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“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 mechan- ics, the quantum mechanical particle is described by a complex-valued wave func- tion (t; ~x). The absolute square j (t; ~x)j2 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 j (t; ~x)j2 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 2 H and multiplies it by a constant, := νφ with ν 2 C such as to satisfy the normalization condition (1.1). Then j (t; ~x)j2 has the interpretation as the probability density (this procedure can be described by saying that only the ray C has a physical interpreta- tion, 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 hφj iH := φ(t; ~x) (t; ~x) ; φ, 2 H (1.2) R3 should be time independent. The dynamics is described by a linear evolution equation on H, which we write as i@t = H ; 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, of the Schr¨odinger equation, 0 = @thφj iH = −i hHφj iH − hφjH iH : (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 parame- ter 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 1 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 hHφj iH = hφjH iH for all φ, 2 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, (t) = e−iHt (0) : 4 F. FINSTER AND J.-H. TREUDE The dynamics of 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, x1.1 andx1.2]. 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 2 M is said to be 9 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 = fξ 2 M j hξ; ξi = 0g, 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 = fξ 2 M j hξ; ξi > 0g. Finally, we introduce the closed light cone J = fξ 2 M j hξ; ξi ≥ 0g. 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 = fy 2 M j (y − x) ≥ 0; (y − x ) ≥ 0g ^ 2 0 0 Jx = fy 2 M j (y − x) ≥ 0; (y − x ) ≤ 0g ; 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 ) = 0 ; (1.5) j where ≡ @j@ is the wave operator. This equation describes a scalar particle (=par- ticle 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 ) = m (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.
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