{'fraction_non_alphanumeric': 0.06574592948859374, 'fraction_numerical': 0.006398093809292679, 'mean_word_length': 3.413631937682571, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 32, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Operator-Valued Tensors on Manifolds: A Framework for Field Quantization

In this paper we try to prepare a framework for field quantization. To this end, we aim to replace the field of scalars R by self-adjoint elements of a commutative C-algebra, and reach an appropriate generalization of geometrical concepts on manifolds. First, we put forward the concept of operator-valued tensors and extend semi-Riemannian metrics to operator valued metrics. Then, in this new geometry, some essential concepts of Riemannian geometry such as curvature ten- sor, Levi-Civita connection, Hodge star operator, exterior derivative, divergence,... will be considered.


Introduction
The aim of the present paper is to extend the theory of semi-Riemannian metrics to operator-valued semi-Riemannian metrics, which may be used as a framework for field quantization. Spaces of linear operators are directly related to C * -algebras, so C *algebras are considered as a basic concept in this article. This paper has its roots in Hilbert C ⋆ -modules, which are frequently used in the theory of operator algebras, allowing one to obtain information about C ⋆ -algebras by studying Hilbert C ⋆ -modules over them. Hilbert C ⋆ -modules provide a natural generalization of Hilbert spaces arising when the field of scalars C is replaced by an arbitrary C ⋆ -algebra. This generalization, in the case of commutative C ⋆ -algebras appeared in the paper of Kaplansky [8], however the non-commutative case seemed too complicated at that time.The general theory of Hilbert C ⋆ -modules appeared in the pioneering papers of W. Paschke [13] and M. Rieffel [15].The theory of Hilbert C ⋆ -modules may also be considered as a non-commutative generalization of the theory of vector bundles and non-commutative geometry [3].
A number of results about geometrical structures of Hilbert C ⋆ -modules and about operators on them have been obtained [6]. Henceforth, Hilbert C ⋆ -modules are generalization of inner product spaces that on the level of manifolds, provide a generalization of semi-Riemannian manifolds,that is the goal of this paper. Due to the physical applications in mind, in this paper, the positive definiteness of the inner product will be replaced by non-degeneracy. Another root of this paper is to provide a framework for field quantization. The main idea for quantization is to replace scalars by operators on a Hilbert space. In the field of quantum mechanics, spectrum of operators plays the role of values of the measurements. So replacements of scalars by operators is the first step for quantization. In this direction, it has been done many works [1] [2], but it seems C *algebras are the best candid to play the role of scalars and we must deal with Modules over C * -algebras. Modules, discussed in this article are free finite dimensional modules, and have bases that simplify many computations.
In this article, we only consider commutative C * -algebras for many reason. Noncommutative algebras can be used in the realm of non-commutative Geometry, and it is not our aim to enter in this realm. Many basic definition such as vector field can not be extended properly for non-commutative C * -algebras, because the set of derivations of an algebra is a module on the center of that algebra. So, extension of vector fields are modules on the center of that C * -algebra and we need center of algebra be equal to the algebra, so the algebra must be commutative. Definition of the inner product encounter the same problem. Bilinear maps over a module whose scalars are non-commutative are very restricted. From Physical point of view, commutativity means operators in the C * -algebra represent quantities that are simultaneously measurable and this assumption is not very restricted.
The content of the present paper is structured as follows: Section 1 contains the preliminary facts about of C ⋆ -algebras needed to explain our concepts, Section 2 covers the definition of extended tangent bundle, operator-valued vector fields, and operator-valued tensors and explains some of their basic properties. The Pettis-integral of operatorvalued volume forms are defined and Stokes's theorem is proved in section 3. Section 4 is devoted to operator-valued connections and curvature, and the covariant derivative of operator-valued vector fields will be defined. In section 5, operator-valued inner product and some of their basic properties will be illustrated. The existence,uniqueness and the properties of the Hodge star operator for operator-valued inner product spaces are the goal of Section 6. In section 7 the concepts of section 5 extend to manifolds. The existence and uniqueness of Levi-Civita connection of operator-valued semi-Riemannian metrics is proved in section 8. In the last section Ricci tensor, scalar curvature, and sectional curvature are deliberated.

