Banach Manifold Structure and Infinite-Dimensional Analysis for Causal Fermion Systems
aa r X i v : . [ m a t h - ph ] J a n BANACH MANIFOLD STRUCTURE ANDINFINITE-DIMENSIONAL ANALYSIS FOR CAUSAL FERMIONSYSTEMS
FELIX FINSTER AND MAGDALENA LOTTNERJANUARY 2021
Abstract.
A mathematical framework is developed for the analysis of causal fermionsystems in the infinite-dimensional setting. It is shown that the regular spacetimepoint operators form a Banach manifold endowed with a canonical Fr´echet-smoothRiemannian metric. The so-called expedient differential calculus is introduced withthe purpose of treating derivatives of functions on Banach spaces which are differ-entiable only in certain directions. A chain rule is proven for H¨older continuousfunctions which are differentiable on expedient subspaces. These results are madeapplicable to causal fermion systems by proving that the causal Lagrangian is H¨oldercontinuous. Moreover, H¨older continuity is analyzed for the integrated causal La-grangian.
Contents
1. Introduction 22. Preliminaries 42.1. Causal Fermion Systems and the Causal Action Principle 42.2. Fr´echet and Gˆateaux Derivatives 52.3. Banach Manifolds 73. Smooth Banach Manifold Structure of F reg L and ℓ along Smooth Curves 25Appendix A. Properties of the Fr´echet Derivative 28Appendix B. The Riemannian Metric in Symmetric Wave Charts 30References 36 Introduction
The theory of causal fermion systems is a recent approach to fundamental physics(see the basics in Section 2, the reviews [11, 12, 16], the textbook [10] or the website [1]).In this approach, spacetime and all objects therein are described by a measure ρ ona set F of linear operators of rank at most 2 n on a Hilbert space ( H , h . | . i H ). Thephysical equations are formulated via the so-called causal action principle , a nonlinearvariational principle where an action S is minimized under variations of the measure ρ .If the Hilbert space H is finite-dimensional , the set F is a locally compact topologicalspace. Making essential use of this fact, it was shown in [9] that the causal actionprinciple is well-defined and that minimizers exist. Moreover, as is worked out in detailin [15], the interior of F (consisting of the so-called regular points ; see Definition 3.1)has a smooth manifold structure. Taking these structures as the starting point, causalvariational principles were formulated and studied as a mathematical generalizationof the causal action principle, where an action of the form S = ˆ F dρ ( x ) ˆ F dρ ( y ) L ( x, y )is minimized for a given lower-semicontinuous Lagrangian L : F × F → R +0 on an(in general non-compact) manifold F under variations of ρ within the class of regularBorel measures, keeping the total volume ρ ( F ) fixed. We refer the reader interested incausal variational principles to [19, Section 1 and 2] and the references therein.This article is devoted to the case that the Hilbert space H is infinite-dimensional and separable. While the finite-dimensional setting seems suitable for describing phys-ical spacetime on a fundamental level (where spacetime can be thought of as beingdiscrete on a microscopic length scale usually associated to the Planck length), aninfinite-dimensional Hilbert space arises in mathematical extrapolations where space-time is continuous and has infinite volume. Most notably, infinite-dimensional Hilbertspaces come up in the examples of causal fermion systems describing Minkowski space(see [10, Section 1.2] or [26]) or a globally hyperbolic Lorentzian manifold (see forexample [11]), and it is also needed for analyzing the limiting case of a classical in-teraction (the so-called continuum limit; see [10, Section 1.5.2 and Chapters 3-5]).A workaround to avoid infinite-dimensional analysis is to restrict attention to locallycompact variations, as is done in [14, Section 2.3]. Nevertheless, in view of the impor-tance of the examples and physical applications, it is a task of growing significance toanalyze causal fermion systems systematically in the infinite-dimensional setting. It isthe task of this paper to put this analysis on a sound mathematical basis.We now outline the main points of our constructions and explain our main results.Extending methods and results in [15] to the infinite-dimensional setting, we endow theset of all regular points of F with the structure of a Banach manifold (see Definition 3.1and Theorem 3.4). To this end, we construct an atlas formed of so-called symmetricwave charts (see Definition 3.3). We also show that the Hilbert-Schmidt norm onfinite-rank operators on H gives rise to a Fr´echet-smooth Riemannian metric on thisBanach manifold. More precisely, in Theorems 3.11 and 3.12 we prove that F reg is asmooth Banach submanifold of the Hilbert space S ( H ) of self-adjoint Hilbert-Schmidtoperators, with the Riemannian metric given by g x : T S x F reg × T S x F reg → R , g x ( A, B ) := tr( AB ) . ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 3
In order to introduce higher derivatives at a regular point p ∈ F , our strategy isto always work in the distinguished symmetric wave chart around this point. Thishas the advantage that we can avoid the analysis of differentiability properties undercoordinate transformations. The remaining difficulty is that the causal Lagrangian L and other derived functions are not differentiable. Instead, directional derivatives existonly in certain directions. In general, these directions do not form a vector space. Asa consequence, the derivative is not a linear mapping, and the usual product and chainrules cease to hold. On the other hand, these computations rules are needed in theapplications, and it is often sensible to assume that they do hold. This motivatesour strategy of looking for a vector space on which the function under consideration isdifferentiable. Clearly, in this way we lose information on the differentiability in certaindirections which do not lie in such a vector space. But this shortcoming is out-weightedby the benefit that we can avoid the subtleties of non-smooth analysis, which, at leastfor most applications in mind, would be impractical and inappropriately technical.Clearly, we want the subspace to be as large as possible, and moreover it should bedefined canonically without making any arbitrary choices. These requirements leadus to the notion of expedient subspaces (see Definition 4.2). In general, the expedientsubspace is neither dense nor closed. On these expedient subspaces, the function isGateaux differentiable, the derivative is a linear mapping, and higher derivatives aremultilinear.The differential calculus on expedient subspaces is compatible with the chain rule inthe following sense: If f is locally H¨older continuous, γ is a smooth curve whose deriva-tives up to sufficiently high order lie in the expedient differentiable subspace of f , thenthe composition f ◦ γ is differentiable and the chain rule holds (see Proposition 4.4),i.e. ( f ◦ γ ) ′ ( t ) = D E f | x γ ′ ( t ) , where the index E denotes the derivative on the expedient subspace. We also provea chain rule for higher derivatives (see Proposition 4.5). The requirement of H¨oldercontinuity is a crucial assumption needed in order to control the error term of thelinearization. The most general statement is Theorem 5.8 where H¨older continuity isrequired only on a subspace which contains the curve γ locally.We also work out how the differential calculus on expedient subspaces applies tothe setting of causal fermion systems. In order to establish the chain rule, we provethat the causal Lagrangian is indeed locally H¨older continuous with uniform H¨olderexponent (Theorem 5.1), and we analyze how the H¨older constant depends on the basepoint (Theorem 5.3). Moreover, we prove that for all x, y ∈ F there is a neighborhood U ⊆ F of y with (see (5.9)) |L ( x, y ) − L ( x, ˜ y ) | ≤ c ( n, y ) k x k k ˜ y − y k n − for all ˜ y ∈ U (where 2 n is the maximal rank of the operators in F ). Relying on these results, wecan generalize the jet formalism as introduced in [17] for causal variational principlesto the infinite-dimensional setting (Section 5.2). We also work out the chain rule forthe Lagrangian (Theorem 5.6) and for the function ℓ obtained by integrating one ofthe arguments of the Lagrangian (Theorem 5.9), ℓ ( x ) = ˆ M L ( x, y ) dρ ( y ) − s (1.1)(where s a positive constant). F. FINSTER AND M. LOTTNER
The paper is organized as follows. Section 2 provides the necessary preliminarieson causal fermion systems and infinite-dimensional analysis. In Section 3 an atlas ofsymmetric wave charts is constructed, and it is shown that this atlas endows the regularpoints of F with the structure of a Fr´echet-smooth Banach manifold. Moreover, it isshown that the Hilbert-Schmidt norm induces a Fr´echet-smooth Riemannian metric.In Section 4 the differential calculus on expedient subspaces is developed. In Section 5,this differential calculus is applied to causal fermion systems. Appendix A gives somemore background information on the Fr´echet derivative. Finally, Appendix B providesdetails on how the Riemannian metric looks like in different charts.We finally point out that, in order to address a coherent readership, concrete appli-cations of our methods and results for example to physical spacetimes have not beenincluded here. The example of causal fermion systems in Minkowski space will beworked out separately in [25]. 2. Preliminaries
Causal Fermion Systems and the Causal Action Principle.
We now recallthe basic definitions of a causal fermion system and the causal action principle.
Definition 2.1. (causal fermion system)
Given a separable complex Hilbert space H with scalar product h . | . i H and a parameter n ∈ N (the “spin dimension” ), we let F ⊆ L( H ) be the set of all selfadjoint operators on H of finite rank, which (counting mul-tiplicities) have at most n positive and at most n negative eigenvalues. On F weare given a positive measure ρ (defined on a σ -algebra of subsets of F ), the so-called universal measure . We refer to ( H , F , ρ ) as a causal fermion system .A causal fermion system describes a spacetime together with all structures and objectstherein. In order to single out the physically admissible causal fermion systems, onemust formulate physical equations. To this end, we impose that the universal measureshould be a minimizer of the causal action principle, which we now introduce.For any x, y ∈ F , the product xy is an operator of rank at most 2 n . However,in general it is no longer a selfadjoint operator because ( xy ) ∗ = yx , and this is dif-ferent from xy unless x and y commute. As a consequence, the eigenvalues of theoperator xy are in general complex. We denote these eigenvalues counting algebraicmultiplicities by λ xy , . . . , λ xy n ∈ C (more specifically, denoting the rank of xy by k ≤ n ,we choose λ xy , . . . , λ xyk as all the non-zero eigenvalues and set λ xyk +1 , . . . , λ xy n = 0). Weintroduce the Lagrangian and the causal action by Lagrangian: L ( x, y ) = 14 n n X i,j =1 (cid:16)(cid:12)(cid:12) λ xyi (cid:12)(cid:12) − (cid:12)(cid:12) λ xyj (cid:12)(cid:12)(cid:17) (2.1) causal action: S ( ρ ) = ¨ F × F L ( x, y ) dρ ( x ) dρ ( y ) . (2.2) ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 5
The causal action principle is to minimize S by varying the measure ρ under thefollowing constraints: volume constraint: ρ ( F ) = const (2.3) trace constraint: ˆ F tr( x ) dρ ( x ) = const (2.4) boundedness constraint: ¨ F × F | xy | dρ ( x ) dρ ( y ) ≤ C , (2.5)where C is a given parameter, tr denotes the trace of a linear operator on H , and theabsolute value of xy is the so-called spectral weight, | xy | := n X j =1 (cid:12)(cid:12) λ xyj (cid:12)(cid:12) . This variational principle is mathematically well-posed if H is finite-dimensional. Forthe existence theory and the analysis of general properties of minimizing measureswe refer to [8, 9, 3]. In the existence theory one varies in the class of regular Borelmeasures (with respect to the topology on L( H ) induced by the operator norm), andthe minimizing measure is again in this class. With this in mind, here we alwaysassume that ρ is a regular Borel measure . Let ρ be a minimizing measure. Spacetime is defined as the support of this measure, M := supp ρ . Thus the spacetime points are selfadjoint linear operators on H . These operatorscontain a lot of additional information which, if interpreted correctly, gives rise tospacetime structures like causal and metric structures, spinors and interacting fields.We refer the interested reader to [10, Chapter 1].The only results on the structure of minimizing measures which will be neededhere concern the treatment of the trace constraint and the boundedness constraint.As a consequence of the trace constraint, for any minimizing measure ρ the localtrace is constant in spacetime, i.e. there is a real constant c = 0 such that (see [10,Proposition 1.4.1]) tr x = c for all x ∈ M .
Restricting attention to operators with fixed trace, the trace constraint (2.4) is equiv-alent to the volume constraint (2.3) and may be disregarded. The boundedness con-straint, on the other hand, can be treated with a Lagrange multiplier. Indeed, asis made precise in [3, Theorem 1.3], for every minimizing measure ρ , there is a La-grange multiplier κ > ρ is a local minimizer of the causal action with theLagrangian replaced by L κ ( x, y ) := L ( x, y ) + κ | xy | , leaving out the boundedness constraint.2.2. Fr´echet and Gˆateaux Derivatives.
We now recall a few basic concepts fromthe differential calculus on normed vector spaces. In what follows, we let ( E, k . k E )and ( F, k . k F ) be real normed vector spaces. The most common concept is that of theFr´echet derivative. F. FINSTER AND M. LOTTNER
Definition 2.2.
Let U ⊆ E be open and f : U → F be an F -valued function on U .The function f is Fr´echet-differentiable in x ∈ U if there is a bounded linearmapping A ∈ L( E, F ) such that f ( x ) = f ( x ) + A ( x − x ) + r ( x ) , where the error term r : U → F goes to zero faster than linearly, i.e. lim x → x ,x = x k r ( x ) k F k x − x k E = 0 . The linear operator A is the Fr´echet derivative , also denoted by Df | x . A functionis Fr´echet-differentiable in U if it is Fr´echet-differentiable at every point of U . The Fr´echet derivative is uniquely defined. Moreover, the concept can be iterated todefine higher derivatives. Indeed, if f is differentiable in U , its derivative Df is amapping Df : U → L( E, F ) . Since L(
E, F ) is a normed vector space (with the operator norm), we can apply Defi-nition 2.2 once again to define the second derivative at a point x by D f | x = D (cid:0) Df (cid:1)(cid:12)(cid:12) x ∈ L (cid:0) E, L( E, F ) (cid:1) . The second derivative can also be viewed as a bilinear mapping from E to F , D f | x : E × E → F , D f | x ( u, v ) := (cid:16) D (cid:0) Df (cid:1)(cid:12)(cid:12) x u, v (cid:17) . It is by definition bounded, meaning that there is a constant c > (cid:13)(cid:13) D f | x ( u, v ) (cid:13)(cid:13) F ≤ c k u k E k v k E for all u, v ∈ E .
By iteration, one obtains similarly the Fr´echet derivatives of order p ∈ N as multilinearoperators D p f | x : E × · · · × E | {z } p factors → F .
A function is
Fr´echet-smooth on U if it is Fr´echet-differentiable to every order. Lemma 2.3.
