Existence of minimizers for causal variational principles on compact subsets of momentum space in the homogeneous setting

We prove the existence of minimizers for the causal action in the class of negative definite measures on compact subsets of momentum space in the homogeneous setting under several side conditions (constraints). The method is to employ Prohorov’s theorem. Given a minimizing sequence of negative definite measures, we show that, under suitable side conditions, a unitarily equivalent subsequence thereof is bounded. By restricting attention to compact subsets, from Prohorov’s theorem we deduce the existence of minimizers in the class of negative definite measures.

In the physical theory of causal fermion systems, spacetime and the structures therein are described by a minimizer of the so-called causal action principle (for an introduction to the physical background and the mathematical context, we refer the interested reader to the textbook [15], the survey articles [17,18] as well as the web platform [1]).Given a causal fermion system (H, F, dρ) together with a non-negative function L : F × F → R + 0 := [0, ∞) (the Lagrangian), the causal action principle is to minimize the action S defined as the double integral over the Lagrangian S(ρ) = ˆF dρ(x) ˆF dρ(y) L(x, y) under variations of the measure dρ within the class of regular Borel measures on F under suitable side conditions.In order to work out the existence theory for minimizers, causal variational principles evolved as a mathematical generalization of the causal action principle [13,19].The aim of the present paper is to give an alternative proof for the existence of minimizers for causal variational principles restricted to compact subsets in the homogeneous setting.
In order to put the present paper into the mathematical context, in [11] it was proposed to formulate physics by minimizing a new type of variational principle in spacetime.The suggestion in [11,Section 3.5] led to the causal action principle in discrete spacetime, which was first analyzed mathematically in [12].A more general and systematic inquiry of causal variational principles on measure spaces was carried out in [13].In [13,Section 3] the existence of minimizers for variational principles in indefinite inner product spaces is proven in the special case that the total spacetime volume as well as the number of particles therein are finite.Under the additional assumption that the kernel of the fermionic projector is homogeneous in the sense that it only depends on the difference of two spacetime points, variational principles for homogeneous systems were considered in [13,Section 4] in order to deal with an infinite number of particles in an infinite spacetime volume.More precisely, the main advantage in the homogeneous setting is that it allows for Fourier methods, thus giving rise to a natural correspondence between position and momentum space.As a consequence, one is led to minimize the causal action by varying in the class of negative definite measures, and the existence of minimizers on bounded subsets of momentum space is proven in [13,Theorem 4.2].The aim of this paper is to give an alternative proof of this existence result for compact subsets.In addition, the result is stated for additional side conditions (see Section 4) which were not considered in [13].
The paper is organized as follows.In Section 2 we first outline some mathematical preliminaries ( §2.1) and afterwards recall causal variational principles in infinite spacetime volume ( §2.2).In order to put the causal variational principles into the context of calculus of variations, in Section 3 we first introduce so-called operator-valued measures ( §3.1); afterwards, we consider variational principles on compact subsets of momentum space in the homogeneous setting ( §3.2).In Section 4, we prove the existence of minimizers for the causal variational principle on compact subsets in the class of negative definite measures (Theorem 4.1).To this end we first show that, under appropriate side conditions, minimizing sequences of negative definite measures are bounded with respect to the total variation ( §4.1).We then state a preparatory result which ensures the existence of weakly convergent subsequences ( §4.2).This allows us to prove our main result ( §4.3).Afterwards we show that the main result also holds in the case that a boundedness constraint is imposed ( §4.4).In this way, we give an alternative proof of [13, Theorem 4.2] (Theorem 4.11).In the appendix we give a possible explanation for the side conditions under consideration (Appendix A).

Mathematical Preliminaries and Notation.
To begin with, let us compile some fundamental definitions being of central relevance throughout this paper.For details we refer the interested reader to [4], [23] and [28].Unless specified otherwise, we always let n ≥ 1 be a given integer.
Definition 2.2.Let V be a finite-dimensional complex vector space, endowed with an indefinite inner product As usual, by L(V ) we denote the set of (bounded) linear operators on a complex (finite-dimensional) vector space V of dimension n ∈ N. The adjoint of A ∈ L(V ) with respect to the Euclidean inner product .| .on V ≃ C n is denoted by A † .On the other hand, whenever (V, ≺ .| .≻) is an indefinite inner product space, unitary matrices and the adjoint A * (with respect to ≺ .| .≻) are defined as follows.