Review of C ⋆ -Algebras
In this section we review some definitions and results from C ⋆ -algebras that we need in the sequel. Definition 1.1. A Banach ⋆-algebra is a complex Banach algebra A with a conjugate linear involution * which is an anti-isomorphism. That is, for all A, B ∈ A and λ ∈ C, For example the space of all bounded linear operators on a Hilbert space of H is a C ⋆ -algebra. This C ⋆ -algebra is denoted by B (H) Remark 1.3. We consider only unital C ⋆ -algebras, and it's unit element is denoted by 1.
By Gelfand-Naimark theorem, all unital commutative C ⋆ -algebras have the form C(X), in which X is a compact Hausdorff space.
For now on, A is a C * -algebra. The set of all self-adjoint elements of A is denoted by A R . Definition 1.5. The spectrum of an element A in a C ⋆ -algebra is the set Theorem 1.6. The spectrum σ(A) of any element A of a C ⋆ -algebra is a non empty compact set and contained in the {z ∈ C : |z| For each normal element A ∈ A there is a smallest C ⋆ -subalgebra C * (A, 1) of A which contains A, 1, and is isomorphic to C(σ(A)).
An element A ∈ A R is said positive if σ(A) ⊆ R + . The set of positive elements of A R is denoted by A R + . For any A ∈ A, A * A is positive. Theorem 1.8. If A ∈ A R + , then there exists a unique element B ∈ A R + such that B 2 = A. [11] We denote by √ A the unique positive element B such that B 2 = A. If A is a self-adjoint element, then A 2 is positive, and we set |A| = √ A 2 , A + = 1 2 (|A| + A), A − = 1 2 (|A| − A). Elements |A|, A + , and A − are positive and A = A + − A − , A + A − = 0. If A, B ∈ A R , then |AB| = |A||B|. (cf. [11]).

Extending Tangent Bundle
Throughout this paper, A is a commutative unital C ⋆ -algebra which according to the Gelfand-Naimark second theorem can be thought as a C ⋆ -subalgebra of some B (H). Let M be a smooth manifold, we set Let A = C(X) for some compact Hausdorff space X. So, any functionf : M −→ A corresponds to a function f : M × X −→ C. Therefore, when A come to scene it means that we transfer from M to M × X as the base manifold. In Physical applications, M is a space-time manifold and X may have variety of interpretations. For example, X may be viewed as internal space of particles that produce quantum effects. In general, X has no intrinsic relation to M. Smooth A-vector fields can be multiplied by smooth A-valued functions and X(M) A is a module over the ⋆-algebra C ∞ (M, A). For a vector field X ∈ X(M) and a function f ∈ C ∞ (M, A), we define X ⊗ f as a A-vector field by These fields are called simple, and any A-vector field can be written locally as a finite sum of simple A-vector fields. We can identify X and . This operation can be extended to A-vector fields, such that for simple elements of If A is non-commutative, this definition is not well-defined. This definition implies that for f, h ∈ C ∞ (M, A) and X ∈ X(M) A we have The Lie bracket of ordinary vector fields can be extended to A-vector fields as the following, if X, Y ∈ X(M) and h, k ∈ C ∞ (M, A) then, The verification of main properties of the Lie bracket are routine. An involution on X(M) A for simple A-vector field, such as X ⊗ f , is defined by Contravariant and mixed A-valued tensors can be defined in a similar way. Alternating covariant A-tensor fields are called A-differential forms, and the set of all A-differential forms of order k is denoted by A k (M, A). A-differential forms and exterior product and exterior derivation of these forms are special case of vector valued differential forms. The only difference is that A maybe infinite dimensional.
Lie derivation of A-tensor fields along A-vector fields is defined naturally. ForX ∈ X(M) A and a covariant A-tensor field of order k, such asT , Lie derivation ofT alongX is also a covariant A-tensor field of order k and defined as follows. ForỸ 1 , · · · ,Ỹ k ∈ X(M) A we have IfT is a differential form, then its Lie derivation along any A-vector field is also a differential form and Cartan formula for its derivation holds .i.e.

