“RELATIVISTIC FERMION SYSTEMS” Contents Preface 2 1. Introduction 3

LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS”

FELIX FINSTER AND JAN-HENDRIK TREUDE

Contents Preface 2 1. Introduction 3 1.1. The Schr¨odingerEquation 3 1.2. The Dirac Equation 4 1.3. A First Glance at Causal Fermion Systems 9 2. Fourier Methods 12 2.1. The Green’s Functions 12 2.2. The Causal Fundamental Solution and Time Evolution 14 2.3. The Lippmann-Schwinger Equation 16 3. The Cauchy Problem, Linear Symmetric Hyperbolic Systems 17 3.1. Finite Propagation Speed and Uniqueness of Solutions 18 3.2. Global Existence of Smooth Solutions 22 3.3. Existence of Causal Green’s Functions 26 4. The Fermionic Projector 28 4.1. The External Field Problem 28 4.2. The Dirac Sea and the Fermionic Projector of the Vacuum 29 4.3. Perturbative Description 32 5. A Non-Perturbative Construction of the Fermionic Projector 33 5.1. The Mass Oscillation Property 34 5.2. A Self-Adjoint Extension of S2 35 5.3. The Operator k and its Extended Domain 37 5.4. The Fermionic Projector as an Operator-Valued Measure 38 5.5. Proof of the Mass Oscillation Property 39 6. Spinors in Curved Space-Time 42 6.1. Curved Space-Time and Lorentzian Manifolds 42 6.2. The Dirac Equation in Curved Space-Time 43 7. Causal Fermion Systems and a Lorentzian Quantum Geometry 50 7.1. Constructing a Causal Fermion System from the Fermionic Projector 50 7.2. Geometric Structures on a Causal Fermion System 51 7.3. The Correspondence to Lorentzian Spin Geometry 57 8. Outlook 59 8.1. Analysis of the Causal Action Principle 59 8.2. The Continuum Limit of Causal Fermion Systems 59 Appendix A. The Free Time Evolution Operator in Momentum Space 60 Appendix B. Uniform L2-Estimates of Derivatives of Dirac Solutions 62 References 63 1 2 F. FINSTER AND J.-H. TREUDE

Preface These notes are based on lectures given at the spring school “Relativistic Fermion Systems” held in Regensburg in April 2013. YOUR NAME CAN We are grateful to . . . for valuable feedback and helpful comments on the manu- APPEAR HERE!!! script.

F.F. and J.-H. T., Regensburg, April 3, 2013 LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 3

1. Introduction 1.1. The Schr¨odingerEquation. In non-relativistic one-particle quantum mechanics, the quantum mechanical particle is described by a complex-valued wave function ψ(t, ~x). The absolute square |ψ(t, ~x)|2 has the interpretation as the probability density of the particle to be at the position ~x. Since the total probability must be one, we need to impose the normalization condition Z |ψ(t, ~x)|2 d3x = 1 . (1.1) R3 In particular, the probability integral must be time independent. Next, the super- position principle tells us that the set of all wave functions form a complex vector space H. For the probabilistic interpretation, one takes a non-zero vector φ ∈ H and multiplies it by a constant, ψ := νφ with ν ∈ C such as to satisfy the normalization condition (1.1). Then |ψ(t, ~x)|2 has the interpretation as the probability density (this procedure can be described by saying that only the ray Cψ has a physical interpretation, but we will not go into this here). For this procedure to work, it is important that the normalization constant ν can be chosen independent of time, meaning that the probability integral must be time independent. By polarizing, we conclude that the L2 scalar product Z hφ|ψiH := φ(t, ~x) ψ(t, ~x) , φ, ψ ∈ H (1.2) R3 should be time independent. The dynamics is described by a linear evolution equation on H, which we write as

i∂tψ = Hψ , where the operator H is the Hamiltonian. As explained above, the scalar product (1.2) should be time independent. This implies that for any solution φ, ψ of the Schrödinger equation, 0 = ∂thφ|ψiH = −i hHφ|ψiH − hφ|HψiH . (1.3) In other words, the operator H should be self-adjoint on H. In the simplest setting without spin, the Hamiltonian is chosen as 1 H = − ∆ + V. 2m 3 Here ∆ is the Laplacian on R , and V (t, ~x) is a real-valued potential. The parameter m > 0 is the rest mass. We always work in units where ~ = c = 1. Since this Hamil- tonian is unbounded, we need to specify a suitable dense domain of definition D(H). 2 3 ∞ 3 For simplicity, we here take the smooth functions, D(H) = L (R ) ∩ C (R ). This operator satisfies the requirement (1.3) because it is formally self-adjoint in the sense that

hHφ|ψiH = hφ|HψiH for all φ, ψ ∈ D(H) . The Schr¨odinger equation can be solved and analyzed with several methods. If the Hamiltonian is time independent, it is useful to extend the domain of H such as to obtain a self-adjoint operator. Then the Schr¨odingerequation can be solved by exponentiating with the spectral theorem, ψ(t) = e−iHt ψ(0) . 4 F. FINSTER AND J.-H. TREUDE

The dynamics of ψ can then be related to spectral properties of the Hamiltonian. An- other method is to consider the Schr¨odingerequation as a parabolic partial diﬀerential equation, and to use the existence theory and a-priori estimates. One advantage is that this also works in the case that the Hamiltonian depends on time. We do not enter the details here, because we will get more familiar with all these methods in the context of the relativistic theory.

1.2. The Dirac Equation. We now give a brief introduction to the Dirac equation, closely following the presentation in [19, §1.1 and§1.2]. In special relativity, space-time is described by Minkowski space (M, h., .i), a real 4-dimensional vector space endowed with an inner product of signature (+ − − −). Thus, choosing a pseudo-orthonormal P3 i basis (ei)i=0,...,3 and representing the vectors of M in this basis, ξ = i=0 ξ ei, the inner product takes the form 3 X j k hξ, ηi = gjk ξ η , (1.4) j,k=0 where gij, the Minkowski metric, is the diagonal matrix g = diag (1, −1, −1, −1). In what follows we usually omit the sums using Einstein’s summation convention (i.e. we sum over all indices which appear twice, once as an upper and once as a lower index). Also, we sometimes abbreviate the Minkowski scalar product by writing ξη := hξ, ηi 2 and ξ := hξ, ξi. A pseudo-orthonormal basis (ei)i=0,...,3 is in physics called a reference frame, because the corresponding coordinate system (xi) of Minkowski space gives the time and space coordinates for an observer in a system of inertia. We also refer to t := x0 as time and denote the spatial coordinates by ~x = (x1, x2, x3). The sign of the Minkowski metric encodes the causal structure of space-time. Namely, a vector ξ ∈ M is said to be  timelike if hξ, ξi > 0  spacelike if hξ, ξi < 0 null if hξ, ξi = 0 .  Likewise, a vector is called non-spacelike if it is timelike or null. The null vectors form the double cone L = {ξ ∈ M | hξ, ξi = 0}, referred to as the light cone. Physically, the light cone is formed of all rays through the origin of M which propagate with the speed of light. Similarly, the timelike vectors correspond to velocities slower than light speed; they form the interior light cone I = {ξ ∈ M | hξ, ξi > 0}. Finally, we introduce the closed light cone J = {ξ ∈ M | hξ, ξi ≥ 0}. The space-time trajectory of a moving object describes a curve q(τ) in Minkowski space (with τ an arbitrary parameter). We say that the space-time curve q is timelike if the tangent vector to q is everywhere timelike. Spacelike, null, and non-spacelike curves are defined analogously. The usual statement of causality that no information can travel faster than with the speed of light can then be expressed as follows, causality: information can be transmitted only along non-spacelike curves. The set of all points which can be joined with a given space-time point x by a non- spacelike curve is precisely the closed light cone centered at x, denoted by Jx := J − x. It is the union of the two single cones ∨ 2 0 0 Jx = {y ∈ M | (y − x) ≥ 0, (y − x ) ≥ 0} ∧ 2 0 0 Jx = {y ∈ M | (y − x) ≥ 0, (y − x ) ≤ 0} , LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 5 which have the interpretation as the points in the causal future and past of x, respec- ∨ ∧ tively. Thus we refer to Jx and Jx as the closed future and past light cones centered ∨ ∧ ∨ ∧ at x, respectively. Similarly, we also introduce the sets Ix , Ix and Lx , Lx . The physical equations should be Lorentz invariant, meaning that they must be formulated in Minkowski space, independent of the reference frame. The simplest relativistic wave equation is the Klein-Gordon equation 2 (− − m ) ψ = 0 , (1.5) j where ≡ ∂j∂ is the wave operator. This equation describes a scalar particle (=particle without spin) of mass m. If the particle has electric charge, one needs to suitably insert the electromagnetic potential A into the Klein-Gordon equation. More precisely, one finds empirically that the equation k k 2 − (∂k − iAk)(∂ − iA ) ψ = m ψ (1.6) describes a scalar particle of mass m and charge e in the presence of an electromagnetic field. In order to describe a particle with spin, it was Dirac’s idea to work with a first order differential operator whose square is the wave operator. One introduces the Dirac matrices γj as 4 × 4-matrices which satisfy the anti-commutation relations 2 gjk 11= {γj, γk} ≡ γjγk + γkγj . (1.7) j Then the square of the operator γ ∂j is 1 (γj∂ )2 = γjγk ∂ ∂ = {γj, γk} ∂ = . (1.8) j j k 2 jk For convenience, we shall always work in the Dirac representation 11 0 0 ~σ γ0 = , ~γ = , (1.9) 0 −11 −~σ 0 where ~σ are the three Pauli matrices 0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = . 1 0 i 0 0 −1 The Dirac equation in the vacuum reads ∂ iγk − m ψ(x) = 0 , (1.10) ∂xk where ψ(x), the Dirac spinor, has four complex components. The leptons and quarks in the standard model are Dirac particles, and thus one can say that all matter is on the fundamental level described by the Dirac equation. Multiplying (1.10) by the j operator (iγ ∂j + m) and using (1.8), one finds that each component of ψ satisfies the Klein-Gordon equation (1.5). In the presence of an electromagnetic field, the Dirac equation must be modified to k iγ (∂k − iAk)ψ = mψ . (1.11) j Multiplying by the operator (iγ (∂j − iAj) + m) and using the anti-commutation relations, we obtain the equation i −(∂ − iA )(∂k − iAk) + F γjγk − m2 ψ = 0 . k k 2 jk 6 F. FINSTER AND J.-H. TREUDE

i j k This differs from the Klein-Gordon equation (1.6) by the extra term 2 Fjkγ γ , which describes the coupling of the spin to the electromagnetic field. We also denote the con- j traction with Dirac matrices by a slash, i.e. u/ = γ uj for u a vector of Minkowski space j and ∂/ = γ ∂j. Acting on a spinor by u/ is often referred to as Clifford multiplication by the vector u. The wave functions at every space-time point are endowed with an indefinite scalar product of signature (2, 2), which we call spin scalar product and denote by 4 X α ∗ α ≺ψ|φ (x) = sα ψ (x) φ (x) , s1 = s2 = 1, s3 = s4 = −1 , (1.12) α=1 where ψ∗ is the complex conjugate wave function (this scalar product is often written as ψφ with the so-called adjoint spinor ψ ≡ ψ∗γ0). By the adjoint A∗ of a matrix A we always mean the adjoint with respect to the spin scalar product as defined via the relations ≺A∗ψ | φ = ≺ψ | Aφ for all ψ, φ. In an obvious way, this definition of the adjoint gives rise to the notions “selfadjoint,” “anti-selfadjoint” and “unitary.” With these notions, the Dirac matrices are selfadjoint, meaning that ≺γlψ | φ = ≺ψ | γlφ for all ψ, φ. To every solution ψ of the Dirac equation we can associate a time-like vector field J by J k = ≺ψ | γk ψ , (1.13) which is called the Dirac current. The Dirac current is divergence-free, k k k k ∂kJ = ∂k ≺ψ | γ ψ = ≺∂kψ | γ ψ + ≺ψ | γ ∂k ψ = i (≺i∂/ψ | ψ − ≺ψ | i∂/ψ ) (1.14) = i ≺(i∂/ + A/ − m)ψ | ψ − ≺ψ | (i∂/ + A/ − m)ψ = 0 . This is referred to as current conservation. Current conservation is closely related to the probabilistic interpretation of the Dirac wave function, as we now explain. Suppose that ψ is a smooth solution of the Dirac equation with suitable decay at spatial infinity. Then current conservation allows us to apply the Gauss divergence theorem to obtain

Z t2 Z 3 k 0 = dt d x ∂k≺ψ | γ ψ (t, ~x) 3 t1 R Z Z (1.15) 0 3 0 3 = ≺ψ | γ ψ (t2, ~x) d x − ≺ψ | γ ψ (t1, ~x) d x R3 R3 (this argument works similarly on a region Ω ⊂ M whose boundary consists of two space-like hypersurfaces). Polarizing, we conclude that for any two solutions φ, ψ of the Dirac equation, the spatial integral Z (φ|ψ) := 2π ≺φ | γ0ψ (t, ~x) d3x (1.16) R3 is time independent. Since the inner product ≺.|γ0. is positive deﬁnite, the integral (1.16) deﬁnes a scalar product. We denote the Hilbert space corresponding to LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 7

2 3 4 this scalar product by H = L (R ) . In analogy to the integrand in (1.2) in non- relativistic quantum mechanics, the quantity ≺ψ|γ0ψ has the physical interpretation as the probability density. As a consequence of current conservation (1.15), the probability integral is time independent. The previous considerations generalize immediately to the situation in the presence of a more general external potential. To this end, we replace the operator A/ in the Dirac equation (1.11) by a multiplication operator B(x), which we assume to be smooth and symmetric with respect to the spin scalar product, i.e. ≺Bψ|φ = ≺ψ|Bφ for all ψ, φ. (1.17) We write the Dirac equation with a Dirac operator D as (D − m) ψ = 0 where D := i∂/ + B . (1.18) Th symmetry assumption (1.17) is needed for current conservation to hold (as one sees immediately if in (1.14) one replaces A/ by B). Mimicking (1.3), we can also rewrite the dynamics with a symmetric operator H. To this end, we multiply the Dirac equation (1.18) by γ0 and bring the t-derivative on a separate side of the equation,

0 i∂tψ = Hψ where H = −γ (i~γ∇~ + B − m) (1.19)

j 0 (note that γ ∂j = γ ∂0 + ~γ∇~ ). We refer to (1.19) as the Dirac equation in the Hamil- tonian form. Now we can again apply (1.3) to conclude that the Hamiltonian is a symmetric operator on H. We remark that in the Hamiltonian formulation, one often combines the prefactor γ0 in (1.19) with the other Dirac matrices and works with the new matrices

α := γ0 and β~ = γ0~γ. This convenient because these new matrices are Hermitian with respect to the standard 4 scalar product on C . In these notes, we will always avoid working with α and β~. We prefer the notation (1.19), because it becomes clearer which parts of the operators are Lorentz invariant. For calculations using α and β~ we refer for example to the monograph by Bernd Thaller [50]. So far, Dirac spinors were introduced in a given reference frame. Let us verify that our deﬁnitions are coordinate independent. A linear transformation of Minkowski space which leaves the form of the Minkowski metric (1.4) invariant is called a Lorentz transformation. The Lorentz transformations form a group, the Lorentz group. The Lorentz transformations which preserve both the time direction and the space orientation form a subgroup of the Lorentz group, the orthochronous proper Lorentz group. We consider two reference frames (xj) and (˜xl) with the same orientation of time and space. Then the reference frames are related to each other by an orthochronous proper Lorentz transformation Λ, i.e. in components ∂ ∂ x˜l = Λl xj , Λl = , j j ∂x˜l ∂xj and Λ leaves the Minkowski metric invariant,

l m Λj Λk glm = gjk . (1.20) 8 F. FINSTER AND J.-H. TREUDE

j Under this change of space-time coordinates, the Dirac operator iγ (∂xj − iAj) transforms to ∂ iγ˜l − iA˜ withγ ˜l = Λl γj . (1.21) ∂x˜l l j l This transformed Dirac operator does not coincide with the Dirac operator iγ (∂x˜l − l iA˜l) as defined in the reference frame (˜x ) because the Dirac matrices have a different form. However, the next lemma shows that the two Dirac operators do coincide up to a suitable unitary transformation of the spinors. Lemma 1.1. For any orthochronous proper Lorentz transformation Λ there is a unitary matrix U(Λ) such that l j −1 l U(Λ) Λjγ U(Λ) = γ . Proof. Since Λ is orthochronous and proper, we can write it in the form Λ = exp(λ), where λ is a suitable generator of a rotation and/or a Lorentz boost. Then Λ(t) := exp(tλ), t ∈ R, is a family of Lorentz transformations, and differentiating (1.20) with respect to t at t = 0, we find that l m λj glk = −gjm λk . Using this identity together with the fact that the Dirac matrices are selfadjoint, it is straightforward to verify that the matrix 1 u := λl γ γk 4 k l is anti-selfadjoint. As a consequence, the family of matrices U(t) := exp (tu) is unitary. We now consider for a fixed index l the family of matrices l j −1 A(t) := U(t) Λ(t)jγ U(t) . Clearly, A(0) = γl. Furthermore, differentiating with respect to t gives d n o A(t) = U Λl u γj − γj u + λj γk U −1 , dt j k and a short calculation using the commutation relations j kj j k γlγk, γ = 2 γl g − δl γ shows that the curly brackets vanish. We conclude that A(1) = A(0), proving the lemma. Applying this lemma to the Dirac operator in (1.21), one sees that the Dirac operator is invariant under the joint transformation of the space-time coordinates and the spinors j j k x −→ Λkx , ψ −→ U(Λ) ψ . Moreover, since the matrix U(Λ) is unitary, the representation of the spin scalar product (1.12) is valid in any reference frame. We conclude that our definition of spinors is indeed coordinate invariant. For what follows, it is important to keep in mind that spin scalar product is Lorentz invariant. In other words, the inner product ≺ψ|φ is a scalar, independent of a choice of reference frame. The combination ψ†φ = ≺ψ|γ0φ , however, is not a scalar; it is the zero component of a Minkowski vector. As a consequence, the integrand of (1.16) LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 9 is not a scalar, but a density. The integral (1.16), on the other hand, is again Lorentz invariant. Out of the Dirac matrices one can form the pseudoscalar matrix ρ by i ρ = γjγkγlγm = iγ0γ2γ2γ3 (1.22) 4! jklm 5 (this matrix in the physics literature is usually denoted by γ ). Here jklm is the totally antisymmetric symbol (i.e. jklm is equal to ±1 if (j, k, l, m) is an even and odd permutation of (0, 1, 2, 3), respectively, and vanishes otherwise). A short calculation shows that the pseudoscalar matrix is anti-selfadjoint and ρ2 = 11. As a consequence, the matrices 1 1 χ = (11 − ρ) , χ = (11+ ρ) (1.23) L 2 R 2 satisfy the relations 2 ∗ χL/R = χL/R , ρ χL = −χL , ρ χR = χR , χL = χR , χL + χR = 11 . They can be regarded as the spectral projectors of the matrix ρ and are called the chiral projectors . The projections χLψ and χRψ are referred to as the left- and right- handed components of the spinor. A matrix is said to be even and odd if it commutes and anti-commutes with ρ, respectively. It is straightforward to verify that the Dirac matrices are odd, and therefore j j γ χL/R = χR/L γ . Using this relation, one can rewrite the Dirac equation (1.11) as a system of equations for the left- and right-handed components of ψ, k k iγ (∂k − iAk) χLψ = m χRψ , iγ (∂k − iAk) χRψ = m χLψ . If m = 0, these two equations decouple, and we get separate equations for the left- and right-handed components of ψ. This observation is the starting point of the 2- component Weyl spinor formalism. We shall not use this formalism here, but will instead describe chiral massless particles (like neutrinos) by the left- or right-handed component of a Dirac spinor. 1.3. A First Glance at Causal Fermion Systems. We now explain a few concepts behind causal fermion systems. We will come back to these concepts in more detail in Section 7. Suppose we consider a system consisting of several Dirac particles in a concrete physical configuration (say an atom or molecule). For simplicity, we here describe the system by the corresponding one-particle wave functions ψ1, . . . , ψf (for the connection to Fock spaces and the Pauli exclusion principle see for example [24]). Here the number f of particles could be finite or infinite. In order to keep the presentation as simple as possible, we now restrict attention to the case f < ∞ and assume that the wave functions ψi are continuous (for an infinite number of particles see Section 7.1 below). In order to work basis independent, we consider the vector space Hparticle := <ψ1, . . . , ψf > spanned by the one-particle wave functions. On Hparticle we again consider the scalar product (1.16), which for clarity we now denote by h.|.iHparticle . We thus obtain the Hilbert space (Hparticle, h.|.iHparticle ). For any space-time point x, we introduce the local correlation operator F (x) as the signature operator of the inner product at x expressed in terms of the Hilbert space scalar product,

ψ(x)φ(x) = −hψ|F (x)φiHparticle for all ψ, φ ∈ Hparticle . 10 F. FINSTER AND J.-H. TREUDE

Taking into account that the inner product at x has signature (2, 2), the local correlation operator is a symmetric operator in L(Hparticle) of rank at most four, which has at most two positive and at most two negative eigenvalues. The local correlation operators encode how the Dirac particles are distributed in space-time and how the wave functions are correlated at the individual space-time points. We want to consider the local correlation operators as the basic objects in space-time, meaning that all geometric and analytic structures of space-time (like the causal and metric structure, connection, curvature, gauge fields, etc.) should be deduced from the operators F (x). The only structure besides the local correlation operators which we want to keep is the volume measure of space-time. In order to implement this concept mathematically, it is useful to identify a space-time point x with the corresponding local correlation operator F (x) ∈ L(Hparticle). Then space- time can be identified with the subset F (M) ⊂ L(Hparticle). Next, we introduce the universal measure ρ = F∗µM as the push-forward of the volume measure on M under −1 p 4 the mapping F (thus ρ(Ω) := µM (F (Ω)), where dµM = | deg g| d x is the standard volume measure). Omitting the subscript “particle” and dropping all the additional structures of space-time, we obtain a causal fermion system of spin dimension two as defined in [31, Section 1.2]:

Definition 1.2. Given a complex Hilbert space (H, h.|.iH) (the “particle space”) and a parameter n ∈ N (the “spin dimension”), we let F ⊂ L(H) be the set of all self-adjoint operators on H of finite rank, which (counting with multiplicities) have at most n positive and at most n negative eigenvalues. On F we are given a positive measure ρ (defined on a σ-algebra of subsets of F), the so-called universal measure. We refer to (H, F, ρ) as a causal fermion system in the particle representation.

