Causal Variational Principles in the Infinite-Dimensional Setting: Existence of Minimizers
aa r X i v : . [ m a t h - ph ] J a n CAUSAL VARIATIONAL PRINCIPLES IN THEINFINITE-DIMENSIONAL SETTING: EXISTENCE OFMINIMIZERS
CHRISTOPH LANGERJANUARY 2021
Abstract.
We provide a method for constructing (possibly non-trivial) measureson non-locally compact Polish subspaces of infinite-dimensional separable Banachspaces which, under suitable assumptions, are minimizers of causal variational prin-ciples in the non-locally compact setting. Moreover, for non-trivial minimizers thecorresponding Euler-Lagrange equations are derived. The method is to exhaust theunderlying Banach space by finite-dimensional subspaces and to prove existence ofminimizers of the causal variational principle restricted to these finite-dimensionalsubsets of the Polish space under suitable assumptions on the Lagrangian. Thisgives rise to a corresponding sequence of minimizers. Restricting the resulting se-quence to countably many compact subsets of the Polish space, by considering theresulting diagonal sequence we are able to construct a regular measure on the Borelalgebra over the whole topological space. For continuous Lagrangians of boundedrange it can be shown that, under suitable assumptions, the obtained measure isa (possibly non-trivial) minimizer under variations of compact support. Under ad-ditional assumptions, we prove that the constructed measure is a minimizer undervariations of finite volume and solves the corresponding Euler-Lagrange equations.Afterwards, we extend our results to continuous Lagrangians vanishing in entropy.Finally, assuming that the obtained measure is locally finite, topological propertiesof spacetime are worked out and a connection to dimension theory is established.
Contents
1. Introduction 22. Physical Background and Mathematical Preliminaries 42.1. Physical Context and Motivation 42.2. Causal Variational Principles in the σ -Locally Compact Setting 53. Causal Variational Principles in the Non-Locally Compact Setting 93.1. Motivation: Infinite-Dimensional Causal Fermion Systems 93.2. Basic Definitions 113.3. Finite-Dimensional Approximation 134. Construction of a Global Measure 144.1. Construction of a Countable Collection of Compact Sets 144.2. Construction of a Regular Global Measure 154.3. Convergence on Relatively Compact Subsets 205. Minimizers for Lagrangians of Bounded Range 235.1. Preliminaries 235.2. Minimizers under Variations of Finite-Dimensional Compact Support 245.3. Existence of Minimizers under Variations of Compact Support 28.4. Existence of Minimizers under Variations of Finite Volume 295.5. Derivation of the Euler-Lagrange Equations 296. Minimizers for Lagrangians Vanishing in Entropy 306.1. Lagrangians Vanishing in Entropy 306.2. Preparatory Results 336.3. Existence of Minimizers 346.4. Derivation of the Euler-Lagrange Equations 367. Topological Properties of Spacetime 367.1. Dimension-Theoretical Preliminaries 377.2. Application to Causal Fermion Systems 38Appendix A. Topological Properties of Causal Fermion Systems 39A.1. Separability 39A.2. Completeness 39Appendix B. Support of Locally Finite Measures on Polish Spaces 44References 451. Introduction
In the physical theory of causal fermion systems, spacetime and the structurestherein are described by a minimizer (for an introduction to the physical backgroundand the mathematical context, we refer the interested reader to § Causal variational principles evolved as a mathe-matical generalization of the causal action principle [16, 23], and were studied in moredetail in [24]. The starting point in [24] is a second-countable, locally compact Haus-dorff space F together with a non-negative function L : F × F → R +0 := [0 , ∞ ) (the Lagrangian ) which is assumed to be lower semi-continuous, symmetric and positive onthe diagonal. The causal variational principle is to minimize the action S defined asthe double integral over the Lagrangian S ( ρ ) = ˆ F dρ ( x ) ˆ F dρ ( y ) L ( x, y )under variations of the measure ρ within the class of regular Borel measures on F ,keeping the (possibly infinite) total volume ρ ( F ) fixed ( volume constraint ). The aim ofthe present paper is to extend the existence theory for minimizers of such variationalprinciples to the case that F is non-locally compact and the total volume is infinite.We also work out the corresponding Euler-Lagrange (EL) equations.In order to put the paper into the mathematical context, in [14] it was proposedto formulate physics by minimizing a new type of variational principle in spacetime.The suggestion in [14, Section 3.5] led to the causal action principle in discrete space-time, which was first analyzed mathematically in [15]. A more general and systematicenquiry of causal variational principles on measure spaces was carried out in [16]. Inthis article, the existence of minimizers is proven in the case that the total volume isfinite. In [23], the setting is generalized to non-compact manifolds of possibly infinitevolume and the corresponding EL equations are analyzed. However, the existence ofminimizers is not proved. This is done in [24] in the slightly more general settingof second-countable, locally compact Hausdorff spaces. In this paper, we extend theresults of [24] by developing the existence theory in the non-locally compact setting. XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 3
The main difficulty in dealing with non-locally compact spaces is that it is no longerpossible to restrict attention to compact neighborhoods. Moreover, it turns out thatwe can no longer assume that the underlying topological space F is σ -compact. Asa consequence, at first sight it is not clear how to construct global measures on thewhole topological space at all. The way out is to introduce a countable collectionof suitable compact subsets, which indeed allows us to construct a global measure ρ on F . For simplicity, we first assume that the Lagrangian is of bounded range (seeDefinition 3.7). In this case, the minimizing property of the measure ρ is proved intwo steps: We first show that ρ is a minimizer under variations of compact support . Ina second step, we extend this result to variations of finite volume under the assumptionthat property (iv) in § x ∈ F ˆ F L ( x, y ) dρ ( y ) < ∞ . Afterwards, we generalize our results to Lagrangians which do not have bounded range,but instead have suitable decay properties. To this end, we consider Lagrangians vanishing in entropy (see Definition 6.2). Introducing spacetime as the support ofthe measure ρ , we finally analyze topological properties of spacetime; moreover, aconnection to dimension theory is established.The paper is organized as follows. In Section 2 we give a short physical motiva-tion ( § § § § § § § § § § § § § vanishing in entropy (see Definition 6.1) which generalize the notion ofLagrangians decaying in entropy (see Definition 2.8). The concept of Lagrangiansvanishing in entropy ( § C. LANGER thus giving rise to a regular measure on the underlying topological space ( § § § § § σ -compact (see Lemma B.2).2. Physical Background and Mathematical Preliminaries
Physical Context and Motivation.
The purpose of this subsection is to out-line a few concepts of causal fermion systems and to explain how the present paperfits into the general physical context and the ongoing research program. The readernot interested in the physical background may skip this section.The theory of causal fermion systems is a recent approach to fundamental physicsmotivated originally in order to resolve shortcomings of relativistic quantum field the-ory (QFT). Namely, due to ultraviolet divergences, perturbative quantum field the-ory is well-defined only after regularization, which is usually understood as a set ofprescriptions for how to make divergent integrals finite (e.g. by introducing a suitable“cutoff” in momentum space). The regularization is then removed using the renormal-ization procedure. However, this concept is not convincing from neither the physicalnor the mathematical point of view. More precisely, in view of Heisenberg’s uncer-tainty principle, physicists infer a correspondence between large momenta and smalldistances. Because of that, the regularization length is often associated to the Plancklength ℓ P ≈ . · − m. Accordingly, by introducing an ultraviolet cutoff in momen-tum space, one disregards distances which are smaller than the Planck length. As aconsequence, the microscopic structure of spacetime is completely unknown. Unfortu-nately, at present there is no consensus on what the correct mathematical model for“Planck scale physics” should be.The simplest and maybe most natural approach is to assume that on the Planckscale, spacetime is no longer a continuum but becomes in some way “discrete.” Thisis the starting point in the monograph [14], where the physical system is described byan ensemble of wave functions in a discrete spacetime. Motivated by the Lagrangianformulation of classical field theory, physical equations are formulated by a variationalprinciple in discrete spacetime. In the meantime, this setting was generalized anddeveloped to the theory of causal fermion systems. It is an essential feature of theapproach that spacetime does not enter the variational principle a-priori, but insteadit emerges when minimizing the action. Thus causal fermion systems allow for thedescription of both discrete and continuous spacetime structures.In order to get the connection to the present paper, let us briefly outline the mainstructures of causal fermion systems. As initially introduced in [19], a causal fermionsystem consists of a triple ( H , F , ρ ) together with an integer n ∈ N , where H denotesa complex Hilbert space, F ⊂ L( H ) being the set of all self-adjoint operators on H offinite rank with at most n positive and at most n negative eigenvalues, and ρ being XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 5 a positive measure on the Borel σ -algebra B ( F ) (referred to as universal measure ).Then for any x, y ∈ F , the product xy is an operator of rank at most 2 n . Denotingits non-trivial eigenvalues (counting algebraic multiplicities) by λ xy , . . . , λ xy n ∈ C , andintroducing the spectral weight | . | of an operator as the sum of the absolute values ofits eigenvalues, the Lagrangian can be introduced as a mapping L : F × F → R +0 , L ( x, y ) = (cid:12)(cid:12) ( xy ) (cid:12)(cid:12) − n | xy | . As being of relevance for this article, we point out that the Lagrangian is a continuousfunction which is symmetric in the sense that L ( x, y ) = L ( y, x ) for all x, y ∈ F . In analogy to classical field theory, one defines the causal action by S ( ρ ) = ¨ F × F L ( x, y ) dρ ( x ) dρ ( y ) . Finally, the corresponding causal action principle is introduced by varying the mea-sure ρ in the class of regular measures on B ( F ) under additional constraints (whichassert the existence of non-trivial minimizers). Given a minimizing measure ρ , space-time M is defined as its support, M := supp ρ . As being outlined in detail in [18], critical points of the causal action give rise toEuler-Lagrange (EL) equations, which describe the dynamics of the causal fermionsystem. In a certain limiting case, the so-called continuum limit , one can establish aconnection to the conventional formulation of physics in a spacetime continuum. Inthis limiting case, the EL equations give rise to classical field equations like the Maxwelland Einstein equations. Moreover, quantum mechanics is obtained in a limiting case,and close connections to relativistic quantum field theory have been established (fordetails see [17] and [21]).In order for the causal action principle to be mathematically sensible, the existencetheory is of crucial importance. If the dimension of the Hilbert space H is finite,the existence of minimizers was proven in [16, Section 2] (based on existence resultsin discrete spacetime [15]), giving rise to minimizing measures ρ on F of finite totalvolume ρ ( F ) < ∞ . For this reason, it remains to extend these existence results bydeveloping the existence theory in the case that H is infinite-dimensional. Then thetotal volume ρ ( F ) is necessarily infinite (for a counter example see [18, Exercise 1.3]).In the resulting infinite-dimensional setting (i.e. dim H = ∞ and ρ ( F ) = ∞ ), the taskis to deal with minimizers of infinite total volume on non-locally compact spaces. Inpreparation, the existence theory of minimizers of possibly infinite total volume ρ ( F )on locally compact spaces is developed in [24] in sufficient generality. The remainingsecond step, which involves the difficulty of dealing with non-locally compact spaces,is precisely the objective of the present paper.2.2. Causal Variational Principles in the σ -Locally Compact Setting. Beforeintroducing causal variational principles on non-locally compact spaces in Section 3below, we now recall the main results in the less general situation of causal variationalprinciples in the σ -locally compact setting [24] which are based on results concerningcausal variational principles in the non-compact setting as studied in [23, Section 2]. C. LANGER
The starting point in [24] is a second-countable, locally compact topological Haus-dorff space F . We let ρ be a (positive) measure on the Borel algebra over F (referredto as universal measure ). Moreover, let L : F × F → R +0 be a non-negative function(the Lagrangian ) with the following properties:(i) L is symmetric, i.e. L ( x, y ) = L ( y, x ) for all x, y ∈ F .(ii) L is lower semi-continuous, i.e. for all sequences x n → x and y n ′ → y , L ( x, y ) ≤ lim inf n,n ′ →∞ L ( x n , y n ′ ) . The causal variational principle is to minimize the action S ( ρ ) = ˆ F dρ ( x ) ˆ F dρ ( y ) L ( x, y ) (2.1)under variations of the measure ρ , keeping the total volume ρ ( F ) fixed ( volume con-straint ). The papers [23, 24] mainly focus on the case that the total volume ρ ( F ) isinfinite. In order to implement the volume constraint and to derive the correspondingEuler-Lagrange equations, in [23] one makes the following additional assumptions:(iii) The measure ρ is locally finite (meaning that any x ∈ F has an open neighbor-hood U ⊂ F with ρ ( U ) < ∞ ).(iv) The function L ( x, . ) is ρ -integrable for all x ∈ F andsup x ∈ F ˆ F L ( x, y ) dρ ( y ) < ∞ . (2.2)By Fatou’s lemma, the integral in (2.2) is lower semi-continuous in the variable x .A measure on the Borel algebra which satisfies (iii) will be referred to as a Borelmeasure (in the sense of [28]), and the set of Borel measures on F shall be denotedby B F . Moreover, the Borel σ -algebra over F is denoted by B ( F ). A Borel measureis said to be regular if it is inner and outer regular (cf. [10, Definition VIII.1.1]). Aninner regular Borel measure is also called a Radon measure [38].In [23, 24] one varies in the following class of measures:
Definition 2.1.
Given a regular Borel measure ρ on F , a regular Borel measure ˜ ρ on F is said to be a variation of finite volume if (cid:12)(cid:12) ˜ ρ − ρ (cid:12)(cid:12) ( F ) < ∞ and (cid:0) ˜ ρ − ρ (cid:1) ( F ) = 0 , (2.3) where the total variation | ˜ ρ − ρ | of two possibly infinite measures ρ and ˜ ρ on B ( F ) isdefined in [24, § as follows: We say that | ˜ ρ − ρ | < ∞ if there exists B ∈ B ( F ) with ρ ( B ) , ˜ ρ ( B ) < ∞ such that ρ | F \ B = ˜ ρ | F \ B . In this case, (˜ ρ − ρ )(Ω) := ˜ ρ ( B ∩ Ω) − ρ ( B ∩ Ω) for any Borel set Ω ⊂ F . Given a regular Borel measure ρ ∈ B F and assuming that (i), (ii) and (iv) hold, forevery variation of finite volume ˜ ρ ∈ B F the difference of the actions as given by (cid:0) S (˜ ρ ) − S ( ρ ) (cid:1) = ˆ F d (˜ ρ − ρ )( x ) ˆ F dρ ( y ) L ( x, y )+ ˆ F dρ ( x ) ˆ F d (˜ ρ − ρ )( y ) L ( x, y ) + ˆ F d (˜ ρ − ρ )( x ) ˆ F d (˜ ρ − ρ )( y ) L ( x, y ) (2.4)is well-defined in view of [23, Lemma 2.1]. For clarity, we point out that condition (iii)is not required in order for (2.4) to hold. XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 7
Note that the assumptions (iii) and (iv) are dropped in [24]. The causal variationalprinciple in the σ -locally compact setting [24] is then defined as follows. Definition 2.2.
Let F be a second-countable, locally compact Hausdorff space, and letthe Lagrangian L : F × F → R +0 be a symmetric and lower semi-continuous function(see conditions (i) and (ii) above). Moreover, we assume that L is strictly positive onthe diagonal, i.e. L ( x, x ) > for all x ∈ F . The causal variational principle on σ -locally compact spaces is to minimizethe causal action (2.1) under variations of finite volume (see Definition 2.1). We point out that (iv) is a sufficient condition for (2.4) to hold. However, since theconditions (iii) and (iv) are not imposed in [24], it is a-priori not clear whether theintegrals in (2.4) exist. For this reason, condition (2.4) is included into the definitionof a minimizer:
Definition 2.3.
A regular Borel measure ρ on F is said to be a minimizer of thecausal action under variations of finite volume [24] if the difference (2.4) is well-defined and non-negative for all regular Borel measures ˜ ρ on F satisfying (2.3) , (cid:0) S (˜ ρ ) − S ( ρ ) (cid:1) ≥ . We denote the support of the measure ρ by M , M := supp ρ = F \ [ (cid:8) Ω ⊂ F (cid:12)(cid:12) Ω is open and ρ (Ω) = 0 (cid:9) (2.5)(thus the support is the set of all points for which every open neighborhood has astrictly positive measure; for details and generalizations see [13, Subsection 2.2.1]).According to Definition 2.1, the condition | ˜ ρ − ρ | < ∞ implies that there exists someBorel set B ⊂ F with ρ ( B ) , ˜ ρ ( B ) < ∞ and ρ | F \ B = ˜ ρ | F \ B . In particular, ρ | B and ˜ ρ | B are finite Borel measures on B ( B ), and thus have support (see [7, Proposition 7.2.9]).Furthermore, the signed measure ˜ ρ − ρ has support.We now recall some results from [24] which will be referred to frequently. The firstexistence results in [24] are based on the assumption that the Lagrangian is of compactrange. For convenience, let us state the definition. Definition 2.4.
The Lagrangian has compact range if for every compact set K ⊂ F there is a compact set K ′ ⊂ F such that L ( x, y ) = 0 for all x ∈ K and y K ′ . Moreover, the definition of minimizers under variations of compact support playsan important role in [24].
Definition 2.5.
A regular Borel measure ρ on F is said to be a minimizer undervariations of compact support [24] of the causal action if for any regular Borelmeasure ˜ ρ on F which satisfies (2.3) such that the signed measure ˜ ρ − ρ is compactlysupported, the inequality (cid:0) S (˜ ρ ) − S ( ρ ) (cid:1) ≥ holds. Assuming that the Lagrangian is of compact range, the main results in [24] can besummarized as follows.
C. LANGER
Theorem 2.6 ( Euler-Lagrange equations ) . Let F be a second-countable, locallycompact Hausdorff space, and assume that L : F × F → R +0 is continuous and ofcompact range. Then there exists a regular Borel measure ρ on F which satisfies theEuler-Lagrange equations ℓ | supp ρ ≡ inf x ∈ F ℓ ( x ) = 0 , (2.6) where ℓ ∈ C ( F ) is defined by ℓ ( x ) := ˆ F L ( x, y ) dρ ( y ) − . (2.7)Combining [24, Theorem 4.9 and Theorem 4.10], we obtain the following result. Theorem 2.7.
