8.9 Spherical Solutions for an Even Number of Particles
In the case that f is an even number, it is possible to assign to a vector on the two-sphere an element inFas we now describe. The variational principle restricted on the obtained subset of F can then be regarded as a variational principle on the sphere, and the minimal action will yield an upper bound of the original variational principle on the ag manifold.
To this purpose, we study the already used spherical harmonics in more detail: For integer l >0 and m=−l, . . . , l the spherical harmonics are explicitly given by
Yl,m(ϑ, ϕ) = (−1)l 2ll!
√
(2l+ 1)! (l+m)!
4π(2l)! (l−m)! eimϕ sin−mϑ dl−m
d(cosϑ)l−msin2lϑ. (8.9.36) If ~L denotes the angular momentum operator, expressed in spherical coordinates, for xed l ∈ N and m = −l, . . . , l, the functions Yl,m are the 2l+ 1 eigenfunctions of L2 to the eigenvalue l(l+ 1) (setting ~= 1). We refer to [26, Chapter 5] for the properties of the induced objects. In the following evaluations, we use the value of the spherical harmonics on the north pole.
Lemma 8.16. On the north pole, the spherical harmonics are evaluated as Yl,0(0,0) =
√2l+ 1
4π , Yl,m(0,0) = 0 for all m∈ {−l, . . . , l}\{0}. (8.9.37) Proof. The Leibniz rule yields
dl−m
d(cosϑ)l−m(1−cos2ϑ)l =l−m∑
ν=0
(l−m
ν
)(−1)ν(l−l!ν)!(1−cosϑ)l−ν(m+ν)!l! (1 + cosϑ)m+ν.
Inserting in (8.9.36), we obtain Yl,m(0,0)6= 0 if and only ifm = 0.
Now let l ∈ N be xed. We introduce the spin spherical harmonics which are dened as the two component wave functions
ϕm(~x) =
√l+1/2+m
2l+1 Yl,m−1
2(~x)
√l+1/2−m
2l+1 Yl,m+1
2(~x)
for ~x∈S2
for m =±12, . . . ,±(l+ 12), see [27, Chapter 8]. These are the 2l+ 2 eigenfunctions of the total angular momentumL~ ·~σ corresponding to the eigenvalue l.
Lemma 8.17. LetΠl be the projector onto the eigenspace of the operator ~L·~σ, thus
Πl(~x, ~y) =
l+1/2∑
m=−(l+1/2)
ϕm(~x)ϕm(~y)† for ~x, ~y∈S2.
110 8 Causal Variational Principles on Flag Manifolds Then for each R∈SO(3) there exists a unique V ∈SU(2) such that
Πl(R ~x, R ~y) = V Πl(~x, ~y)V−1 for all ~x, ~y∈S2. (8.9.38) In particular,
Πl(~x, ~x) = l+ 1
4π 12 for all ~x∈S2. (8.9.39) Proof. The rst statement follows from the fact that the operatorL~·~σis spherically symmetric and that SO(3) ' SU(2)/{±1}. In order to calculate Πl(~x, ~x), we can consequently choose~x=e~3, and use formula (8.9.37).
With this Lemma, we can construct a mapping on the family of matrices with prescribed eigenvalues:
Proposition 8.18. Leta, b∈Rand~x∈S2. ThenF(~x)∈Mat(
(2l+2)×(2l+2),C) dened as
F(~x) =((
ϕm(~x)(a12+b ~x·~σ)ϕn(~x))
C2
)
m,n=−(l+1/2),...,(l+1/2) (8.9.40) is hermitian of rank at most two. Its non-vanishing eigenvalues are given by
l+ 1
4π (a±b). (8.9.41)
Proof. According to Formula (8.9.40), the matrix F = F(~x) is hermitian with rk(F) ≤ 2. Using the invariance of the trace under cyclic permutations and for-mula (8.9.39), we obtain
Tr(F) = Tr(
(a12+b ~x·~σ) Πl(~x, ~x))
= l+ 1
4π Tr(a12+b ~x·~σ) = l+ 1 2π a and
Tr(F2) = Tr(
(a12+b ~x·~σ) Πl(~x, ~x) (a12 +b ~x·~σ) Πl(~x, ~x))
=(l+1
4π
)2
Tr(a212+ab ~x·~σ+b212) =(l+1
4π
)2
(2a2 + 2b2),
concluding that the non-vanishing eigenvalues ofF, which are uniquely determined byTr(F) and Tr(F2), are independent of~x.
We apply this Proposition to obtain a subset ofFin the case f = 2(l+ 1). Thus we demand that the eigenvalues of F(~x)are given byα and−β. According to (8.9.41), we determine the real parameters a, bby
a= 2π
l+ 1(α−β), b= 2π
l+ 1(−α−β). (8.9.42) The function Dand thus the Lagrangian restricted on the subset {F(~x) : ~x∈S2} can be regarded as a function onS2×S2. Let
DS :S2×S2 →R, DS(~x, ~y) = D(F(~x), F(~y)), whereD is the function on F dened in (8.3.10).
8.9 Spherical Solutions 111
0.5 1.0 1.5 2.0 2.5 3.0
J
-10 10 20 30
DHJL
Τ= 2
Τ=1
Τ=2
Figure 8.3: The function Din the spheric symmetric setting.
