Coherent Spaces

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The notion of a coherent space is a nonlinear version of the notion of a complex Hilbert space: The vector space axioms are dropped while the notion of inner product, now called a coherent product, is kept.

Every coherent space can be uniquely embedded into a Hilbert space, its completed quantum space, by suitably extending the coherent product to an inner product. In the interesting examples, the coherent space is an extended classical phase space, and there is a quantization functor that turns the symmetries of the coherent space into unitary operators in the corresponding quantum space. Thus the quantum space is a representation space for quantum dynamics.

This provides a universal framework for quantization, extending the traditional geometric quantization of finite-dimensional symplectic manifolds to more general situations, and in particular to the quantization of certain classical field theories.

Fields defined by linear field equations can be quantized by means of Klauder spaces, a class of coherent spaces discussed by Neumaier and Ghani Farashahi in Anal. Math. Phys. 12 (2022), 1-47.

We describe another quantization of linear field equations in terms of symplectic and orthogonal Hua spaces (for bosons and fermions), a new class of coherent spaces based on the geometric analysis by Loo-Keng Hua in Trans. Amer. Math. Soc. 57 (1945), 441-481. Their symmetry groups are infinite-dimensional metaplectic or metagonal (spin) groups. They allow one to describe the full quantum scattering behavior of linear field equations in terms of classical scattering and Maslov corrections for the phase of the S-matrix.

joint work with Arash Ghaani Farashahi

This lecture introduces coherent spaces, coherent manifolds, and their quantization. Connections to other fields of mathematics, statistics, and physics are pointed out. Moreover, causality properties and their use in quantum field theory are mentioned.

This lecture introduces coherent spaces. The main connections to some other fields of mathematics and physics are pointed out.
(Superseded by the lecture ''Invitation to Coherent Spaces'' mentioned above.)

This lecture gives an introduction to reproducing kernel Hilbert spaces and their basic properties. To illustrate their power it is shown how to derive simple error estimates for numerical integration.

Reproducing kernel Hilbert spaces and the associated coherent states have applications in complex analysis and group theory, but also many other fields of mathematics, statistics, and physics.

Coherent spaces abstract the essential geometric properties needed to define a reproducing kernel Hilbert space.

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Formal papers

The notion of a coherent space is a nonlinear version of the notion of a complex Euclidean space: The vector space axioms are dropped while the notion of inner product is kept.

Coherent spaces provide a setting for the study of geometry in a different direction than traditional metric, topological, and differential geometry. Just as it pays to study the properties of manifolds independently of their embedding into a Euclidean space, so it appears fruitful to study the properties of coherent spaces independent of their embedding into a Hilbert space.

Coherent spaces have close relations to reproducing kernel Hilbert spaces, Fock spaces, and unitary group representations, and to many other fields of mathematics, statistics, and physics.

This paper is the first of a series of papers and defines concepts and basic theorems about coherent spaces, associated vector spaces, and their topology. Later papers in the series discuss symmetries of coherent spaces, relations to homogeneous spaces, the theory of group representations, C^*-algebras, hypergroups, finite geometry, and applications to quantum physics. While the applications to quantum physics were the main motiviation for developing the theory, many more applications exist in complex analysis, group theory, probability theory, statistics, physics, and engineering.

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:

This paper is one of a series of papers on coherent spaces and their applications, defined in the recent book 'Coherent Quantum Mechanics' by the first author.

The paper studies coherent quantization -- the way operators in the quantum space of a coherent space 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 operator that associates homomorphically with every coherent map a linear operator from the quantum space into itself. This operator generalizes to general symmetry groups of coherent spaces the second quantization procedure for free classical fields. The latter is obtained by specialization to Klauder spaces, whose quantum spaces are the bosonic Fock spaces. A coordinate-free derivation is given of the basic properties of creation and annihilation operators in Fock spaces.

Coherent spaces figure prominently in Chapters 3-7 of my recent book

This book introduces mathematicians, physicists, and philosophers to a new, coherent approach to theory and interpretation of quantum physics, in which classical and quantum thinking live peacefully side by side and jointly fertilize the intuition. The formal, mathematical core of quantum physics is cleanly separated from the interpretation issues.

The book demonstrates that the universe can be rationally and objectively understood from the smallest to the largest levels of modeling. The thermal interpretation featured in this book succeeds without any change in the theory. It involves one radical step, the reinterpretation of an assumption that was virtually never questioned before - the traditional eigenvalue link between theory and observation is replaced by a q-expectation link:

Objective properties are given by q-expectations of products of quantum fields and what is computable from these. Averaging over macroscopic spacetime regions produces macroscopic quantities with negligible uncertainty, and leads to classical physics.

A justification of the thermal interpretation from first principles is given in my paper

Starting from a new principle inspired by quantum tomography rather than from Born's rule, this paper gives a self-contained deductive approach to quantum mechanics and quantum measurement. A suggestive notion for what constitutes a quantum detector and for the behavior of its responses leads to a logically impeccable definition of measurement. Applications to measurement schemes for optical states, position measurements and particle tracks demonstrate the applicability to complex realistic experiments without any idealization.

The various forms of quantum tomography for quantum states, quantum detectors, quantum processes, and quantum instruments are discussed. The traditional dynamical and spectral properties of quantum mechanics are derived from a continuum limit of quantum processes, giving the Lindblad equation for the density operator of a mixing quantum system and the Schrödinger equation for the state vector of a pure, nonmixing quantum system. Normalized density operators are shown to play the role of quantum phase space variables, in complete analogy to the classical phase space variables position and momentum. A slight idealization of the measurement process leads to the notion of quantum fields, whose smeared quantum expectations emerge as reproducible properties of regions of space accessible to measurements.

The new approach is closer to actual practice than the traditional foundations. It is more general, and therefore more powerful. It is simpler and less technical than the traditional approach, and the standard tools of quantum mechanics are not difficult to derive. This makes the new approach suitable for introductory courses on quantum mechanics.

A variety of quotes from the literature illuminate the formal exposition with historical and philosophical aspects.

Many more papers will be placed here in the near future. In particular, some in some draft form already existing papers of this series will discuss

• the quantization of coherent spaces with a compatible manifold (or stratification) structure and related quantum dynamics,
• relations between coherent spaces and C^*-algebras,
• finite coherent spaces related to finite geometries and hypergroups, and
• the classical limit in coherent spaces and related semiclassical expansions.

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