Assume that L : F × F → R +0 is continuous and of compact range.Then there is a regular Borel measure ρ on F which is a minimizer under variationsof compact support [24] (see Definition 2.5). Under the additional assumptions thatthe Lagrangian L is bounded and condition (iv) is satisfied (see (2.2) ), the measure ρ is a minimizer under variations of finite volume [24] (see Definition 2.3). In [24, Section 5] it was shown that the assumption that the Lagrangian L is ofcompact range can be weakened. To this end, we recall that every second-countable,locally compact Hausdorff space can be endowed with a Heine-Borel metric (for detailswe refer to the explanations in [24, § § F , for any r > x ∈ F the closed ball B r ( x ) is compact, and hence can be covered by finitely many balls ofradius δ >
0. The smallest such number is denoted by E x ( r, δ ) and is called entropy .This gives rise to Lagrangians decaying in entropy, being defined as follows (cf. [24,Definition 5.1]). Definition 2.8.
Assume that F is endowed with an unbounded Heine-Borel metric d .The Lagrangian L : F × F → R +0 is said to decay in entropy if the following conditionsare satisfied: (a) c := inf x ∈ F L ( x, x ) > . (b) There is a compact set K ⊂ F such that δ := inf x ∈ F \ K sup n s ∈ R : L ( x, y ) ≥ c for all y ∈ B s ( x ) o > . (c) The Lagrangian has the following decay property: There is a monotonically de-creasing, integrable function f ∈ L ( R + , R +0 ) such that L ( x, y ) ≤ f (cid:0) d ( x, y ) (cid:1) C x (cid:0) d ( x, y ) , δ (cid:1) for all x, y ∈ F with x = y , where C x ( r, δ ) := C E x ( r + 2 , δ ) for all r > , and the constant C is given by C := 1 + 2 c < ∞ . We point out that the above definition of Lagrangians decaying in entropy as intro-duced in [24, Section 5] requires an unbounded
Heine-Borel metric. For a more generaldefinition we refer to § XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 9
For clarity we note that, if ( H , F , ρ ) is a causal fermion system with dim( H ) < ∞ ,the space L( H ) of bounded linear operators on H is finite-dimensional. Combiningthe fact that all norms on finite-dimensional vector spaces are equivalent with theHeine-Borel theorem yields that the Fr´echet metric induced by the operator norm onthe vector space L( H ) is an unbounded Heine-Borel metric on F .Let us now recall the main results in [24] under the assumption that the Lagrangiandecays in entropy. Theorem 2.9.
Let F be a second-countable, locally compact Hausdorff space, andassume that L : F × F → R +0 is continuous and decays in entropy. Then there exists aregular Borel measure ρ on F which satisfies the Euler-Lagrange equations ℓ | supp ρ ≡ inf x ∈ F ℓ ( x ) = 0 , where ℓ ∈ C ( F ) is defined by (2.7) . The following theorem ensures the existence of minimizing Borel measures.
Theorem 2.10.
Assume that L : F × F → R +0 is continuous and decays in entropy.Then there is a regular Borel measure ρ on F which is a minimizer under variationsof compact support [24] . Under the additional assumptions that the Lagrangian L isbounded and condition (iv) is satisfied (see (2.2) ), the measure ρ is a minimizer undervariations of finite volume [24] . The goal of this paper is to extend the above results to the infinite-dimensionalsetting.3.
Causal Variational Principles in the Non-Locally Compact Setting
Motivation: Infinite-Dimensional Causal Fermion Systems.
As explainedin Section 1, causal variational principles evolved as a mathematical generalization ofthe causal action principle in order to develop the existence theory for causal fermionsystems. In order to point out the connection to causal variational principles in thenon-locally compact setting, let us briefly recall the basic structures of causal fermionsystems (for details cf. [18, § H , an integer s ∈ N (the so-called spindimension ) and a measure ρ defined on the Borel σ -algebra B ( F ), where F ⊂ L( H )consists of all self-adjoint operators on H which have at most s positive and at most s negative eigenvalues. This gives rise to a triple ( H , F , ρ ). The set F can be endowedwith the topology induced by the operator norm on L( H ), thus becoming a topologi-cal space. More precisely, denoting the Fr´echet metric induced by the operator normon L( H ) by d , the space ( F , d ) is a separable complete metric space (Theorem A.1).Whenever dim( H ) < ∞ , the topological space F ⊂ L( H ) is locally compact. Onthe contrary, whenever H is an infinite-dimensional Hilbert space, the correspondingset F ⊂ L( H ) is non-locally compact (see Lemma 3.3 below). In preparation, let usfirst state the following results. Proposition 3.1.
Any locally compact Banach space X is finite-dimensional.Proof. Let x ∈ X with k x k = 1. Given x , . . . , x r ∈ X linearly independent unitvectors (i.e. k x i k = 1 for all i = 1 , . . . , r ), the space G r = span { x , . . . , x r } is an r -dimensional subspace of X . Since G r is finite-dimensional, it is closed. If G r ( E ,there exists a unit vector x r +1 ∈ X with k x r +1 − x i k ≥ / i = 1 , . . . , r . If we assume that X is infinite-dimensional, this holds for every r ∈ N , thus endingup with an infinite sequence ( x r ) r ∈ N of unit vectors satisfying k x p − x q k ≥ / p = q . In particular, the sequence ( x r ) r ∈ N admits no convergent subsequence incontradiction to the assumption that E is locally compact. (cid:3) Corollary 3.2.
Any infinite-dimensional Banach space X is non-locally compact. Thesame holds true for open subsets of X .Proof. This is an immediate consequence of Proposition 3.1. (cid:3)
Lemma 3.3.
Let H be an infinite-dimensional, separable complex Hilbert space, andlet F reg ⊂ L( H ) be the set of self-adjoint operators which have exactly s positive andexactly s negative eigenvalues for some s ∈ N . Then F reg is non-locally compact.Proof. Since F reg is a Banach manifold (for details see [25]), it can be covered by anatlas ( U α , φ α ) α ∈ A for some index set A (cf. [43, Chapter 73]). In particular, everypoint x ∈ U reg is contained in some open set U α , whose image V α := φ α ( U α ) is open insome infinite-dimensional Banach space X α . Due to Corollary 3.2, the set V α is non-locally compact. As the mapping φ α is a homeomorphism, we deduce that U α ⊂ F reg is non-locally compact for any α ∈ A , which proves the claim. (cid:3) Considering an infinite-dimensional , separable complex Hilbert space H , then theset F ⊂ L( H ) as introduced in [18] is non-locally compact and Polish (see Lemma 3.3and Theorem A.1). Our goal in the following is to prove the existence of a regular(possibly non-locally finite) measure ρ on the Borel algebra B ( F ) such that ρ is aminimizer of the corresponding causal action principle, giving rise to a causal fermionsystem ( H , F , ρ ). Instead of immediately delving into the corresponding causal actionprinciple (see [18, § H , which canbe viewed as generalizations of the causal action principle (as introduced in [16], [23]and considered in more detail in [24]). Corresponding results concerning the causalaction principle are then obtained as a special case. With this in mind, it suffices toprove the existence of minimizers of the causal variational principle (3.2) under theconstraints (2.3) in the non-locally compact setting as introduced in Definition 3.4.In order to motivate the basic definitions in § F ⊂ K ( H ),where by K ( H ) ⊂ L( H ) we denote the set of compact operators on H . Since H isa separable, infinite-dimensional complex Hilbert space, let us point out that K ( H )is a Banach space (see e.g. [40, Satz II.3.2]) and separable in view of [31, § K ( H ) ⊃ F is infinite-dimensional. This allows us to approximate K ( H ) by finite-dimensional subspaces. More precisely, we may apply [2, Lemma 7.1] to deduce thatthere is a sequence of finite-dimensional subspaces ( L n ) n ∈ N in K ( H ) with L n ⊂ L n +1 for all n ∈ N such that S n ∈ N L n is dense in K ( H ). From the fact that subspaces oflocally compact spaces are again locally compact we conclude that F ( n ) := F ∩ L n islocally compact for every n ∈ N . Denoting by d the Fr´echet metric induced by theoperator norm on L( H ), the space ( F , d ) is Polish (cf. Theorem A.1), i.e. a separablemetric space. As a consequence, the subsets F ( n ) are separable for every n ∈ N dueto [2, Lemma 2.16] or [1, Corollary 3.5]. Together with the fact that separable metricspaces are second-countable this yields that the set F ( n ) is a second-countable, locallycompact Hausdorff space for every n ∈ N . Moreover, from Lemma 3.3 we concludethat F ⊂ L( H ) is non-locally compact. XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 11
In order to treat the corresponding causal variational principle in sufficient gener-ality, it seems reasonable to vary in the class of regular, not necessarily locally finitemeasures on the Borel σ -algebra B ( F ) (as intended in the textbook [18, § ρ ( F ) isfinite and the Hilbert space H is infinite-dimensional. However, the causal actionprinciple does make mathematical sense in the so-called infinite-dimensional setting where H is infinite-dimensional and the total volume is infinite, i.e. ρ ( F ) = ∞ . Theseconsiderations motivate causal variational principles in the infinite-dimensional (ornon-locally compact) setting as defined in the next subsection.3.2. Basic Definitions.
Let us first state the causal variational principle in the non-locally compact setting and discuss its difficulties afterwards.
Definition 3.4.
Assume that X is a separable, infinite-dimensional Banach space,and let F ⊂ X be a non-locally compact Polish space. Moreover, assume that theLagrangian L : F × F → R +0 is a symmetric and lower semi-continuous function (seeconditions (i) and (ii) in § L ( x, x ) > for all x ∈ F . (3.1) The causal variational principle in the non-locally compact setting is tominimize S ( ρ ) := ˆ F ˆ F dρ ( x ) dρ ( y ) L ( x, y ) (3.2) under variations of finite volume (see Definition 3.5 below) in the class of all regularmeasures on B ( F ) (in the sense of [28] , cf. [24] ) with ρ ( F ) = ∞ . The condition (3.1) is needed in order to avoid trivial minimizers supported at x ∈ F with L ( x, x ) = 0 (see [26, Section 1.2]). Furthermore, condition (3.1) is a plausibleassumption in view of [18, Exercise 1.2]. Namely, given a minimizing measure ρ ofthe causal action principle (3.2), there exists a real constant c such that tr( x ) = c forall x ∈ supp ρ according to [18, Proposition 1.4.1]. Under the reasonable assumptionthat c = 0 (cf. [18, § L ( x, x ) > x ∈ F inview of [18, Exercise 1.2]. This motivates as well as justifies the assumption that theLagrangian is strictly positive on the diagonal.Dropping the assumption that the measures under consideration are locally finite,we slightly adapt the definition of a minimizer of the causal action as follows. Definition 3.5.
A regular measure ρ on B ( F ) is said to be a minimizer of the causalaction under variations of finite volume if the difference (2.4) is well-defined andnon-negative for all regular measures ˜ ρ on B ( F ) satisfying (2.3) , (cid:0) S (˜ ρ ) − S ( ρ ) (cid:1) ≥ . Given a measure ρ on the Borel algebra B ( F ) and assuming that ˜ ρ is a variation offinite volume, in view of Definition 3.5 there exists B ∈ B ( F ) such that ρ | F \ B = ˜ ρ | F \ B .In particular, the measures ρ | B and ˜ ρ | B are finite. Henceforth, whenever ρ is locallyfinite, then the same holds true for the measure ˜ ρ | F \ B . From the fact that ˜ ρ | B is afinite measure we conclude that ˜ ρ | B is locally finite. Consequently, the measure ˜ ρ islocally finite if ρ is so. For this reason, Definition 3.5 can be viewed as a generalizationof Definition 2.3 (cf. [24, Definition 2.1]). The same holds for Definition 3.6 below. For clarity we point out that “causal variational principles in the non-locally compact setting”and “causal variational principles in the infinite-dimensional setting” are used synonymously.
Definition 3.6.
A regular measure ρ on B ( F ) is said to be a minimizer undervariations of compact support of the causal action if for any regular measure ˜ ρ on B ( F ) which satisfies (2.3) such that the signed measure ˜ ρ − ρ is compactly supported,the inequality (cid:0) S (˜ ρ ) − S ( ρ ) (cid:1) ≥ holds. Let us now point out some difficulties regarding causal variational principles onnon-locally compact spaces. First of all, let us recall that a topological space is calledhemicompact if there is a sequence ( K n ) n ∈ N of compact subsets of X such that anycompact set K ⊂ X is contained in K n for some n ∈ N (see [41, 17I]). Since F isfirst-countable and non-locally compact, by virtue of [11, Exercise 3.4.E] we concludethat F cannot be hemicompact.Next, by contrast to the σ -locally compact setting as worked out in [24], it is ingeneral not even possible to assume that F is σ -compact, as the following argumentshows: Every Polish space (as well as every locally compact Hausdorff space) is Baireaccording to [31, Theorem (8.4)]. In view of [41, 25B], a σ -compact topologicalspace X is Baire if and only if the set of points at which X is locally compact isdense in X . Given an infinite-dimensional Hilbert space H , and defining F ⊂ K ( H ) inanalogy to [18] (see § F is a Polish space (see Appendix A). Consequently,the assumption that F is σ -compact implies that there exists x ∈ F being containedin a compact neighborhood K ⊂ F with K ◦ = ∅ . From this we conclude that theintersection K reg := K ∩ F reg is a compact set with non-empty interior, where theBanach manifold F reg ⊂ F is defined in Lemma 3.3 (for details see [25]). Givenan atlas ( U α , φ α ) α ∈ A of F reg (cf. [43]) and making use of the fact that each φ α is ahomeomophism mapping to some infinite-dimensional Banach space X α , we deducethat the image of K reg is a compact subset with non-empty interior in contradictionto [32, Exercise 14.3]. For this reason, it is not possible to assume that the space F is σ -compact (by contrast to the setting in [24]).Next, it is no longer possible to assume that the Lagrangian L : F × F → R +0 is simultaneously lower semi-continuous and of compact range (see Definition 2.4)as introduced in [24, Definition 3.3]. Namely, due to lower semi-continuity of theLagrangian, the latter assumption already implies that F is locally compact.Finally, it is not possible to assume that the Lagrangian decays in entropy in thesense of [24, Definition 5.1] (see Definition 2.8); indeed, this assumption requires aHeine-Borel metric on F , which clearly does not exist in non-locally compact spaces(otherwise each x ∈ F is contained in a corresponding ball with compact closure).In view of these difficulties in the non-locally compact setting, let us begin bygeneralizing the assumption that L is of compact range in the following way. Definition 3.7.
Let ( F , d ) be a metric space. The Lagrangian L : F × F → R +0 issaid to be of bounded range if every bounded set B ⊂ F is contained in a boundedneighborhood B ′ ⊂ F such that L ( x, y ) = 0 for all x ∈ B and y / ∈ B ′ . On proper metric spaces (that is, on spaces with the Heine-Borel property), thisdefinition clearly implies that L is of compact range (see Definition 2.4) as defined For clarity, we recall that a topological space X is said to be Baire if the intersection of eachcountable family of dense open sets in X is dense (see e.g. [41, Definition 25.1]). XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 13 in [24]. For this reason, Definition 3.7 provides a good starting point for dealing withcausal variational principles on non-locally compact spaces. As we shall see below, theassumption that the Lagrangian is of bounded range can be weakened (see § Finite-Dimensional Approximation.
In the infinite-dimensional setting (seeDefinition 3.4), the space X is assumed to be a separable, infinite-dimensional Banachspace. Hence we may apply [2, Lemma 7.1] to deduce that there exists a sequenceof finite-dimensional subspaces ( X n ) n ∈ N in X with X n ⊂ X n +1 for all n ∈ N suchthat S n ∈ N X n is dense in X . This allows us to introduce the topological spaces F ( n ) := F ∩ X n for every n ∈ N . Since finite-dimensional topological vector spaces are locally compact (see e.g. [33, § X n ⊂ X is locally compact for all n ∈ N . For easein notation, we shall denote the restriction of the Lagrangian to F ( n ) × F ( n ) by L ( n ) .Thus for every n ∈ N , we are given a second-countable, locally compact Hausdorffspace F ( n ) ⊂ F together with a symmetric, lower semi-continuous Lagrangian L ( n ) : F ( n ) × F ( n ) → R +0 , which is strictly positive on the diagonal. Henceforth for every n ∈ N we are exactlyin the σ -locally compact setting as worked out in [24].In the following, we additionally assume that L : F × F → R +0 is continuous and ofbounded range (see Definition 3.7). We again consider the above exhaustion ( X n ) n ∈ N of X by finite-dimensional subsets with X n ⊂ X n +1 for all n ∈ N . Let us point outthat each ( X n , k . k ) is a finite-dimensional normed vector space, and all norms on X n are equivalent. Due to the Heine-Borel theorem [2, Bemerkungen 2.6 and Satz 2.9],each closed ball B r ( x ) ⊂ X n is compact for all r > x ∈ X n . As a consequence,each bounded set A ⊂ F ( n ) is contained in some compact ball B := B r ( x ) ⊂ F ( n ) .Definition 3.7 yields the existence of a compact set B ′ ⊂ F ( n ) such that L ( n ) ( x, y ) = 0for all x ∈ B and y / ∈ B ′ . These considerations show that, whenever L is continuousand of bounded range, for every n ∈ N the restricted Lagrangian L ( n ) is continuousand of compact range (see [24, Definition 3.3] or Definition 2.4). As a consequence,by virtue of Theorem 2.6 (also see [24, Theorem 4.2]), for each n ∈ N there exists aregular Borel measure ρ n on F ( n ) such that the following EL equations hold, ℓ n | supp ρ n ≡ inf x ∈ F ( n ) ℓ n ( x ) = 0 , (3.3)where ℓ n ∈ C ( F ) = C ( F , R ) is defined by ℓ n ( x ) := ˆ F ( n ) L ( n ) ( x, y ) dρ n ( y ) − . (3.4)According to Theorem 2.7 (cf. [24, Theorem 4.10]), each Borel measure ρ n ∈ B F ( n ) isa minimizer of the corresponding causal variational principleminimize S ( n ) := ˆ F ( n ) ˆ F ( n ) L ( n ) dρ ( x ) dρ ( y )under variations of compact support [24] in the class of regular Borel measures on F ( n ) with respect to the constraints (2.3).We extend the measures ρ n by zero on the whole topological space F , ρ [ n ] ( A ) := ρ n ( A ∩ F ( n ) ) for all A ∈ B ( F ) . (3.5) Thus ℓ [ n ] | supp ρ [ n ] ≡ inf x ∈ F ( n ) ℓ [ n ] ( x ) = 0 , (3.6)where the function ℓ [ n ] ∈ C ( F ) is defined by ℓ [ n ] ( x ) := ˆ F L ( x, y ) dρ [ n ] ( y ) − . (3.7)This gives rise to a sequence of regular Borel measures ( ρ [ n ] ) n ∈ N on F . In particular,whenever condition (iv) is satisfied for ρ [ n ] (see (2.2)), that issup x ∈ F ˆ F L ( x, y ) dρ [ n ] < ∞ for all n ∈ N , (3.8)each measure ρ [ n ] is a minimizer on F ( n ) under variations of finite volume [24] (seeDefinition 2.1 and Definition 2.3). In virtue of Theorem 2.10, the same holds true ifthe Lagrangian L ( n ) decays in entropy for any n ∈ N , provided that condition (3.8) issatisfied. 4. Construction of a Global Measure
In the following, let X be an infinite-dimensional, separable complex Banach space,and let F ⊂ X be a non-locally compact Polish space endowed with a correspondingmetric d such that ( F , d ) is a separable, complete metric space. By O ( F ) and P ( F )we denote the collection of open subsets of F and the power set of F , respectively.Moreover, the collection of all compact subsets of F is represented by K ( F ).The goal of this section is to construct a global measure ρ based on the sequenceof regular Borel measures ( ρ [ n ] ) n ∈ N as obtained in § D ⊂ K ( F ) consisting of compact subsets of F ( § D in order to obtain a measure ρ on F ( § § F , the measure ρ is the weak limit of a subsequenceof ( ρ [ n ] ) n ∈ N .4.1. Construction of a Countable Collection of Compact Sets.
