On the Mathematical Foundations of Causal Fermion Systems in Minkowski Spacetime

The emergence of the concept of a causal fermion system is revisited and further investigated for the vacuum Dirac equation in Minkowski spacetime. After a brief recap of the Dirac equation and its solution space, in order to allow for the effects of a possibly nonstandard structure of spacetime at the Plank scale, a regularization by a smooth cutoff in momentum space is introduced, and its properties are discussed. Given an ensemble of solutions, we recall the construction of a local correlation function, which realizes spacetime in terms of operators. It is shown in various situations that the local correlation function maps spacetime points to operators of maximal rank and that it is a closed homeomorphism to its image. It is inferred that the corresponding causal fermion systems are regular and have a smooth manifold structure. The cases considered include a Dirac sea vacuum and systems involving a finite number of particles and anti-particles.


Introduction
In most attempts to quantum gravity, a widely accepted principle is the existence of a minimal observable length, generally associated to the Plank length l P ∼ 10 −35 m. In order to understand this problem in a simple way, imagine to probe the microscopic structure of spacetime down to the Planck scale. Then the uncertainty principle would lead to energy densities which are large enough to change the structure of spacetime itself drastically. This argument poses severe constraints to any attempt of defining a notion of localization.
Sticking to the simplest case of Minkowski spacetime, one possible way to implement the existence of a minimal length is to introduce a cutoff regularization in momentum space. Roughly speaking, if the spacetime uncertainty is believed to be bounded from below by l P , then one would expect an upper bound in the momentum uncertainty of the order of l −1 P . This can be realized by taking out the momenta of the wave functions of interest which lie above such an upper scale (for mathematical simplicity we stick to a smooth cutoff in this paper, see Section 3.2). This procedure is reminiscent of the usual ultraviolet regularization frequently used in the renormalization program in quantum field theory as a technical tool to remove divergences. However, in our setting the regularization has a physical significance as it effectively describes the microscopic structure of physical spacetime in the presence of a minimal length.
As shown in [8], the first consequence of implementing the minimal length as a momentum cutoff is the existence of a natural realization of spacetime in terms of finite rank operators on the one-particle Hilbert space (the so-called local correlation function, see Theorem 4.7) which depends upon a choice of an ensemble of solutions of the Dirac equation. The structure consisting of an ensemble of solutions together with the corresponding local correlation function and a Borel measure on its image is called a causal fermion system. The fermionic wave functions encode information on the spacetime structure and, once suitably rich families of solutions are chosen (see regular systems, Definition 4.15), the topological and differentiable structures are recovered: Minkowski spacetime is diffeomorphic to a four-dimensional manifold consisting of bounded operators (see Theorem 6.8 and Corollary 6.9). Moreover these operators also encode the spinorial structure of the wave functions (see Section 4.3) There is a distinguished ensemble of solutions which provides us with a full realization of spacetime and is invariant under spacetime translation and as such gives a sensible candidate for a vacuum structure: the family of negative energy solutions of the Dirac equation. This resembles the original idea of Dirac of the vacuum as a quantum system where all the negative energy states are filled. Although in the standard framework of quantum field theory such a concept is no longer used, in our setting Dirac's idea finds a new possibility of interpretation. The addition of particles corresponds then to an extension of the solution ensemble within the positive energy subspace, while the addition of antiparticles is realized by a restriction of such a family within the negative one. In this formalism there is an asymmetry between matter and anti-matter. Indeed, the addition of positive energy solutions simply adds information to the vacuum configuration and the corresponding local correlation function still gives a full realization of spacetime. On the other hand, removing negative energy solutions from the vacuum ensemble might cause a critical loss of information, ending up in a too poor local correlation function. This can be prevented by removing solutions which are sufficiently "spread out" in space and do not vary too much on the microscopic scale (see Sections 5.2 and 6). In the general case, a representation is still possible, but some differences emerge: some spacetime events, for instance, might be no longer distinguishable (see the beginning of Section 6.1). Although this might be an interesting matter of study, it will be not discussed in this paper.
Clearly, the full construction presented here (and originally introduced in [8]) depends heavily on the choice of the regularization cutoff. Nevertheless, in the theory of causal fermion systems, the belief is that a distinguished physically meaningful regularization really exists and arises naturally as a minimizer of an action principle (see the causal action principle in [8], Section 1.1.1). An attempt to construct such "optimal" regalarizations can be found in [7]. Here we do not enter such constructions, because our focus is to analyze the analytic properties of the local correlation function for a simple class of regularizations.

Standard results on the Dirac Equation
In the present work we assume that an inertial reference frame has been assigned and Minkowski space realised accordingly as the Euclidean space R 4 together with the standard inner product η, where we adopt the signature convention (+, −, −, −). In addition, for the sake of notation simplicity, we use natural units = c = 1.

The equation and its solutions space
The aim of this section is to introduce the basic notions about the Dirac equation and its solution spaces which are relevant for the theory of causal fermion systems. We will focus exclusively on the case of a strictly positive mass m > 0.
The starting point is the first-order linear partial differential Dirac operator: The space of smooth solutions of D is denoted by ker D, i.e. the set of f ∈ C ∞ (R 4 , C 4 ) s.t.: iγ µ ∂ µ f = mf. 1 (2.1) Equation (2.1) is a symmetric hyperbolic system of partial differential equations and as such it admits unique global solutions if regular initial data are given on a fixed Cauchy surface Σ t := {(t, x) ∈ R 4 | x ∈ R 3 } (for details see for example [6] or [10,Section 5.3]).
Theorem 2.1 Referring to equation (2.1), the following statements hold.
(i) Let f, g ∈ ker D such that f↾ Σt = g↾ Σt for some t ∈ R, then f = g. 1 In the remainder of the paper we will make use of Feynmann notation / a := aµγ µ for a ∈ C 4 and / ∂ := γ µ ∂µ.
(ii) For any t ∈ R and ϕ ∈ C ∞ c (R 3 , C 4 ) 2 there exists 3 E t (ϕ) ∈ ker D with E t (ϕ)↾ Σt = ϕ. Points (i) and (ii) in the theorem above guarantee that, for every t ∈ R, the function is a well-defined injective linear mapping whose image is the set: 3) It can be proved that the support of the generated solution E t (ϕ) is contained in the causal propagation of the support of ϕ, as we expect from finite propagation speed.
Proposition 2.2 For every s, t ∈ R and ϕ ∈ C ∞ c (R 3 , C 4 ) the following statements hold: ) and E s (E t (ϕ)↾ Σs ) = E t (ϕ). Proof. Point (i) is a direct consequence of the fact that two solutions of (2.1) which coincide on the past boundary of a lens-shaped region do coincide on the whole region (see the correponding section in [6]; also see [3, Section 2.1]). More precisely, any solution whose restriction to Σ t has compact support must coincide with the trivial solution on the set complement of {t} × supp ϕ + {x ∈ R 4 | η(x, x) ≥ 0}. The proof of (ii) follows from (i) and from (i) of Theorem 2.1.
From this proposition we see that the choice of a specific Cauchy surface does not actually play any role in the definition of (2.3). More precisely: ) for any s, t ∈ R. Definition 2.3 The image of the operators E t is called the space of smooth solutions of the Dirac equation with spatially compact support and denoted by H sc m .
On this space of solutions it is possible to introduce a Hermitean inner product and make it a pre-Hilbert space. Fix a t ∈ R and consider the sesquilinear function 4 : Again, the choice of the Cauchy surface does actually play no role, due to current conservation. More precisely we have the following result.
Proposition 2.4 For any t ∈ R the function (·|·) t defines a Hermitian inner product -i.e. it gives H sc m a pre-Hilibert space structure -and makes the isomorpshim Proof. The positive-definiteness of (·|·) t follows directly from the corresponding feature of (·|·) L 2 and the uniqueness of the solutions of (2.1) given data at time t ∈ R. The function E t is clearly an isometry by construction. Finally, the independence of the inner product from the time variable can be proved by differentiating under the integral sign, expressing the time derivative in terms of spatial derivatives by means of equation (2.1) and eventually applying the divergence theorem (note that the involved functions have spatially compact support).
Having in mind the goal of defining a one-particle Hilbert space, the natural next step would consist in taking the completion of H sc m with respect to this inner product. Anyway, in the general case where no background (complete) Hilbert space is given, the completion of a pre-Hilbert space is constructed in a totally abstract way, by taking as linear space the set of equivalence classes of Cauchy sequences and extending the inner product to it by continuity. Since the goal is to build a quantum theory of wave functions, it would be useful to show that even the limit points realise measurable functions on spacetime.
A natural space where to embed our space of smooth solutions is the set L 2 loc (R 4 , C 4 ) of locally square-integrable functions. It can be proved that such a space can be made a complete metric space (see for example Lemma 5.17 in [14]). More precisely, the notion of convergence is given by: In the following we will make use of the set R T := [−T, T ] × R 3 .
Lemma 2.5 Given the pre-Hilbert topology of H sc m , the identity H sc m ֒→ L 2 loc (R 4 , C 4 ) is Cauchy continuous, in particular it is continuous.
Proof. See Appendix.
At this point, bearing in mind the way the abstract completion of a pre-Hilbert space is constructed, we can characterize the completion of H sc m with respect to (·|·) 0 by assigning to every Cauchy sequence within H sc m the corresponding limit within L 2 loc (R 4 , C 4 ) and extending the inner product (·|·) 0 by continuity in the obvious way. To support our choice of H m as completion of H sc m , we also state the following technical result, whose proof can be found in the Appendix.
Lemma 2.7 Let {f n } n be a Cauchy sequence in H sc m , then the following statements hold.
(i) The function u = lim n→∞ f n ∈ H m fulfills u↾ R T ∈ L 2 (R T , C 4 ) for any T > 0 and (ii) There exists a subsequence {f σ(n) } n which converges to u pointwise a.e.
In taking the completion space, the elements of H m can be still interpreted as solutions of a partial differential equation, even though in a weak sense, for they do not define regular functions in general. where D * := i(γ µ ) † ∂ µ + mI 4 is the formal adjoint of D.
Proof. See Appendix.
(b) The space L 2 (R 3 , C 4 ) can be interpreted as the set of generalized initial data for (2.1).
(c) Due to unitary equivalence, both L 2 (R 3 , C 4 ) and H m can be taken as one-particle Hilbert spaces. The elements of the former are called wave functions.
To summarize, the functions E t allow us to interpret the solutions of the Dirac equation in two equivalent ways: either as functions globally defined on spacetime R 4 , i.e. the elements of H m , or in terms of evolving wave functions within L 2 (R 3 , C 4 ), i.e. as a path: (2.8) for arbitrary initial data ψ ∈ L 2 (R 3 , C 4 ). This latter description fits better to the standard formulation of quantum mechanics. In the next section we will study the feature of this evolution map.

The Hamiltonian operator and its spectral decomposition
In both classical and quantum mechanics, with due mathematical differences, the Hamiltonian is defined as the generator of time evolution and it is generally (but not always) identified with an observable physical quantity of the system: the energy. In our framework, the time evolution is given in terms of a strongly continuous one-parameter group of unitary operators and as such it admists a unique self-adjoint generator thanks to Stone Theorem: this is the Hamiltonian we are looking for. The time evolution operators were defined in the previous section in (2.8). We need to prove that they do define a strongly continuous one-parameter group of unitary operators.
Definition 2.10 The time-evolution operator is defined for every t ∈ R by U t := E −1 t E 0 .
Restricting to the dense subspace of compactly supported smooth functions, the action of such a mapping is given by: As expected, the family {U t } t∈R fulfills all the properties of a linear unitary evolution.

Proposition 2.11
The function R ∋ t → U t is a strongly continuous one-parameter group of unitary operators. The corresponding self-adjoint generator (the Hamiltonian) is given by Proof. See Appendix.
The spectral properties of the Hamiltonian operator are easier to understand if analised in momentum space through the (unitary) Fourier Transform 5 where the index was added just to make clear the distinction between position (x) and momentum (p) representations.
The Fourier Transform is an isometric isomorphism on the space of (spinor-valued) Schwartz [15]): We can apply this transformation to our operator H and work directly in momentum space. In what follows we will make use of the energy function: Theorem 2.13 The operator H is mapped to the multiplication operatorĤϕ = h · ϕ with h : In particular the associated one-parameter group reads: Proof. The proof of the first part follows from the properties of the Fourier Transform (see [20], Section 1.4.4). The last statement follows directly from Stone Theorem and the uniqueness of the self-adjoint generator. 5 In this paper the Fourier Transform on R 3 is defined with respect to the Euclidean inner product: while the Fourier Transform on R 4 is carried out with respect to the Minkowski inner product: which fits better to a relativistic setting.
The spectral features of the Hamiltonian H are far easier to analyse in momentum representation than in position representation, in that in this setting we merely have to study the matrix h.
First of all, notice that, for any choice of k, the matrix h(k) is symmetric (with respect to the Euclidean inner product of C 4 ) and has eigenvalues ±ω(k), both two-fold degenerate (for details follow the discussion in [4, Section 9.2] with the appropriate modifications; see also [3,Section 2.2]). In particular the linear space C 4 decomposes into two orthogonal subspaces: which are the images of the following orthogonal projections over C 4 : Proposition 2.14 Referring to (2.13), for every k ∈ R 3 it holds that: Proof. The result can be proved by direct inspection, bearing in mind the properties of the Dirac matrices: (γ µ ) † = η νν γ µ and {γ µ , γ ν } = 2η µν I 4 .
For every k ∈ R 3 , an orthogonal basis of C 4 is given by the four vectors: where e ↑ = (1, 0) t and e ↓ = (0, 1) t . The upwards and downwards arrows are chosen in connection with the physical interpretation of these vectors as the spinors carried by the spin up and spin down plane-wave solutions (see also Proposition 2.30).
Proof. The proof follows from the corresponding projection features of the matrices p ± (k). Just notice that the operatorsP ± are well-defined by virtue of the boundedness of p ± .
At this point, we can decompose our Hilbert space in terms of two orthogonal subspaces: Lemma 2.17 The following statements are true: Proof. The proof follows directly from the density of the Schwartz space within the space of square-integrable functions and the fact thatP ± (S p (R 3 , C 4 )) ⊂ S p (R 3 , C 4 ). This last inclusion follows from the second identity in (2.13), together with the fact that the Schwartz space is closed under multiplication by polynomials and Lemma 8.1 It is now possible to explicit the action of the Hamiltonian on these orthogonal subspaces. Notice that the domain S p (R 3 , C 4 ) is a dense invariant core forĤ and on its elements the Hamiltonian acts as a multiplication operator.
, then the following holds.
Proof. The proof of point (i) follows immediately from the fact thatP ± (ϕ) ∈ S p (R 3 , C 4 ) ⊂ D(Ĥ) and the definition ofĤ. Similarly, point (ii) can be proved exploiting the explicit action of e −itĤ . Point (iii) can be found in [20,Theorem 1.1].
At this point, we may wonder how the momentum distributions in S p (R 3 , C 4 ) look like when represented within H m through the unitary operator E 0 • F −1 . We have: More precisely, for Schwartz functions we have the following result.
Proof. See Appendix.
As a conclusion of this section, bearing in mind the identities of Theorem 2.18, we see that the subspacesP ± (S p (R 3 , C 4 )) can be rightly interpreted as the positive and negative energy eigenspaces of the Hamiltonian. Sticking to H m as our favourite realization of the one-particle Hilbert space, we can consider the projectors: and give the following definition.
Definition 2.20 The subspaces P ± (H m ) are called the positive and negative energy subspaces and denoted by H ± m . Their elements are called the positive and negative energy physical solutions of (2.1).
Remark 2.21 A few remarks follow.
(i) The elements in the corresponding orthogonal subspaces P ± (L 2 x (R 3 , C 4 )) will be referred to as the positive and negative energy wave functions.