Definition 2.3.Let ≺ .| .≻ be an indefinite inner product on V ≃ C n , and let S be the associated invertible hermitian matrix determined by [23, eq.(2.1.1)], Then for every A ∈ L(V ), the adjoint of A (with respect to We remark that every non-negative matrix (with respect to ≺ .| .≻) is self-adjoint (with respect to ≺ .| .≻) and has a real spectrum (cf.[23,Theorem 5.7.2]).Moreover, the adjoint A * of A ∈ L(V ) satisfies the relation in view of [23, eq. (4.1.3)](where A † denotes the adjoint with respect to .| .and A * the adjoint with respect to ≺ .| .≻).For details concerning self-adjoint operators (with respect to ≺ .| .≻) we refer to [28] and the textbook [4].
In the remainder of this paper we will restrict attention exclusively to indefinite inner product spaces (V, ≺ .| .≻) with V ≃ C 2n for some n ∈ N. It is convenient to work with a fixed pseudo-orthonormal basis (e i ) i=1,...,2n of V in which the inner product has the standard representation with a signature matrix S, where .| .C 2n denotes the standard inner product on C 2n .The signature matrix can be regarded as an operator on V , where Symm V denotes the set of symmetric matrices on V with respect to the spin scalar product (also cf.[13, proof of Lemma 3.4]).Without loss of generality we may assume that e i = (0, . . ., 0, 1, 0, . . ., 0) T for all i = 1, . . ., 2n.
In what follows, we denote Minkowski space by M ≃ R 4 and momentum space by M ≃ R 4 .Identifying M with Minkowski space M, the Minkowski inner product (of signature (+, −, −, −)) can be considered as a mapping ., .
In the remainder of this paper, let K ⊂ M be a compact subset.By B( K) we denote the Borel σ-algebra on K.The class of finite complex measures on K is denoted by M C ( K).By C c ( M) we denote the set of continuous functions on M with compact support, whereas C b ( M) and C 0 ( M) indicate the sets of continuous functions on M which are bounded or vanishing at infinity, respectively.Since K is compact, the sets C c ( K) and C b ( K) coincide.By L 1 loc (M) we denote the set of locally integrable functions on M with respect to Lebesgue measure, denoted by dµ.Unless otherwise specified, we always refer to locally finite measures on the Borel σ-algebra as Borel measures in the sense of [22].A Borel measure is said to be regular if it is inner and outer regular.Inner regular Borel measures are referred to as Radon measures [8].
2.2.Variational Principles in Infinite Spacetime Volume.Before entering variational principles in infinite spacetime volume, let us briefly recall the concept of a Dirac sea as introduced by Paul Dirac in his paper [6].In this article, he assumes that "(...) all the states of negative energy are occupied except perhaps a few of small velocity.(...) Only the small departure from exact uniformity, brought about by some of the negative-energy states being unoccupied, can we hope to observe.(...) We are therefore led to the assumption that the holes in the distribution of negative-energy electrons are the [positrons]."Dirac made this picture precise in his paper [7] by introducing a relativistic density matrix R(t, x; t ′ , x ′ ) with (t, x), In analogy to Dirac's original idea, in [10] the kernel of the fermionic projector is introduced as the sum over all occupied wave functions for spacetime points x, y ∈ M as outlined in [14].A straightforward calculation shows that (see e.g.[16, §4.1]) the kernel of the fermionic projector takes the form (where δ denotes Dirac's delta distribution and Θ is the Heaviside function).We refer to P (x, y) as the (unregularized) kernel of the fermionic projector of the vacuum (cf.[15, eq.(1.2.20) and eq.(1.2.23)] as well as [11, eq. (4.1.1)];this object already appears in [9]).We also refer to (2.3) as a completely filled Dirac sea.The kernel of the fermionic projector (2.3) is the starting point for the analysis in [13,Section 4].