Causal fermion systems provide a general abstract mathematical framework for the formulation of relativistic physical theories. The setting is so general that it allows for the description of continuum space-times (like Minkowski space or a Lorentzian manifold), discrete space-times (like a space-time lattice) and even so-called “quantum space-times” which have no simple classical correspondence (for examples we refer to [8, 29] and [32]). It is a remarkable fact that in causal fermion systems there are many inherent geometric and analytic structures. Namely, starting from a general causal fermion system, one can deduce space-time together with geometric structures which generalize the setting of spin geometry (see [29]). Moreover, the causal action principle gives interesting analytic structures (see [23]). In particular, one obtains a background-free formulation of quantum ﬁeld theory in which ultraviolet divergences of standard QFT are avoided. Since the setting also works in curved space-time, causal fermion systems are also a promising approach for quantum gravity. We refer the interested reader to the review article [29] as well as to the preprints [22, 26, 27]. We now outline how to deduce the geometric and analytic structures which will be relevant in this paper. Starting from a causal fermion system (H, F, ρ), one deﬁnes space-time as the support of the universal measure, M := supp ρ. On M, we consider the topology induced by F ⊂ L(H). The causal structure is encoded in the spectrum of the operator products xy:

Deﬁnition 1.3. For any x, y ∈ F, the product xy is an operator of rank at most 2n. xy xy We denote its non-trivial eigenvalues by λ1 , . . . , λ2n (where we count with algebraic xy multiplicities). The points x and y are called timelike separated if the λj are all real. LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 11

xy They are said to be spacelike separated if all the λj are complex and have the same absolute value. In all other cases, the points x and y are said to be lightlike separated. Since the operators xy and yx are isospectral (this follows from the matrix identity det(BC − λ11)= det(CB − λ11);see for example [21, Section 3]), this definition is symmetric in x and y. The main analytic tool are the causal variational principles defined as follows. For two points x, y ∈ F we define the spectral weight |.| of the operator products xy and (xy)2 by 2n 2n X xy 2 X xy 2 |xy| = |λi | and (xy) = |λi | . i=1 i=1 We also introduce the 1 Lagrangian L(x, y) = |(xy)2| − |xy|2 . (1.24) 2n For a given universal measure ρ on F, we define the non-negative functionals ZZ action S[ρ] = L(x, y) dρ(x) dρ(y) F×F ZZ constraint T [ρ] = |xy|2 dρ(x) dρ(y) . F×F Our action principle is to minimize S for fixed T , under variations of the universal measure. These variations should keep the total 0 volume unchanged, which means that a variation (ρ(τ))τ∈(−ε,ε) should for all τ, τ ∈ (−ε, ε) satisfy the conditions 0 0 ρ(τ) − ρ(τ ) (F) < ∞ and ρ(τ) − ρ(τ ) (F) = 0 (where |.| denotes the total variation of a measure; see [37, §29]). Depending on the application, one may impose additional constraints: (C1) The trace constraint: Z Tr(x) dρ(x) = f . F (C2) The identity constraint: Z x dρ(x) = 11H . F (C3) Prescribing 2n eigenvalues: We denote the non-trivial eigenvalues of x counted with multiplicities by ν1, . . . , ν2n and order them such that

ν1 ≤ · · · ≤ νn ≤ 0 ≤ νn+1 ≤ ... ≤ ν2n .

For given constants c1, . . . , c2n, we impose that

νj(x) = cj for all x ∈ M and j = 1,..., 2n . Moreover, one may prescribe properties of the universal measure by choosing a measure space (M,ˆ µˆ) and restricting attention to universal measures which can be represented as the push-forward ofµ ˆ,

ρ = F∗µˆ with F : Mˆ → F measurable . 12 F. FINSTER AND J.-H. TREUDE

One then minimizes the action under variations of the mapping F . The Lagrangian (1.24) is compatible with our notion of causality in the following sense. Suppose that two points x, y ∈ F are spacelike separated (see Defini- xy tion 1.3). Then the eigenvalues λi all have the same absolute value, so that the Lagrangian (1.24) vanishes. Thus pairs of points with spacelike separation do not enter the action. This can be seen in analogy to the usual notion of causality where points with spacelike separation cannot influence each other. At this stage, it is not at all obvious whether physics can be described in terms of causal fermion systems. Moreover, it is not obvious how the geometric and analytic structures in a causal fermion systems are related to the usual structures in space-time. In preparation for analyzing these questions, we need to get a better understanding of the solutions of the Dirac equations. Therefore, we now proceed by developing the necessary tools for analyzing the Dirac equation. Nevertheless, we give here a naive argument which explains why in the Minkowski vacuum, the causal structure of Definition 1.3 indeed coincides with the usual notion of causality. Suppose that x, y ∈ M. Constructing a corresponding causal fermion system by taking Hparticle as all the negative energy solutions of the Dirac equation (the “Dirac sea”), the local correlation matrices satisfy the relations F (x) F (y) ' a(ξ) /ξ + b(ξ) 11 with real-valued functions a and b and ξ := y − x. Here ' means that this identity holds in a suitable basis, dropping the irrelevant kernel of the operator on the left. Such a relation was to be expected in view of Lorentz invariance. We shall see later that it is indeed valid in a suitable limit when an ultraviolet regularization is removed. Subtracting the trace and taking the square, the eigenvalues of xy are computed by p 2 2 p 2 2 λ+ = b + a ξ and λ− = b − a ξ . It follows that the eigenvalues of xy are real if ξ2 > 0, whereas they form a complex conjugate pair if ξ2 < 0. This shows that the causal structure of Definition 1.3 agrees with the usual causal structure in Minkowski space.

2. Fourier Methods 2.1. The Green’s Functions. The Green’s function of the free Dirac equation is characterized by the distributional equation 4 (i∂/x − m) sm(x, y) = δ (x − y) , (2.1) Taking the Fourier transform of (2.1), Z d4k s (x, y) = s (k) e−ik(x−y) , (2.2) m (2π)4 m we obtain the algebraic equation

(k/ − m) sm(k) = 11 . (2.3) The Green’s function is not unique, as we now briefly recall (for details see [4]). Namely, multiplying by k/ + m and using the identity (k/ − m)(k/ + m) = k2 − m2, one sees that if k2 6= m2, the matrix k/ − m is invertible. If conversely k2 = m2, we have the relation (k/ − m)2 = −2m(k/ − m), showing that the matrix k/ − m is diagonalizable and that its eigenvalues are either −2m or zero. Taking the trace, Tr(k/ − m) = −4m, it follows that the matrix k/ − m has a two-dimensional kernel. We conclude that the LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 13 matrix k/ − m is singular on the mass shell. More systematically, the defining equation for the Green’s function (2.1) determines sm only up to a solution of the homogeneous Dirac equation. A convenient method for solving the equation (2.3) for sm(k) is to use a ±iε- regularization on the mass shell. Common choices are the advanced and the retarded Green’s functions defined by

∨ k/ + m ∧ k/ + m sm(k) = lim and sm(k) = lim , (2.4) ε&0 k2 − m2 − iεk0 ε&0 k2 − m2 + iεk0 respectively (with the limit ε & 0 taken in the distributional sense). Computing their Fourier transform (2.2), one sees that they are causal in the sense that their supports lie in the upper and lower light cone, respectively, ∨ ∨ ∧ ∧ supp sm(x, .) ⊂ Jx , supp sm(x, .) ⊂ Jx . (2.5) Mathematically, the formulas in (2.4) deﬁne the Green’s functions in momentum space as tempered distributions. Taking their Fourier transform (2.2), the advanced and retarded Green’s functions are tempered distributions in the variable ξ := y − x. We also regard these distributions as integral kernels of corresponding operators on the wave functions, i.e. Z 4 (sm(ψ))(x) := sm(x, y) ψ(y) d y . M We thus obtain operators ∧ ∨ ∞ ∞ sm, sm : C0 (M,SM) → Csc (M,SM) . (2.6) ∞ Here C0 (M,SM) denote the smooth functions with compact support in M, taking ∞ values in the spinors. Likewise, Csc denotes the smooth functions with spatially compact support. More details? Another common choice is the Feynman propagator

F k/ + m sm(k) := lim . ε&0 k2 − m2 + iε Taking the Fourier transform (2.2) with residues, one ﬁnds i Z F / −ik(x−y) sm(x, y) = 3 (k + m) e dµ~p 3 k=((t) ω(~p), ~p) (2π) R with t := (y − x)0 and p 2 2 ω(~p) := |ω(~p,)| = |~p| + m , dµ~p := d~p/(2ω(~p)) . Here denotes the step function (x) = 1 for x ≥ 0 and (x) = −1 otherwise. Thus for positive t we get the integral over the upper mass shell, whereas for negative t we integrate over the lower mass shell. As a consequence, the Feynman propagator is not causal, but it is instead characterized by the frequency conditions that it is for positive and negative time t composed only of positive and negative frequencies, respectively. Feynman [13] gave the frequency conditions in the Feynman propagator the physical interpretation as “positive-energy states moving to the future” and “negative-energy states moving to the past”. Identifying “negative-energy states moving to the past” with “antiparticle states” he concluded that the Feynman propagator is distinguished from all other Green’s functions in that it takes into account the particle/anti-particle interpretation of the Dirac equation in the physically correct way. The problem is that the frequency conditions are not invariant under general coordinate and gauge 14 F. FINSTER AND J.-H. TREUDE transformations (simply because such transformations “mix” positive and negative frequencies), and therefore Feynman’s method is not compatible with the equivalence principle and the local gauge principle. This is not a problem for most calculations in physics, but it is not satisfying conceptually.

2.2. The Causal Fundamental Solution and Time Evolution. We now introduce the causal fundamental solution km as the diﬀerence of the advanced and the retarded Green’s function, 1 k (x, y) := s∨(x, y) − s∧(x, y) . (2.7) m 2πi It is a distribution which is causal in the sense that it vanishes if x and y have spacelike separation. Moreover, it is a distributional solution of the homogeneous Dirac equation,

(i∂/x − m) sm(x, y) = 0 . We also consider km as the integral kernel of a corresponding operator ∞ ∞ km : C0 (M,SM) → Csc (M,SM) . (2.8) Using the formulas for the advanced and retarded Green’s functions (2.4), the fundamental solution can be computed in momentum space by 1 1 1 km(p) = (p/ + m) lim − 2πi ε&0 p2 − m2 − iεp0 p2 − m2 + iεp0 1 1 1 = (p/ + m) lim − (p0) 2πi ε&0 p2 − m2 − iε p2 − m2 + iε (where again denotes the step function). Using the distributional equation 1 1 lim − = 2πi δ(x) , ε&0 x − iε x + iε we obtain the simple formula 2 2 0 km(p) = (p/ + m) δ(p − m ) (p ) . (2.9) We now explain how the causal fundamental solution can be used to solve the Cauchy problem. We use a proof which immediately generalizes to the situation in an external ﬁeld or in curved space-time. To this end, we denote the initial time surface t = t0 by N, let ν = ∂t be the future-directed normal on N, and write the corresponding j Cliﬀord multiplication as ν/ = γ νj. In the Cauchy problem, one seeks for a solution of the Dirac equation with initial data ψN prescribed on a given Cauchy surface N. Thus in the smooth setting, ∞ (i∂/ − m) ψ = 0 , ψ|N = ψN ∈ C (N,SM) . (2.10) Lemma 2.1. The solution of the Cauchy problem (2.10) has the representation Z ψ(x) = 2π km(x, y) ν/ ψN (y) dµN (y) , N where km(x, y) is the integral kernel of the operator km, i.e. Z (kmφ)(x) = km(x, y) φ(y) dµM (y) . M LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 15

Proof. It suffices to consider a point x in the future of N, in which case (2.10) simplifies in view of (2.7) to Z ∧ ψ(x) = i sm(x, y) ν/(y) ψN (y) dµN (y) . N This identity is derived as follows: We let η ∈ C∞(M) be a function which is identically equal to one at x and on N, but such that the function ηψ has compact support (for Put in figure? ∞ example, in a foliation (Nt)t∈R one can take η = χ(t) with χ ∈ C0 (R)). Then

(∗) ∧ ∧ j ψ(x) = (ηψ)(x) = sm (D − m)(ηψ) = sm iγ (∂jη) ψ) , (2.11) where we used (2.1) and the fact that ψ is a solution of the Dirac equation. In (∗) we applied the identity

∧ ∞ ψ = s (D − m)ψ for ψ ∈ C0 (M,SM) , which follows either using integration by parts or abstractly from the uniqueness of the solution of the Cauchy problem (which will be established in Section 3.1), noting that the function ψ − s∧((D − m)ψ) satisﬁes the Dirac equation and vanishes in the past of the support of ψ. To conclude the proof, as the function η in (2.11) we choose a sequence η` which converges in the distributional sense to the function which in the future and past of N is equal to one and zero, respectively (more speciﬁcally, in the case that N is the hypersurface t = t0, we can choose η` as η`(t) = η((t − t0)/`), ∞ where η ∈ C (R) is a smooth non-negative function with η|R− ≡ 0 and η|[1,∞) ≡ 1).

The unique solvability of the Cauchy problem allows us to introduce the time evolution operator as follows. For clarity, we denote the Hilbert space H of square integrable spinors at time t with the scalar product (1.16) by (Ht, (.|.)H). Moreover, we denote a wave function ψ at time t by ψ|t (it can be the restriction of a wave function in space-time, but it could just as well be a function deﬁned only on the hyperplane of constant t). Solving the Cauchy problem with initial data at time t and evaluating the solution at some other time t0 gives rise to a mapping

t0,t U : Ht → Ht0 .

Since the scalar product (1.16) is time independent, the operator U t0,t is unitary. Since the Cauchy problem can be solved forwards and backwards in time, the unitary time evolution operators form a representation of the group (R, +). More precisely,

0 0 00 00 U t,t = 11 and U t,t U t ,t = U t,t .

Lemma 2.1 immediately gives the following representation of U t0,t with integral kernel, Z t0,t t0,t 3 U ψ|t (~y) = (U ψ|t)(~y, ~x) ψ(t, ~x) d x , (2.12) R3 t0,t 0 0 U ψ|t (~y, ~x) = 2π km (t , ~y), (t, ~x) γ . (2.13)

In Appendix A, this integral kernel is computed and estimated in more detail. 16 F. FINSTER AND J.-H. TREUDE

2.3. The Lippmann-Schwinger Equation. The Green’s functions and the time evolution operator are very useful for a perturbative treatment of the Dirac equation. In order to illustrate these methods, we briefly explain how the Cauchy problem in an external field can be treated. We thus consider the initial value problem ∞ 3 (i∂/ + B − m) ψ = 0 , ψ|t0 = ψ0 ∈ C (R ,SM) (2.14) with B as in (1.18). The following identity will be very useful in Section 5 for getting estimates. Proposition 2.2. (Lippmann-Schwinger equation) The Cauchy problem (2.14) has a solution ψ which satisfies the equation Z t t,t0 t,τ 0 ψ|t = U ψ0 − U γ B ψ |τ dτ . (2.15) t0 Proof. Obviously, the wave function ψ defined by (2.15) has the correct initial conditions at t = t0. Thus it remains to show that ψ satisfies the Dirac equation. To this end, we write the Dirac equation in the Hamiltonian form (1.19) and separate the free Hamiltonian H0 from the term involving the external potential, 0 0 0 (i∂t − H0) ψ = −γ B ψ with H0 = −iγ ~γ∇~ + γ m . (2.16)

Applying the operator (i∂t − H0) to (2.15) and using that the time evolution operator maps to solutions of the free Dirac equation, we obtain t,τ 0 0 (i∂t − H0) ψ|t = −U γ B ψ|τ τ=t = −γ B ψ|t , so that (2.16) is indeed satisfied. Note that the Lippmann-Schwinger equation involves the unknown solution both on the left and on the right. By iteratively applying the Lippmann-Schwinger equation, one gets a formal solution of the Cauchy problem in terms of the Dyson series, Z t t,t0 t,τ 0 τ,t0 ψ(t) = U ψ0 − U γ B|τ U ψ0 dτ t0 t t Z Z 0 0 0 t,τ 0 τ,τ 0 τ ,t0 + dτ dτ U γ B|τ U γ B|τ 0 U ψ0 + ··· . t0 t0 The combination U .,. γ0 appearing here can be written in view (2.13) in terms of the causal fundamental solution as t0,t 0 0 U ψ|tγ (~y, ~x) = 2π km (t , ~y), (t, ~x) . If one also combines the time integrals with the spatial integrals in (2.12) to integrals over Minkowski space, the Dyson series can be written in a manifestly covariant form. This is the covariant perturbation theory as used extensively for computations in perturbative quantum field theory. Unfortunately, the convergence of the Dyson series is not known in general. Moreover, the perturbative methods do not tell us about existence and uniqueness of solutions of the Cauchy problem. The existence and uniqueness as well as finite speed of propagation can be proved most conveniently in the framework of symmetric hyperbolic systems, which we introduce in the next section. LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 17