Four momentum representation and the fermionic projectors
Bearing in mind the distributional identity the (positive and negative components of the) physical solution of the Dirac equation with three dimensional momentum distribution ϕ ∈ S p (R 3 , C 4 ) can be restated aŝ In the above equations: x = (t, x) ∈ R 4 andφ is any function of S p (R 4 , C 4 ) whose value on the mass-shell is: for some arbitrary f ∈ C ∞ c (R, R + ) with supp f ⊂ [−m/2, m/2] and f ≡ 1 on [−m/4, m/4]. This function belongs to S p (R 4 , C 4 ), as can be checked using Lemma 8.1.
The final identity obtained above suggests a more compact way to denote these solutions.
Proposition 2.23 For any f ∈ S x (R 4 , C 4 ), the function is a smooth solution of (2.1) and has the following representation: More precisely, the following identification holds: Proof. See Appendix.
Proposition 2.23 shows that it is possible to represents all the physical solutions u ϕ with ϕ ∈ S p (R 3 , C 4 ) via the formula P ± ( · , f ), with f ranging within S x (R 4 , C 4 ). This association defines a linear operator.
Definition 2.24 The (unregularized) fermionic projectors onto the positive and negative spectrum are defined as the linear mappings: Their difference generates the so-called (unregularized) causal propagator P c : More precisely, the causal propagator fulfills P c (S x (R 4 , C 4 )) =Ê(S p (R 3 , C 4 )).
Proof. The first statement follows directly from the second one, which can be proved using (2.16) and (2.20).
It is possible to represent the (unregularized) fermionic projectors as kernel operators: even though just in a distributional sense.
Proposition 2.26 The kernel of the (unregularized) fermionic projector P ± is given by

22)
and has the following representation: For any y ∈ R 4 and a ∈ C 4 , the function P ± ( · , y)a is a smooth solution of the Dirac equation in the open complement of the light cone centered at y and it diverges on it.
Proof. The facts that P ± (x, y) defines a kernel for P ± and its explicit form can be proved by direct inspection. Concerning the last statements, see (1.2.25) and Lemma 1.2.9 in [8].
Proposition 2.27 The kernel of the (unregularized) causal propagator 7 is given by Remark 2.28 Some remarks follow.
(i) The term "causal" comes from the fact that the corresponding kernel vanishes for spatially separated spacetime points. This is not true for the single terms P ± (x, y) (compare with [8, Section 2.1.3]).
(ii) The operator P nc := P − + P + fulfills analogous properties, but is not causal. Its correponding kernel 8 is given by (iii) The solutions P − (x, y) and −P + (x, y) correspond to the negative and positive energy components of P c (x, y), respectively, and should not be confused with the advanced and retarded Green functions (see [8,Section 2.1.3]).
(v) The function P ± ( · , (0, y))a can be interpreted as the the evolution in time of the given initial data in Σ 0 : The integrals in (2.24) are actually ill-defined and have a meaning only as tempered distributions. Also, the second identity, which is based on Accordingly, the corresponding distributional solution P c ( · , (0, y))a is generated by the initial data: Ω(a, y; · ) := Ω − (a, y; · ) − Ω + (a, y; · ) = −(2π) −1 γ 0 a δ y , (2.25) which corresponds to a localized state at position y ∈ R 3 . The distributional solutions arising from (2.24) correspond then to its negative and (minus) its positive energy components, respectively. These correspond also to the negative and positive energy (distributional) solutions which are as far as possible localized in space.
As will be shortly discussed in Section 2.4, it is not possible to construct physical (normalizabile) solutions within the positive or negative spectrum which have compactly supported initial data.
(vi) It should be mentioned that the solutions in point (iv) differ from the eigenstates of the Newton-Wigner position operator by a factor 2ω · (ω + m) −1 in the momentum distribution (see for example [18,Section 2]).
(vii) In the reminder of the paper we will sometimes use the following notation: As distributions: We already know that to every physical solution in H m it is possible to assigne a three dimensional momentum distribution via the functionÊ −1 0 (see Definition 2.12). The above discussion shows that, at least for the smooth cases considered, this can be done also in the four dimensional momentum space, even though in a distributional sense.
Definition 2.29 For every f ∈ S x (R 4 , C 4 ), the tempered distributionP c · F(f ) is called the four dimensional momentum distribution of P c ( · , f ).
To conclude this section, we want to point out that there exists an additional and more detailed representation of the discussed solutions. Indeed, making use of Proposition 2.19 and the basis vectors (2.14), we see that any positive or negative energy solution can be written in terms of spin up and spin down solutions.
Proof. Take any λ ± ↑,↓ ∈ S p (R 3 , C), then, reasoning as in the proof of Lemma 2.17, it is possible to show that the functions λ ↑,↓ χ ± ↑,↓ belong to S p (R 3 , C 4 ), more precisely toP ± (S p (R 3 , C 4 )). Together with Proposition 2.19, this proves that the functions in (2.26) do define elements of E 0 (P ± (S p (R 3 , C 4 ))). In order to prove the other inclusion, let us stick to the positive energy case (the negative one being analogous) and take any u ϕ ∈Ê 0 (P + (S p (R 3 , C 4 ))) for some ϕ ∈P + (S p (R 3 , C 4 )) ⊂ S p (R 3 , C 4 ). Since the vectors χ + ↑,↓ (k) define a basis of W + k at every k ∈ R 3 , we can write , all the components of ϕ must be Schwartz functions in the ordinary sense and therefore, in particular, λ + ↑,↓ ∈ S p (R 3 , C). This concludes the proof.
In the appendix two estimates concerning these functions are carried out: see Lemma 8.2. These will turn out to be useful later on.

A few words on the decay properties of Dirac solutions
In the analysis of the smooth features of the vacuum causal fermion system the decay properties of the smooth solutions of (2.1) will become crucial.
As a first remark we want to point out that the smooth solutions of the Dirac equation which lie within either H − m or H + m are spread all over spacetime: in particular they cannot be localized in bounded regions. This is made mathematically precise by the following theorem whose proof can be found in [20,Corollary 1.7].
In particular this theorem has the following corollary: Indeed, if this were not the case, then it would be possible to find a non-empty open set Ω ⊂ R 4 such that u↾ Ω = 0. As a consequence, there must exist some t ∈ R such that the open set Ω t := Σ t ∩ Ω is not empty and u↾ Ωt = 0, contradicting Theorem 2.31.
Notice that this theorem does not mean that the solutions inÊ 0 (P ± (S p (R 3 , C 4 ))) vanish nowhere, but that they vanish at most on the boundary of a dense open subset of R 4 (or R 3 for the restriction to a Cauchy surface).
Nevertheless, even though these solutions hardly vanish on spacetime, it can be proved that they decay pretty fast at space and time infinities, as one could expect. In order to see this, consider the Dirac operator D := i / ∂ − m defined in (2.1) and fix any element f ∈ ker D. Using the identity / ∂ 2 = 9 , wee that the components of f satisfy the Klein-Gordon equation: and therefore we can apply the results of [9, Section 7.2] and prove the following important result on the asymptotic behaviour of the solutions of (2.1).
Proof. Focus on solutions (2.26) with negative energy (the positive energy case is analogous).
and therefore the form of the four components of the functions u − ↑,↓ match with the one of the solutions (7.2.3) in [9]. The inequality (7.2.4) is clearly fulfilled for any negative integer N ,φ being a Schwartz function. Moreover if we take N = −3, then N < −(n + 1)/2 with n = 3 and if we choose M − = 0 then M + = 1/2 and equation (7.2.5) becomes (for |t| + |x| ≥ 1): The same argument applies to the positive energy solutions. Since any solution can be written as linear composition of the solutions u ± ↑,↓ , the thesis follows.

Microscopic investigation: regularization
In this chapter we introduce a regularization of the physical solutions. As anticipated in the introduction, led by the conjecture that the nature of spacetime is not continuous on a microscopic scale or, being more cautious, that it looks differently from what we expect from a macroscopic point of view, we can make an attempt and modify the theory on this scale by restricting its domain of validity up to a scale of the order of the Planck length. For the sake of generality, we denote this critical microscopic scale by ε and let it vary within some interval (0, ε max ).

Some quantitative assumptions on the regularization parameter
Let us denote by m P the Planck mass whose value in natural units 10 reads: As a comparison, the rest masses of the electron, the proton and the top quark are, respectively: m e ∼ 5.11 × 10 5 eV, m p ∼ 9.38 × 10 8 eV, m t ∼ 1.73 × 10 11 eV, The corresponding mass ratios are given by Directly related with the Planck mass, we can define the Planck length l P by the relation The length l P is a good candidate for the microscopic scale ε, at least concerning the order of magnitude. Nevertheless, just to stay on the safe side, we can make the following weaker assumption, which will be taken for granted in the rest of the paper, if not specified otherwise.
The microscopic scale ε is bounded from above by mε max = 10 −15 .

