Introduction to coherent spaces

Question

Referee this paper: arXiv:1804.01402 by Arnold Neumaier

Please use comments to point to previous work in this direction, and reviews to referee the accuracy of the paper. Feel free to edit this submission to summarise the paper (just click on edit, your summary will then appear under the horizontal line)

(Is this your paper?)

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Abstract: 

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.

Comments

For background about the role that should be played by coherent spaces in my vision for the foundations of physics, see my preprint ''Coherent foundations for quantum mechanics'' and my book

• Coherent Quantum Physics, de Gruyter, Berlin 2019.

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@ArnoldNeumaier

Dear Pr. Neumaier, in all your papers involving the new concept of coherent space, you define a "euclidean space" as a complex vector space with a hermitian scalar product.

In the literature, "euclidean" seems to be reserved for the real case, whereas what your call "euclidean space" is usually referred to as "inner product space", "pre-hilbert space" or "hermitian space".

Could you explain why you depart from the standard definition?

Thank you in advance
 

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@newlog: I defined it this way because in quantum physics one always needs the complex base field. There is no possibility of confusion.

Inner product space and Hermitian space do not exclude (at least not in their name) the case of an indefinite scalar product.

Pre-Hilbert space is inappropriate since while the Euclidean structure implies the Hilbert space structure of the completion, it gives more structure, namely a distinguished dense subspace. This is important for the applications in quantum physics, where all operators of interest should have a common dense domain, in order to eliminate all sorts of problems with domains.

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@ArnoldNeumaier

Dear Pr. Neumaier, thank you for your reply.

Unfortunately, the standard terminology differs from yours, which may turn confusing.

Here is what most textbooks agree about.

Inner-product space
    • Real OR Complex vector space
    • Inner-product <x,y>
        Real OR complex-valued
        Symmetric in the real case: <y,x> = <x,y> OR hermitian (= conjugate-symmetric) in the complex case: <y,x> = <x,y>*
        Linear wrt the 1st argument (in mathematics) OR wrt the 2nd argument (in physics)
            Note that linearity and symmetry/conjugate-symmetry imply <x,0> = <0,x> = 0
            Mathematical terminology: (x,y) -> <x,y> is a (real) symmetric bilinear form (= quadratic form) OR a (complex) hermitian sesquilinear form (= hermitian form)
        Positive-definite: <x,x> > 0 for x != 0 and <x,x> = 0 => x = 0

Special cases
    (regarding dimension)
    • Euclidean space = Finite-dimensional REAL inner-product space
    • Hermitian space = Finite-dimensional COMPLEX inner-product space
    • Unitary spaces = (not necessarily finite-dimensional) COMPLEX inner-product space

    (regarding completion)
    • Pre-Hilbert space = (not necessarily complete) inner-product space
    • Hilbert space = complete inner-product space
        Cauchy completion theorem: any pre-Hilbert space can be densely embedded in a Hilbert space, unique up to hilbert-space isomorphism.

Extensions
    Semi-definite inner-product: <x,x> > 0 for x != 0
    Non-degenerate (= indefinite) inner-product: <x,y> = 0 for all y => x = 0
        Exemple: Pseudo-euclidean space = Finite-dimensional REAL non-degenerate inner-product space
        Minkowski space (= R^4 with metric signature +--- o -+++) is a pseudo-euclidean space

So for your readers, including me, it is easier to translate your "Euclidean space" as a "Complex Inner-product space ", or "Hermitian space" in the finite-dimensional case.

Best regards.

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@newlog:  Of course, you may always read ''complex positive definite inner product space'' in place of my ''Euclidean space'', if that helps you. But terminology is a matter of convenience. It is quite common that a term traditionally used in a more narrow way is later extended to mean something more general. Compare the use of the term ''number'', which was for the Greek a positive real number, with negative numbers and complex numbers allowed only much later. 

Entering "complex Euclidean space" (including the quotation marks) into scholar.google.com produces nearly 4000 references to complex Euclidean spaces in the sense I defined it (and nearly 1000 since 2014). A much cited example is:

• E. Bishop, Differentiable manifolds in complex Euclidean space. Duke Mathematical Journal, 32 (1965), 1-21.