To begin with,separability of F yields the existence of a countable dense subset E := { x j : j ∈ N } such that, for every n ∈ N the set E ( n ) := E ∩ F ( n ) is dense in F ( n ) . We denoteits elements by x ( n ) j ∈ E ( n ) with j, n ∈ N . Moreover, since F ( n ) is locally compact,for all j, k, n ∈ N there is a compact neighborhood V ( n ) j,k ⊂ F ( n ) of x ( n ) j ∈ E ( n ) suchthat V ( n ) j,k ⊂ B /k ( x ( n ) j ) and each V ( n ) j,k being the closure of its interior (in the topologyof F ( n ) , where the interior of a set V shall be denoted by V ◦ ). This gives rise to theset V (1) := n V ( n ) j,k : j, k, n ∈ N o . Denoting the union of V (1) and the empty set ∅ by ˜ D (1) , and making use of the factthat a countable union of countable sets is countable (see e.g. [27, Section 2]), we Since F is separable, there exists a countable set E (0) ⊂ F being dense in F . Similarly, foreach i ∈ N there are countable sets E ( i ) which are dense in F ( i ) . As a consequence, the set E := S ∞ i =0 has the desired properties. For simplicity, one may consider V ( n ) j,k = B / (2 k ) ( x ( n ) j ) ∩ F ( n ) for all j, k, n ∈ N . XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 15 conclude that ˜ D (1) is countable. Therefore, applying Cantor’s diagonal argument andproceeding iteratively, we conclude that˜ D ( i ) := n D ∪ ˜ D : D, ˜ D ∈ ˜ D ( i − o is countable for every i ∈ N with i ≥
2. As a consequence, the set D := ∞ [ i =1 ˜ D ( i ) (4.1)is countable; we denote its members by ( D m ) m ∈ N . In particular, each D ∈ D is acompact subset of F . Moreover, for every n ∈ N we introduce D ( n ) ⊂ P ( F ( n ) ) by D ( n ) := n D ∈ D : D ⊂ F ( n ) o . (4.2)4.2. Construction of a Regular Global Measure.
In order to construct a globalmeasure on F , we proceed similarly to [24] by selecting suitable subsequences of thesequence ( ρ [ n ] ) n ∈ N restricted to compact subsets D ∈ D . This allows us to constructa regular measure ρ on the whole space F (see Theorem 4.3 below). In Section 5 wewill show that, under suitable assumptions, the measure ρ is indeed a minimizer ofthe causal variational principle. In analogy to [24, Lemma 4.1], let us first state thefollowing result. Lemma 4.1.
Assume that the Lagrangian L : F × F → R +0 is lower semi-continuousand strictly positive on the diagonal (3.1) . Furthermore, let ( ρ [ n ] ) n ∈ N be a sequence ofmeasures ρ [ n ] : B ( F ) → [0 , ∞ ] such that, for every x ∈ supp ρ [ n ] , ˆ F L ( x, y ) dρ [ n ] ( y ) = 1 for all n ∈ N . Then for every compact subset K ⊂ F there is a constant C K > such that ρ [ n ] ( K ) ≤ C K for all n ∈ N . Proof.
This statement is proven exactly as [24, Lemma 4.1]. (cid:3)
Next, we apply Lemma 4.1 to the compact sets D ∈ D . More precisely, restrict-ing the sequence ( ρ [ n ] ) n ∈ N as obtained in (3.5) (cf. § D ∈ D ,the resulting sequence ( ρ [ n ] | D ) n ∈ N is bounded (due to Lemma 4.1) as well as uni-formly tight (for the definition see [7, Definition 8.6.1]). Since compact subsets ofPolish spaces are again Polish, Prohorov’s theorem (see for instance [7, Theorem 8.6.2]or [10, Satz VIII.4.23]) implies that a subsequence of ( ρ [ n ] | D ) n ∈ N converges weaklyon D . Denoting the corresponding subsequence by ( ρ [1 ,n k ] ) k ∈ N and considering itsrestriction to D ∈ D , the same arguments as before yield the existence of a weaklyconvergent subsequence ( ρ [2 ,n k ] ) k ∈ N on D . Proceeding iteratively, we denote the re-sulting diagonal sequence by ρ ( k ) := ρ [ k,n k ] for all k ∈ N . (4.3)Thus by construction, for every m ∈ N the sequence ( ρ ( k ) | D m ) k ∈ N converges weakly tosome measure ρ D m : B ( D m ) → [0 , ∞ ), ρ ( k ) | D m ⇀ ρ D m . (4.4) In particular, lim k →∞ ρ ( k ) ( D m ) = ρ K m ( D m ) for all m ∈ N . We point out that each measure ρ ( k ) is a minimizer on F ( n k ) . For this reason, werestrict attention to the finite-dimensional exhaustion ( F ( k ) ) k ∈ N , where for notationalsimplicity by F ( k ) we denote the sets F ( n k ) for all k ∈ N . Note that the sequenceconstructed in (4.3) above in general does not converge weakly on arbitrary compactsubsets, but only restricted to compact sets D ∈ D (cf. (4.4)). In [24], this problemwas resolved by deriving vague convergence of the sequence ( ρ ( n ) ) n ∈ N to some globalmeasure ρ . In order to obtain a similar situation, let us state the following result. Proposition 4.2.
The set function ϕ : D → [0 , ∞ ) defined by ϕ ( D ) := lim k →∞ ρ ( k ) ( D ) < ∞ for any D ∈ D (4.5) has the following properties: (1) ϕ ( D ) ≤ ϕ ( D ) for all D , D ∈ D with D ⊂ D , (2) ϕ ( D ∪ D ) ≤ ϕ ( D ) + ϕ ( D ) for all D , D ∈ D , and (3) ϕ ( D ∪ D ) = ϕ ( D ) + ϕ ( D ) for all D , D ∈ D with D ∩ D = ∅ .Proof. Given D , D ∈ D with D ⊂ D , property (1) follows from ϕ ( D ) = lim k →∞ ˆ F dρ ( k ) | D ≤ lim k →∞ ˆ F dρ ( k ) | D = ϕ ( D ) . Next, for all D , D ∈ D , construction of D yields D ∪ D ∈ D . Thus property (2) isa consequence of ϕ ( D ∪ D ) = lim k →∞ ˆ F dρ ( k ) | D ∪ D ≤ lim k →∞ ˆ F dρ ( k ) | D + lim k →∞ ˆ F dρ ( k ) | D = ϕ ( D ) + ϕ ( D ) . Similarly, for all D , D ∈ D with D ∩ D = ∅ we obtain ϕ ( D ∪ D ) = lim k →∞ ρ ( k ) ( D ∪ D ) = lim k →∞ ρ ( k ) ( D ) + lim k →∞ ρ ( k ) ( D ) = ϕ ( D ) + ϕ ( D ) , which proves property (3). (cid:3) In order to construct a global measure ρ on F , we proceed in analogy to the proofof [10, Satz VIII.4.22]. We point out that, since the underlying topological space F isnon-locally compact, we cannot employ the Riesz representation theorem as in [24], andRiesz representation theorems on more general Hausdorff spaces as presented in [32,Section 16] do not seem applicable at this stage. Nevertheless, we obtain the followingresult. Theorem 4.3.
Introducing the set function ϕ : D → [0 , ∞ ) by (4.5) and defining theset functions µ : O ( F ) → [0 , ∞ ] and η : P ( F ) → [0 , + ∞ ] by µ ( U ) := sup { ϕ ( D ) : D ⊂ U , D ∈ D } for all U ⊂ F open ,η ( A ) := inf { µ ( U ) : A ⊂ U , U ⊂ F open } for any A ∈ P ( F ) , (4.6) then the restriction ρ := η | B ( F ) (4.7) XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 17 defines a (possibly non-trivial) measure on the Borel σ -algebra B ( F ) . In particular, ρ ( D ) = ϕ ( D ) = lim k →∞ ρ ( k ) ( D ) for any D ∈ D . (4.8) Proof.
Let us first point out that, by construction, the set function ϕ : D → [0 , ∞ )defined by (4.5) has the following properties:(1) ϕ ( D ) ≤ ϕ ( D ) for all D , D ∈ D with D ⊂ D ,(2) ϕ ( D ∪ D ) ≤ ϕ ( D ) + ϕ ( D ) for all D , D ∈ D , and(3) ϕ ( D ∪ D ) = ϕ ( D ) + ϕ ( D ) for all D , D ∈ D with D ∩ D = ∅ .Indeed, properties (1)–(3) are a consequence of Proposition 4.2. Moreover, ϕ ( ∅ ) = 0(since ∅ ∈ D ).Next, similarly to the proof of [10, Satz VIII.4.22], our goal is to show that η is anouter measure, and that every Borel set B ∈ B ( F ) is η -measurable. This shall bedone in the following by proving that η is an outer measure, and each closed set A ⊂ F is η -measurable . (4.9)Denoting the σ -algebra of η -measurable sets by A η , the statement (4.9) implies thatthe Borel σ -algebra B ( F ) is contained in A η , i.e. B ( F ) ⊂ A η . Therefore, in view ofCarath´eodory’s theorem (see e.g. [10, Satz II.4.4]), the restriction ρ := η | B ( F ) defines a measure on the Borel σ -algebra B ( F ). Thus it suffices to prove (4.9), whichshall be done in several steps in the remainder of the proof.(a) Let A ⊂ U ⊂ F with A closed and U open. Whenever A ⊂ D for some D ∈ D ,then there exists E ∈ D with A ⊂ E ⊂ U . Proof: Since each D ∈ D is compact, the closed set A ⊂ D is compact as well.Moreover, D ⊂ F ( n ) for sufficiently large n ∈ N . Since F ( n ) is locally compact, forevery x ∈ A there exists V x ∈ D such that x ∈ V ◦ x ⊂ V x ⊂ U . Since A is compact, theset E := S Nj =1 V x j ∈ D for some integer N = N ( A ) has the desired property.(b) Whenever
U, V ⊂ F open, µ ( U ∪ V ) ≤ µ ( U ) + µ ( V ) . Proof: Without loss of generality, let U = F = V and µ ( U ) , µ ( V ) < ∞ (otherwisethe inequality is true). For this reason, let U, V ⊂ F be open sets with U c = ∅ = V c and D ⊂ U ∪ V for D ∈ D . We then consider the closed sets A := { x ∈ D : d ( x, U c ) ≥ d ( x, V c ) } ⊂ D ,B := { x ∈ D : d ( x, U c ) ≤ d ( x, V c ) } ⊂ D .
Obviously, A ⊂ U and B ⊂ V . Assuming conversely that x ∈ A \ U , we concludethat x ∈ V , and therefore d ( x, U c ) = 0 < d ( x, V c ) because V c is closed, giving rise tothe contradiction that x / ∈ A . Similarly, we conclude that B ⊂ V . Since A ⊂ D , by Given a set X , a set function η : P ( X ) → R := [ −∞ , + ∞ ] is said to be an outer measure if it hasthe following properties (see e.g. [10, Definition II.4.1]):(i) η ( ∅ ) = 0.(ii) For all A ⊂ B ⊂ X holds η ( A ) ≤ η ( B ) (monotonicity).(iii) For every sequence ( A n ) n ∈ N of subsets of X holds η (cid:0) S ∞ n =1 A n (cid:1) ≤ P ∞ n =1 η ( A n ) ( σ -subadditivity). virtue of (a) there exists E ∈ D with A ⊂ E ⊂ U . Similarly, there exists F ∈ D suchthat B ⊂ F ⊂ V , and D = A ∪ B ⊂ E ∪ F . Hence (1) and (2) yield ϕ ( D ) ≤ ϕ ( E ∪ F ) ≤ ϕ ( E ) + ϕ ( F ) ≤ µ ( U ) + µ ( V ) . Taking the supremum over all D ∈ D with D ⊂ U ∪ V gives (b).(c) For all n ∈ N and U n ⊂ F open, µ (cid:0) S ∞ n =1 U n (cid:1) ≤ P ∞ n =1 µ ( U n ) . Proof: Let D ∈ D with D ⊂ S ∞ n =1 U n . Then by compactness of D there exists p ∈ N such that D ⊂ S pn =1 U n . Applying (b) inductively, we conclude that ϕ ( D ) ≤ µ p [ n =1 U n ! ≤ p X n =1 µ ( U n ) ≤ ∞ X n =1 µ ( U n ) . Since D ∈ D with D ⊂ S ∞ n =1 U n is arbitrary, we obtain (c).(d) η is an outer measure. Proof: As seen before, ϕ ( ∅ ) = 0, and monotonicity of η is a consequence of (1)–(3)and (4.6). In order to prove σ -subadditivity, let ε > M n ⊂ F with η ( M n ) < ∞ for all n ∈ N . In view of (4.6), for every n ∈ N there exists an open set U n ⊃ M n with µ ( U n ) ≤ η ( M n ) + 2 − n ε . Making use of (4.6) and applying (c) yields η ∞ [ n =1 M n ! ≤ µ ∞ [ n =1 U n ! ≤ ∞ X n =1 µ ( U n ) ≤ ∞ X n =1 η ( M n ) + ε . Since ε >
Each closed set A ⊂ F is η -measurable. Proof: By definition of measurability (cf. [10, Definition II.4.2]), we need to show that,for all Q ⊂ F , η ( Q ) ≥ η ( Q ∩ A ) + η ( Q ∩ A c ) . (4.10)Without loss of generality we may assume that η ( Q ) < ∞ . We first prove (4.10) inthe case that Q = U ⊂ F is open . To this end, let ε > A ⊂ F closed, the set U ∩ A c is open and µ ( U ∩ A c ) = η ( U ∩ A c ) < ∞ . In view of (4.6),there exists D ∈ D with ϕ ( D ) ≥ µ ( U ∩ A c ) − ε . Next, since U ∩ D c is open, we maychoose E ∈ D with E ⊂ U ∩ D c and ϕ ( E ) ≥ µ ( U ∩ D c ) − ε . Since D , E are disjointand D ∪ E ⊂ U , from (1), (3), (4.6) and the fact that U ∩ D c ⊃ U ∩ A we concludethat µ ( U ) ≥ ϕ ( D ∪ E ) = ϕ ( D ) + ϕ ( E ) ≥ µ ( U ∩ A c ) + µ ( U ∩ D c ) − ε ≥ η ( U ∩ A ) + µ ( U ∩ A c ) − ε . Since ε > Q = U open.Given arbitrary Q ⊂ F with η ( Q ) < ∞ , for ε > U ⊃ Q openwith η ( Q ) ≥ η ( U ) − ε according to (4.6). Then the latter inequality yields η ( Q ) ≥ η ( U ) − ε ≥ η ( U ∩ A ) + η ( U ∩ A c ) − ε ≥ η ( Q ∩ A ) + η ( Q ∩ A c ) − ε , proving (4.10). XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 19
As a consequence, the set function η is an outer measure according to (d), and eachclosed set A ⊂ F is η -measurable in view of (e). This yields (4.9), which completes theproof. (cid:3) The next result shows that the measure ρ given by (4.7) is regular [10]. Lemma 4.4.
Let ρ : B ( F ) → [0 , ∞ ] be the measure defined by (4.7) . Then every opensubset of F is inner regular. Moreover, the measure ρ is regular.Proof. Let us first prove that every open subset of F is inner regular in view of (4.6).Namely, considering arbitary U ∈ O ( F ) and K ∈ K ( U ), according to (4.6) and (4.8)we obtain ρ ( K ) ≤ ρ ( U ) = sup { ϕ ( D ) : D ∈ D , D ⊂ U } = sup { ρ ( D ) : D ∈ D , D ⊂ U }≤ sup { ρ ( K ) : K ⊂ U compact } . Taking the supremum on the left hand side yields ρ ( U ) = sup { ρ ( K ) : K ⊂ U compact } for all U ∈ O ( F ) . From this we conclude that every open set U ∈ O ( F ) is inner regular (in the senseof [10, Definition VIII.1.1]).In view of (4.6), we are given ρ ( U ) = µ ( U ) for any U ∈ O ( F ), and the aboveconsiderations show that every open set is inner regular. From this we conclude thatthe measure ρ is regular in the sense of [10, Definition VIII.1.1]. (cid:3) As a matter of fact, in general it seems possible the regular measure ρ obtained inTheorem 4.3 to be zero. Nevertheless, the following remark gives a sufficient conditionfor the measure ρ defined by (4.7) to be non-zero. Remark 4.5.