Regularization: momentum cutoff or spacetime mollification?
As discussed in the previous sections, to every solution u ∈ H m a three dimensional momentum distributionÊ −1 0 (u) ∈ L 2 p (R 3 , C 4 ) can be associated (see Definition 2.12). Similarly, restricting to the dense subspace spanned by the solutions P( · , f ), also a four dimensional momentum distribution can be defined (see Definition 2.29). These representations of the solutions allow us to better understand how the minimal length can be implementend.
Roughly speaking, if the spacetime dimension below the scale ε is unreachable by our theory, i.e. some lower bound ∆x ε exists, then we may expect an upper bound in momentum space of the order ∆k ε −1 . This can be accomplished by introducing a cutoff in momentum space. More precisely, the idea is to multiply the momentum distributions by a smooth function which decays fast enough to take out -or at least weaken -the contribution made by the momenta larger than ε −1 . This can be carried out in both the three and four dimensional momentum spaces.
It is possible to make some assumptions on these cutoff functions, led by physical intuition. For example, it is sensible to assume that these functions are spherically symmetric in the three momentum coordinates, for there should be no distinguished direction. Similarly, in the four dimensional space, it sounds reasonable to assume that these functions are symmetric under inversion of the energy sign. We want now to discuss this in detail.
Let us start with the following definition.
is called a four momentum cutoff. for all k ∈ R 4 .
As a matter of fact (see for example 2.16)), the four momenta which contribute to the Fourier expansion of the solutions of (2.1) are exclusively the four vectors lying on the mass shell |k 0 | = ω(k). This means that we could in fact focus on the three momenta k ∈ R 3 . More precisely, we can define the following function which selects the values of the cutoff on the mass shell. Then, the following properties hold.
Proposition 3.4 Referring to (3.1), the function g belongs to S p (R 3 , R) and satisfies Proof. Points (i) and (ii) follow directly from the analogous properties of G. The fact that g is a Schwartz function follows from the fact that G is a Schwartz function itself and from Lemma 8.1.
Of course, in order not to get a trivial momentum regularization, it is mandatory to assume that such a function does not vanish entirely. This can be achieved, for example, if G is strictly positive on a ball which intersect the mass shell. So far, we have shown that a four momentum cutoff induces a three momentum cutoff. As a matter of fact, also the other way around is possible, although not in a unique way. This shows that the two concepts are indeed interchangeable. Proposition 3.6 Let g be a three momentum cutoff, then there exists a four momentum cutoff G such that g(k) = G(ω(k), k) for all k ∈ R 3 .
In particular, G does not vanish identically on the mass shell.
Proof. Let g be as in Definition 3. So far, we have not discussed the role played by the microscopic length scale ε or, more precisely, how it enters the definition of a cutoff. The leading idea is that a cutoff function should be concentrated around a ball of radius ε −1 . However, before making this mathematically more precise, we need to make some considerations.
Even though from a physical point of view such a microscopic scale should be fixed a priori, from a purely mathematical perspective it is useful not to fix the value of the cutoff parameter ε, but instead to leave some freedom in the choice of the microscopic scale. This allows us to consider the limit ε → 0 of the various regularized quantities: this is the socalled continuum limit (not analysed here, see [8]). Therefore, in what follows we will merely assume that (compare with Assumption 3.1): The physical microscopic scale ε ranges within (0, ε max ).
Bearing this in mind, the above ideas can be implemented as follows.
Suppose we are given a four dimensional cutoff function G which, roughly speaking, is concentrated around the unit ball. For example, we may consider a Gaussian function centered at the origin with unit variance, or a compactly supported function which equals the identity on the unit ball. If now we define G ε (k) := G(ε · k), we obtain a new cutoff function which is concentrated on the strechted set B(0, ε −1 ). Led by this idea, we can give the following general definition.
It should be mentioned that there is an analogous way to implement a regularization cutoff family. One could, indeed, fix a three momentum cutoff g, define g ε (k) := g(ε · k) for every ε ∈ (0, ε max ) and then take as G ε any extension of g ε as in Proposition 3.6. This construction, thought, is generally not equivalent to the one introduced in Definition 3.7, because of the non-linearity of ω. More precisely, a cutoff {G ε } ε constructed in this way can generally not be written as {G(ε · k)} ε for some fixed G as in Definition 3.7. In this paper we will focus mainly on the construction of Definition 3.7.
Remark 3.8 A few remarks follow.
(i) Since what really matters if the value of G ε on the mass shell, we can interchangeably talk of G ε or g ε as regularization cutoff in momentum space.
(ii) The intuitive picture is that G ε provides a smooth approximation of the characteristic function of the sphere B(0, ε −1 ). There is no need to make this assumption mathematically more precise, though, for we will mainly focus on the properties of the Fourier transform of G ε , as will be discussed shortly.
At this point, we are ready to regularize the functions of H m . As anticipated at the beginning of this section, the idea is to take out the large momenta in the momentum distributions of the solutions in H m by multiplying them with cutoff functions.
So, take a regularization family {G ε } ε and fix ε ∈ (0, ε max ). Every element u of H m can be written as: The idea is then to replace the momentum distribution ψ by the modified one g ε ψ.
Proposition 3.9 Given a regularization family {g ε } ε , define the regularization operators: The following properties are fulfilled: . In particular, if g ε vanish almost nowhere, then ker R ε is trivial.
Proof. Point (i) follows from the continuity of ψ → g ε ψ on L 2 p (R 3 , C 4 ) and the fact that E 0 is a unitary operator. Point (iii) is a direct consequence of the injectivity ofÊ 0 . Let us prove point (ii). Exploiting identity (2.15), we can see that R ε (Ê 0 (P ± (S p (R 3 , C 4 )))) ⊂ H ± m , for g ε is a scalar function and does not change the spectral properties of the momentum distributions. The statement follows from the density ofÊ 0 (P ± (S p (R 3 , C 4 ))) within H ± m (see Lemma 2.17 and the definition ofÊ 0 ) and the continuity of R ε .
Remark 3.10 Depending on the choice of the regularization family, the regularization operators can be injective or not. By now, we prefer not to make any assumption on the shape of the cutoff functions. The set ker R ε is to be interpreted as the set of high momenta solutions of H m , and as such they become unphysical once the regularization is assumed to have physical meaning. In the specific case of a mollification regularization that will be introduced soon (see Theorem 3.15 and Assumption 3.16), the kernel is indeed trivial.
As should be, this regularization procedure is equivalent to modifying the four dimensional momentum distribution by means of G ε , when this is possible. More precisely, if we restrict our attention to the smooth elements u ϕ = P c ( · , f ) (see Proposition 2.23), it is not difficult to see that (compare with (2.16)): Proposition 3.11 Referring to Proposition 3.9 and Proposition 2.23, it holds that: for any f ∈ S x (R 4 , C 4 ) and x ∈ R 4 .
A well-known feature of the Fourier transform is that it maps products into convolution and viceversa 11 . This allows for a simple representation of the regularized operator in position space, at least for these smooth solutions. Indeed, take any f ∈ S x (R 4 , C 4 ), then: Remark 3.12 It is important to notice that identities (3.2) and (3.3) depend only on the values attained by G ε on the mass shell. In particular, if two regularization functions coincide on the mass shell, their Fourier Transform might be different, but their convolution with the solutions would provide the same result.
Proposition 3.13 The following statements are true.
(i) Let {G ε } ε be a regularization family and h ε : In this case the set of zeros of g ε is a Lebesgue null set and ker R ε = {0}.
Proof. Let us start with point (i). The first equality in (3.4) was proven above, while the second equality comes from the fact that G ε ·F(f ) = F(f * h ε ). From Proposition 3.9 we know that R ε (P ± ( · , f )) ∈ H ± m . Moreover, as the convolution of smooth functions is smooth, we get R ε (P ± ( · , f )) ∈ C ∞ (R 3 , C 4 ), concluding the proof of (1).
Let us pass to the proof of point (ii). Consider any function . It can be proven by direct inspection that the smooth function h := h 1 * h 1 is supported in the set B(0, 1), it takes values within the positive real line and has the same symmetries of h 1 . Moreover, by means of a suitable normalization, we can always suppose that h L 1 = 1. Now, consider its Fourier Transform G := (2π) 2 F(h). From the invariance of h 1 with respect to spacetime inversion x → −x, it follows that F(h 1 ) is realvalued and, therefore, since G = (2π) 4 F(h 1 ) · F(h 1 ), we see that the function G takes values within R + . Finally, it can be checked that the symmetries of h are preserved in transforming it into G. Therefore, G defines a four dimensional cutoff function and we can consider the regularization G ε (k) := G(εk). Now, it can be seen by direct inspection that the function To conclude we need to prove the last statement. Since h ε is compactly supported, Paley-Wiener Theorem for several variables (see for example Theorem 4.9 of [19]) ensures that the Fourier transform G ε is the restriction to R 4 of an entire function on C 4 and, as such, it must be real analytic. Now, consider the function G ∈ C ∞ ((0, ∞), R) defined by G(r) := G ε (ω(re 1 ), re 1 ) = g ε (re 1 ). This is also real analytic, it being the composition of the real analytic function G ε and the real analytic function (0, ∞) ∋ r → ( √ r 2 + m 2 , r, 0, 0) ∈ R 4 (see Proposition 2.2.8 of [12]). As such, the function G can be extended analytically to a complex analytic (homeomorphic) function on an open neighbourhood of (0, ∞) in the complex plane. It is well-known that such functions admit only isolated zeroes and therefore the same must hold true for the restriction G on (0, ∞), as well. In particular, the set N of the zeroes of G is at most countable. At this point, bearing in mind the symmetries of g ε (or equivalently G ε ), it follows that such a function vanishes exactly on the spheres B(0, R) with R ∈ N (and maybe at k = 0). The union of all these spheres is a Lebesgue null measure set of R 3 . Exploiting point (iii) of Proposition 3.9, we see that ker R ε = {0} The properties in (ii)-(1) of the previous proposition are exactly the ones which characterize the mollifiers on R 4 (see [21,Section 1.6]). Let us denote the set of such functions by M (R 4 ).
Lemma 3.14 Let h ∈ M (R 4 ), then the following statements are true: At this point, if we stick to regularization families as in (ii) of Proposition 3.13, it is possible to complete the discussion started with equation (3.3). Indeed, equation (3.4) shows how the regularization operator acts in position space only for smooth solutions like P( · , f ). What about the remaining elements of H m ?
In the special case where the Fourier Transform of the regularization cutoff is a mollifier, it is possible to extend this result to the whole space H m . Indeed, since we already know that R ε f = f * h ε on H sc m and both R ε and · * h ε are linear continuous operators, they must coincide on the whole Hilbert space.
Theorem 3.15 Let g ε be is as in point (ii) of Proposition 3.13, then R ε u = u * h ε for any u ∈ H m . In this case the regularization is said to be of mollification type.
Assumption 3. 16 In the remainder of the paper we will always assume that a regularization family is fixed. Moreover, if not stated otherwise, we will always assume that it is of mollification type and thus of the form presented in Theorem 3.15.
Of course the regularization should return the original function in the limit ε → 0. This is true, even though in a distributional sense. In particular, if we apply the regularization operator to smooth functions which do not vary much on the scale ε, then we expect the modification of the function to be negligible.
In order to make this mathematically more precide define for any u ∈ C ∞ (R 4 , C 4 ): The following estimates then hold.
Proposition 3.17 The following properties hold for u ∈ H m 12 : Point (ii) shows that R ε defines a continuous functional at every point x ∈ R 4 .
Proof. The proof of points (i) can be carried out and (ii) (with minor adjustements) can be found in Example 1.2.4 of [8]. So, let us prove point (iii).
where we applied a multivariable version of the mean value theorem (see [1], Example 2 after Theorem 12.9) .

The regularized fermionic projectors
Referring to point (i) of Proposition 3.13, we can now define a regularized version of the fermionic projectors introduced in Definition 2.24.
Definition 3.18 The regularized fermionic projectors onto the positive and negative spectrum are defined as the linear mappings: Similarly, we define the regularized causal propagator by P c,ε := P −,ε − P +,ε .
As in the unregularized case, it is possible to represent the regularized fermionic projectors as kernel operators, that is: where, in this case, the kernel does define a regular function on R 4 × R 4 .

Proposition 3.19
The kernel of the regularized fermionic projector P ε,± is given by and has the following representation: Moreover, for every y ∈ R 4 and a ∈ C 4 it holds that P ±,ε ( · , y)a ∈ H ± m ∩ C ∞ (R 4 , C 4 ) and where Proof. The fact that P ε,± (x, y) does define a kernel for P ±,ε can be proved by direct inspection. The smoothness of P ±,ε ( · , y) and the fact that it solves the Dirac equation follow from the fact that G ε belongs to S p (R 4 , C 3 ) together with Proposition 2.19. Finally, identity (3.8) follows directly by plugging F(T y (h ε a))(k) = (2π) −2 G ε (k)a e iη(y,k) into (2.18).
Remark 3.20 If we compare the proposition above with point (v) of Remark 2.28, we see that the "maximally localized" distributional solutions of (2.1) are now replaced by the regular solutions determined by the initial data in Σ 0 : Note that Ω ± ε (a, y; · ) ∈ S p (R 3 , C 4 ). The corresponding solutions P ±,ε ( · , y)a are very picked around the light cone centered at (0, y), but are regular and do not diverge anywhere.

The doubly-regularized kernel of the fermionic projector
Given the importance in the rest of the paper, we also introduce the following object, which consists in a double regularization of the kernel P ± (x, y).
Let us analyse some features of these matrices: Proposition 3.22 Referring to Definition 3.21 the following properties hold.
Proof. Point (i) can be easily proved exploiting the definitions given so far. Let us pass to point (ii) and suppose that P ±,ε 2 ( · , y)a = 0 for some a ∈ C 4 . In particular if we evaluate this function at y, i.e. P ±,ε 2 (y, y)a = 0, we can get rid of the exponential in (3.10). At this point, making use of the three dimensional expression for the distributional integrals (see (2.16)) and noticing that (γ 0 a) † p ± (k) γ 0 a ≥ 0 as well as G ε (k) 2 ≥ 0, we conclude that: Now, w.l.o.g we can suppose that g ε = 0 on an open neighbourhood B(0, r) (g ε must be strictly non-zero up to a set of isolated points as proved in Proposition 3.13 and now we assume this is true at k = 0, but the proof would apply with minor changes also to any other point). As a consequence, we get (γ 0 a) † p ± (k)(γ 0 a) = 0 for any k ∈ B(0, r). Since p ± is a projector, we see from here that p ± (k)(γ 0 a) = 0, or equivalently γ 0 a ∈ W ∓ k , for any k ∈ B(0, r). Now, focus on the −-case, the other one being equivalent. Exploiting (2.14) we see that there exist scalars λ ↑,↓ such that: Comparing the 0th and 1st components we get λ ↑,↓ (k) = λ ↑,↓ (0) =: λ ↑,↓ and therefore the scalars do not depend on the point k. At this point, choosing for example k = (r/2, 0, 0) and comparing the 2nd and 3rd components we get λ ↑,↓ = 0. This gives γ 0 a = 0 and therefore a = 0, concluding the proof.
To conclude this section, in view of what will follow, it is useful to evaluate the explicit form of the regularized kernel on the diagonal. Exploiting (2.16) it is easy to see that At this point, using the fact that the function g ε is assumed to be rotationally invariant, the term involving the matrices γ i vanishes and we get the following result.
Proposition 3.23 For any x ∈ R 3 the doubly-regularized kernel has the following form: In particular, its spectrum consists of two elements, both with multiplicity two. More precisely: Proof. The explicit form of the eigenvalues follow directly from the diagonal expression of the kernel. In order to check the sign of λ − , notice first that 1 > mω −1 > 0 on R 3 \ {0} and therefore g 2 > 0 almost everywhere, this implies that g vanishes almost everywhere and therefore g = 0, by continuity. This is not possible.
Remark 3.24 A few remarks follow.
(i) Note that the limit ε → 0 is generally ill-defined as the integrals diverge.
(ii) In the intuitive picture of g ε as the characteristic function of B(0, ε −1 ), the kernel (3.12) and the eigenvalues (3.13) can be calculated explicitly by means of: Expanding the first term of in powers of mε (compare with Assumption 3.1), the leadingorder terms of the above quantities are: (3.14) These approximate indentites give some rough intuition on the dependence of the doublyregularized kernel on the microscopic regularization length. It should be kept in mind, though, that in this work the regularization function g ε is chosen as a smooth rapidly decaying function, and not as a brutally discontinuous cutoff.
As a conclusion of this section we want to point out another result, which will become important later on. The leading question is: How does the spectrum of P −,ε 2 (x, x) deviate from (3.13) when a bunch of positive (negative) energy physical solutions are added to (deleted from) the system?
The full meaning of this question will become clear later on, after the concept of a vacuum causal fermion system has been introduced. For now, let us state the following result, whose proof can be found in the Appendix and will be proved in the general case of positive or negative projectors.
for every couple of spacetime points x, y ∈ R 4 . In particular, we have: Now, let us go back to the fermionic projector onto the negative spectrum. The addition of positive (deletion of negative) energy physical solutions consists in the addition (subtraction) of terms like −(2π) −1 Ru(x) Ru(x) in (3.15) (See Section 4.5, in particular Proposition 4.28).
with the perturbation Proof. Exploiting identity (3.12) we see that the matrix P −,ε 2 (x, x) is symmetric, more precisely it is real and diagonal with two doubly-degenerate non-vanishing eigenvalues λ ± −,ε 2 . The thesis follows from Bauer-Fike Theorem (see Theorem IIIa in [2]).
Notation 3.27 For the sake of notational simplicity, we will drop the pedix − from P − from now one, as we will focus exclusively on the negative spectrum.