For a recent physics paper, see, e.g.,

• Dorkin, S. M., Kaptari, L. P., Hilger, T., & Kämpfer, B.  Analytical properties of the quark propagator from a truncated Dyson-Schwinger equation in complex Euclidean space. Physical Review C, 89 (2014)., 034005.

Since I define everything precisely and all my spaces are complex, no confusion is possible.

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@ArnoldNeumaier

Dear Pr. Neumaier, thank you very much for your reply.

I guess things are now clear.

Perhaps your explaination/motivation might be included in later editions of your book, or in your forthcoming papers on the subject.
At least for "old-school" readers like me. :)

Best regards.

Answer 1

Review: Introduction to coherent spaces by Arnold Neumaier

Coherent spaces provide a unified geometric description of a fundamental structure underlying quantum and classical physical theories. The theory of coherent spaces offers an alternative to both fundamental descriptions of physical systems embodied in the theories of geometric quantization on one hand and operator algebras on the other.

1. Introduction and motivation

In this work [1], the author develops the new geometric notion of coherent spaces, upon which quantum and classical theories can be based.

To motivate the construction and make a connection to the traditional picture of mechanics, coherent spaces can be thought of as state spaces, i.e., each point of a coherent space may be thought as a state of a mechanical system. The most known examples of state spaces are coherent state manifolds which parametrize coherent states. These spaces are equipped with reproducing kernels responsible for the resolution of unity property of the coherent states. Said differently, the reproducing kernels serve as evaluation functionals [2] of the coherent states reproducing kernel Hilbert spaces.

The main idea of this work is to promote the reproducing kernels to coherent products, which are binary complex functions of positive type on a space, and at the same time strip the space from any other structure.  Thus, at the most basic level, coherent spaces $Z$ are sets of points equipped with a coherent product $Z \times Z \rightarrow \mathbb{C}$. (The positivity of the coherent product is essential to constructions needed for the applications). The properties of the coherent states, which motivated this construction, are derivable from this basic definition. However, coherent spaces will be equipped with additional structures, such as smooth structure, in order to define dynamics and connect to realistic physical systems.

This general definition allows for applications outside physics, however, this will not be treated in this review.

A coherent space generalizes the notion of a Euclidean space equipped with a Hermitian product with the linearity property dropped, the same way as a metric space generalizes the notion of a Euclidean space equipped with a Euclidean metric with the linearity waved out. Just as metric spaces can be naturally equipped by isometries, coherent spaces can be naturally equipped with coherent maps. In quantum and classical systems, the coherent maps generate both symmetries and (integrable) dynamics on the coherent spaces.

Returning to the coherent states motivational example; Coherent states live on the interface between the quantum and the classical. On one hand they are vectors on some quantum Hilbert space, and on the other hand they are parametrized by a classical phase space.  The reflection of this property in the concept of coherent spaces is that every coherent space Z is equipped with a virtually unique quantum space $\mathbb{Q}(Z)$ which is a Euclidean coherent space equipped with the standard Hermitian coherent product. Each point in the coherent space has a corresponding vector in the Euclidean space, however, theses vectors are not orthonormal but their scalar product equals to the coherent product of the underlying points. This construction is of course inspired by the coherent state case. There are two immediate consequences of this fact:

• In the theory of coherent spaces, the notion of quantization of a space, i.e., the assignment of a Hilbert space to a phase space is trivial, as the quantum space can be regarded as the quantization of its coherent space. This quantization is functorial, i.e., this theory has a determined recipe of quantization, (just as in Weyl or Berezin quantization).
• However, there exists a notion of quantization of coherent maps (which will be explained in detail the second article of this series), which is a map $\Gamma : \mathbb{Q}(Z) \rightarrow \mathbb{Q}(Z')$ needed to make the following diagram commutative:

$$\mathbb{Q}(Z)  \xrightarrow{\Gamma_A}  \mathbb{Q}(Z')$$

$$\uparrow \qquad   \qquad  \uparrow$$

$$Z \quad  \xrightarrow{A} \quad Z'$$

This coherent map quantization is also functorial. It generalizes the second quantization map.

