Operator-Valued Tensors on Manifolds: A Framework for Field Quantization
aa r X i v : . [ m a t h - ph ] J a n Operator-Valued Tensors on Manifolds:A Framework for Field Quantization
H. Feizabadi , N. Boroojerdian ∗ , Department of pure Mathematics, Faculty of Mathematics and Computer Science,Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran.
Abstract
In this paper we try to prepare a framework for field quantization. To thisend, we aim to replace the field of scalars R by self-adjoint elements of a commu-tative C ⋆ -algebra, and reach an appropriate generalization of geometrical conceptson manifolds. First, we put forward the concept of operator-valued tensors andextend semi-Riemannian metrics to operator valued metrics. Then, in this newgeometry, some essential concepts of Riemannian geometry such as curvature ten-sor, Levi-Civita connection, Hodge star operator, exterior derivative, divergence,...will be considered. Keywords:
Operator-valued tensors, Operator-Valued Semi-Riemannian Met-rics, Levi-Civita Connection, Curvature, Hodge star operator
MSC(2010):
Primary: 65F05; Secondary: 46L05, 11Y50.
The aim of the present paper is to extend the theory of semi-Riemannian metrics tooperator-valued semi-Riemannian metrics, which may be used as a framework for fieldquantization. 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 rootsin 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 ⋆ -modulesover them. Hilbert C ⋆ -modules provide a natural generalization of Hilbert spaces arisingwhen 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 ∗ Corresponding author.
E-mail addresses: [email protected](N. Boroojerdian), hassan [email protected](H. Feizabadi).2010 Mathematics Subject Classification. Primary: 65F05; Secondary: 46L05, 11Y50. he non-commutative case seemed too complicated at that time.The general theory ofHilbert 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-commutativegeneralization of the theory of vector bundles and non-commutative geometry [3].A number of results about geometrical structures of Hilbert C ⋆ -modules and aboutoperators on them have been obtained [6]. Henceforth, Hilbert C ⋆ -modules are general-ization of inner product spaces that on the level of manifolds, provide a generalizationof semi-Riemannian manifolds,that is the goal of this paper. Due to the physical ap-plications in mind, in this paper, the positive definiteness of the inner product will bereplaced by non-degeneracy. Another root of this paper is to provide a framework forfield quantization. The main idea for quantization is to replace scalars by operators ona Hilbert space. In the field of quantum mechanics, spectrum of operators plays the roleof values of the measurements. So replacements of scalars by operators is the first stepfor 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 Modulesover 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. Non-commutative algebras can be used in the realm of non-commutative Geometry, and it isnot our aim to enter in this realm. Many basic definition such as vector field can not beextended properly for non-commutative C ∗ -algebras, because the set of derivations ofan algebra is a module on the center of that algebra. So, extension of vector fields aremodules on the center of that C ∗ -algebra and we need center of algebra be equal to thealgebra, so the algebra must be commutative. Definition of the inner product encounterthe same problem. Bilinear maps over a module whose scalars are non-commutativeare very restricted. From Physical point of view, commutativity means operators in the C ∗ -algebra represent quantities that are simultaneously measurable and this assumptionis not very restricted.The content of the present paper is structured as follows: Section 1 contains the pre-liminary facts about of C ⋆ -algebras needed to explain our concepts, Section 2 covers thedefinition of extended tangent bundle, operator-valued vector fields, and operator-valuedtensors and explains some of their basic properties. The Pettis-integral of operator-valued volume forms are defined and Stokes’s theorem is proved in section 3. Section 4is devoted to operator-valued connections and curvature, and the covariant derivative ofoperator-valued vector fields will be defined. In section 5, operator-valued inner prod-uct and some of their basic properties will be illustrated. The existence,uniqueness andthe properties of the Hodge star operator for operator-valued inner product spaces arethe goal of Section 6. In section 7 the concepts of section 5 extend to manifolds. Theexistence and uniqueness of Levi-Civita connection of operator-valued semi-Riemannianmetrics is proved in section 8. In the last section Ricci tensor, scalar curvature, andsectional curvature are deliberated. 2 Review of C ⋆ -Algebras In this section we review some definitions and results from C ⋆ -algebras that we need inthe sequel. Definition 1.1.