Lemma 8.19. The function DS(~x, ~y) just depends on the angle ϑ ∈[0, π] given by cos(ϑ) = ~x·~y between the points ~x, ~y∈S2.
Proof. Let R ∈SO(3). Since SO(3) 'SU(2)/{±12}, there exists a unique unitary matrix V ∈ SU(2) such that both formulas (3.1.6) and (8.9.38) are satised. We obtain
Tr(
F(R~x)F(R~y))
= Tr(
(a+b(R~x)·~σ) Πl(R~x, R~y) (a+b(R~y)·~σ) Πl(R~y, R~x))
= Tr(
V(a+b(~x)·~σ) Πl(~x, ~y) (a+b (~y)·~σ) Πl(~y, ~x)V−1)
= Tr(
F(~x)F(~y)) . Similarly, one obtains Tr(
(F(R~x)F(R~y))2)
= Tr(
(F(~x)F(~y))2)
. We conclude that D(F(R~x), F(R~y)) =D(F(~x), F(~y)).
For the calculation of the function DS(~x, ~y), we can consequently assume that
~x=~e3 and ~y∈S2 is arbitrary.
The minimizer of the variational principle on S2 with respect to the function DS
yields an upper bound on the minimal action Smin of the variational principle on the whole ag manifold F,
inf
ρ∈M(F)S[ρ]≤ inf
ρ∈M(S2)S[ρ].
Exemplarily, we now consider the case f = 4 resp. l= 1. We again prescribe the eigenvalues α,−β as 1±τ for τ ≥1. In this case, the function DS is calculated as
DS(ϑ) = 18τ2(1 + cosϑ)(1−3 cosϑ)2 (
2(1 + 3 cos2ϑ) +τ2(cosϑ−1)(1 + 3 cosϑ)2 ) (8.9.43) Typical plots are shown in Figure 8.3. This example diers from the already exam-ined examples on the sphere in its causal structure. If τ > 1, the function DS has ve zeros, namely at ϑ = arccos(1/3), ϑ=π and the three zeros of the polynomial
term (
9τ2cos3ϑ−3(
τ2−2)
cos2ϑ−5τ2cosϑ−τ2+ 2) ,
112 8 Causal Variational Principles on Flag Manifolds
Figure 8.4: Numerical minima for the weighted counting measure on the circle.
denoted in ascending order as ϑ1 < arccos(1/3) < ϑ2 < arccos(−1/3) < ϑ3. In
For a better understanding, we rst consider the variational principle on the circle.
Using the same numerical routine as in Section 7.1, we obtain the results shown in Figure 8.4. We now apply the structural results of Chapter 6. Letµagain denote the Lebesgue measure on the circle. The eigenvalues of the operator Dµ are calculated in general as The eigenvalue corresponding to the constant function, given by
ν0 = 1 Applying Proposition 6.17, there cannot exist generically timelike minimizers ifτ >
2
√59
53 ≈2.11017. The measure supported at the set X8 given by (7.1.1) with equal weights is a generically timelike minimizer for τ <
[ yields that in this range every minimizer is generically timelike. The numerics show that indeed the generically timelike minimizer is for τ ≈ 1 not unique, but a statement similar to Lemma 7.1 does not hold. If τ > 2√
5 10+√
2, the numerics
8.9 Spherical Solutions 113
1 1.5 2 2.5
0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
S
τ
10 20 30 40
50
ν0
Figure 8.5: Numerical minima for the weighted counting measure on the sphere.
suggest that there does not exist a generically timelike minimizer. We assume that the critical value of τ, where a phase transition occurs, is given as
τc = 2
√ 5 10 +√
2 ≈1.32371.
But a proof of this assumption lacks.
Theorem 6.19 yields that for τ > τc it is supp◦ ρ = ∅. Additionally the numerics suggest that for eachx∈suppρthere existsy∈suppρ∩ K(x). The boundaryK(x) consists of several orbits, but no orbit is stressed out. Thus it is not possible to dene a chain as done in Denition 7.3.
We now consider the function DS dened in (8.9.43) as a function on S2 ×S2, where for x, y ∈ S2 we set cosϑ = x·y. The eigenvalues are now calculated in general as
νk = 1 2
∫π 0
DS(ϑ)Pk(cosϑ) sinϑ dϑ for k ∈N, (8.9.46) where Pk denotes the k-th Legendre polynomial. The eigenvalue corresponding to the constant function, given by
ν0 =τ2 (6
5− 38 105τ2
)
, (8.9.47)
is positive for τ ∈ [1,3
√7
19]. In this range, it is and ν1, . . . , ν6 > 0, νk = 0 for k >6, concluding that there cannot exist generically timelike minimizers in the case τ > 3
√7
19 ≈ 1.82093. The numerical results have to be very precise in order to decide whether the minimizer is indeed generically timelike or not. But it seems that the critical value of τ now is dierent since the equality Smin =ν0 only holds
114 8 Causal Variational Principles on Flag Manifolds
1 1.2 1.4 1.6 1.8 2
0.6 0.8 1 1.2 1.4 1.6 1.8 2
S
τ
ν0 Ss(20)
S[µ]
S(20)
S(16)
Figure 8.6: Numerical minima for the weighted counting measure on the ag mani-fold.
if τ < 1.1, as can be seen in Figure 8.5. Additionally, the plots suggest that again the minimizing measure is a discrete measure supported at only a nite number of points.