The author continues to define subfamilies of coherent spaces which are especially important for physics applications:

• Normal coherent spaces to which any coherent space can be "normalized" by a trivial modification of its coherent product. Belonging to the same normalization class should be a physically transparent equivalence relation.
• Nondegenerate coherent spaces in which the coherent product induces a metric. These spaces are important to physics applications because they include the Kähler case and also because a metric is essential if we want to include fermions.
• Projective coherent spaces: As very well known, quantum mechanical state spaces are projective. The projective coherent spaces replace, in the coherent space formalism, the line bundles of geometric quantization.

The coherent space theory is constructed solely by means of objects of the coherent category (spaces and maps), expressing the aspiration of formulating a fundamental mathematical theory describing physical reality.

The next few sections very briefly describe the concept of coherent spaces as given in the article.

2. Coherent spaces

Definition: A coherent space is a nonempty set Z equipped with a coherent product $K: Z \times Z \rightarrow \mathbb{C}$ of positive type.

Remark: K is of positive type iff:

$$ \sum_{ij}\bar{u}_iK(z_i, z_j)u_j \ge 0, \forall z_i, z_j \in Z, u_i, u_j \in \mathbb{C}$$

Definition: A coherent space is Euclidean coherent space, iff it is a complex vector space $\mathbb{H}$ and its coherent product is given by:

$$K(u_i, u_j)= u_i^*u_j,  u_i, u_j\in \mathbb{H}$$

3. Coherent maps

Definition:  A map $ A: (Z', K') \rightarrow (Z, K)$  is called a coherent map if there exists an adjoint map: $A^*: (Z, K) \rightarrow (Z', K')$  such that:

$$K(z, Az') = K'(A^*z, z'), \forall z\in Z, z'\in Z'$$

4. Quantum spaces

Definition: To every coherent space $(Z, K)$, we associate a Eulidean coherent space $\mathbb{Q}(Z)$ named a quantum space , by associating to each point $z \in Z$ a vector $q_z \in \mathbb{Q}(Z)$, such that:

$$q_z^*q_{z'} = K(z,z')$$

5. Nondegenerate coherent spaces

Definition: A coherent space $(Z, K)$, is called nondegenerate if $K(z, z') = K(z'', z')$ for all $z' \in Z$ implies that $z''=z$.

Proposition: On a nondegenerate coherent space, the distance function:

$$d(z,z') = \sqrt{K(z,z)+K(z', z')-2K(z,z')}$$

is a metric. This is always possible, and the quantum space is unique up to isometry.

6. Projective coherent spaces

Definition: A coherent space $(Z, K)$, is called projective if there exists an action $\mathbb{C} \times Z \rightarrow Z:  (\lambda, z)  \mapsto \lambda z$   such that:  $K(\lambda z, z') = \lambda^e K(z, z')$ .

7. Conclusions

The author defined the concept of coherent spaces which together with their natural operations can provide a candidate for a foundational mathematical theory of quantum and classical physics.

This theory is inspired by the properties of spaces of coherent states; hence it was named coherent spaces. However, its scope is much more general that coherent states per se.

This article will be followed by a series of articles which will include further  physics application of the theory. Preprints of two additional articles of this series were already completed [3], [4].

The article is clearly written with strict mathematical rigor. All propositions were followed by detailed mathematical proofs. The paper shows a deep knowledge in both the traditional mathematical theory of quantum mechanics and in addition in the mathematical basis required for the new concept: The theory of reproducing kernel Hilbert spaces and other functional analytical subjects. The material of the latter subject was clearly summarized in chapter 5. The paper shows a deep knowledge of the applications of this theory outside physics, however these subjects are not covered in this review.

The paper includes an abundance of examples, describing familiar physical models using the new concept of coherent spaces. The examples ease the comprehension for the readership with physics background.

1. [1] Neumaier, A., 2018. Introduction to coherent spaces. arXiv preprint arXiv:1804.01402.

2. [2] Moore, R.E. and Cloud, M.J., 2007. Computational functional analysis. Elsevier.

   [3] Neumaier, A. and Farashahi, A.G., 2018. Introduction to coherent quantization. arXiv preprint arXiv:1804.01400.

   [4] Neumaier, A., 2018. Coherent foundations for quantum mechanics. http://arnold-neumaier.at/ms/cohFound.pdf