A Banach ⋆ -algebra is a complex Banach algebra A with a conjugatelinear involution ∗ which is an anti-isomorphism. That is, for all A, B ∈ A and λ ∈ C ,( A + B ) ∗ = A ∗ + B ∗ , ( λA ) ∗ = λA ∗ , A ∗∗ = A, ( AB ) ∗ = B ∗ A ∗ . Definition 1.2. A C ⋆ -algebra A , is a Banach ⋆ -algebra with the additional norm con-dition. For all A ∈ A k A ∗ A k = k A k . 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 . 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. Definition 1.4.
An element A of a C ⋆ -algebra is said to be self-adjoint if A ∗ = A , normal if A ∗ A = AA ∗ , and unitary if A ∗ A = AA ∗ = 1.For now on, A is a C ∗ -algebra. The set of all self-adjoint elements of A is denotedby A R . Definition 1.5.
The spectrum of an element A in a C ⋆ -algebra is the set σ ( A ) = { z ∈ C : z − A is not invertible } . Theorem 1.6.
The spectrum σ ( A ) of any element A of a C ⋆ -algebra is a non emptycompact set and contained in the { z ∈ C : | z | k A k} , if A ∈ A R , then σ ( A ) ⊆ R . ([7],[16])Remark . For any A ∈ A , if λ ∈ σ ( A ), then | λ | ≤ k A k .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 = A . [11] We denote by √ A the unique positive element B such that B = A . If A is aself-adjoint element, then A is positive, and we set | A | = √ A , A + = ( | A | + A ), A − = ( | 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]).3 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 T M A = ∪ p ∈ M ( T p M ⊗ R A ), so T M A is a bundleof free A -modules over M . Smooth functions from M to A can be defined and the setof them is denoted by C ∞ ( M, A ) . Addition, scalar multiplication, and multiplication offunctions in C ∞ ( M, A ) are defined pointwise. The involution of A can be extended to C ∞ ( M, A ) as follows: ∗ : C ∞ ( M, A ) −→ C ∞ ( M, A ) f f ∗ where f ∗ ( x ) = f ( x ) ∗ .Let A = C ( X ) for some compact Hausdorff space X . So, any function ˜ f : M −→ A corresponds to a function f : M × X −→ C . Therefore, when A come to scene it meansthat 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 . Definition 2.1. A A -vector field ˜ X over M is a section of the bundle T M A .The set of all smooth A -vector fields on M is denoted by X ( M ) A , in fact X ( M ) A = X ( M ) ⊗ C ∞ ( M ) C ∞ ( M, A ) . Smooth A -vector fields can be multiplied by smooth A -valued functions and X ( M ) A isa 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( X ⊗ f ) p = X p ⊗ f ( p ) . These fields are called simple, and any A -vector field can be written locally as a finite sumof simple A -vector fields. We can identify X and X ⊗
1, so X ( M ) is a C ∞ ( M )-subspaceof X ( M ) A .A smooth vector field X ∈ X ( M ) defines a derivation on C ∞ ( M, A ) by f Xf.
In fact, for any integral curve α : I −→ M of X , we have, ( Xf )( α ( t )) = dd t f ( α ( t )).This operation can be extended to A -vector fields, such that for simple elements of X ( M ) A such as X ⊗ h we have( X ⊗ h )( f ) = h. ( Xf ) f ∈ C ∞ ( M, A )If A is non-commutative, this definition is not well-defined. This definition implies thatfor f, h ∈ C ∞ ( M, A ) and e X ∈ X ( M ) A we have( h e X )( f ) = h. ( e Xf ) . A -vector fields as thefollowing, if X, Y ∈ X ( M ) and h, k ∈ C ∞ ( M, A ) then,[ X ⊗ h , Y ⊗ k ] = [ X, Y ] ⊗ ( hk ) + Y ⊗ ( hX ( k )) − X ⊗ ( kY ( h )) . 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( X ⊗ f ) ∗ = X ⊗ f ∗ . For e X, e Y ∈ X ( M ) A , f ∈ C ∞ ( M, A ), we have e X ( f ) ∗ = e X ∗ ( f ∗ ) , h e X, e Y i ∗ = h e X ∗ , e Y ∗ i . (2.1) Definition 2.2. A A -valued covariant tensor field of order k on M is an operator T : X ( M ) A × · · · × X ( M ) A −→ C ∞ ( M, A ) such that is k − C ∞ ( M, A )-linear.Contravariant and mixed A -valued tensors can be defined in a similar way. Al-ternating 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 ex-terior product and exterior derivation of these forms are special case of vector valueddifferential forms. The only difference is that A maybe infinite dimensional.Lie derivation of A -tensor fields along A -vector fields is defined naturally. For ˜ X ∈ X ( M ) A and a covariant A -tensor field of order k , such as ˜ T , Lie derivation of ˜ T along ˜ X isalso a covariant A -tensor field of order k and defined as follows. For ˜ Y , · · · , ˜ Y k ∈ X ( M ) A we have ( L ˜ X ˜ T )( ˜ Y , · · · , ˜ Y k ) = ˜ X ( ˜ T ( ˜ Y , · · · , ˜ Y k )) − k X i =1 ˜ T ( ˜ Y , · · · , [ ˜ X, ˜ Y i ] , · · · , ˜ Y k )If ˜ T is a differential form, then its Lie derivation along any A -vector field is also adifferential form and Cartan formula for its derivation holds .i.e. L ˜ X ˜ T = d ( i ˜ X ˜ T ) + i ˜ X ( d ˜ T ) A -Valued Volume Forms and Stokes’sTheorem 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 Lebesguemeasure. Definition 3.1.