If the function f : U ⊆ E → F is p times Fr´echet-differentiable in x ∈ U , then its p th Fr´echet derivative is symmetric, i.e. for any u , . . . , u p ∈ E and anypermutation σ ∈ S p , D p f | x (cid:0) u , . . . , u p (cid:1) = D p f | x (cid:0) u σ (1) , . . . , u σ ( p ) (cid:1) . We omit the proof, which can be found for example in [5, Section 4.4]. For theFr´echet derivative, most concepts familiar from the finite-dimensional setting carryover immediately. In particular, the composition of Fr´echet-differentiable functions isagain Fr´echet-differentiable. Moreover, the chain and product rules hold. We refer forthe details to [5, Sections 2.2 and 2.3] and [6, Chapter 8] and Appendix A.A weaker concept of differentiability which we will use here is Gˆateaux differentia-bility. In this reference, everything is worked out in the case of Banach spaces, but the completeness isnot needed for these results.
ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 7
Definition 2.4.
Let U ⊆ E be open and f : U → F be an F -valued function on U .The function f is Gˆateaux differentiable in x ∈ U in the direction u ∈ E if thelimit of the difference quotient exists, d u f ( x ) := lim h → ,h =0 f ( x + hu ) − f ( x ) h . The resulting vector d u f ( x ) ∈ F is the Gˆateaux derivative . By definition, the Gˆateaux derivative is homogeneous of degree one, i.e. d λu f ( x ) = λ d u f ( x ) for all λ ∈ R . Moreover, if f is Fr´echet-differentiable in x , then it is also Gˆateaux differentiable inany direction u ∈ E and d u f ( x ) = Df | x u . However, the converse is not true because, even if the Gˆateaux derivatives exist forany u ∈ E , it is in general not possible to represent them by a bounded linear operator.As a consequence, the chain and product rules in general do not hold for Gˆateauxderivatives. We shall come back to this issue in Section 5.2.3. Banach Manifolds.
We recall the basic definition of a smooth Banach manifold(for more details see for example [29, Chapter 73]).
Definition 2.5.
Let B be a topological Hausdorff space and ( E, k . k E ) a Banach space.A chart ( U, φ ) is a pair consisting of an open subset U ⊆ B and a homeomorphism φ of U to an open subset V := φ ( U ) of E , i.e. φ : U open ⊆ B → V open ⊆ E . A smooth atlas A = (cid:0) φ i , U i ) i ∈ I is a collection of charts (for a general index set I )with the properties that the domains of the charts cover B , B = [ i ∈ I U i and that for any i, j ∈ I , the transition map φ j ◦ φ − i : φ i (cid:0) U i ∩ U j (cid:1) ⊆ E → φ j (cid:0) U i ∩ U j (cid:1) is Fr´echet-smooth. The triple ( B, E, A ) is referred to as a smooth Banach manifold . Definition 2.6.
Just as in the case of finite-dimensional manifolds, we call a function f : U ⊆ A → B between two Banach manifolds A and B (with U ⊆ A open) n -times (Fr´echet) differentiable (resp. smooth ) if for any two charts φ a : U a → V a of A and φ b : U b → V b of B with f ( U a ) ⊆ U b the mapping φ b ◦ f ◦ φ − a : V a → V b is n -times(Fr´echet) differentiable (resp. smooth). Smooth Banach Manifold Structure of F reg In the definition of causal fermion systems, the number of positive or negativeeigenvalues of the operators in F can be strictly smaller than n . This is importantbecause it makes F a closed subspace of L( H ) (with respect to the norm topology),which in turn is crucial for the general existence results for minimizers of the causalaction principle (see [9] or [18]). However, in most physical examples in Minkowskispace or in a Lorentzian spacetime, all the operators in M do have exactly n positive F. FINSTER AND M. LOTTNER and exactly n negative eigenvalues. This motivates the following definition (see also [10,Definition 1.1.5]). Definition 3.1.
An operator x ∈ F is said to be regular if it has the maximal possiblerank, i.e. dim x ( H ) = 2 n . Otherwise, the operator is called singular . A causal fermionsystem is regular if all its spacetime points are regular.In what follows, we restrict attention to regular causal fermion systems. Moreover, itis convenient to also restrict attention to all those operators in F which are regular, F reg := (cid:8) x ∈ F | x is regular (cid:9) . F reg is a dense open subset of F (again with respect to the norm topology on L( H )).3.1. Wave Charts and Symmetric Wave Charts.
We now choose specific chartsand prove that the resulting atlas endows F reg with the structure of a smooth Banachmanifold (see Definition 2.5). In the finite-dimensional setting, these charts were in-troduced in [15]. We now recall their definition and generalize the constructions to theinfinite-dimensional setting.Given x ∈ F reg we denote the image of x by I := x ( H ). We consider I as a 2 n -dimensional Hilbert space with the scalar product induced from h . | . i H . Denoting itsorthogonal complement by J := I ⊥ , we obtain the orthogonal sum decomposition H = I ⊕ J .
This also gives rise to a corresponding decomposition of operators, like for exampleL( H , I ) = L( I, I ) ⊕ L( J, I ) . (3.1)Given an operator ψ ∈ L( H , I ), we denote its adjoint by ψ † ∈ L( I, H ); it is defined bythe relation h u | ψ v i I = h ψ † u | v i H for all u ∈ I and v ∈ H . We now form the operator R x ( ψ ) := ψ † x ψ ∈ L( H ) . (3.2)By construction, this operator is symmetric and has at most n positive and at most n negative eigenvalues. Therefore, it is an operator in F . Using (3.1), we concludethat R x is a mapping R x : L( I, I ) ⊕ L( J, I ) → F . (3.3)Before going on, it is useful to rewrite the operator R x ( ψ ) in a slightly different way.On I one can also introduce the indefinite inner product ≺ . | . ≻ x : S x × S x → C , ≺ u | v ≻ x = −h u | xv i H , (3.4)referred to as the spin inner product . For conceptual clarity, we denote I endowedwith the spin inner product by ( S x , ≺ . | . ≻ x ) and refer to it as the spin space at x (formore details on the spin spaces we refer for example to [10, Section 1.1]). It is anindefinite inner product space of signature ( n, n ). We denote the adjoint with respectto the spin inner product by a star. More specifically, for a linear operator A ∈ L( S x ),the adjoint is defined by ≺ φ | A ˜ φ ≻ x = ≺ A ∗ φ | ˜ φ ≻ x for all φ, ˜ φ ∈ S x . Using again the definition of the spin inner product (3.4), we can rewrite this equationas −h φ | X A ˜ φ i H = −h A ∗ φ | X ˜ φ i H , ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 9 where we introduced the short notation X := x | S x : S x → S x . (3.5)Taking adjoints in the Hilbert space H gives −h X − A † Xφ | X ˜ φ i H = −h A ∗ φ | X ˜ φ i H (note that the operator X is invertible because S x is by definition its image). We thusobtain the relation A ∗ = X − A † X . (3.6)Using such transformations, one readily verifies that, identifying the image of ψ witha subspace of S x , the right side of (3.2) can be written as − ψ ∗ ψ (for details see [15,Lemma 2.2]). Thus, with this identification, the operator R x can be written insteadof (3.2) and (3.3) in the equivalent form R x : L( I, S x ) ⊕ L( J, S x ) → F , R x ( ψ ) = − ψ ∗ ψ , (3.7)where ψ ∗ is the adjoint with respect to the corresponding inner products, i.e. ≺ φ | ψ u ≻ x = h ψ ∗ φ | u i H for u ∈ H and φ ∈ S x . We want to use the operator R x in order to construct local parametrizations of F reg .The main difficulty is that the operator R x is not injective. For an explanation of thispoint in the context of local gauge freedom we refer to [15]. Here we merely explainhow to arrange that R x becomes injective. We let Symm( S x ) ⊆ L( S x ) be the realvector space of all operators A on S x which are symmetric with respect to the spininner product, i.e. ≺ φ | A ˜ φ ≻ x = ≺ Aφ | ˜ φ ≻ x for all φ, ˜ φ ∈ S x . We now restrict the operator R x in (3.3) and (3.7) to R symm x := R x | Symm( S x ) ⊕ L( J,S x ) : Symm( S x ) ⊕ L( J, S x ) → F , R x ( ψ ) = − ψ ∗ ψ. (3.8)We write the direct sum decomposition as ψ = ψ I + ψ J with ψ I ∈ Symm( S x ) , ψ J ∈ L( J, S x ) . Extending the analysis in [15, Section 6.1] to the infinite-dimensional setting, onefinds that this mapping is a local parametrization of F reg : Theorem 3.2.
There is an open neighborhood W x of ( id S x , ∈ Symm( S x ) ⊕ L( J, S x ) such that the restriction of R symm x maps to an open subset Ω x := R symm x ( W x ) of F reg , R symm x | W x : W x → Ω x open ⊆ F reg , and is a homeomorphism to its image (always with respect to the topology induced bythe operator norm on L( H ) ).Proof. The estimate k R symm x ( ψ ) − R symm x ( ˜ ψ ) k L( H ) = (cid:13)(cid:13) ψ ∗ ψ − ˜ ψ ∗ ˜ ψ (cid:13)(cid:13) L( H ) ≤ (cid:13)(cid:13) ψ ∗ ψ − ψ ∗ ˜ ψ (cid:13)(cid:13) L( H ) + (cid:13)(cid:13) ψ ∗ ˜ ψ − ˜ ψ ∗ ˜ ψ (cid:13)(cid:13) L( H ) ≤ k ψ ∗ k L( H ) (cid:13)(cid:13) ψ − ˜ ψ (cid:13)(cid:13) L( H ) + (cid:13)(cid:13) ˜ ψ ∗ − ˜ ψ ∗ (cid:13)(cid:13) L( H ) k ˜ ψ k L( H ) (3.9)shows that R symm x is continuous. Since the point R symm x (id S x ,
0) = x ∈ F reg is regular,by continuity we may choose an open neighborhood W x of (id S x ,
0) such that R x mapsto F reg . In order to show that R symm x is bijective, we begin with the formula for φ x as derivedin [15, Proposition 6.6], which will turn out to be the inverse of R symm x . It has the form φ x ( y ) = (cid:0) P ( x, x ) − A xy P ( x, x ) − (cid:1) − P ( x, x ) − P ( x, y ) Ψ( y ) ∈ L( H , S x ) , (3.10)where P ( x, y ) (the kernel of the fermionic projector ) and A xy (the closed chain ) aredefined by P ( x, y ) := π x y | S y : S y → S x , A xy := P ( x, y ) P ( y, x ) : S x → S x . (3.11)Our task is to show that for a sufficiently small neighborhood Ω x of x , this formuladefines a continuous mapping φ x : Ω x ⊆ F reg → Symm( S x ) ⊕ L( J, H ) , and that the compositions φ x ◦ R symm x | W x and R symm x ◦ φ x (3.12)are both the identity (showing that φ x is indeed the inverse of R symm x ).In preparation, we rewrite the formula (3.10) as φ x ( y ) = (cid:16) X − π x yπ y xX − (cid:17) − X − π x yπ y = (cid:16) X − π x y | S x (cid:17) − X − π x y , (3.13)where we again used the notation (3.5). Choosing y = x , the operator X − π x y | S x isthe identity on S x . We choose Ω x so small such that for any y ∈ Ω x , (cid:13)(cid:13) id S x − X − π x y | S x (cid:13)(cid:13) L( H ) < . (3.14)Then the square root as well as the inverse square root are well-defined by the respectivepower series, A := ∞ X n =0 ( − n (cid:18) / n (cid:19) (id S x − A ) n , A − := ∞ X n =0 ( − n (cid:18) − / n (cid:19) (id S x − A ) n , with the generalized binomial coefficients given for β ∈ R and n ∈ N by (cid:18) βn (cid:19) := ( n ! β · ( β − · · · ( β − n + 1) if n >
00 if n = 0(note that for both power series the radius of convergence equals one). We concludethat the mapping φ x is well-defined on Ω x .In order to verify that φ x maps into Symm( S x ) ⊕ L( J, S x ), we restrict φ x ( y ) to S x , φ x ( y ) (cid:12)(cid:12) I = (cid:16)(cid:16) X − π x y (cid:12)(cid:12) S x (cid:17) − X − π x y (cid:17)(cid:12)(cid:12)(cid:12) I = (cid:16) X − π x y (cid:12)(cid:12) S x (cid:17) − / X − π x y | S x = (cid:16) X − π x y π x (cid:12)(cid:12) S x (cid:17) . (3.15)A direct computation using (3.6) shows that the operator X − π x yπ x | S x , and hencealso its square root, are symmetric on S x . ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 11
It remains to compute the compositions in (3.12). First, φ x ◦ R symm x ( ψ ) = φ x ( ψ † Xψ ) = (cid:16) X − π x ψ † X | {z } = ψ † I X ψ | S x |{z} ψ I (cid:17) − X − π x ψ † X | {z } = ψ † I X ψ = (cid:16) X − ψ † I X | {z } = ψ I ψ I (cid:17) − X − ψ † I X | {z } = ψ I ψ = (cid:0) ψ I (cid:1) − ψ I ψ = ψ , where in the last line we applied (3.6) and used that ψ I is symmetric on S x . Moreover, R symm x ◦ φ x ( y ) = φ x ( y ) † Xφ x ( y )= y π x X − (cid:16) π x yπ x X − (cid:17) − X (cid:16) X − π x yπ x | S x (cid:17) − X − π x y . Since the spectral calculus is invariant under similarity transformations, we know thatfor any invertible operator B on S x , X − B − X = (cid:16) X − BX (cid:17) − . Hence R symm x ◦ φ x ( y ) = y π x (cid:16) X − π x yπ x | S x (cid:17) − (cid:16) X − π x yπ x | S x (cid:17) − X − π x y = y π x (cid:16) X − π x yπ x | S x (cid:17) − X − π x y = y π x (cid:16) π x yπ x | S x (cid:17) − π x y = y x (cid:16) π x yx | S x (cid:17) − π x y = y P ( y, x ) (cid:16) P ( x, y ) P ( y, x ) (cid:17) − P ( x, y ) = y (note that P ( x, y ) : S y → S x is invertible in view of (3.14)). This concludes theproof. (cid:3) The mapping φ x , which already appeared in the proof of the previous lemma, canalso be introduced abstractly to define the chart. Definition 3.3.