3. The Cauchy Problem, Linear Symmetric Hyperbolic Systems In this section, we want to prove that the Cauchy problem (2.14) in the presence of an external potential has a unique global solution. Moreover, we want to show that the finite speed of propagation as postulated by special relativity is indeed respected by the solutions of the Dirac equation. For later purposes, it is preferable to include an inhomogeneity. Thus in generalization of (2.14), we consider the Cauchy problem ∞ ∞ 3 (i∂/ + B − m) ψ = φ ∈ C (M,SM) , ψ|t0 = ψ0 ∈ C (R ,SM) for given φ and ψ0. In order to make the standard methods available, we multiply the equation by −iγ0, 0 0 11C4 ∂tψ + γ ~γ∇ψ = −iγ φ . (3.1) Now the matrices in front of the derivatives are all Hermitian (with respect to the 4 standard scalar product on C ). Moreover, the matrix in front of the time derivative is positive definite. Kurt Otto Friedrichs [35] (who, by the way, also invented the Friedrichs extension which we will use in Section 5.2) observed that these properties are precisely what is needed in order to get a well-posed Cauchy problem. He combined these properties in the notion of a symmetric hyperbolic system. We now give their general definition. More specifically, we consider a system of N complex-valued m equations with spatial coordinates ~x ∈ R and time t in an interval [0,T ] with T > 0. The initial data will always be prescribed at time t = 0. For notational clarity, we denote partial derivatives in spatial direction by ∇. Definition 3.1. A linear system of differential equations of the form m 0 X α A (t, ~x) ∂tu(t, ~x) + A (t, ~x) ∇αu(t, ~x) + B(t, ~x) u(t, ~x) = w(t, ~x) (3.2) α=1 with 0 α ∞ m N ∞ m N A ,A ,B ∈ C [0,T ] × R , L(C ) , w ∈ C [0,T ] × R , C . is called symmetric hyperbolic if (i) The matrices A0 and Aα are Hermitian, (A0)† = A0 and (Aα)† = Aα N (where † denotes the adjoint with respect to the canonical scalar product on C ). (ii) The matrix A0 is uniformly positive definite, i.e. there is a positive constant C such that 0 m A (t, ~x) > C 11CN for all (t, ~x) ∈ ([0,T ] × R ) . In the case w ≡ 0, the linear system is called homogeneous. A good reference for linear symmetric hyperbolic systems is the book by Fritz John [39, Section 5.3] (who was Friedrichs’ colleague at the Courant Institute). Our presentation was also influenced by [45]. We remark that the concept of symmetric hyperbolic systems can be extended to nonlinear equations of the form m 0 X α A (t, ~x,u) ∂tu(t, ~x) + A (t, ~x,u) ∇αu + B(t, ~x,u) = 0 , α=1 where the matrices A0 and Aα should again satisfy the above conditions (i) and (ii). For details we refer to [49, Section 16] or [46, Section 7]. Having the Dirac equation in 18 F. FINSTER AND J.-H. TREUDE mind, we always restrict attention to linear systems. We also note that an alternative method for proving existence of fundamental solutions is to work with the so-called Riesz distributions (for a good textbook see [1]). It is a remarkable fact that all partial differential equations in relativistic physics as well as most wave-type equations can be rewritten as a symmetric hyperbolic system. As an illustration, we now explain this reformulation in the example of a scalar hyperbolic equation. Example 3.2. Consider a scalar hyperbolic equation of the form m m X X ∂ttφ(t, ~x) = aαβ(t, ~x) ∇αβφ + bα(t, ~x) ∇αφ + c(t, ~x) ∂tφ + d(t, ~x) φ (3.3) α,β=1 α=1 with (aαβ) a symmetric, uniformly positive matrix (in the case aαβ = δαβ and b, c, d = 0, one gets the scalar wave equation). Now the initial data prescribes φ and its first time derivatives,

∞ ∞ φ|t=0 = φ0 ∈ C (M) , ∂tφ|t=0 = φ1 ∈ C (M) . (3.4) In order to rewrite the equation as a symmetric hyperbolic system, we introduce the vector u with k := m + 2 components by

u1 = ∇1φ, . . . , um = ∇mφ, um+1 = ∂tφ, um+2 = φ . (3.5) Then the system  m m X X  a ∂ u − a ∇ u = 0  αβ t β αβ β m+1  β=1 β=1  m m X X (3.6) − a ∇ u − b u +∂ u − c u −d u = 0  αβ β α α α t m+1 m+1 m+2   α,β=1 α=1  0 −um+1 +∂tum+2 = 0 is symmetric hyperbolic (as one verifies by direct inspection). Also, a short calculation shows that if φ is a smooth solution of the scalar equation (3.3), then the corresponding vector u is a solution of the system (3.6). Conversely, assume that u is a smooth solution of (3.6) which satisfies the initial condition u|t=0 = u0, where u0 is determined by φ0 and φ1 via (3.5). Setting φ = um+1, the last line in (3.6) shows that um+2 = ∂tφ. Moreover, the first line in (3.6) implies that ∂tuβ = ∇βum+2 = ∂t∇βφ. Integrating over t and using that the relation uβ = ∇βφ holds initially, we conclude that this relation holds for all times. Finally, the second line in (3.6) yields that φ satisfies the scalar hyperbolic equation (3.4). In this sense, the Cauchy problem for the system (3.6) is equivalent to that for the scalar equation (3.4). ♦ 3.1. Finite Propagation Speed and Uniqueness of Solutions. For what follows, it is convenient to combine the time and spatial coordinates to a space-time vector x = m (t, ~x) ∈ [0,T ] × R . We denote the space-time dimension by n = m + 1. Moreover, setting ∂0 ≡ ∂t, we use latin space-time indices i ∈ {0, . . . , m} and use the Einstein summation convention. Then our linear system (3.2) can be written in the compact form j A (x) ∂ju(x) + B(x) u(x) = w(x) . (3.7) LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 19

m+1 Next, a direction in space-time can be described by a vector ξ = (ξi)i=0,...,m ∈ R . Contracting with the matrices Aj(x), we obtain the Hermitian N × N-matrix j A(x, ξ) := A (x) ξj , referred to as the characteristic matrix. Note that in the example of the Dirac equation (3.1), the index i is a vector index in Minkowski space, and ξ should be regarded as a covector (i.e. a vector in the cotangent bundle). One should keep in mind that, despite the suggestive notation, the equation (3.7) should not be considered as being covariant, because it corresponds to the Hamiltonian formulation (3.1), where a time manifestly covariant? direction is distinguished. The determinant of the characteristic matrix is referred to as the characteristic polynomial, being a polynomial in the components ξi. For our purposes, it is most helpful to consider whether the characteristic matrix is positive or negative definite. If the vector ξ = (τ,~0) points in the time direction, then A(x, ξ) = τA0, which in view of Definition 3.1 is a definite matrix. By continuity, A(x, ξ) is definite if the spatial components of ξ are sufficiently small. In the example of the Dirac equation (3.1), the fact that 0 0 A(x, ξ) = 11ξ0 + γ ~γξ~ has eigenvalues ξ ± |ξ~| (3.8) shows that A(x, ξ) is definite if and only if ξ is a timelike vector. Moreover, it is positive definite if and only if ξ is future-directed and timelike. This suggests that the causal properties of the equation are encoded in the positivity of the characteristic matrix. We simply use this connection to define the causal structure for a general symmetric hyperbolic system. m+1 Definition 3.3. The vector ξ ∈ R is called timelike at the space-time point x if the characteristic matrix A(x, ξ) is definite. A timelike vector is called future- directed if A(x, ξ) is positive definite. If the characteristic polynomial vanishes, then m the vector ξ is called lightlike. A hypersurface H ⊂ [0,T ] × R with normal ν is called space-like if the matrix A(x, ν) is positive definite for all x ∈ H. The notion “normal” used here requires an explanation. The simplest method is to represent the hypersurface locally as the zero set of a function φ(x). Then the normal can be defined as the gradient of φ. In this way, the gradient is a co-vector, so that j j the contraction A νj = A ∂jφ is well-defined without referring to a scalar product. In particular, the last definition is independent of the choice of a scalar product on n space-time vectors in R . We always choose the normal to be future-directed and m+1 we normalize it with respect to the Euclidean scalar product on R , but these are merely a conventions. We shall now explain why and in which sense the solutions of symmetric hyperbolic systems comply with this notion of causality. Definition 3.4. Let u be a smooth solution of the linear symmetric hyperbolic system (3.7). A subset K of the initial value surface {t = 0} determines the solution at a m space-time point x ∈ [0,T ]×R if every smooth solution of the system which coincides on K with u, also coincides with u at x. The domain of dependence of K is the set of all space-time points which are determined by K. m Definition 3.5. An open subset L ⊂ (0,T )×R is called a lens-shaped region if L n is relatively compact in R and if its boundary ∂L is contained in the union of two m smooth hypersurfaces S0 and S1 whose intersection with [0,T ] × R is space-like. We 20 F. FINSTER AND J.-H. TREUDE set (∂L)+ = ∂L ∩ S1 and (∂L)− = ∂L ∩ S0, where we adopt the convention that (∂L)+ lies to the future of (∂L)−.

A typical example of a lens-shaped region is obtained by choosing S0 = {t = 0} as the initial data surface; see Figure 1. Moreover, it is often convenient to write the

( L) +

S ( L)− 0 S 1

Figure 1. A lens-shaped region hypersurface as a graph S = {(t, ~x) | t = f(~x)}. In this case, S is the zero set of the function φ(t, ~x) = t − f(~x), and the normal ν is the gradient of this function, i.e.

(νj)j=0,...,m = (1, ∇1f, . . . , ∇mf) . We ﬁrst consider the homogeneous equation

j (A ∂j + B) u = 0 . (3.9) The idea for analyzing the domain of dependence is to multiply this equation by a suitable test function and to integrate over a lens-shaped region. More precisely, we consider the equation Z −Kt j n 0 = e 2Rehu, (A ∂j + B)ui d x , (3.10) L N where h., .i denotes the canonical scalar product on C , and K > 0 a positive parameter to be determined later. Since the Aj are Hermitian, we have

j j j ∂jhu, A ui = 2 Re hu, A ∂jui + hu, (∂jA )ui , (3.11) and using this identity in (3.10) gives Z −Kt j ∗ j n 0 = e ∂jhu, A ui + u, B + B − (∂jA ) u d x . (3.12) L In the ﬁrst term we integrate by parts with the Gauß divergence theorem, Z Z −Kt j n −Kt 0 n e ∂jhu, A ui d x = K e hu, A ui d x L L Z Z (3.13) −Kt j −Kt j + e hu, νjA ui dµ∂L+ − e hu, νjA ui dµ∂L− . (∂L)+ (∂L)− LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 21

We now use (3.13) in (3.12) and solve for the surface integral over (∂L)+, Z Z −Kt j −Kt j e hu, νjA ui dµ∂L+ = e hu, νjA ui dµ∂L− (∂L)+ (∂L)− Z (3.14) −Kt ∗ j n + e u, − K − B − B + (∂jA ) u d x . L This identity is the basis for the following uniqueness results.

Theorem 3.6. Let u1 and u2 be two smooth solutions of the linear symmetric hyperbolic system (3.2) which coincide on the past boundary of a lens-shaped region L,

u1|(∂L)− = u2|(∂L)− .

Then u1 and u2 coincide in the whole set L.

Proof. The function u := u1 − u2 is a solution of the homogeneous system (3.9) with u|(∂L)− = 0. Hence (3.14) simpliﬁes to Z Z −Kt j −Kt ∗ j e hu, νjA ui dµ∂L+ = e u, − K − B − B + ∂jA u . (∂L)+ L Assume that u does not vanish identically in L. By choosing K suﬃciently large, we can then arrange that the right side becomes negative. However, since ∂L+ is a space-like hypersurface, the left side is non-negative. This is a contradiction.

As an immediate consequence, we obtain the following uniqueness result for solutions of the Cauchy problem.

Corollary 3.7. Let u1 and u2 be two smooth solutions of the linear symmetric hyperbolic system (3.2) with the same initial at t = 0. Then u1 ≡ u2 in a neighborhood of the initial data surface. j m If the matrices A are uniformly bounded, then u1 ≡ u2 in [0,T ] × R . Proof. The local uniqueness result follows immediately by covering the initial data surface by lens-shaped regions. For the global uniqueness, for any x0 = (t0, ~x0) ∈ Put in a ﬁgure? More m [0,T ] × R our task is to choose a lens-shaped region which contains x0 and whose details, explain the j geometry. past boundary S0 is contained in the surface {t = 0}. Using that the matrices A are uniformly bounded, there is ε > 0 such that the inequality k∇fk ≤ ε implies that the hypersurface S1 = {(t = f(~x), ~x)} is spacelike. Possibly after decreasing ε, we may choose p 2 f(~x) = t0 + ε 1 − 1 + k~x − ~x0k .

This concludes the proof.

By a suitable choice of lens-shaped region one can get an upper bound for the maximal propagation speed. For the Dirac equation, where the causal structure of Deﬁnition 3.3 coincides with that of Minkowski space in view of (3.8), one can choose for S1 a family of space-like hypersurfaces which converge to the boundary of a light cone. This shows that the maximal propagation speed for Dirac waves is indeed the speed of light. Explain more details, maybe more ﬁgures? 22 F. FINSTER AND J.-H. TREUDE

3.2. Global Existence of Smooth Solutions. We now write the linear system (3.7) as j Lu = w with L := A ∂j + B, where we again sum over j = 0, . . . , m. Going back to the formula for the divergence (3.11) and using the equation, we obtain j ∂jhu, A ui + hu, Cui = 2 Re hu, wi, (3.15) ∗ j C := B + B − (∂jA ) . (3.16) In what follows, for any λ ∈ [0,T ] we consider the time strip m Rλ = [0, λ] × R . j We assume that the functions A , B and w are smooth and uniformly bounded in Rλ. Moreover, we assume that w has spatially compact support (meaning that w(t, .) ∈ ∞ m C0 (R ) for all t ∈ [0, λ]). We denote the s-times continuously diﬀerential functions s on Rλ with spatially compact support by C (Rλ). The function spaces s s C (Rλ) and C (Rλ) are deﬁned as the functions which in addition vanish at t = 0 and t = λ, respectively. We want to solve the Cauchy problem ∞ m Lu = w , u|t=0 = u0 ∈ C0 (R ) (3.17) s in C (RT ). First of all, we may restrict attention to the case u ≡ 0,

Lu = w , u|t=0 ≡ 0 . (3.18) To see this, let u be a solution of the above Cauchy problem. Choosing a function v ∈ ∞ C (RT ) which at t = 0 coincides with u0. Then the functionu ˜ := (u − v) satisfies the j equation Lu˜ =w ˜ withw ˜ = w + A ∂jv + Bv) and vanishes at t = 0. If converselyu ˜ is a solution of the corresponding Cauchy problem with zero initial data, then u :=u ˜ + v solves is a solution of the original problem (3.17). We first derive so-called energy estimates. To this end, we integrate (3.15) over Rλ, integrate by parts with the Gauss divergence theorem and use that the initial values at t = 0 vanish. We thus obtain Z Z λ Z E(λ) := hu, A0ui dmx = dt 2 Re hu, wi − hu, Cui dmx . (3.19) t=λ 0 Rm Since the matrix C is uniformly bounded and A0 is uniformly positive, there is a constant K > 1 such that |hu, Cui| ≤ Khu, A0ui . Moreover, the linear term in u can be estimated with the Schwarz inequality by 1 1 2 Re hu, wi ≤ µ hu, ui + hw, wi ≤ hu, A0ui + hw, A0wi µ µ2 with a suitable constant µ > 0. Applying these estimates in (3.19) gives Z λ Z 1 0 n E(λ) ≤ (K + 1) E(t) dt + 2 hw, A wi d x . 0 µ Rλ Writing this inequality as Z λ Z d −(K+1)λ −(K+1)λ 1 0 n e E(t) dt ≤ e 2 hw, A wi d x , dλ 0 µ RT LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 23 we can integrate over λ to obtain Z T (K+1)T Z e − 1 1 0 n E(λ) dλ ≤ 2 hw, A wi d x . 0 K + 1 µ RT Finally, we apply the mean value theorem and use that the exponential function is monotone to conclude that Z T Z T (K+1)T 0 n E(λ) dλ ≤ 2 e hw, A wi d x . (3.20) 0 µ RT This is the desired energy estimate. Before going on, we point out that the notion of “energy” used for the quantity E(λ) does in general not coincide with the physical energy. In fact, for the Dirac equation (3.1), E(λ) has the interpretation as the electric charge. Following Example 3.2, for the scalar wave equation φ = 0, we find Z 2 2 2 m E(λ) = |∂tφ| + |∇φ| + |φ| d x . (3.21) Rm This differs from the physical energy by the last summand |φ|2 (and an overall factor 2). The name “energy” for E(λ) was motivated by the fact, considering only the highest derivative terms, the expression (3.21) is indeed the physical energy. We point out that, in contrast to the physical energy, the quantity E(λ) does in general depend on time. The point is that (3.20) gives an a-priori control of the energy in terms of the inhomogeneity. The exponential factor in (3.20) can be understood in analogy to a Grönwall estimate. 1 For the following construction, it is convenient to introduce on C (RT ) the scalar product Z (u, v) = hu, A0vi dnx . (3.22) RT We denote the corresponding norm by k · k. Setting furthermore T Γ2 = e(K+1)T , µ2 the energy estimate can be written in the compact form (u, u) ≤ Γ2 (w, w) . This inequality holds for every solution u of the differential equation Lu = w which 1 vanishes at t = 0. Noting that every function u ∈ C (RT ) is a solution of this differential equation with inhomogeneity w := Lu, we obtain 1 kuk ≤ Γ kLuk for all u ∈ C (RT ) . (3.23) This is the form of the energy estimates suitable for an abstract existence proof. In preparation, we want to introduce the notion of a weak solution. As the space of test 1 functions we choose C (RT ); this guarantees that integrating by parts does not yield 1 boundary terms at t = T . For a classical solution u ∈ C (RT ), we obtain

1 (v, w) = (Lv,˜ u) for all v ∈ C (RT ) , (3.24) where L˜ is the formal adjoint of L with respect to the scalar product (3.22), i.e. j j 0 −1 j 0 0 −1 † j 0 L˜ = −A˜ ∂j + B˜ with A˜ = (A ) A A , B˜ = (A ) B − (∂jA ) A . 1 If conversely (3.24) holds for all u ∈ C (RT ), then u is indeed a solution of the Cauchy problem (3.18) (the relation u|t=0 ≡ 0 is verified by considering the boundary terms 24 F. FINSTER AND J.-H. TREUDE obtained after integrating by parts). Hence we can use (3.24) as the definition of a weak solution of the Cauchy problem. Note that the operator −L˜ is again symmetric hyperbolic and has the same boundedness and positivity properties as L. Hence, repeating the above arguments, we obtain similar to (3.23) the “dual estimate” 1 kvk ≤ Γ˜ kLv˜ k for all v ∈ C (RT ) . (3.25) We now want to show the existence of weak solutions with the help of the Fréchet- Riesz theorem (for basics on functional analysis see [44] or [42]). To this end, we first 1 introduce on C (RT ) the scalar product hv, v0i = (Lv,˜ Lv˜ ) . (3.26) This scalar product is indeed positive definite, because for any v 6= 0, hv, vi = (Lv,˜ Lv˜ ) ≥ Γ˜2(v, v) 6= 0 , where in the last step we applied (3.25). Forming the completion, we obtain the Hilbert space (H, h., .i). We denote the corresponding norm by ||| . |||. In view of (3.25) 2 n and (3.26), we know that every vector v ∈ H is a function in L (RT , d x). Moreover, 2 n we know from (3.26) that Lv˜ is also in L (RT , d x). We remark that in functional 1,2 analytic language, the space H can be identified with the Sobolev space W (RT ), but we do not need this here. 0 1 We now consider for w ∈ C (RT ) und v ∈ C (RT ) the linear functional (v, w). In view of the estimate

(v, w) ≤ kvkkwk ≤ Γ˜kwk ||| v ||| , this functional is continuous in v ∈ H. The Fr´echet-Riesz theorem shows that there is U ∈ H with

2 ˜ ˜ hv, wiL (RT ) = hv, Ui = (Lv, LU) for all v ∈ H . 2 n Hence the function u := LU˜ ∈ L (RT , d x) satisfies the equation (3.24) and is thus the desired weak solution. Note that all our methods apply for arbitrarily large T . We have thus proved the global existence of weak solutions. We next want to show that the solutions are smooth. Thus our task is to show that s our constructed weak solution u is of the class C (Rλ), where s ≥ 1 can be chosen arbitrarily large. We first show that a linear symmetric hyperbolic system can be “enlarged” to include the partial derivatives of φ. j Lemma 3.8. Suppose that the system A ∂ju+Bu = w is symmetric hyperbolic. Then there is a symmetric hyperbolic system of the form j A˜ ∂jΨ + B˜Ψ =w ˜ (3.27) (n+1)N for the vector Ψ := (∂tu, ∇1u, . . . , ∇mu, u) ∈ C . Proof. Let i be a fixed space-time index. We differentiate the equation Lu = w, j ∂iw = ∂iLu = L∂iu + (∂iA ) ∂ju + (∂iB) u . This equation can be written as n j X ˜j A ∂jΨi + Bi Ψj + (∂iB) u =w ˜i , j=1 where we set ˜j j j Bi = B δi + (∂iA ) andw ˜i = ∂iw . LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 25

Combining these equations with the equation Lu = w, we obtain a system of the form (3.27), where the matrices Aj are block diagonal in the sense that ˜j ˜j α ˜j α j α A = (A )β α,β=0,...,m+1 with (A )β = A δβ . Obviously, this system is again symmetric hyperbolic. Iterating this lemma, we obtain (at least in principle) a symmetric hyperbolic system for u and all its partial derivatives up to any given order s. Since the corresponding 2 weak solution is in L (RT ), we conclude that u and all its weak partial derivatives are square integrable. The next lemma, which is a special case of the general Sobolev embedding theorems (see for example [12, Section II.5.] or [48, Section 4]), gives smoothness of the solution. m m Lemma 3.9. Let s = [ 2 ] + 1. If a function g on R is s-times weakly differentiable and Z |∇αg|2dx < C for all |α| = s , Rm s m then g is pointwise bounded, g ∈ C (R ). Proof. We apply the Schwarz inequality to the Fourier transform, Z m 2 2 d k −ikx g(x) = gˆ(k) e (2π)m Z 2 dk 2 − s 2 s −ikx = (1 + |k| ) 2 (1 + |k| ) 2 gˆ(k) ∈ e (2π)m Z dk ≤ c (1 + |k|2)s |gˆ(k)|2 , m (2π)m where the constant cm is finite due to our choice of s, Z dk c = (1 + |k|2)−s < ∞ . m (2π)m Using the Plancherel formula together with the fact that a factor k2 corresponds to a Laplacian in position space, we obtain s Z dmk X j (1 + |k|2)s |gˆ(k)|2 = k∇sgk2 < c . (2π)m n L2(Rm) j=0 √ ∞ Hence cm c is an L -bound for g. More precisely, in order to apply this lemma, we fix a time t and consider the solution u(λ, .). The identity (3.19) implies that E(λ) is controlled in terms of kwk and kuk. After interatively applying Lemma 3.8, we conclude the weak derivatives of u(λ, .) ex- 2 m ist to any order and are in L (R ). It follows that u(λ, .) is smooth. Finally, one uses the equation to conclude that u is also smooth in the time variable. The results of this section can be summarized as follows. Theorem 3.10. Consider the Cauchy problem m 0 X i (A ∂t + A ∇i + B)u = w , u|t=0 = u0 , i=1 26 F. FINSTER AND J.-H. TREUDE

∞ m 0 j where u0 and w(t, ·) ∈ C0 (R ). Assume that the matrices A , A and B as well as the functions w and u0 are smooth. Moreover, assume that all these functions as well as as all their partial derivatives are uniformly bounded on RT (where the bound may depend on the order of the derivatives). Then the Cauchy problem has a smooth solution on RT . This theorem also applies in the case T = ∞, giving global existence of a smooth solution. 3.3. Existence of Causal Green’s Functions. We now return to the Dirac equation in an external potential B. Then the previous existence and uniqueness results can be stated in a slightly stronger version. Theorem 3.11. Consider the Cauchy problem for the Dirac equation (2.14) for a smooth potential B. Then there is a unique global solution ψ ∈ C∞(M,SM).