Suppose V ∈ B n . A measurable function f : V −→ A is called5i) weakly integrable if Λ( f ) is Lebesgue integrable for every Λ ∈ A ∗ (ii) Pettis integrable if there exists x ∈ A such that Λ( x ) = R V Λ( f ) d µ , for every Λ ∈ A ∗ .If f is Pettis-integrable over V ∈ B n then x is unique and is called Pettis-integralof f over V . We use the notations R V f d µ or ( P ) R V f d µ 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]). Theorem 3.2 (Change of Variables) . Suppose D and D ′ are open domains of integrationin R n , and G : D −→ D ′ is a diffeomorphism . For every Pettis-integrable function f : D ′ −→ A , Z D ′ f d µ = Z D ( f ◦ G ) | det( DG ) | d µ. Proof.
Applying the Pettis-integral’s definition and using classical change of variablestheorem, one can conclude the desired result .
Definition 3.3. An A -valued n -form on M ( n = dim M ) is called an A -valued volumeform on M .In the canonical coordinate system, an A -valued volume form ˜ ω on R n is written asfollows: e ω = f d x ∧ · · · ∧ d x n , (3.1)where f ∈ C ∞ ( R n , A ) . Definition 3.4.
Let ˜ ω be a compactly supported A -valued n -form on R n . Define theintegral of ˜ ω over R n by, Z R n e ω = ( P ) Z R n f d x · · · d x n where ˜ ω is as (3.1). Definition 3.5.
For Λ ∈ A ∗ , define the operator Λ : A k ( M, A ) −→ A k ( M ) by(Λ ˜ ω )( X , ..., X k ) = Λ( e ω ( X , ..., X k )) , where e ω ∈ A k ( M, A ), X i ∈ X ( M ) . Definition 3.6.
Let M be an oriented smooth n -manifold, and let ˜ ω be a A -valued n -form on M . First suppose that the support of ˜ ω is compact and include in the domainof a single chart ( U, ϕ ) which is positively oriented. We define the integral of ˜ ω over Mas Z M e ω = ( P ) Z ϕ ( U ) ( ϕ − ) ∗ ( e ω ) . (3.2)6y using the change of variable’s theorem one can prove that R M e ω does not dependon the choice of chart whose domain contains supp( e ω ) . To integrate an arbitrary compactsupport A -valued n -form, we can use partition of unity as the same as ordinary n -forms. Lemma 3.7.