Setting φ x := R symm x (cid:12)(cid:12) − W x : Ω x → Symm( S x ) ⊕ L( J, S x ) , we obtain a chart ( φ x , Ω x ) , referred to as the symmetric wave chart about thepoint x ∈ F reg . We remark that more general charts can be obtained by restricting R x to anothersubspace of L( I, S x ) ⊕ L( J, S x ), i.e. in generalization of (3.8), R Ex := R x | E ⊕ L( J,S x ) : E ⊕ L( J, S x ) → F , R ( ψ ) = − ψ ∗ ψ , where E is a subspace of L( S x ) which has the same dimension as Symm( S x ). Theresulting charts φ Ex are obtained by composition with a unitary operator U x on S x , i.e. φ Ex = U x ◦ φ x with U x ∈ U( S x )(for details and the connection to local gauge transformations see [15, Section 6.1]).Since linear transformations are irrelevant for the question of differentiability, in whatfollows we may restrict attention to symmetric wave charts. A Fr´echet Smooth Atlas.
The goal of this section is to prove that the sym-metric wave charts ( φ x , Ω x ) form a smooth atlas of F reg . Theorem 3.4 ( Symmetric wave atlas).
The collection of all symmetric wave chartson F reg defines a Fr´echet-smooth atlas of F reg , endowing F reg with the structure of asmooth Banach manifold (see Definition 2.5).Proof. We first verify that for any x ∈ F reg , the vector space Symm( S x ) ⊕ L( J, S x )together with the operator norm of L( H , I ) = L( H , S x ) is a Banach space. To thisend, we note that this vector space coincides with the kernel of the mapping ψ ( X − ψ † π x X − ψ | I ) on L( H , I ). Since this mapping is continuous on L( H , I ) (as oneverifies by an estimate similar to (3.9)), its kernel is closed. As a consequence, thevector space Symm( S x ) ⊕ L( J, S x ) is a closed subspace of L( H , I ) and thus indeed aBanach space.We saw in Theorem 3.2 that for any x ∈ F reg , ( φ x , Ω x ) defines a chart on F reg . Sincethe Ω x clearly cover F reg , it remains to show that all crossover mappings are Fr´echet-smooth. To this end, we first note that for any x, y ∈ F reg and ψ ∈ φ x (Ω x ∩ Ω y ), φ y ◦ φ − x ( ψ ) = φ y (cid:0) ψ † X ψ (cid:1) = (cid:16) Y − π y ψ † X ψ | S y (cid:17) − Y − π y ψ † X ψ .
Next, we define the mappings B xy : Symm( S x ) ⊕ L( J, S x ) → L( H , S y ) , ψ Y − π y ψ † X ψ , ˜ B xy : Symm( S x ) ⊕ L( J, S x ) → L( S y ) , ψ Y − π y ψ † X ψ | S y ,W : B (0) ⊆ L( S y ) → L( S y ) , B (1 + B ) − = ∞ X n =0 ( − n (cid:18) − / n (cid:19) B n (where the radius of the ball B / (0) is taken with respect to the operator norm).Recall that in the proof of Theorem 3.2 (more precisely (3.14)) we chose Ω y so smallthat the operator k id S y − Y − π y z | S y k < / z ∈ Ω y . Thus, since for any ψ ∈ φ x (Ω x ∩ Ω y ) we have ψ † Xψ = φ − x ( ψ ) ∈ Ω y , we obtain ˜ B xy ( φ x (Ω x ∩ Ω y )) ⊆ B / (id S y ).Therefore, we can write the crossover mapping φ y ◦ φ − x as φ y ◦ φ − x ( ψ ) = W (cid:0) id S y − ˜ B xy ( ψ ) (cid:1) ◦ B xy ( ψ ) . Now note that for the Fr´echet derivative, we consider all vector spaces here as a real
Banach spaces, but still with the canonical operator norm induced by k . k H . In viewof the chain rule for Fr´echet derivatives (for details see Lemma A.2 in Appendix A)and the properties of the Fr´echet derivative A.1 in Appendix A, it remains to showthat the mappings W , B xy and ˜ B xy are Fr´echet-smooth (note that the compositionoperator of R -linear mappings is also always Fr´echet-smooth as it defines a bounded R -bilinear map and the map L( S y ) ∋ y id S y − y ∈ L( S y ) is clearly Fr´echet-smoothas well). For W this is clear due to [21, p. 40–42] (note that L( S y ) obviously defines afinite-dimensional unital Banach-algebra). Moreover, the mappings B xy and ˜ B xy areobviously R -bilinear and bounded and thus Fr´echet-smooth. (cid:3) The Tangent Bundle.
Having endowed F reg with a canonical smooth Banachmanifold structure, the next step is to consider its tangent bundle. For finite-di-mensional manifolds, the tangent space can be defined either by equivalence classesof curves or by derivations, and these two definitions coincide (see for example [24, ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 13
Chapter 2]). In infinite dimensions, however, this does no longer be the case: In gen-eral, the derivation-tangent vectors (usually called operational tangent vectors ) form alarger class of than the curve-tangent vectors (called kinematic tangent vectors ). Theremight even be operational tangent vectors that depend on higher-order derivatives ofthe inserted function (while the kinematic tangent vectors interpreted as directionalderivatives only involve the first derivatives); for details on such issues see for exam-ple [22, Sections 28 and 29] or [2, p. 3–6]. It turns out that for our applications inmind, it is preferable to define tangent vectors as equivalence classes of curves. Indeed,as we shall see, with this definition the usual computation rules remain valid. Morespecifically, the tangent vectors of F reg are compatible with the Fr´echet derivative, andeach fiber of the corresponding tangent bundle can be identified with the underlyingBanach space V x := Symm( S x ) ⊕ L( J, S x )with respect to the chart φ x .Following [22, p. 284], we begin with the abstract definition of the (kinematic)tangent bundle, which makes it easier to see the topological structure. Afterward, wewill show that this notion indeed agrees with equivalence classes of curves. Given x ′ ∈ F reg , we consider the set Ω x ′ × V x ′ × { x ′ } (endowed with the topology inherited fromthe direct sum of Banach spaces). We take the disjoint union [ x ′ ∈ F reg Ω x ′ × V x ′ × { x ′ } and introduce the equivalence relation( x, v , x ′ ) ∼ ( y, w , y ′ ) ⇐⇒ x = y and ( φ x ′ ◦ φ − y ′ ) ′ | φ y ′ ( x ) w = v . For clarity, we point out that the first entry represents the point of the Banach mani-fold F reg , whereas the third entry labels the chart. Definition 3.5.
We define the tangent bundle T F reg as the quotient space with respectto this equivalence relation, T F reg := (cid:16) [ x ′ ∈ F reg Ω x ′ × V x ′ × { x ′ } (cid:17). ∼ . The canonical projection is given by π : T F reg → F reg , π ([ x, v , x ′ ]) = x . For every x ∈ F reg the tangent space at x is defined by T x F reg := π − ( x ) . Note that each T x F reg has a canonical vector space structure in the following sense:Since all equivalence classes in T x F reg have a representative of the form [ x, v , x ], thisrepresentative can be identified with v ∈ V x . In this way, we obtain an identificationof T x F reg with V x .The tangent bundle is again a Banach manifold, as we now explain. For any x ∈ F reg ,the mapping( φ x , Dφ x ) : π − ( W x ) → Ω x × V x , [ y, v , z ] (cid:16) φ x ( y ) , D (cid:0) φ x ◦ φ − z (cid:1)(cid:12)(cid:12) φ z ( y ) v (cid:17) has the inverse( φ x , Dφ x ) − : Ω x × V x → π − ( W x ) , ( ψ, v ) [ φ − x ( ψ ) , v , x ] . On T F reg we choose the coarsest topology with the property that the natural projec-tions of these mappings to Ω x and V x are both continuous (where on Ω x and V x wechoose the topology induced by the norm topology of L( H )). With this topology, themapping ( φ x , Dφ x ) defines a chart of T F reg . For any ( ψ, v ) ∈ ( φ y , Dφ y ) (cid:0) π − (Ω x ) ∩ π − (Ω y ) (cid:1) , the crossover mappings are given by( φ x , Dφ x ) ◦ ( φ y , Dφ y ) − ( ψ, v ) = ( φ x , Dφ x )([ φ − y ( ψ ) , v , y ])= (cid:16) ( φ x ◦ φ − y )( ψ ) , D (cid:0) φ x ◦ φ − y (cid:1)(cid:12)(cid:12) ψ v (cid:17) . Proposition 3.6. T F reg is again a Banach-manifold.Proof. We need to show that crossover maps are Fr´echet-smooth. This is clear for thefirst component because the crossover mappings φ x ◦ φ − y are Fr´echet-smooth. Thesecond component can be considered as the composition of the insertion mapL( V y , V x ) × V y ∋ ( A, v ) A ( v ) ∈ V x (which is obviously continuous and bilinear and thus Fr´echet-smooth, for details seeLemma A.1 in Appendix A) with the mapping W y × V y ∋ ( ψ, v ) (( φ x ◦ φ − y ) ′ | ψ , v ) ∈ L( V x , V y ) × V y , which is Fr´echet-smooth due to the Fr´echet-smoothness of the crossovermappings. (cid:3) In what follows, we will sometimes use the notation Dφ x ([ y, v , z ]) := D (cid:0) φ x ◦ φ − z (cid:1)(cid:12)(cid:12) φ z ( y ) v ∀ x ∈ F reg , [ y, v , z ] ∈ π − (Ω x ) , which also clarifies the independence of the choice of representatives. Lemma 3.7.
For any x ∈ F reg , the mapping ψ x : Ω x × V x → π − (Ω x ) , ( y, v ) [ y, v , x ] is a local trivialization.Proof. We need to verify the properties of a local trivialization. Clearly, the opera-tor π ◦ ψ x is the projection to the first component, and for fixed y ∈ Ω x , the mapping v ψ x ( y, v ) = [ y, v , x ] = [ y, ( φ y ◦ φ − x ) ′ | φ x ( x ) v , y ] corresponds to v ( φ y ◦ φ − x ) ′ | φ x ( x ) v (by the identification of T y F reg with V y from before), which is obviously an isomorphismof vector spaces in view of Lemma A.1 (vi). (cid:3) To summarize, the Banach manifold F reg has similar properties as in the finite-dimen-sional case.We now explain how the above definition of tangent vectors relates to the equivalenceclasses of curves (following [22, p. 285]): Remark 3.8. (equivalence classes of curves)
On curves γ, ˜ γ ∈ C ∞ ( R , F reg ), weconsider the equivalence relation γ ∼ ˜ γ defined by the conditions that γ (0) = ˜ γ (0) andthat in a chart φ x with γ (0) ∈ Ω x , the relation ( φ x ◦ γ ) ′ | = ( φ x ◦ ˜ γ ) ′ | holds. Notethat if the last relation holds in one chart, then it also holds in any other chart φ y with γ (0) ∈ Ω y because, due to the chain rule,( φ y ◦ γ ) ′ | = ( φ y ◦ φ − x ◦ φ x ◦ γ ) ′ | = ( φ y ◦ φ − x ) ′ | φ x ( γ (0)) ( φ x ◦ γ ) ′ | = ( φ y ◦ φ − x ) ′ | φ x ( γ (0)) ( φ x ◦ ˜ γ ) ′ | = ( φ y ◦ φ − x ◦ φ x ◦ ˜ γ ) ′ | = ( φ y ◦ ˜ γ ) ′ | . ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 15
Now we can identify C ∞ ( R , F reg ) / ∼ with T F reg via the mapping C ∞ ( R , F reg ) / ∼ → T F reg [ γ ] h γ (0) , ( φ γ (0) ◦ γ ) ′ | , γ (0) i , (3.16)which bijective with inverse (for details see [22, p. 285])[ x, v , x ′ ] h t φ − x ′ (cid:16) φ x ′ ( x ) + t ξ v ( t ) v (cid:17)i , where ξ v ∈ C ∞ ( R ) is a smooth cutoff function with 0 ≤ ξ v ≤
1. Moreover, supp( ξ v ) ⊆ ( − ε, ε ) and ξ v | ( − ε/ ,ε/ ≡ ε > B ε k v k (cid:0) φ x ′ ( x ) (cid:1) ⊆ W x ′ . Note that in (3.16) the tangent vector at γ (0) was expressed in the specific chart( φ γ (0) , Ω γ (0) ). However, the tangent vector can also be represented in another chartas follows. Let x ∈ F reg and [ x, v , z ] ∈ T x F reg be arbitrary. We say that a curve γ ∈ C ∞ ( R , F reg ) represents [ x, v , z ] if in one chart φ y with x ∈ Ω y (and thus anychart, as one can show using the chain rule just as before) it holds that[ x, v , z ] = [ γ (0) , ( φ y ◦ γ ) ′ | , y ] . (3.17)In order to show independence of y , let w ∈ F reg with x ∈ Ω w . Then( φ w ◦ γ ) ′ | = ( φ w ◦ φ − y ◦ φ y ◦ γ ) ′ | = ( φ w ◦ φ − y ) ′ | φ y ( x ) ( φ y ◦ γ ) ′ | , and thus [ γ (0) , ( φ w ◦ γ ) ′ | , w ] = [ γ (0) , ( φ y ◦ γ ) ′ | , y ] = [ x, v , z ] . Hence if (3.17) holds in one chart, it also holds in any other chart around x . ♦ Remark 3.9. (directional derivatives)
Let γ ∈ C ∞ ( R , F reg ) be a curve thatrepresents [ x, v , z ]. We define the directional derivative of a Fr´echet-differentiablefunction f : F reg → R at x in the direction [ x, v , z ] as D [ x, v ,z ] f | x := ddt ( f ◦ γ ) | t =0 . This definition is independent of the choice of the curve γ . Indeed, for any chart φ w around x , we have ddt ( f ◦ γ ) | t =0 = ( f ◦ φ − w ◦ φ w ◦ φ z ◦ φ − z γ ) ′ (0)= D ( f ◦ φ − w ) | φ w ( x ) D ( φ w ◦ φ − z ) | φ z ( x ) ( φ z ◦ γ ) ′ (0)= D ( f ◦ φ − w ) | φ w ( x ) D ( φ w ◦ φ − z ) | φ z ( x ) v = D ( f ◦ φ − w ) | φ w ( x ) Dφ w ([ x, v , z ]) . ♦ We close this subsection with one last definition:
Definition 3.10. (Tangent vector fields)
A tangent vector field on a Banach man-ifold is – similar to the finite-dimensional case – a Fr´echet-smooth map v : F reg → T F reg such that v ( x ) ∈ T x F reg (i.e. π ( v ( x )) = x ) for all x ∈ F reg . We denote the setof all tangent vectors fields of F reg by Γ( F reg , T F reg ) . A Riemannian Metric.