Integration of A-Valued Volume Forms and Stokes's Theorem
We remind the integral of vector valued functions, called Pettis-integral [4]. The Borel σ-algebra over R n is denoted by B n = B(R n ), and suppose that µ is the Lebesgue measure.
If f is Pettis-integrable over V ∈ B n then x is unique and is called Pettis-integral of f over V . We use the notations V f dµ or (P ) V f dµ to show the Pettis-integral of f over V . It is proved that each function f ∈ C c (R n , A) is Pettis integrable over any V ∈ B n (cf. [16]).
Proof. Applying the Pettis-integral's definition and using classical change of variables theorem, one can conclude the desired result .
In the canonical coordinate system, an A-valued volume formω on R n is written as follows: Definition 3.4. Letω be a compactly supported A-valued n-form on R n . Define the integral ofω over R n by, whereω is as (3.1).
Definition 3.6. Let M be an oriented smooth n-manifold, and letω be a A-valued n-form on M. First suppose that the support ofω is compact and include in the domain of a single chart (U, ϕ) which is positively oriented. We define the integral ofω over M as By using the change of variable's theorem one can prove that M ω does not depend on the choice of chart whose domain contains supp( ω). To integrate an arbitrary compact support A-valued n-form, we can use partition of unity as the same as ordinary n-forms.
To prove (ii) Suppose thatω is compactly supported in the domain of a single chart (U, ϕ) that is positively oriented, thus The general case follows from the above result and using partition of unity.
Every ordinary connection on M can be extended uniquely to an A-connection on M. For an ordinary connection on M such as ∇, its extension as an A-connection is defined as follows. If X, Y ∈ XM and h, k ∈ C ∞ (M, A), then define The torsion tensor of a A-connection ∇ is defined by  Proof. The proof is similar to ordinary connections(cf. [14]).
If ∇ 1 , ∇ 2 are A-connections on M, then, the operator T = ∇ 1 − ∇ 2 is a (1, 2) Atensor and the space of A-connections on M is an affine space which is modeled on the space of (1, 2) A-tensors.

Operator-Valued Inner Product
In this section, we introduce the notion A-valued inner products on A-modules, and investigate some of their basic properties . For the case of Hilbert C ⋆ -modules refer to ([9], [10], [13]). For an A-module V , denote the collection of all A-linear mappings from V into A by V ♯ . This space is also an A-module. If V is a free module, then V ♯ is also a free module.
Definition 5.1. Let V be a A-module. A mapping ·, · : V × V −→ A is called an A-valued inner product on V if for all a ∈ A, and x, y, z ∈ V the following conditions hold: Conditions (iii) and (iv) are called non-degeneracy of the inner product. Note that for each x ∈ V the mappingx : V −→ A, defined byx(y) = y, x belongs to V ♯ . One can see that the mapping x −→x from V into V ♯ is conjugate A-linear and non-degenracy of the inner product is equivalent to the bijectivity of this map.
Theorem 5.2. Let V be a finite dimensional free A-module and ·, · : V × V −→ A satisfies (i) and (ii). If ·, · is non-degenerate then for any basis {e i } of V , det( e i , e j ) is invertible in A. Conversely, If for some basis {e i } of V , det( e i , e j ) is invertible, then ·, · is non-degenerate.
Proof. First, assume that for some basis {e i }, det( e i , e j ) is invertible in A. Set g ij = e i , e j , so (g ij ) is an invertible A-valued matrix. Denote the inversion of this matrix by (g ij ). Set e i = g ij e j ; {e i } is also a basis and is called the reciprocal base of {e i }. The characteristic property of the reciprocal base is that e i , e j = δ j i , so for every x ∈ V , if x = λ i e i , then λ i = x, e i . To prove non-degenracy, suppose for all w ∈ V we have w, x = 0. So, for all index i we have x, e i = 0 that implies λ i = 0, so x = 0. For any T ∈ V ♯ , set λ i = T (e i ) so, for x =λ i e i we havex = T .
Conversely, suppose that ·, · is non-degenerate. Let {e i } be an arbitrary basis and set g ij = e i , e j . For each i, consider the A-linear map T i : V −→ A defined by T i (e j ) = δ i j . By non-degenarcy, the map x −→x is bijective and there exists u i ∈ V such that u i = T i , so e j , u i = δ i j . Any u i can be written as u i = a ij e j for some a ij ∈ A. An easy computation shows that the matrix A = (a ij ) is the inverse of the matrix D = (g ij ), In fact, δ i j = e j , u i = u i , e j = a ik e k , e j = a ik e k , e j = a ik g kj Therefore AD = I, so det(A) det(D) = 1 and det(D) is invertible.
Let V be an A module. V , may have an involution that is a conjugate A-linear isomorphism * : V −→ V such that * 2 = 1 V . An inner product ·, · on V is called compatible with the involution if for all x, y ∈ V x * , y * = x, y * In these spaces, if x and y are self-conjugate elements of V , then x, y is self-adjoint. Example 5.3. Let W be a finite dimensional real vector space and ·, · : W × W −→ A R be a symmetric bilinear map such that for some basis {e i } of W , det( e i , e j ) is invertible in A. The tensor product W A = W ⊗ R A is a free A-module and ·, · can be extended uniquely to a A-valued inner product on it as follows. For all The A-module W A has a natural involution that is (u ⊗ a) * = u ⊗ a * and the extended inner product on W A is compatible with this involution. In fact, any inner product on W A which is compatible with this involution is obtained by the above method.
For an inner product on a free finite dimensional A-modules, we define an element of A as its signature. In the ordinary cases, signature is a scalar that is ±1 and is defined by orthonormal bases. Here, we must define signature by arbitrary bases.
Theorem 5.4. Let V be a free finite dimensional A-module and ·, · be an inner product on it. If {e i } is a basis on V , set g ij = e i , e j and g = det(g ij ). Then, g is self-adjoint and |g| g dose not depend on the choice of the base. Proof. Put G = (g ij ), since g * ij = g ji we find t G = G * , therefore g = det(G) = det( t G) = det(G * ) = det(G) * = g * Now, suppose that {e ′ i } is another basis. Set g ′ ij = e ′ i , e ′ j and G ′ = (g ′ ij ) and g ′ = det(G ′ ). For some matrix A = (a j i ) we have e ′ i = a j i e j , so g ′ ij = e ′ i , e ′ j = a k i e k , a l j e l = a k i a l * j e k , e l = a k i a l * j g kl This equality implies that The value ν = |g| g which does not depend on the choice of the base, is called the signature of the inner product. Note that ν −1 = ν and σ(ν) ⊆ {−1, 1}. In the ordinary metrics, ν is exactly (−1) q where q is the index of the inner product.
Note that in free finite dimensional A modules such as V that has an A inner product, V and V ♯ are naturally isomorphic and this isomorphism induces an inner product on V ♯ . So, all results about V , can be considered for V ♯ too.