Proof. It remains to show that we do not need to assume that B and ψ0 as well as all their partial derivatives are uniformly bounded. The reason is that, as explained at the end of Section 3.1, the propagation speed is the speed of light. Therefore, in the proof of Corollary 3.7, we can choose the lens-shaped region independent of B, making it unnecessary to impose bounds on B and its partial derivatives. For the existence of solutions, in order to construct ψ(x) it suﬃces to consider the equation in ∧ the compact region J (x)∩{t ≥ t0}. On this compact region, smoothness immediately gives uniform bounds for B, ψ and all their partial derivatives. Next, we explain how the previous existence and uniqueness results give rise to the existence of causal Green’s functions, being deﬁned as distributions. Our main tool is the Schwartz kernel theorem (see [38, Section 5.2] or [48, Section 4.6]). For clarity, we denote the objects in the external potential with an additional tilde. We begin with the causal fundamental solution and generalize Lemma 2.1. Theorem 3.12. Assume that the external potential B is smooth and that B and all its partial derivatives are uniformly bounded in Minkowski space. Then for any t, t0 ˜ 0 3 3 there is a unique distribution km(t, .; t0,.) ∈ D (R × R ) such that the solution of the Cauchy problem (2.14) has the representation Z 0 3 ψ(t, ~x) = 2π k˜m(t, ~x; t0, ~y) γ ψ0(~y) d y . (3.28) N 0 The integral kernel km is also a distribution in space-time, km ∈ D (M × M) It is a distributional solution of the Dirac equation, ˜ (i∂/x + B − m) km(x, y) = 0 . (3.29) Proof. The energy estimates combined with the Sobolev embedding of Lemma 3.9 showed that there is k ∈ N and a constant C = C(t, t0, ~x, B) such that the solution ψ(t, .) of the Cauchy problem is bounded in terms of the initial data by

|ψ(t, ~x)| ≤ C |ψ0|Ck , (3.30) where |ψ|2 := ≺ψ|γ0ψ , and the Ck-norm is defined by β |ψ0|Ck = max sup |∇ ψ0(~x)| . |β|≤k 3 ~x∈R Moreover, this estimate is locally unform in ~x, meaning that for any compact set K ⊂ 3 R , there is a constant C such that (3.30) holds for all ~x ∈ K. This makes it possible to LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 27 apply the the Schwartz kernel theorem [38, Theorem 5.2.1], showing that k˜m(t, .; t0,.) ∈ 0 3 3 D (R × R ). Next, we note that the constant C in (3.30) can also be chosen locally uniformly in t and t0. Thus, after evaluating weakly in t and t0, we may again apply the Schwartz 0 kernel theorem to obtain that k˜m ∈ D (M × M). Finally, the distributional equation (3.29) follows immediately from the fact that (3.28) is satisfies the Dirac equation for any choice of ψ0. Having defined the causal fundamental solution, we can introduce the causal Green’s functions by decomposing k˜m in time in such a way that the relation (2.7) extends to the setting in an external potential. Namely, for any t, t0 we introduce the distribu- ∨ ∧ 0 3 3 tions ˜m(t, .; t0,.), s˜m(t, .; t0,.) ∈ D (R × R ) by ( s˜∨ (t, .; t ,.) = 2πi k˜ (t, .; t ,.) Θ(t − t) m 0 m 0 0 (3.31) ∧ ˜ s˜m(t, .; t0,.) = −2πi km(t, .; t0,.) Θ(t − t0) (where Θ denotes the Heaviside function). Theorem 3.13. Assume that the external potential B is smooth and that B and all its partial derivatives are uniformly bounded in Minkowski space. Then there are unique distributions ∨ ∧ 0 s˜m, s˜m ∈ D (M × M) which satisfy the distributional equations 4 (i∂/x + B − m) sm(x, y) = δ (x − y) (3.32) and are supported in the upper respectively lower light cone, ∨ ∨ ∧ ∧ supps ˜m(x, .) ⊂ Jx , supps ˜m(x, .) ⊂ Jx . (3.33) Proof. It is clear by construction and the fact that the constant C in (3.30) can be chosen locally uniformly in x and y that the causal Green’s functions are well-defined distributions in D0(M × M). The support property (3.33) follows immediately from finite propagation speed as explained at the end of Section 3.1. The uniqueness of More details? the Green’s functions is clear from the uniqueness of solutions of the Cauchy problem. In order to derive the distributional equations (3.32), we only consider the retarded Green’s function (the argument for the advanced Green’s function is analogous). Then, according to (3.28) and (3.31), Z ∧ 0 3 Θ(t − t0) ψ(t, ~x) = i s˜m(t, ~x; t0, ~y) γ ψ0(~y) d y , N where ψ is the solution of the corresponding Cauchy problem. Applying the Dirac operator in the distributional sense yields Z 0 ∧ 0 3 iγ δ(t − t0) ψ0(t, ~x) = i(Dx − m) s˜m(t, ~x; t0, ~y) γ ψ0(~y) d y . N We now choose the initial values as the restriction of a test function in space-time, ∞ ψ0 = φ|t=t0 with φ ∈ C0 (M,SM). Then we can integrate over t0 to obtain This argument could Z be made a bit cleaner. 0 ∧ 0 4 iγ φ(x) = (Dx − m) s˜m(x, y) iγ φ(y) d y . M This gives the result. 28 F. FINSTER AND J.-H. TREUDE

4. The Fermionic Projector 4.1. The External Field Problem. Shortly after the formulation of the Dirac equation, it was noticed that this equation has solutions of negative energy, which have no obvious physical interpretation and lead to conceptual and mathematical difficulties. Dirac suggested to solve this problem by assuming that in the physical vacuum all states of negative energy are occupied by electrons forming the so-called Dirac sea [9, 10]. Since this many-particle state is homogeneous and isotropic, it should not be accessible to measurements. Due to the Pauli exclusion principle, additional particles must occupy states of positive energy, thus being observable as electrons. Moreover, the concept of the Dirac sea led to the prediction of anti-particles. Namely, by taking out particles of negative energy, one can generate “holes” in the Dirac sea, which are observable as positrons. Today, Dirac’s intuitive concept of a “sea of interacting particles” is often not taken literally. In perturbative quantum field theory, the problem of the negative-energy solutions is bypassed by a formal replacement and re-interpretation of the creation and annihilation operators of the negative-energy states of the free Dirac field, giving rise to a positive definite Dirac Hamiltonian on the fermionic Fock space. In the subsequent perturbation expansion in terms of Feynman diagrams, the Dirac sea no longer appears. This procedure allows to compute the S-matrix in a scattering process and gives rise to the loop corrections, in excellent agreement with the high-precision tests of quantum electrodynamics. One shortcoming of the perturbative approach is that the particle interpretation of a quantum state gets lost for intermediate times. This problem becomes apparent already in the presence of a time-dependent external field. Namely, as first observed by Fierz and Scharf [14], the Fock representation must be adapted to the external field as measured by a local observer. Thus the Fock representation becomes time and observer dependent, implying that also the distinction between particles and anti- particles loses its invariant meaning. The basic problem can be understood already in the one-particle picture: Mathematically, the distinction between particles and anti- particles corresponds to a splitting of the solution space of the Dirac equation into two subspaces. In the vacuum, or more generally in the presence of a static external field B(~x), in the Dirac equation (i∂/ + B(~x) − m) ψ(x) = 0 one can separate the time dependence with the plane-wave ansatz ψ(t, ~x) = e−iωt ψ(~x) . The separation constant ω, having the interpretation as the energy of the state, gives a natural splitting of the solution space into solutions of positive and negative energy. The Dirac sea can be introduced by occupying all states of negative energy. However, if the external field is time dependent, (i∂/ + B(t, ~x) − m) ψ(x) = 0 , the separation ansatz no longer works, corresponding to the fact that the energy of the Dirac states is no longer conserved. Hence the concept of positive and negative energy solutions breaks down, and the natural splitting of the solution space seems to get lost. LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 29

Another shortcoming of the standard reinterpretation of the free Dirac states of negative energy is that this procedure leads to inconsistencies when the interaction is taken into account on a non-perturbative level. For example, in [36] the vacuum state is constructed for a system of Dirac particles with electrostatic interaction in the Bogoliubov-Dirac-Fock approximation. In simple terms, the analysis shows that the interaction “mixes” the states in such a way that it becomes impossible to distinguish between the particle states and the states of the Dirac sea. Thus the only way to obtain a well-deﬁned mathematical setting is to take into account all the states forming the Dirac sea, with a suitable ultraviolet regularization. The framework of the fermionic projector is an approach to formulate quantum ﬁeld theory in such a way that the above-mentioned problems disappear. Out of all the states of the Dirac sea we build up the so-called fermionic projector, which puts Dirac’s idea of a “sea of interacting particles” on a rigorous mathematical basis. The fermionic projector gives a global (i.e. observer-independent) interpretation of particles and anti-particles even in the time-dependent setting at intermediate times.

4.2. The Dirac Sea and the Fermionic Projector of the Vacuum. We first explain the basic idea behind the construction of the fermionic projector in the Minkowski vacuum. Then the Dirac equation (1.10) can be solved in terms of plane wave solutions, ~ ψ(t, ~x) = e−iωt+ik~xχ , q where ω = ± |~k|2 + m2 and the constant spinor χ satisfies the algebraic equation (k/ − m) χ = 0 . (4.1) The sign of ω gives a distinction between solutions of positive and negative energy. As explained above, the sign of the frequency is no longer a well-defined concept if a time- dependent external field is present. The key point for what follows is the observation that, instead of using the sign of the frequency, we can just as well work with the sign of the spin scalar product ≺χ|χ of the spinor χ. Namely, the Dirac equation (4.1) implies that 1 ≺χ|χ = ≺χ|kχ/ . m The inner product ≺.|k./ is positive definite if the vector k is future directed, whereas it is negative definite if the vector k is past directed (this can either be verified by direct computation, or else it can be understood immediately from Lorentz invariance and the fact that the inner product ≺.|γ0. is positive definite). We conclude that ≺χ|χ > 0 if ω > 0 (4.2) ≺χ|χ < 0 if ω < 0 . Hence the frequency of the plane-wave solution is also encoded in the sign of the spin scalar product. This observation is very useful because the sign of an inner product is a well-defined concept even in situations when the frequency of solutions is not. In preparation for these generalizations, we point out that the inner product ≺ψ|ψ (x) will in general depend on the space-time point x. In order to obtain a real number, it is natural to integrate the spin scalar product over space-time to obtain the inner product Z <ψ|φ> = ≺ψ|φ (x) d4x . (4.3) M 30 F. FINSTER AND J.-H. TREUDE

This inner product is obviously Lorentz invariant. It is well-defined on wave functions in Minkowski space (which need not be solutions of the Dirac equation), provided that these wave functions decay so fast at infinity that the integral in (4.3) is finite. For example, it can be defined as the bilinear form ∞ ∞ <.|.> : C (M,SM) × C0 (M,SM) → C . When evaluating the inner product (4.3) for the plane wave solutions in Minkowski space, we again obtain the same signs as in (4.2), albeit with a divergent prefactor due to the fact that the volume of space-time is infinite. Therefore, before we can use the sign of the inner product (4.3) for a splitting of the solutions space into two subspaces, we need to find a way for dealing with the divergences of the space-time integral in (4.3). Another problem is how to use the sign of the inner product (4.3) to obtain a splitting of the solution space into two subspaces. We now explain how to resolve these difficulties in the Minkowski vacuum. Basically, we need to work with the causal fundamental solution in a specific way. Recall that the causal fundamental solution is a multiplication operator in momentum space (2.9). This operator involves the factor (p/ + m), which in view of the identity (p/ + m)2 = 2 2 p + 2pm/ + m = 2m(p/ + m) (where we used that km is supported on the mass shell p2 = m2) has non-negative eigenvalues (in fact, it has the eigenvalues 2m and 0, both with multiplicity two). In view of the step function (p0), we find that the operator km has a positive spectrum on the upper mass shell, whereas its spectrum is negative on the lower mass shell. Hence the sign of the spectrum of km coincides with the sign of the inner products in (4.2). In fact, the operator km is even the signature operator of the inner product h.|.i, as we now make precise. ∞ ∞ Proposition 4.1. For any ψ ∈ H ∩ Csc (M,SM) and φ ∈ C0 (M,SM),

(ψ | km φ) = <ψ|φ> . (4.4) Proof. Similar as in Lemma 2.1, we use a notation which generalizes to general Cauchy surfaces and to curved space-time. We choose Cauchy surfaces N+ and N− lying in Put in figure? the future and past of supp φ, respectively. Let Ω be the space-time region between these two Cauchy surfaces, i.e. ∂Ω = N+ ∪ N−. Then, according to (2.7), i (ψ | k φ) = (ψ | k φ) = (ψ | s∧ φ) m m N+ 2π m N+ i h i = (ψ | s∧ φ) − (ψ | s∧ φ) 2π m N+ m N− Z j ∧ = i ∇j≺ψ | γ smφ x dµ(x) , Ω where in the last line we applied the Gauß divergence theorem and used (1.16). Using that ψ satisfies the Dirac equation, a calculation similar to (1.14) yields Z Z ∧ (2.1) (ψ | km φ) = ≺ψ | (D − m) smφ x dµ(x) = ≺ψ|φ x dµ(x) . Ω Ω As φ is supported in Ω, we can extend the last integration to all of M, giving the result. According to this proposition, we can split up the solution space H into a positive and a negative definite subspace with respect to <.|.> by taking the positive and negative spectral subspace of the operator km. In order to define such spectral subspaces, LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 31 we must make sense of polynomials of km (like the characteristic polynomial or the polynomials used in the functional calculus). Hence we need to give powers of the operator km a mathematical meaning. In naive calculations, the problem arises that Maybe explain products of the distribution in (2.9) are ill-defined. However, this problem can be problem: The image bypassed if we work with a variable mass parameter and use a δ-normalization in the is not contained in mass parameter. For example, by formally manipulating the δ-distributions with the the domain of definition. usual calculating rules, we obtain for any m, m0 > 0, 2 2 0 0 2 0 2 0 km(p) km0 (p) = (p/ + m) δ(p − m ) (p )(p/ + m ) δ(p − (m ) ) (p ) = (p2 + (m + m0)p/ + mm0) δ(m2 − (m0)2) δ(p2 − m2) 1 (4.5) = (p2 + (m + m0)p/ + mm0) δ(m − m0) δ(p2 − m2) 2m = δ(m − m0)(p/ + m) δ(p2 − m2) . Hence 0 km km0 = δ(m − m ) pm , (4.6) where pm is the distribution 2 2 pm(p) = (p/ + m) δ(p − m ) .

The distribution pm again satisﬁes the Dirac equation. It diﬀers from km only by the fact that the factor (k0) in (2.9) is missing. Similarly, we obtain the calculation rules 0 pm pm0 = δ(m − m ) pm (4.7) 0 pm km0 = km0 pm = δ(m − m ) km . (4.8)

Combining (4.6), (4.7) and (4.8), we ﬁnd that the negative spectral subspace of km is precisely the image of the operator P sea given by 1 P sea = (p − k ) . (4.9) 2 m m In momentum space, P sea is the operator of multiplication by the distribution P sea(p) = (p/ + m) δ(p2 − m2) Θ(−p0) . Note that this distribution is supported on the lower mass shell, meaning that its image really consists precisely of all negative-energy solutions of the Dirac equation. We refer to P sea as the fermionic projector of the vacuum. Computing its Fourier transform, the resulting distribution P sea(x, y) turns out not to be causal, meaning that it is non-zero even in space-like directions. Similar to (2.6) and (2.8), we can consider P sea(x, y) as the integral kernel of an operator, sea ∞ ∞ P : C0 (M,SM) → C (M,SM) (note that, as P sea is not causal, this operator no longer maps to functions with spacelike compact support). The fermionic projector is idempotent if we again normalize with a δ-distribution in the mass parameter, sea sea 0 sea Pm Pm0 = δ(m − m ) Pm (4.10) (where we denoted the mass parameter by a subscript). Moreover, a direct computation shows that it is symmetric with respect to the space-time inner product (4.3), i.e. sea sea ∞

= <ψ|P φ> for all ψ, φ ∈ C0 (M,SM) . (4.11) 32 F. FINSTER AND J.-H. TREUDE

The relations (4.10) and (4.11) were the motivation for the name fermionic projector. In the remainder of Section 4, we shall generalize the above construction to the setting with a time-dependent external ﬁeld. We ﬁrst give a brief outline of a perturbative treatment (Section 4.3) and then give a non-perturbative construction (Section 5). This analysis will also give a mathematically satisfying treatment of the formal manipulation of distributions in (4.5).

4.3. Perturbative Description. We brieﬂy outline the construction in [15, 28]; see also [19, Chapter 2]. We again consider the Dirac equation in an external potential (1.18), which we always assume to be symmetric (1.17). From the existence and uniqueness results of Section 3 we know that the causal Green’s functions exist. In order to distinguish them from the Green’s functions in the vacuum, we denote them ∨ ∧ bys ˜m ands ˜m. They are expressed perturbatively by ∞ ∞ ∨ X ∨ n ∨ ∧ X ∧ n ∧ s˜m = (−sm B) sm , s˜m = (−sm B) sm , (4.12) n=0 n=0 where the operator products involving the potential B are deﬁned as follows, Z ∨ ∨ 4 ∨ ∨ (sm B sm)(x, y) := d z sm(x, z) B(z) sm(z, y) .