In order to deal with systems containing an infinite number of particles in an infinite spacetime volume, the main simplification in [13] is to assume that the kernel of the fermionic projector (2.3) is homogeneous in the sense that P (x, y) only depends on the difference vector y − x for all spacetime points x, y ∈ M. The underlying homogeneity assumption P (x, y) = P (y − x) for all x, y ∈ M is referred to as "homogeneous regularization of the vacuum" (cf.[11, eq.(4.1. 2)] and the explanations thereafter; also see [15,Assumption 3.3.1]).Introducing ξ = ξ(x, y) := y − x for all x, y ∈ M and for all k ∈ M, the fermionic projector (2.3) can be written as a Fourier transform, (for details concerning the Fourier transform we refer to [21]).In order to arrive at a measure-theoretic framework, it is convenient to regard P (k) d 4 k/(2π) 4 as a Borel measure dν on M, taking values in L(V ).In particular, the measure has the remarkable property that −dν is positive in the sense that with respect to the "spin scalar product" ≺ .| .≻ on C 4 introduced in §2.1. 1 In order to avoid ultraviolet problems, caused by measures of the form (2.4), one is led to restrict attention to compact subsets of momentum space [13].Moreover, generalizing (C 4 , ≺ .| .≻) to some indefinite inner product space (V, ≺ .| .≻) 1 In order to see this, we make use of the fact that the Dirac matrices anti-commute, i.e.
Thus for every k ∈ M with k = (k 0 , k), the operators p±( k) given by [30, eq.(2.13)] satisfy Therefore, positivity (2.5) is a consequence of the corresponding behavior of the operator (/ k + m).
Definition 2.4.A vector-valued Borel measure dν on a compact set K ⊂ M taking values in L(V ) is called a negative definite measure on K with values in L(V ) whenever d ≺ u | −ν u ≻ is a positive finite measure for all u ∈ V .By Ndm we denote the class of negative definite measures on K taking values in L(V ).
In terms of a negative definite measure dν, the kernel of the fermionic projector is then introduced by for all ξ ∈ M .
In order to clarify the dependence on dν, we also write P [ν].For every ξ ∈ M, the closed chain is defined by A(ξ) := P (ξ) P (−ξ).In order to emphasize that the closed chain depends on dν, we also write A[ν].According to [13, eq.(3.7)], the spectral weight |A| of an operator A ∈ L(V ) is given by the sum of the absolute values of the eigenvalues of A, where by λ i we denote the eigenvalues of A, counted with algebraic multiplicities.In analogy to [13, eq.(3.8)], for every ξ ∈ M the Lagrangian is introduced via Defining the action S according to [13, eq.(4.5)] by the causal variational principle in the homogeneous setting is to minimize S(ν) by suitably varying dν in Ndm .
Introducing the functional T by the main result in [13, Section 4] can be stated as follows (see [13,Theorem 4.2]): Theorem 2.5.Let (dν k ) k∈N be a sequence of negative definite measures on the bounded set K ⊂ M such that the functional T is bounded by some constant C > 0, i.e.
Then there is a subsequence (dν k ℓ ) ℓ∈N as well as a sequence of unitary transformations converge weakly to a negative definite measure dν with the properties Theorem 2.5 is stated as a compactness result.Applying it to a minimizing sequence yields statements similar to [13, Theorem 2.2 and Theorem 2.3], asserting that the functional S attains its minimum.
Given a negative definite measure dν, the complex measure we define integration with respect to negative definite measures as follows: Definition 2.6.Let (V, ≺ .| .≻) be an indefinite inner product space and let dν be a negative definite measure.Moreover, let f : K → C be a bounded Borel measurable function.For all u, v ∈ V , integration with respect to dν is defined by A similar definition in terms of operator-valued measures is stated below (see Definition 3.6).For a connection to spectral theory we refer to [29,Chapter 31].

Causal Variational Principles in the Homogeneous Setting
3.1.Operator-Valued Measures.In order to deal with causal variational principles in the homogeneous setting in sufficient generality, this subsection is devoted to put the definition of negative definite measures (see Definition 2.4) into the context of calculus of variations.More precisely, as explained in §2.2, the variational principle as introduced in [13, Section 4] is to minimize the causal action S in the class of negative definite measures.Unfortunately, in view of (2.5), the set of negative definite measures does not form a vector space, whereas in calculus of variations one usually considers functionals on a real, locally convex vector space (for details we refer to [33,Section 43.2]).Hence in order to obtain a suitable framework, we first introduce operator-valued measures, which can be regarded as a generalization of negative definite measures, thus providing the basic structures required for the calculus of variations (see Lemma 3.3 below).Concerning the connection to vector-valued measures we refer to [5].
Then operator-valued measures on a compact subset K ⊂ M with values in L(V ) are introduced as a generalization of negative definite measures (see Definition 2.4) in the following way: Whenever K and V are understood, the class of operator-valued measures on K with values in L(V ) shall be denoted by Ovm.