In this section we show that the Hilbert-Schmidt scalarproduct gives rise to a canonical Riemannian metric on F reg . For the constructions,it is most convenient to recover F reg as a Banach submanifold of the real Hilbertspace S ( H ) of all selfadjoint Hilbert-Schmidt operators on H endowed with the scalarproduct ( S because of the second Schatten class; for details see [7, Section XI.6]) h A, B i S ( H ) := tr (cid:0) AB (cid:1) . Theorem 3.11. F reg is a smooth Fr´echet submanifold of S in the following sense.Given x ∈ F reg , we choose ψ ∈ Symm( S x ) ⊕ L( J, I ) with x = − ψ ∗ ψ . Then themapping R : (cid:0) Symm( S x ) ⊕ L( J, I ) (cid:1) ⊕ S ( J ) → S ( H )( ψ , B )
7→ − ψ ∗ ψ + (cid:18) B (cid:19) (where the last matrix denotes a block operator on H = I ⊕ J ) is a local Fr´echet-diffeomorphism at ( ψ , . Its local inverse takes the form Φ := ( R | ˆ W ) − : S ( H ) ∩ ˆ W → ˆΩ x ⊆ (cid:0) Symm( S x ) ⊕ L( J, I ) (cid:1) ⊕ S ( J ) E (cid:18) φ x ( π x E ) , π J (cid:16) E + (cid:0) φ x ( π x E ) (cid:1) ∗ φ x ( π x E ) (cid:17)(cid:12)(cid:12)(cid:12) J (cid:19) , where ˆ W = W x ⊕ S ( J ) (with W x as in Theorem 3.2), and φ x ( π x E ) is defined inanalogy to (3.13) by φ x (cid:0) π x E (cid:1) := (cid:16) X − π x E | S x (cid:17) − X − π x E ∈ Symm( S x ) ⊕ L( J, I ) (the fact that the this maps to the symmetric operators on S x is verified as in (3.15) ).Proof. A direct computation shows that R and Φ are inverses of each other. Themappings are Fr´echet-smooth because for operators of finite rank (namely rank atmost 2 n ), the operator norm is equivalent to the Hilbert-Schmidt norm. Indeed, foran operator A : H → I mapping to a finite-dimensional Hilbert space I , k A k ≤ k A † A k ≤ tr( A † A ) = k A k S ( H ,I ) ≤ dim( I ) k A k . This concludes the proof. (cid:3)
We consider a smooth curve γ : ( − δ, δ ) → F reg with γ (0) = x .ddτ (cid:0) φ y ◦ γ ( τ ) (cid:1)(cid:12)(cid:12) τ =0 = v ∈ V y . The corresponding equivalence class defines a tangent vector [ x, v , y ] ∈ T x F reg . On theother hand, considering γ as a curve in S , it has the tangent vector dγ ( τ ) dτ (cid:12)(cid:12)(cid:12) τ =0 ∈ S . In the chart φ x and setting ψ = φ x ( x ), the curve is parametrized by ψ ( τ ) := φ x ◦ γ ( τ )with γ ( τ ) = φ − x ◦ ψ ( τ ) = − ψ ( τ ) ∗ ψ ( τ ) ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 17 and thus dγ ( τ ) dτ (cid:12)(cid:12)(cid:12) τ =0 = Dφ − x | ψ v = − v ∗ ψ − ψ ∗ v with v ∈ V x . As ψ = φ x ( x ) = π x , a direct computation (for details see the proof of Lemma B.6 inAppendix B) that the map V x ∋ v
7→ − v ∗ ψ − ψ ∗ v = − v ∗ π x − π ∗ x v is injective.Thismakes it possible to write the tangent space as T x F reg ≃ T S x F reg := (cid:8) − ψ ∗ ψ − ψ ∗ ψ (cid:12)(cid:12) ψ ∈ Symm( S x ) ⊕ L( J, I ) (cid:9) ⊆ S ( H ) . (3.18) Theorem 3.12.
Using the identification (3.18) , the mapping g x : T S x F reg × T S x F reg → R , g x ( A, B ) := tr( AB ) . defines a Fr´echet-smooth Riemannian metric on F reg . Moreover, the topology on F reg induced by the operator norm coincides with the topology induced by the Riemannianmetric.Proof. Follows immediately because g x is the restriction of the Hilbert space scalarproduct to the smooth Fr´echet submanifold F reg . (cid:3) We finally remark that the symmetric wave charts are related to Gaussian charts(see the formulas in [15, Sections 5 and 6.2], which apply to the infinite-dimensionalcase as well). Detailed computations for the Riemannian metric in symmetric wavecharts are given in Appendix B.4.
Differential Calculus on Expedient Subspaces
If all functions arising in the analysis were Fr´echet-smooth, all the methods and no-tions from the finite-dimensional setting could be adapted in a straightforward way tothe infinite-dimensional setting. However, this procedure is not sufficient for our pur-poses, because the Lagrangian is not Fr´echet-smooth. Therefore, we need to develop adifferential calculus on Banach spaces for functions which are only H¨older continuous.Clearly, in general such functions are not even Fr´echet-differentiable, but the Gˆateauxderivative may exist in certain directions. The disadvantage of Gˆateaux derivatives isthat the differentiable directions in general do not form a vector space. As a conse-quence, the usual computation rules like the linearity of the derivative or the chain andproduct rules cease to hold. Our strategy for preserving the usual computation rules isto work on suitable linear subspaces of the star-shaped set of all Gˆateaux-differentiabledirections, referred to as the expedient differentiable subspace .4.1.
The Expedient Differentiable Subspaces.
In this section E and F denoteBanach spaces. Definition 4.1.
Let U ⊆ E be open and f : U → F an F -valued function. Moreover,let V be a subspace of E . The function f is k times V -differentiable at x ∈ U if for every finite-dimensional subspace H ⊆ V , the restriction of f to the affinesubspace H + x denoted by g H : H → F , g H ( h ) = f ( x + h ) is k -times continuously differentiable at h = 0 . If this condition holds, the subspace V is called k -admissible at x . Thus a function f is once V -differentiable at x if for every finite-dimensional sub-space H ⊆ V , for every h in a small neighborhood of the origin, g H ( h ) = g H ( h ) + Dg H | h ( h − h ) + o ( h − h ) for all h ∈ H , and if Dg H | h is continuous in the variable h at h = 0. Equivalently, choosing abasis e , . . . , e L of H , this condition can be stated that all partial derivatives ∂∂α i g H (cid:0) α e + · · · + α L e L (cid:1) exist and are continuous at α , . . . , α L = 0. The higher differentiability of g H can bedefined inductively or, equivalently, by demanding that all partial derivatives up tothe order k , i.e. all the functions ∂ p ∂α i · · · α i p g H ( α e + · · · + α L e L )with i , . . . , i p ∈ { , . . . , L } and p ≤ k , exist and are continuous at α , . . . , α L = 0.An admissible subspace V is maximal if there are no admissible proper exten-sions ˜ V ) V . The existence of maximal admissible subspaces is guaranteed by Zorn’slemma, but maximal subspaces are in general not unique. In order to obtain a canon-ical subspace, we take the intersection of all maximal admissible subspaces: Definition 4.2.
The expedient k -differentiable subspace E k ( f, x ) of f at x isdefined as the intersection E k ( f, x ) := \ (cid:8) V (cid:12)(cid:12) V ⊆ E k -admissible at x and maximal (cid:9) . Since the expedient differentiable subspace is again admissible at x , we obtain acorresponding derivative as follows. Given k ∈ N and vectors h , . . . , h k ∈ E ( f, x ), wechoose H as a finite-dimensional subspace which contains these vectors. We set D k, E f | x ( h , . . . , h k ) := D k g H | ( h , . . . , h k ) (4.1)(where again g H ( h ) := f ( x + h )). Lemma 4.3.
This procedure defines D k, E f | x canonically as a symmetric, multilinearmapping D k, E f | x : E k ( f, x ) × · · · × E k ( f, x ) | {z } k factors → F .
Proof.
In order to show that D k, E f | x is well-defined, let H and ˜ H be two finite-dimensional subspaces of E ( f, x ) which contain the vectors h , . . . , h k . Then, express-ing the partial derivatives in terms of partial derivatives, it follows that D k g H | ( h , . . . , h k ) = ∂ p ∂α · · · α k f ( x + α h + · · · + α k h k ) (cid:12)(cid:12)(cid:12) α = ··· = α k =0 = D k g ˜ H | ( h , . . . , h k ) . This shows that the definition (4.1) does not depend on the choice of H .The symmetry and homogeneity follow immediately from the corresponding prop-erties of D k g H in (4.1). In order to prove additivity, we let h , . . . , h k ∈ E k ( f, x )and ˜ h , . . . , ˜ h k ∈ E k ( f, x ). We let H be the span of all these vectors and use thatthe corresponding operator D k g H | in (4.1) applied to h + ˜ h , . . . , h k + ˜ h k is multilin-ear. (cid:3) ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 19
Note that the operator D k, E f | x is in general not bounded. Moreover, E k ( f, x ) willin general not be a closed subspace of E , nor will it in general be dense.4.2. Derivatives Along Smooth Curves.
We now analyze under which assump-tions directional derivatives exist. To this end, we let I be an interval and γ : I → E asmooth curve (here the notions of Fr´echet and Gˆateaux smoothness coincide). More-over let t ∈ I with x := γ ( t ) ∈ U and U ⊆ E open. Given a function f : U → F ,we consider the composition f ◦ γ : I → F .
Proposition 4.4. (chain rule)
Assume that f is locally H¨older continuous at x ,meaning that there is a neighborhood V ⊆ U of x as well as constants α, c > suchthat k f ( x ) − f ( x ′ ) k F ≤ c k x − x ′ k αE for all x, x ′ ∈ V . (4.2)
Moreover, assume that all the derivatives of γ at x up to the order p := (cid:24) α (cid:25) (4.3) (where ⌈·⌉ is the ceiling function) lie in the expedient differentiable subspace at x , i.e. γ ( n ) ( t ) ∈ E ( f, x ) for all n ∈ { , . . . , p } . Then the function f ◦ γ is differentiable at t and ( f ◦ γ ) ′ ( t ) = D E f | x γ ′ ( t ) . Proof.
We consider the polynomial approximation of γγ p ( t ) := p X n =0 γ ( n ) ( t ) n ! ( t − t ) n . (4.4)By assumption, this curve lies in the affine subspace E ( f, x ) + x . Using that therestriction of f to this subspace is continuously differentiable, it follows that( f ◦ γ p ) ′ ( t ) = D E f | x γ ′ ( t ) . It remains to control the error term of the polynomial approximation. Using that f is locally H¨older continuous, we know that (cid:13)(cid:13) ( f ◦ γ )( t ) − ( f ◦ γ p )( t ) (cid:13)(cid:13) F ≤ c k γ ( t ) − γ p ( t ) k αE . Using that γ is smooth, it follows that (cid:13)(cid:13) ( f ◦ γ )( t ) − ( f ◦ γ p )( t ) (cid:13)(cid:13) F ≤ (cid:13)(cid:13) o (cid:0) ( t − t ) p (cid:1)(cid:13)(cid:13) αE = o (cid:0) ( t − t ) αp (cid:1) . (4.5)According to (4.3), we know that αp ≥
1. Therefore, the error term is of the order o ( t − t ), which shows that also the function t ( f ◦ γ )( t ) − ( f ◦ γ p )( t ) is differentiable withvanishing derivative. This proves the desired result. (cid:3) This result immediately generalizes to higher derivatives:
Proposition 4.5. (higher order chain rule)
Assume that f is locally H¨older con-tinuous at x (see (4.2) ). Moreover, assume that all the derivatives of γ at x up tothe order p := (cid:24) qα (cid:25) (4.6) lie in the expedient differentiable subspace at x , i.e. γ ( n ) ( t ) ∈ E q ( f, x ) for all n ∈ { , . . . , p } . Then the function f ◦ γ is q -times differentiable at t , and the derivative can be computedwith the usual product and chain rules (formula of Fa`a di Bruno).Proof. We again consider f along the polynomial approximation γ p (4.4) of the curve γ .By assumption, this curve lies in a finite-dimensional subspace of the affine space E q ( f, x ) + x ⊂ F .
Using that the restriction of f to this subspace is continuously differentiable, we knowthat f ◦ γ p is q times continuously differentiable at t = t , and the derivatives can becomputed with the formula of Fa`a di Bruno,( f ◦ γ p ) ( q ) ( t ) = D E ,q f | x (cid:0) γ ′ ( t ) , . . . , γ ′ ( t ) (cid:1) + q ( q − D E ,q − f | x (cid:0) γ ′′ ( t ) , γ ′ ( t ) , . . . , γ ′ ( t ) (cid:1) + · · · . Using (4.5) and (4.6), we conclude that( f ◦ γ )( t ) − ( f ◦ γ p )( t ) = o (cid:0) ( t − t ) q (cid:1) . It follows that also this function is q -times differentiable and that all its derivativesvanish. This concludes the proof. (cid:3) Application to Causal Fermion Systems in Infinite Dimensions
Local H¨older Continuity of the Causal Lagrangian.
The goal of this sectionis to prove the following result.
Theorem 5.1.
The Lagrangian is locally H¨older continuous in the sense that forall x, y ∈ F there is a neighborhood U ⊆ F of y and a constant c > such that (cid:12)(cid:12) L ( x, y ) − L ( x, ˜ y ) (cid:12)(cid:12) ≤ c k y − ˜ y k n − for all y, ˜ y ∈ U , (5.1) where n is the spin dimension. Moreover, the integrand of the boundedness constraintis locally Lipschitz continuous in the sense that (cid:12)(cid:12)(cid:12) | xy | − | x ˜ y | (cid:12)(cid:12)(cid:12) ≤ c k y − ˜ y k n for all y, ˜ y ∈ U . (5.2)We begin with a preparatory lemma.