Using the support property (2.5), one verifies inductively that every summand of the perturbation expansion (4.12) is again supported in the future respectively past light cone. Moreover, using (2.1), it is straightforward to verify that the perturbation series (4.12) indeed satisfy the defining relations for the advanced Green’s function ∨ 4 (i∂/x + B(x) − m)s ˜m(x, y) = δ (x − y) . The structure of the perturbation series (4.12) can be understood in analogy to a Neumann series. Namely, if we consider the Green’s function as the inverse of the −1 Dirac operator, sm = (i∂/ − m) , a formal manipulation gives −1 −1 s˜m = (i∂/ − m) + B = (i∂/ − m) (11+ sm B) ∞ −1 X n = (11+ sm B) sm = (−sm B) sm . n=0 But clearly, this calculation does not go beyond a formal level; in particular, it does not tell us whether the intermediate factors should be the advanced or retarded Green’s functions. Physically, the summands of (4.12) can be regarded as Feynman tree diagrams. Mathematically, the perturbation series (4.12) is defined in the following sense. The operator products in (4.12) are well-defined tempered distributions, provided that B is smooth and decays so fast at infinity that the functions B(x), xiB(x), and xixjB(x) are all integrable (see [19, Lemma 2.2.2]). Thus every summand of the perturbation series is well-defined. However, it is not clear whether or in which sense the perturbation series converge. For this reason, one considers (4.12) merely as a formal power series in B. Having uniquely introduced the causal Green’s functions, we can introduce the causal fundamental solution in analogy to (2.7) by 1 k˜ = s˜∨ − s˜∧ . m 2πi m m LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 33

Substituting the perturbation series for the Green’s functions and using (2.7), we obtain for k˜m an operator product expansion of the form

∞ αmax(k) ˜ X X km = km + cα C1,α B C2,α B ··· B Ck+1,α , k=1 α=0 where the factors Cl,α are the advanced or Green’s functions in the vacuum, and the cα are combinatorial factors. Similar to (4.6), one also has rules to compute operator products which involve the Green’s functions. Using these rules, one obtains ∞ αmax(k) ˜ ˜ 0 X X km km0 = δ(m − m ) p˜m + cα C1,α B C2,α B ··· B Ck+1,α , k=1 α=0 where the factors Cl,α are now the Green’s functions or the fundamental solutions pm or km, and the cα are more complicated combinatorial factors. In this way, one can introduce a functional calculus for the operator k˜m on the level of formal power series, making it possible to give a unique perturbation expansion for the fermionic projector P sea in the presence of the external potential B,

∞ αmax(k) sea X X P = cα C1,α B C2,α B ··· B Ck+1,α . k=0 α=0 Exactly as explained for the Green’s functions, every summand of this perturbation expansion is a well-defined tempered distribution. But it is not known whether and in which sense the perturbation expansion converges. The fermionic project is again idempotent (4.10) and symmetric (4.11). The outlined methods and the resulting expansions are referred to as causal perturbation theory. To avoid confusion about the notation, we point out that our causal perturbation theory does not seem to be related to Scharf’s “causal approach” to quantum electrodynamics [47]. Similar to the Epstein-Glaser method [11], Scharf uses causality to avoid the ultraviolet divergences of loop diagrams. The so-regularized loop diagrams are then unique up to a finite number of free parameters. In other setting, however, one only considers an external potential, such that the resulting Feynman diagrams are all finite tree diagrams. The point of our causal perturbation expansion is that it uniquely fixes the choice of Green’s functions in the perturbation expansion. As a further difference, Scharf only considers a scattering process, whereas our goal is to describe the dynamics also for intermediate times. We finally remark that the above operator product expansions are also very useful for analyzing the fermionic projector in position space. Namely, it turns out that the fermionic projector P sea(x, y) has singularities on the boundary of the light cone. The so-called light-cone expansion gives a systematic method for analyzing this singular behavior in explicit detail. We refer the interested reader to [17, 18] and [19, §2.5].

5. A Non-Perturbative Construction of the Fermionic Projector In this section, we shall give a functional analytic construction of the fermionic projector in an external potential. We ﬁrst explain our strategy. Our starting point is that, due to the existence and uniqueness results in Section 3, we have causal fundamental solutions in the presence of an external potential given as operators ∨ ∧ ∞ ∞ sm, sm : C0 (M,SM) → Csc (M,SM) . 34 F. FINSTER AND J.-H. TREUDE

Hence we can again introduce the causal fundamental solution by 1 k := s∨ − s∧ : C∞(M,SM) → C∞(M,SM) . (5.1) m 2πi m m 0 sc The basic difficulty in a functional analytic treatment is related to the fact that the operator km acts on the wave functions in space-time, which do not form a Hilbert space. More specifically, km is symmetric with respect to the Lorentz invariant inner product (4.3). But as (4.3) is not positive definite, the corresponding function space merely is a Krein space. There is a spectral theorem in Krein spaces (see for example [5, 40]), but this theorem only applies to so-called definitizable operators. The operator km, however, is not known to be definitizable, making it impossible to apply spectral methods in indefinite inner product spaces. In order to overcome this difficulty, we make essential use of the fact that, according to Proposition 4.1, the operator km is the signature operator of the space-time inner product (note that Proposition 4.1 holds just as well in the presence of an external potential). To explain the idea, let us consider what happened if (4.4) were true ∞ for all ψ, φ ∈ H ∩ Csc (M,SM). Then, taking the complex conjugate of (4.4) and exchanging ψ with φ, we could conclude that

∞ (ψ | km φ) = (km ψ | φ) for all ψ, φ ∈ H ∩ Csc (M,SM) .

Thus km would be a symmetric operator with respect to the scalar product (.|.). Thus we would have a symmetric operator on a Hilbert space, making the powerful methods of spectral theory in Hilbert spaces available. Unfortunately, the relation (4.4) does ∞ not hold for all ψ, φ ∈ H ∩ Csc (M,SM), because the time integral in the space-time inner product (4.3) may diverge. Our method for dealing with this problem is inspired by the δ-normalization in the mass parameter used in (4.6). Namely, we shall work with a variable mass parameter m and consider families of solutions of the Dirac equation. The so-called mass oscillation property will make sense of the space-time integral in (4.3) after integrating over the mass parameter.

5.1. The Mass Oscillation Property. We consider the mass parameter in a bounded open interval, m ∈ I := (ma, mb) with ma, mb > 0. For a given Cauchy surface N, we consider a function ψN (x, m) ∈ SxM with x ∈ N and m ∈ I. We assume that this wave ∞ function is smooth and has compact support in both variables, ψN ∈ C0 (N × I,SM). For every m ∈ I, we consider the solution of the Cauchy problem, taking ψN (., m) as the initial data,

(Dx − m) ψ(x, m) = 0 , ψ(x, m) = ψN (x, m) ∀x ∈ N. (5.2) Since the solution of the Cauchy problem is smooth and depends smoothly on parameters, we know that ψ ∈ C∞(M × I,SM). Moreover, due to ﬁnite propagation speed, ψ(., m) has spatially compact support. Finally, the solution is clearly com- pactly supported in the mass parameter m. We summarize these properties of ψ in the notation ∞ ψ ∈ Csc,0(M × I,SM) .

We often denote the dependence on m by a subscript, ψm(x) := ψ(x, m). In particular, for any ﬁxed m we can introduce the scalar product (1.16). As it is independent of the choice of the Cauchy surface, we can again denote it by (.|.)m. On solutions ψ, φ ∈ LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 35

∞ Csc,0(M × I,SM) of (5.2), we can thus introduce the scalar product Z (ψ|φ) = (ψm|φm)m dm (5.3) I (where dm is the Lebesgue measure). Forming the completion gives the Hilbert space (H, (.|.)). It consists of measurable functions ψ(x, m) such that for almost all m ∈ I, the function ψ(., m) is a weak solution of the Dirac equation which is square integrable over any Cauchy surface. Moreover, the integral (1.16) is integrable Explain in detail over m ∈ I, so that the scalar product (5.3) is well-defined. We denote the norm on H why ψ(., m) is a weak by k.k. solution? Our motivation for considering a variable mass parameter is that integrating over this mass parameter should improve the decay properties of the wave function for large times. The naive picture is that ψ(x, .) oscillates as a function of m, with a frequency which increases for large times (note that for plane wave solutions√ in Minkowski space, this oscillatory behavior is described by the phase factor e−i m2+|~k|2t). Thus when integrating over m, contributions with different phases should compensate each other, giving rise to better decay properties. This decay for large times should also ensure that solutions for different masses should be orthogonal with respect to the inner product (6.27). Our next definition makes precise what we need for the construction of the fermionic projector. We introduce the operator of multiplication by m,

T : H → H , (T ψ)m = m ψm . Obviously, this operator maps preserves the support properties, and thus ∞ ∞ T | ∞ : C (M × I,SM) → C (M × I,SM) . Csc,0(M×I,SM) sc,0 sc,0 Moreover, it is a symmetric operator, and it is bounded because the interval I is, T ∗ = T ∈ L(H) . (5.4) Integrating over m gives the operation Z ∞ ∞ p : Csc,0(M × I,SM) → Csc (M,SM) , pψ = ψm dm . (5.5) I Deﬁnition 5.1. The Dirac operator D = i∂/ + B has the mass oscillation property in the interval I ⊂ R if the following conditions hold: ∞ (a) For every ψ, φ ∈ Csc,0(M × I,SM), the function ≺pφ|pψ is integrable on M. Moreover, there is a constant c = c(ψ) such that ∞ || ≤ c kφk ∀φ ∈ Csc,0(M × I,SM) . (5.6) ∞ (b) For all ψ, φ ∈ Csc,0(M × I,SM), = . (5.7) 5.2. A Self-Adjoint Extension of S2. In view of the inequality (5.6), every ψ ∈ ∞ Csc,0(M × I,SM) gives rise to a bounded linear functional on H, which by the Riesz representation theorem we can identify with a vector of H. We thus obtain the mapping ∞ S : Csc,0(M × I,SM) → H , (ψ|Sφ) = ∀φ ∈ H . (5.8) This operator is symmetric, because ∞ (ψ|Sφ) = = (Sψ|φ) ∀φ, ψ ∈ Csc,0(M × I,SM) . 36 F. FINSTER AND J.-H. TREUDE

Moreover, (5.7) implies that the operators S and T commute, ∞ ST = TS : Csc,0(M,SM) → H . (5.9) We would like to construct a self-adjoint extension of the operator S which commutes with T . A general method for constructing self-adjoint extensions of symmetric operators is provided by the Friedrichs extension (see for example [44, 42]). However, as this method only applies to semi-bounded operators, we are led to working with the operator S2. We thus introduce the scalar product ∞ ∞ hψ|φiS2 = (ψ|φ) + (Sψ|Sφ): Csc,0(M × I,SM) × Csc,0(M × I,SM) → C . The corresponding norm is bounded from below by the norm k.k. Thus forming the completion gives a subspace of H,

∞ h.|.iS2 H0 := Csc,0(M × I,SM) ⊂ H . We set 2 D(S ) = u ∈ H0 such that hu|φiS2 ≤ c(u) kφk ∀φ ∈ H0 . Proposition 5.2. Introducing the operator S2 with domain of deﬁnition D(S2) by 2 2 2 S : D(S ) ⊂ H → H , (S ψ|φ) = hψ|φiS2 − (ψ|φ) ∀φ ∈ H0 , (5.10) this operator is self-adjoint. The operator T maps D(S2) to itself and commutes with S2, S2 T = TS2 : D(S2) → H . (5.11) 2 2 2 Add more details in Proof. Let us show that T (D(S )) ⊂ D(S ). Thus let u ∈ D(S ). Then u ∈ H0, so ﬁnal version? ∞ that there exists a series un ∈ Csc,0(M × I,SM) which converges to u in the topology given by h.|.iS2 . Since the operator T is bounded on H, the series T un converges to T u. ∞ This shows that T u ∈ H0. Next, for any φ ∈ Csc,0(M × I,SM), it follows from (5.4) and (5.9) that

hT u|φiS2 ≤ c(u) kT φk ≤ c(u) kT k kφk . We conclude that T u ∈ D(S2). To prove (5.11), we first evaluate the operator product to un. Then we know 2 2 from (5.12) and (5.4) that S T un = TS un. Taking the limit n → ∞ gives the result. Now the spectral theorem for commuting operators (see [44]) yields a spectral mea- Add more details in sure E on σ(S2) × I such that final version? Z 2 S T = λ m dEλ,m . σ(S2)×I Using this spectral measure, we can also extend the operator S to the natural domain of definition D(S) = D(|S|). Lemma 5.3. The operator S in (5.8) can be extended by continuity to the domain of definition Z

D(S) := u ∈ H |λ| d(u|Eλ,mu) < ∞ . σ(S2)×I LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 37

Proof. Suppose that u ∈ D(S). Then u is in the domain of deﬁnition of the opera- ∞ tor |S|, and thus there is a sequence un ∈ Csc,0(M,SM) with

un → u and |S|(un − u) → 0 . According to (5.10), we know that

2 2 i,j→∞ kS(ui − uj)k = |S|(ui − uj) −−−−→ 0 . Hence we can deﬁne Su by Su = limn→∞ Sun.

5.3. The Operator k and its Extended Domain. Acting with the operator km as ∞ deﬁned in (5.1) for each m separately gives an operator on C0 (M × I,SM), ∞ ∞ k : C0 (M × I,SM) → Csc,0(M × I,SM) ∩ H , (kψ)m = kmψm . (5.13) The goal of this section is to extend the domain of deﬁnition such as to obtain a surjective operator k : D(k) → H . (5.14)

For notational clarity, we denote the Hilbert space of wave functions on Nt with the scalar product (1.16) with N = Nt by (Ht, (.|.)t). 2 Deﬁnition 5.4. A section ψ ∈ Lloc(M,SM) is called timelike compact if the following conditions hold: (a) There is a constant C = C(ψ) > 0 such that for almost all t ∈ R, the restriction ψ|t to the hypersurface Nt is a measurable function and ψ|t | ψ|t t < C . (5.15)

(b) The function ψ|t vanishes identically for t outside a compact set.

Proposition 5.5. We choose D(km) as the vector space generated by all timelike compact sections in SM. Then the operator km, (5.1), extends to a well-deﬁned operator

km : D(km) → Hm . This operator is surjective. 2 Proof. The Hilbert space Hm consists of all weak solutions in Lloc(M,SM) of the Dirac equation with the property that their restriction to the Cauchy surfaces is in Ht for almost all t ∈ R. The unique solvability of the Cauchy problem gives rise to the unitary time evolution operator (for details see [32, Section 3.4]) t,t0 U : Ht0 → Ht0 .

This time evolution gives a natural identiﬁcation of the Hilbert spaces (Ht)t∈R with Hm. According to Lemma 2.1 (which holds just as well in the presence of an external potential), we can express the operator km, (5.1), in terms of the time evolution operator Ut,t0 by Z ∞ 1 0 0 (kmφ)(t, x) = Ut,t0 γ φ|t0 (x) dt . (5.16) 2π −∞ This equation can be used to extend the operator km to the domain D(km). Namely, suppose that ψ has timelike compact support. Then the restriction ψ|t is in Ht for almost all t, so that the integrand in (5.16) is well-deﬁned for almost all t0. Moreover, in view of (5.15), the integrand is even uniformly bounded. Using property (b) of 38 F. FINSTER AND J.-H. TREUDE

Deﬁnition 5.4, the integrand vanishes for t0 outside a compact set. We conclude that 0 the t -integral in (5.16) exists and is ﬁnite, giving a vector kmφ ∈ Hm. ∞ In order to prove that km is surjective, we choose a function χ ∈ C (R) such that χ|[0,∞) ≡ 1 and χ|(−∞,−1] ≡ 0. For a given ψm ∈ Hm, we introduce the function φ by i 1 φ(t, x) := D − mχ(t) ψ (t, x) = − γ0χ0(t) ψ (t, x) , 2π m 2π m where in the last step we used that the Dirac operator can be written as D = 0 γ (i∂t − Ht) with a purely spatial operator Ht. Clearly, the function φ is timelike compact. Moreover, the unique solvability of the inhomogeneous Cauchy problem implies that Z i ∧ (χψm)(t, x) = s (t, x; τ, y) φ(τ, y) dµM (τ, y) . 2π M Evaluating this equation for t > 0, the factor χ can be left out. Adding a corresponding term with the advanced Green’s function, this term vanishes if t > 0. Using (5.1), we thus obtain the equation Z ψm(t, x) = km(t, x; τ, y) φ(τ, y) dµM (τ, y) , M valid if t > 0. Since both sides of this equation are solutions of the Dirac equation, the unique solvability of the Cauchy problem (for initial data chosen for example at t = 2) implies that this equation holds for all (t, x) ∈ M. This concludes the proof. Using this proposition, we can extend the operator k, (5.13), to a surjective operator of the form (5.14) by setting D(k) = ψ(x, m) with ψm ∈ D(km) for a.a. m ∈ I and kψ ∈ H . 5.4. The Fermionic Projector as an Operator-Valued Measure. After these 0 preparations, we can introduce P± as a spectral measure on I, i.e. for any f ∈ C (I), Z Z p 1 −1 2 f(m) dP±(m) = f(m) S ± |λ| 2 λ dEλ,m k : D(f) → Lloc(M,SM) , I 2 σ(S2)×I where

D(f) = ψ ∈ D(k) with kψ ∈ Df(T ) |S|−1 . Then the spectral theorem applied to S gives the following result. Proposition 5.6. For all f ∈ C0(I) and φ, ψ ∈ D(f), the operator-valued measure dP± has the properties Z Z f(m) = f(m) <φ | dP±(m) ψ> . (5.17) I I Moreover, for all f, g ∈ C0(I) and φ ∈ D(f), ψ ∈ D(g), ZZ Z 0 0 f(m) g(m ) = (fg)(m) <φ | dP±(m) ψ> (5.18) I×I I ZZ 0 0 f(m) g(m ) = 0 . (5.19) I×I LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 39

The relations (5.18) mean qualitatively that the operators dP±(m) act “locally” for fixed m. This property can be used to show that dP±(m) does not depend on the choice of the interval I (provided, of course, that m ∈ I). The fermionic projector dP (m) is defined by dP (m) = dP−(m). Clearly, defining the fermionic projector as the spectral measure dP (m) is not the end of the story. More specifically, one would like to define P for fixed m as a distribution. This can indeed be accomplished using the Schwarz kernel theorem, but it requires additional estimates which we cannot present here. Moreover, one would like to analyze the behavior of P in position space with a Hadamard or light-cone expansion. Due to the lack of time, we cannot enter these topics here. Instead, we conclude this section by explaining how the mass oscillation property can be proved in Minkowski space under suitable decay assumptions of B at infinity.