In what follows, the variation of an operator-valued measure plays a central role: Definition 3.2.Given an operator-valued measure dω ∈ Ovm, the variation of dω, denoted by d|ω|, is defined by where d | .| denotes the variation of a complex measure.Moreover, the total variation of dω, denoted by d ω , is given by We point out that the variation as given by Definition 3.2 crucially depends on the pseudo-orthogonal (e i ) i=1,...,2n basis of V .Nevertheless, the set of operator-valued measures Ovm is a Banach space with respect to the total variation: d ≺ e i | (ω k − ω m ) e j ≻ = 0 for all i, j = 1, . . ., 2n .
Consequently, each sequence (d ≺ e i | ω k e j ≻) k∈N is a Cauchy sequence of complex measures in M C ( K) for all i, j ∈ {1, . . ., 2n}.Since M C ( K) is a complex Banach space with respect to the total variation d • in virtue of [8, Aufgabe VII.1.7],there is a complex measure dω i,j ∈ M C ( K), being the unique limit of (d ≺ e i | ω k e j ≻) k∈N for all i, j ∈ {1, . . ., 2n}.
Next, for all i, j ∈ {1, . . ., 2n}, the complex measures dω i,j in M C ( K) give rise to an operator-valued measure dω on K with values in L(V ) in such a way that, for all i, j = 1, . . ., 2n, we are given d ≺ e i | ω e j ≻ = dω i,j .More precisely, defining the operator ω(Ω) ∈ L(V ) for any Ω ∈ B( K) by we obtain a mapping dω : for all i, j ∈ {1, . . ., 2n}.Since (e i ) i=1,...,2n is a basis of V , for any Ω ∈ B( K) and arbitrary elements u = 2n i=1 α i e i , v = 2n j=1 β j e j ∈ V we arrive at The fact that M C ( K) is a complex Banach space implies that This shows that dω ∈ Ovm is an operator-valued measure in view of Definition 3.1.Thus (Ovm, d • ) is a complex Banach space with respect to the norm d • defined by (3.1).Since each norm induces a corresponding Fréchet metric, (Ovm, d • ) can be regarded as a metric space.In particular, each complex vector space is a real one, and each Banach space is locally convex.This completes the proof.
Remark 3.4.The set of negative definite measures Ndm clearly is a subset of the vector space Ovm.However, Ndm itself is not a vector space (see [27,Remark 5.6]), but a cone, i.e. a closed subset under multiplication with positive real numbers.
Next, let us introduce the support of operator-valued measures as follows: Definition 3.5.We define the support of an operator-valued measure dω in Ovm as the support of its variation measure d|ω|, i.e.
Since d|ω| is a locally finite measure on a locally compact Polish space, we conclude that d|ω| is regular and has support, d|ω|( K \ supp d|ω|) = 0.In a similar fashion, following [3, Definition 7.1.5],an operator-valued measure dω is called regular if and only if d|ω| is regular.Moreover, the measure dω is said to be tight if for every ε > 0 there is a compact set K ε ⊂ K such that d|ω|( K \ K ε ) < ε (cf.[3,Definition 7.1.4]).Clearly, whenever K ⊂ M is compact, every operator-valued measure on K is tight.Definition 3.6.In analogy to negative definite measures (see Definition 2.6), for any bounded Borel measurable function f : K → C we define integration with respect to operator-valued measures dω by Let us finally state the definition of weak convergence of operator-valued measures, which will be required later on (see §4.3 below).Definition 3.7.We shall say that a sequence of operator-valued measures (dω k ) k∈N in Ovm converges weakly to some operator-valued measure dω if and only if We write symbolically dω k ⇀ dω.
Whenever dν ∈ Ndm is a negative definite measure, we recall that, for all u, v ∈ V , the complex measure d ≺ u | ν v ≻ in M C ( K) is defined by polarization (2.6).Thus a sequence of negative definite measures (dν) k∈N converges weakly to some negative definite measure dν ∈ Ndm if and only if By polarization (2.6) we then conclude that in accordance with Definition 3.7.
Note that, with the very same reasoning, the definitions and results stated in this section can be generalized to operator-valued measures on whole momentum space.
