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.

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

[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://www.mat.univie.ac.at/~neum/ms/cohFound.pdf