For any Λ ∈ A ∗ we have,(i) if f : M −→ N is a smooth map, then for each e ω ∈ A n ( N, A ) , Λ f ∗ ( e ω ) = f ∗ (Λ e ω ) ,(ii) for a compactly supported A -valued volume n -form e ω on M , Λ( R M e ω ) = R M Λ( e ω ) ,(iii) for e ω ∈ A k ( M, A ) , Λ(d e ω ) = d(Λ e ω ) .Proof. To prove ( i ) suppose that X , · · · , X k ∈ X M , then f ∗ (Λ ˜ ω )( X , · · · , X k ) =(Λ ˜ ω )( f ∗ ( X ) , · · · , f ∗ ( X k )) = Λ( e ω ( f ∗ ( X ) , · · · , f ∗ ( X k ))=Λ(( f ∗ e ω )( X , · · · , X k )) = (Λ( f ∗ e ω ))( X , · · · , X k ) . To prove ( ii ) Suppose that ˜ ω is compactly supported in the domain of a single chart( U, ϕ ) that is positively oriented, thusΛ( Z M e ω ) = Λ( Z R n ( ϕ − ) ∗ e ω ) = Z R n Λ(( ϕ − ) ∗ e ω ) = Z R n ( ϕ − ) ∗ (Λ ˜ ω ) = Z M Λ ˜ ω. The general case follows from the above result and using partition of unity.( iii ) Note that in the special case k = 0, A ( M, A ) = C ∞ ( M, A ) and the continuityof Λ implies ( iii ) for the elements of C ∞ ( M, A ). The general case follows from thisone. Theorem 3.8 (Stokes’s Theorem) . Let M be an oriented smooth n -manifold with bound-ary and orientations of M and ∂M are compatible, and let e ω be a compactly supportedsmooth valued ( n − -form on M .Then Z M d e ω = Z ∂M e ω. Proof.
By Lemma (3.7) for each Λ ∈ A ∗ we haveΛ( Z M d e ω ) = Z M Λ(d e ω ) = Z M d(Λ ˜ ω ) = Z ∂M (Λ ˜ ω ) = Λ( Z ∂M e ω )Therefore, Han-Banach’s theorem yields that R M d e ω = R ∂M e ω The notion of covariant derivation of A -vector fields can be defined as follows.7 efinition 4.1. An A -connection ∇ on M is a bilinear map ∇ : X ( M ) A × X ( M ) A −→ X ( M ) A such that for all ˜ X, ˜ Y ∈ X ( M ) A and any f ∈ C ∞ ( M, A ),( i ) ∇ f ˜ X ˜ Y = f ∇ ˜ X ˜ Y ( ii ) ∇ ˜ X f ˜ Y = ˜ X ( f ) ˜ Y + f ∇ ˜ X ˜ Y .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 isdefined as follows. If X, Y ∈ X M and h, k ∈ C ∞ ( M, A ), then define ∇ X ⊗ h ( Y ⊗ k ) = h (( ∇ X Y ) ⊗ k + Y ⊗ X ( k ))The torsion tensor of a A -connection ∇ is defined by T ( ˜ X, ˜ Y ) = ∇ ˜ X ˜ Y − ∇ ˜ Y ˜ X − [ ˜ X, ˜ Y ] ˜ X, ˜ Y ∈ X ( M ) A If T = 0, then ∇ is called torsion-free. Definition 4.2.
Let ∇ be an A -connection on M . The function R : X ( M ) A × X ( M ) A × X ( M ) A −→ X ( M ) A given by R ( ˜ X, ˜ Y )( ˜ Z ) = ∇ ˜ X ∇ ˜ Y ˜ Z − ∇ ˜ Y ∇ ˜ X ˜ Z − ∇ [ ˜ X, ˜ Y ] ˜ Z is a (1 , A -tensor on M and is called the curvature tensor of ∇ . Proposition 4.3 (The Bianchi identities) . If R is the curvature of a torsion free A -connection ∇ on M , then for all ˜ X, ˜ Y , ˜ Z ∈ X ( M ) A , ( i ) R ( ˜ X, ˜ Y ) ˜ Z + R ( ˜ Y , ˜ Z ) ˜ X + R ( ˜ Z, ˜ X ) ˜ Y = 0 ; ( ii ) ( ∇ ˜ X R )( ˜ Y , ˜ Z ) + ( ∇ ˜ Y R )( ˜ Z, ˜ X ) + ( ∇ ˜ Z R )( ˜ X, ˜ Y ) = 0 . Proof.
The proof is similar to ordinary connections(cf. [14]).If ∇ , ∇ are A -connections on M , then, the operator T = ∇ − ∇ is a (1 , A -tensor and the space of A -connections on M is an affine space which is modeled on thespace of (1 , A -tensors. 8 Operator-Valued Inner Product
In this section, we introduce the notion A -valued inner products on A -modules, andinvestigate 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 alsoa free module. Definition 5.1.