3.2.Causal Variational Principles on Compact Subsets.After these technical preliminaries, let us now return to causal variational principles in the homogeneous setting.Motivated by (2.3), the fermionic projector P (x, y) in the homogeneous setting takes the form for all x, y ∈ M, where the measure dν is given by (2.4).Generalizing dν according to §2.In order to emphasize the dependence on the operator-valued measure dω, we also write P [ω](x, y).As P (x, y) is supposed to be homogeneous, only the difference of two spacetime points x, y ∈ M matters; denoting the difference vector by ξ = y − x ∈ M, the kernel of the fermionic projector reads The first step in order to set up the variational principle is to form the closed chain, which (as motivated by [11, §3.5]) for any ξ ∈ M is defined as the mapping We also write A[ω](ξ) in order to clarify the dependence of the closed chain on the operator-valued measure dω.Next, given a linear operator A : V → V , we define the spectral weight by where by (λ i ) i=1,...,2n we denote the eigenvalues of the operator A, counted with algebraic multiplicities.In this way, the spectral weight furnishes a connection between endomorphisms and scalar functionals.
In order to set up a real-valued variational principle on the set of operator-valued measures, for every dω ∈ Ovm we introduce the Lagrangian Defining the causal action S : the causal variational principle in the homogeneous setting is to minimize S(ν) by suitably varying dν in Ndm .
In order to exclude trivial minimizers, we impose the trace constraint for some c > 0. Additionally, for f > 0 we shall either introduce the constraint In agreement with [33,Definition 43.4], we define a minimizer for S as follows: Definition 3.9.A negative definite measure dν ∈ N is said to be a minimizer for the causal variational principle (3.7) if and only if
For further details concerning the calculus of variations we refer to [33,Chapter 43].

Existence of Minimizers on Compact Subsets
This section is devoted to developing the existence theory for minimizers of the causal action principle (3.3) for given c, f > 0 either with respect to the constraints or with respect to the side conditions The main result of this section can be stated as follows: Theorem 4.1.Let (dν (j) ) j∈N be a minimizing sequence of negative definite measures in Ndm of the causal variational principle (3.3) with respect to the constraints (4.1) or (4.2), respectively.Then there exists a sequence of unitary operators (U j ) j∈N on V (with respect to ≺ .| .≻) and a subsequence (dν (j k ) ) k∈N such that (U j k dν (j k ) U −1 j k ) k∈N converges weakly to some non-trivial negative definite measure dν = 0.Moreover, S(ν) ≤ lim inf k→∞ S(ν (j k ) ) , and the limit measure dν ∈ Ndm satisfies the side conditions or respectively (with positive constants c, f > 0).In particular, the limit measure dν is a non-trivial minimizer of the causal variational principle (3.3) with respect to the side conditions (4.1) or (4.2), respectively.A fortiori, the above statements remain true in case that "≤" in (4.1), (4.2) and (4.3), (4.4) is replaced by "=".
The remainder of this section is devoted to the proof of Theorem 4.1.The key idea for proving Theorem 4.1 is essentially to apply Prohorov's theorem (see e.g.[3,Section 8.6]).To this end, we proceed in several steps.Given a minimizing sequence of negative definite measures which satisfies the side conditions (4.1) or (4.2), we first prove boundedness of a unitarily equivalent subsequence thereof ( §4.1).The proof of Theorem 4.1 is completed afterwards ( §4.3).Once this is accomplished, we show that Theorem 4.1 also applies in the case that a boundedness constraint is imposed ( §4.4).

Boundedness of Minimizing Sequences.
Let us assume that (dν (k) ) k∈N is a sequence of negative definite measures in Ndm, either satisfying for all k ∈ N and some positive constant f > 0 (and | • | denotes the spectral weight).The aim of this subsection is to show that in both cases, there exists a sequence of unitary matrices (U k ) k∈N in L(V ) (with respect to ≺ .| .≻) such that the resulting sequence (U k dν (k) U −1 k ) k∈N is bounded in Ndm (with respect to the norm (3.1)).In particular, whenever the first condition is imposed, it eventually turns out that one can choose U k = 1 1 V for all k ∈ N. In preparation, let us state the following results: Proof.Introducing the kernel of the fermionic projector by (3.2) and making use of Definition 2.6, for all u, w ∈ V and ξ ∈ M we obtain for any negative definite measure dν ∈ Ndm and any unitary matrix U (with respect to ≺ .| .≻).Thus non-degeneracy of the indefinite inner product implies that Henceforth, employing Lemma 4.3, we deduce that the spectral weight of the closed chain A is unaffected by unitary similarity, i.e.
Analogously, for every ξ ∈ M we obtain for all ξ ∈ M as well as S(U ν U −1 ) = S(ν).This completes the proof.