Lemma 5.2. (H¨older continuity of roots)
Let P ( λ ) := λ g + c g − λ g − + · · · + c = g Y i =1 ( λ − λ i ) be a complex monic polynomial of degree g with roots λ , . . . , λ g . Then there are con-stants C, ε > such that any complex monic polynomial ˜ P ( λ ) = λ g +˜ c g − λ g − + · · · +˜ c of degree g which is close to P in the sense that k ˜ P − Pk := max ℓ ∈{ ,...,g − } (cid:12)(cid:12) ˜ c ℓ − c ℓ (cid:12)(cid:12) < ε can be written as ˜ P ( λ ) = Q gi =1 ( λ − ˜ λ i ) with | λ i − ˜ λ i | ≤ C k ˜ P − Pk pi for all i = 1 , . . . , g , ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 21 where p i is the multiplicity of the root λ i . This lemma is proven in a more general context in [4, Theorem 2]. For self-consistencywe here give a simple proof based on Rouch´e’s theorem:
Proof of Lemma 5.2.
After the rescaling λ → νλ and λ i → νλ i with ν >
0, we canassume that all the roots λ i are in the unit ball. Then the polynomial ∆ P := ˜ P − P is bounded in the ball of radius two by | ∆ P ( λ ) | ≤ g g k ∆ Pk for all λ with | λ | ≤ . (5.3)We denote the minimal distance of distinct eigenvalues by D := min λ i = λ j | λ i − λ j | . Since there is a finite number of roots, it clearly suffices to prove the lemma for one ofthem. Given i ∈ { , . . . , g } we choose δ = (cid:18) g g − p i +1 D g − p i k ∆ Pk (cid:19) pi . (5.4)Next, we choose ε so small that δ < D/
2. We consider the ball Ω = B δ ( λ i ). Then forany λ ∈ ∂ Ω, the polynomial P satisfies the bound |P ( λ ) | ≥ ( D/ g − p i δ p i ≥ g g +1 k ∆ Pk > | ∆ P ( λ ) | , where we used (5.4) and (5.3). Therefore, Rouch´e’s theorem (see for example [27,Theorem 10.36]) implies that the polynomials P and ˜ P have the same number of rootsin the ball Ω. Thus, after a suitable ordering of the roots, | λ i − ˜ λ i | ≤ δ . Using (5.4) gives the result. (cid:3)
Proof of Theorem 5.1.
Let x, y ∈ F . Since both operators x and y vanish on the or-thogonal complement of the span their images combined, J := span( S x , S y ), it sufficesto compute the eigenvalues on the finite-dimensional subspace J . Choosing an or-thonormal basis of S x = x ( H ) and extending it to an orthonormal basis of J , thematrix xy | J − J has the block matrix form (cid:18) xyπ x − λ ∗ − λ (cid:19) . Therefore, its characteristic polynomial is given bydet J ( xy − J ) = ( − λ ) dim J − dim x ( H ) det x ( H ) (cid:0) xyπ x − λ x ( H ) (cid:1) . This consideration shows that it suffices to analyze the operators xyπ x and simi-larly x ˜ yπ x on the finite-dimensional Hilbert space x ( H ). We denote the correspondingcharacteristic polynomials by P and ˜ P , respectively. They are monic polynomials ofdegree g := dim x ( H ). The difference of these polynomials can be estimated in termsof operator norms on L( H ) as follows, k ˜ P − Pk ≤ c (cid:0) g, k x k , k y k (cid:1) (cid:13)(cid:13) x ˜ yπ x − xyπ x (cid:13)(cid:13) ≤ c ′ (cid:0) g, k x k , k y k (cid:1) (cid:13)(cid:13) ˜ y − y (cid:13)(cid:13) , valid for all ˜ y with k ˜ y k ≤ k y k . According to Lemma 5.2, for sufficiently small k y − ˜ y k the eigenvalues of these matrices can be arranged to satisfy the inequalities | λ i − ˜ λ i | ≤ C k ˜ P − Pk pi ≤ C ′ (cid:0) x, y (cid:1) (cid:13)(cid:13) ˜ y − y (cid:13)(cid:13) pi . In order to prove (5.2), we consider the estimate (cid:12)(cid:12)(cid:12) | xy | − | x ˜ y | (cid:12)(cid:12)(cid:12) ≤ g X i =1 (cid:12)(cid:12)(cid:12) | λ i | − | ˜ λ i | (cid:12)(cid:12)(cid:12) ≤ g X i =1 | λ i − ˜ λ i | (cid:0) | λ i | + | ˜ λ i | (cid:1) ≤ ˜ C ( x, y ) (cid:13)(cid:13) ˜ y − y (cid:13)(cid:13) g (5.5)and use that g ≤ n .It remains to prove (5.1). In the case g < n , a simple estimate similar to (5.5)gives the result. In the remaining case g = 2 n , using the abbreviation ∆ λ i := ˜ λ i − λ i ,we obtain (cid:12)(cid:12) L ( x, ˜ y ) − L ( x, y ) (cid:12)(cid:12) ≤ g g X i,j =1 (cid:12)(cid:12)(cid:12) | ˜ λ i − ˜ λ j | − | λ i − λ j | (cid:12)(cid:12)(cid:12) ≤ g g X i,j =1 (cid:16) | ∆ λ i − ∆ λ j | | λ i − λ j | + | ∆ λ i − ∆ λ j | (cid:17) ≤ c ( x, y ) g X i,j =1 (cid:16)(cid:13)(cid:13) ˜ y − y (cid:13)(cid:13) max (cid:0) pi , pj (cid:1) | λ i − λ j | + (cid:13)(cid:13) ˜ y − y (cid:13)(cid:13) g (cid:17) ≤ c ( x, y ) g X i,j =1 (cid:16)(cid:13)(cid:13) ˜ y − y (cid:13)(cid:13) g − + (cid:13)(cid:13) ˜ y − y (cid:13)(cid:13) g (cid:17) , where in the last step we used that, whenever λ i = λ j , the multiplicities of both rootsare at most g −
1. The inequality2 g = 1 n ≥ n − g − , yields the desired H¨older inequality with exponent 1 / (2 n − y . This concludes theproof. (cid:3) In the case of spin dimension one, the Lagrangian is Lipschitz continuous, in agree-ment with the findings in [20]. If the spin dimension is larger, one still has H¨oldercontinuity, but the H¨older exponent becomes smaller if the spin dimension is increased.This can be understood from the fact that the higher the spin dimension is, the higherthe degeneracies of the eigenvalues of xy can be.We next prove a global H¨older continuity result. Theorem 5.3. (Global H¨older continuity)
There is a constant c ( n ) which dependsonly on the spin dimension such that for all x, y ∈ F with y = 0 there is a neighborhood U ⊆ F of y with |L ( x, y ) − L ( x, ˜ y ) | ≤ c ( n ) k y k − n − k x k k ˜ y − y k n − for all ˜ y ∈ U . (5.6)
Proof.
Without loss of generality we can assume that x = 0. Moreover, using thatboth sides of the inequality (5.6) have the same scaling behavior under the rescaling x → x k x k , y → y k y k , ˜ y → ˜ y k y k , it suffices to consider the case that k x k = k y k = 1. ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 23
Next, choosing a fixed 4 n -dimensional subspace of I ⊆ H , we can always find aunitary transformation U : H → H such that U xU − ( H ) , U yU − ( H ) ⊆ I . Since theLagrangian and the operator norms are invariant under such joint unitary transforma-tions (as they leave the eigenvalues of xy invariant), we can assume that both x and y map into the fixed finite dimensional subspace I .After these transformations, the operators x and y can be considered as operatorsin L( I ). Therefore, they lie in the compact set B (0) ⊆ L( I ). Since the H¨older con-stant for the local H¨older continuity depends continuously on x and y , a compactnessargument shows that we can choose the H¨older constant uniformly in x and y . (cid:3) Remark 5.4. (1) Since the Lagrangian is symmetric, Theorem 5.3 also gives riseto global H¨older continuity with respect to the other argument. Thus for all x, y ∈ F with x = 0 there is a neighborhood U ⊆ F of x such that |L ( x, y ) − L (˜ x, y ) | ≤ c ( n ) k x k − n − k y k k ˜ x − x k n − . (5.7)(2) As explained in the proof of Theorem 5.3, the Lagrangian L ( x, y ) depends onlyon the non-zero eigenvalues of xy and these coincide with the eigenvalues of xyπ x .Thus denoting J := span( S x , S ˜ x ) , we immediately obtain the following strengthened version of (5.7), |L ( x, y ) − L (˜ x, y ) | = |L ( x, π J y π J ) − L (˜ x, π J y π J ) |≤ c ( n ) k x k − n − k π J y π J k k ˜ x − x k n − . (5.8)This estimate will be needed for the proof of the chain rule for the integratedLagrangian ℓ in Theorem 5.9.(3) In the case y = 0, a direct estimate of the eigenvalues shows that one has H¨oldercontinuity with the improved exponent two, (cid:12)(cid:12) L ( x, ˜ y ) (cid:12)(cid:12) ≤ c ( n ) k x k k ˜ y k . This inequality can be combined with the result of Theorem 5.3 to the statementthat for all x, y there is a neighborhood U ⊆ F of y with |L ( x, y ) − L ( x, ˜ y ) | ≤ c ( n, y ) k x k k ˜ y − y k n − for all ˜ y ∈ U . (5.9)Likewise, (5.8) generalizes to |L ( x, y ) − L (˜ x, y ) | ≤ c ( n, x ) k π J y π J k k ˜ x − x k n − . (5.10)This inequality will be used in the proof of Theorem 5.9. ♦ Definition of Jet Spaces.
For the analysis of causal variational principles, thejet formalism was developed in [17]; see also [13, Section 2]. We now generalize thedefinition of the jet spaces to causal fermion systems in the infinite-dimensional setting.Our method is to work with the expedient subspaces, where for convenience derivativesat x are always computed in the corresponding chart φ x . For example, for analyzingthe differentiability of a real-valued function f at a point x ∈ F reg , we consider thecomposition f ◦ φ − x : Ω x ⊆ Symm( S x ) ⊕ L( J, I ) → R . We introduce Γ diff ρ as the linear space of all vector fields for which the directional de-rivative of the function ℓ exists in the sense of expedient subspaces (see Definition 4.2),Γ diff ρ = n u ∈ C ∞ ( M, T F reg ) (cid:12)(cid:12) u ( x ) ∈ E (cid:0) ℓ ◦ φ − x , φ x ( x ) (cid:1) for all x ∈ M o . This gives rise to the jet space J diff ρ := C ∞ ( M, R ) ⊕ Γ diff ρ ⊆ J ρ . (5.11)We choose a linear subspace J test ρ ⊆ J diff ρ with the property that its scalar and vectorcomponents are both vector spaces, J test ρ = C test ( M, R ) ⊕ Γ test ρ ⊆ J diff ρ , (5.12)and the scalar component is nowhere trivial in the sense thatfor all x ∈ M there is a ∈ C test ( M, R ) with a ( x ) = 0 . It is convenient to consider a pair u := ( a, u ) consisting of a real-valued function a on M and a vector field u on T F reg along M , and to denote the combination of multiplicationand directional derivative by ∇ u ℓ ( x ) := a ( x ) ℓ ( x ) + (cid:0) D u ℓ (cid:1) ( x ) . (5.13)For the Lagrangian, being a function of two variables x, y ∈ F reg , we always workin charts φ x and φ y , giving rise to the mapping L ◦ (cid:0) φ − x × φ − y (cid:1) = L (cid:0) φ − x ( . ) , φ − y ( . ) (cid:1) : Ω x × Ω y ⊆ E → R , (5.14)where E is the Cartesian product of Banach spaces E := (cid:0) Symm( S x ) ⊕ L( J x , I x ) (cid:1) × (cid:0) Symm( S y ) ⊕ L( J y , I y ) (cid:1) with the norm k ( ψ x , ψ y ) k E := max (cid:0) k ψ x k L( H ) , k ψ y k L( H ) (cid:1) (where the subscripts x and y clarify the dependence on the base points, i.e. I x = x ( H ), J x = I ⊥ x ⊆ H and similarly at y ). We denote partial derivatives acting on the firstand second arguments by subscripts 1 and 2, respectively. Throughout this paper, weuse the following conventions for partial derivatives and jet derivatives: ◮ Partial and jet derivatives with an index i ∈ { , } , as for example in (5.15), onlyact on the respective variable of the function L . This implies, for example, thatthe derivatives commute, ∇ , v ∇ , u L ( x, y ) = ∇ , u ∇ , v L ( x, y ) . ◮ The partial or jet derivatives which do not carry an index act as partial derivativeson the corresponding argument of the Lagrangian. This implies, for example, that ∇ u ˆ F ∇ , v L ( x, y ) dρ ( y ) = ˆ F ∇ , u ∇ , v L ( x, y ) dρ ( y ) . Definition 5.5.
For any ℓ ∈ N ∪ {∞} , the jet space J ℓρ ⊆ J ρ is defined as the vectorspace of test jets with the following properties: (i) The directional derivatives up to order ℓ exist in the sense that J ℓρ ⊆ n ( b, v ) ∈ J ρ (cid:12)(cid:12)(cid:12) (cid:0) v ( x ) , v ( y ) (cid:1) ∈ Γ ℓρ ( x, y ) for all y ∈ M and x in an open neighborhood of M ⊆ F reg o , ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 25 where Γ ℓρ ( x, y ) := E ℓ (cid:16) L ◦ (cid:0) φ − x × φ − y (cid:1) , (cid:0) φ x ( x ) , φ y ( y ) (cid:1)(cid:17) . The higher jet derivatives are defined by using (5.13) and multiplying out, keepingin mind that the partial derivatives act only on the Lagrangian, i.e. ∇ p, E L ◦ (cid:0) φ − x × φ − y (cid:1)(cid:12)(cid:12) ( φ x ( x ) ,φ y ( y )) (cid:16)(cid:0) v ( x ) , v ( y ) (cid:1) , . . . , (cid:0) v p ( x ) , v p ( y ) (cid:1)(cid:17) := D p, E L ◦ (cid:0) φ − x × φ − y (cid:1)(cid:12)(cid:12) ( φ x ( x ) ,φ y ( y )) (cid:16)(cid:0) v ( x ) , v ( y ) (cid:1) , . . . , (cid:0) v p ( x ) , v p ( y ) (cid:1)(cid:17) + (cid:0) b ( x ) + b ( y ) (cid:1) D p − , E L ◦ (cid:0) φ − x × φ − y (cid:1)(cid:12)(cid:12) ( φ x ( x ) ,φ y ( y )) × (cid:16)(cid:0) v ( x ) , v ( y ) (cid:1) , . . . , (cid:0) v p ( x ) , v p ( y ) (cid:1)(cid:17) + (cid:0) b ( x ) + b ( y ) (cid:1) D p − , E L ◦ (cid:0) φ − x × φ − y (cid:1)(cid:12)(cid:12) ( φ x ( x ) ,φ y ( y )) × (cid:16)(cid:0) v ( x ) , v ( y ) (cid:1) , (cid:0) v ( x ) , v ( y ) (cid:1) , . . . , (cid:0) v p ( x ) , v p ( y ) (cid:1)(cid:17) + · · · + (cid:0) b ( x ) + b ( y ) (cid:1) · · · (cid:0) b p ( x ) + b p ( y ) (cid:1) L ( x, y ) . (ii) The functions (cid:0) ∇ , v + ∇ , v (cid:1) · · · (cid:0) ∇ , v p + ∇ , v p (cid:1) L ( x, y ):= ∇ p, E L ◦ (cid:0) φ − x × φ − y (cid:1)(cid:12)(cid:12) ( φ x ( x ) ,φ y ( y )) (cid:16)(cid:0) v ( x ) , v ( y ) (cid:1) , . . . , (cid:0) v p ( x ) , v p ( y ) (cid:1)(cid:17) (5.15) are ρ -integrable in the variable y , giving rise to locally bounded functions in x .More precisely, these functions are in the space L ∞ loc (cid:16) M, L (cid:0) M, dρ ( y ) (cid:1) ; dρ ( x ) (cid:17) . (iii) Integrating the expression (5.15) in y over M with respect to the measure ρ , the re-sulting function g (defined for all x in an open neighborhood of M ) is continuouslydifferentiable in the direction of every jet u ∈ J test ρ , i.e. Γ test x ⊆ E ( g, x ) for all x ∈ M .