A comment on Lorentz invariance
As a conclusion of this chapter, we want to point out that introducing a regularization in momentum space, as presented in this work, unavoidably breaks the Lorentz invariance of the theory. More precisely, all the constructions carried out in this paper are referencedependent. This is evident, for example, if we have a look at the kernel of the regularized fermionic projector. Nevertheless, the invariance under spacetime translations is preserved.
Proposition 3.28 For any a ∈ R 4 the operator defined on the solutions space H m by Proof. The proof can be carried out working on the dense subspace H sc m and then taking the unique continuous extension.
It should be mentioned that the loss of Lorentz invariance is neither worrisome nor surprising, for the microscopic nature of physical spacetime is completely unknown, and there is no reason why Lorentz symmetry should be preserved on scales of the order of l P .

The emergence of causal fermion systems
Before entering the construction of the causal fermion system for Minkowski spacetime we need to introduce some preliminary definitions and results.

The general mathematical set-up of the theory
In this section we introduce the basic set-up of the theory, together with all the main mathematical objects of interest. It should be clear that the set F (H ) does not inherit any linear structure, the only operation we are allowed to take is the multiplication by scalars. Proof. 15 The fact that F (H ) is a double-cone is obvious, as σ(λF ε (x)) = λσ(F ε (x)) for any λ ∈ R. So, let us go to the proof of the closedness. First, notice that the set of operators with rank at most four is closed in the WOT topology and therefore if we have any sequence F n ∈ F (H ) which converges to F ∈ B(H ) in the uniform topology, then ran F ≤ 4. Now, let 0 < ǫ < 1 be arbitrarily small and take N ∈ N such that F − F n < ǫ for any n ≥ N . If we use the notation T := F , A := F n − F and S = T + A = F n , then we can follow the discussion in Chapter V, Section 4.3 of [11]. First of all, notice that the operator A is T -bounded with vanishing T -bound: In order to fit with the notation in [11], we prefer not to repeat the eigenvalues according to their algebraic multiplicity. Therefore we suppose by contradiction that there exists 0 < k ≤ 4 strictly positive eigenvalues λ 1 , . . . , λ k of T = F with algebraic multiplicities m i such that m 1 + · · · + m k ≥ 3. Let d i > 0 denote the isolation distance (defined in [11]) for the eigenvalue λ i . Choosing ǫ sufficiently small we see that the inequality (4.11) in [11] is trivially satisfied in our case (notice that the distance d i depends only on F and not on ǫ). The discussion therein shows that the total algebraic multiplicity of the eigenvalues of S = F n which lies in the interval (   The support of the measure ̺ plays an important role in the theory of causal fermion systems, in that it is believed to describe physical spacetime when the measure arises as a minimizer of a specific action (for more information see the causal action principle in 14 A double-cone of a linear space V is a subset which is closed under multiplication by real scalars.+ 15 The proof of this theorem is based on a discussion in Chapter V Section 4.3 of [11]. I would like to thank Christoph Langer for pointing this out. For a concise analysis on this topic and its connection with the theory of causal fermion systems the interested reader is referred to his forthcoming PhD thesis [13] [8]). In this paper we will not enter the details of this, but we will simply show that there exists a natural example of measure whose support realises a smooth manifold which is diffeomorphic to Minkowski spacetime.
In the following sections we will deal with causal fermion systems constructed out of closed subspaces of H m . As will become clear later, it is useful to study the relations which occur among such objects. More precisely: given two Hilbert spaces H 0 ⊂ H 1 , how are the corresponding spaces F related to each other? Notice that we can always write H 1 as Bearing this in mind, we can state and prove the following result. Lemma 4.5 Let H 0 be a closed subspace of a Hilbert space H 1 , then the function is a well-defined, one-to-one, norm-preserving (thus continuous) closed map. In particular, by means of this identification, the set F (H 0 ) defines a closed subset of F (H 1 ) and the map ι is an homeomorphism with onto its image.
Proof. It is trivial to prove that the map is one-to-one and norm-preserving. In particular the map is continuous with respect to the uniform topology. Now we prove that the map is also closed. If this is true, then the last statement is trivial, the set F (H 0 ) being closed within F (H 0 ) itself. So, suppose first that C is a closed subset of F (H 0 ) in the uniform topology and let {ι(x n )} ⊂ ι(C) be any sequence converging to T ∈ F (H 1 ) in the uniform topology. It being convergent, the sequence {ι(x n )} is of Cauchy type within F (H 1 ) and so the same holds true also for {x n }, the map ι being norm-preserving. Since C is closed there must exist some x ∈ C such that x n → x. Since the map ι is continuous we get T = lim n→∞ ι(x n ) = ι(x) and so T ∈ ι(C), i.e. ι(C) is closed. The last point follows immediately from the fact that ι is injective, continuous and closed.
Lemma 4.5 suggests that one can stick his attention to the larger space F (H 1 ) and this is indeed the case even in presence of a Borel measure, as we are going to explain. As already mentioned, not every operator in F is of physical interest: what is relevant for the theory is what is contained in the support of some specific Borel measure and this information is not loss when doing the identifications of Lemma 4.5. Let us discuss this in more detail.
Consider a causal fermion system on the Hilbert spaces H 0 with a measure ̺ 0 defined on the Borel σ-algebra of F (H 0 ). Suppose that H 0 identifies itself as a Hilbert subspace of H 1 . We know by Lemma 4.5 that F (H 0 ) can be embedded within F (H 1 ), but what about the measure? How can we read ̺ 0 as a measure on F (H 1 )? The most natural thing to do is to pushforward it by means of the map ι: Notice that this is well-defined, as the map ι is continuous and therefore measurable.
where we used the fact that ι is injective. By definition of a support of a measure, this means that all the points belonging to F (H ) \ ι(supp ̺ 0 ) cannot belong to the support of ι * ̺ 0 , the given set being open and of vanishing measure. More precisely we have F (H ) \ ι(supp ̺ 0 ) ⊂ F (H ) \ supp ι * ̺ 0 or, equivalently, supp ι * ̺ 0 ⊂ ι(supp ̺ 0 ). On the contrary, take any ι(x) ∈ ι(supp ̺ 0 ) and let A ⊂ F (H ) be any open set such that ι(x) ∈ A. Since the map ι is continuous, the set ι −1 (A) must be an open neighbourhood of x in F (H 0 ). Thus ι * ̺ 0 (A) = ̺ 0 (ι −1 (A)) = 0. Since the set A is arbitrary it must hold that ι(x) ∈ supp ι * ̺ 0 .
This results makes the identification complete: also the measure can be embedded as a measure on the larger space and no information is lost. In what follows we will always make use of these identifications, if not stated otherwise.

The emergence of causal fermion systems
At this point we are ready to see how such structures arise in relativistic quantum mechanics, when a regularization is introduced. The fundamental result is the following: Every ensemble of physical solutions of (2.1) gives rise to a representation of spacetime in terms of finite-rank self-adjoint operators.
Theorem 4.7 Let H be a closed subspace of H m , then for every x ∈ R 4 there exists a unique operator F ε (x) ∈ F (H ) such that Moreover the function F ε : Proof. The proof of existence and continuity can be found in [8], respectively in Sections 1.2.2 and 1.2.3. The uniqueness follows directly from the arbitrariness of u, v and the nondegeneracy of the scalar product.
Remark 4.8 The operator F ε (x) gives information on the densities and correlations of the physical solutions at the spacetime point x ∈ R 4 . For this reason the operator F ε (x) is called local correlation operator, while the function F ε will be referred to as the local correlation function.
The function F ε gives a realization of the spacetime events in terms of operators which collect in their image a few distinguished physical solutions which are relevant at the said location. As will become clear later on, for special choices of the closed subspace H , the corresponding local correlation functions are even homeomorphisms onto their image F ε (R 4 ). In such a way, spacetime realizes itself within F not only as a set, but also as a differentiable manifold.
In order to understand everything in terms of causal fermion systems, it is necessary to identify suitable measures ̺ on F which realize F ε (R 4 ) as their support. A possible and sensible choice for this measure is obtained by push-forwarding the Lebesgue-Borel 16 measure µ on R 4 to F through F ε : Notice that the function F ε is continuous and therefore the push-forward is well-defined. We can therefore give the following definition. As will be discussed at the end of this paper (see Proposition 6.10), the choice of (F ε ) * µ as the physical measure is a sensible choice in the case of H = H − m , which is believed to represent Minkowski vacuum. However in presence of particles or antiparticles, this choice might not be optimal. Nevertheless, it should be remarked that matter about which measure to choose is not important within the purposes of this paper.
Remark 4.10 A few remarks follow.
(i) From the continuity of F ε and the definition of (F ε ) * µ it is easy to see that: where the closure can be taken indifferently on F (H ) or B(H ), the former being a closed subset of the latter (see Theorem 4.2). In the special cases we will treat later on (e.g. the vacuum), the function F ε is injective and closed and, therefore, showing that the support of the measure is indeed a realization of Minkowski spacetime.
(ii) The construction of F ε depends heavily on the chosen regularization. This might be a problem if the regularization is seen just as a regularity tool which has to be removed afterwards, but it is not if the regularization is believed to play a physical role itself, as might be in modeling the microscopic structure of spacetime.
(iii) The function F ε depends on the chosen closed subspace H ⊂ H m and as such the representation of Minkowski spacetime through F ε depends on a specific choice of solutions of the Dirac equation. Such solutions carry information about spacetime itself and when specific and sufficiently large ensembles of them are gathered together the said representation becomes indeed faithful, as we we will see in Section 6.1.
(iv) Referring to Section 4.1, everything can be understood in the larger set F (H m ), and this will always be the case from now on. For simplicity of notation the set F (H m ) will be denoted simply by F .
(v) For the sake of compactness, we can denote the (regularized) causal fermion system simply by (H , F ε ), for the measure is always understood as (F ε ) * µ. Also, notice that the function F ε depends uniquely on the choice of the subspace H and therefore the CFS depends uniquely on H , too.
At this point, it is useful to analyse the relations between the causal fermion systems which arise from different ensambles of physical solutions. Consider two different Hilbert subspaces H 0 ⊂ H 1 of H m . As established with Theorem 4.7, we can construct two causal fermion systems, by means of the functions: These two functions are strictly connected to each other. Indeed, consider the orthogonal projector Π 0 onto the subspace H 0 , then for every u, v ∈ H 0 we have: Since the restriction of (·|·) to any closed subspace still defines a Hermitean inner product and u, v ∈ H 0 are arbitrary, we have just proven the following result.
To summarize, to every spacetime point x ∈ R 4 we have associated a bounded operator which encodes information on the densities and correlations of the physical solutions generating H at the spacepoint x ∈ R 4 .
A related and fundamental concept is given by the kernel of the fermionic operator which carries information about the correlations among the wave functions at different spacetime points x, y ∈ R 4 . Of course the two concepts have to agree when x = y. The most natural way to do this is consider the orthogonal projection of the local correlation operator at one point onto the image of the other.
Definition 4.12 Let (H , F ε ) be a CFS and x, y ∈ R 4 . The associated kernel of the fermionic operator is defined as the operator: with π x the orthogonal projector on ran F ε (x). In particular P ε (x, x) = F ε (x) for every x ∈ R 4 .
As the name suggests, this operators is strictly connected to the kernel of the fermionic projector of Definition 2.26, more precisely with the doubly-regularized one of Proposition 3.21. This will be analysed in detail in Section 4.5 where the motivation behind the choice for the definition of P will become more evident.

