We dene the operator Ψby
Ψ = ⊕
x∈M
Ψx :Cf →H, u7→ ∑
x∈M
Ψxu.
According to the identity constraint (F1), we obtain for u, v ∈Cf hΨu|Ψvi=∑m
x=1
Ψxu ∑m
y=1
Ψyv
=
∑m x=1
hΨxu|Ψxvi=−
∑m x=1
(u|Fxv) =−(u|v).
Consequently,Ψis a fermion matrix. By construction, the local correlation matrices of the corresponding fermionic projector P =−ΨΨ∗ coincide with the given family of matrices (Fx).
The question of uniqueness is treated in [33]. If the hermitian matrices Fx are all non-zero, the local fermion matrix Ψx which satises −Ψ∗xΨx = Fx is unique up to gauge transformation Ψx → UxΨx with Ux ∈ U(Ex(H)). But a zero local correlation matrix yields local fermion matrices and thus fermionic projectors which are not gauge equivalent.
If the family of matrices (Fx)x∈M satisfy (E) and the trace constraint (F3), a similar construction as in the above proof yields an operator of classPf.
2.4 General Remarks
We nally compare the two approaches. The approach in the particle representation is easier accessible. The invariance under the non-compact gauge groupGis replaced by a compact U(f)-invariance. But since there are fermionic projectors belonging to the same local correlation matrices which are not gauge equivalent, there occurs a loss of information.
The fermion matrix and thus the space-time representation is more appropriate for solving the variational principle numerically because the constraints can be imple-mented more conveniently. For a comparison of the dierent solutions and in order to decide whether the minimizers are gauge equivalent, it is reasonable to analyze the local correlation matrices corresponding to the minimizing fermionic projector.
We nally state properties which are valid for both settings, and start with a remark on the notion of causality:
Lemma 2.26. If the fermionic projector P solves the variational principle (2.1.8), each space-time point is timelike separated from itself.
Proof. Let(Fx)x∈M be the corresponding family of local correlation matrices. Since the eigenvalues of the hermitian matrixFx2 are real, the space-time pointxis either lightlike or timelike separated from itself. Assume there existsy∈M which is light-like separated from itself. Then the discrete kernel P(y, y) is nilpotent, implying
16 2 Causal Variational Principles on Fermion Systems that the corresponding local correlation matrix Fy vanishes. The family (Fx)x∈M of local correlation matrices thus reduces to a set of only m−1 matrices and conse-quently corresponds to a fermionic projector in a space-time of m−1 space-time points, in contradiction to (2.1.10).
We now state an important transformation of the Lagrangian. Since the La-grangian of the closed chainAxy and ofFx, Fy coincide, the lemma similarly applies in both settings.
Lemma 2.27. Let L be the Lagrangian dened in (2.2.17). ThenL is given as the positive part of a smooth function,
L[Fx, Fy] = max (
0,D[FxFy]
) where D[Fx, Fy] = Tr (
(FxFy)2 )− 1
2 Tr (FxFy)2. (2.4.30) In the case f = 2, the function D can be expressed as
D[Fx, Fy] = 1
2Tr(FxFy)2−2 det(FxFy). (2.4.31) Proof. Let x, y ∈ M be xed. After a unitary transformation (2.2.20), we can assume that Fx = diag(αx,−βx,0. . . ,0) with αx, βx ≥ 0. Let λ± denote the non-trivial eigenvalues of FxFy. Let F˜y denote the 2×2 leading principal submatrix, thus F˜y = (gij)i,j=1,2 for Fy = (gij)i,j=1,...,f. Then the eigenvalues of F˜xF˜y coincide with the non-trivial eigenvalues of FxFy. Since the matrix F˜y is indenite, it is det( ˜Fy)≤0. We obtain
Tr(FxFy)∈R and det( ˜FxF˜y) =−αxβxdet( ˜Fy)≥0
and conclude that either λ+ and λ− are real and have the same sign or else build a complex conjugate pair. For the Lagrangian given by (2.2.17), this yields to the form
L[Fx, Fy] = max (
0,1
2(λ+−λ−)2 )
, where the non-trivial argument can be expressed as
1
2(λ+−λ−)2 = Tr (
(FxFy)2 )− 1
2 Tr (FxFy)2. Finally one veries the basic identity
Tr(A2) = Tr(A)2−2 det(A) for A∈Mat(2×2,C).