5.5. Proof of the Mass Oscillation Property. We ﬁrst state the main result of this section. We denote spatial derivatives by ∇ and use the notation with multi-indices, α i.e. ∇ ≡ ∂α1 ··· ∂αp , where p = |α| is the length of the multi-index. We denote the spatial L2-Sobolev norms as usual by

2 X β 2 kψ(t)kW p,2 = k∇ ψ(t)k , (5.20) |β|≤p where ψ(t, ~x) is a Dirac wave function and kψk its norm corresponding to the Hilbert space scalar product (1.16). For the potential B we work with spatial Ck-norms defined by β |B(t)|Ck = max sup |∇ B(t, ~x)| , (5.21) |β|≤k 3 ~x∈R where |.| denotes any matrix norm. Theorem 5.7. Assume that the external potential B is smooth and for large times decays faster than quadratically for large times in the sense that c |B(t)| 2 ≤ (5.22) C 1 + |t|2+ε for suitable constants c, ε > 0. Then the Dirac operator ∂/ + B has the mass oscillation property (see Definition 5.1). Before coming to the proof, we point out that the decay assumption (5.22) is probably not optimal. Also, using that Dirac solutions dissipate, we expect that the pointwise decay in time could be replaced or partially compensated by a suitable spatial decay assumption. Also, one could probably work with different norms. Thus we do not aim for largest generality. Instead, our goal is to explain the basic mechanism in the simplest possible setting. The proof of Theorem 5.7 is based on the following basic estimate. Proposition 5.8. Under the decay assumptions (5.22) on the external potential, there ∞ are constants c, ε > 0 such that for every family ψ ∈ Csc,0(M × I,SM) of solutions of the Dirac equation,

c X b kpψ(t)k ≤ sup k∂ ψ (0)k 2,2 . (5.23) 1 + |t|1+ε m m W m∈I b≤2 40 F. FINSTER AND J.-H. TREUDE

This estimate shows that after integrating over the mass parameter (see (5.5)), the Dirac wave function decays in time. This decay can be understood qualitatively from the fact that the wave function ψm(t, ~x) oscillates when m is varied, so that integrating over m leads to cancellations. Since the frequency of these oscillations increases with |t|, the cancellations in the integral give rise to a decay in time. This “oscillation argument” was the motivation for the name “mass oscillation property.” The above proposition quantiﬁes this argument. For clarity, we ﬁrst explain why Proposition 5.8 implies the mass oscillation property Proof Theorem 5.7 under the assumption that Proposition 5.8 holds. To prove (5.6), we estimate as follows, Z ∞ Z ∞

|| ≤ (pψ | pφ)(t) dt ≤ sup kpφ(t)k kpψ(t)k dt . −∞ t −∞ The last integral is ﬁnite by Proposition 5.8. The supremum can be bounded by the Hilbert space norm using the H¨olderinequality,

1 Z Z Z 2 p 2 p kpφ(t)k = φm(t) dm ≤ kφm(t)k dm ≤ |I| kφm(t)k dm = |I| kφk . I I I The identity (5.7) follows by integrating the Dirac operator in space-time by parts, Z = = = ≺Dpψ|pφ (x) d4x M (∗) Z = ≺pψ|Dpφ (x) d4x = = . M In (∗) we used that the Dirac operator is formally self-adjoint with respect to <.|.>. Moreover, we do not get boundary terms in view of the time decay in Proposition 5.8. The remainder of this section is devoted to the proof of Proposition 5.8. One ingredient of our proof is the Lippmann-Schwinger equation (2.15), Z t t,0 t,τ 0 ψm(t) = Um ψm(0) − Um γ B ψm |τ dτ , (5.24) 0 where for clarity we denoted the m-dependence of the time evolution operator by a subscript m. The free time evolution operator appearing here was estimated in Appendix A in momentum space. Keeping in mind that a factor ~k corresponds to a derivative −i∇~ in position space, the result of Lemma A.3 can readily be translated to position space. It gives the representation 2 0 1 ∂ 0 ∂ 0 0 U t,t = At,t + Bt,t + Ct,t , (5.25) m (t − t0)2 ∂m2 m ∂m m m with operators t,t0 t,t0 t,t0 2,2 3 2 3 Am ,Bm ,Cm : W (R ,SM) → L (R ,SM) , which are bounded uniformly in time by t,t0 t,t0 t,t0 kAm (φ)k + kBm (φ)k + kCm (φ)k ≤ c kφkW 2,2 , (5.26) where c is a constant which depends only on m. LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 41

We now explain how the representation (5.25) and (5.26) can be used to obtain time decay of pψ. For clarity, we treat the two summands in (5.24) separately. In the first summand, we apply (5.25) and integrate by parts in m to obtain Z Z t,0 1 2 t,0 t,0 t,0 Um ψm(0) dm = 2 ∂mAm + ∂mBm + Cm ψm(0) dm I t I Z 1 t,0 2 t,0 t,0 = 2 Am (∂mψm(0)) − Bm (∂mψm(0)) + Cm ψm(0) dm . t I Using (5.26) gives the estimate Z 2 t,0 c |I| X b U ψm(0) dm ≤ sup k∂ ψm(0)k 2,2 . (5.27) m t2 m W I m∈I b=0 Note that the right side depends only on the wave function at time t = 0. Hence (5.27) gives quadratic decay in time. We next consider the integrand of the second summand in (5.24). Again using (5.25) and integrating by parts, we obtain Z Z t,τ 0 1 t,0 2 t,0 t,0 0 Um γ B ψm |τ dm = 2 Am ∂m − Bm ∂m + Cm γ B ψm |τ dm I (t − τ) I and thus Z 2 t,τ 0 c |I| X b U γ B ψm |τ dm ≤ sup B(τ) ∂ ψm(τ) 2,2 m (t − τ)2 m W I m∈I b=0 2 c |I| X b ≤ kB(τ)k 2 sup ∂ ψm(τ) 2,2 . (t − τ)2 C m W m∈I b=0 We now bound B(τ) with the help of (5.22). Moreover, we estimate the Sobolev b norm ∂mψm(τ) W 2,2 at time τ with the help of Lemma B.1 proved in Appendix B. This gives the estimate Z 2 2 2 t,τ 0 c C |I| 1 + |t| X p U γ B ψm |τ dm ≤ sup k∂ ψ(0)k 2,2 . m (t − τ)2 1 + |t|2+ε m W I m∈I p=0 This estimate gives the desired decay provided that τ and t are sufficiently close to each other. More precisely, we shall apply it in the case |τ| ≤ |t|/2. Then the estimate simplifies to Z ˜ 2 t,τ 0 C X p U γ B ψm |τ dm ≤ sup k∂ ψ(0)k 2,2 if |τ| ≤ |t|/2 . (5.28) m t2+ε m W I m∈I p=0 t,τ In the remaining case |τ| > |t|/2, we simply use the unitarity of Um to obtain Z t,τ 0 Um γ B ψm |τ dm ≤ |I| kB(τ)kC0 sup kψmk . I m∈I Now we apply (5.22) and use that |τ| > |t|/2. This gives Z ˜ t,τ 0 C Um γ B ψm |τ dm ≤ 2+ε sup kψmk if |τ| > |t|/2 . (5.29) I t m∈I

This again decays for large t because τ is close to t and kB(τ)kC0 decays for large τ. 42 F. FINSTER AND J.-H. TREUDE

Comparing (5.28) and (5.29), we ﬁnd that the inequality in (5.28) even holds for all τ. Thus we may integrate τ from 0 to t to obtain the following estimate for the second summand in (5.24),

Z Z t 0 2 t,τ 0 C X p dm U γ B ψm |τ dτ ≤ sup k∂ ψ(0)k 2,2 . m t1+ε m W I 0 m∈I p=0 Combining this inequality with the estimate (5.27) of the ﬁrst summand in (5.24), we obtain the desired inequality (5.23) (possibly after decreasing ε such that ε < 1). This concludes the proof of Proposition 5.8.

6. Spinors in Curved Space-Time We give a brief introduction to spinors in curved space-time. We closely follow the presentation in [19, §1.1 and §1.5].

6.1. Curved Space-Time and Lorentzian Manifolds. The starting point for general relativity is the observation that a physical process involving gravity can be understood in different ways. Consider for example an observer at rest on earth looking at a freely falling person (e.g. a diver who just jumped from a diving board). The observer at rest argues that the earth’s gravitational force, which he can feel himself, also acts on the freely falling person and accelerates him. The person at free fall, on the other hand, does not feel gravity. He can take the point of view that he himself is at rest, whereas the earth is accelerated towards him. He then concludes that there are no gravitational fields, and that the observer on earth merely feels the force of inertia corresponding to his acceleration. Einstein postulated that these two points of view should be equivalent descriptions of the physical process. More generally, it depends on the observer whether one has a gravitational force or an inertial force. In other words, equivalence principle: no physical experiment can distinguish between gravitational and inertial forces. In mathematical language, observers correspond to coordinate systems, and so the equivalence principle states that the physical equations should be formulated in general (i.e. “curvilinear”) coordinate systems, and should in all these coordinate systems have the same mathematical structure. This means that the physical equations should be invariant under diffeomorphisms, and thus space-time is to be modeled by a Lorentzian manifold (M, g). A Lorentzian manifold is “locally Minkowski space” in the sense that at every space- time point p ∈ M, the corresponding tangent space TpM is a vector space endowed with a scalar product h., .ip of signature (+ − − −). Therefore, we can distinguish between spacelike, timelike and null tangent vectors. Defining a non-spacelike curve q(τ) by the condition that its tangent vector u(τ) ∈ Tq(τ)M be everywhere non-spacelike, our above definition of light cones and the notion of causality immediately carry over to a Lorentzian manifold. In a coordinate chart, the scalar product h., .ip can be represented in the form (1.4) where gjk is the so-called metric tensor. In contrast to Minkowski space, the metric tensor is not a constant matrix but depends on the space-time point, gjk = gjk(p). Its ten components can be regarded as the relativistic analogue of Newton’s gravitational potential. For every p ∈ M there are coordinate LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 43 systems in which the metric tensor coincides with the Minkowski metric up to second order,

gjk(p) = diag(1, −1, −1, −1) , ∂jgkl(p) = 0 . (6.1) Such Gaussian normal coordinates correspond to the reference frame of a “freely falling observer” who feels no gravitational forces. However, it is in general impossible to arrange that also ∂jkglm(p) = 0. This means that by going into a suitable reference frame, the gravitational field can be transformed away locally (=in one point), but not globally. With this in mind, a reference frame corresponding to Gaussian normal coordinates is also called a local inertial frame. The physical equations can be carried over to a Lorentzian manifold by the prescrip- tion that they should in a local inertial frame have the same form as in Minkowski space; this is referred to as the strong equivalence principle. It amounts to replacing all partial derivatives by the corresponding covariant derivatives ∇ of the Levi-Civita connection; we write symbolically ∂ −→ ∇ . The gravitational field is described via the curvature of space-time. More precisely, the Riemannian curvature tensor is defined by the relations i l i i Rjkl u = ∇j∇ku − ∇k∇ju . (6.2) i Contracting indices, one obtains the Ricci tensor Rjk = Rjik and scalar curvature j R = Rj . The relativistic generalization of Newton’s gravitational law are the Einstein equations 1 R − R g = 8πκ T , jk 2 jk jk where κ is the gravitational constant. Here the energy-momentum tensor Tjk gives the distribution of matter and energy in space-time.

6.2. The Dirac Equation in Curved Space-Time. Dirac spinors are often formulated on a manifold using frame bundles, either an orthonormal frame [2, 34] or a Newman-Penrose null frame [43, 7]. We here outline an equivalent formulation of spinors in curved space-time in the framework of a U(2, 2) gauge theory (for details see [16]). We restrict attention to the Dirac operator in local coordinates; for global issues like topological obstructions to the existence of spin structures see e.g. [41]. We let M be a 4-dimensional manifold (without Lorentz metric) and define the spinor 4 bundle SM as a vector bundle over M with fibre C . The fibres are endowed with a scalar product ≺.|. of signature (2, 2), which is again referred to as the spin scalar product. Sections in the spinor bundle are called spinors or wave functions. In local coordinates, a spinor is represented by a 4-component complex function on space-time, usually denoted by ψ(x). Choosing at every space-time point a pseudo-orthonormal basis (eα)α=1,...,4 in the fibres,

≺eα|eβ = sα δαβ , s1 = s2 = 1, s3 = s4 = −1 α and representing the spinors in this basis, ψ = ψ eα, the spin scalar product takes again the form (1.12). Clearly, the basis (eα) is not unique, but at every space-point can be transformed according to −1 β eα −→ (U )α eβ , 44 F. FINSTER AND J.-H. TREUDE where U is an isometry of the spin scalar product, U ∈ U(2, 2). Under this basis transformation the spinors behave as follows, α α β ψ (x) −→ Uβ (x) ψ (x) . (6.3) In view of the analogy to gauge theories, we interpret this transformation of the wave functions as a local gauge transformation with gauge group G = U(2, 2). We refer to a choice of the spinor basis (eα) as a gauge. Our goal is to formulate classical Dirac theory in such a way that the above U(2, 2) gauge transformations correspond to a physical symmetry, the U(2, 2) gauge symmetry. To this end, we shall introduce the Dirac operator as the basic object on M, from which we will later deduce the Lorentz metric and the gauge potentials. We define a differential operator D of first order on the wave functions by the requirement that in a chart and gauge it should be of the form ∂ D = iGj(x) + B(x) (6.4) ∂xj with suitable (4×4)-matrices Gj and B. This definition does not depend on coordinates and gauge, although the form of the matrices Gj and B clearly does. More precisely, under a change of coordinates xi → x˜i the operator (6.4) transforms into ∂x˜j ∂ i Gk(˜x) + B(˜x) , (6.5) ∂xk ∂x˜j whereas a gauge transformation ψ → Uψ yields the operator ∂ UDU −1 = i UGjU −1 + UBU −1 + iUGj(∂ U −1) . (6.6) ∂xj j We define the Dirac operator by the requirement that by choosing suitable coordinates and gauge, one can arrange that the matrices Gj in front of the partial derivatives “coincide locally” with the Dirac matrices of Minkowski space. Definition 6.1. A differential operator D of first order is called Dirac operator if i for every p ∈ M there is a chart (x ,U) around p and a gauge (eα)α=1,...,4 such that D is of the form (6.4) with Gj(p) = γj , (6.7) where the γj are the Dirac matrices of Minkowski space in the Dirac representation (1.9). It may seem unconventional that we defined the Dirac operator without having a connection. We shall now construct from the Dirac operator a gauge-covariant derivative D, also referred to as spin derivative. To this end, we first write the transformation law (6.3) in the shorter form ψ(x) −→ U(x) ψ(x) (6.8) with U ∈ U(2, 2). Clearly, partial derivatives of ψ do not behave well under gauge transformations because we pick up derivatives of U. This problem disappears if instead of partial derivatives we consider gauge-covariant derivatives

Dj = ∂j − iAj , (6.9) provided that the gauge potentials transform according to −1 −1 Aj −→ UAjU + iU (∂jU ) . (6.10) LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 45

Namely, a short calculation shows that the gauge-covariant derivative behaves under gauge transformations according to −1 Dj −→ UDj U , (6.11) and thus the gauge-covariant derivatives of ψ obey the simple transformation rule

Djψ −→ UDjψ .

Our task is to find matrices Aj which transform under local gauge transformations according to (6.10). This construction will also reveal the structure of the matrix B, and this will finally lead us to the definition of the so-called physical Dirac operator, which involves precisely the gravitational and electromagnetic fields. In the chart and gauge where (6.7) holds, it is obvious from (1.7) that the anti- commutator of the matrices Gj(p) gives the Minkowski metric. Using the transformation rules (6.5) and (6.6), one sees that in a general coordinate system and gauge, their anti-commutator defines a Lorentz metric, 1 gjk(x) 11= {Gj(x),Gk(x)} . (6.12) 2 In this way, the Dirac operator induces on the manifold a Lorentzian structure. We refer to the matrices Gj as the Dirac matrices in curved space-time. Since we can arrange that these matrices coincide locally with the Dirac matrices of Minkowski space, all manipulations of Dirac matrices can be performed at any given space-time point in an obvious way. In particular, the pseudoscalar matrix (1.22) now takes the more general form i ρ(x) = ε GjGkGlGm , 4! jklm where the anti-symmetric tensor εjklm differs from the anti-symmetric symbol jklm p by the volume density, εjklm = | det g| jklm. The pseudoscalar matrix gives us again the notion of even and odd matrices and of chirality (1.23). Furthermore, we introduce the bilinear matrices σjk by i σjk(x) = [Gj,Gk] . 2 As in Minkowski space, the matrices Gj , ρGj , 11 , iρ , σjk (6.13) form a basis of the 16-dimensional (real) vector space of selfadjoint matrices (with respect to ≺.|. ). The matrices Gj and ρGj are odd, whereas 11, iρ and σjk are even. For the construction of the spin connection we must clearly consider derivatives. The Lorentzian metric (6.12) induces the Levi-Civita connection ∇, which defines the covariant derivative of tensor fields. Taking covariant derivatives of the Dirac matri- j j j l ces, ∇kG = ∂kG + Γkl G , we obtain an expression which behaves under coordinate transformations like a tensor. However, it is not gauge covariant, because a gauge transformation (6.8) yields contributions involving first derivatives of U. More precisely, according to (6.6), j j −1 j −1 j −1 j −1 ∇kG −→ ∇k(UG U ) = U(∇kG )U + (∂kU)G U + UG (∂kU ) j −1 −1 j −1 = U(∇kG )U − U(∂kU ),UG U . (6.14) We can use the second summand in (6.14) to partially fix the gauge. 46 F. FINSTER AND J.-H. TREUDE

Lemma 6.2. For every space-time point p ∈ M there is a gauge such that j ∇kG (p) = 0 (6.15) (for all indices j, k). Proof. We start with an arbitrary gauge and construct the desired gauge with two subsequent gauge transformations:

(1) The matrix ∂jρ is odd, because

0 = ∂j11= ∂j(ρρ) = (∂jρ)ρ + ρ(∂jρ) .

As a consequence, the matrix iρ(∂jρ) is selfadjoint. We can thus perform a 1 gauge transformation U with U(p) = 11, ∂jU(p) = 2 ρ(∂jρ). In the new gauge the matrix ∂jρ(p) vanishes, 1 ∂ ρ −→ ∂ (UρU −1) = ∂ ρ + [ρ(∂ ρ), ρ] = ∂ ρ − ρ2(∂ ρ) = 0 . j |p j |p j |p 2 j |p j |p j |p j j Differentiating the relation {ρ, G } = 0, one sees that the matrix ∇kG|p is odd. We can thus represent it in the form j j m j m ∇kG|p = Λkm G|p + Θkm ρG (6.16) j j with suitable coefficients Λkm and Θkm. This representation can be further simplified: According to Ricci’s Lemma, jk ∇ng = 0. Expressing the metric via the anti-commutation relations and differentiating through with the Leibniz rule, we obtain j k j k 0 = {∇nG ,G } + {G , ∇nG } j mk j mk k mj k mj = 2Λnm g − Θnm 2iρσ + 2Λnm g − Θnm 2iρσ (6.17) and thus j mk k mj Λnm g|p = −Λnm g|p . (6.18) j In the case j = k 6= m, (6.17) yields that Θnm = 0. For j 6= k, we obtain j jk k kj j k Θnj σ + Θnk σ = 0 and thus Θnj = Θnk (j and k denote fixed indices, no summation is performed). We conclude that there are coefficients Θk with j j Θkm = Θk δm . (6.19) (2) We perform a gauge transformation U with U(p) = 11and 1 i ∂ U = − Θ ρ − Λm gnl σ . k 2 k 4 kn ml Using the representation (6.16) together with (6.18) and (6.19), the matrix j ∇kG transforms into j j j ∇kG −→ ∇kG + [∂kU, G ] i = Λj Gm + Θ ρGj − Θ ρGj − Λm gnl [σ ,Gj] km k k 4 kn ml i = Λj Gm − Λm gnl 2i (G δj − G δj ) km 4 kn m l l m 1 1 = Λj Gm + Λm gnj G − Λj Gm = 0 . km 2 kn m 2 km LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 47 We call a gauge satisfying condition (6.15) a normal gauge around p. In order to analyze the remaining gauge freedom, we let U be a transformation between two normal −1 j −1 gauges. Then according to (6.14) and (6.15), the commutator [U(∂kU ),UG U ] vanishes at p or, equivalently, −1 j [i(∂kU ) U, G ]|p = 0 . As is easily verified in the basis (6.13) using the commutation relations between the Dirac matrices, a matrix which commutes with all Dirac matrices is a multi- −1 ple of the identity matrix. Moreover, the matrix i(∂jU ) U is selfadjoint because −1 ∗ −1 −1 −1 −1 (i(∂jU )U) = −iU (∂jU) = −i∂j(U U)+i(∂jU )U = i(∂jU )U. We conclude −1 that the matrix i(∂jU ) U is a real multiple of the identity matrix, and transforming −1 it unitarily with U we see that it also coincides with the matrix iU (∂jU ). Under this strong constraint for the gauge transformation it is easy to find expressions with the required behavior (6.10) under gauge transformations. Namely, setting 1 a = Re Tr(G B) 11 , (6.20) j 4 j where “Tr” denotes the trace of a 4 × 4-matrix, one sees from (6.6) that 1 a −→ a + Re Tr G Gk i(∂ U −1) U 11= a + iU(∂ U −1) . j j 4 j k j j

We can identify the aj with the gauge potentials Aj and use (6.9) as the definition of the spin connection. Definition 6.3. The spin derivative D is defined by the condition that it behaves under gauge transformations (6.8) according to (6.11) and in normal gauges around p has the form ∂ D (p) = − ia (6.21) j ∂xj j with the potentials aj according to (6.20). In general gauges, the spin derivative can be written as ∂ D = − iE − ia (6.22) j ∂xj j j with additional matrices Ej(x), which involve the Dirac matrices and their first derivatives. A short calculation shows that the trace of the matrix Ej does not change under gauge transformations, and since it vanishes in normal gauges, we conclude that the matrices Ej are trace-free. A straightforward calculation yields that they are explicitly given by i i i E = ρ (∂ ρ) − Tr(Gm ∇ Gn) G G + Tr(ρG ∇ Gm) ρ . j 2 j 16 j m n 8 j m In the next two theorems we collect the basic properties of the spin connection. Theorem 6.4. The spin derivative satisfies for all wave functions ψ, φ the equations j j l [Dk,G ] + Γkl G = 0 (6.23) ∂j ≺ψ | φ = ≺Djψ | φ + ≺ψ | Djφ . (6.24) 48 F. FINSTER AND J.-H. TREUDE