Derivatives of L and ℓ along Smooth Curves. In this section we use the chainrule in Proposition 4.4 in order to differentiate the Lagrangian L and the function ℓ along smooth curves. Theorem 5.6.
Let γ and γ be two smooth curves in F reg , γ , γ ∈ C ∞ (( − δ, δ ) , F reg ) . Setting x = γ (0) and y = γ (0) , we assume that the tangent vectors up to the order p =2 n − denoted by v (1)1 := ( φ x ◦ γ a ) ′ (0) , . . . , v ( p )1 := ( φ x ◦ γ a ) ( p ) (0) v (1)2 := ( φ y ◦ γ a ) ′ (0) , . . . , v ( p )2 := ( φ y ◦ γ a ) ( p ) (0) are in the expedient differentiable subspace of the Lagrangian, i.e. (cid:0) v (1)1 , v (1)2 (cid:1) , . . . , (cid:0) v ( p )1 , v ( p )2 (cid:1) ∈ Γ ρ ( x, y ) . Then the function L ( γ ( τ ) , γ ( τ )) is τ -differentiable at τ = 0 and the chain rule holds,i.e. ddτ L (cid:0) γ ( τ ) , γ ( τ ) (cid:1)(cid:12)(cid:12) τ =0 = D E (cid:0) L ◦ (cid:0) φ − x × φ − y (cid:1)(cid:1)(cid:12)(cid:12) ( φ x ( x ) ,φ y ( y )) (cid:0) v , v (cid:1) ≡ (cid:0) D ,γ ′ (0) + D ,γ ′ (0) (cid:1) L ( x, y ) . Proof.
We again consider the Lagrangian in the charts φ x and φ y , (5.14). In order toshow that this function is locally H¨older continuous on E , we begin with the estimate (cid:12)(cid:12) L (cid:0) φ − x ( ˜ ψ x ) , φ − y ( ˜ ψ y ) (cid:1) − L ( x, y ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) L (cid:0) φ − x ( ˜ ψ x ) , φ − y ( ˜ ψ y ) (cid:1) − L (cid:0) φ − x ( ˜ ψ x ) , y (cid:1)(cid:13)(cid:13) + (cid:13)(cid:13) L (cid:0) φ − x ( ˜ ψ x ) , y (cid:1) − L ( x, y ) (cid:12)(cid:12) ≤ c (cid:16) k φ − x ( ˜ ψ x ) − x k α L( H ) + k φ − y ( ˜ ψ y ) − y k α L( H ) (cid:17) . Noting that the function φ − x ( ˜ ψ x ) = − ˜ ψ ∗ x ˜ ψ x is bilinear and therefore Fr´echet-smooth, it follows that (cid:12)(cid:12) L (cid:0) φ − x ( ˜ ψ x ) , φ − y ( ˜ ψ y ) (cid:1) − L ( x, y ) (cid:12)(cid:12) ≤ cC (cid:16) k ˜ ψ x − ψ x k α L( H ,I ) + k ˜ ψ y − ψ y k α L( H ,I ) (cid:17) ≤ cC (cid:13)(cid:13) ( ˜ ψ x , − ˜ ψ y ) − ( ψ x , ψ y ) (cid:13)(cid:13) αE , where ψ x := φ − x ( x ) and ψ y := φ − y ( y ). This proves local H¨older continuity on E .Applying Proposition 4.4 gives the result. (cid:3) We remark that, using Proposition 4.5, the above method could be generalized in astraightforward manner to higher derivatives.
Definition 5.7.
We call ℓ H¨older continuous with H¨older exponent α along asmooth curve γ : I → F (with I an open interval) if for any t ∈ I with x = γ ( t ) there exists a subspace E ⊆ Symm S x ⊕ L ( J x , I x ) and δ > such that the mapping γ x : ( t − δ, t + δ ) → E , t φ x ◦ γ ( t ) − (11 , , is well defined and locally H¨older continuous with H¨older exponent α . Theorem 5.8.
Let γ : I → F be a smooth curve and ℓ H¨older continuous along γ withH¨older exponent α . For t ∈ I with x = γ ( t ) we set ℓ x : E → R , ℓ x ( x ) = ℓ ◦ φ − x (cid:0) x + (11 , (cid:1) . If for any x ∈ I the derivatives of γ x up to the order p := ⌈ q/α ⌉ lie in the expedientdifferentiable subspace at x , i.e. ( γ x ) ( n ) ( t ) ∈ E q (cid:16) ℓ x , (cid:17) for all n ∈ { , . . . , p } , then the function ℓ ◦ γ = ℓ x ◦ γ x is q -times differentiable at t . Moreover, the usualproduct and chain rules hold (keeping the chart φ x fixed).Proof. Applying 4.5 to ℓ x and γ x yields the claim as the assumptions for this theoremare clearly fulfilled. (cid:3) ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 27
Theorem 5.9.
Let γ be a smooth curve in F with ˆ M (cid:13)(cid:13) P ( γ ( τ ) , y ) (cid:13)(cid:13) (cid:13)(cid:13) Y − (cid:13)(cid:13) dρ ( y ) < C for all τ ∈ ( − δ, δ ) , where P ( x, y ) is again the kernel of the fermionic projector (3.11) and Y is (similarto (3.5) ) the invertible operator Y := y | S y : S y → S y . Then the integrated Lagrangian ℓ defined by (1.1) is H¨older continuous along γ withH¨older exponent n − .Proof. The idea of the proof is to integrate the estimate (5.10) over M . To this end,it is crucial to estimate the factor k π J yπ J k . We let ( ˜ φ i ) i ∈ ,...m be an orthonormalbasis of J and denote the orthogonal projection on span( ˜ φ i ) by π i . Since on thefinite-dimensional vector space L ( J ) all norms are equivalent, we can work with theHilbert-Schmidt norm of π J yπ J , i.e. for a suitable constant C = C ( n ), k π J y π J k ≤ C k π J y π J k HS = C tr (cid:0) π J y π J y π J (cid:1) = C m X i =1 h ˜ φ i | y π J y ˜ φ i i = C m X i,j =1 h ˜ φ i | y ˜ φ j ih ˜ φ j | y ˜ φ i i = C m X i,j =1 (cid:12)(cid:12) h ˜ φ i | y ˜ φ j i (cid:12)(cid:12) = C m X i,j =1 (cid:12)(cid:12) h ˜ φ i | π i y Y − y π j ˜ φ j i (cid:12)(cid:12) ≤ C m X i,j =1 (cid:13)(cid:13) π i y (cid:13)(cid:13) (cid:13)(cid:13) Y − (cid:13)(cid:13) (cid:13)(cid:13) y π j (cid:13)(cid:13) . (5.16)In order to simplify the sums, using that ( ˜ φ i ) is an orthonormal basis of I , for any ψ ∈ H we obtain the identity (cid:13)(cid:13) π J y ψ (cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13) m X i =1 ˜ φ i h ˜ φ i | y ψ i (cid:13)(cid:13)(cid:13)(cid:13) = m X i =1 (cid:12)(cid:12) h ˜ φ i | y ψ i (cid:12)(cid:12) = m X i =1 (cid:13)(cid:13) π i y ψ (cid:13)(cid:13) . Taking the supremum over all unit vectors ψ , it follows that k π J y k = m X i =1 (cid:13)(cid:13) π i y (cid:13)(cid:13) . Using this identity in (5.16), we conclude that k π J y π J k ≤ C (cid:13)(cid:13) π J y (cid:13)(cid:13) (cid:13)(cid:13) Y − (cid:13)(cid:13) . Next, (cid:13)(cid:13) π J yψ (cid:13)(cid:13) ≤ (cid:0)(cid:13)(cid:13) π x yψ (cid:13)(cid:13) + (cid:13)(cid:13) π ˜ x yψ (cid:13)(cid:13)(cid:1) ≤ (cid:13)(cid:13) π x y ψ (cid:13)(cid:13) + 2 (cid:13)(cid:13) π ˜ x y ψ (cid:13)(cid:13) , Thus we can proceed in our previous estimate to obtain k π J y π J k ≤ C ( n ) (cid:16)(cid:13)(cid:13) π x y ψ (cid:13)(cid:13) + (cid:13)(cid:13) π ˜ x y ψ (cid:13)(cid:13) (cid:17) (cid:13)(cid:13) Y − (cid:13)(cid:13) ≤ C ( n ) (cid:13)(cid:13) Y − (cid:13)(cid:13) (cid:16)(cid:13)(cid:13) π x y ψ (cid:13)(cid:13) + (cid:13)(cid:13) π ˜ x y ψ (cid:13)(cid:13) (cid:17) = 4 C ( n ) (cid:13)(cid:13) Y − (cid:13)(cid:13) (cid:16)(cid:13)(cid:13) P ( x, y ) ψ (cid:13)(cid:13) + (cid:13)(cid:13) P (˜ x, y ) ψ (cid:13)(cid:13) (cid:17) . Using this estimate when integrating (5.10) over M and noting that φ − x is locallyLipschitz (since it is Fr´echet-smooth) yields the claim. (cid:3) Appendix A. Properties of the Fr´echet Derivative
This appendix lists a set of properties and computation rules for Fr´echet derivativeswhich are needed for the direct computations in appendix B. It turns out that mostderivation rules known from the finite dimensional case generalize to Fr´echet derivativesin a straightforward way.
Lemma A.1. (Properties of the Fr´echet derivative)
Let
V, W, Z be real normedvector. Then the following Fr´echet derivative rules hold: (i)
Let U ⊆ V open and f : V → W Fr´echet-differentiable at x ∈ U , then f iscontinuous at x and Df | x is well defined. (ii) Let f ∈ L( V, W ) be linear and bounded, then it is Fr´echet-smooth at any x ∈ V and Df | x = f . (iii) A continuous bilinear map B : V × W → Z is Fr´echet-smooth at any ( v, w ) ∈ V × W and DB | ( v,w ) ( h v , h w ) = B ( v, h w ) + B ( h v , w ) , ∀ ( h v , h w ) ∈ V × W . (iv)
Chain rule: Let U V ⊆ V and U W ⊆ W open, f : U V → W , g : U W → Z such that f ( U V ) ⊆ U W . If f is Fr´echet-differentiable at x ∈ U V and g in f ( x ) ∈ U W ,then also g ◦ f is Fr´echet-differentiable in x and D ( g ◦ f ) | x = Dg | f ( x ) ◦ Df | x . (v) Let W , ..., W n be real normed vector spaces, W := W × W × ... × W n theproduct space, U ⊆ V open and f = ( f , ..., f n ) : U → W with f i : V → f i for i = 1 , . . . , n . Then f is Fr´echet-differentiable at x ∈ U if and only ifeach f i is Fr´echet-differentiable at x . Moreover, in this case, we have Df | x =( Df | x , ..., Df n | x ) . (vi) Let U V ⊆ V and U W ⊆ W be open and f : U V → U W a homeomorphism withinverse g : U W → U V . If f is Fr´echet-differentiable at x ∈ U V and g is Fr´echet-differentiable in y = f ( x ) ∈ U W , then Df | x is an isomorphism with inverse Dg | y .Proof. (i) See [5, Prop. 2.2, Chapter 2.2].(ii) f is clearly Fr´echet-differentiable with Df | x = f for any x ∈ U as k f ( x + h ) − f ( x ) − f h k W = 0 for all x, h ∈ V (see also [6, p. 149-150] and note that thecompleteness of the vector spaces is not needed for this result). Moreover, as Df : U → L ( V, W ) is constant it is clear that all higher Fr´echet-derivatives of f vanish (and in particular f is Fr´echet-smooth).(iii) B is Fr´echet differentiable with the stated Fr´echet derivative as k B ( v + h v , w + h w ) − B ( v, w ) − B ( v, h w ) − B ( h v , w ) k Z = k B ( h v , h w ) k Z ≤ C k h v k · k h w k ≤ C (cid:0) max( k h v k , k h w k ) (cid:1) , for a fixed C > B is continuous and bilinear), see also [6, p. 149-150] (againthe completeness in not needed). And since DB : V × W → L ( V × W, L ( V × W, Z ))( v, w ) (cid:16) ( h v , h w ) B ( v, h w ) + B ( h v , w ) (cid:17) , is clearly bounded linear, B is Fr´echet-smooth due to part (ii).(iv) See [5, Theorem 2.1, Chapter 2.3]. ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 29 (v) See [6, p. 149–151] (again completeness is not needed).(vi) This follows immediately from the chain rule and part (ii) sinceid V ( ii ) = D (id V ) | x = D ( g ◦ f ) | x ( iv ) = Dg | y ◦ Df | x , and similarly id W = Df | x ◦ Dg | y . (cid:3) Lemma A.2.