Correspondence to the four dimensional spinor space
In the previous sections we saw how spacetime can be represented in terms of physical solutions of the Dirac equation. More precisely, the solutions are incorporated in the spacetime points themselves through the operators F ε (x). At this point it is natural to wonder whether or not (and how, in case) the spinorial structure of the solutions can be retrieved in terms of the features of the operators F ε (x). This is indeed possible to some extent and is the content of the this section.
Let us fix a closed subspace H of H m together with its local correlation function F ε and bear in mind that everything is embedded within the larger space H m , also F ε .
Let us start our analysis with the linear space C 4 (where the spinors take values), equipped with the spin inner product ab := a † γ 0 b. The closest object to C 4 that we have at our disposal is the set ran F ε (x), whose dimension is indeed not larger than four.
From the self-adjointness of F(x) we have the orthogonal decomposition.
Moreover it also holds that F ε (x)(S(x)) ⊂ S(x). The set S(x) is a good candidate to provide a realization of C 4 , but in order to reach a full identification we need to endow it with a inner product of signature (2, 2) and find some canonical unitary mapping connecting the two spaces. The first claim is indeed always true, as shown in the next proposition. The second statement is true only under suitable assumptions.
Proposition 4.13 Let x ∈ R 4 , then the linear space S(x) can be equipped with a nondegenerate indefinite inner product 18 of signature (2, 2) by defining: The couple (S(x), ·|· x ) is called the spin space at x.
Proof. It is clear that ·|· x is a sesquilinear form. Also, u|v x = −(u|F ε (x)v) = −(F ε (x)v|u) = −(v|F ε (x)u) = v|u x . Now, let us check the non-degeneracy. Fix any u ∈ S(x) and suppose that v|u x = 0 for any v ∈ S(x). Consider any ω ∈ H , then ω = s + t, with s ∈ S(x) and t ∈ N(x), and so (ω|F(x)u) = (s|F(x)u) = 0. Since ω is arbitrary, this implies F ε (x)u = 0. Therefore we have u ∈ ran F(x) ∩ ker F(x) = {0}. The signature of the inner product follows from the assumptions on F .
Notation 4.14 The adjoint in this inner product space will be denoted by * .
The realization of a unitary mapping between S(x) and C 4 is unfortunately not always possible, as the dimension of S(x) is not necessarily maximal. This may happen, for example, when the causal fermion systems is generated by an ensemble of physical solutions which is not large enough or, more simply, is not carrying enough information (more on this later on). On the other hand, when the dimension happens to be maximal, the system turns out to be rich of interesting features. In the remainder of the paper we will focus mainly on such cases. Given this definition we can now prove the following theorem: It is always possible to embed (canonically and) isometrically the space S(x) within C 4 (as non-degenerate indefinite inner-product spaces), but unitarily only for regular points.
Theorem 4. 16 For any x ∈ R 4 , referring to the corresponding indefinite inner products, the function is a linear isometry. The point is regular if and only if this function is surjective.
Now, the non-degeneracy of ·|· x guarantees that Φ x is injective. The last statement follows from the rank-nullity theorem.
Lemma 4.17 Let (H , F ε ) be a causal fermion system, then for any spacetime point x ∈ R 4 the following statements hold.
(ii) If x is regular, then the set inclusion in point (i) can be substituted by an equality.
(v) The point x is regular for every causal fermion system (H 1 , F ε 1 ) such that H ⊂ H 1 .
Proof. Let us start with point (i). Consider any u ∈ H such that R ε u(x) = 0, then we have the identity (v|F(x)u) = −R ε v(x)R ε u(x) = 0 for any v ∈ H . Since the vector v is arbitrary, we get F(x)u = 0, i.e. u ∈ N(x). Now, let us prove point (ii). Suppose the spacetime point is regular and consider u ∈ N(x) ∩ H . Since the point is regular, we can always find vectors v µ ∈ S(x) ⊂ H such that R ε v µ (x) = e µ . Now we have R ε u(x)e µ = R ε u(x)R ε v µ (x) = −(u|F(x)v µ ) = −(F(x)u|v µ ) = 0, which gives R ε u(x) = 0. Point (iii) is obvious by the linearity of R ε and points (i),(ii). Let us now prove point (iv). If the point is regular then the thesis follows immediatetly, just take u µ := Φ −1 x (e µ ). On the contrary, suppose there are functions u µ ∈ H as in the assumptions. Without loss of generality we can always assume that R ε u µ (x) = γ 0 e µ . Take any v ∈ N(x) ∩ H , then (e µ ) † R ε v(x) = R ε u µ (x)R ε v(x) = −(u µ |F ε (x)v) = 0, which implies R ε v(x) = 0. At this point notice that u µ = π x u µ + n µ for some n µ ∈ N(x) ∩ H (see (4.7)) and therefore γ 0 e µ = R ε u µ (x) = R ε (π x u µ )(x) + R ε n µ (x) = R ε (π x u µ )(x). Since π x u µ ∈ S(x), the thesis follows by definition of the isometry Φ x . To conclude, notice that point (v) follows directly from point (iv).
At this point, fix a causal fermion system (H , F ε ) and consider the closed subspace generated by all the spin spaces S(x). Exploiting the properties of the orthogonal operation we have: If a vector v ∈ H belongs to the orthogonal of S, then F ε (x)v = 0 for every x ∈ R 4 and, therefore, it would be of no use. Therefore, only the former space should be interpreted as the (effective) physical Hilbert space of the system. Proof. Let v ∈ H be orthogonal to every subspace S(x), then for any x ∈ R 4 : Since the CSF is regular, there always exist u µ ∈ H such that R ε u µ (x) = e µ for µ = 0, 1, 2, 3. This implies R ε v(x) = 0. As x is arbitrary, we have v ∈ ker R ε = {0}, i.e. v = 0.
Remark 4.20 It should be mentioned at this point that in more general cases where ker R ε is not trivial, the physical space S does not coincide with the original space H . Not even in the case of regular systems.

Some further correpondences to spinors for regular systems
Fix a causal fermion system (H , F ε ) and a regular point x ∈ R 4 . In the previous section we saw that the function Φ x defined in (4.8) determines an isometry between S(x) and C 4 as indefinite inner product spaces. What can be said if we endow C 4 with the Euclidean positive definite inner product (λ, σ) → λ † σ? Remember that the two inner products are related bȳ λ σ = λ † γ 0 σ where γ 0 is the zeroth Dirac matric which satisfies Also, we are equipped with other three Dirac matrices γ i , which fulfill The matrices γ µ can be lifted to operators on S(x) as follows: It is clear that this linear map satisfies This linear map can always be understood as an operator on the whole space H , just by rewriting it asΓ µ (x) := Γ µ (x) π x . At this point we can consider a new set of operators, namely F ε,µ (x) := F ε (x)Γ µ (x).
Proposition 4.21 Let x ∈ R 4 be regular for (H , F ε ), then the mapsΓ µ (x) satisfy: Proof. Point (i) follows directly from the definition. To prove point (ii) choose (dropping the index ε for simplicity) any u, v ∈ H : , from which point (iii) follows. Point (iv) follows immediately by noticing that Γ µ (x) and F ε (x) are bijections on S(x) and that F µ (x) is self-adjoint.
At this point we can state the following important representation result, which plays as counterpart of Theorem 4.7, Proposition 4.13 and Theorem 4.16.
Proposition 4.22 Let x ∈ R 4 be regular, the for every couple u, v ∈ H it holds that: In particular S(x) can be endowed with a positive definite inner product by defining: which makes Φ x a unitary mapping onto C 4 equipped with the Euclidean inner product.
Proof. Let us start with the first equality. So, take any u, v ∈ H , then: Now, if we show that (·|·) x defines a positive definite inner product on S(x), then the last statement will follow directly from (4.11). The sesquilinearity and Hermiticity come from the analogous properties of (·|·) and the self-adjointness of F ε,0 (x). The positivity follows from (4.11) and the positivity of the Euclidean inner product.

Some physical interpretations
Consider a finite dimensional subspace U ⊂ H m and let {u i } i=1,...,n , {v i } i=1,...,n be two arbitrary orthonormal bases of its, then the corresponding alternating tensor products (the particle are fermions) are connected by: where M is the unitary matrix transforming one basis into the other, in particular det M ∈ U(1). Therefore, being equal up to a phase, the multiparticle states are physically equivalent. This shows that there exists a one-to-one mapping between n-particle (pure products) states and n-dimensional subspaces of H m .
Bearing this in mind, while a wedge products of infinitely many orthonormal states has no mathematical meaning, one could imagine to describe an infinitely large ensamble of fermions by means of the infinite dimensional subspace of H m generated by the corresponding orthonormal physical solutions. Following the original idea of Dirac, one could then interpret the whole subset H − m as the "multiparticle state" containing all the negative energy particle states and interpret it as the vacuum. The addition of positive energy states and the removal of negative energy states correspond instead to the presence of particles or antiparticles. This interpretation is revived in the context of causal fermion systems.
For example, H − m is the smallest (non-trivial) ensamble of physical solutions whose causal fermion system carries a unitary representation of the translation group: see Proposition 3.28 and Proposition 5.4. This is no longer true when states are added or deleted from the system, i.e. when the ensamble is modified (in a non trivial way). Therefore, the causal fermion system arising from H − m is a sensible choice for describing a vacuum (the same could be said for H + m , but the two construction would be unitarily equivalent). For the sake of mathematical tidiness, we introduce the following notation. Given any finite dimensional subspaces U + ⊂ H + m and U − ⊂ H − m , we define: Similarly, If we have a combined system with some states in the negative spectrum and some states in the positive spectrum, we write: Remark 4.24 It should be noticed that the theory here presented does not encompass the description of entangled states, in that the identifications discussed above hold only for pure products of single-particle states.
The following result follows directly from Proposition 4.21, bearing in mind that the trace does not depend on the chosen Hilbert basis and that F ε (x) vanishes on the orthogonal of S(x). It should be mentioned at this point (it will be proved in Section 5.1) that the causal fermion system arising from H − m is indeed regular at any point x ∈ R 4 . Proposition 4.25 Let (H , F ε ) be a CFS, x ∈ R 4 be regular and let N x , N be Hilbert bases of S(x) and H , respectively, then (4.14) In particular the left-hand series always exists and does not depend on the chosen basis.
The result above has a direct physical interpretation. Sticking to the standard interpretation of quantum mechanics, given a normalized smooth physical solution ψ, the value J ψ (t, x) := ψ(t, x)γ µ ψ(t, x) defines the current density at time t ∈ R 3 . A regularization consists in the substitution which has to be interpreted as the (regularized) current density at time t. If we are given a system with more then one particle, the total current density consists of the sum of the single densities. In particular the series in the left-hand side of (4.14) can be understood as the current density of the system where all the states (a Hilbert basis to more precise) in H are occupied. Notice that this is finite also for infinite dimensional subspaces like H − m . Nevertheless, in the limit ε → 0, this sum generally diverges. In order to get a finite quantity, one can proceed as follows.
Theorem 4.26 Let U ± ⊂ H ± m be finite dimensional subspaces with Hilbert bases N ± , then where F is the causal fermion system relative to a(U − , U + ) and and the thesis follows.
Remark 4.27 After the realization of R 4 in terms of local correlation operators F ε (x), the particle current density is addressed as a property of spacetime alone, no longer of the physical solutions (or wave functions).
To conclude this section we want to show how the kernel of the fermionic operator looks like when lifted to C 4 through the action of the isometries Φ x .
Consider an arbitrary Hilbert basis {u n } n∈N of H , then the completeness relation gives: n∈N π x u n π y u n |π y · y (4.16) From this equation we see that the correlation between the points x, y comes as the result of the contribution made by all the the basis elements at the given points. More precisely, exploiting the definition of the isometry (4.8) we have the following.
With this result, the statement above is clearer, in that all the physical solutions generating H give their contribution in the correlation between x, y ∈ R 4 . We will go back to this representation in Section 5.4.
Remark 4.29 Equation (4.17) shows that the choice for P ε made in Definition 4.12 was indeed sensible, as it does realize a correlation function between two points, once represented within C 4 .

On the regularity of causal fermion systems
In section 4.3 we introduced the concept of a regular causal fermion system (see Definition 4.15). It is interesting, now, to study in which situations this is indeed the case. As one could expect, the vacuum realises a regular causal fermion system . Moreover this feature is not lost when additional positive energy physical solutions are added to the system: in line with the concept of a Dirac see one could interpret this process as an enrichment of the system and therefore it is reasonable that the regularity is preserved. On the other way around, the removal of negative energy solutions might break the regularity as we will see in an straightforward example. Only in very special cases the regularity is preserved, namely when the physical solutions do not vary too much on the microscopic scale ε. In this terms we see an asymmetry between particles and antiparticles, at least within the limits of validity of this interpretation.