The Hodge Star Operator
To define the Hodge star operator, first we need the notion of orientation of free modules. Let V be a free finite dimensional A-module. By definition two ordered bases of V have same orientation if determinant of the transition matrix between them is a positive element of A. This is an equivalence relation between the bases of V , but there exist many equivalence classes. We choose one of these classes as an orientation for V and we call it an orientation for V . In this case, we call V an orientated space and every basis in the orientation, is called a proper base of V . Note that there are many orientations on V and it is not appropriate to call some of them positive. Definition 6.1. Let V be an oriented free n-dimensional A-module that has an A-inner product. For each proper base {e i } with reciprocal base {e i }, and g = det(g ij ), set Ω = |g| e 1 ∧ · · · ∧ e n . This tensor is called the canonical volume form of the inner product. g ij = e i , e j , u ij = u i , v j , g = det(g ij ), u = det(u ij ), then, u = det A det A * g = (detA) 2 g. Positivity of det A yields |u| = detA. |g|, henceforth |g| e 1 ∧ · · · ∧ e n = |g| g e 1 ∧ · · · ∧ e n = |g| g Any A-inner product on the free n-dimensional A-module V , in a natural way, can be extended to the space of each exterior powers Λ k V . For α = u 1 ∧ · · · ∧ u k and β = v 1 ∧ · · · ∧ v k , set α, β = det( u i , v j ). If {e i } is a base of V with reciprocal base {e j } , then it is straightforward to check that {e i 1 ∧ · · · ∧ e i k } 1≤i 1 <...<i k ≤n is a basis of Λ k V and its reciprocal base is {e i 1 ∧ · · · ∧ e i k } 1≤i 1 <...<i k ≤n . Remark 6.3. Note that Ω, Ω = ν.
We now define the operation ⋆, called the Hodge star operator which is similar to the Hodge star operator for ordinary metrics. This is a conjugate linear map of Λ k V onto Λ n−k V . The operator depends on the inner product and also on the orientation of V . For each β ∈ Λ k V , µ β ∈ (Λ n−k V ) ♯ is defined by Because of the non-degeneracy of the inner product on Λ n−k V , it follows that there is a unique ⋆β ∈ Λ n−k V such that µ β (α) = α, ⋆β , that is; α, ⋆β = ν β ∧ α, Ω . So The above equation shows that ⋆ is a conjugate linear map, that is i.e.
Since β ∧ α is a multiple of Ω, and the coefficient is ν β ∧ α, Ω , we find that We now summarize the properties of the operator ⋆ in the following theorem.
Proof. The proof is a straightforward computation and is similar to the ordinary case.
All these results, also hold for V ♯ .