Let V be a A -module. A mapping h· , ·i : V × V −→ A is called an A -valued inner product on V if for all a ∈ A , and x, y, z ∈ V the following conditionshold:( i ) h y, x i = h x, y i ∗ ( ii ) h ax + y, z i = a h x, z i + h y, z i ( iii ) ∀ w ∈ V h w, x i = 0 ⇒ x = 0( iv ) ∀ T ∈ V ♯ ∃ x ∈ V ∀ y ∈ V ( T ( y ) = h y, x i )Conditions ( iii ) and ( iv ) are called non-degeneracy of the inner product. Note thatfor each x ∈ V the mapping ˆ x : V −→ A , defined by ˆ x ( y ) = h y, x i belongs to V ♯ . One cansee that the mapping x ˆ x from V into V ♯ is conjugate A -linear and non-degenracyof the inner product is equivalent to the bijectivity of this map. Theorem 5.2.
Let V be a finite dimensional free A -module and h· , ·i : V × V −→ A satisfies ( i ) and ( ii ) . If h· , ·i is non-degenerate then for any basis { e i } of V , det( h e i , e j i ) is invertible in A . Conversely, If for some basis { e i } of V , det( h e i , e j i ) is invertible, then h· , ·i is non-degenerate.Proof. First, assume that for some basis { e i } , det( h e i , e j i ) is invertible in A . Set g ij = h e i , e j i , 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 } . Thecharacteristic property of the reciprocal base is that h e i , e j i = δ ji , so for every x ∈ V ,if x = λ i e i , then λ i = h x, e i i . To prove non-degenracy, suppose for all w ∈ V we have h w, x i = 0. So, for all index i we have h x, e i 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 have ˆ x = T .Conversely, suppose that h· , ·i is non-degenerate. Let { e i } be an arbitrary basis andset g ij = h e i , e j i . For each i , consider the A -linear map T i : V −→ A defined by T i ( e j ) = δ ij . By non-degenarcy, the map x ˆ x is bijective and there exists u i ∈ V such that b u i = T i , so h e j , u i i = δ ij . 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 thematrix D = ( g ij ), In fact, δ ij = h e j , u i i = h u i , e j i = h a ik e k , e j i = a ik h e k , e j i = a ik g kj Therefore AD = I , so det( A ) det( D ) = 1 and det( D ) is invertible.9et V be an A module. V , may have an involution that is a conjugate A -linearisomorphism ∗ : V −→ V such that ∗ = 1 V . An inner product h· , ·i on V is calledcompatible with the involution if for all x, y ∈ V h x ∗ , y ∗ i = h x, y i ∗ In these spaces, if x and y are self-conjugate elements of V , then h x, y i is self-adjoint. Example 5.3.
Let W be a finite dimensional real vector space and h· , ·i : W × W −→ A R be a symmetric bilinear map such that for some basis { e i } of W , det ( h e i , e j i ) is invertible in A . The tensor product W A = W ⊗ R A is a free A -moduleand h· , ·i can be extended uniquely to a A -valued inner product on it as follows. For all u, v ∈ V and a, b ∈ A set h u ⊗ a , v ⊗ b i = a b ∗ h u , v i . The A -module W A has a natural involution that is ( u ⊗ a ) ∗ = u ⊗ a ∗ and the extendedinner 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 ± Theorem 5.4.
Let V be a free finite dimensional A -module and h· , ·i be an inner producton it. If { e i } is a basis on V , set g ij = h e i , e j i and g = det( g ij ) . Then, g is self-adjointand | 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 = h e ′ i , e ′ j i and G ′ = ( g ′ ij ) and g ′ =det( G ′ ). For some matrix A = ( a ji ) we have e ′ i = a ji e j , so g ′ ij = h e ′ i , e ′ j i = h a ki e k , a lj e l i = a ki a l ∗ j h e k , e l i = a ki a l ∗ j g kl This equality implies that G ′ = AG ( t A ∗ ), so g ′ = g det( A )det( A ) ∗ . Since det( A )det( A ) ∗ is a positive self-adjoint element of A we deduced that | g ′ | = | g | det( A )det( A ) ∗ . Conse-quently, | g ′ | g ′ = | g | det( A )det( A ) ∗ g det( A )det( A ) ∗ = | g | g The value ν = | g | g which does not depend on the choice of the base, is called thesignature of the inner product. Note that ν − = ν and σ ( ν ) ⊆ {− , } . In the ordinarymetrics, ν is exactly ( − 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.10 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 havesame orientation if determinant of the transition matrix between them is a positiveelement of A . This is an equivalence relation between the bases of V , but there existmany equivalence classes. We choose one of these classes as an orientation for V and wecall it an orientation for V . In this case, we call V an orientated space and every basisin the orientation, is called a proper base of V . Note that there are many orientationson 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 -innerproduct. For each proper base { e i } with reciprocal base { e i } , and g = det( g ij ), setΩ = p | g | e ∧ · · · ∧ e n . This tensor is called the canonical volume form of the innerproduct. Theorem 6.2.