Let ( F ( n ) ) n ∈ N be a finite-dimensional approximation of F (see § ρ ( n ) we are given supp ρ ( n ) ⊂ F ( n ) for every n ∈ N . Assumingthat the Lagrangian is bounded and of bounded range, for every x ∈ F and δ > there exists B x ⊂ F bounded and closed such that L (˜ x, y ) = 0 for all ˜ x ∈ B δ ( x ) andall y / ∈ B x . Furthermore, in view of boundedness of the Lagrangian we introduce theupper bound C < ∞ by C := sup x,y ∈ F L ( x, y ) > . Thus for any n ∈ N we deduce that L ( n ) (˜ x, y ) = 0 for all ˜ x ∈ B ( n ) δ ( x ) := B δ ( x ) ∩ F ( n ) and all y / ∈ B ( n ) x := B x ∩ F ( n ) . Hence the EL equations (3.6) and (3.7) imply that ≤ ˆ F ( n ) L ( n ) (˜ x, y ) dρ ( n ) = ˆ B ( n ) x L ( n ) (˜ x, y ) dρ ( n ) ≤ sup y ∈ B ( n ) x L ( n ) (˜ x, y ) ρ ( n ) ( B ( n ) x ) for every n ∈ N . Thus positivity (3.1) yields ρ ( n ) ( B ( n ) x ) ≥ C − > for all n ∈ N . For each n ∈ N and arbitrary ε > , by regularity of ρ ( n ) there exists D n ∈ D suchthat ρ ( n ) ( D n ) > C − − ε . Moreover, ˆ D N := S Nn =1 D n ∈ D for every N ∈ N . Wheneverthere exists N ∈ N such that ρ ( n ) ( ˆ D N ) ≥ c for almost all n ∈ N and some c > , thenthe measure ρ defined by (4.7) is non-zero. If this holds true for an infinite number ofdisjoints sets ( ˆ D N i ) i ∈ N , the measure ρ possibly has infinite total volume. Next, in agreement with [32, Theorem 16.7] and the remark thereafter, it is notclear whether ρ as given by (4.7) is a locally finite measure. Nevertheless, the followingresults provide sufficient conditions for ρ as obtained in (4.7) to be locally finite. Lemma 4.6.
Let ρ : B ( F ) → [0 , ∞ ] be defined by (4.7) . Assuming that ρ ( K ) < ∞ for all K ∈ K ( F ) , then the measure ρ is locally finite and thus a Borel measure in thesense of [28] . In this case, ρ is regular and moderate.Proof. We point out that the space F is first-countable. Thus under the assumptionthat ρ ( K ) < ∞ for all K ∈ K ( F ), the statement that ρ is locally finite is a consequenceof Lemma 4.4 and [10, Folgerungen VIII.1.2 (d)]. The last statement follows fromUlam’s theorem [10, Theorem VIII.1.16]. (cid:3) Remark 4.7.
We point out that, if F in Definition 3.4 were locally compact, thenthe measure ρ constructed in the proof of Theorem 4.3 would be locally finite, i.e. aBorel measure in the sense of [28] . Namely, whenever x ∈ F , there exists a compactneighborhood V of x . Hence Lemma 4.1 implies that ρ ( n ) ( V ) ≤ C V for all n ∈ N andsome C V > . Choosing U x ⊂ V open with x ∈ U , we conclude that ρ ( U x ) = sup { ϕ ( D ) : D ⊂ U x , D ∈ D } ≤ sup n lim n →∞ ρ ( n ) ( D ) : D ⊂ V , D ∈ D o ≤ C V as desired. In the remainder of this subsection, we discuss the properties (iii) and (iv) in § § Lemma 4.8.
Assume that the Lagrangian L : F × F → R +0 is lower semi-continuous,symmetric and strictly positive on the diagonal (3.1) , and let ρ be a measure on B ( F ) .Under the assumption that condition (iv) in § (2.2) ), i.e. sup x ∈ F ˆ F L ( x, y ) dρ ( y ) < ∞ , the measure ρ is locally finite (i.e. condition (iii) in § Assume conversely that there exists x ∈ F such that ρ ( U ) = ∞ for any openneighborhood U of x . Then L ( x, x ) > U x of x such that L ( x, y ) > L ( x, x ) / > y ∈ U x . Consequently, ˆ F L ( x, y ) dρ ( y ) ≥ ˆ U x L ( x, y ) dρ ( y ) > L ( x, x ) / ρ ( U x ) = ∞ in contradiction to condition (iv) in § (cid:3) Convergence on Relatively Compact Subsets.
In Section 5 below our goal isto show that, under suitable assumptions, the measure ρ as given by (4.7) is a minimizerunder variations of finite volume (see Definition 3.5). To this end, we provide someuseful tools which shall be worked out in the remainder of this section. For clarity,we point out that for every D ∈ D there exists some n ′ ∈ N such that D ◦ = ∅ in therelative topology of F ( n ) for all n ≤ n ′ and D ◦ = ∅ in the relative topology of F ( n ) for all n > n ′ . Considering the restriction ρ | D ◦ shall always be understood in thesense of a restriction to the interior of D in the relative topology of F ( n ′ ) . In order to XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 21 derive weak convergence on suitable relatively compact sets (see Lemma 4.11 below),we require some properties of the measures ρ D with D ∈ D as obtained in (4.4).Given D, E ∈ D with D ⊂ E and denoting the interior of D by D ◦ , we claim that ˆ D f dρ D = ˆ E f dρ E for all f ∈ C c ( D ◦ ) . (4.11)Namely, in view of C c ( D ◦ ) ⊂ C b ( D ) ∩ C b ( E ), weak convergence (4.4) yields ˆ D f dρ D = lim k →∞ ˆ D f dρ ( k ) | D ( ⋆ ) = lim k →∞ ˆ E f dρ ( k ) | E = ˆ E f dρ E for all f ∈ C c ( D ◦ ), where in ( ⋆ ) we made use of the fact that supp f ⊂ D ◦ ⊂ E .This allows us to prove the following result. Proposition 4.9.
Whenever D ∈ D and Ω ⊂ D ◦ open, then sup n ρ ˜ D | D ◦ ( ˜ D ) : ˜ D ⊂ Ω , ˜ D ∈ D o = sup n ρ D | D ◦ ( ˜ D ) : ˜ D ⊂ Ω , ˜ D ∈ D o . (4.12) Proof.
In order to prove (4.12), we need to show that for all
E, F ∈ D with E, F ⊂ Ωthere exist sets ˜ E, ˜ F ∈ D with ˜ E, ˜ F ⊂ Ω such that ρ E | D ◦ ( E ) ≤ ρ D | D ◦ ( ˜ E ) and ρ D | D ◦ ( F ) ≤ ρ ˜ F | D ◦ ( ˜ F ) . Whenever E ⊂ Ω compact, there exists η ∈ C c (Ω; [0 , η | E ≡ E ∈ D with supp η ⊂ ˜ E ◦ ⊂ Ω, and by weak convergence (4.4) we are given ρ E | D ◦ ( E ) = ˆ E dρ E = lim n →∞ ˆ F dρ ( n ) | E ≤ lim n →∞ ˆ F η dρ ( n ) | D = ˆ F η dρ D ≤ ρ D | D ◦ ( ˜ E ) . On the other hand, whenever F ∈ D with F ⊂ Ω, there exists η ∈ C c (Ω; [0 , η | F ≡ F ∈ D with supp η ⊂ ˜ F ◦ and ˜ F ◦ ⊂ Ω. Thus by (4.11) weobtain ρ D | D ◦ ( F ) = ˆ F dρ D ≤ ˆ D η dρ D = ˆ ˜ F η dρ ˜ F ≤ ρ ˜ F | D ◦ ( ˜ F ) , which completes the proof. (cid:3) Lemma 4.10.
For every D ∈ D , the measures ρ | D ◦ and ρ D | D ◦ coincide. Moreover, ρ ( n ) | D ◦ v → ρ | D ◦ vaguely . (4.13) Proof.
According to (4.4) and (4.5), the measure ρ D | D ◦ : B ( D ◦ ) → [0 , ∞ ) is finitefor any D ∈ D . As a consequence, it is locally finite and thus Borel in the senseof [28]. Moreover, since open subsets of Polish spaces are Polish (see [4, § ρ | D ◦ for any D ∈ D . Thus forany D ∈ D , we may approximate arbitrary Borel sets A ∈ B ( D ◦ ) by compact setsfrom inside, ρ D | D ◦ ( A ) = sup { ρ D | D ◦ ( K ) : K ⊂ A compact } ,ρ | D ◦ ( A ) = sup { ρ | D ◦ ( K ) : K ⊂ A compact } . Whenever D ∈ D and Ω ⊂ D open, for each K ⊂ Ω compact there exists ˜ D ∈ D suchthat K ⊂ ˜ D ⊂ Ω by construction of D . From this we conclude that ρ D | D ◦ (Ω) = sup { ρ D | D ◦ ( K ) : K ⊂ Ω compact } = sup n ρ D | D ◦ ( ˜ D ) : ˜ D ⊂ Ω, ˜ D ∈ D o ,ρ | D ◦ (Ω) = sup { ρ | D ◦ ( K ) : K ⊂ Ω compact } = sup n ρ | D ◦ ( ˜ D ) : ˜ D ⊂ Ω, ˜ D ∈ D o . Moreover, for any ˜ D ∈ D , by (4.4) and (4.8) we obtain ϕ ( ˜ D ) = lim k →∞ ρ ( k ) ( ˜ D ) = lim k →∞ ρ ( k ) | ˜ D ( ˜ D ) = ρ ˜ D ( ˜ D ) . (4.14)Given D ∈ D and Ω ⊂ D ◦ open, we conclude that ρ | D ◦ as well as ρ D | D ◦ are regularfinite Borel measures on B ( D ◦ ), implying that ρ | D ◦ (Ω) = sup n ϕ ( ˜ D ) : ˜ D ⊂ Ω, ˜ D ∈ D o (4.14) = sup n ρ ˜ D ( ˜ D ) : ˜ D ⊂ Ω, ˜ D ∈ D o = sup n ρ ˜ D | D ◦ ( ˜ D ) : ˜ D ⊂ Ω, ˜ D ∈ D o (4.12) = sup n ρ D | D ◦ ( ˜ D ) : ˜ D ⊂ Ω, ˜ D ∈ D o = ρ D | D ◦ (Ω) . As a consequence, the measures ρ | D ◦ and ρ D | D ◦ coincide on all open sets Ω ⊂ D ◦ .Making use of [7, Lemma 7.1.2], we conclude that ρ | D ◦ and ρ D | D ◦ already coincide onall Borel sets, i.e. ρ | D ◦ = ρ D | D ◦ for all D ∈ D . (4.15)Given f ∈ C c ( D ◦ ), we thus obtainlim n →∞ ˆ F f dρ ( n ) | D ◦ = lim n →∞ ˆ F f dρ ( n ) | D (4.4) = ˆ F f dρ D = ˆ F f dρ D | D ◦ (4.15) = ˆ F f dρ | D ◦ . Since f ∈ C c ( D ◦ ) was arbitrary, we obtain vague convergence ρ ( n ) | D ◦ v → ρ | D ◦ . This completes the proof. (cid:3)
Having proved vague convergence on open subsets of D ∈ D , the following resulteven yields weak convergence on suitable relatively compact subsets V ⊂ F (so-called continuity sets , cf. [7, Section 8.2]). Lemma 4.11.
For every D ∈ D there exists E ∈ D with D ⊂ E ◦ as well as a relativelycompact, open subset V ⊂ E ◦ with D ⊂ V such that ρ ( n ) | V ⇀ ρ | V weakly . (4.16) Similarly, whenever D ∈ D ( n ) and U ⊃ D open, there exists a relatively compact, opensubset V ⊂ F ( n ) with D ⊂ V ⊂ U such that (4.16) holds.Proof. Given D ∈ D , by construction of D we know that D is compact and thuscontained in the interior of some E ∈ D with ρ ( E ) < ∞ in view of (4.5) and (4.8).Therefore, ρ | E : B ( E ) → [0 , ∞ ) is a nonnegative finite Borel measure, where B ( E )denotes the Borel σ -algebra on the topological space E . Since E is metrizable, it iscompletely regular, implying that the class Γ ρ | E of all Borel sets in E with boundariesof ρ | E -measure zero contains a base (consisting of open sets) of the topology of E (for details see [7, Proposition 8.2.8]). Since E is compact, the set D ⊂ E can becovered by finitely many relatively compact, open sets V , . . . , V N ⊂ E ◦ in Γ ρ | E . By XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 23 construction, the closure of the set V := S Ni =1 V i ⊂ E ◦ is compact. Considering therestriction ρ ( n ) | V , for any f ∈ C c ( V ) we obtainlim n →∞ ˆ V f dρ ( n ) | V = lim n →∞ ˆ F f dρ ( n ) | E ◦ (4.13) = ˆ F f dρ | E ◦ = ˆ V f dρ | V , proving vague convergence ρ ( n ) | V v → ρ | V . (4.17)Since Γ ρ | E is a subalgebra in B ( E ) (see [7, Proposition 8.2.8]), the set V is also con-tained in Γ ρ | E , implying that ρ ( ∂V ) = ρ | E ( ∂V ) = 0. Note that the measures ( ρ ( n ) | E ◦ )as well as ρ | E ◦ are regular Borel measures and thus Radon [38]. Therefore, making useof vague convergence (4.13) and applying [4, Theorem 30.2], for the relatively compact,open set V ⊂ E ◦ and the compact set V ⊂ E ◦ we obtain ρ ( V ) = ρ ( V ) ≥ lim sup n →∞ ρ ( n ) ( V ) ≥ lim sup n →∞ ρ ( n ) ( V ) ≥ lim inf n →∞ ρ ( n ) ( V ) ≥ ρ ( V ) , proving that ρ ( V ) = lim n →∞ ρ ( n ) ( V ) . (4.18)Let us point out that, for each n ∈ N , the measure ρ ( n ) | V /ρ ( n ) ( V ) is normalized inthe sense that ρ ( n ) | V ( V ) /ρ ( n ) ( V ) = 1 (cf. [24, § f ∈ C c ( V ) we are givenlim n →∞ ˆ V f dρ ( n ) | V /ρ ( n ) ( V ) = ˆ V f dρ | V /ρ ( V ) . As a consequence, the sequence of normalized measures ( ρ ( n ) | V /ρ ( n ) ( V )) n ∈ N convergesvaguely to the normalized measure ρ | V /ρ ( V ), and from [4, Corollary 30.9] we deducethat ( ρ ( n ) | V /ρ ( n ) ( V )) n ∈ N converges weakly to the normalized measure ρ | V /ρ ( V ). Thusin view oflim n →∞ ˆ V f dρ ( n ) | V = lim n →∞ ρ ( n ) ( V ) ˆ V f dρ ( n ) | V /ρ ( n ) ( V ) = ρ ( V ) ˆ V f dρ ( n ) | V /ρ ( V )= ˆ V f dρ ( n ) | V for any f ∈ C b ( V ), we finally obtain weak convergence ρ ( n ) | V ⇀ ρ | V . (cid:3) By contrast to [24], it is not reasonable to consider vague convergence ρ ( n ) v → ρ inview of [32, Exercise 14.4].5. Minimizers for Lagrangians of Bounded Range
Preliminaries.
This section is devoted to the proof that, under suitable assump-tions, the measure ρ as defined in (4.7) is a minimizer of the causal variational prin-ciple (3.2) under variations of finite volume (see Definition 3.5). This is accomplishedin § ρ obtained in Theorem 4.3 (see (5.1)). After-wards we prove that ρ is a minimizer on suitable compact subsets (see § § ρ = 0 is locally finite, we finally show that ρ satisfies correspond-ing Euler-Lagrange (EL) equations (see § EL equations obtained in [24]. Throughout this section, we shall assume that theLagrangian is of bounded range (see Definition 3.7).In order to prove that ρ is a minimizer, we impose the following condition:(B) For any ε > B ⊂ F bounded, there exists N ∈ N such that ρ ( B \ B ( n ) ) < ε for all n ≥ N , (5.1)where B ( n ) := B ∩ F ( n ) . Lemma 5.1.
Assume that the measure ρ defined by (4.7) satisfies condition (B). Thenthe measure ρ is locally finite, and any bounded subset of F has finite ρ -measure.Proof. Assuming that B ⊂ F is bounded, in view of condition (B) there exists N ∈ N such that ρ ( B \ B ( n ) ) < ε for all n ≥ N , where B ( n ) := B ∩ F ( n ) is relatively compact inview of [2, Bemerkungen 2.9]. For this reason, B ( n ) can be covered by a finite numberof compact sets D , . . . , D L with D i ∈ D ( n ) for all i = 1 , . . . , L , where D ( n ) is givenby (4.2). From (4.5) we obtain ρ ( B ( n ) ) < ∞ , implying that ρ ( B ) = ρ ( B \ B ( n ) ) + ρ ( B ( n ) ) < ∞ . Since each compact set K ⊂ F is bounded, the measure ρ is locally finite in view ofLemma 4.6. (cid:3) Minimizers under Variations of Finite-Dimensional Compact Support.
Before proving our first existence result, we point out that the restricted Lagrangian L ( n ) = L| F ( n ) × F ( n ) : F ( n ) × F ( n ) → R +0 is of compact range (see Definition 2.4 and § n ∈ N . Therefore, forall j, k, n ∈ N , there exist compact subsets ( V ( n ) j,k ) ′ ⊂ F ( n ) such that L ( x, y ) = 0 forall x ∈ V ( n ) j,k and y ∈ F ( n ) \ ( V ( n ) j,k ) ′ . By construction of D (see § V ( n ) j,k ) ′ canbe covered by a finite number of sets ( V ( n ) j ′ ℓ ,k ′ ℓ ) ℓ =1 ,...,L in D , whose union is also containedin D . For this reason, we may assume that ( V ( n ) j,k ) ′ ∈ D for all j, k, n ∈ N .After these preparations, we are now in the position to state our first existenceresult. Proposition 5.2.