The regularity of the vacuum
The aim of this section is to show that the causal fermion system associated with H − m , i.e. the vacuum, is regular.
(a) Let us start by considering the smooth function G (a) ∈ S p (R 3 , C) defined by: The L 2 and L 1 norms can be easily computed: Now, let us substitute λ ↑,↓ = (2π) 3/2 G (a) into (2.26), then, taking into account (2.14) and the fact that the Gaussian and energy functions are even, the corresponding physical solutions u Let us discuss G (b) L 2 first, which will become useful later on. Exploiting the inequalities (k 3 ) 2 ≤ |k| 2 and m ≤ ω(k) + m ≤ 2ω(k), a direct -but tedious -calculation leads us to: Now, as we did in point (a), we substitute λ ↑,↓ = (2π) 3/2 G (b) into (2.26) and evaluate the correponsing physical solutions u ↓ at x = x 0 . Exploiting the form of the fundamental solutions (2.14) it is not difficult to see that the first, second and third components of u (b) ↑ as well as the zeroth, second and third ones of u (b) ↓ vanish, for the function in the integral is odd with respect to k 3 . We are left with the integral More precisely: u Concluding, we see that the vectors u (a),(b) ↓,↑ (x 0 ) are orthogonal to each other and, therefore, linearly independent. We want to show that this feature is preserved if we regularize the corresponding functions, that is the vectors R ε u ↓,↑ (x 0 ) are linearly independent.
In order to do this, we need to carry out some upper estimate on the derivatives of the wave functions. Exploiting the inequalities in (ii) of Lemma (8.2), the inequality |k µ | ≤ |k|+m which holds for every µ = 0, 1, 2, 3 and (ω(k) + m)(|k| + m)|k 3 | ≤ (|k| + 2m) 2 |k| a tedious (not long, tough) calculation gives for any x ∈ R 4 : Exploiting these estimates we are know ready to state the final result.
Proposition 5.1 Let u (a) , u (b) be the special solutions defined above with σ = m, then ↓ (x 0 ) are linearly independent vectors of C 4 .
In order to prove this, we need a technical result whose proof is left to the reader.  Since the point x 0 ∈ R 4 was chosen arbitrarily, the following theorem follows.
Theorem 5.3 The causal fermion system associated with H − m (i.e. the vacuum) is regular.
To conclude this section we prove the following result, which shows how the operators F ε (x) for the vacuum system do not only have full-rank, but are also unitary equivalent to each other.
Proof. First notice that U a (H − m ) ⊂ H − m as proved in Proposition 3.28. Now, fix any u ∈ H − m and a ∈ R 4 , then by definition we have: Therefore it follows that:

The presence of holes in the Dirac sea
Before studying this case in detail, we want to show that in the general case of a removal of negative energy solutions the regularity might be lost. Indeed, consider the causal fermion system generated by the vacuum (H − m , F ε vac ) and choose any spacetime point x ∈ R 4 . We already know that the linear subspace S(x) has dimension four, because of regularity. So, we can think of removing exactly the physical solutions corresponding to the finite dimensional subspace S(x) itself: and realise its causal fermion system (a − (S(x)), F ε 0 ). From Proposition 4.11 we know that where Π 0 is the orthogonal projector onto a − (S(x)). Since the operator F ε vac (x) is self-adjoint, it satisfies ker F ε vac (x) = ran F vac (x) which implies F ε 0 (x) = 0. Indeed, let u ∈ H m be any arbitrary vector, then Π 0 u ∈ a − (S(x)) ⊂ ker F ε vac (x) and therefore F ε 0 (x)u = Π 0 F ε vac (x) Π 0 u = 0.
In particular the causal fermion system (a − (U 0 ), F ε 0 ) is not regular at x. The removal of the physical solutions which are relevant for the given spacetime point creates a critical defect in the local correlation function.
Remark 5.5 Note that this argument does not apply to the positive energy case. Indeed, consider the causal fermion system constructed out of a + (U ) for some U ⊂ H + m and fix any x ∈ R 4 , then the spin-space S(x) is not a subset of H + m in general, but of the whole space H − m ⊕ U : its vectors have components also in the negative energy spectrum. Therefore their removal from a + (U ) does not correspond to the annihilation of positive energy solutions.
This simple example shows that the removal of even a finite number of particles from the Dirac sea can perturb the system in such a way that it is no longer regular. Nevertheless, the functions belonging to S(x) are quite peculiar: they are very picked around the light cone centered at x and they diverge on it in the limit ε → 0. However, one might wonder if the removal of slowly varying solutions might preserve the regularity. The discussion in the following section shows that this is indeed the case and provide some sufficient conditions under which the creation of holes in the Dirac sea does not perturb the regularity.
Mathematically speaking, we are interested in the causal fermion system constructed out of a finite dimensional closed subspace U ⊂ H − m like: The strategy is the same as above: we need to find four elements in a − (U ) whose values at x 0 are linearly independent. This is fairly more complicated then the vacuum case, since the constraint u ∈ U ⊥ is tricky to control while carrying out an argument like above. On the other hand, given their convenience, we do not want to give up Gaussian-like solutions, although they generally do not belong to a − (U ). In what follows we show how all this can be put together.
(1) In this first part we exploit the density of the Schwartz functions to construct a set of orthonormal smooth vectors which span a subspace which is "as close as possible" to U .
Remark 5.7 A few remarks follow.
With the symbol |ψ(x)| we denote the C 4n -norm of the vector ψ(x), more precisely: where ̺ ψ i is the probability density of ψ i . The quantity ̺ ψ is to be understood as the number of particle density associated with the multiparticle state ψ.
(2) In this second part we construct smooth solutions which are orthogonal to U . More specifically we define a specific way of projecting the spaceÊ 0 (P − (S p (R 3 , C 4 )) onto the spaceÊ 0 (P − (S p (R 3 , C 4 )) ∩ a − (U ).
(3) In this third part we estimate the norm of the vector λ(ϕ) ∈ C n appearing in (5.9).
Given the finite-dimensional subspace U of H − m , it is clear that such states ψ i give good smooth approximations of the states within U . We can then give a new definition.
Definition 5.10 Let U be a finite dimensional subspace of H − m , then a family of physical solutions {ψ 1 , . . . , ψ} ⊂Ê 0 (P − (S p (R 3 , C 4 ))) as in Proposition 5.9 is called an approximating set of smooth states for U .
We are ready to discuss the regularity of the causal fermion system in presence of some holes. So, fix an approximating set of smooth states for U . The features of the operator Ψ and the boundedness of the scalars λ i allow us to mimic the discussion we did for the vacuum in the previous section. The idea is to consider the functions Ψ[u] with u the solutions defined in (a) and (b) of Section 5.1, apply the regularization operator and find some suitable σ which makes their values at x 0 linearly independent.
For simplicity of notation, let u α with α = 0, 1, 2, 3 denote the solutions u (a),(b) ↑,↓ defined in Section 5.1 (which depend on a parameter σ), C α (σ) the associated scalar β(σ) or ±γ(σ) and χ α the corresponding correction term − n i=1 λ i (u α )ψ i given in Proposition 5.8. We need to check if the estimate is satisfied for any α. If this holds true, then Lemma 5.2 would conclude the argument.
The quantity between parentheses plays an important role in the following discussion and deserves its own notation.
Definition 5.11 Let {ψ 1 , . . . , ψ n } ⊂ H m ∩ C ∞ (R 3 , C 4 ) be any orthonormal set and x ∈ R 4 , then the quantity is called the microscopic behaviour distribution of the multiparticle state.
Remark 5.12 This quantity gives information about the local behaviour of the particle number distribution and about its variation on the ε-scale.
At this point, reasoning as in the derivation of the inequalities (5.6), setting σ = λm (with λ arbitrary dimensionless parameter) and making use of (5.2) and (5.4), we get -after some tedious calculations and simplifications: If we choose λ = (ε 0 /ε) 1/2 with some arbitrary length parameter ε 0 , the right-hand sides above become: At this point, we see that the desired condition A < 1/4 could be obtained by taking ε small enough. Nevertheless, this argument is not legitimate in that we required the length parameter ε to freely vary within the interval (0, ε max ). Only at the very end a specific choice for ε should be made, which is based upon physical arguments and cannot depend on the specific choice of the physical solutions. The problematic term is the addend containing the microscopic behaviour function, which might be pretty large. The remaining terms involve only positive powers of the microscopic scale ε and are generally pretty small. Therefore we need to make some assumptions on the quantity E(ψ, ε, x 0 ).
Let us start with the limiting case E(ψ, ε, x 0 ) = 0. If we choose ε 0 = ε, then the above inequalities can be simplified to: Therefore, if we take mε < 10 −15 as in Assumption 3.1, we safely get the estimate A < 1/4. Unfortunately, the condition E(ψ, ε, x 0 ) = 0 is generally impossible to achieve: indeed, as discussed briefly in Section 2.4, smooth negative energy solutions vanish basically nowhere at every fixed time t ∈ R, more precisely: supp u(t, ·) = R 3 for all u ∈Ê 0 (P − (S p (R 3 , C 4 )).
Physically speaking, we require a low particle number density around the point x 0 and a low variation of the physical solutions compared to the microscopic scale ε.
Proposition 5.13 Let u (a) , u (b) be the special solutions defined in Section 5.1 with σ = 10 8 m and suppose that U admits an approximating smooth set of states {ψ 1 , . . . , ψ n } that fulfills E(ψ, ε, x 0 ) < 10 9 · m 3/2 . (5.13) Then the following vectors are linearly independent: Remark 5.14 Condition (5.13) can be understood as a "macroscopic" behaviour of the physical solutions ψ i at the spacetime point x 0 with respect to the microscopic scale ε.
This suggest the following definition: Definition 5.15 A finite dimensional subspace U ⊂ H − m is said to be ε-macroscopic at x ∈ R 4 if it admits an approximating smooth set of states which fulfills the macroscopic condition (5.13).
Concluding we have the following result: m be a finite dimensional subspace which is ε-macroscopic at x ∈ R 4 , then the associated causal fermion system (a − (U ), F ε ) is regular at x.

The general case of particle and antiparticles
The addition of positive energy particles to the vacuum -or to a regular system with holesmerely add information to the system and, as such, it should not affect the regularity of the causal fermion system. This is indeed the case, as we now go to prove.
Suppose we are given a finite-dimensional subspace U − ⊂ H − such that a − (U − ) is regular, such as as a ε-macroscopic system or the vacuum itself. If now we add some positive energy physical solutions, the relevant Hilbert space changes into a(U + , U − ) = a − (U − ) ⊕ U + for some finite dimensional subspace U + ⊂ H + m . Since this Hilbert space contains a − (U − ), we can apply Lemma 4.17-(v) and get the following.
Corollary 5.17 Let U ± ⊂ H ± m be finite dimensional subspaces and suppose that a − (U − ) is regular at x ∈ R 4 , then a(U − , U + ) is also regular at x.
In particular the above result apply in the special case of U − = {0}, i.e. when only particles (and not antiparticle) are added to the system. Corollary 5.17 shows that no assumptions need to be made in order to get a regular system, differently from the antiparticle case. This shows an asymmetry in the concepts of particle and antiparticle, at least within this setting.

Characterization of the vacuum local correlation operators
In this section we stick to the vacuum, i.e. to the causal fermion system generated by H − m and conclude the discussion started at the end of Section 4.5. In the previous section we checked that the vacuum indeed defines a regular system and therefore every spin space S(x) is isometric to C 4 by means of the function Φ x (see Theorem 4.16). We can then resume the discussion with the following important result.
Proof. The proof comes from Proposition 3.25 and Equation (4.17).
At this point, we want to make use of this identifications and study the spectral properties of the operators F ε (x). From Proposition 5.4 we already know that the local correlation operators F ε (x) all have the same spectrum σ vac . Thanks to the definition of P ε (x, y) and Theorem 5.18 it is clear that the spectrum of F ε (x)| S(x) coincides with the spectrum of 2πP ε 2 (x, x). Therefore we can refer our analysis to this matrix and exploit Proposition 3.23.
Proposition 5.19 Referring to Proposition 3.23, for every x ∈ R 4 it holds that: Going even further, we can now make use of the above representation and investigate the properties of the physical solutions which generate S(x) for any fixed x ∈ R 4 . In particular, we focus on the eigenvector basis of the local correlation operator F ε (x).
Proposition 5.20 For any spacetime point x ∈ R 4 the following statements hold.
(i) A vector u ∈ H − m belongs to S(x) if and only if there exists a ∈ C 4 such that u = P ε ( ·, x)a. (5.14) More precisely, for every u ∈ S(x) there exists a unique a ∈ C 4 such that (5.14) holds.
(ii) The action of the local correlation operator on u ∈ H − m is given by (iii) In particular, a linear basis for S(x) is given by the four linearly independent functions e x,µ := P ε ( · , x)e µ (5.15) Moreover, it holds that U a (e x,µ ) = e x+a,µ for every a ∈ R 4 .

.16)
Proof. Point (i). First, notice that P ε ( · , x)a ∈ H − m for any a ∈ C 4 as proved in Proposition 3.19. Now, by definition a vector u ∈ H − m belongs to S(x) if it can be written as u = F(x)w for some element w in H − m . Therefore, at every other spacetime point y ∈ R 4 we have: where we made use of Proposition 3.22. Exploiting Lemma 4.17 and 4.19 we have: . On the other hand, take a ∈ C 4 and consider the solution P ε ( ·, x)a ∈ H − m , then by regularity there always exists some w ∈ H − m such that R ε w(x) = a and therefore we have: Reasoning as above, the arbitrariness of y ensures that 2π P ε ( ·, x)a = F ε (x)w ∈ S(x). To conclude point (i), notice that the uniqueness of a in (5.14) follows easily by first applying R ε to both sides of (5.14) and then making use of Proposition 3.22. Now, let us prove point (ii). Take u ∈ H − m . Thanks to point (i) there exists unique a ∈ C 4 such that F ε (x)u = P ε ( · , x)a. At this point, exploiting the isometry Φ x and F ε (x) = P ε (x, x) we get: At this point, arguing as in the proof of point (ii) of Lemma 3.22, we see that a = 2π Ru(x). The proof of point (iii) can be checked easily, by applying the regularization operator, exploiting lemma 3.22 and noticing that S(x) is four dimensional. To conclude, let us prove point (iv). Exploiting points (ii) and Proposition 3.23 we have: The proof is complete.
Remark 5.21 At every spacetime point x ∈ R 4 , the spin space S(x) of the vacuum collects together all the "maximally localized" physical solutions P ε ( · , x)a at the given point.
As a side result we can discuss how the eigenvalues of this operator change when some particles or antiparticles are added to the system. Consider finite dimensional subspaces U + ⊂ H + m and U − ⊂ H − m , then the local correlation operators of a ± (U ± ) are self-adjoint on the corresponding spin spaces and as such they can be diagonalized with real eigenvalues. For simplicity, we consider the case of smooth physical solutions. First, notice that, in the case of regular systems, the kernel of the fermionic operator lifts to C 4 as the following matrix (see equation (4.17)): where {e ± i } i=1,...,n ± are Hilbert bases of U ± . Theorem 5.22 Let U ± ⊂Ê 0 (P ± (S p (R 3 , C 4 ))) be finite dimensional and suppose that the causal fermion system a(U − , U + ) is regular at x ∈ R 4 , then the corresponding kernel P ε (x, x) have four non-vanishing real eigenvalues {λ i } i=1,2,3,4 (counting algebraic (=geometric) multiplicities) which satisfy where e = e + ∪ e − , with e ± any Hilbert basis of U ± .
Proof. The thesis follows from Proposition 3.17, Proposition 3.26 and the following: where we used AB 2 ≤ A 2 B 2 and γ 0 2 = 1.