Let V , W and Z be real normed spaces, n ∈ N arbitrary and U V ⊆ V, U W ⊆ W open subsets, f : U V → W n -times Fr´echet-differentiable (Fr´echet-smooth)and g : U W → Z n -times Fr´echet-differentiable (Fr´echet-smooth) such that f ( U V ) ⊆ U W . Then g ◦ f is also n -times Fr´echet-differentiable (respectively Fr´echet-smooth).Proof. We show the result by induction over n following [6, p. 183]: The case n = 1follows from the chain rule. Now let n ≥ n −
1. Then the induction hypothesis yields that the mapping x D ( f ◦ g ) | x = Df | g ( x ) ◦ Dg | x is ( n − Df , g and Dg are atleast ( n − ◦ is even Fr´echet-smooth(as it is bounded linear). Thus f ◦ g is n -times Fr´echet-differentiable.The smoothness result follows immediately from the result for n -times differentia-bility. (cid:3) The following lemma gives a useful computation rule for higher Fr´echet derivatives(see also [6, p. 179, 181]):
Lemma A.3.
Let
V, W be real normed spaces, U ⊆ V open and f : U → W n -timesdifferentiable. Then for any x ∈ U and v , · · · , v n ∈ V , D (cid:0) D ( n − f | . ( v , · · · , v n − ) (cid:1)(cid:12)(cid:12) x v n = D ( n ) f | x ( v , · · · , v n ) . (A.1) In particular, the map U ∋ x D ( n − f | x ( v , · · · , v n − ) ∈ W is Fr´echet-differentiable.Proof. We follow the idea of the proof given in [6, p. 179, 181] and also make use ofthe symmetry result in Lemma 2.3. We first fix v , . . . , v n ∈ V and define a linear mapby E v ,...,v n − : L( V, W ) n − → WA ( . . . (( Av ) v ) . . . ) v n − , which simply inserts all the v , . . . , v n − in an A ∈ L( V, W ). Note that E v ,...,v n − isclearly bounded linear and thus Fr´echet-smooth. So if we use the representation of D ( n − f as element of L( V, L( V, . . . L( V | {z } ( n − , W ) . . . )) , the composition E v ,...,v n − ◦ D ( n − f is also Fr´echet-differentiable at any x ∈ U with (cid:16) E v ,...,v n − ◦ D ( n − f (cid:17) ′ (cid:12)(cid:12)(cid:12) x v n = E v ,...,v n − ◦ D n f | x v n = ( . . . (( D n f | x v n ) v ) . . . ) v n − = D n f | x ( v n , v , . . . , v n − ) = D n f | x ( v , . . . , v n − , v n ) , where in the first step we used the chain rule and Lemma A.1 (ii). In the second step,we used the definition of E v ,...,v n − , whereas in the third step we re-identified D n f | x with the corresponding multilinear mapping from V n to W . Finally, in the last stepwe used the symmetry of D n f | x . (cid:3) We finally state one last computation rule:
Lemma A.4.
Let V , W and Z be normed vector spaces, U ⊆ V open, f : U → W n -times Fr´echet-differentiable and A ∈ L( W, Z ) . Then also the function A ◦ f is n -timesFr´echet-differentiable and D n ( A ◦ f ) | x ( v , · · · , v n ) = A (cid:16) D n f | x ( v , · · · , v n ) (cid:17) ∀ x ∈ U, v , · · · , v n ∈ V . (A.2)
Proof.
The Fr´echet-differentiability follows immediately from Lemma A.2, using that A is Fr´echet-smooth. We show the identity (A.2) by induction over n : The case n = 1follows immediately by the chain rule and Lemma A.1 (ii). Now let n ≥ n −
1. Using the previous lemma (first step), the inductionhypothesis (i.e. (A.2) for the ( n − f ( n ), for all x ∈ U and v , · · · , v n ∈ V we obtain D n ( A ◦ f ) | x ( v , · · · , v n ) (A.1) = D (cid:16) D ( n − ( A ◦ f ) | . ( v , · · · , v n − ) (cid:17)(cid:12)(cid:12)(cid:12) x v n ( IH ) = D (cid:16) A (cid:16) D ( n − f | . ( v , · · · , v n − ) (cid:17)(cid:17)(cid:12)(cid:12)(cid:12) x v n = A (cid:16) D n f | x ( v , · · · , v n − , v n ) (cid:17) . (cid:3) Appendix B. The Riemannian Metric in Symmetric Wave Charts
In this subsection we give a detailed computation of the Riemannian metric intro-duced in Section 3.4 in terms of the symmetric wave charts. Hereby we adapt themethods in [15, Section 4] to the infinite-dimensional setting.We begin by defining a distance function on F reg by d : F reg × F reg → R +0 , ( x, y ) p tr(( x − y ) ) . The trace operator involved here is well-defined and can be expressed in any orthonor-mal basis ( e i ) i ∈ N of H by (for details see for example [23, Section 30.2])tr( A ) = ∞ X i =1 h e i | Ae i i H for A ∈ L( H ) of finite rank . (B.1)Moreover note that d does indeed define a distance function on F reg as for any two x, y ∈ F reg d ( x, y ) = k x − y k S ( H ) , where k . k S ( H ) denotes the Hilbert-Schmidt norm(see for example [7, Section XI.6] or [28, p. 321–322, 309–310]).The following remark is a reminder of a calculation rule for the trace operator actingon operators with finite rank. Remark B.1.
Let A ∈ L( H ) be of finite rank and V ⊆ H a finite-dimensionalsubspace V ⊆ H containing the image of A , i.e. A ( H ) ⊆ V . Moreover, let ( e i ) ≤ i ≤ k be an orthonormal basis of v and (˜ e i ) i ∈ N an orthonormal basis of V ⊥ . Then we obtainan orthonormal basis (ˆ e i ) i ∈ N of H by setting ˆ e i := e i for i = 1 , · · · , k and ˆ e k + j := ˜ e j for j ∈ N . Using this basis in B.1 the trace of A reduces to:tr( A ) = k X i =1 h ˆ e i | A ˆ e i i H = k X i =1 h e i | Ae i i H . ♦ ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 31
The next lemma is mostly based on [28, Satz VI.5.8] and states some more propertiesof the trace operators.
Lemma B.2. (Properties of the trace) (1)
Linearity: The trace operator tr is linear. (2) Boundedness: For a finite-dimensional subspace V ⊆ H consider the correspond-ing subspace V L := { A ∈ L( H ) | A ( H ) ⊆ V } ⊆ L( H ) . Then tr | V L is bounded. (3) Cyclic Permutation: For x, y ∈ L( H ) with x of finite rank it holds that: tr( xy ) = tr( yx ) . (4) Trace of adjoint: For any x ∈ L( H ) of finite rank also x † is of finite rank and: tr( x † ) = tr( x ) . Proof. (i): Follows from the definition of tr by (B.1), see also [28, Satz VI.5.8 (a)].(iii) and (iv): See [28, Satz VI.5.8 (c),(b)].(ii): Let A ∈ V L . Then, as explained in Remark B.1, choosing an orthonormal basis( e i ) ≤ i ≤ k of V (so dim( V ) = k ), we can estimate: | tr( A ) | = (cid:12)(cid:12)(cid:12) k X i =1 h e i | Ae i i H (cid:12)(cid:12)(cid:12) ≤ k X i =1 |h e i | Ae i i| H ≤ k X i =1 k A k H = k k A k H . This concludes the proof. (cid:3)
In the following lemma we consider differentiability properties of a mapping E whichcorresponds to the square of the distance function d . Later we want to use it tointroduce the Riemannian metric as second Fr´echet-derivative of E . Lemma B.3.
The mappings: E : F reg × F reg → R , ( x, y ) tr(( x − y ) ) , and for any fixed x ∈ F reg : E x : F reg → R , y tr(( x − y ) ) , are Fr´echet-smooth. Moreover, for all x, y ∈ F reg with x ∈ Ω y and all u , v ∈ V y , D (cid:0) E x ◦ φ − y (cid:1)(cid:12)(cid:12) φ y ( x ) = 0 and (B.2) D (cid:0) E x ◦ φ − y (cid:1)(cid:12)(cid:12) φ y ( x ) ( v , u ) = 4 Re (cid:16) tr( yφ y ( x ) v † yφ y ( x ) u † ) + tr( y uv † yφ y ( x ) φ y ( x ) † ) (cid:17) . Proof.
First we have to show that E ◦ ( φ − x , φ − y ) is Fr´echet-smooth for all x, y ∈ F reg .To this end first consider the following calculation for arbitrary ϕ ∈ W x , ψ ∈ W y : E ◦ ( φ − x , φ − y ) | ( ϕ,ψ ) = tr (cid:0) ( ϕ † xϕ − ψ † yψ ) (cid:1) = tr( ϕ † xϕϕ † xϕ ) − tr( ϕ † xϕψ † yψ ) − tr( ψ † yψϕ † xϕ ) + tr( ψ † yψψ † yψ )= tr( xϕϕ † xϕϕ † ) − tr( xϕψ † yψϕ † ) − tr( yψϕ † xϕψ † ) + tr( yψψ † yψψ † ) , = tr | S x L ( xϕϕ † xϕϕ † ) − tr | S x L ( xϕψ † yψϕ † ) − tr | S y L ( yψϕ † xϕψ † ) + tr | S y L ( yψψ † yψψ † ) , where in the second step we used the linearity of the trace and in the third stepthe cyclic permutation property (which can be applied as all factors and summandsobviously have finite rank). The last line is clearly a sum of composition of Fr´echet-smooth mappings in ( ϕ, ψ ), which proves the Fr´echet-smoothness of E ◦ ( φ − x , φ − y ). For calculating the Fr´echet derivative of E x consider the expansion E x ◦ φ − y ( ψ ) = tr(( x − ψ † yψ ) )= tr( x ) − tr( xψ † yψ ) − tr( ψ † yψx ) + tr( ψ † yψψ † yψ )= tr( x ) − xψ † yψ ) + tr( yψψ † yψψ † )= tr | S x L ( x ) − | S x L ( xψ † yψ ) + tr | S y L ( yψψ † yψψ † ) , which is again a sum of compositions of Fr´echet-smooth functions showing that also E x ◦ φ − y is Fr´echet-smooth.Applying the computation rule from Lemma A.4 together with the Fr´echet derivativerule for bilinear functions in Lemma A.1 (iii) (multiple times and together with thechain rule) we obtain: D (cid:0) E x ◦ φ − y (cid:1)(cid:12)(cid:12) ψ v = − | S x L ( x v † yψ ) − | S x L ( xψ † y v ) + tr | S y L ( y v ψ † yψψ † )+ tr | S y L ( yψ v † yψψ † ) + tr | S y L ( yψψ † y v ψ † ) + tr | S y L ( yψψ † yψ v † ) . Using that Lemma B.2 (iii) and (iv) this simplifies to D (cid:0) E x ◦ φ − y (cid:1)(cid:12)(cid:12) ψ v = − x v † yψ ) − xψ † y v ) + 2tr( ψψ † y v ψ † y ) + 2tr( yψ v † yψψ † )= 4 · Re (cid:16) tr | S y L ( yψψ † yψ v † ) − tr | S y L ( x v † yψ ) (cid:17) . (B.3)= 4 · Re (cid:16) tr( ψ † yψ v † yψ ) − tr( x v † yψ ) (cid:17) (B.4)In the case ψ † yψ = φ − y ( ψ ) = x the terms in (B.4) cancel each other, showing that D ( E x ◦ φ − y (cid:1)(cid:12)(cid:12) φ y ( x ) = 0 . Moreover, proceeding from (B.3) a straightforward computation using the propertiesof the Fr´echet derivative and the trace operator as before gives D (cid:0) E x ◦ φ − y (cid:1)(cid:12)(cid:12) ψ ( v , u )= 4 · Re (cid:16) tr | S y L ( y u ψ † yψ v † ) + tr | S y L ( yψ u † yψ v † ) + tr | S y L ( yψψ † y uv † ) − tr | S x L ( x v † y u ) (cid:17) (B.5)= 4 · Re (cid:16) tr( ψ † yψ v † y u ) + tr( yψ u † yψ v † ) + tr( yψψ † y uv † ) − tr( x v † y u ) (cid:17) . (B.6)As for ψ † yψ = φ − y ( ψ ) = x the first and the last term cancel each other we obtain D (cid:0) E x ◦ φ − y (cid:1)(cid:12)(cid:12) φ y ( x ) ( v , u )= 4 · Re (cid:16) tr( yφ y ( x ) u † yφ y ( x ) v † ) + tr( yφ y ( x ) φ y ( x ) † y uv † ) (cid:17) , which concludes the proof. (cid:3) Lemma B.4. D ( E x ◦ φ − y ) | φ y ( x ) is independent of the choice of chart (i.e. the choiceof y ) as long as y ∈ F reg is chosen such that x ∈ Ω y . Moreover, for all tangent vectorfields v , u ∈ Γ ( F reg , T F reg ) and any y ∈ F reg with x ∈ Ω y D v ( x ) (cid:16) D u ( . ) E x ( . ) (cid:17) = D (cid:0) E x ◦ φ − y (cid:1)(cid:12)(cid:12) φ y ( x ) (cid:16) Dφ y ( u ( x )) , Dφ y ( v ( x )) (cid:17) , (B.7) ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 33 where the derivatives act on the arguments containing a dot.
This Lemma also shows, that the the order of differentiation of E x with respect tothe two vector fields does not matter. The proof shows that this is due to the fact thatthe first derivative of E x vanishes. Proof.
Let v , u ∈ Γ( F reg , T F reg ) and x, y ∈ F reg with x ∈ Ω y be arbitrary. As we haveseen before, for the first directional derivative we have D u (˜ x ) E x = D (cid:0) E x ◦ φ − y (cid:1)(cid:12)(cid:12) φ y (˜ x ) Dφ y ( u (˜ x )) . It follows for the second directional derivative that D v (˜ x ) (cid:16) D u ( . ) E x ( . ) (cid:17) = D (cid:0) E x ◦ φ − y (cid:1)(cid:12)(cid:12) φ y (˜ x ) (cid:16) Dφ y ( v (˜ x )) , Dφ y ( u ( x )) (cid:17) + D (cid:0) E x ◦ φ − y (cid:1)(cid:12)(cid:12) φ y (˜ x ) D (cid:16) Dφ y (cid:0) u ( φ − y ( . )) (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) φ y (˜ x ) Dφ y ( v (˜ x )) , where we applied the Fr´echet derivative rule for R -bilinear maps in Lemma A.1 (iii)together with the chain rule. Evaluating this expression at ˜ x = x , the second summandvanishes in view of (B.2). We thus obtain D v (˜ x ) (cid:16) D u ( . ) E x ( . ) (cid:17)(cid:12)(cid:12)(cid:12) ˜ x = x = D (cid:0) E x ◦ φ − y (cid:1)(cid:12)(cid:12) φ y ( x ) (cid:16) Dφ y ( v ( x )) , Dφ y ( u ( x )) (cid:17) . Using the symmetry of the second Fr´echet derivatives gives the result. (cid:3)
Remark B.5.