The canonical volume form does not depend on the choice of the properbase.Proof.
Assume that { u i } is another proper base with reciprocal base { u j } . For somematrix A = ( a ji ), we have u i = a ji e j . The same orientation of { e i } and { u i } implies thatdet A is positive. If A − = ( β ji ), then e i = β ji u j . According to the proof of Theorem(5.4) if g ij = h e i , e j i , u ij = h u i , v j i , g = det( g ij ) , u = det( u ij ) , then, u = det A det A ∗ g = (det A ) g . Positivity of det A yields p | u | = det A. p | g | ,henceforth p | g | e ∧ · · · ∧ e n = p | g | g e ∧ · · · ∧ e n = p | g | g ( β i u i ) ∧ · · · ∧ ( β i n n u i n )= p | g | g det A − u ∧ · · · ∧ u n = p | g | g det A ( u i u i ) ∧ · · · ∧ ( u ni n u i n )= p | g | ug det A u ∧ · · · ∧ u n = p | g | u det Ag (det A ) u ∧ · · · ∧ u n = p | u | u ∧ · · · ∧ u n . Any A -inner product on the free n -dimensional A -module V , in a natural way, canbe extended to the space of each exterior powers Λ k V . For α = u ∧ · · · ∧ u k and β = v ∧ · · · ∧ v k , set h α, β i = det( h u i , v j i ). If { e i } is a base of V with reciprocal base11 e j } , then it is straightforward to check that { e i ∧ · · · ∧ e i k } ≤ i <...
The proof is a straightforward computation and is similar to the ordinary case.All these results, also hold for V ♯ . 12 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 asmooth map h ., . i : T M A ⊕ T M A −→ A such that for each p ∈ M , h ., . i 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 isan 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 h ., . i be a semi-Riemannian A -valuedmetric on it. If ν is the signature function of the metric, then ν is constant.Proof. First, consider the following subset of A . N = { a ∈ A | a ∗ = a , a = 1 } Clearly, N is nonempty and the values of ν are in N . We can prove that the distance ofany two distinct elements of N is greater than 2. suppose a, b ∈ N and a = b . So, k a − b k = k ( a − b ) k = k a + b − ab k = 2 k (1 − ab ) k Since ( ab ) = a b = 1, we have σ ( ab ) ⊂ {− , } . -1 must be in σ ( ab ), otherwise σ ( ab ) = { } that implies ab = 1, hence a = b that is contrary to assumption. So,2 ∈ σ (1 − ab ) that implies k (1 − ab ) k ≥
2. Consequently, k a − b k = 2 k (1 − ab ) k ≥ ⇒ k a − b k ≥ N implies that the induced topology on N is discrete, on the otherhand 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 orientedbasis in T p M to define an orientation for T p M A . Let Ω p be the canonical volume formon T p M A , this gives rise to a globally defined A -valued volume form e Ω over M . In apositive coordinate system ( U, x , · · · , x n ), if we put g ij = h ∂∂x i , ∂∂x j i and g = det( g ij ),then e Ω = p | g | d x ∧ · · · ∧ d x n . We could also define the Hodge star operator on A -valued differential forms. Here, foreach p ∈ M we should consider the induced inner product on ( T p M A ) ♯ .13 efinition 7.3. The Hodge star operator ⋆ : A k ( M, A ) → A n − k ( M, A ) maps any k -form α ∈ A k ( M, A ) to the ( n − k )-form ⋆α ∈ A n − k ( M, A ) such that for any β ∈ A n − k ( M, A ), α ∧ β = h β, ⋆α i e Ω . Our previous results now imply that ⋆ (1) = ν e Ω , ⋆ ( e Ω) = 1 , α ∧ ⋆β = ( β ∧ ⋆α ) ∗ . For each α ∈ A k ( M, A ), Also we have the identity ⋆ ⋆ ( α ) = ( − k ( n − k ) ν α. (7.1)Using the Hodge star operator, one can define a new operator, called the coderivative,denoted by δ . Definition 7.4.