Assume that the Lagrangian
L ∈ C b ( F × F ; R +0 ) is of bounded range,and that condition (3.8) holds. Moreover, assume that the measure ρ defined by (4.7) satisfies condition (B) in Section 5. Then ρ is a minimizer under variations in D in the sense that (cid:0) S (˜ ρ ) − S ( ρ ) (cid:1) ≥ whenever ˜ ρ satisfying (2.3) is a regular measure on B ( F ) with supp(˜ ρ − ρ ) ∈ D .Proof. Assume that ˜ ρ : B ( F ) → [0 , ∞ ] is a measure on the Borel σ -algebra of F with D := supp(˜ ρ − ρ ) ∈ D such that (2.3) is satisfied, i.e.0 < ˜ ρ ( D ) = ρ ( D ) < ∞ . Since D ∈ D is compact, the difference (2.4) is well-defined in view of [24, § S (˜ ρ ) − S ( ρ ) = ˆ F d (˜ ρ − ρ )( x ) ˆ F dρ ( y ) L ( x, y ) + ˆ F dρ ( x ) ˆ F d (˜ ρ − ρ )( y ) L ( x, y )+ ˆ F d (˜ ρ − ρ )( x ) ˆ F d (˜ ρ − ρ )( y ) L ( x, y ) . XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 25
Making use of the symmetry of the Lagrangian and applying Fubini’s theorem, we canwrite this expression as S (˜ ρ ) − S ( ρ ) = 2 ˆ F d (˜ ρ − ρ )( x ) ˆ F dρ ( y ) L ( x, y )+ ˆ F d (˜ ρ − ρ )( x ) ˆ F d (˜ ρ − ρ )( y ) L ( x, y ) . Since D ∈ D is a bounded subset of F , Definition 3.7 yields the existence of somebounded set W ⊂ F such that L ( x, y ) = 0 for all x ∈ D and y ∈ F \ W . Forarbitrary ˜ ε >
0, by virtue of condition (B) in Section 5 (see (5.1)) there exists someinteger n ′ = n ′ ( D ) such that ρ ( W \ W ( n ) ) < ˜ ε for all n ≥ n ′ . We choose n ′ ∈ N sufficiently large and let ˜ D ∈ D with ˜ D ⊂ F ( n ′ ) and ˜ D ⊃ D . As explained at thebeginning of § D ′ ∈ D with ˜ D ′ ⊂ F ( n ′ ) and L ( x, y ) = 0 for all x ∈ ˜ D and y ∈ F ( n ′ ) \ ˜ D ′ . In particular, L ( x, y ) = 0 for all x ∈ D and y ∈ F ( n ′ ) \ ˜ D ′ . In viewof Lemma 4.11, there exist relatively compact open sets V, V ′ ⊂ F ( n ′ ) with V ⊃ ˜ D and V ′ ⊃ ˜ D ′ such that L ( x, y ) = 0 for all x ∈ V and y ∈ F ( n ′ ) \ V ′ , and ρ ( n ) | V ⇀ ρ | V as well as ρ ( n ) | V ′ ⇀ ρ | V ′ . These considerations give rise to S (˜ ρ ) − S ( ρ ) = 2 ˆ V d (˜ ρ − ρ )( x ) ˆ V ′ dρ ( y ) L ( x, y )+ 2 ˆ V d (˜ ρ − ρ )( x ) ˆ F \ V ′ dρ ( y ) L ( x, y ) + ˆ V d (˜ ρ − ρ )( x ) ˆ V d (˜ ρ − ρ )( y ) L ( x, y ) , where the expression ˆ V d (˜ ρ − ρ )( x ) ˆ F \ V ′ dρ ( y ) L ( x, y ) = ˆ D d (˜ ρ − ρ )( x ) ˆ W \ W ( n ′ ) dρ ( y ) L ( x, y )can be chosen arbitrarily small for sufficiently large n ′ ∈ N . Thus for ε > S (˜ ρ ) − S ( ρ ) ≥ ˆ V d (˜ ρ − ρ )( x ) ˆ V ′ dρ ( y ) L ( x, y )+ ˆ V d (˜ ρ − ρ )( x ) ˆ V d (˜ ρ − ρ )( y ) L ( x, y ) − ε . Next, in analogy to the proof of [24, Theorem 4.9], for any n ∈ N we introduce themeasures ˜ ρ n : B ( F ) → [0 , ∞ ] by˜ ρ n := (cid:26) c n ˜ ρ on Vρ ( n ) on F \ V with c n := ρ ( n ) ( V )˜ ρ ( V ) for all n ∈ N . Since ρ and ˜ ρ coincide on F \ D , by virtue of (4.18) (see Lemma 4.11) we obtainlim n →∞ c n = lim n →∞ ρ ( n ) ( V )˜ ρ ( V ) = ρ ( V )˜ ρ ( V ) = ρ ( V \ D ) + ρ ( D )˜ ρ ( V \ D ) + ˜ ρ ( D ) = 1 . Making use of the fact that V ⊂ F is separable (see for instance [1, Corollary 3.5])and applying [6, Theorem 2.8] in a similar fashion to the proof of [24, Theorem 4.9], we thus arrive at S (˜ ρ ) − S ( ρ ) ≥ lim n →∞ (cid:20) ˆ V d (cid:0) c n ˜ ρ − ρ ( n ) (cid:1) ( x ) ˆ V ′ dρ ( n ) ( y ) L ( x, y )+ ˆ V d (cid:0) c n ˜ ρ − ρ ( n ) (cid:1) ( x ) ˆ V d (cid:0) c n ˜ ρ − ρ ( n ) (cid:1) ( y ) L ( x, y ) (cid:21) − ε . In view of the fact that ˜ ρ n and ρ ( n ) coincide on F ( n ) \ V for sufficiently large n ∈ N and L ( x, y ) = 0 for all x ∈ V and y / ∈ V ′ , the difference S (˜ ρ ) − S ( ρ ) can finally beestimated by S (˜ ρ ) − S ( ρ ) ≥ lim n →∞ (cid:20) ˆ F ( n ) d (cid:0) ˜ ρ n − ρ ( n ) (cid:1) ( x ) ˆ F ( n ) dρ ( n ) ( y ) L ( x, y )+ ˆ F ( n ) d (cid:0) ˜ ρ n − ρ ( n ) (cid:1) ( x ) ˆ F ( n ) d (cid:0) ˜ ρ n − ρ ( n ) (cid:1) ( y ) L ( x, y ) (cid:21) − ε . Since ρ ( n ) is a minimizer on F ( n ) for every n ∈ N (see § § (cid:0) S F ( n ) (˜ ρ n ) − S F ( n ) ( ρ ( n ) ) (cid:1) ≥ n ∈ N . (5.2)Taking the limit n → ∞ on the left hand side of (5.2), one obtains exactly the aboveexpression in square brackets for S (˜ ρ ) − S ( ρ ). Since ε > S (˜ ρ ) − S ( ρ )) ≥ . Hence ρ is a minimizer under variations in D . (cid:3) Our next goal is to extend the previous result to minimizers under variations offinite-dimensional compact support, which is defined as follows.
Definition 5.3.
A regular measure ρ on B ( F ) is said to be a minimizer undervariations of finite-dimensional compact support if the inequality (cid:0) S (˜ ρ ) − S ( ρ ) (cid:1) ≥ holds for any regular measure ˜ ρ on B ( F ) satisfying (2.3) with the following property:There exists n ′ ∈ N such that, for all n ≥ n ′ , supp (cid:0) ˜ ρ − ρ (cid:1) ⊂ F ( n ) compact and supp (cid:0) ˜ ρ − ρ (cid:1) ∩ F \ F ( n ) = ∅ . Based on this definition, we may state the following existence result.
Proposition 5.4.
Assume that
L ∈ C b ( F × F ; R +0 ) is of bounded range, and assumethat condition (3.8) holds. Furthermore, assume that the measure ρ defined by (4.7) satisfies condition (B) in Section 5. Then ρ is a minimizer under variations of finite-dimensional compact support.Proof. Assuming that ˜ ρ is a variation of finite-dimensional compact support, thereexists n ′ ∈ N such that K ♯ := supp (cid:0) ˜ ρ − ρ (cid:1) ⊂ F ( n ) is compact for all n ≥ n ′ , implying that ˜ ρ ( K ♯ ) = ρ ( K ♯ ) < ∞ (in virtue of Lemma 5.1).According to Lemma 4.4, the measure ρ given by (4.7) is regular, and by regularityof ρ and ˜ ρ , for ˜ ε > U ⊃ K ♯ open such that ρ ( U \ K ♯ ) < ˜ ε and ˜ ρ ( U \ K ♯ ) < ˜ ε . XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 27
By construction of D , there is some compact set D ∈ D such that K ♯ ⊂ D ⊂ U ( n ′ ) ,where U ( n ′ ) := U ∩ F ( n ′ ) . In particular, ρ ( D \ K ♯ ) < ˜ ε and ˜ ρ ( D \ K ♯ ) < ˜ ε . Applying Lemma 4.11 yields the existence of some relatively compact set V ⊂ F with D ⊂ V ⊂ U ( n ′ ) such that (4.16) holds. Moreover, ρ ( V \ K ♯ ) < ˜ ε and ˜ ρ ( V \ K ♯ ) < ˜ ε . Since ˜ ρ − ρ is a signed measure of finite total variation and compact support, thedifference (2.4) is well-defined (cf. [24, § S (˜ ρ ) − S ( ρ )) = 2 ˆ K ♯ d (˜ ρ − ρ )( x ) ˆ F dρ ( y ) L ( x, y )+ ˆ K ♯ d (˜ ρ − ρ )( x ) ˆ K ♯ d (˜ ρ − ρ )( y ) L ( x, y ) . By adding and subtracting the terms2 ˆ V \ K ♯ d (˜ ρ − ρ )( x ) ˆ F dρ ( y ) L ( x, y ) + ˆ V \ K ♯ d (˜ ρ − ρ )( x ) ˆ K ♯ d (˜ ρ − ρ )( y ) L ( x, y )as well as ˆ V d (˜ ρ − ρ )( x ) ˆ V \ K ♯ d (˜ ρ − ρ )( y ) L ( x, y ) , one easily verifies that (cid:0) S (˜ ρ ) − S ( ρ ) (cid:1) = 2 ˆ V d (˜ ρ − ρ )( x ) ˆ F dρ ( y ) L ( x, y )+ ˆ V d (˜ ρ − ρ )( x ) ˆ V d (˜ ρ − ρ )( y ) L ( x, y ) − " ˆ V d (˜ ρ − ρ )( x ) ˆ V \ K ♯ d (˜ ρ − ρ )( y ) L ( x, y )+ 2 ˆ V \ K ♯ d (˜ ρ − ρ )( x ) ˆ F dρ ( y ) L ( x, y ) + ˆ V \ K ♯ d (˜ ρ − ρ )( x ) ˆ K ♯ d (˜ ρ − ρ )( y ) L ( x, y ) . Since the Lagrangian L is of bounded range (see Definition 3.7), its restriction L ( n ′ ) is of compact range, implying that L ( x, y ) = 0 for all x ∈ V and y / ∈ V ′ for somerelatively compact, open set V ′ ⊂ F ( n ′ ) such that (4.16) holds. Choosing the openset U ⊃ K ♯ suitably, one thus can arrange that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ V \ K ♯ d (˜ ρ − ρ )( x ) ˆ F dρ ( y ) L ( x, y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup x,y ∈ V ′ L ( x, y ) ρ ( V ′ ) | {z } < ∞ (cid:0) | ˜ ρ ( V \ K ♯ ) | + | ρ ( V \ K ♯ ) | (cid:1)| {z } < ˜ ε is arbitrarily small. Applying similar arguments to all summands of the above term insquare brackets, one obtains the estimate (cid:0) S (˜ ρ ) − S ( ρ ) (cid:1) ≥ (cid:26) ˆ V d (˜ ρ − ρ )( x ) ˆ F dρ ( y ) L ( x, y )+ ˆ V d (˜ ρ − ρ )( x ) ˆ V d (˜ ρ − ρ )( y ) L ( x, y ) (cid:27) − ε for any given ε >
0. Proceeding in analogy to the proof of Proposition 5.2 by applyingweak convergence (4.16) together with [6, Theorem 2.8] (for details we refer to the proof of [24, Theorem 4.9]), one can show that the term in curly brackets is greaterthan or equal to zero, up to an arbitrarily small error term. Since ε > (cid:0) S (˜ ρ ) − S ( ρ ) (cid:1) ≥ , which proves the claim. (cid:3) Existence of Minimizers under Variations of Compact Support.
Havingderived the above preparatory results in § ρ is a variation of compact support of the measure ρ satisfying (2.3), the condition | ˜ ρ − ρ | < ∞ yields ρ ( K ) = ˜ ρ ( K ) < ∞ , where thecompact set K ⊂ F is defined by K := supp(˜ ρ − ρ ) (for details see Definition 2.1 andthe explanations in § Lemma 5.5.
Assume that
L ∈ C b ( F × F ; R +0 ) is of bounded range, and assume thatcondition (3.8) holds. Moreover, assume that the measure ρ defined by (4.7) satisfiescondition (B) in Section 5. Then ρ is a minimizer under variations of compact support.Proof. Let ˜ ρ be a variation of compact support. Then the set K := supp(˜ ρ − ρ ) ⊂ F is compact, and ρ ( K ) = ˜ ρ ( K ) < ∞ according to (2.3). Given ˜ ε > ρ , ˜ ρ there exists U ⊃ K open such that ρ ( U ) , ˜ ρ ( U ) < ∞ and ρ ( U \ K ) < ˜ ε and ˜ ρ ( U \ K ) < ˜ ε . Moreover, in view of (4.6), there exists D ∈ D such that ρ ( U \ D ) < ˜ ε and ˜ ρ ( U \ D ) < ˜ ε . Since K ⊂ F is compact, the difference (2.4) is well-defined (cf. [24, § S (˜ ρ ) − S ( ρ )) = 2 ˆ K d (˜ ρ − ρ )( x ) ˆ F dρ ( y ) L ( x, y )+ ˆ K d (˜ ρ − ρ )( x ) ˆ K d (˜ ρ − ρ )( y ) L ( x, y ) . Note that, by definition of D , each D ∈ D is the finite union of finite-dimensionalsubsets of ( F ( n ) ) n ∈ N (cf. § D ∈ D there exists n ′ ∈ N such that D ⊂ F ( n ) for all n ≥ n ′ . Moreover, by construction of D there exists E ∈ D such that D ⊂ E ◦ ⊂ U ( n ) , where U ( n ) = U ∩ F ( n ) , and according to Lemma 4.11 thereexists a relatively compact, open set V ⊂ E ◦ such that D ⊂ V . In particular, ρ ( U \ V ) < ˜ ε , ˜ ρ ( U \ V ) < ˜ ε . Thus adding and subtracting the terms2 ˆ U \ K d (˜ ρ − ρ )( x ) ˆ F dρ ( y ) L ( x, y ) + ˆ U \ K d (˜ ρ − ρ )( x ) ˆ K d (˜ ρ − ρ )( y ) L ( x, y )as well as ˆ U d (˜ ρ − ρ )( x ) ˆ U \ K d (˜ ρ − ρ )( y ) L ( x, y ) , and proceeding in analogy to the proof of Proposition 5.4, we conclude that (cid:0) S (˜ ρ ) − S ( ρ ) (cid:1) ≥ . Hence ρ is indeed a minimizer under variations of compact support. (cid:3) XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 29
Existence of Minimizers under Variations of Finite Volume.
We nowproceed similarly to [24, § § x ∈ F ˆ F L ( x, y ) dρ ( y ) < ∞ . Then we can state the following result.
Theorem 5.6.
Assume that
L ∈ C b ( F × F ; R +0 ) is of bounded range, and assumethat condition (3.8) holds. Furthermore, assume that the measure ρ defined by (4.7) satisfies condition (B) in Section 5. Then ρ is a minimizer under variations of finitevolume.Proof. Assume that ˜ ρ be a variation of finite volume satisfying (2.3). Introducing theset B := supp(˜ ρ − ρ ), we thus obtain ρ ( B ) = ˜ ρ ( B ) < ∞ . Given ˜ ε > ρ , ˜ ρ there exists U ⊃ B open such that ρ ( U \ B ) < ˜ ε and ˜ ρ ( U \ B ) < ˜ ε . Making use of the additional assumption that condition (iv) in § (cid:0) S (˜ ρ ) − S ( ρ ) (cid:1) ≥ , which implies that ρ is a minimizer under variations of finite volume. (cid:3) The remainder of this section is devoted to the derivation of the corresponding ELequations for minimizers under variations of finite volume ( § Derivation of the Euler-Lagrange Equations.
The strategy in [24] was toderive the EL equations in order to prove the existence of minimizers under variationsof finite volume. However, proceeding similar to [24, § § ρ given by (4.7) is a minimizer under variations of finite volume (see Definition 3.5).Under these assumptions, Lemma 4.8 implies that ρ is locally finite. This allows usto proceed similarly to the proof of [23, Lemma 2.3], thus giving rise to correspondingEL equations. For convenience, let us state the latter result in greater generality. Theorem 5.7 ( The Euler-Lagrange equations ) . Let F be topological Hausdorffspace, let ρ be a Borel measure on F (in the sense of [28] , i.e. a locally finite measure)and assume that L : F × F → R +0 is symmetric and lower semi-continuous. If ρ = 0 isa minimizer of the causal variational principle (3.2) , (2.3) under variations of finitevolume, then the Euler-Lagrange equations ℓ | supp ρ ≡ inf x ∈ F ℓ ( x ) (5.3) hold, where the mapping ℓ : F → [0 , ∞ ) is defined by ℓ ( x ) := ˆ F L ( x, y ) dρ ( y ) − s (5.4) for some parameter s ∈ R .Proof. Proceed in analogy to the proof of [23, Lemma 2.3]. (cid:3)
For clarity, we point out that Theorem 5.7 requires that ρ is locally finite. Choosingthe parameter s suitably, one can arrange that the infimum in (5.3) vanishes: Lemma 5.8.
