinates in the tangent fibre bundle over complex projective Hilbert state space CP(N−1)CP(N-1)CP(N−1) are used instead of space-time coordinates. The fate of quantum system modeled by the generalized coherent states is rooted in this manifold. Dynamical (state-dependent) space-time arises only at the stage of the quantum "yes/no" measurement. The quantum measurement of the gauge ``field shell'' of the generalized coherent state is described in terms of the affine parallel transport of the local dynamical variables in CP(N−1)CP(N-1)CP(N−1).
An attempt to build quantum theory of field (extended) objects without a priori space-time geometry has been represented. Space-time coordinates are replaced by the intrinsic coordinates in the tangent fibre bundle over complex projective Hilbert state space CP(N−1)CP(N-1)CP(N−1). The fate of a quantum system modeled by the generalized coherent states is rooted in this manifold. Dynamical (state-dependent) space-time arises only at the stage of the quantum "yes/no" measurement. The quantum measurement of the gauge ``field shell'' of the generalized coherent state is described in terms of the affine parallel transport of the local dynamical variables in CP(N−1)CP(N-1)CP(N−1).
New construction of 4D dynamical space-time (DST) has been proposed in the framework of unification of relativity and quantum theory. Such unification is based solely on the fundamental notion of generalized coherent state (GCS) of N-level system and the geometry of unitary group SU(N) acting in state space CNC^NCN. Neither contradictable notion of quantum particle, nor space-time coordinates (that cannot be a priori attached to nothing) are used in this construction. Morphogenesis of the "field shell"-lump of GCS and its dynamics have been studied for N=2 in DST. The main technical problem is to find non-Abelian gauge field arising from conservation law of the local Hailtonian vector field. The last one may be expressed as parallel transport of local Hamiltonian in projective Hilbert space CP(N−1)CP(N-1)CP(N−1). Co-movable local "Lorentz frame" being attached to GCS is used for qubit encoding result of comparison of the parallel transported local Hamiltonian in infinitesimally close points. This leads to quasi-linear relativistic field equations with soliton-like solutions for "field shell" in emerged DST. The terms "comparison" and "encoding" resemble human's procedure, but here they have objective content realized in invariant quantum dynamics. The dynamical motion of the lump in DST may be associated with "kinesis" time whereas the evolution parameter describing morphogenesis of GCS evolving in CP(N−1)CP(N-1)CP(N−1), may be naturally identified with "metabole" time.
The reconstruction of the unitary symmetry \cite{TFD} under non-linear dynamical mapping Hilbert space of action amplitudes CNC^NCN onto projective Hilbert space CP(N−1)CP(N-1)CP(N−1) \cite{Le1} has been applied here to the quantum dynamics of elementary vacuum excitations. The "vacuum manifold of virtual action states" is represented here by CP(N−1)CP(N-1)CP(N−1) whereas its tangent vectors define local dynamical variables (LDV's) describing "matter". The conservation laws of LDV's express self-conservation of the "material particles" during continuous evolution being expressed as the affine parallel transport agrees with Fubuni-Study metric, create the "affine gauge potential" as the solution of the partial differential equations. Such procedure embeds the quantum dynamics into dynamical space-time whose state-dependent coordinates arose due to encoding results of quantum measurement by the qubit spinor whose components subjected to Lorentz transformations of "quantum boosts" and "quantum rotations". Thereby, in the framework of super-relativity, the objective character of the quantum measurement is inherently related to the dynamical space-time structure that replaces the notion of "observer".
In the present article I propose a non-linear relativistic 4-d field model originated by the internal dynamics in CP(N-1). There is no initially distinction between `particle' and `field', and the space-time manifold is derivable. The main idea is to base the theory on the relative amplitudes solely. Quantum measurements will be described in terms of the parallel transport of the local dynamical variables and a specific gauge reduction of the full state vector to the Qubit coherent state. I will discuss here field equations of quantum particle arising in the dynamical space-time.
A non-linear relativistic 4D field model of a quantum particle which emerges from the internal dynamics in the quantum phase space CP(N−1)CP(N-1)CP(N−1) is proposed. In this model there is no distinction between `particle' and its `surrounding field', and the space-time manifold emerges from the description of the quantum state. The quantum observables of the `quantum particle field' are described in terms of the affine parallel transport of the local dynamical variables in CP(N−1)CP(N-1)CP(N−1).
Quantum mechanics is traditionally formulated on complex Hilbert spaces, whereas gauge theories are formulated on fiber bundles equipped with connections and curvature. Despite the central role of geometry in modern physics, the Hilbert-space structure of quantum theory is usually postulated rather than derived. In this work, we investigate a differential-geometric formulation in which quantum states are represented as points on a coherence manifold characterized by a radial coherence coordinate and a holonomy-induced phase coordinate. The resulting geometry naturally admits the structure of a fiber bundle whose base space is associated with a coherence lattice and whose fiber corresponds to a phase circle. Within this framework, Hilbert spaces emerge as function spaces defined on the coherence manifold, while Schrödinger dynamics arise from the Laplace–Beltrami operator associated with the underlying geometry. The construction unifies concepts from geometric quantum mechanics, fiber-bundle gauge theory, holonomy, and coherence geometry within a common differential-geometric framework.