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arXiv:0808.3172

Reconstruction of the unitary symmetry in super-relativity

Peter Leifer

Published Aug 23, 2008• 3 citations
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Abstract

The reconstruction of the unitary symmetry \cite{TFD} under non-linear dynamical mapping Hilbert space of action amplitudes $C^N$ onto projective Hilbert space $CP(N-1)$ \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 $CP(N-1)$ 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".

DOI
10.48550/arxiv.0808.3172

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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".
Gauge fields frequently used as an independent construction additional to so-called wave fields of matter. This artificial separation is of course useful in some applications (like Berry's interactions between the "heavy" and "light" sub-systems) but it is restrictive on the fundamental level of "elementary" particles and entangled states. It is shown that the linear superposition of action states and non-linear dynamics of the local dynamical variables form an oscillons of energy representing non-local particles - "lumps" arising together with their "affine gauge potential" agrees with Fubini-Study metric. I use the conservation laws of local dynamical variables (LDV's) during affine parallel transport in complex projective Hilbert space CP(N−1)CP(N-1)CP(N−1) for twofold aim. Firstly, I formulate the variation problem for the ``affine gauge potential" as system of partial differential equations \cite{Le1}. Their solutions provide embedding quantum dynamics into dynamical space-time whose state-dependent coordinates related to the qubit spinor subjected to Lorentz transformations of "quantum boosts" and "quantum rotations". Thereby, the problem of quantum measurement being reformulated as the comparison of LDV's during their affine parallel transport in CP(N−1)CP(N-1)CP(N−1), is inherently connected with space-time emergences. Secondly, the important application of these fields is the completeness of quantum theory. The EPR and Schrödinger's Cat paradoxes are discussed from the point of view of the restored Lorentz invariance due to the affine parallel transport of local Hamiltonian of the soliton-like field.
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).
The relativity to the measuring device in quantum theory, i.e. the covariance of local dynamical variables relative transformations to moving quantum reference frame in Hilbert space, may be achieved only by the rejection of super-selection rule. In order to avoid the subjective nuance, I emphasis that the notion of "measurement" here, is nothing but the covariant differentiation procedure in the functional quantum phase space CP(N−1)CP(N-1)CP(N−1), having pure objective sense of evolution. Transition to the local moving quantum reference frame leads to some particle-like solutions of quasi-linear field PDE in the dynamical space-time. Thereby, the functionally covariant quantum dynamics gives the perspective to unify the Einstein relativity and quantum principles which are obviously contradictable under the standard approaches.
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).
New method of shaping quantum "particle - unparticle" vacuum excitations has been proposed in the framework of unification of relativity and quantum theory. Such unification is based solely on the notion of generalized coherent state (GCS) of N-level system and the geometry of unitary group SU(N) acting in state space CNC^NCN. Initially, neither contradictable notion of quantum particle, nor space-time coordinates (that cannot be a priori attached to nothing) are used in this construction. Quantum measurement of local dynamical variables (LDV) leads to the emergence of 4D dynamical space-time (DST). Morphogenesis of the "field shell" of GCS and its dynamics have been studied for N=2 in DST.
In conventional quantum mechanics, all unitary evolution takes place within the space-time Hilbert space Hxt=L2(Mxt)\mathcal H_{xt}=L^2(\mathcal M_{xt})Hxt​=L2(Mxt​), with time as the sole evolution parameter. The momentum-energy representation φ(k,E)φ(k,E)φ(k,E) is treated merely as a Fourier re-expression of the same state-kinematically equivalent but dynamically inert. Here we restore the fundamental symmetry between the conjugate pairs (x,t)(x,t)(x,t) and (k,E)(k,E)(k,E) by extending the quantum theory to an enlarged Hilbert space Htotal=Hxt⊕HkE\mathcal H_{\text{total}} = \mathcal H_{xt} \oplus \mathcal H_{kE}Htotal​=Hxt​⊕HkE​, within which the momentum-energy sector HkE=L2(MkE)\mathcal H_{kE}=L^2(\mathcal M_{kE})HkE​=L2(MkE​) carries its own autonomous unitary evolution generated by a self-adjoint operator T^\hat{\mathcal T}T^. The resulting structure establishes a meta-symmetry: a symmetry between two conjugate dynamical projections of a single global quantum state. It produces a dual-manifold geometry in which each domain is locally complete yet globally open, with divergent limits in one mapping onto extended regions in the other. Remarkably, the dual-manifold symmetry alone reproduces both the uniform dark-energy background and the exponential boundary mapping near black-hole horizons that underlies Hawking radiation. This framework thus opens a quantum-theoretic route to cosmological phenomena that are ordinarily treated within general relativity.