Assume that the measure ρ given by (4.7) is non-zero. Then, under theassumptions of Theorem 5.6, for a suitable choice of s ≥ in (5.4) the Euler-Lagrangeequations (5.3) read ℓ | supp ρ ≡ inf x ∈ F ℓ ( x ) = 0 . (5.5) Proof.
Assuming that ρ = 0, we conclude that supp ρ = ∅ . Moreover, under theassumptions of Theorem 5.6, by Lemma 4.8 we know that ρ is locally finite. Next,from Theorem 5.6 and Theorem 5.7 we deduce that ρ satisfies the EL equations (5.3).By assumption, condition (iv) in § ≤ ˆ F L ( x, y ) dρ ( y ) < ∞ for every x ∈ F . This allows us to choose s ≥ (cid:3) Lemma 5.8 generalizes the results of [23, Section 2] and [24, Section 4] to the infinite-dimensional setting. It remains an open task to prove the existence of a Lagrangian ofbounded range such that the measure ρ in (4.7) is non-zero and satisfies condition (iv)in § Minimizers for Lagrangians Vanishing in Entropy
In Section 5 the results from [24, Section 4] were generalized to the non-locallycompact setting. This raises the question whether it is possible also to weaken theassumption that the Lagrangian is of bounded range (see Definition 3.7) similarlyto Lagrangians decaying in entropy as introduced in [24, Section 5]. It is preciselythe objective of this section to analyze this question in detail. To this end, we firstgeneralize the notion of Lagrangians decaying in entropy ( § ρ obtained in Theorem 4.3 is a minimizer under variations of compactsupport and variations of finite volume (see Definition 3.6 and Definition 3.5).6.1. Lagrangians Vanishing in Entropy.
This subsection is devoted to generalizethe notion of Lagrangians decaying in entropy as introduced in [24, Section 5]. Moreprecisely, in order for the constructions in [24, Section 5] to work, the definition ofLagrangians decaying in entropy (see Definition 2.8 and [24, Definition 5.1]) requiresan unbounded
Heine-Borel metric. On the other hand, any Heine-Borel space (thatis, a topological space endowed with a Heine-Borel metric) is σ -compact and locallycompact, see [42]. In particular, every separable Heine-Borel space X is a second-countable, locally compact Hausdorff space, and hemicompact (see [41, Problem 17I])in view of [11, Exercise 3.8.C]. Accordingly, there is a sequence ( K n ) n ∈ N of compactsubsets of X with K n ⊂ K ◦ n +1 for every n ∈ N such that any compact set K ⊂ X is contained in K n for some n ∈ N and X = S ∞ n =1 K n (also see [4, Lemma 29.8]).Moreover, in view of [4, Theorem 31.5], the space X is Polish. These considerationsmotivate the following procedure. XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 31
For any second-countable, locally compact Hausdorff topological space X , assumethat the Lagrangian L : X × X → R +0 is continuous, symmetric and positive on thediagonal (3.1). Moreover, assume that the measure ˜ ρ on B ( X ) is obtained similarlyto the constructions in [24, § x ∈ X and arbitrary ε >
0, there exists K x,ε ⊂ X compact such that ˆ X \ K x,ε L ( x, y ) d ˜ ρ ( y ) < ε . The results in [42] imply that for any separable, locally compact metric space (
X, d )there is a Heine-Borel metric d HB on X which generates the same topology. In orderto also allow for bounded Heine-Borel metrics in [24, Section 5], it seems preferableto not specify the set K x,ε in [24, eq. (5.2)] and the calculations thereafter in termsof a possibly bounded Heine-Borel metric d HB . For this reason, it seems preferableto formulate the calculations in [24, § Y , we denote the set of allfunctions f : Y → R by F ( Y ), and let F + ( Y ) be the subset of non-negative suchfunctions. Given a second-countable, locally compact Hausdorff space X , by F +0 ( X )we denote the subset of non-negative functions vanishing at infinity in the sense that,for any ε >
0, there exists K ⊂ X compact with f | X \ K < ε (for continuous functionsvanishing at infinity we refer to [10, § VIII.2]).This allows us to generalize the definition of Lagrangians decaying in entropy byrestating condition (c) in Definition 2.8 in the following way: Given the compactexhaustion ( K m ) m ∈ N of X with K m ⊂ K ◦ m +1 for every m ∈ N and X = S ∞ m =1 K m , forevery x ∈ X let N = N ( x ) be the least integer such that x ∈ K m for all m ≥ N . Wenow introduce the sets ( K m ( x )) m ∈ N by K m ( x ) := K m + N − for all m ∈ N . (6.1)Introducing entropy E x ( K m ( x ) , δ ) according to § δ > K m ( x ), and replacing (c) in Definition 2.8 by (c’), we defineLagrangians vanishing in entropy as follows. Definition 6.1.
Let ( X, d ) be a second-countable, locally compact metric space. Thenthe Lagrangian L : X × X → R +0 is said to vanish in entropy if the followingconditions are satisfied: (a) c := inf x ∈ X L ( x, x ) > . (b) There is a compact set K ⊂ X such that δ := inf x ∈ X \ K sup n s ∈ R : L ( x, y ) ≥ c for all y ∈ B s ( x ) o > . (c’) The Lagrangian has the following decay property: Given an exhaustion of X bycompact subsets ( K m ) m ∈ N , there exists f : X × X → R +0 with f ( x, · ) ∈ F +0 ( X ) forevery x ∈ X such that, for every x ∈ X and all m ∈ N , L ( x, y ) ≤ − m f ( x, y ) C x ( m, δ ) for all y ∈ K m ( x ) , where ( K m ( x )) m ∈ N is defined by (6.1) , C x ( m, δ ) := C E x ( K m +2 ( x ) , δ ) for all x ∈ F , m ∈ N and δ > (with entropy E x ( K m ( x ) , δ ) as introduced in § C is given by C := 1 + 2 c < ∞ . As mentioned in [24], we may assume that δ = 1 (otherwise we suitably rescale thecorresponding metric on X ). Let us point out that Definition 6.1, by contrast toDefinition 2.8 (see [24, Definition 5.1]), does not require a Heine-Borel metric and thusallows for more general applications. Definition 2.8 can be considered as a special caseof Definition 6.1.Under the assumptions (a), (b), (c’), for any x ∈ X and ε > K x,ε ⊂ X compact such that ˆ X \ K x,ε L ( x, y ) d ˜ ρ ( y ) < ε . To see this, we make use of the fact that K n ⊂ K ◦ n +1 for all n ∈ N . Given x ∈ X andarbitrary ε >
0, there exists ˜ K x,ε ⊂ X compact with f ( x, y ) < ε/ y / ∈ ˜ K x,ε .Since X is hemicompact, there exists n ∈ N with ˜ K x,ε ⊂ K n . We denote the least suchinteger by N = N ( x, ε ). Then the compact set (cf. [24, eq. (5.2)]) K x,ε := K N ⊂ X (6.2)has the desired property: ˆ X \ K N L ( x, y ) d ˜ ρ ( y ) = ∞ X m = N ˆ K m +1 \ K m L ( x, y ) d ˜ ρ ( y ) ≤ ∞ X m = N sup y ∈ K m +1 L ( x, y ) ˜ ρ ( K m +1 \ K m ) | {z } ≤ C x ( m, ≤ sup y ∈ X \ K N f ( x, y ) ∞ X m = N − m < ε/ . By definition of C x ( m, δ ), we are given ˆ X \ K x,ε L (˜ x, y ) d ˜ ρ ( y ) < ε/ x in a sufficiently small neighborhood of x .Assuming that the Lagrangian is continuous, we proceed similarly to [24] to provethat the same is true for the measures ˜ ρ ( n ) as given by [24, eq. (4.5)] (where themeasures ˜ ρ ( n ) originate in the same manner as in [24, § x ∈ X and ε >
0, we introduce the compact sets A m ( x ) ⊂ X by A m ( x ) := K m +1 ( x ) \ K m ( x ) for all m ≥ N = N ( x, ε ) . Next, regularity of ˜ ρ yields the existence of open sets U m ( x ) ⊃ A m ( x ) with U m ( x ) ⊂ K m +1 ( x ) \ K m − ( x ) such that˜ ρ (cid:0) U m \ (cid:0) K m +1 ( x ) \ K m ( x ) (cid:1)(cid:1) < − m − ε/ m ≥ N In view of [1, Lemma 2.92], for every m ≥ N there exists η m ∈ C c ( U m ( x ); [0 , η m | A m ( x ) ≡
1, implying that L ( x, · ) η m ∈ C c ( U m ( x )) for all m ≥ N . Repeatingthe arguments in [24], we finally arrive at [24, eq. (5.6)], i.e. ˆ X \ K x,ε L (˜ x, y ) d ˜ ρ ( n ) ( y ) < ε and ˆ X \ K x,ε L (˜ x, y ) d ˜ ρ ( y ) < ε (6.4) XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 33 for all ˜ x in a small neighborhood of x and sufficiently large n ∈ N . As a consequence,all results in [24, Section 5] remain valid for Lagrangians decreasing in entropy.The advantage of Definition 6.1 is that it applies to arbitrary second-countable,locally compact metric spaces. In particular, by contrast to Definition 2.8, it neednot be endowed with an unbounded Heine-Borel metric. Furthermore, the concept ofLagrangians vanishing in entropy carries over to possibly non-locally compact metricspaces in the following way. Definition 6.2.
Given a metric space ( X, d ) , the Lagrangian L : X × X → R +0 issaid to vanish in entropy if, for any second-countable, locally compact Hausdorffspace Y ⊂ X , its restriction L| Y × Y : Y × Y → R +0 vanishes in entropy with respect tothe induced metric d Y := d | Y × Y (see Definition 6.1). Preparatory Results.
After these preliminaries we return to causal variationalprinciples in the non-locally compact setting (see Definition 3.4). Accordingly, let X be a separable infinite-dimensional complex Banach space and assume that F ⊂ X is anon-locally compact Polish subspace (with respect to the Fr´echet metric d induced bythe norm on X ). Then ( F , d ) is a separable, complete metric space. In what followswe assume that the Lagrangian L : F × F → R +0 vanishes in entropy (see Definition 6.2and Definition 6.1). Considering the finite-dimensional exhaustion F ( n ) ⊂ F endowedwith the induced metric d n := d | F ( n ) × F ( n ) for all n ∈ N , the explanations in § F ( n ) (with respect to d n ) is compact. Moreover, therestricted Lagrangians L ( n ) : F ( n ) × F ( n ) → R +0 vanish in entropy (see Definition 6.1).As outlined in § n ∈ N , there is some regular Borel measure ρ [ n ] on F ( n ) whichis a minimizer of the corresponding action S ( n ) := S F ( n ) under variations of compactsupport, where S E ( ρ ) := ˆ E dρ ( x ) ˆ E dρ ( y ) L ( x, y )for any E ∈ B ( F ) (cf. [24, § n ∈ N the following Euler-Lagrange equations hold, ℓ [ n ] | supp ρ [ n ] ≡ inf x ∈ F ℓ [ n ] ( x ) = 0 , where the mapping ℓ [ n ] : F → R is defined by ℓ [ n ] ( x ) := ˆ F L ( x, y ) dρ [ n ] ( y ) − . We point out that Lemma 4.1 is applicable to the sequence ( ρ [ n ] ) n ∈ N , implying that ρ [ n ] ( K ) ≤ C K for all n ∈ N . For this reason, we may proceed in analogy to Section 4 by introducing a countableset D ⊂ K ( F ) (see § § ρ [ n ] ) n ∈ N to D m ⊂ F compact with D m ∈ D for all m ∈ N and denotethe resulting diagonal sequence by ( ρ ( k ) ) k ∈ N (cf. (4.3)). Defining the correspondingset function ϕ : D → [0 , ∞ ) by (4.5) and proceeding in analogy to the proof ofTheorem 4.3, we obtain a (possibly trivial) measure ρ on the Borel σ -algebra B ( F ). Lemma 4.4 yields that the resulting measure ρ : B ( F ) → [0 , + ∞ ] is regular. Moreover,the useful results Lemma 4.10 and Lemma 4.11 still apply.In analogy to Remark 4.5, the following remark yields a sufficient condition for themeasure ρ obtained in Theorem 4.3 to be non-zero. Remark 6.3.
Let ( F ( n ) ) n ∈ N be a finite-dimensional approximation of F (see § x ( n ) ∈ F ( n ) and < ε < there exists K ( n ) x,ε ⊂ F ( n ) compact such that ˆ F \ K ( n ) x,ε L ( x ( n ) , y ) dρ ( n ) ( y ) < ε . In view of boundedness of the Lagrangian we introduce the upper bound C < ∞ by C := sup x,y ∈ F L ( x, y ) > . Then the EL equations (3.6) and (3.7) yield ≤ ˆ F L ( x ( n ) , y ) dρ ( n ) = ˆ K ( n ) x,ε L ( x ( n ) , y ) dρ ( n ) + ˆ F \ K ( n ) x,ε L ( x ( n ) , y ) dρ ( n ) , implying that < − ε C ≤ ρ ( n ) ( K ( n ) x,ε ) for sufficiently large n ∈ N . Without loss of generality we may assume that K ( n ) x,ε ∈ D for every n ∈ N . Moreover,we are given K x,ε ( N ) := S Nn =1 K ( n ) x,ε ∈ D for every N ∈ N . Therefore, whenever thereexists N ∈ N such that ρ ( n ) ( K x,ε ( N )) ≥ c for almost all n ∈ N and some c > , themeasure ρ as defined by (4.7) is non-zero. If this holds true for an infinite number ofdisjoints sets ( K x i ,ε ( N i )) i ∈ N , the measure ρ possibly has infinite total volume. The remainder of this section is devoted to the proof that the measure ρ definedby (4.7) is, under suitable assumptions, a minimizer under variations of compact sup-port as well as under variations of finite volume. For non-trivial minimizers, we shallderive the corresponding EL equations (see § Existence of Minimizers.
The aim of this subsection is to prove that, undersuitable assumptions, the measure ρ defined by (4.7) is a minimizer of the causalvariational principle (3.2), (2.3) under variations of finite volume (see Definition 3.5).To this end, we first show that ρ is a minimizer of the causal action under variationsof compact support (see Definition 3.6).In order to show that the measure ρ obtained in Theorem 4.3 is a minimizer undervariations of compact support, we need to assume that condition (iv) in § § x ∈ F ˆ F L ( x, y ) dρ ( y ) < ∞ . Under the additional assumption that the measure ρ obtained in Theorem 4.3 alsosatisfies condition (B) in Section 5, we obtain the following existence result. Lemma 6.4.
Assume that the Lagrangian
L ∈ C b ( F × F ; R +0 ) vanishes in entropy,and that condition (3.8) holds. Moreover, assume that the measure ρ obtained in (4.7) satisfies condition (B) in Section 5, and that condition (iv) in § ρ is aminimizer under variations of compact support. XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 35
Proof.
Assume that ˜ ρ : B ( F ) → [0 , ∞ ] is a regular Borel measure satisfying (2.3) suchthat K := supp(˜ ρ − ρ ) is a compact subset of F . Then the difference of actions (2.4)as given by ( S (˜ ρ ) − S ( ρ )) = 2 ˆ K d (˜ ρ − ρ )( x ) ˆ F dρ ( y ) L ( x, y )+ ˆ K d (˜ ρ − ρ )( x ) ˆ K d (˜ ρ − ρ )( y ) L ( x, y )is well-defined (see the explanations in [24, § § x ∈ F and ˜ ε > R = R ( x, ˜ ε ) such that ˆ F \ B R ( x ) L ( x, y ) dρ ( y ) < ˜ ε/ . (6.5)By continuity of the Lagrangian, there is an open neighborhood U x of x such that ˆ F \ B R ( x ) L ( z, y ) dρ ( y ) < ˜ ε for all z ∈ U x . (6.6)Proceeding in analogy to the proof of [24, Theorem 5.8] by covering the compactset K ⊂ F by a finite number of such neighborhoods U x , . . . , U x L and introducing thebounded set B K := S Lj =1 B R ( x j ), we conclude that ˆ F \ B K L ( x, y ) dρ ( y ) < ˜ ε for all x ∈ K . (6.7)This implies that, by choosing ˜ ε > S (˜ ρ ) − S ( ρ )) = (cid:20) ˆ K d (˜ ρ − ρ )( x ) ˆ B K dρ ( y ) L ( x, y )+ ˆ K d (˜ ρ − ρ )( x ) ˆ K d (˜ ρ − ρ )( y ) L ( x, y ) (cid:21) + 2 ˆ K d (˜ ρ − ρ )( x ) ˆ F \ B K dρ ( y ) L ( x, y )is arbitrarily small. For this reason, it remains to consider the term in square bracketsin more detail. Combining the facts that ρ satisfies condition (B) in Section 5 andthat B K ⊂ F is bounded, Lemma 5.1 implies that ρ ( B K ) < ∞ . Proceeding similarlyto the proof of Lemma 5.5, we deduce that the term in square bracket is bigger thanor equal to zero, up to an arbitrarily small error term. This gives rise to( S (˜ ρ ) − S ( ρ )) ≥ , which proves the claim. (cid:3) Proceeding similarly to the proof of Lemma 6.4, we obtain the following result.
Theorem 6.5.
Assume that the Lagrangian
L ∈ C b ( F × F ; R +0 ) vanishes in entropy,and that condition (3.8) holds. Moreover, assume that the measure ρ obtained in (4.7) satisfies condition (B) in Section 5, and that condition (iv) in § ρ is aminimizer under variations of finite volume.Proof. Assume that ˜ ρ : B ( F ) → [0 , ∞ ] is a regular measure satisfying (2.3). By virtueof Lemma 5.1, the measure ρ is locally finite, implying that ˜ ρ is also a locally finitemeasure (see the explanations after Definition 3.5). Introducing B := supp(˜ ρ − ρ ), we are given ρ ( B ) = ˜ ρ ( B ) < ∞ . Since condition (iv) in § S (˜ ρ ) − S ( ρ )) = 2 ˆ B d (˜ ρ − ρ )( x ) ˆ F dρ ( y ) L ( x, y )+ ˆ B d (˜ ρ − ρ )( x ) ˆ B d (˜ ρ − ρ )( y ) L ( x, y ) . Making use of regularity of ρ and ˜ ρ , for arbitrary ˜ ε > U ⊃ B open such that ρ ( U \ B ) < ˜ ε and ˜ ρ ( U \ B ) < ˜ ε . Approximating U from inside by compact sets K such that ρ ( U \ K ) < ˜ ε and ˜ ρ ( U \ K ) < ˜ ε and proceeding in analogy to the proof of [24, Theorem 5.9] and Lemma 6.4, we finallymay deduce that ( S (˜ ρ ) − S ( ρ )) ≥ , which proves the claim. (cid:3) Theorem 6.5 concludes the existence theory in the non-locally compact setting.6.4.