Proof. The left side of (6.23) behaves under gauge transformations according to the adjoint representation . → U.U −1 of the gauge group. Thus it suffices to check (6.23) in a normal gauge, where i [D ,Gj] + Γj Gl = ∇ Gj − Re Tr(G B) [11,Gj] = 0 . k kl k 4 j Since both sides of (6.24) are gauge invariant, it again suffices to consider a normal gauge. The statement is then an immediate consequence of the Leibniz rule for partial derivatives and the fact that the spin derivative differs from the partial derivative by an imaginary multiple of the identity matrix (6.21). The identity (6.23) means that the coordinate and gauge invariant derivative of the Dirac matrices vanishes. The relation (6.24) shows that the spin connection is compatible with the spin scalar product. We define torsion T and curvature R of the spin connection as the following 2-forms, i i T = ([D ,G ] − [D ,G ]) , R = [D ,D ] . jk 2 j k k j jk 2 j k Theorem 6.5. The spin connection is torsion-free. Curvature has the form 1 1 R = R σmn + (∂ a − ∂ a ) , jk 8 mnjk 2 j k k j where Rmnjk is the the Riemannian curvature tensor and the aj are given by (6.20). Proof. The identity (6.23) yields that l l l m m [Dj,Gk] = [Dj, gkl G ] = (∂jgkl) G − gkl Γjm G = Γjk Gm and thus, using that the Levi-Civita connection is torsion-free, i T = (Γm − Γm ) G = 0 . jk 2 jk kj m Again using (6.23), we can rewrite the covariant derivative as a spin derivative, l l Gl ∇ku = [Dk,Glu ] . Iterating this relation, we can express the Riemann tensor (6.2) by i l l l Gi Rjkl u = [Dj, [Dk,Glu ]] − [Dk, [Dj,Glu ]] l l = [[Dj,Dk],Glu ] = −2i [Rjk,Glu ] . This equation determines curvature up to a multiple of the identity matrix, 1 R (x) = R σmn + λ 11 . jk 8 mnjk jk Thus it remains to compute the trace of curvature, 1 1 1 Tr(R ) 11= Tr(∂ A − ∂ A ) 11= (∂ a − ∂ a ) , 4 jk 8 j k k j 2 j k k j where we used (6.22) and the fact that the matrices Ej are trace-free. We come to the physical interpretation of the above construction. According to Lemma 6.2 we can choose a gauge around p such that the covariant derivatives of the Dirac matrices vanish at p. Moreover, choosing normal coordinates and making a j global (=constant) gauge transformation, we can arrange that G(p) = γ and ∂jgkl(p) = LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 49

0. Then the covariant derivatives at p reduce to partial derivatives, and we conclude that j j j G (p) = γ , ∂kG (p) = 0 . (6.25) These equations have a large similarity with the conditions for normal coordinates (6.1), only the role of the metric is now played by the Dirac matrices. Indeed, differentiating (6.12) one sees that (6.25) implies (6.1). Therefore, (6.25) is a stronger condition which not only gives a constraint for the coordinates, but also for the gauge. We call a coordinate system and gauge where (6.25) is satisfied a normal reference frame around p. In a normal reference frame, the Dirac matrices, and via (6.12) also the metric, are the same as in Minkowski space up to the order o(x − p). According to the strong equivalence principle, the Dirac equation at p should coincide with that in Minkowski space. This implies that there should be a normal gauge such that all gauge potentials vanish at p, and thus the Dirac operator at p should coincide with the free Dirac operator i∂/. This physical argument allows us to specify the zero order term in (6.4). Definition 6.6. A Dirac operator D is called physical Dirac operator if for any p ∈ M there is a normal reference frame around p such that B(p) = 0. Equivalently, the physical Dirac operator could be defined as a differential operator of first order (6.4) with the additional structure that for any p ∈ M there is a coordinate chart and gauge such that the following three conditions are satisfied, j j j G (p) = γ , ∂kG (p) = 0 ,B(p) = 0 . This alternative definition has the disadvantage that it is a-priori not clear whether j the second condition ∂kG (p) = 0 can be satisfied for a general metric. This is the reason why we preferred to begin with only the first condition (Definition 6.1), then showed that the second condition can be arranged by choosing suitable coordinates and gauge, and satisfied the third condition at the end (Definition 6.6). In general coordinates and gauge, the physical Dirac operator can be written as j j D = iG Dj = iG (∂j − iEj − iaj) , where D is the spin connection of Definition 6.3. The matrices Ej take into account the gravitational field and are called spin coefficients, whereas the aj can be identified with the electromagnetic potential (compare (1.11)). We point out that the gravitational field cannot be introduced into the Dirac equation by the simple replacement rule ∂ → D, because gravity has an effect on both the Dirac matrices and the spin coefficients. But factorizing the gauge group as U(2, 2) = U(1) × SU(2, 2), the SU(2, 2)-gauge j transformations are linked to the gravitational field because they influence G and Ej, whereas the U(1) can be identified with the gauge group of electrodynamics. In this sense, we obtain a unified description of electrodynamics and general relativity as a U(2, 2) gauge theory. The Dirac equation (D − m) ψ = 0 describes a Dirac particle in the gravitational and electromagnetic field. According to Theorem 6.5, the curvature of the spin connection involves both the Riemann tensor and the electromagnetic field tensor. We can write down the classical action in terms of these tensor fields, and variation yields the classical Einstein-Dirac-Maxwell equations. 50 F. FINSTER AND J.-H. TREUDE

For the probabilistic interpretation of the Dirac equation in curved space-time, we choose a space-like hypersurface N (corresponding to “space” for some observer) and consider in generalization of (1.16) on solutions of the Dirac equation the scalar product Z j hψ|φiH = ≺ψ | G νj φ dµN , (6.26) N where ν is the future-directed normal on N and dµN is the invariant measure on the Riemannian manifold N. Then (ψ|ψ) is the normalization integral, which we again normalize to one. Its integrand has the interpretation as the probability density. In analogy to (1.13) the Dirac current is introduced by J k = ≺ψ | Gkψ . Using Theorem 6.4 one k sees similar as in Minkowski space that the Dirac current is divergence-free, ∇kJ = 0. From the Gauss divergence theorem one obtains that the scalar product (6.26) does not depend on the choice of the hypersurface N. In analogy to (4.3), we can introduce the inner product Z <ψ|φ> := ≺ψ|φ x dµM . (6.27) M in which the wave functions (which need not satisfy the Dirac equation but must have a suitable decay at infinity) are integrated over the whole space-time. We finally remark that, using Theorem 6.4 together with Gauss’ theorem, one easily verifies that the physical Dirac operator is symmetric with respect to this inner product.

7. Causal Fermion Systems and a Lorentzian Quantum Geometry 7.1. Constructing a Causal Fermion System from the Fermionic Projector. We now explain how one can construct a causal fermion system from a given fermionic projector P (x, y) as constructed in Section 5. We closely follow the presentation in [32, Section 4]. Our task is to introduce an ultraviolet regularization. This is done most conveniently with so-called regularization operators.

Deﬁnition 7.1. A family (Rε)ε>0 of bounded linear operators on Hm are called regularization operators if they have the following properties: (i) Solutions of the Dirac equation are mapped to continuous solutions, 0 Rε : Hm → C (M,SM) ∩ Hm (ii) For every ε > 0 and x ∈ M, there is a constant c > 0 such that

k(Rεψm)(x)k ≤ c kψmk ∀ ψm ∈ Hm . (7.1)

(where the norm on the left is any norm on SxM). (iii) In the limit ε & 0, the regularization operators go over to the identity with ∗ strong convergence of Rε and Rε, i.e. ∗ ε&0 Rεψm, Rεψm −−−→ ψm in Hm ∀ ψm ∈ Hm . There are many possibilities to choose regularization operators. As a typical exam- (n) ∞ ple, one can choose finite-dimensional subspaces H ⊂ Csc (M,SM) ∩ Hm which are (0) (1) (n) an exhaustion of Hm in the sense that H ⊂ H ⊂ · · · and Hm = ∪nH . Set- ting n(ε) = max([0, 1/ε] ∩ N), we can introduce the operators Rε as the orthogonal projection operators to H(n(ε)). An alternative method is to choose a Cauchy hypersurface N, to mollify the restriction ψm|N to the Cauchy surface on the length scale ε, LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 51 and to define Rεψm as the solution of the Cauchy problem for the mollified initial data. Given regularization operators Rε, for any ε > 0 we introduce the particle space

(Hparticle, h.|.iHparticle ) as the Hilbert space ⊥ Hparticle = ker Rε χ(−∞,0)(S) , h.|.iHparticle = (.|.)|Hparticle×Hparticle . Next, for any x ∈ M we consider the bilinear form

b : Hparticle × Hparticle → C , b(ψm, φm) = −≺(Rε ψm)(x) | (Rε φm)(x) x . This bilinear form is bounded in view of (7.1). The local correlation operator F ε(x) is defined as the signature operator of this bilinear form, i.e. ε b(ψm, φm) = hψm | F (x) φmiHparticle for all ψm, φm ∈ Hparticle . Taking into account that the spin scalar operator has signature (2, 2), the local correlation operator is a symmetric operator in L(Hparticle) of rank at most four, which has at most two positive and at most two negative eigenvalues. Finally, we introduce the ε universal measure ρ = F∗ dµM as the push-forward of the volume measure on M under ε ε −1 the mapping F (thus ρ(Ω) := µM ((F ) (Ω))). Omitting the subscript “particle”, we thus obtain a causal fermion system of spin dimension (see Definition 1.2). 7.2. Geometric Structures on a Causal Fermion System. We now outline constructions from [29] which give general notions of a connection and curvature (see Theorem 7.10, Definition 7.12 and Definition 7.13). We closely follow the presentation in [31]. 7.2.1. Construction of the Spin Connection. Let (H, F, ρ) be a causal fermion system of spin dimension n in the particle representation. We now construct additional objects, leading us to the more familiar space-time representation. First, on F we consider the topology induced by the operator norm kAk := sup{kAukH with kukH = 1}. For every x ∈ F we define the spin space Sx by Sx = x(H); it is a subspace of H of dimension at most 2n. On Sx we introduce the spin scalar product ≺.|. x by

≺u|v x = −hu|xuiH (for all u, v ∈ Sx) ; (7.2) it is an indefinite inner product of signature (p, q) with p, q ≤ n. We define space- time M as the support of the universal measure, M = supp ρ. It is a closed subset of F, and by restricting the causal structure of F to M, we get causal relations in space- time. A wave function ψ is defined as a function which to every x ∈ M associates a vector of the corresponding spin space,

ψ : M → H with ψ(x) ∈ Sx for all x ∈ M. (7.3) On the wave functions we introduce the indefinite inner product Z <ψ|φ> = ≺ψ(x)|φ(x) x dρ(x) . (7.4) M In order to ensure that the last integral converges, we also introduce the norm ||| . ||| by Z 2 ||| ψ ||| = hψ(x)| |x| ψ(x)iH dρ(x) M (where |x| is the absolute value of the operator x on H). The one-particle space K is defined as the space of wave functions for which the norm ||| . ||| is finite, with the topology induced by this norm, and endowed with the inner product <.|.>. Then (K, <.|.>) 52 F. FINSTER AND J.-H. TREUDE is a Krein space (see [5]). Next, for any x, y ∈ M we define the kernel of the fermionic operator P (x, y) by P (x, y) = πx y : Sy → Sx , (7.5) where πx is the orthogonal projection onto the subspace Sx ⊂ H. The closed chain is defined as the product

Axy = P (x, y) P (y, x): Sx → Sx .

As it is an endomorphism of Sx, we can compute its eigenvalues. The calculation Axy = (πxy)(πyx) = πx yx shows that these eigenvalues coincide precisely with the non-trivial xy xy eigenvalues λ1 , . . . , λ2n of the operator xy as considered in Deﬁnition 1.3. In this way, the kernel of the fermionic operator encodes the causal structure of M. Choosing a suitable dense domain of deﬁnition1 D(P ), we can regard P (x, y) as the integral kernel of a corresponding operator P , Z P : D(P ) ⊂ K → K , (P ψ)(x) = P (x, y) ψ(y) dρ(y) , (7.6) M referred to as the fermionic operator. We collect two properties of the fermionic operator: (A) P is symmetric in the sense that

= <ψ|P φ> for all ψ, φ ∈ D(P ): According to the deﬁnitions (7.5) and (7.2),

≺P (x, y) ψ(y) | ψ(x) x = −h(πx y ψ(y)) | x φ(x)iH

= −hψ(y) | yx φ(x)iH = ≺ψ(y) | P (y, x) ψ(x) y . We now integrate over x and y and apply (7.6) and (7.4). (B) (−P ) is positive in the sense that <ψ|(−P )ψ> ≥ 0 for all ψ ∈ D(P ): This follows immediately from the calculation ZZ <ψ|(−P )ψ> = − ≺ψ(x) | P (x, y) ψ(y) x dρ(x) dρ(y) M×M ZZ = hψ(x) | x πx y ψ(y)iH dρ(x) dρ(y) = hφ|φiH ≥ 0 , M×M where we again used (7.4) and (7.5) and set Z φ = x ψ(x) dρ(x) . M The space-time representation of the causal fermion system consists of the Krein space (K, <.|.>), whose vectors are represented as functions on M (see (7.3), (7.4)), together with the fermionic operator P in the integral representation (7.6) with the above properties (A) and (B). Having Dirac spinors in a four-dimensional space-time in mind, from now on we assume that the spin dimension n = 2. Moreover, we only consider space-time points x ∈ M which are regular in the sense that the corresponding spin spaces Sx have the maximal dimension four.

1For example, one may choose D(P ) as the set of all vectors ψ ∈ K satisfying the conditions Z φ := x ψ(x) dρ(x) ∈ H and ||| φ ||| < ∞ . M LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 53

An important structure from spin geometry missing so far is Clifford multiplication. To this end, we need a Clifford algebra represented by symmetric operators on Sx. For convenience, we first consider Clifford algebras with the maximal number of five generators; later we reduce to four space-time dimensions (see Definition 7.15 below). We denote the set of symmetric linear endomorphisms of Sx by Symm(Sx); it is a 16-dimensional real vector space.

Definition 7.2. A five-dimensional subspace K ⊂ Symm(Sx) is called a Clifford subspace if the following conditions hold: (i) For any u, v ∈ K, the anti-commutator {u, v} ≡ uv + vu is a multiple of the identity on Sx. (ii) The bilinear form h., .i on K defined by 1 {u, v} = hu, vi 11 for all u, v ∈ K 2 is non-degenerate and has signature (1, 4). In view of the situation in spin geometry, we would like to distinguish a specific Clifford subspace. In order to partially fix the freedom in choosing Clifford subspaces, it is useful to impose that K should contain a given so-called sign operator. 2 Definition 7.3. An operator v ∈ Symm(Sx) is called a sign operator if v = 11and if the inner product ≺.|v . : Sx × Sx → C is positive definite. Definition 7.4. For a given sign operator v, the set of Clifford extensions T v is defined as the set of all Clifford subspaces containing v, T v = {K Clifford subspace with v ∈ K} .

Considering x as an operator on Sx, this operator has by deﬁnition of the spin dimension two positive and two negative eigenvalues. Moreover, the calculation

(7.2) 2 ≺u|(−x) u x = hu|x uiH > 0 for all u ∈ Sx \{0} shows that the operator (−x) is positive definite on Sx. Thus we can introduce a unique sign operator sx by demanding that the eigenspaces of sx corresponding to the eigenvalues ±1 are precisely the positive and negative spectral subspaces of the operator (−x). This sign operator is referred to as the Euclidean sign operator. A straightforward calculation shows that for two Clifford extensions K, K˜ ∈ T v, there is a unitary transformation U ∈ eiRv such that K˜ = UKU −1 (for details see [29, Section 3]). By dividing out this group action, we obtain a five-dimensional vector space, endowed with the inner product h., i. Taking for v the Euclidean signature operator, we regard this vector space as a generalization of the usual tangent space.

Definition 7.5. The tangent space Tx is defined by sx Tx = Tx / exp(iRsx) . It is endowed with an inner product h., .i of signature (1, 4). We next consider two space-time points, for which we need to make the following assumption. Definition 7.6. Two points x, y ∈ M are said to be properly time-like separated if the closed chain Axy has a strictly positive spectrum and if the corresponding eigenspaces are definite subspaces of Sx. 54 F. FINSTER AND J.-H. TREUDE

This definition clearly implies that x and y are time-like separated (see Definition 1.3). Moreover, the eigenspaces of Axy are definite if and only if those of Ayx are, showing that Definition 7.6 is again symmetric in x and y. As a consequence, the spin space + − can be decomposed uniquely into an orthogonal direct sum Sx = I ⊕ I of a positive + − definite subspace I and a negative definite subspace I of Axy. This allows us to introduce a unique sign operator vxy by demanding that its eigenspaces corresponding to the eigenvalues ±1 are the subspaces I±. This sign operator is referred to as the directional sign operator of Axy. Having two sign operators sx and vxy at our disposal, we can distinguish unique corresponding Clifford extensions, provided that the two sign operators satisfy the following generic condition. Definition 7.7. Two sign operators v, v˜ are said to be generically separated if their commutator [v, v˜] has rank four.

Lemma 7.8. Assume that the sign operators sx and vxy are generically separated. (y) sx vxy Then there are unique Cliﬀord extensions Kx ∈ T and Kxy ∈ T and a unique (y) operator ρ ∈ Kx ∩ Kxy with the following properties:

(i) The relations {sx, ρ} = 0 = {vxy, ρ} hold. iρ (ii) The operator Uxy := e transforms one Cliﬀord extension to the other, (y) −1 Kxy = Uxy Kx Uxy .

(iii) If {sx, vxy} is a multiple of the identity, then ρ = 0.

The operator ρ depends continuously on sx and vxy.

We refer to Uxy as the synchronization map. Exchanging the roles of x and y, we also have two sign operators sy and vyx at the point y. Assuming that these sign operators vyx are again generically separated, we also obtain a unique Cliﬀord extension Kyx ∈ T . After these preparations, we can now explain the construction of the spin connection D (for details see [29, Section 3]). For two space-time points x, y ∈ M with the above properties, we want to introduce an operator

Dx,y : Sy → Sx

(generally speaking, by the subscript xy we always denote an object at the point x, whereas the additional comma x,y denotes an operator which maps an object at y to an object at x). It is natural to demand that Dx,y is unitary, that Dy,x is its inverse, and that these operators map the directional sign operators at x and y to each other, ∗ −1 Dx,y = (Dy,x) = (Dy,x) (7.7)

vxy = Dx,y vyx Dy,x . (7.8) The obvious idea for constructing an operator with these properties is to take a polar decomposition of P (x, y); this amounts to setting

1 − 2 Dx,y = Axy P (x, y) . (7.9) This deﬁnition has the shortcoming that it is not compatible with the chosen Cliﬀord extensions. In particular, it does not give rise to a connection on the corresponding tangent spaces. In order to resolve this problem, we modify (7.9) by the ansatz

− 1 iϕxy vxy 2 Dx,y = e Axy P (x, y) (7.10) LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 55 with a free real parameter ϕxy. In order to comply with (7.7), we need to demand that

ϕxy = −ϕyx mod 2π ; (7.11) then (7.8) is again satisﬁed. We can now use the freedom in choosing ϕxy to arrange that the distinguished Cliﬀord subspaces Kxy and Kyx are mapped onto each other,

Kxy = Dx,y Kyx Dy,x . (7.12) π It turns out that this condition determines ϕxy up to multiples of 2 . In order to fix ϕxy π uniquely in agreement with (7.11), we need to assume that ϕxy is not a multiple of 4 . This leads us to the following definition. Definition 7.9. Two points x, y ∈ M are called spin connectable if the following conditions hold: (a) The points x and y are properly timelike separated (note that this already implies that x and y are regular as defined in Section 7.2.1). (b) The Euclidean sign operators sx and sy are generically separated from the directional sign operators vxy and vyx, respectively. (c) Employing the ansatz (7.10), the phases ϕxy which satisfy condition (7.12) are π not multiples of 4 . We denote the set of points which are spin connectable to x by I(x). It is straightforward to verify that I(x) is an open subset of M. Under these assumptions, we can fix ϕxy uniquely by imposing that π π π π ϕ ∈ − , − ∪ , , (7.13) xy 2 4 4 2 giving the following result (for the proofs see [29, Section 3.3]). Theorem 7.10. Assume that two points x, y ∈ M are spin connectable. Then there is a unique spin connection Dx,y : Sy → Sx of the form (7.10) having the properties (7.7), (7.8), (7.12) and (7.13). 7.2.2. A Time Direction, the Metric Connection and Curvature. We now outline a few further constructions from [29, Section 3]. First, for spin connectable points we can distinguish a direction of time. Definition 7.11. Assume that the points x, y ∈ M are spin connectable. We say that y lies in the future of x if the phase ϕxy as defined by (7.10) and (7.13) is positive. Otherwise, y is said to lie in the past of x. According to (7.11), y lies in the future of x if and only if x lies in the past of y. By distinguishing a direction of time, we get a structure similar to a causal set (see for example [6]). However, in contrast to a causal set, our notion of “lies in the future of” is not necessarily transitive. The spin connection induces a connection on the corresponding tangent spaces, as we now explain. Suppose that uy ∈ Ty. Then, according to Definition 7.5 and (x) sy Lemma 7.8, we can consider uy as a vector of the representative Ky ∈ T . By applying the synchronization map, we obtain a vector in Kyx, −1 uyx := Uyx uy Uyx ∈ Kyx . According to (7.12), we can now “parallel transport” the vector to the Clifford subspace Kxy, uxy := Dx,y uyx Dy,x ∈ Kxy . 56 F. FINSTER AND J.-H. TREUDE

Finally, we apply the inverse of the synchronization map to obtain the vector −1 (y) ux := Uxy uxy Uxy ∈ Kx . (y) As Kx is a representative of the tangent space Tx and all transformations were unitary, we obtain an isometry from Ty to Tx. Deﬁnition 7.12. The isometry between the tangent spaces deﬁned by

∇x,y : Ty → Tx : uy 7→ ux is referred to as the metric connection corresponding to the spin connection D. We next introduce a notion of curvature. Deﬁnition 7.13. Suppose that three points x, y, z ∈ M are pairwise spin connectable. Then the associated metric curvature R is deﬁned by

R(x, y, z) = ∇x,y ∇y,z ∇z,x : Tx → Tx . (7.14) The metric curvature R(x, y, z) can be thought of as a discrete analog of the holonomy of the Levi-Civita connection on a manifold, where a tangent vector is parallel trans- ported along a loop starting and ending at x. On a manifold, the curvature at x is immediately obtained from the holonomy by considering the loops in a small neighborhood of x. With this in mind, Deﬁnition 7.13 indeed generalizes the usual notion of curvature to causal fermion systems. The following construction relates directional sign operators to vectors of the tangent space. Suppose that y is spin connectable to x. By synchronizing the directional sign operator vxy, we obtain the vector −1 (y) yˆx := Uxy vxy Uxy ∈ Kx . (7.15)

(y) sx As Kx ∈ T is a representative of the tangent space, we can regardy ˆx as a tangent vector. We thus obtain a mapping

I(x) → Tx : y 7→ yˆx .