We are now in the position to prove the following result: Lemma 4.5.Let f > 0 and assume that (dν (k) ) k∈N is a sequence in Ndm such that (where S denotes the signature matrix).Then there exists a positive constant C > 0 in such a way that d ν (k) ≤ C for all k ∈ N, where d • denotes the total variation according to Definition 3.2.
Proof.For convenience, we fix an arbitrary integer k ∈ N and let dν = dν (k) .Next, we let (e i ) i=1,...,2n be a pseudo-orthonormal basis of V with signature matrix S such that (2.1) is satisfied.Then d ≺ e i | ν e j ≻ is a finite complex measure in M C ( K) for every i, j ∈ {1, . . ., 2n} according to Definition 3.1, i.e.
Employing the definition of the total variation of complex measures and applying the Schwarz inequality (see e.g.[27,Lemma A.13] or [23, ineq.(2.3.9)]),we obtain where the supremum is taken over all partitions (E n ) n∈N of K (cf.[32,Chapter 6]).Applying Young's inequality (see e.g.[2, §1]), for all i, j ∈ {1, . . ., 2n} we arrive at Due to the fact that d ≺ e i | −ν e i ≻ is a positive measure for each i ∈ {1, . . ., 2n}, the total variation d ≺ e i | ν e j ≻ is bounded by for all i, j ∈ {1, . . ., 2n}.The last expression can be estimated by thus completing the proof.
In the case that the spectral weight is bounded (in analogy to [12, Theorem 6.1]), we obtain the following result: Proof.The basic idea is to make use of [13,Lemma 4.4].For convenience, we fix an arbitrary integer k ∈ N and let dν = dν (k) .Moreover, let (e i ) i=1,...,2n be a pseudoorthonormal basis of V with signature matrix S such that (2.1) is satisfied (see for instance [23, §2.3] or [27, §3.3]).Since V is a finite-dimensional vector space, all norms on L(V ) are equivalent, and one of these norms is given by for any A ∈ L(V ), where |•| denotes the absolute value.Moreover, for any unitary matrix U in L(V ) (with respect to ≺ .| .≻), we may introduce another pseudoorthonormal basis (f j ) j=1,...,2n by Making use of U * = U −1 , for all i, j = 1, . . ., 2n we obtain Since dν is a negative definite measure, the operator −ν( K) is positive (2.5).Thus in view of [13,Lemma 4.4], for any ε > 0 there is a unitary matrix U = U (ε) in L(V ) (with respect to ≺ .| .≻) so that U ν( K) U −1 is diagonal, up to an arbitrarily small error term ∆ν( K) with ∆ν( K) 1 < ε.Since k ∈ N is arbitrary, we thus obtain a sequence of negative definite measures (U k dν (k) U −1 k ) k∈N .Next, in order to prove that (U k dν (k) U −1 k ) k∈N is bounded with respect to the total variation defined by (3.1), for each k ∈ N we consider the basis (f i ) i=1,...,2n given by (4.7) with respect to the unitary matrix for all i, j = 1, . . ., 2n .
Employing the definition of the total variation of complex measures and applying the Schwarz inequality in analogy to the proof of Lemma 4.5, we obtain where the supremum is taken over all partitions (E n ) n∈N of K (cf.[32,Chapter 6]).Applying Young's inequality in analogy to the proof of Lemma 4.5, we arrive at for all i, j ∈ {1, . . ., 2n}.Since S = S −1 and U * = S −1 U † S in view of [23, eq.(4.1. 3)] (where U † denotes the adjoint with respect to .| .and U * the adjoint with respect to ≺ .| .≻), for all i = 1, . . ., 2n we obtain where we made use of Taken the previous results together, by (4.8) we obtain the inequality for all i = 1, . . ., 2n.Thus it only remains to find an upper bound for U ν( K) U −1 1 in terms of f by establishing a connection to the spectral weight |ν( K)|.To this end we exploit the fact that U ν( K) U −1 is diagonal according to [13,Lemma 4.4], up to an arbitrarily small error term ∆ν( K), Denoting the eigenvalues of U ν( K) U −1 by λ i (U ) for all i = 1, . . ., 2n, by choosing the error term ∆ν( K) sufficiently small we can arrange that Since the off-diagonal elements ∆ν( K) 1 < ε are arbitrarily small, we thus obtain Hence in view of Definition 3.2 and (4.9), we finally obtain This completes the proof.