Derivation of the Euler-Lagrange Equations.
Under the assumptions ofTheorem 6.5, for non-trivial measures ρ = 0 we are able to deduce the correspondingEuler-Lagrange equations. More precisely, in analogy to [23, Lemma 2.3] we obtainthe following result. Theorem 6.6.
Assume that the Lagrangian
L ∈ C b ( F × F ; R +0 ) vanishes in entropy(see Definition 6.2), and that condition (3.8) holds. Moreover, assume that the regularmeasure ρ obtained in (4.7) is non-zero and satisfies condition (B) in Section 5 as wellas condition (iv) in § ℓ | supp ρ ≡ inf x ∈ F ℓ ( x ) = 0 , (6.8) where ℓ ∈ C ( F ) is defined by (5.4) for a suitable parameter s ∈ R +0 .Proof. Under the assumptions of Theorem 6.6, the measure ρ constructed in (4.7) islocally finite in view of Lemma 4.8 and a minimizer under variations of finite volume.Assuming that ρ = 0 and arguing similarly to the proof of Lemma 5.8, Theorem 5.7gives rise to (6.8). (cid:3) Theorem 6.6 generalizes the results of [24, Section 5] to the infinite-dimensionalsetting. It remains an open task to prove the existence of Lagrangians vanishing inentropy such that the measure ρ given by (4.7) is non-zero and satisfies condition (iv)in § Topological Properties of Spacetime
The goal of this section is to derive topological properties of spacetime and to workout a connection to dimension theory. To this end, we let F be a non-locally compactPolish space in the non-locally compact setting (see Definition 3.4). Under suitableassumptions on the Lagrangian (see Theorem 5.6 and Theorem 6.5), the measure ρ obtained in (4.7) is a minimizer of the corresponding variational principle (3.2). Inorder to obtain dimension-theoretical statements on its support, let us first recall XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 37 some basic results from dimension theory ( § § Dimension-Theoretical Preliminaries.
To begin with, let us first point outthat there are several notions of “dimension” of a topological space, among themthe small inductive dimension ind, the large inductive dimension
Ind, the coveringdimension dim, the
Hausdorff dimension dim H and the metric dimension µ dim (fordetails we refer to [3], [12], [29] and [35]). For a separable metric space X , the relationdim X ≤ dim H X (7.1)holds in view of [29, Section VII.4]. Moreover, for every separable metrizable space X we have ind X = Ind X = dim X (see [12, Theorem 4.1.5]), and µ dim Y = dim Y forevery compact metric space Y (see e.g. [3]).For a metric space X , the local dimension dim loc : X → [0 , ∞ ] is given bydim loc ( x ) = inf { dim H ( B ε ( x )) : ε > } (see [9, § H ( A ) = inf { s ≥ H s ( A ) = 0 } is the Hausdorff dimension of A ⊂ X , and H s is the s -dimensional Hausdorff measure(the interested reader is referred to [13, Section 2.10], [29, Chapter VII] and [37]).Whenever X is a separable metric space, then one can show thatdim H ( X ) = sup x ∈ X dim loc ( x ) . Moreover, if X is compact then the supremum is attained (cf. [9, Proposition 2.7]). Definition 7.1.
A normal space X is locally finite-dimensional if for every x ∈ X there exists a normal open subspace U of X such that x ∈ U and dim U < ∞ . See [12,Section 5.5] . In order to apply the above preliminaries to minimizers of the causal variationalprinciple, let us summarize some general topological properties of the support of alocally finite measure µ on a Polish space F in the next statement. Lemma 7.2.
Let X be a Polish space, and assume that µ is a locally finite measureon B ( X ) . Then supp µ ⊂ X is σ -compact, and there exists a locally finite-dimensionalsubspace F being dense in supp µ . Whenever supp µ is hemicompact, then supp µ islocally compact and thus locally finite-dimensional. Moreover, in the latter case thereexists a (Heine-Borel) metric on supp µ such that each bounded subset in supp µ isfinite-dimensional.Proof. According to Lemma B.2, supp µ is a σ -compact separable metric space, andthere is a dense subset F ⊂ supp µ such that each x ∈ F is contained in a compactneighborhood N x . In view of [9, Proposition 2.7] we conclude that dim H N x < ∞ forevery x ∈ supp µ . Thus Definition 7.1 together with (7.1) gives the first statement.Whenever supp µ is hemicompact, it is locally compact according to Lemma B.2.Thus each x ∈ supp µ is contained in a compact neighborhood N x . Making use of [9,Proposition 2.7] we deduce that dim H N x < ∞ . Since x ∈ supp µ is arbitrary andthe interior of N x is open, from Definition 7.1 we obtain that supp µ is locally finite-dimensional. Moreover, due to Lemma B.2, the space supp µ can be endowed with aHeine-Borel metric. Accordingly, whenever B ⊂ supp µ is bounded (with respect to the Heine-Borel metric), its closure is compact. Covering the resulting compact set bya finite number of compact neighborhoods, we conclude that dim H ( B ) < ∞ . (cid:3) Application to Causal Fermion Systems.
In the remainder of this section,we finally apply the previous results to the case of causal fermion systems. To thisend, let H be an infinite-dimensional separable complex Hilbert space. For a givenspin dimension n ∈ N , the set F ⊂ L( H ) (for details see [18, Definition 1.1.1]) is anon-locally compact Polish space (see Theorem A.1 and Lemma 3.3). Assuming thatthe Lagrangian L : F × F → R +0 is symmetric, lower semi-continuous and strictlypositive on the diagonal (3.1), we are exactly in the non-locally compact setting asintroduced in § ρ : B ( F ) → [0 , ∞ ]. Assuming in addition that the Lagrangianis continuous, bounded and of bounded range and that condition (B) in Section 5 issatisfied, the measure ρ is a minimizer of the causal variational principle (3.2), (2.3)under variations of compact support by virtue of Lemma 5.5. Under the additionalassumption that condition (iv) in § ρ is a minimizer ofthe causal variational principle under variations of finite volume due to Theorem 5.6.Under these assumptions, the same is true for Lagrangians vanishing in entropy (seeTheorem 6.4 and Theorem 6.5). As a consequence, we are given a causal fermionsystem ( H , F , ρ ), and spacetime M is defined as the support of the universal measure ρ , M := supp ρ . Combining the results of Lemma 7.2 and Lemma B.2, we arrive at the following mainresults of this section.
Theorem 7.3.
Assume that
L ∈ C b ( F × F ; R +0 ) is of bounded range or vanishes inentropy. Moreover, assume that the measure ρ defined by (4.7) satisfies condition (B) in Section 5. Then spacetime M is σ -compact and contains a locally finite-dimensionaldense subspace. Under the additional assumption that spacetime M is hemicompact,it is a locally finite-dimensional, σ -locally compact Polish space.Proof. Assuming that condition (B) in Section 5 is satisfied, the measure ρ is locallyfinite in view of Lemma 5.1. Henceforth the statement is a consequence of Lemma B.2and Lemma 7.2. (cid:3) In [5] the question is raised whether the support of minimizing measures alwaysis compact. Theorem 7.3 indicates that the support should in general at least be σ -compact.Under the assumption that the measure ρ is locally finite (for sufficient conditionssee Lemma 4.6, Lemma 4.8 and Lemma 5.1), we obtain the following result. Theorem 7.4.
Assume that the measure ρ : B ( F ) → [0 , ∞ ] given by (4.7) is locallyfinite. Then the interior of spacetime M = supp ρ is empty (in the topology of F ).Proof. Assume that M ◦ = ∅ in the topology of F . Then U reg := M ◦ ∩ F reg is openin the relative topology. Since F reg is a Banach manifold (see [25]), it can be coveredby an atlas ( U α , φ α ) α ∈ A for some index set A (cf. [43, Chapter 73]). In particular,each x ∈ U reg is contained in some open set U α , whose image V α := φ α ( U α ) is openin some infinite-dimensional Banach space X α . From Lemma B.2 we know that thereexists a dense subset F ⊂ supp ρ such that each x ∈ F has a compact neighborhood.Given x ∈ F and choosing a compact neighborhood N x ⊂ U α for some α ∈ A , from XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 39 the fact that the mapping φ α is a homeomorphism we conclude that φ α ( N x ) ⊂ X α isa compact neighborhood of φ α ( x ) ∈ X α which contains a non-empty open subset incontradiction to [32, Exercise 14.3]. This gives the claim. (cid:3) Theorem 7.4 generalizes [26, Theorem 3.16] to the infinite-dimensional setting.
Appendix A. Topological Properties of Causal Fermion Systems
The goal of this appendix is to prove the following result:
Theorem A.1.
Let ( H , F , ρ ) be a causal fermion system. Then F is a Polish space. Throughout this section we assume that ( H , F , ρ ) is a causal fermion system ofspin dimension s ∈ N . More precisely, we consider a (possibly infinite-dimensional)separable complex Hilbert space H endowed with a scalar product h . | . i H . Denotingthe set of all bounded linear operators on H by L( H ), we let F ⊂ L( H ) be the subsetconsisting of those operators A ∈ L( H ) which are self-adjoint with respect to the scalarproduct h . | . i H on H and have at most s positive and at most s negative eigenvalues(see [18, § F is separable ( § A.1). Afterwards, we prove that F is completely metrizable( § A.2). The result can be immediately generalized to the case of operators which haveat most p positive and at most q negative eigenvalues.A.1. Separability.
In order to prove separability of F , we employ the following ar-gument: Given an infinite-dimensional, separable complex Hilbert space H , the setof linear operators on H , denoted by L( H ), is a non-separable Banach space (see [31, § § H is bounded, from [31, § K ( H ) ⊂ L( H ) is separable (as well as aBanach space according to [40, Satz II.3.2]). Applying the previous results to a causalfermion system ( H , F , ρ ) shows that K ( H ) is separable. Since each A ∈ F has finiterank (see e.g. [34, Chapter 15]), from [40, Section II.3] we conclude that A is compact,implying that F ⊂ K ( H ). Since L( H ) is metrizable by the Fr´echet metric induced bythe operator norm on L( H ), the set K ( H ) is metrizable, and hence F is separable inview of [1, Corollary 3.5].A.2. Completeness.
The aim of this subsection is to show that F is completelymetrizable with respect to the Fr´echet metric induced by the operator norm on L( H ).To this end, we proceed as follows. Given a sequence of operators ( A n ) n ∈ N in F , ourtask is to prove that its limit A ∈ K ( H ) is self-adjoint (with respect to the scalarproduct h . | . i H on H ) and has at most n positive and at most n negative eigenvalues.A.2.1. Self-Adjointness.
We start by proving that A is self-adjoint in the case of ageneral Hilbert space H . Lemma A.2.
Let ( H, h . | . i H ) be a Hilbert space, and let ( A n ) n ∈ N be a sequence ofself-adjoint operators in L( H ) converging in norm to some A ∈ L( H ) . Then A isself-adjoint. More precisely, endowed with the Fr´echet metric d induced by the operator norm on L( H ), thespace ( F , d ) is a separable, complete metric space. Proof.
For any u, v ∈ H , applying the Cauchy-Schwarz inequality and making use ofthe fact that A n is self-adjoint for every n ∈ N yields |h u | A ∗ v i H − h u | A v i H | = |h A u | v i H − h u | A v i H | = |h A u | v i H − h A n u | v i H + h A n u | v i H − h u | A v i H |≤ k A − A n k L( H ) k u k k v k → n →∞ . This completes the proof. (cid:3)
A.2.2.
Operators, Resolvents and Spectra.
The remainder of this section is dedicatedto the proof that the limit A ∈ L( H ) of a sequence ( A n ) n ∈ N in F (with respect to theoperator norm) has at most s positive and at most s negative eigenvalues. To this end,we will essentially make use of results in [30], which we now briefly recall.For Banach spaces X and Y , by B ( X, Y ) and C ( X, Y ) we denote the set of allbounded and closed operators from X to Y , respectively. Then B ( X, Y ) is a Banachspace, and we let B ( X ) := B ( X, X ) and C ( X ) := C ( X, X ). We denote the domain ofan operator T from X to Y by D ( T ), and its graph G ( T ) is by definition the subsetof X × Y consisting of all elements of the form ( u, T u ) with u ∈ D ( T ). Note that G ( T )is a closed linear subspace of X × Y if and only if T ∈ C ( X, Y ) (see [30, III- § X , Y be complex Banach spaces, and let H be a complex Hilbertspace. For ζ ∈ C and T ∈ C ( X ), we introduce the operator T ζ := T − ζ . Then the resolvent set ρ ( T ) is defined to consist of all ξ ∈ C for which T ζ has aninverse, denoted by R ( ζ ) = R ( ζ, T ) := ( T − ζ ) − . We call R ( ζ, T ) the resolvent of T (see [30, III- §
6] and [36, Definition 8.38]). The spectrum σ ( T ) of T is given by the complementary set of the resolvent set in thecomplex plane, σ ( T ) := C \ ρ ( T ). Note that the spectrum of a compact operator T in a Banach space X has a simple structure analogous to that of an operator in afinite-dimensional space. Namely, for compact operators, each non-zero eigenvalue isof finite multiplicity: Theorem A.3.
Let T ∈ B ( X ) be compact. Then σ ( T ) is a countable set with noaccumulation point different from zero, and each nonzero λ ∈ σ ( T ) is an eigenvalueof T with finite multiplicity.Proof. See [30, Theorem III-6.26]. (cid:3)
Moreover, the spectrum σ ( T ) of a selfadjoint operator T in H is a subset of the realaxis.An isolated point of the spectrum is referred to as isolated eigenvalue [30, III- § T on X , we may state the following remark. Remark A.4.
Let T be a compact operator. Then every complex number λ = 0 belongsto ρ ( T ) or is an isolated eigenvalue with finite multiplicity. See [30, Remark III-6.27] . XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 41
A.2.3.
Projection and Decomposition.
Let
X, Y be Banach spaces, and let M ⊂ X bea linear subspace (or “manifold” in the terminology of [30]). As usual, an idempotentoperator P ∈ B ( X ) ( P = P ) is called a projection , giving rise to the decomposition X = M ⊕ N , (A.1)where M = P X and N = (1 − P ) X are closed linear subspaces of X which are referredto as complementary (see [30, III- § x ∈ X can be uniquely expressedin the form u = u ′ + u ′′ with u ′ ∈ M and u ′′ ∈ N . The vector u ′ is called the projectionof u on M along N , and P is called the projection operator (or simply the projection ) on M along N . Accordingly, the operator 1 − P is the projection on N along M .The range of P is M and the null space of P is N . For convenience we often writedim P for dim M = dim R ( P ), where R ( P ) denotes the range of P . Since P u ∈ M forevery u ∈ X , we have P P u = P u , implying that P is idempotent : P = P .Next, a linear subspace M is said to be invariant under an operator T ∈ B ( X )if T M ⊂ M . In this case, T induces a linear operator T M on M to M , definedby T M u = T u for u ∈ M . the operator T M is called the part of T in M . If there aretwo invariant linear subspaces M, N for T such that X = M ⊕ N , the operator T issaid to be decomposed (or reduced ) by the pair M, N .The notion of the decomposition of T by a pair M, N of complementary subspaces(see (A.1), [30, III-(3.14)]) can be extended in the following way. An operator T issaid to be decomposed according to X = M ⊕ N if P D ( T ) ⊂ D ( T ) , T M ⊂ M , T N ⊂ N , (A.2)where P is the projection on M along N . When T is decomposed as above, the parts T M , T N of T in M, N , respectively, can be defined. Then T M is an operator inthe Banach space M with D ( T M ) = D ( T ) ∩ M such that T M u = T u ∈ M , and T N isdefined similarly.A.2.4. Generalized Convergence.
Let us briefly recall the definition of convergence inthe generalized sense:
Definition A.5.
Let
T, T n ∈ B ( X, Y ) for all n ∈ N . (i) The convergence of ( T n ) n ∈ N to T in the sense of k T n − T k → is called uniformconvergence or convergence in norm . (ii) Given closed operators
T, S ∈ C ( X, Y ) , their graphs G ( T ) , G ( S ) are closed linearsubspaces of the product space X × Y . For two closed linear subspaces M, N of aBanach space Z we let δ ( M, N ) be the smallest number δ such that dist( u, N ) ≤ δ k u k for all u ∈ M .We call ˆ δ ( T, S ) := δ ( G ( T ) , G ( S )) the gap between T and S . If ˆ δ ( T n , T ) → , weshall also say that the operator T n converges to T (or T n → T ) in the generalizedsense .See [30, Chapter III, § and [30, Chapter IV, § . The following theorem establishes a connection between convergence in the gener-alized sense and uniform convergence.
Theorem A.6.
Let
T, T n ∈ C ( X, Y ) for all n ∈ N . If T ∈ B ( X, Y ) , then T n → T inthe generalized sense iff T n ∈ B ( X, Y ) for sufficiently large n and k T n − T k → .Proof. See [30, Theorem IV-2.23]. (cid:3)
A.2.5.
Separation of the Spectrum.
Sometimes it happens that the spectrum σ ( T ) of aclosed operator T contains a bounded part σ ′ separated from the rest σ ′′ in such a waythat a rectifiable, simple closed curve Γ (or, more generally, a finite number of suchcurves) can be drawn so as to enclose an open set containing σ ′ in its interior and σ ′′ inits exterior. Under such a circumstance, we have the following decomposition theorem . Theorem A.7.