We refer toy ˆx as the directional tangent vector of y in Tx. As vxy is a sign operator and the transformations in (7.15) are unitary, the directional tangent vector is a timelike unit vector with the additional property that the inner product ≺.|yˆx. x is positive definite. We finally explain how to reduce the dimension of the tangent space to four, with the desired Lorentzian signature (1, 3). Definition 7.14. The fermion system is called chirally symmetric if to every x ∈ M we can associate a spacelike vector u(x) ∈ Tx which is orthogonal to all directional tangent vectors, hu(x), yˆxi = 0 for all y ∈ I(x) , and is parallel with respect to the metric connection, i.e.

u(x) = ∇x,y u(y) ∇y,x for all y ∈ I(x) . Deﬁnition 7.15. For a chirally symmetric fermion system, we introduce the reduced red tangent space Tx by red ⊥ Tx = huxi ⊂ Tx . LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 57

Clearly, the reduced tangent space has dimension four and signature (1, 3). Moreover, the operator ∇x,y maps the√ reduced tangent spaces isometrically to each other. The local operator γ5 := −iu/ −u2 takes the role of the pseudoscalar matrix. 7.3. The Correspondence to Lorentzian Spin Geometry. We also explain how these notions correspond to the usual objects of differential geometry in Minkowski space (Theorem 7.16) and on a globally hyperbolic Lorentzian manifold (Theorem 7.17). We closely follow the presentation in [31, Section 3.3]. We let (M, g) be a time-oriented Lorentzian spin manifold with spinor bundle SM (thus SxM is a 4-dimensional complex vector space endowed with an inner product ≺.|. x of signature (2, 2)). Assume that γ(t) is a smooth, future-directed and timelike curve, for simplicity parametrized by the arc length, defined on the interval [0,T ] with γ(0) = y and γ(T ) = x. Then the parallel transport of tangent vectors along γ with respect to the Levi-Civita connection ∇LC gives rise to the isometry LC ∇x,y : Ty → Tx . In order to compare with the metric connection ∇ of Definition 7.12, we subdivide γ (for simplicity with equal spacing, although a non-uniform spacing would work just as well). Thus for any given N, we define the points x0, . . . , xN by nT x = γ(t ) with t = . n n n N N We define the parallel transport ∇x,y by successively composing the parallel transport between neighboring points, N ∇x,y := ∇xN ,xN−1 ∇xN−1,xN−2 · · · ∇x1,x0 : Ty → Tx . Our first theorem gives a connection to the Minkowski vacuum. For any ε > 0 we regularize on the scale ε > 0 by inserting a convergence generating factor into the integrand of the Dirac sea in Minkowski space (4.9), 4 Z d k 0 P ε(x, y) = (k/ + m) δ(k2 − m2) Θ(−k0) eεk e−ik(x−y) . (7.16) (2π)4 This function can indeed be realized as the kernel of the fermionic operator (7.5) corresponding to a causal fermion system (H, F, ρε). Here the measure ρε is the push- forward of the volume measure in Minkowski space by an operator F ε, being an ultraviolet regularization of the operator F , similar as constructed in Section 7.1. Moreover, for technical convenience we assume that the manifold coincides with Minkowski space to the past of any Cauchy hypersurface (for details see [29, Section 4]). Theorem 7.16. For given γ, we consider the family of regularized fermionic pro- ε jectors of the vacuum (P )ε>0 as given by (7.16). Then for a generic curve γ and for every N ∈ N, there is ε0 such that for all ε ∈ (0, ε0] and all n = 1,...,N, the points xn and xn−1 are spin connectable, and xn+1 lies in the future of xn (according to Definition 7.11). Moreover, LC N ∇x,y = lim lim ∇x,y . N→∞ ε&0 By a generic curve we mean that the admissible curves are dense in the C∞-topology (i.e., for any smooth γ and every K ∈ N, there is a sequence γ` of admissible curves k k such that D γ` → D γ uniformly for all k = 0,...,K). The restriction to generic curves is needed in order to ensure that the Euclidean and directional sign operators 58 F. FINSTER AND J.-H. TREUDE are generically separated (see Definition 7.9 (b)). The proof of the above theorem is given in [29, Section 4]. LC Clearly, in this theorem the connection ∇x,y is trivial. In order to show that our connection also coincides with the Levi-Civita connection in the case with curvature, in [29, Section 5] a globally hyperbolic Lorentzian manifold is considered. For technical simplicity, we assume that the manifold is flat Minkowski space in the past of a given Cauchy hypersurface. Theorem 7.17. Let (M, g) be a globally hyperbolic manifold which is isometric to Minkowski space in the past of a given Cauchy-hypersurface N . For given γ, we ε ε consider the family of regularized fermionic projectors (P )ε>0 such that P (x, y) coincides with the distribution (7.16) if x and y lie in the past of N . Then for a generic curve γ and for every sufficiently large N, there is ε0 such that for all ε ∈ (0, ε0] and all n = 1,...,N, the points xn and xn−1 are spin connectable, and xn+1 lies in the future of xn (according to Definition 7.11). Moreover, N LC ∇R scal lim lim ∇x,y − ∇x,y = O L(γ) 1 + O , N→∞ ε&0 m2 m2 where R denotes the Riemann curvature tensor, scal is scalar curvature, and L(γ) is the length of the curve γ. Thus the metric connection of Definition 7.12 indeed coincides with the Levi-Civita connection, up to higher order curvature corrections. For detailed explanations and the proof we refer to [29, Section 5]. We conclude this section by pointing to a few additional constructions in [29] which cannot be explained consistently in this short survey article. First, there is the subtle point that the unitary transformation U ∈ exp(iRsx) which is used to identify two −1 representatives K, K˜ ∈ Tx via the relation K˜ = UKU (see Definition 7.5) is not unique. More precisely, the operator U can be transformed according to

U → −U and U → sx U. As a consequence, the metric connection (see Definition 7.12) is defined only up to the transformation ∇x,yu → sx (∇x,yu) sx . Note that this transformation maps representatives of the same tangent vector into each other, so that ∇x,yu ∈ Tx is still a well-defined tangent vector. But we get an ambiguity when composing the metric connection several times (as for example in the expression for the metric curvature in Definition 7.13). This ambiguity can be removed by considering parity-preserving systems as introduced in [29, Section 3.4]. At first sight, one might conjecture that Theorem 7.17 should also apply to the spin connection in the sense that LC N Dx,y = lim lim Dx,y , N→∞ ε&0 where DLC is the spin connection on SM induced by the Levi-Civita connection and N Dx,y := DxN ,xN−1 DxN−1,xN−2 ··· Dx1,x0 : Sy → Sx (7.17) (and D is the spin connection of Theorem 7.10). It turns out that this conjecture is false. But the conjecture becomes true if we replace (7.17) by the operator product

N (xN |xN−2) (xN−1|xN−3) (x2|x0) D(x,y) := DxN ,xN−1 UxN−1 DxN−1,xN−2 UxN−2 ··· Ux1 Dx1,x0 . LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 59

(.|.) Here the intermediate factors U. are the so-called splice maps given by (z|y) −1 Ux = Uxz VUxy , where Uxz and Uxy are synchronization maps, and V ∈ exp(iRsx) is an operator which identiﬁes the representatives Kxy,Kxz ∈ Tx (for details see [29, Section 3.7 and Section 5]). The splice maps also enter the spin curvature R, which is deﬁned in analogy to the metric curvature (7.14) by

(z|y) (x|z) (y|x) R(x, y, z) = Ux Dx,y Uy Dy,z Uz Dz,x : Sx → Sx .

8. Outlook We now mention a few directions which cannot be covered in the lectures and give a brief outlook.

8.1. Analysis of the Causal Action Principle. The causal action principle was introduced at the beginning of these lectures (see Section 1.3). But we never came back to it. Indeed, entering the analysis of this variational principle would have been a separate topic. Here we can only mention a few works in this direction. In the papers [21, 23], the existence of minimizers is proved in various situation. In simple terms, these results show that the causal action principle makes mathematical sense. The important question is how the minimizers look like. In [8] a few simple numerical examples are discussed. In [20] an effect of spontaneous symmetry breaking elaborated. This can be interpreted as a manifestation of a more general effect of “spontaneous structure formation” (see [31, Section 3]). In [33, 3] the Euler-Lagrange equations corresponding to the causal action principle are worked out. The analysis in [3] reveals a tendency of minimizers to be discrete. For a physical discussion of this effect we refer to [31, Section 4]. Finally, the paper [30] is devoted to the question how an initial value problem can be posed for causal variational principles, and whether it has a unique solution. Except for these few results, nothing is known on the structure of the minimizers. It is an interesting open problem to get a better understanding of the structure of minimizers of causal variational principles. It is planned to study the causal action principle in more advanced explicit examples in the near future.

8.2. The Continuum Limit of Causal Fermion Systems. It was the main motivation early on to get an alternative description of quantum field theory. This connection can indeed be made by considering the so-called continuum limit. Similar as in Section 7.1, one introduces an ultraviolet regularization on a microscopic scale ε. But then one analyzes the Euler-Lagrange equations corresponding to the causal action principle in the limit as ε & 0 when the regularization is removed. A subtle point is that the results do depend on how the regularization is introduced. In order to analyze this effect in detail, one considers a large class of regularizations (“method of variable regularization”). It turns out that the structure of the effective equations obtained in the continuum limit are indeed regularization independent; only the coupling constants and the bosonic masses depend on the details of the regularizations. Due to these complications, it does not seem suitable to enter the details of the continuum limit analysis in a lecture series. But we the interested reader to the survey article [25] and the references therein. In the more recent paper [26], the continuum 60 F. FINSTER AND J.-H. TREUDE limit is worked out for a system involving neutrinos, giving rise to an effective interaction described by a left-handed, massive SU(2) gauge field and a gravitational field. The next step is to work out a model involving leptons and quarks [27], which should combine gravity with all the interactions of the standard model. The plan is to publish the three papers [22, 26, 27] together with a nice general introduction as a book. We finally point out that it is an important open problem to work out the detailed expansion of the effective interaction in terms of Feynman diagrams (including all loop diagrams) and to compare it to the standard expansion of perturbative quantum field theory. This is a major project for the future.

Appendix A. The Free Time Evolution Operator in Momentum Space For explicit computations, it is most convenient to transform the spatial coordinates to momentum space, i.e.

0 Z 0 ~ U t,t (~k) := U t,t (0, ~y) eik~x d3x . R3 Using (2.13) and (2.9), we obtain Z ∞ t,t0 ~ / 0 2 2 −iω(t−t0) U (k) = (k + m) γ δ(k − m ) k=(ω,~k) (ω) e dω . −∞ Carrying out the ω-integral, we get

t,t0 X ∓iω(t−t0) U (~k) = Π±(~k) e , (A.1) ± where we set ~ s 0 Πs(k) = (k/s + m) γ , (A.2) 2ω(~k) q 2 2 ω(~k) = |~k| + m and k± = (±ω(~k),~k) .

Using Plancherel’s theorem, the scalar product (1.16) can be written in momentum space as Z (φ|ψ) = (2π)−2 ≺φ|γ0ψ (t,~k) d3k . R3 The unitarity the time evolution operator implies that the matrix U t,t0 (~k) is unitary (with respect to the scalar product (.|.)), meaning that its eigenvalues are on the unit circle and the corresponding eigenspace are orthogonal. Comparing with (A.1), one concludes that the operators Π±(~k) must be the projection operators to the eigenspaces corresponding to the eigenvalues e∓iω(t−t0). It is instructive to verify by an explicit calculation that the operators Π±(~k) are indeed projection operators, and that their images are orthogonal:

Lemma A.1. The operators Π±(~k) satisfy the relations ~ † ~ ~ ~ ~ Πs(k) = Πs(k) and Πs(k)Πs0 (k) = δs,s0 Πs(k) . LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 61

Proof. Omitting the argument ~k, a short calculation using the anti-commutation relations yields † 0 ∗ 0 Πs = γ Πsγ = Πs 0 ss 0 0 Π Π 0 = (k/ + m) γ (k/ 0 + m) γ s s 4ω2 s s ss0 = (sωγ0 + ~k~γ + m) γ0(s0ωγ0 + ~k~γ + m) γ0 4ω2 ss0 = (sωγ0 + ~k~γ + m)(s0ωγ0 − ~k~γ + m) 4ω2 ss0 = ss0ω2 + |~k|2 + m2 + (s + s0) ω (~k~γ) γ0 + (s + s0) mωγ0 . 4ω2 In the case s = −s0, the last expression vanishes due to the dispersion relation ω2 − |~k|2 = m2. In the other case s = s0, we obtain

1 2 2 2 0 0 Π Π 0 = ω + |~k| + m + 2sω (~k~γ) γ + 2smωγ s s 4ω2 1 s = ω − s(~k~γ) γ0 + smγ0 = sωγ0 + (~k~γ) + m γ0 = Π , 2ω 2ω s where in the last line we again used the dispersion relation. The next two lemmas involve a derivatives with respect to the mass parameter m. They will be essential for proving the mass oscillation property in Section 5.5. For clarity, we denote the m-dependence of the operators by a subscript m. Lemma A.2. The time evolution operator satisﬁes the relation

0 ∂ 0 0 (t − t0) U t,t (~k) = V t,t (~k) + W t,t (~k) , (A.3) m ∂m m m where

0 X i 0 V t,t (~k) = (k/ + m)γ0 e∓iω(t−t ) m 2m ± ± 0 X i is 0 W t,t (~k) = k/ γ0 ∓ e∓iω(t−t ) . m 2m2 ± 2ω ± t,t0 The operators Vm are estimated uniformly by 0 0 |~k| kV t,t (~k)k + kW t,t (~k)k ≤ C 1 + , (A.4) m m m where the constant C is independent of m, ~k and t, t0. Proof. First, we generate the factor t − t0 by diﬀerentiating the exponential in (A.1) with respect to ω,

0 X ∂ 0 (t − t0) U t,t (~k) = Π (~k) ± i e∓iω(t−t ) m ± ∂ω ± Next, we want to rewrite the ω-derivative as a derivative with respect to m. Diﬀeren- tiating the identity ω2 − |~k|2 = m2 with respect to m and ω, one ﬁnds that ∂ ω ∂ = . (A.5) ∂ω m ∂m 62 F. FINSTER AND J.-H. TREUDE

Hence 0 X ω ∂ 0 (t − t0) U t,t = Π ± i e∓iω(t−t ) m ± m ∂m ± ∂ X ω 0 X ∂ h ω i 0 = ±i Π e∓iω(t−t ) − ±i Π e∓iω(t−t ) . ∂m m ± ∂m m ± ± ± Computing the operators in the round brackets using (A.2) gives the result. This method can be iterated to generate more factors of t − t0, as illustrated in the next lemma. Lemma A.3. The time evolution operator satisﬁes the relation 2 0 ∂ 0 ∂ 0 0 (t − t0)2 U t,t (~k) = At,t (~k) + Bt,t (~k) + Ct,t (~k) , (A.6) m ∂m2 m ∂m m m t,t0 t,t0 t,t0 where the operators Am , Bm and Cm are bounded by 2 0 0 0 C |~k| |~k| kAt,t (~k)k + kBt,t (~k)k + kCt,t (~k)k ≤ 1 + + , (A.7) m m m m m m2 with a numerical constant C. Proof. The lemma follows by a straightforward computation using exactly the same methods as in Lemma A.2. The additional 1/m in the prefactor (compared to (A.4)) can be understood from scaling dimensions, because the additional factor t−t0 in (A.6) (compared to (A.3)) brings in an additional dimension of length. The additional summand |~k|2/m2 in (A.7) can be understood from the fact applying (A.5) generates ~ ~ a factor of ω/m which for large |k| scales like |k|/m.

Appendix B. Uniform L2-Estimates of Derivatives of Dirac Solutions We now derive a few estimates which will be needed for the proof of the mass oscillation property in Section 5.5. We use standard methods of the theory of partial diﬀerential equations and adapt them to the Dirac equation. Lemma B.1. We are given non-negative parameter a and b. We assume that B is smooth. In the case a > 0, we assume furthermore that B decays for large times faster than linearly in the sense that c |∇B(t)| a−1 ≤ (B.1) C 1 + |t|1+ε for suitable constants c, ε > 0. Then there is a constant C = C(c, ε, a, b) such that ∞ every family of solutions ψ ∈ HCsc,0(M,SM) of the Dirac equation (1.18) can be estimated for all times in terms of the boundary values at t = 0 by b b b X p k∂mψ(t)kW a,2 ≤ C 1 + |t| k∂mψ(0)kW a,2 . p=0 Proof. We choose a multi-index α of length a := |α| and a non-negative integer b. Diﬀerentiating the Dirac equation in the external potential (1.18) gives α b α b−1 α b α b (i∂/ + B − m) ∇ ∂mψ = b ∇ ∂m ψ − ∇ B ∂mψ + B ∇ ∂mψ . LECTURE NOTES ON “RELATIVISTIC FERMION SYSTEMS” 63

Introducing the abbreviations α b α b−1 α b α b Ξ = ∇ ∂mψ and φ = b ∇ ∂m ψ − ∇ B ∂mψ + B ∇ ∂mψ , we write this equation as the inhomogeneous Dirac equation (D − m) Ξ = φ . A calculation similar to current conservation (1.14) gives

∂t(Ξ|Ξ) = ≺(D − m)Ξ | Ξ − ≺Ξ | (D − m)Ξ

= ≺φ | Ξ − ≺Ξ | φ ≤ 2 kΞk kφk and thus

∂tkΞ(t)k ≤ kφ(t)k . Substituting the speciﬁc form of Ξ and φ and using the Schwarz and triangle inequal- ities, we obtain the estimate α b α b−1 α b ∂tk∇ ∂mψ(t)k ≤ b ∇ ∂m ψ(t) + c a |∇B(t)|Ca−1 ∇ ∂mψ(t) W a−1,2 , (B.2) where we again used the notation (5.20) and (5.21). We now proceed inductively in the total order a + b of the derivatives. In the case a = b = 0, the claim follows immediately in view of the unitarity of the time evolution. In order to prove the induction step, we note that on the right of (B.2), the order of diﬀerentiation of the wave function is at least by one smaller than on the left. In the case a = 0, the induction hypothesis yields the inequality b−1 b b−1 b−1 X p ∂tk∂mψ(t)k ≤ b ∂m ψ(t) ≤ b C 1 + |t| k∂mψ(0)k , p=0 and integrating this inequality from 0 to t gives the result. In the case a 6= 0, we apply (B.1) and the induction hypothesis to obtain b−1 b b−1 X p ∂tk∂mψ(t)k W a,2 ≤ b C 1 + |t| k∂mψ(0)kW a,2 p=0 b b 1 + |t| X p + c C ∂ ψ(0) a−1,2 . 1 + |t|1+ε m W p=0 Integrating again over t gives the result.

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