The major simplification when restricting attention to compact subsets is that any minimizing sequence is uniformly tight a priori.As a consequence, we may apply Prohorov's theorem to each component, thereby obtaining the desired minimizer.4.2.Preparatory Result.Given a sequence of negative definite measures which is bounded and uniformly tight, we employ Prohorov's theorem to prove that a subsequence thereof converges weakly (see Definition 3.7) to a negative definite measure: Lemma 4.7.Let (dν k ) k∈N be a sequence of negative definite measures in Ndm with the following properties: (a) There is a constant C > 0 such that d|ν k |( K) ≤ C for all k ∈ N. (b) The sequence (dν k ) k∈N is uniformly tight in the sense that, for every ε > 0, there is a compact subset K ε ⊂ K such that d|ν k |( K \ K ε ) < ε for all k ∈ N. Then a subsequence of (dν k ) k∈N converges weakly to some negative definite measure dν.
Proof.The main idea is to apply Prohorov's theorem.More precisely, let (e i ) i=1,...,2n be a pseudo-orthonormal basis of V satisfying (2.1), and for every k ∈ N we denote by d|ν k | the corresponding variation of dν k according to Definition 3.
Following the proof of Lemma 3.3, we introduce the operator-valued measure dν for every Ω ∈ B( K) by The measure dν has the property that, for all i, j ∈ {1, . . ., 2n}, is a complex measure.For elements u = 2n m=1 α j (u) e j and v = 2n m=1 α j (v) e j in V , by linearity we conclude that d ≺ u | ν v ≻ ∈ M C ( K) for all u, v ∈ V .Hence dν is an operator-valued measure in the sense of Definition 3.1, and by linearity we arrive at for all f ∈ C b ( K) and u, v ∈ V .This yields weak convergence dν k ⇀ dν of operatorvalued measures in the sense of Definition 3.7.In particular, d ν < ∞.
It remains to show that dν is indeed negative definite.To this end, we need to prove that d ≺ u | −ν u ≻ is a positive measure for all u ∈ V .We point out that, by assumption, the measures d ≺ u | −ν k u ≻ are positive for each u ∈ V and all k ∈ N. Assume now, for some u ∈ V , that dµ u := d ≺ u | −ν u ≻ is a signed measure with In this case, there is Ω ∈ B( K) with the property that µ + u (Ω) < µ − u (Ω) (assuming conversely that µ + u (Ω) ≥ µ − u (Ω) for all Ω ∈ B( K), then the measure dµ u is non-negative, implying that dµ − u = 0).In virtue of Ulam's theorem we know that as well as For proving the last assertion, we require the next lemma: Proof of Proposition 4.9.By weak convergence, the first two equalities can be verified as follows: and analogously lim j→∞ In order to prove the remaining equality, we essentially make use of the fact that the spectral weight is continuous.More precisely, by continuity of the absolute value and weak convergence we obtain (where . 1 is given by (4.6)).Denoting the eigenvalues of ν( K) by (λ i ) i=1,...,2n and those of ν (j) ( K) for every j ∈ N by (λ i ) i=1,...,2n , by applying Lemma 4.10 together with the inverse triangle inequality we thus arrive at This completes the proof.
After these preliminaries we are finally in the position to prove Theorem 4.1.
Proof of Theorem 4.1.Let us first assume that the side conditions (4.2) are satisfied.In this case, Lemma 4.6 yields a sequence of unitary operators (U j ) j∈N in L(V ) (with respect to ≺ .| .≻) as well as a constant C > 0 such that for all j ∈ N .
Since K ⊂ M is compact, the sequence of measures (dν (j) ) j∈N is uniformly tight.As a consequence, we may apply Lemma 4.7 in order to conclude that a subsequence of (U j dν (j) U −1 j ) j∈N converges weakly to some negative definite measure dν ∈ Ndm, Making use of (4.5), from Proposition 4.8 we deduce that In the case that the constraints (4.1) are imposed, the above arguments remain valid by applying Lemma 4.5 instead of Lemma 4.6 and choosing U j = 1 1 V for all j ∈ N.
Thus it only remains to prove that the measure dν satisfies the conditions (4.3) or (4.4), respectively.In both cases, this follows readily from Proposition 4.9.In particular, the limit measure dν is non-trivial, which completes the proof.