The co-differential of α ∈ A k ( M, A ) is δα ∈ A k − ( M, A ) defined by δα = ( − n ( k +1)+1 ν ( ⋆ d ⋆ α ) . Note that the point-wise inner product, induces an A -valued inner product on A kc ( M, A )by ( α, β ) = Z M α ∧ ⋆β = Z M ν h α, β i e Ω . The next theorem states that with respect to this inner product, the co-differentialoperator is adjoint to the differential operator.
Theorem 7.5.
For any α ∈ A kc ( M, A ) and β ∈ A k − c ( M, A ) , (d β, α ) = ( β, δα ) Proof.
We have(d β, α ) = Z M d β ∧ ⋆α = Z M d( β ∧ ⋆α ) − ( − k − β ∧ d ⋆ α = ( − k Z M β ∧ d ⋆ α, On the other hand,( β, δα ) = ( β, ( − n ( k +1)+1 ν ( ⋆ d ⋆ α ))= Z M β ∧ ( − n ( k +1)+1 ν ( ⋆ ⋆ d ⋆ α )= ( − n ( k +1)+1 ν ( − ( n − k +1)( k − ν Z M β ∧ d ⋆ α = ( − k Z M β ∧ d ⋆ α. Using the operator δ , we can define Laplacian operator ∆ on A -valued differentialforms, as follows ∆ = (d δ + δ d) : A k ( M, A ) −→ A k ( M, A ) . In particular, for a function f (0-form) we find∆ f = − ν ⋆ d ⋆ d f = − p | g | X ∂ i ( g ij p | g | ∂ j f ) . The Levi-Civita Connection of Operator-valuedMetrics
Suppose that M is an A -valued semi-Riemannian manifold, We say that a A -connection ∇ is compatible with h ., . i if for any three vector fields ˜ X, ˜ Y , ˜ Z ∈ X ( M ) A .˜ X h ˜ Y , ˜ Z i = h∇ ˜ X ˜ Y , ˜ Z i + h ˜ Y , ∇ ˜ X ∗ ˜ Z i (8.1)This equality holds iff it is hold for ordinary vector fields in X M . Lemma 8.1.
Suppose that M is an A -valued semi-Riemannian manifold. For e V ∈ X ( M ) A , let e V ♭ be the A -valued one-form on M given by e V ♭ ( ˜ X ) = h e V , ˜ X ∗ i , for all ˜ X ∈ X ( M ) A . Then the map e V −→ e V ♭ is a C ∞ ( M, A ) -module isomorphism from X ( M ) A onto A ( M, A ) .Proof. Nondegeneracy of the metric give the result.
