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De Gruyter eBooks

6. Coherent quantum physics

Arnold Neumaier

Published Oct 24, 2019• 0 citations
Publisher

Abstract

Coherent quantum physics is quantum physics in terms of a coherent space and an appropriate coherent product. The kinematical structure and the meaning of the quantities are given by the symmetries (invertible coherent maps) of the coherent space. If Z is a coherent manifold, a canonical symplectic form is often canonically determined by the coherent product through the coherent action principle discussed in Section 6.1. In a quantum mechanical context, Z is indeed a classical phase space or an extended phase space-typically a symplectic manifold, a Poisson manifold, or a circle or line bundle over such a manifold that incorporates the classical action variable (encoding the Berry phase under quantization). For example, the Aharonov-Bohm effect [3] needs the bundle formulation. A coherent space and its quantum space provide a classical and a quantum view of the same physical system, discussed in Section 6.2. Section 6.3 shows that symmetries of coherent spaces can be quantized, that is, promoted to linear operators on the quantum space. This leads to quantum dynamics, which in special (completely integrable) situations can be solved in closed form in terms of classical motions on the underlying coherent space, if the latter has a compatible manifold structure. Close relations to concepts from geometric quantization and Kähler manifolds are pointed out in Section 6.4. In order that the quantum space of a coherent space Z can describe physics, one needs not only a distinguished class of coherent states but a more general concept of state. In Sections 6.5-6.6, we give a general abstract setting for states in Lie ∗-algebras, emphasize the essential mathematical features and the close analogy between classical and quantum physics. We illustrate these generalities with a number of examples in Section 6.7. The coherent action principle is put to use in Section 6.8 for numerical quantum physics and an application to quantum chaos. Spectral issues can, in favorable cases, be handled in terms of dynamical Lie algebras, discussed in Section 6.9.