Remark 5.23
The above result shows that in presence of particles or antiparticles the eigenvalues of the fermionic projector change within an interval which is determined by the local behaviour of the wave functions around the point.

On the smooth manifold structure of regular systems
In this final section we finally prove how, and under which assumptions, regular causal fermion systems realize differentible manifolds within F . More precisely the goal is to show that F ε : R 4 → F is a closed function which is a homeomorphism on its image. If this is proven to be true, then F ε (R 4 ) = supp ̺ and F ε itself realises a global chart which makes it a smooth manifold. The proof is divided into two steps.

Step 1: injectivity of the local correlation function
The function F : R 4 → F can be seen as a representation of spacetime in terms of operators. The question is: to which extent are we still able to distinguish between spacetime points in R 4 ? Do we loose information in representing R 4 as operators? In other words: Is the local correlation function F ε : R 4 → F faithful?
The general answer is negative. As we already mentioned before, choosing poor ensembles of physical solutions might end up in a critical loss of information, as in the case of regularity for example. The same goes for the injectivity and here we show a simple example of this.
Consider the vacuum causal fermion system (H − m , F ε vac ) and take any couple of different spacetime points x = y. If we now remove the solutions corresponding to the subspace U 0 := S(x) + S(y), i.e. we construct the causal fermion system for the set a − (U 0 ), the corresponding function F ε 0 satisfies (see Lemma 4.11): This equality can be proved with an argument analogous to the one used in proving the loss of regularity in presence of holes. At this point it is clear that the injectivity is lost.
Neverthelessm for special subspaces of H m this is instead true. The first example is of course the vacuum. Then we will show that the addition of positive energy solutions does not affet injectivity (as we expect). On the other hand, in creating holes we have to choose the solutions very carefully in order to preserve the faithfulness. This is the content of the current section.

The Vacuum
Let us start with the vacuum. In order to prove the injectivity we make use of special solutions, Gaussian wave-packets: It is well-known that these functions converge to the Dirac delta in the limit σ → 0: p .
Now, consider the negative energy solution u At this point, performing the regularization of this function, we obtain: Exploiting the properties of G After these preliminaries, we are ready to prove the most important result of this section: Proof. Suppose there exists two points x, y such that F ε vac (x) = F ε vac (y). Applying the definition, we see that Now, consider the special case of u = u (σ) q for arbitrary p ∈ R 3 , then, exploiting (6.2), we get Without loss of generality we can assume that g = 0 on a neighbourhood B δ (0) ⊂ R 3 (the cutoff function must be non-zero everywhere up to a set of isolated points and we assume this is true at 0 for simplicity: for a different point the proof still applies with minor changes). Therefore, if we take p within this set we can divide the above equations by g ε (0) and g ε (p) and get: Now, notice that the function (0, δ) ∋ t → ω(p t ) − m = √ t 2 + m 2 − m is continuous and strictly monotonically incresing. From this it is not difficult to see that t x and t y must coincide.
At this point, going back to equation (6.4), we can get rid of the time component and reduce it to the identity: e ip·(x−y) = 1 for every p ∈ B δ (0). Taking p = te i with t ∈ (0, δ), the above equality gives e it(x i −y i ) = 1 for all t ∈ (0, δ), i = 1, 2, 3 Reasoning as above we can easily prove that x i = y i for any i = 1, 2, 3. This concludes the proof.

The general case of particles and antiparticles
Now the question: what happens if we add particles or antiparticles? As in the case of regularity, the faithfulness of F ε is preserved if additional positive energy physical solutions are inserted into the system, while the creation of holes is much more a delicate matter. Again, we need to find suitable assumptions in order not to lose too much information.
In the case of regular systems, a sufficient condition is presented in the following theorem. Remember that a sufficient condition for a system with holes to be regular is the assumption of ε-macroscopicity (see Theorem 5.16). Again, for simplicity we stick to the smooth case. In particular, we assume that the solutions describing the particles and antiparticles have compactly supported momentum distributions.
) be finite dimensional subspaces and suppose the causal fermion system a − (U − ) is regular, then the local correlation function F ε : R 4 → F of the causal fermion system a(U − , U + ) is an injective mapping.
Proof. It is sufficient to focus on the case U + = {0}. Indeed, once we have proven that a − (U − ) admits an injective local correlation function, then the injectivity for a(U − , U + ) would follow directly from Proposition 4.11. So, let us consider the purely negative case. By F ε vac we denote the local correlation funcion associated with H − m , i.e. the vacuum. So, fix any two different points x, y ∈ R 4 and suppose by contradiction that F ε (x) = F ε (y). This can be rewritten as Π(F ε vac (x) − F ε vac (y))Π = 0, where Π is the orthogonal projector on a − (U − ) ⊂ H − m . Since the CFS is assumed to be regular, there must exist at least one element u ∈ a − (U − ) such that R ε u(x) = e 0 (this will be used later). The contradiction hypothesis implies that (F ε vac (x) − F ε vac (y))u ∈ U − and therefore, exploiting Proposition 5.20: Since the Fourier transform is a bijection on S (R 3 , C 4 ), the above identity implies At this point, notice that the function g ε cannot be of compact support, as we know that it vanishes at most on a countable family of spheres which are isolated from each other (see the proof of Proposition 3.13). Therefore, bearing in mind that g ε is rotationally symmetric, there must exists some annulus B(r, R) ⊂ R 3 (0 < r < R) on which g ε > 0 and ϕ ≡ 0 and therefore: which implies γ 0 R ε u(x)e iη(k,x−y) − R ε u(y) ∈ W + k for all k ∈ B(r, R) (see Section 2.2), or equivalently (exploiting the form of the gamma matrix γ 0 ): for some continuous functions λ + ↑,↓ . At this point, we finally exploit R ε u(x) = e 0 which gives: From the third identity, taking k = se 1 and k = se 2 with arbitrary r < s < R, we see that sλ ↓ = −isλ ↓ which implies λ ↓ = 0. The last identity ,then, reduces to: At this point, choosing k = se 1 we get [R ε u(y)] 2 = 0 and therefore λ ↑ (k)k 3 = 0 for any k ∈ B(r, R). The continuity of λ ↑ yields λ ↑ (k) = 0 on B(r, R). In particular, the first identity in (6.6) becomes: At this point we can reason as in the proof of Theorem 6.1 and get t x = t y and x = y.
Remark 6.3 It should be remarked that the assumption of momentum distributions with compact support was crucial only for the negative energy solutions and it was not used for the positive energy solutions. In fact, this choice was based merely upon a taste for simplicity and the latter could be chosen arbitrarily within H + m . Again, this reveals an asymmetry between particles and antiparticles.
Again, we see that our choice of H − m as representing the vacuum is sensible. Adding positive energy solutions to H − m we enrich our system by putting in more information and the injectivity (as well as the regularity) is not lost. On the other way around, the removal of negative energy solutions might end up in a critical loss of information which degenerates in a loss of regularity and/or injectivity. This is prevented if the removed solutions are not relevant in the construction of the causal fermion system: in other word if their distribution on spacetime is regular enough.
Remark 6.4 Notice that the regularity of a − (U − ) in the assumptions of Theorem 6.2 implies that the causal fermion system a(U − , U + ) is regular as well (see Corollary 5.17). In this and in the next section we will always focus on regular systems.

Step 2: closedness of the local correlation function
So far we have showed that the local correlation function is continuous and, under suitable assumptions, also injective. However we have no clue, yet, on the continuity of its inverse. If this could be proved, then we would have a global homeomorphism from R 4 to the image F ε (R 4 ), which would give the latter a smooth manifold structure.
Another critical property that we would like to prove is the closedness of the local correlation function. In such a way, the image F ε (R 4 ) would be a closed set and therefore coincide with F ε (R 4 ) = supp (F ε ) * µ, showing in this way that the causal fermion system detects exactly Minkowski spacetime as the support of its measure. In order to prove all this we need some techinical results.
A function f : X → Y between two topological spaces is called proper if the inverse image of any compact set is compact. In [5,Section 3.7] (see in particular Theorem 3.7.18) it is proved that every such a function is also closed if Y is a k-space. First-countable Hausdorff spaces are always of this kind (see Theorem 3.3.20): in particular this applies to F .

The Vacuum
Let us start with the vacuum first.
Theorem 6.5 The local correlation function F ε vac : R 4 → F of the vacuum is closed.
Proof. Let K ⊂ F be compact (hence closed) and consider any sequence {x n } n ⊂ H := (F ε vac ) −1 (K) ⊂ R 4 : we want to prove that there exists some subsequence which converges to some element in H. This gives the compactness of H.
Since K is compact and {F ε vac (x n )} n ⊂ K, there must exist some subsequence {y n } n and some A ∈ K such that F ε vac (y n ) → A. Notice that A * = A. Now we want to prove that A = 0. So, suppose by contradiction that A = 0, then F ε vac (y n ) → 0. Let λ ∈ σ vac \ {0} and take u ∈ H − m such that F ε vac (y 1 )u = λu, then u n := U y 1 −yn u satisfies F ε vac (y n )u n = λu n , as follows from Proposition 5.4. At this point we have |λ| u = |λ| u n = F ε vac (y n )u n ≤ F ε vac (y n ) u → 0, which is impossible, it being u = 0. Now, notice that the setÊ 0 (P − (S p (R 3 , C 4 ))) is dense within H − m , as proved in Lemma 2.17, and its elements are exactly given by the functions with ϕ ∈P − (S p (R 3 , C 4 )) (see Proposition 2.19). Therefore, as A is bounded, self-adjoint and different from zero, there must be at least one ϕ ∈P − (S p (R 3 , C 4 )) with (u ϕ |Au ϕ ) = 0. At this point we have two possibilities: either {y n } n is bounded or it is not. If {y n } n is unbounded, then we can always extract some subsequence {z n } n which diverges to infinity: |z n | → ∞. Since ϕg ε ∈ S p (R 3 , C 4 ), Theorem 2.32 implies that R ε u ϕ (z n ) → 0 in the limit n → ∞ and therefore for all ϕ ∈ S p (R 3 , C 4 ), which is a contradiction. Therefore, {y n } n must be bounded and so there exists some subsequence {w n } n which converges to some x ∈ R 4 . To conclude, notice that H is closed, it being the continuous inverse of K, and therefore {w n } n ⊂ H and w n → x imply x ∈ H. The proof is complete.

The general case of particles and antiparticles
Exploiting the closedness of the vacuum local correlation function, it is possible to show that the closedness is preserved in the presence of particles or antiparticles, under suitable hypotheses for the latter case, as for the injectivity. Again, we stick to the case of smooth solutions, though we do not need to put any restriction on their support this time.
Theorem 6.6 Let U ± ⊂Ê 0 (P ± (S p (R 3 , C 4 ))) be finite dimensional subspaces and suppose that a − (U − ) is regular. Moreover suppose that there exists a Hilbert basis e − of U − such that then the local correlation function F ε : R 4 → F of the causal fermion system a(U − , U + ) is a closed mapping.
Proof. Let us start with the purely-negative energy case first, i.e. U + = {0}. We proceed similarly to the proof of Theorem 6.5. So, let K ⊂ F be compact (hence closed) and consider any sequence {x n } n ⊂ H := (F ε ) −1 (K) ⊂ R 4 : we want to prove that there exists some subsequence which converges to some element in H. This gives the compactness of H.
Since K is compact and {F ε (x n )} n ⊂ K, there must exist some subsequence {y n } n and some A ∈ K such that F ε (y n ) → A. Notice that A * = A. Moreover A↾ a − (U − ) ⊥ = 0, as the same holds for any F ε (y n ). Nevertheless, we want to show that A = 0. So, suppose that A = 0, then in particular F ε (y n ) → 0. This implies that m n := max |σ(F ε (y n ))| = F ε (y n ) → 0 and therefore all the eigenvalues λ i (n) of F ε (y n ) will converge to zero, since |λ i (n)| ≤ m n . Nevertheless, this is not possible, since Theorem 5.22 would imply that, for any i = 1, 2, 3, 4: which is a contradiction. At this point, notice that the orthogonal projector over a − (U − ) is given exactly by the function Ψ defined in Proposition 5.8 -with ψ = u = e − any Hilbert basis of . At this point, the proof follows as in Theorem 6.5, substituting H − m with a − (U − ) andÊ 0 (P − (S (R 3 , C 4 )) with Ψ Ê 0 (P − (S (R 3 , C 4 )) . Now, suppose we add some positive energy solutions to the systems, i.e. we consider the causal fermion system a(U − , U + ) with U + = {0}. Again, consider any compact set K ⊂ F and let {x n } n ⊂ H := (F ε ) −1 (K) ⊂ R 4 . Since K is compact, there exists a subsequence {y n } n of {x n } n and some A ∈ K such that F ε (y n ) → A. Now, let us denote by F ε − the local correlation function associated to the causal fermion system a − (U − ) analysed in the first part of the proof. Thanks to Proposition 4.11, we know that F ε − (z) = Π − F ε (z) Π − =: A − (with Π − the orthogonal projector onto a − (U − )) for every z ∈ R 4 and therefore F ε − (y n ) → Π − AΠ − . At this point, the argument continues as in the first part of the proof, substituting F ε with F ε − and A with A − .
Remark 6.7 Just to get some intuition, referring to the brutal approximation carried out in (3.14), the upper bound in (6.8) is roughly given by which is very large in the assumption mε < mε max = 10 −15 if compared with (5.13).
We are ready to state the most important result of this section.
Theorem 6.8 Under the hypotheses of Theorems 6.2 and 6.6 the following statements hold: Proof. The proof follows directly from the injectivity, continuity and closedness of F ε .
It is actually possible to get even more, for the canonical foliation of R 4 into space and time is preserved. Let us introduce the following symbol: then the following result can be proved as the previous theorem. Corollary 6.9 Under the hypotheses of Theorems 6.2 and 6.6 the following statements hold for every t ∈ R: Thus supp (F ε ) * µ admits a smooth foliation in terms of 3-dimensional submanifolds {Σ t } t∈R .
As a conclusion of this chapter, we want to conclude the discussion we started right after Definition 4.9. We make use of Theorem 6.8.
Consider the set H − m of negative energy physical solutions and let F ε vac be its local correlation function, as usual. We want now to give some arguments which support the choice of (F ε vac ) * µ as the Borel measure on F which realizes F ε vac (R 4 ) as its support. Proposition 3.28 and Proposition 5.4 show that the vacuum causal fermion system carries a unitary representation of the translation group. It is sensible to assume that any Borel measure ̺ which is meant to describe the vacuum is invariant under such transformations: ̺((U a ) † Ω U a )) = ̺(Ω) for all Ω ∈ Bor(F ) and a ∈ R 4 . (6.10) This assumption leaves no chance, as the following result shows.
Proof. For simplicity of notation we drop the indices ε and vac from the local correlation function. Now, to start, we show that the push-forward measure does satisfy condition (6.10). First, notice that Proposition 5.4 implies U † a F(R 4 ) U a = F(R 4 ) for every a ∈ R 4 . Therefore, given supp F * µ = F(R 4 ) and the unitarity of U a , in order to prove (6.10) we can stick to the Borel subsets of F(R 4 ). Indeed, let Ω be any Borel set of F , then: Thus, we can stick to the Borel sets of F(R 4 ). Let Ω be any of them, then: and, therefore, as the Lebesgue measure is invariant under translations, we get: To conclude the proof we have to show the uniqueness (up to a non-negative factor) of such a measure. So, suppose there exists another measure ̺ as in the hypotheses. Since supp ̺ = F ε vac (R 4 ), again we can stick our analysis to the Borel subsets of F(R 4 ), which coincide with the images of the Borel subsets of R 4 through F, the latter being a homeomorphism. Now, fix a Borelian set ∆ ⊂ R 4 and a vector a ∈ R 4 , then we have Now, as F is a homeomorphism onto its image, the function ∆ → ̺(F(∆)) defines a Borel measure on R 4 , which is translation invariant and is finite on compact subsets. It is a wellknown fact (see [16,Theorem 2.20]) that every Borel measure fulfilling these properties is a non-negative multiple of the Lebesgue-Borel measure µ. Therefore, there exists λ ≥ 0 such that, for any Borel set Ω ⊂ F(R 4 ), it holds that λ F * µ(Ω) = λ µ(F −1 (Ω)) = ̺(F(F −1 (Ω))) = ̺(Ω), and the proof is complete.