Theorem 8.2. If M is an A -valued semi-Riemannian manifold, then there exists aunique torsion free connection ∇ that is compatible with the metric.Proof. Fix ˜ X, ˜ Y ∈ X ( M ) A , and define µ ˜ X, ˜ Y : X ( M ) A −→ C ∞ ( M, A ) by µ ˜ X, ˜ Y ( ˜ Z ) = ˜ X h ˜ Y , ˜ Z ∗ i + ˜ Y h ˜ Z, ˜ X ∗ i − ˜ Z h ˜ X, ˜ Y ∗ i + h [ ˜ X, ˜ Y ] , ˜ Z ∗ i − h [ ˜ Y , ˜ Z ] , ˜ X ∗ i + h [ ˜ Z, ˜ X ] , ˜ Y ∗ i . A straightforward computation shows that the map ˜ Z µ X,Y ( ˜ Z ) is C ∞ ( M, A )-linearand is a A -valued one-form. By the lemma 8.1, there is an unique A -vector field, denotedby 2 ∇ ˜ X ˜ Y , such that µ ˜ X, ˜ Y ( ˜ Z ) = 2 h∇ ˜ X ˜ Y , ˜ Z ∗ i for all ˜ Z ∈ X ( M ) A . Now, standardargument shows that ∇ is the unique torsion free A -connection that is compatible withthe metric. This connection is called the Levi-Civita connection of the metric. Proposition 8.3. If ˜ X, ˜ Y , ˜ Z, ˜ V ∈ X ( M ) A , then for the A -curvature tensor of the Levi-Civita connection, we have(i) hR ( ˜ X, ˜ Y )( ˜ Z ) , ˜ V i = −hR ( ˜ X , ˜ Y )( ˜ V ∗ ) , ˜ Z ∗ i ;(ii) hR ( ˜ X, ˜ Y )( ˜ Z ) , ˜ V i = hR ( ˜ Z, ˜ V ∗ )( ˜ X ) , ˜ Y ∗ i .Proof. The proof is similar to the method of Semi-Riemannian manifolds (cf. [12]).On A -valued semi-Riemannian manifolds we can straightly generalize differentialoperators such as gradient, and divergence.15 efinition 8.4. The gradient of a function f ∈ C ∞ ( M, A ) is A -vector field that isequivalent to the differential d f ∈ A ( M, A ), Thus h∇ f, ˜ X ∗ i = ˜ 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 deriva-tive Hess( f ) = ∇ ( d f ).The Hessian of f is a symmetric (0 , A -tensor field and its operation on vectorfields ˜ X, ˜ Y ∈ X ( M ) A is as followsHess( f )( ˜ X, ˜ Y ) = ˜ X ( ˜ Y f ) − h∇ f, ∇ ˜ X ∗ ˜ Y ∗ i = h∇ ˜ X ( ∇ f ) , ˜ Y ∗ i . Definition 8.6. If e X is an A -vector field, the contraction of its covariant differential iscalled divergence of e X and is denoted by div( e X ) ∈ C ∞ ( M, A ). In a coordinate system,div( e X ) = g ij h∇ ∂ i e X, ∂ j i . Theorem 8.7.
Let M be an oriented A -valued semi-Riemannian manifold, and ˜Ω be itscanonical volume form. Then, for any A -vector field ˜ X ∈ X ( M ) A we have L ˜ X ˜Ω = div( ˜ X ) ˜Ω Proof.
Computations, as are done in [17] show that the equality holds in the case ofscalar metrics. But, all these computations, without any change, are also valid in thecase of A -valued metrics, except that the functions we encountered are A -valued. In the past sections, we have presented the basic notions and facts about the curvatureof the Levi-Civita connection of a given A -valued semi-Riemannian manifold. We beginto consider some invariants that truly characterize curvature. In this section M is an A -valued semi-Riemannian manifold with the A -Levi-Civita connection ∇ . Definition 9.1.
For each p ∈ M , the Ricci curvature tensor, R ic p : T p M A × T p M A −→ A is given by R ic p (˜ u, ˜ v ) = trace (cid:0) ˜ w −→ R ( ˜ w, ˜ u )˜ v (cid:1) , and the scalar curvature S is the trace of R ic.16n coordinate systems, R ic( e X, e Y ) = g ij hR ( ∂∂x i , e X ) e Y , ∂∂x j i , S = g ij R ic( ∂∂x i , ∂∂x j )Thus, R ic is a symmetric (0 ,
2) tensor on M. A two-dimensional free A -submodule Π of T p M A is called an A -tangent plane to M at p . For p ∈ M , ˜ u, ˜ v ∈ T p M A define, Q (˜ u, ˜ v ) = h ˜ u, ˜ u ih ˜ v, ˜ v i − h ˜ u, ˜ v i h ˜ u, ˜ v i ∗ . In fact, Q (˜ u, ˜ v ) = h ˜ u ∧ ˜ v, ˜ u ∧ ˜ v i . Definition 9.2. A A -tangent plane Π to M is called non-degenerate if for some base { ˜ u, ˜ v } of Π, Q (˜ u, ˜ v ) is invertible in A .The invertiblility of Q (˜ u, ˜ v ) does not depend on the choice of the base. If { ˜ u, ˜ v } is abase T p M A , then K (˜ u, ˜ v ) := hR (˜ u, ˜ v )˜ v ∗ , ˜ u iQ (˜ u, ˜ v )is well-defined and only depends on the 2-dimensional submodule determined by ˜ u and˜ v. Definition 9.3.
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