DOI
10.1515/9783110667387-006

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Coherent quantum physics is quantum physics in terms of a coherent space and an appropriate coherent product. The kinematical structure and the meaning of the quantities are given by the symmetries (invertible coherent maps) of the coherent space. If Z is a coherent manifold, a canonical symplectic form is often canonically determined by the coherent product through the coherent action principle discussed in Section 6.1. In a quantum mechanical context, Z is indeed a classical phase space or an extended phase space-typically a symplectic manifold, a Poisson manifold, or a circle or line bundle over such a manifold that incorporates the classical action variable (encoding the Berry phase under quantization). For example, the Aharonov-Bohm effect [3] needs the bundle formulation. A coherent space and its quantum space provide a classical and a quantum view of the same physical system, discussed in Section 6.2. Section 6.3 shows that symmetries of coherent spaces can be quantized, that is, promoted to linear operators on the quantum space. This leads to quantum dynamics, which in special (completely integrable) situations can be solved in closed form in terms of classical motions on the underlying coherent space, if the latter has a compatible manifold structure. Close relations to concepts from geometric quantization and Kähler manifolds are pointed out in Section 6.4. In order that the quantum space of a coherent space Z can describe physics, one needs not only a distinguished class of coherent states but a more general concept of state. In Sections 6.5-6.6, we give a general abstract setting for states in Lie ∗-algebras, emphasize the essential mathematical features and the close analogy between classical and quantum physics. We illustrate these generalities with a number of examples in Section 6.7. The coherent action principle is put to use in Section 6.8 for numerical quantum physics and an application to quantum chaos. Spectral issues can, in favorable cases, be handled in terms of dynamical Lie algebras, discussed in Section 6.9.
This paper defines coherent manifolds and discusses their properties and their application in quantum mechanics. Every coherent manifold with a large group of symmetries gives rise to a Hilbert space, the completed quantum space of ZZZ, which contains a distinguished family of coherent states labeled by the points of the manifold. The second quantization map in quantum field theory is generalized to quantization operators on arbitrary coherent manifolds. It is shown how the Schrödinger equation on any such completed quantum space can be solved in terms of computations only involving the coherent product. In particular, this applies to a description of Bosonic Fock spaces as completed quantum spaces of a class of coherent manifolds called Klauder spaces.
This paper is a programmatic article presenting an outline of a new view of the foundations of quantum mechanics and quantum field theory. In short, the proposed foundations are given by the following statements: * Coherent quantum physics is physics in terms of a coherent space consisting of a line bundle over a classical phase space and an appropriate coherent product. * The kinematical structure of quantum physics and the meaning of the fundamental quantum observables are given by the symmetries of this coherent space, their infinitesimal generators, and associated operators on the quantum space of the coherent space. * The connection of quantum physics to experiment is given through the thermal interpretation. The dynamics of quantum physics is given (for isolated systems) by the Ehrenfest equations for q-expectations.
Abstract This paper studies coherent quantization, the way operators in the quantum space of a coherent space—defined in the recent book ’Coherent Quantum Mechanics’ by the first author—can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties. The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization of coherent maps. The importance of coherent maps for quantum mechanics is due to the fact that there is a quantization map that associates homomorphically with every coherent map a linear operator from the quantum space into itself. The quantization map generalizes the second quantization procedure for free classical fields to symmetry groups of general coherent spaces. Field quantization is obtained by specialization to Klauder spaces, whose quantum spaces are the bosonic Fock spaces. Implied by the new approach is a short, coordinate-free derivation of all basic properties of creation and annihilation operators in Fock spaces.
Mapping methods are often used for the numerical simulation of nonadiabatic systems by propagating classical mapping variable trajectories. A recently popularised mapping method is spin-mapping, whose mapping variables arise from quantum mechanical operators with symmetries described by a Lie-Poisson algebra. Simulating the classical-like dynamics of spin-mapping systems accurately is generally challenging, with many methods unable to preserve the underlying geometric structure of the symplectic form. The Spin-MInt algorithm is a recently proposed algorithm propagating spin-mapping variables, with a direct proof of symplecticity existing only for 2 electronic states. Here, we directly prove the symplecticity of the Spin-MInt algorithm for a general KKK electronic states. A review of the symplectic nature of coadjoint orbits of the su(K)\mathfrak{su}(K)su(K) Lie-Poisson algebra provides the framework needed to understand symplecticity of the Spin-MInt algorithm in this general case. The symplecticity of the method on the associated coadjoint orbit is then shown for what we believe to be the first time via an explicit verification of the symplecticity condition MJMT=J\mathbf{MJ}\mathbf{M}^\textrm{T}=\mathbf{J}MJMT=J exploiting the Lie-Poisson structure of the system. To our knowledge, this is the first time the monodromy matrix for the Spin-MInt algorithm has been explicitly stated using canonical coordinates on the coherent state manifold for a general number of states. We hope that this will assist the development of classical-like spin-mapping methods which might utilise elements of the monodromy matrix, and inform future work on similar symplectic algorithms for coupled and uncoupled Lie-Poisson systems.
It is well known from the work of Berezin [24] (Commun. Math. Phys. 40:153–174, 1975) in 1975 that the quantization problem of a classical mechanical system is closely related with coherent states. In particular coherent states help to understand the limiting behaviour of a quantum system when the Planck constant ℏ\hbar ℏ becomes negligible in macroscopic units. This problem is called the semi-classical limit problem. In this chapter we discuss properties of quantum systems when the configuration space is the Euclidean space Rn\mathbb R^nRn. So that in the Hamiltonian formalism, the phase space is Rn×Rn\mathbb R^n\times \mathbb R^nRn×Rn with its canonical symplectic form σ\sigma σ. The quantization problem has many solutions, so we choose a convient one, introduced by Weyl [259] (The Classical Groups, 1997) and Wigner [261] (Group Theory and Its Applications to Quantum Mechanics of Atomic Spectra, 1959). We study the symmetries of Weyl quantization, the operationnal calculus and applications to propagation of observables. We show that Wick quantization is a natural bridge between Weyl quantization and coherent states. Applications are given to the semi-classical limit after introducing an efficient modern tool: the semi-classical measures. We illustrate the general results proved in this chapter by explicit computations for the harmonic oscillator. More applications will be given in the following chapters, in particular concerning propagators and trace formulas for a large class of quantum systems.
Spin quantization has always been an interesting intrinsic feature in quantum mechanics. This thesis discussed the holomorphic polarization method, motivated by geometric quantization, in a new formulation of canonical group quantization on noncotangent bundle phase space to produce spin quantization. The first part focuses on determining the one-dimensional complex projective space CP1 as a compact phase space and a special unitary group of degree two SU(2) as its canonical group that is not in the semi-direct product form. The emergence of the hidden discrete symmetry which is not deducible from the Lie algebraic structure of SU(2) indicates that it is the double-covering group. Thus its global structure is determined through the lifting SU(2) action on the fibre bundle over phase space. The second part focuses on the quantization process with the holomorphic wavefunction determined through the holomorphic local section of the fibre bundle and the natural polarization arises through the unitary irreducible representation of SU(2) that does not follow Mackey’s induced representation theory. From the representation operators, a set of spin angular momentum operators is generated as complex differential operators associated with a connection-type term l from action on holomorphic wavefunctions. Such representation operators’ matrix elements and characters are determined as Jacobi polynomials and its application in describing the single-qubit pure state is discussed. In conclusion, it is shown that the holomorphic polarization naturally emerged in the canonical group quantization on non-cotangent bundle phase space and has its application in quantum information theory which arises geometrically from the quantization problem.