Some concluding remarks
In constructing a causal fermion systems (H , F ε ), we always refer to Minkowski spacetime R 4 and represent it in F through the map F ε . In addition, we push-forward the Lebesgue measure µ of R 4 to a measure ̺ = (F ε ) * µ on F , which satisfies supp ̺ = F ε (R 4 ).
If now we focus on the special cases treated in the previous chapter, for which the corresponding local correlation function defines a closed homeomorphism onto its image, then it is possible to bypass Minkowski spacetime and discuss the relations existing between different causal fermion systems without referring to it. In this way, we are able to point out some underlying structures which might hold in more general settings than Minkowski spacetime.
Let us go more into details. Consider some finite-dimensional subspaces U (i) ± ⊂ H ± m , i = 1, 2 as in the previous chapter and construct the corresponding causal fermion systems. Also, consider the vacuum one (this is, actually, a special case of the former with U ± = {0}, but it might be useful to consider it separately in this discussion). So, we have: where we drop the index ε, for simplicity of notation. The first pair of casual fermion systems describe two possible configurations of matter, with a few particles and a few antiparticles. The second one represents the total absence of matter.
By definition, the corresponding measures are defined as the push-forward of the Lebesgue measure onto F through the local correlation functions: We also introduce the following notation for the support of these measures: At this point, it is possible to put all these mathematical structures together in a commutative diagram, namely: where we defined the homeomorphisms: By construction, every set appearing in the diagram above is a smooth manifold and every function is a diffeomorphism. Moreover, it is possible to rewrite every measure as the pushforward of one another, as the defining functions are homeomorphisms. More precisely, it is clear (by definition of push-forward) that: The identities above translate into the mathematical language of causal fermion systems what is generally known as the action of a creation or annihilation operator. More precisely, we can give the following interpretations.
(i) The identity ̺ 1 = (f 1 ) * ̺ vac defines the creations of the particles of U In the general case, when the local correlation function is not a homeomorphism with its image, the discussion presented here may no longer apply. However, in some special cases it is possible to get similar results, even though the functions involved might be only measurable. This, though, goes beyond the scope of this paper and will not be discussed here.
It is worth stressing once more that the multiparticle states considered so far are exclusively those who can generally described as pure tensor products of single particle states (see Remark 4.24). A notion of an entanglement between states can be defined, but it still a matter of investigation.
Notice that x → ϕ x (k) is smooth for every fixed k ∈ R 3 . Moreover, differentiating with respect to x, we get for any multiindex α ∈ N 4 : Notice that k α ϕ is an element of S p (R 3 , C 4 ) for any multiindex α. This follows from the fact that the Schwartz space is closed under multiplication by polynomials and by Lemma 8.1. At this point, as k α ϕ ∈ L 1 p (R 3 , C 4 ) and does not depend on x, we can apply [15, Theorem 1.88] and conclude that the function is differentiable to every order and that partial derivatives and integral can be switched. In particular, if we apply the Dirac operator D to (8.2), it is not difficult to see that (8.2) solves the Dirac equation. The claim is proved. We now proceed with the proof.
In the special case of functions ϕ ∈ F(C ∞ c,x (R 3 , C 4 )), the functionũ ϕ must coincide with u ϕ = E 0 • F −1 (ϕ) ∈ H sc m , for they are both smooth solutions of (2.1) and coincide at t = 0. Let us go back to general functions ϕ ∈ S p (R 3 , C 4 ). It holds that The first statement follows from the fact thatũ(t, ·) is, by definition, the inverse Fourier transform of ϕ + e −iωt +ϕ − e iωt which is a Schwartz function (again, use Lemma 8.1). In order to prove the second identity, notice once more thatũ(t, ·) is the inverse Fourier transform of ϕ + e −iωt + ϕ − e iωt and, therefore, using Parseval identity (see [15,Proposition 3.105]) and the fact that the subspaces W ± k are orthogonal to each other (see Proposition 2.14) we have: |ϕ + (k) e −iω(k)t + ϕ − (k) e iω(k)t | 2 d 3 k = At this point, reasoning as in (8.1) and making use of the identity between parentheses right above, it follows that: u ϕ ↾ R T ∈ L 2 (R T , C 4 ) and ũ ϕ ↾ R T L 2 = √ 2T ũ ϕ (0, ·) L 2 = √ 2T ϕ L 2 for every T > 0.
Since F(C ∞ c,x (R 3 , C 4 )) is dense within S p (R 3 , C 4 ) in the L 2 -norm, there must exist a sequence {ϕ n } n ⊂ F(C ∞ c,x (R 3 , C 4 )) such that ϕ n − ϕ L 2 → 0 The sequence {u ϕn } n ⊂ H sc m is then of Cauchy type, as {ϕ n } n is Cauchy and E 0 • F −1 is an isometry. Now, take an open bounded set B ⊂ R 4 and T > 0 such that B ⊂ R T , then, bearing in mind thatũ ϕn = u ϕn , we have: which implies that u ϕn →ũ ϕ within L 2 loc (R 4 , C 3 ). By definition, this means thatũ ϕ ∈ H m (notice thatũ ϕ clearly is locally square-integrable, it being continuous) and ũ ϕ = lim n→∞ u ϕn = lim n→∞ ϕ n L 2 = ϕ L 2 .
Lemma 8.2 Referring to Proposition 2.30 the following estimates hold true.
Proof of Lemma 3.14. Let us start with point (i). Take any h ∈ M (R 4 ) with supp h ⊂ B(0, δ) and let f ∈ H sc m be arbitrary. The function f * h is smooth (see [21, Theorem 1.6.1]). From Proposition 2.2 we know that supp f is contained in the causal propagation of the support of the initial data. By definition of convolution, it holds that supp (f * h) ⊂ supp f + B(0, δ) and therefore supp (f * h| Σ 0 ) must be compact. If we manage to prove that f * h belongs to ker D, then the proof of (i) is finished. This is true. Indeed, applying again [21, Theorem 1.6.1], we get: The only non-obvious step in the above chain of equalities identity (*). In order to understand it, take any y ∈ R n \B δ , then by definition |y−x| > δ for every x ∈ B and therefore h(x−y) = 0 for every x ∈ B, giving no contribution to the integral before (*).
We can go back to the proof of point (ii). So, let u ∈ H m and {f n } n ⊂ H sc m be a Cauchy sequence converging to u within L 2 loc (R 4 , C 4 ) as in the definition of H m . First, exploiting Lemma 2.7, Lemma 8.3 and the fact that f n * h ∈ H sc m , we get for any T > 0: and therefore {f n * h} n is a Cauchy sequence in H sc m . Second, exploiting Lemma 8.3, for any bounded open set B ⊂ R 4 , we see that f n * h↾ B −u * h↾ B L 2 ≤ f n ↾ B δ −u↾ B δ L 2 → 0 where we used the fact that B δ is open and bounded. By definition of H m , this implies that u * h ∈ H m .
To prove point (iii), fix any T > 0 and notice that (see Lemma 2.7) √ 2T u * h = u * h↾ R T L 2 ≤ u↾ R T +δ L 2 = 2(T + δ) u , for every u ∈ H m . This gives the continuity.
Exploiting these inqualities we conclude the proof of point (a). Indeed, let i = 1, . . . , n, then Now, fix any ǫ ′′ > 0, then we can always choose ǫ ′ small enough so that ϕ i − u i < ǫ ′′ for any i = 1, . . . , n. To conclude define the orthonormal set ψ i = ϕ i / ϕ i for every i = 1, . . . , n, then it is not difficult to see that for sufficiently small ǫ ′′ it holds that u i − ψ i < ǫ. The second statement of point (i) can be proved easily exploiting the first one. Now, let us prove point (ii). Fix any 0 < ǫ < n −1/2 as in the hypothesis and take a corresponding set {ψ 1 , . . . , ψ n } as in the proof of point (a), then we want to show that there is no non-vanishing n-ple {λ 1 , . . . , λ n } ⊂ C such that i=1 λ i ψ i ⊥ U . In this sense the space spanned by the ψ i is "close enough" to U . Suppose by contradiction that this is instead the case. Thus, we have: where A must be strictly positive, as we assumed that at least some of the scalars λ i do no vanish and that the vectors ψ i are linearly independent. At the same time, using Hölder inequality, we have Putting all together we have A + B ≤ nǫ 2 B from which 1 ≤ 1 + A B ≤ nǫ 2 < 1, which is a contradiction. This proves that n i=1 λ i ψ i ⊥ U implies λ i = 0 for all i = 1, . . . , n.
Proof of Proposition 5.8. Let {ψ 1 , . . . , ψ n } be an approximating set for an orthonormal basis {u 1 , . . . , u n } of U as in Proposition 5.6 and consider an element ϕ ∈Ê 0 (P − (S p (R 3 , C 4 ))). We want to find scalars {λ 1 , . . . , λ n } ⊂ C such that the linear combination is orthogonal to the subspace U and show that these are uniquely determined by the function ϕ. This is equivalent to the requirement: λ i (u j |ψ i ) ∀j = 1, . . . , n. (8.9) Suppose first that ϕ ⊥ U , then it must be n i=1 λ i ψ i ⊥ U , which is only possible if λ i = 0 for any i = 1, . . . , n, as follows from (b) in Proposition 5.6. On the other way around, it is clear that λ i = 0 for any i = 1, . . . , n implies ϕ ⊥ U . This proves the last statement of the proposition. Now, suppose on the contrary that ϕ ⊥ U , then there must exist at least one basis element, say u 1 , such that (ϕ|u 1 ) = 0. Equation (8.9) is equivalent to the linear system        n i=1 λ i (u 1 |ψ i ) = (u 1 |ϕ) . . . The matrix M in the right-hand side is non singular. Indeed, suppose there is some n-tuple (λ 1 , . . . , λ n ) in its kernel, then we would have λ i ψ i for all j = 1, . . . , n, which implies n i=1 λ i ψ i ⊥ U and so, as above, it must be λ i = 0 for all i = 1, . . . , n. Thus, the linear system (8.10) has one and only one solution given by Proof of Proposition 5.9. Let us fix an orthonormal basis {u 1 , . . . , u n } of U and fix for any 0 < ǫ < n −1/2 an approximating set {ψ ǫ 1 , . . . , ψ ǫ n } of it as in Proposition 5.6 (where the dependence on ǫ has been made explicit). Then, in particular, it holds that |(u i |ψ ǫ j ) − δ ij | < ǫ for any i, j = 1, . . . , n and therefore 21 At this point, exploiting the continuity of the inverse matrix function, it is possible to choose ǫ small enough so that M −1 ǫ −I n 2 < 1. Therefore, we get M −1 ǫ 2 ≤ M −1 ǫ −I n 2 + I n 2 < 2 and so, exploiting (8.11), we get: 1 |ϕ), . . . , (u n |ϕ))| ≤ 2 n i=1 |(u i |ϕ)| 2 ≤ 2 ϕ where we exploited Bessel inequality in the last inequality. 21 It is possible to prove that A 2 ≤ n i,j=1 |aij | 2 =: A F (Frobenius norm) for any A ∈ M(4, C).