Let σ ( T ) be separated into two parts σ ′ , σ ′′ in the way described above.Then we have a decomposition of T according to a decomposition X = M ′ ⊕ M ′′ of thespace (in the sense of (A.2) , cf. [30, III- § ) in such a way that the spectra of theparts T M ′ , T M ′′ coincide with σ ′ , σ ′′ respectively and T M ′ ∈ B ( M ′ ) .Proof. See [30, Theorem III-6.17]. (cid:3)
The proof of Theorem A.7 makes use of the so-called eigenprojection P [ T ] = − πi ˆ Γ R ( ζ, T ) dζ ∈ B ( X ) , (A.3)which is a projection on M ′ = P [ T ] X along M ′′ = (1 − P [ T ]) X . Remark A.8 ( Finite system of eigenvalues ) . Suppose that the spectrum σ ( T ) of T ∈ C ( X ) has an isolated point λ . Obviously σ ( T ) is divided into two separateparts σ ′ , σ ′′ where σ ′ consists of the single point λ ; any closed curve enclosing λ butno other point of σ ( T ) may be chosen as Γ . Then the spectrum of the operator T M ′ described in [30, Theorem III-6.17] (see Theorem A.7) consists of the single point λ .If M ′ is finite-dimensional, λ is an eigenvalue of T . In fact, since λ belongs tothe spectrum of the finite-dimensional operator T M ′ , it must be an eigenvalue of T M ′ and hence of T . In this case, dim M ′ is called the (algebraic) multiplicity of theeigenvalue λ of T .For brevity, a finite collection λ , . . . , λ s of eigenvalues with finite multiplicities willbe called a finite system of eigenvalues . A.2.6.
Continuity of the Spectrum.
This paragraph is devoted that proof that thespectrum of a sequence of operators converging in the generalized sense behaves con-tinuously.
Theorem A.9.
Let T ∈ C ( X ) and let σ ( T ) be separated into two parts σ ′ ( T ) , σ ′′ ( T ) by a closed curve Γ as in § A.2.5 (cf. [30, III- § ). Let X = M ′ ( T ) ⊕ M ′′ ( T ) bethe associated decomposition of X . Then there exists δ > , depending on T and Γ ,with the following properties. Any S ∈ C ( X ) with ˆ δ ( S, T ) < δ has spectrum σ ( S ) likewise separated by Γ into two parts σ ′ ( S ) , σ ′′ ( S ) ( Γ itself running in ρ ( S ) ). Inthe associated decomposition X = M ′ ( S ) ⊕ M ′′ ( S ) , M ′ ( S ) and M ′′ ( S ) are isomor-phic with M ′ ( T ) and M ′′ ( T ) , respectively. In particular, dim M ′ ( S ) = dim M ′ ( T ) , dim M ′′ ( S ) = dim M ′′ ( T ) and both σ ′ ( S ) and σ ′′ ( S ) are nonempty if this is true for T .The decomposition X = M ′ ( S ) ⊕ M ′′ ( S ) is continuous in S in the sense that theprojection P [ S ] of X onto M ′ ( S ) along M ′′ ( S ) tends to P [ T ] in norm as ˆ δ ( S, T ) → .Proof. See [30, Theorem IV-3.16]. (cid:3)
Lemma A.10.
Let ( T n ) n ∈ N be a sequence of compact operators in K ( X ) , and supposethat T n → T in norm for some operator T ∈ K ( X ) . Moreover, let σ ′ ( T ) be a finitesystem of m eigenvalues, separated from the rest σ ′′ ( T ) of σ ( T ) by a closed curve Γ inthe manner of [30, III- § . Then σ ( T n ) is separated by Γ into σ ′ ( T n ) , σ ′′ ( T n ) such XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 43 that each σ ′ ( T n ) also consists of m eigenvalues of T n , provided that n is sufficientlylarge.Proof. Since T is a compact operator, Remark A.4 states that each non-zero eigenvalueof T is isolated (see [30, Remark III-6.27]). For this reason, each non-zero eigenvalueof T as well as all positive or all non-zero eigenvalues of T can be enclosed by aclosed curve Γ running in ρ ( T ) as described in § A.2.5 (cf. [30, III- § σ ′ ( T ), and the set of eigenvalues without Γby σ ′′ ( T ); thus Γ encloses a finite system of eigenvalues. By virtue of Theorem A.7,we have decomposition of T according to X = M ′ ⊕ M ′′ such that the spectra ofthe parts T M ′ , T M ′′ coincide with σ ′ ( T ), σ ′′ ( T ). Since T n , T ∈ B ( X ) for all n ∈ N ,Theorem A.6 ensures that T n → T in the generalized sense, i.e. ˆ δ ( T n , T ) → n → ∞ . Hence we can apply Theorem A.9, giving rise to some N ∈ N such that Γseparates the spectra σ ( T n ) into two parts σ ′ ( T n ), σ ′′ ( T n ) for all n ≥ N . Consideringthe corresponding decomposition X = M ′ ( T n ) ⊕ M ′′ ( T n ) for all n ≥ N , by virtue ofTheorem A.9 there exist isomorphisms M ′ ( T n ) ≃ M ′ ( T ) for all n ≥ N . In particular,we obtain dim M ′ ( T n ) = dim M ′ ( T ) for all n ≥ N , and the decomposition X = M ′ ( T n ) ⊕ M ′′ ( T n ) is continuous in n in the sense that the projection P [ T n ] (see (A.3))of X onto M ′ ( T n ) along M ′′ ( T n ) tends to P [ T ] in norm as ˆ δ ( T n , T ) → T within Γ coincides with the algebraic multiplicity of eigenvalues of T n within Γ forall n ≥ N . (cid:3) Making use of continuity of the spectrum according to Lemma A.10, we may derivethe following result.
Lemma A.11.
Let H be a Hilbert space, and let ( A n ) n ∈ N be a sequence of self-adjointcompact operators in K ( H ) such that each operator A n has at most s positive and atmost s negative eigenvalues. If ( A n ) n ∈ N converges in norm to some A ∈ L( H ) , then A is also a self-adjoint compact operator which has at most s positive and at most s negative eigenvalues.Proof. Given a sequence ( A n ) n ∈ N in K ( H ) with A n → A in L( H ), the fact that K ( H )is a closed subspace of L( H ) implies that the limit A is a compact operator. Thusin view of Theorem A.3 and Remark A.4, each non-zero eigenvalue of A is isolatedand has finite multiplicity, and according to Remark A.8 the non-zero eigenvalues of A form a finite system of isolated eigenvalues. In particular, there is a closed curve Γ asdescribed in Paragraph A.2.5 (cf. [30, III- § A . Now assume that A has m > s positive (negative) eigenvalues. ThenLemma A.10 yields the existence of some N ∈ N such that the spectrum σ ( A n ) isseparated by Γ into two parts σ ′ ( A n ) within Γ and σ ′′ ( A n ) without Γ for all n ≥ N . Asa consequence, σ ′ ( A n ) consists of m > s positive (negative) eigenvalues for all n ≥ N in contradiction to the fact that A n has at most s positive and at most s negativeeigenvalues for all n ∈ N . Hence A ∈ K ( H ) is a selfadjoint operator which has atmost s positive and at most s negative eigenvalues. This concludes the proof. (cid:3) A.2.7.
Application to Causal Fermion Systems.
After these preparations, we finallyare in the position to prove Theorem A.1.
Proof of Theorem A.1.
Let ( H , F , ρ ) be a causal fermion system. Separability of F follows from § A.1. By virtue of Lemma A.11, we conclude that F ⊂ L( H ) is closed. Since L( H ) is a complete metric space with respect to the Fr´echet metric induced bythe operator norm, we conclude that F is completely metrizable. Taken together, F isa separable, completely metrizable space, and thus Polish [31, Definition (3.1)]. (cid:3) More precisely, the space ( F , d ) is a complete metric space, where d is the Fr´echetmetric induced by the operator norm on L( H ) (cf. [2, § Appendix B. Support of Locally Finite Measures on Polish Spaces
In this section we derive useful topological properties concerning the support oflocally finite measures (or Borel measures in the sense of [28]) on Polish spaces (seeLemma B.2 below). To begin with, let us recall the following preparatory result.
Proposition B.1.
Let X be Polish and µ a finite measure on B ( X ) . Then A ⊂ X is µ -measurable if and only if there exists a σ -compact set F ⊂ A with µ ( A \ F ) = 0 .Proof. See [31, Theorem (17.11)]. (cid:3)
Moreover, based on [8, Chapter IX], a Borel measure (in the sense of [28]) on atopological Hausdorff space X is said to be moderated if X is the union of countablymany open subsets of finite µ -measure (see [10, Chapter VIII]). (Since open sets aremeasurable, every moderated measure is σ -finite.) We point out that, due to Ulam’stheorem [10, Satz VIII.1.16], every Borel measure on a Polish space is regular andmoderated. (Due to Meyer’s theorem, the same is true for Borel measures on Souslinspaces, see [10, Satz VIII.1.17].) As a consequence, we may derive useful properties ofthe support of Borel measures on Polish spaces, as the following lemma shows. Lemma B.2.
Let X be a Polish space, and assume that µ is a Borel measure on B ( X ) .Then supp µ ⊂ X is a σ -compact topological space. Moreover, there exists a densesubset F ⊂ supp µ such that each x ∈ F has a compact neighborhood in supp µ .Whenever supp µ is hemicompact, one can arrange that supp µ is a Polish space whichhas the Heine-Borel property. The corresponding Heine-Borel metric can be chosenlocally identical to a complete metric on X (in the relative topology of supp µ ).Proof. We make essentially use of the fact that the measure µ is moderated in view ofUlam’s theorem. As a consequence, there is a sequence of sets ( U n ) n ∈ N with U n ⊂ X open and µ ( U n ) < ∞ for all n ∈ N such that S n ∈ N U n = X . From the fact thatopen subsets of Polish spaces are Polish (in the relative topology, see [4, §
26] or [31,Theorem (3.11)]), we conclude that each set U n ⊂ X is Polish. Thus each µ | U n is afinite Borel measure on a Polish space. Due to [7, Proposition 7.2.9], every (finite)Borel measure on a separable metric space has support, implying that µ | U n ( U n \ supp µ | U n ) = 0 . In view of Proposition B.1, we conclude that supp µ | U n is contained in a σ -compactset F n ⊂ U n , and thus supp µ | U n is σ -compact for every n ∈ N . From this we deducethat S n ∈ N supp µ | U n is σ -compact. Making use of the fact that subsets of σ -compactspaces are σ -compact, we conclude thatsupp µ ⊂ [ n ∈ N supp µ | U n is σ -compact(where µ has support in view of [10, Lemma VIII.2.15]). Since supp µ ⊂ X is closed(see [10, § VIII.2.5]), the support supp µ is Polish and thus Baire (cf. § XISTENCE OF MINIMIZERS IN THE INFINITE-DIMENSIONAL SETTING 45 F ⊂ supp µ such that each x ∈ F has a compact neighborhood in supp µ .Assuming that supp µ is hemicompact (see for instance [41, 17I]), then supp µ islocally compact in view of [11, Exercise 3.4.E] and thus a complete σ -locally compactspace (in the sense of [39]). In this case, due to [42, Theorem 2’] and the explanationsin [24, Section 3], the space supp µ is metrizable by a Heine-Borel metric which is(Cauchy) locally identical to a complete metric on X ; endowed with such a metric,the space supp µ has the Heine-Borel property, i.e. each closed bounded subset (withrespect to the Heine-Borel metric) is compact. (cid:3) Acknowledgments:
C. L. gratefully acknowledges support by the “Studienstiftung desdeutschen Volkes.”
References
1. C.D. Aliprantis and K.C. Border,
Infinite Dimensional Analysis: A hitchhiker’s guide , third ed.,Springer, Berlin, 2006.2. H.W. Alt,
Lineare Funktionalanalysis. Eine anwendungsorientierte Einf¨uhrung , F¨unfte,¨uberarbeitete Auflage, Springer-Verlag, 2006.3. A.V. Arkhangel’skii and V.V. Fedorchuk,
General Topology I: Basic Concepts and ConstructionsDimension Theory , vol. 17, Springer Science & Business Media, 2012.4. H. Bauer,
Measure and Integration Theory , De Gruyter Studies in Mathematics, vol. 26, Walterde Gruyter & Co., Berlin, 2001, Translated from the German by Robert B. Burckel.5. Y. Bernard and F. Finster,
On the structure of minimizers of causal variational principles in thenon-compact and equivariant settings , Advances in Calculus of Variations (2014), no. 1, 27–57.6. P. Billingsley, Convergence of Probability Measures , second ed., Wiley Series in Probability andStatistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999, A Wiley-Interscience Publication.7. V.I. Bogachev,
Measure Theory. Vol. I, II , Springer-Verlag, Berlin, 2007.8. N. Bourbaki,
Integration. II. Chapters 7–9 , Elements of Mathematics (Berlin), Springer-Verlag,Berlin, 2004, Translated from the 1963 and 1969 French originals by Sterling K. Berberian.9. J. Dever,
Local Hausdorff measure , arXiv preprint arXiv:1610.00078 (2016).10. J. Elstrodt,
Maß- und Integrationstheorie , fourth ed., Springer-Lehrbuch. [Springer Textbook],Springer-Verlag, Berlin, 2005, Grundwissen Mathematik [Basic Knowledge in Mathematics].11. R. Engelking,
General Topology , second ed., Sigma Series in Pure Mathematics, vol. 6, Helder-mann Verlag, Berlin, 1989, Translated from the Polish by the author.12. ,
Theory of Dimensions Finite and Infinite , Sigma Series in Pure Mathematics, vol. 10,Heldermann Verlag, Lemgo, 1995.13. H. Federer,
Geometric Measure Theory , Die Grundlehren der mathematischen Wissenschaften,Band 153, Springer-Verlag New York Inc., New York, 1969.14. F. Finster,
The Principle of the Fermionic Projector , AMS/IP Studies in Advanced Mathematics,vol. 35, American Mathematical Society, Providence, RI; International Press, Somerville, MA,2006.15. ,
A variational principle in discrete space-time: existence of minimizers , Calc. Var. PartialDifferential Equations (2007), no. 4, 431–453.16. , Causal variational principles on measure spaces , J. Reine Angew. Math. (2010),141–194.17. ,
Perturbative quantum field theory in the framework of the fermionic projector , J. Math.Phys. (2014), no. 4, 042301, 53.18. , The Continuum Limit of Causal Fermion Systems , Fundamental Theories of Physics,vol. 186, Springer, 2016, From Planck scale structures to macroscopic physics.19. F. Finster, A. Grotz and D. Schiefeneder,
Causal fermion systems: a quantum space-time emergingfrom an action principle , (2012), 157–182.
20. F. Finster and M. Jokel,
Causal fermion systems: An elementary introduction to physical ideasand mathematical concepts , Progress and Visions in Quantum Theory in View of Gravity (2020),63–92.21. F. Finster and N. Kamran,
Complex structures on jet spaces and bosonic Fock space dynamicsfor causal variational principles , arXiv preprint arXiv:1808.03177 (2018).22. F. Finster and J. Kleiner,
Causal fermion systems as a candidate for a unified physical theory ,Journal of Physics: Conference Series (2015), 012020.23. ,
A Hamiltonian formulation of causal variational principles , Calc. Var. Partial DifferentialEquations (2017), no. 3, Paper No. 73, 33.24. F. Finster and C. Langer, Causal variational principles in the sigma-locally compact setting:Existence of minimizers , arXiv preprint arXiv:2002.04412 (2020).25. F. Finster and M. Lottner,
Banach manifold structure and jet spaces for infinite-dimensionalcausal fermion systems , in preparation.26. F. Finster and D. Schiefeneder,
On the support of minimizers of causal variational principles ,Arch. Ration. Mech. Anal. (2013), no. 2, 321–364.27. S.A. Gaal,
Point Set Topology , Pure and Applied Mathematics, Vol. XVI, Academic Press, NewYork-London, 1964.28. R.J. Gardner and W.F. Pfeffer,
Borel measures , Handbook of set-theoretic topology (1984), 961–1043.29. W. Hurewicz and H. Wallman,
Dimension Theory , Princeton Mathematical Series, v. 4, PrincetonUniversity Press, Princeton, N. J., 1941.30. T. Kato,
Perturbation Theory for Linear Operators , Classics in Mathematics, Springer-Verlag,Berlin, 1995, Reprint of the 1980 edition.31. A.S. Kechris,
Classical Descriptive Set Theory , Graduate Texts in Mathematics, vol. 156,Springer-Verlag, New York, 1995.32. H. K¨onig,
Measure and Integration: An Advanced Course in Basic Procedures and Applications ,Springer Science & Business Media, 2009.33. G. K¨othe,
Topological Vector Spaces. I , Translated from the German by D. J. H. Garling. DieGrundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag New York Inc., NewYork, 1969.34. R. Meise and D. Vogt,
Introduction to Functional Analysis , Oxford Graduate Texts in Mathe-matics, vol. 2, The Clarendon Press, Oxford University Press, New York, 1997.35. J.R. Munkres,
Topology , second ed., Prentice Hall, Inc., Upper Saddle River, NJ, 2000.36. M. Renardy and R.C. Rogers,
An Introduction to Partial Differential Equations , Texts in AppliedMathematics, vol. 13, second ed., Springer-Verlag, New York, 2004.37. C.A. Rogers,
Hausdorff Measures , Cambridge Mathematical Library, Cambridge University Press,Cambridge, 1998, Reprint of the 1970 original, With a foreword by K. J. Falconer.38. L. Schwartz,
Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures , Pub-lished for the Tata Institute of Fundamental Research, Bombay by Oxford University Press,London, 1973, Tata Institute of Fundamental Research Studies in Mathematics, No. 6.39. L.A. Steen and J.A. Seebach, Jr.,
Counterexamples in Topology , Dover Publications, Inc., Mineola,NY, 1995, Reprint of the second (1978) edition.40. D. Werner,
Funktionalanalysis , extended ed., Springer-Verlag, Berlin, 2000.41. S. Willard,
General Topology , Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills,Ont., 1970.42. R. Williamson and L. Janos,
Constructing metrics with the Heine-Borel property , Proc. Amer.Math. Soc. (1987), no. 3, 567–573.43. E. Zeidler,
Nonlinear Functional Analysis and its Applications. IV: Applications to mathematicalphysics , Springer-Verlag, New York, 1988, Translated from the German and with a preface byJuergen Quandt.
Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, D-93040 Regensburg, Germany
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