In the case y=x, the function L and D coincide since L[Fx, Fx] =D[Fx, Fx] = 1
2
(α2x−βx2)2
≥0. (2.4.32)
With regard to Denition 2.10, the space-time points x, y ∈ M are spacelike sepa-rated if and only if D[Fx, Fy] < 0, lightlike separated if and only if D[Fx, Fy] = 0, and timelike separated if and only if D[Fx, Fy] > 0. Thus the sign of the smooth function D determines the causal structure. The fact that L is the positive part of a smooth function causes interesting eects, and will be treated in a more general context in Chapter 6.
3 Geometry of Causal Fermion Systems
3.1 Identication with Vectors on the Sphere in the Case of two Particles
In order to analyze the structure of a fermion system in discrete space-time, it is helpful to visualize the fermionic projector. For this purpose, it is most convenient to work in the particle representation and regard the corresponding local correlation matrices.
In a system with only two particles, each local correlation matrix can be visualized as a vector inR3, as we now describe. We start with introducing the Pauli-matrices
σ1 =
(0 1 1 0 )
, σ2 =
(0 −i i 0
)
, σ3 =
(1 0 0 −1
) ,
and refer to [26, Chapter 9] for their properties. As the matrices 12, σ1, σ2, σ3 are linearly independent, anyF ∈Mat(2×2,C)can be written as
F = ρ1+~c·~σ =ρ12+c1σ1+c2σ2+c3σ3 with ρ∈C, ~c∈C3. (3.1.1) Using the product identity for the Pauli matrices
σkσl=δkl1+i
∑3 m=1
εklmσm for k, l= 1,2,3 (3.1.2) (where εklm denotes the antisymmetric symbol) and that the Pauli matrices are traceless, the coecients are given by
ρ= 1
2Tr(F) and ck = 1
2Tr(F σk) for k = 1,2,3.
The determinant and the eigenvalues λ± of F are then calculated as
det(F) =ρ2−
∑3 i=1
c2i and λ± =ρ± vu ut∑3
i=1
c2i. (3.1.3) We apply these considerations to hermitian matrices and obtain:
17
18 3 Geometry of Causal Fermion Systems Lemma 3.1. If F ∈Mat(2×2,C) is hermitian, F can be expanded as
F = ρ12+~c·~σ with ρ∈R, ~c∈R3, (3.1.4) where the vector~c is called Bloch vector. The eigenvalues λ± of F are given by
λ± =ρ± k~ck. (3.1.5)
Finally we examine transformations of the objects in (3.1.4): The group SU(2)is the universal covering group of SO(3), SO(3)'SU(2)/{±12},where the Pauli matrices can be used to construct the twofold covering map, (see [24, Chapter 1]). Thus for eachV ∈SU(2) there exists a unique R∈SO(3) such that
V(ρ12+~c·~σ)V−1 =ρ12+ (R~c)·~σ for all ρ∈R, ~c∈R3, (3.1.6) concluding that a unitary transformation of a hermitian matrixF causes a rotation of the corresponding Bloch vector. Conversely, a rotation of the Bloch vector causes a unitary transformation of the matrix.
Now let(H,h.|.i,(Ex)x∈M, P)be a fermion system in discrete space-time with two particles. According to Lemma 3.1, each local correlation matrixFx ∈Mat(2×2,C) can be assigned the parameter ρx and the Bloch vector~cx. Since each Fx has non-positive determinant we get the relation
k~cxk ≥ |ρx| for all x∈M. (3.1.7) The completeness of the family(Fx) yields
∑
x∈M
ρx = 1 and ∑
x∈M
~cx = 0. (3.1.8)
According to formula (3.1.6), a unitary transformation (2.2.20) of the family of cor-relation matrices corresponds to a rotation of all Bloch vectors. If there is a system of parameters ρx ∈ R and vectors ~cx ∈ R3 which satisfy (3.1.8) and (3.1.7), then using Proposition 2.25 there exists a fermionic projector such that the corresponding local fermion matricesFx realizeρx and~cx. This fermionic projector, however, may not be unique, see [5, Example 4.3].
We now express the function Ddened by (2.4.31) and thus the Lagrangian using the local traces and Bloch vectors of the local correlation matricesFx:
Lemma 3.2. For x, y ∈M let Fx, Fy be decomposed as in (3.1.4), Fx = ρx12+~cx ·~σ and Fy = ρy12+~cy ·~σ . Then the function D[Fx, Fy] dened in (2.4.30) is calculated as
D[Fx, Fy] = 2 [(
ρxρy+~cx·~cy )2
−(
ρ2x− k~cxk2)(
ρ2y− k~cyk2)]
. (3.1.9)
3.2 Fermion Systems with Prescribed Eigenvalues 19