As worked out in the next subsection, Theorem 4.1 also holds in the case that the side conditions (3.5) and (3.Given C > 0, the corresponding boundedness constraint reads In analogy to Theorem 4.1 we then obtain the following existence result: Theorem 4.11.Assume that (dν (j) ) j∈N is a minimizing sequence of negative definite measures in Ndm for the causal variational principle (3.3) with respect to the side conditions (3.4) and (4.10) for some positive constants c, C > 0. Then there exists a sequence of unitary operators (U j ) j∈N on V (with respect to ≺ .| .≻) as well as a subsequence (dν (j k ) ) k∈N such that the sequence (U j k dν (j k ) U −1 j k ) k∈N converges weakly to some non-trivial negative definite measure dν = 0.Moreover, S(ν) ≤ lim inf k→∞ S(ν (j k ) ) , and the limit measure dν ∈ Ndm satisfies the side conditions Tr V (ν( K)) = c and T (ν) ≤ C .
In particular, the limit measure dν is a non-trivial minimizer of the causal variational principle (3.3) with respect to the side conditions (3.4) and (4.10).
Since ν( K) < ∞, the absolute values of λi are bounded for all i = 1, . . ., 2n; from this we conclude that the spectrum of diag λ2 1 , . . ., λ2 2n coincides with the spectrum of A[ν](0), up to an arbitrarily small error term (where we applied the fact that the spectra of A[ν](0) and U A[ν](0) U −1 coincide according to Lemma 4.3).In a similar fashion, one can show that the spectra of ν( K) and − diag( λ1 , . . ., λ2n ) coincide, up to an arbitrarily small error term.Neglecting the error terms in what follows, we thus can arrange that Since the fermionic projector P (0) = ν( K) can be diagonalized (up to an arbitrarily small error term) according to [13,Lemma 4.4], in order to develop the existence theory of minimizers in the homogeneous setting it seems promising to demand that constraint (3.6) is satisfied.On the other hand, following the original ideas in [11] and its modifications in [13], it is natural to impose a boundedness constraint (4.10).The arguments in §4.4 show that (4.10) already implies condition (3.6).Let us finally discuss the remaining side condition (3.5).Since working with the spectral weight as appearing in the constraint (3.6) may be awkward, it might seem preferable to work with a similar condition which is more easy to handle.Bearing in mind that the operator ν( K) may be diagonalized (up to an arbitrarily small error term) in virtue of [13,Lemma 4.4] in such a way that its diagonal entries are ordered according to [13, eq. (2.6)], the specific form of the signature matrix S (see (2.2)) suggests to replace condition (3.6) by (3.5), The same arguments as before illustrate that Tr V (−Sν( K)) is a density; we refer to this quantity as particle density.
with respect to ≺ .| .≻) if and only if A = A * .In a similar fashion, an operator U ∈ L(V ) is said to be unitary (with respect to ≺ .| .≻) if it is invertible and U −1 = U * (see[23, Section 4.1]).

2
and §3.1 to operator-valued measures, for a given operator-valued measure dω on K with values in L(V ) and all x, y ∈ M we introduce the kernel of the fermionic projector by P (x, y) : V → V, P (x, y) := ˆK e ik(y−x) dω(k) .

Lemma 4 . 6 . 1 k≤
Let f > 0 and assume that (dν(k) ) k∈N is a sequence in Ndm such that |ν (k) ( K)| ≤ f for all k ∈ N (where | • | denotes the spectral weight).Then there is a sequence (U k ) k∈N of unitary operators on V (with respect to ≺ .| .≻) as well as a positive constant C > 0 such that d U k ν (k) U −C for all k ∈ N (where d • denotes the total variation according to Definition 3.2).
2. Decomposing the complex measure d ≺ e i | −ν k e j ≻ into its real and imaginary part, d ≺ e i | −ν k e j ≻ = Re d ≺ e i | −ν k e j ≻ + i Im d ≺ e i | −ν k e j ≻ , and introducing the (positive) measures dℜ ± [i,j],k := Re d ≺ e i | −ν k e j ≻ ± and dℑ ± [i,j],k := Im d ≺ e i | −ν k e j ≻ ± by applying the Jordan decomposition [24, §29], we arrive at d

Lemma 4 . 10 .
Let W be a finite-dimensional vector space and let T ∈ L(W ).Then for any sequence (T n ) n∈N of operators in L(W ) with T n − T → 0 as n → ∞ (where .denotes any norm on W ), the eigenvalues of T n converge to those of T .Proof.See[26,  Chapter II, §5-1].