Operator-valued Metrics on Manifolds
We now consider operator valued metrics on manifolds.
Definition 7.1. A A-valued semi-Riemannian metric on an smooth manifold M is a smooth map ., . : T M A ⊕ T M A −→ A such that for each p ∈ M, ., . restricts to a A-valued inner product on T p M A that is compatible with its natural involution.
This definition implies that inner product of any two ordinary vector fields on M is an A R -valued function on M.
For each p ∈ M, denote the signature of the inner product on T p M by ν p . The map ν : p → ν p is called the signature function of the metric. This function is continuous on M and we can prove that it is constant on the each connected components of M.
Theorem 7.2. Let M be a connected manifold and ., . be a semi-Riemannian A-valued metric on it. If ν is the signature function of the metric, then ν is constant.
Proof. First, consider the following subset of A.
Clearly, N is nonempty and the values of ν are in N. We can prove that the distance of any two distinct elements of N is greater than 2. suppose a, b ∈ N and a = b. So, Since (ab) 2 = a 2 b 2 = 1, we have σ(ab) ⊂ {−1, 1}. -1 must be in σ(ab), otherwise σ(ab) = {1} that implies ab = 1, hence a = b that is contrary to assumption. So, 2 ∈ σ(1 − ab) that implies (1 − ab) ≥ 2. Consequently, This property of N implies that the induced topology on N is discrete, on the other hand ν(M) must be connected, so ν(M) is a singleton and ν is constant.
If M is an oriented manifold, then for each p ∈ M we can use any positive oriented basis in T p M to define an orientation for T p M A . Let Ω p be the canonical volume form on T p M A , this gives rise to a globally defined A-valued volume form Ω over M. In a positive coordinate system (U, x 1 , · · · , x n ), if we put g ij = ∂ ∂x i , ∂ ∂x j and g = det(g ij ), then Ω = |g| dx 1 ∧ · · · ∧ dx n .
We could also define the Hodge star operator on A-valued differential forms. Here, for each p ∈ M we should consider the induced inner product on (T p M A ) ♯ .

The Levi-Civita Connection of Operator-valued Metrics
Suppose that M is an A-valued semi-Riemannian manifold, We say that a A-connection ∇ is compatible with ., . if for any three vector fieldsX,Ỹ ,Z ∈ X(M) A .
This equality holds iff it is hold for ordinary vector fields in XM.
Proof. Nondegeneracy of the metric give the result. A straightforward computation shows that the mapZ → µ X,Y (Z) is C ∞ (M, A)-linear and is a A-valued one-form. By the lemma 8.1, there is an unique A-vector field, denoted by 2 ∇XỸ , such that µX ,Ỹ (Z) = 2 ∇XỸ ,Z * for allZ ∈ X(M) A . Now, standard argument shows that ∇ is the unique torsion free A-connection that is compatible with the metric. This connection is called the Levi-Civita connection of the metric. Proof. The proof is similar to the method of Semi-Riemannian manifolds (cf. [12]).
On A-valued semi-Riemannian manifolds we can straightly generalize differential operators such as gradient, and divergence. Definition 8.4. The gradient of a function f ∈ C ∞ (M, A) is A-vector field that is equivalent to the differential df ∈ A 1 (M, A), Thus ∇f,X * =X(f ) ∀X ∈ X(M) A in terms of a coordinate system, ∇f = ∂f ∂x i g ij ∂ ∂x j .
Definition 8.5. The Hessian of a function f ∈ C ∞ (M, A) is its second covariant derivative Hess(f ) = ∇( df ).
Definition 8.6. If X is an A-vector field, the contraction of its covariant differential is called divergence of X and is denoted by div( X) ∈ C ∞ (M, A). In a coordinate system, div( X) = g ij ∇ ∂ i X, ∂ j .
Theorem 8.7. Let M be an oriented A-valued semi-Riemannian manifold, andΩ be its canonical volume form. Then, for any A-vector fieldX ∈ X(M) A we have LXΩ = div(X)Ω Proof. Computations, as are done in [17] show that the equality holds in the case of scalar metrics. But, all these computations, without any change, are also valid in the case of A-valued metrics, except that the functions we encountered are A-valued.
9 Ricci, Scalar Curvature, and Sectional Curvature In the past sections, we have presented the basic notions and facts about the curvature of the Levi-Civita connection of a given A-valued semi-Riemannian manifold. We begin to consider some invariants that truly characterize curvature. In this section M is an A-valued semi-Riemannian manifold with the A-Levi-Civita connection ∇. In coordinate systems, Thus, Ric is a symmetric (0, 2) tensor on M.