Equation (B.7) also shows that D v ( x ) ( D u ( . ) E x ( . )) only depends on thevalue of the vector fields u and v at the point x . Moreover, since to arbitrary x ∈ F reg and u , v ∈ T x F reg one can always find a smooth tangent vector field with v ( x ) = v , u ( x ) = u (e.g. using a suitable bump function in a chart around x ), we can considerthe expression D E x | x ( u , v ) := D ( E x ◦ φ − y ) | φ y ( x ) (cid:16) Dφ y ( u ) , Dφ y ( v ) (cid:17) , (B.8)as a well defined, coordinate invariant – in the sense that the right hand side of equation(B.8) returns the same values for any y ∈ F reg with x ∈ Ω y – and symmetric bilinearform. ♦ Now it seems convenient to compute B.8 in the cart φ x . Note that then we have φ x ( x ) = π x and thus we obtain for any u , v ∈ V x : D (cid:0) E x ◦ φ − x (cid:1) | φ x ( x ) ( v , u ) = 4 · Re (cid:16) tr( xπ x u † xπ x v † ) + tr( xπ x π † x x uv † ) (cid:17) = 4 · Re (cid:16) tr( x u † xv † ) + tr( x uv † ) (cid:17) = 4 · Re (cid:16) tr( x v x u ) + tr( x uv † x ) (cid:17) . Motivated by this for any x ∈ F reg we set:˜ g x : V x × V x → R , ( u , v ) · Re (cid:16) tr( x v x u ) + tr( x uv † x ) (cid:17) . Due to the properties of the trace operator, ˜ g x defines a symmetric, real-valued bilinearform on V x , which is even positive-definite as the following lemma shows: Lemma B.6.
The symmetric bilinear form ˜ g x is positive definite and thus defines areal valued inner product on V x . Proof.
Let u ∈ V x be arbitrary, choose an orthonormal basis ( e i ) i =1 , ··· ,k of the finite-dimensional vector-space ( S x + u † ( S x )) and compute: tr( x u x u ) = k X i =1 h e i | x u x u e i i H = k X i =1 h u † xe i | x u e i i H , ⇒ Re (cid:0) tr( x u x u ) (cid:1) = 12 k X i =1 (cid:16) h u † xe i | xue i i H + h u † xe i | x u e i i H (cid:17) = 12 k X i =1 (cid:16) h u † xe i | x u e i i H + h x u e i | u † xe i i H (cid:17) , tr( x uu † x ) = k X i =1 h e i | x uu † xe i i H = k X i =1 h u † xe i | u † xe i i H , tr( x uu † x ) = tr( u † x u ) = k X i =1 h e i | u † x u e i i H = k X i =1 h x u e i | x u e i i H , ⇒ Re (cid:0) tr( x uu † x ) (cid:1) = Re (cid:16) k X i =1 (cid:16) h u † xe i | u † xe i i H + h x u e i | x u e i i H (cid:17)(cid:17) = 12 k X i =1 (cid:16) h u † xe i | u † xe i i H + h x u e i | x u e i i H (cid:17) , where in the last step we used that h u † xe i | u † xe i i H = k u † x k H and h x u e i | x u e i i H = k x u k H are already real (for i = 1 , . . . , k ), so we can leave out the ”Re”.Combining this we obtain:Re (cid:16) tr( x u x u ) + tr( x uu † x ) (cid:17) = 12 k X i =1 (cid:16) h ( u † x + x u ) e i | x u e i i H + h ( x u + u † x ) e i | u † xe i i H (cid:17) = 12 k X i =1 h ( u † x + x u ) e i | ( u † x + x u ) e i i H ≥ . This shows the positive semi-definiteness of ˜ g x . Moreover we see that ˜ g x ( u , u ) vanishesif and only if 0 = (cid:0) u † x + x u (cid:1) | S x + u † ( S x ) . But as ( u † x + x u ) is obviously selfadjoint and its image is contained in S x + u † ( S x ),it vanishes on the orthogonal complement of S x + u † ( S x ) anyhow, so the previousequation is equivalent to 0 = u † x + x u . (B.9)Moreover, denoting π I := π x as the orthogonal projection on S x = I and π J as theorthogonal projection on J = ( S x ) ⊥ , we can write u = u π I + u π J . As explained in Remark B.1, the trace operator for the finite-rank operators x u x u , x uu † x and u † x u can indeed be calculated like that as they all map into ( S x + u † ( S x )). ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 35
Plugging this in equation (B.9) yields:0 = ( u π I + u π J ) † x + x ( u π I + u π J ) = π I u † x + x u π I + π J u † x + x u π J , (B.10)Using u | I ∈ Symm( S x ) we conclude X − π I u † X = X − ( u | I ) † X = u | I , ⇒ π I u † X = X u | I . As x is self adjoint this also yields π I u † x = x u π I . Inserting this in (B.10) gives:0 = 2 x u π I + π J u † x + x u π J . (B.11)Using a block operator notation for the orthogonal decomposition H = I ⊕ ⊥ J , thisequation can be visualized as (cid:18) (cid:19) = (cid:18) x u π I x u π J π J u † x (cid:19) . This notation can be justified by “testing” equation (B.11) with ( v, , (0 , w ) ∈ H = I ⊕ ⊥ J with v ∈ I and w ∈ J arbitrary.Thus we see that each of the operators 2 x u π I , π J u † x and x u π J must vanish indi-vidually. Furthermore as x | I has full rank and u maps into S x = I , this yields u π I = 0 , u π J = 0 , and therefore also u = u ( π I + π J ) = 0 . This proves the positive definiteness of ˜ g x ( u , u ) = Re (cid:16) tr( x u x u ) + tr( x uu † x ) (cid:17) . (cid:3) Now we can finally introduce a Riemannian metric on F reg : Lemma B.7.
Setting pointwise for any x ∈ F reg : g x : T x F reg × T x F reg → R , ( u , v ) D E x | x ( u , v ) , we obtain a well defined Riemannian metric on F reg . Proof.
First of all g x is well defined due to lemma B.4 as explained in remark B.5.Moreover, choosing representatives [ x, u , x ], [ x, v , x ] ∈ T x F reg we have: g x ([ x, u , x ] , [ x, v , x ]) = D E x | x ([ x, u , x ] , [ x, v, x ])= D ( E x ◦ φ − x ) | φ x ( x ) (cid:16) Dφ x ([ x, u , x ]) | {z } = u , Dφ x ([ x, v, x ]) | {z } = v (cid:17) = ˜ g x ( u , v ) , and since we have already seen that for any x ∈ F reg , ˜ g x defines a symmetric positive-definite bilinear form, so does g x .Thus it only remains to show that g is Fr´echet-smooth. But due to the (coordi-nate invariant) definition of D E x | x in B.8 this follows immediately from the Fr´echet-smoothness of D ( E x ◦ φ − y ) (for this see lemma B.3, in particular equation (B.5)). Where a Riemannian metric on a Banach-manifold is defined just as in the finite-dimensional casebut with smoothness with respect to the Fr´echet derivative.
More precisely, since for any two smooth vector fields u , v ∈ Γ( F reg , T F reg ) for anychart φ y with x ∈ Ω y also Dφ y ◦ u ◦ φ − y and Dφ y ◦ v ◦ φ − y are smooth, we have g φ − y ( ψ ) ( u ◦ φ − y ( ψ ) , v ◦ φ − y ( ψ )) = D E x | x ( u , v )= D ( E x ◦ φ − y ) | ψ (cid:16) Dφ y (cid:0) u ◦ φ − y ( ψ ) (cid:1) , Dφ y (cid:0) v ◦ φ − y ( ψ ) (cid:1)(cid:17) , ∀ ψ ∈ W y , which is Fr´echet-smooth as composition of Fr´echet-smooth maps. More precisely, in-troducing the mappings B : L( V y , R ) × V y → L( V y , R ) , ( A, v ) A v ,B : L( V y , R ) × V y → R , ( A ′ , v ′ ) A ′ v ′ , which are both obviously R -bilinear and continuous (and thus Fr´echet-smooth), wecan rewrite the previous equation to g φ − y ( ψ ) ( u ◦ φ − y ( ψ ) , v ◦ φ − y ( ψ ))= B (cid:16) B (cid:16) ( d ( x, φ − y ( . ))) (2) | ψ , Dφ y ( u ◦ φ − y ( ψ )) (cid:17) , Dφ y ( v ◦ φ − y ( ψ )) (cid:17) , which is now clearly a composition of Fr´echet-smooth maps. (cid:3) Acknowledgments:
We are grateful to Olaf M¨uller, Marco Oppio and Johannes Wurmfor helpful discussions. M.L. acknowledges support by the Studienstiftung des deutschenVolkes.
References [1] .[2] D. Beltit¸˘a, T. Goli´nski, and A.-B. Tumpach,
Queer Poisson brackets , arXiv:math-ph/1710.03057[math.FA], J. Geom. Phys. (2018), 358–362.[3] Y. Bernard and F. Finster,
On the structure of minimizers of causal variational principles in thenon-compact and equivariant settings , arXiv:1205.0403 [math-ph], Adv. Calc. Var. (2014), no. 1,27–57.[4] D. Brink, H¨older continuity of roots of complex and p -adic polynomials , Comm. Algebra (2010),no. 5, 1658–1662.[5] R. Coleman, Calculus on Normed Vector Spaces , Universitext, Springer, New York, 2012.[6] J. Dieudonn´e,
Foundations of Modern Analysis , Academic Press, New York-London, 1969, En-larged and corrected printing, Pure and Applied Mathematics, Vol. 10-I.[7] N. Dunford and J.T. Schwartz,
Linear Operators. Part II: Spectral theory. Self adjoint operatorsin Hilbert space , With the assistance of William G. Bade and Robert G. Bartle, IntersciencePublishers John Wiley & Sons New York-London, 1963.[8] F. Finster,
A variational principle in discrete space-time: Existence of minimizers ,arXiv:math-ph/0503069, Calc. Var. Partial Differential Equations (2007), no. 4, 431–453.[9] , Causal variational principles on measure spaces , arXiv:0811.2666 [math-ph], J. ReineAngew. Math. (2010), 141–194.[10] ,
The Continuum Limit of Causal Fermion Systems , arXiv:1605.04742 [math-ph], Funda-mental Theories of Physics, vol. 186, Springer, 2016.[11] ,
Causal fermion systems: A primer for Lorentzian geometers , arXiv:1709.04781 [math-ph], J. Phys.: Conf. Ser. (2018), 012004.[12] F. Finster and M. Jokel,
Causal fermion systems: An elementary introduction to physical ideas andmathematical concepts , arXiv:1908.08451 [math-ph], Progress and Visions in Quantum Theory inView of Gravity (F. Finster, D. Giulini, J. Kleiner, and J. Tolksdorf, eds.), Birkh¨auser Verlag,Basel, 2020, pp. 63–92.[13] F. Finster and N. Kamran,
Complex structures on jet spaces and bosonic Fock space dynamicsfor causal variational principles , arXiv:1808.03177 [math-ph], to appear in Pure Appl. Math. Q.(2020).
ANACH MANIFOLD STRUCTURE OF CAUSAL FERMION SYSTEMS 37 [14] F. Finster, N. Kamran, and M. Oppio,
The linear dynamics of wave functions in causal fermionsystems , arXiv:2101.08673 [math-ph] (2021).[15] F. Finster and S. Kindermann,
A gauge fixing procedure for causal fermion systems ,arXiv:1908.08445 [math-ph], J. Math. Phys. (2020), no. 8, 082301.[16] F. Finster and J. Kleiner, Causal fermion systems as a candidate for a unified physical theory ,arXiv:1502.03587 [math-ph], J. Phys.: Conf. Ser. (2015), 012020.[17] ,
A Hamiltonian formulation of causal variational principles , arXiv:1612.07192 [math-ph],Calc. Var. Partial Differential Equations (2017), no. 3, 33.[18] F. Finster, J. Kleiner, and J.-H. Treude,
An Introduction to the Fermionic Projector and CausalFermion Systems
Causal variational principles in the σ -locally compact setting: Existenceof minimizers , arXiv:2002.04412 [math-ph], to appear in Adv. Calc. Var. (2020).[20] F. Finster and D. Schiefeneder, On the support of minimizers of causal variational principles ,arXiv:1012.1589 [math-ph], Arch. Ration. Mech. Anal. (2013), no. 2, 321–364.[21] J. Hilgert and K.-H. Neeb,
Structure and Geometry of Lie Groups , Springer Monographs inMathematics, Springer, New York, 2012.[22] A. Kriegl and P.W. Michor,
The Convenient Setting of Global Analysis , Mathematical Surveysand Monographs, vol. 53, American Mathematical Society, Providence, RI, 1997.[23] P.D. Lax,
Functional Analysis , Pure and Applied Mathematics (New York), Wiley-Interscience[John Wiley & Sons], New York, 2002.[24] J.M. Lee,
Riemannian Manifolds: An Introduction to Curvature , Graduate Texts in Mathematics,vol. 176, Springer-Verlag, New York, 1997.[25] M. Oppio,
H¨older continuity of the integrated causal Lagrangian in Minkowski space , in prepara-tion.[26] ,
On the mathematical foundations of causal fermion systems in Minkowski space ,arXiv:1909.09229 [math-ph], to appear in Ann. Henri Poincar´e (2020).[27] W. Rudin,
Real and Complex Analysis , third ed., McGraw-Hill Book Co., New York, 1987.[28] D. Werner,
Funktionalanalysis , eighth ed., Springer-Verlag, Berlin, 2018.[29] E. Zeidler,
Nonlinear Functional Analysis and its Applications. IV , Springer-Verlag, New York,1988, Applications to mathematical physics, Translated from the German and with a preface byJ¨urgen Quandt.
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