7.2 The Variational Principles on the Sphere
7.2.2 Estimates of the Action
5
3 if ϑ∈[0,arccos(0.8)],
35
9 if ϑ∈[arccos(0.4),arccos(0.2)]
40
9 if ϑ∈[arccos(−0.5),arccos(−0.7)], 0 otherwise.
Then ifτ < 1.00157, a straightforward calculation shows thatf has the properties (a) and (b) of Corollary 6.16. Thus the measure dρ=f dµ is a minimizing generically
timelike measure with supp◦ ρ6=∅. ♦
7.2.2 Estimates of the Action
As not even the solution of the Tammes problem is explicitly known, we cannot expect to nd explicit minimizers for general τ. Therefore, we need good estimates of the action from above and below. We now explain dierent methods for getting estimates, which are all compiled in Figure 7.5.
Estimates from above can be obtained simply by computing the action for suitable test measures. For example, the action of the normalized volume measure is
S[µ] = 1 4π
∫ 2π
0
dϕ
∫ ϑmax
0
dϑ sinϑ D(ϑ) = 4− 4
3τ2 ≥ Smin.
86 7 Applications of the Structural Results As one sees in Figure 7.5, this estimate is good ifτ is close to one. Another example is to take the measure supported at the Tammes distribution for K points, with equal weights. We denote the corresponding action by STK. We then obtain the estimate
Smin ≤ ST := min
K STK .
One method is to compute ST numerically using the tables in [28]. This gives quite good results (see Figure 7.5), with the obvious disadvantage that the estimate is not given in closed form. Moreover, the Tammes distribution, see 5.2, is useful for analyzing the asymptotics for largeτ. For given τ >1 we choose K ∈ N such that τK−1 ≤τ < τK. Then using the estimate (5.2.14) we obtain
Constructing a lower bound is more dicult. From (7.2.13) it is obvious that the operatorDµis positive semi-denite ifτ ≤√
3. Thus we can apply Proposition 6.15 to obtain
Smin ≥ν0 if τ ≤√ 3. If τ ≤ √
2, this lower bound is even equal to Smin according to Corollary 7.9. As shown in Figure 7.5, the estimate is no longer optimal if τ >√
2.
Another method to obtain lower bounds is based on the following observation:
Proposition 7.11. Assume that Kµ is an integral operator on Hµ with integral kernel K ∈C0(S2×S2,R) with the following properties:
(a) K(x, y)≤ L(x, y) for all x, y ∈S2.
(b) The operator Kµ is positive semi-denite.
Then the minimal action satises the estimate Smin ≥
∫∫
S2×S2
K(x, y)dµ(x)dµ(y). (7.2.14) Proof. For anyρ∈M, assumption (a) gives rise to the estimate
S[ρ] =
Next, using property (b), we can apply Proposition 6.6 to conclude that the volume measureµ is a minimizer of the variational principle corresponding to K, i.e.
∫∫ Combining these inequalities gives the result.
7.2 The Variational Principles on the Sphere 87
Figure 7.6: The Lagrangian L and the function K in the heat kernel estimate for τ = 2.
In order to construct a suitable kernel, we rst consider the heat kernel ht on S2, ht(x, y) =(
The heat kernel has the advantage that condition (b) is satised, but condition (a) is violated. This leads us to choosingK as the dierence of two heat kernels,
K(x, y) =λ(
By direct inspection one veries that condition (a) is satised (see Figure 7.6 for a typical example). The eigenvalues of the operator Kµ are computed to be
λ(e−t1l(l+1)−δ e−t2l(l+1)),
showing that the operator Kµ is indeed positive semi-denite. Thus we can apply Proposition 7.11. Using that we obtain the heat kernel estimate
Smin ≥SK =λ(1−δ).
In this estimate, we are still free to choose the parameters t1 and t2. By adjusting these parameters, one gets the lower bound shown in Figure 7.5. Thus the heat kernel estimate diers from the minimal action only by an error of about20%, and describes the qualitative dependence onτ quite well. But of course, it does not take into account the discreteness of the minimizers. We nally remark that for smallϑ the function Dcan be expanded as
D(ϑ) = 8τ2−2ϑ2( concluding that forτ > √
3 +√
10 =τd Corollary 6.29 can be applied.
8 Causal Variational Principles on Flag Manifolds
8.1 Preliminaries
We now investigate the causal variational principles introduced in Section 2.2 on the space F of hermitian f ×f-matrices with prescribed eigenvalues α and −β in higher dimension with regard to the structural results of Chapter 6. As mentioned in Section 3.2, this space can be identied with the ag manifold F1,2(Cf).
For the following calculations, we introduce the Dirac delta function δ:C0∞(R)→R, f 7→f(0),
which may formally be written as ∫∞
−∞f(x)δ(x)dx = f(0). Using the Heaviside function
Θ :R→R, Θ(x) =
0 if x <0
1
2 if x= 0 1 if x >0,
the basic properties of theδ-function (see [7, Chapter 2.3]) can be written as follows:
Proposition 8.1. Let g ∈ C0∞(R) be a function with only a nite number of zeros (xi)i∈I which are all simple, and let a∈R∪ {−∞}. Then for all f ∈C0∞(R) it is
∫ ∞
a
δ( g(x))
f(x)dx=∑
i∈I
f(xi)
|g0(xi)| Θ(xi−a). (8.1.1) In the case a=−∞, the relation ∫∞
−∞δ( g(x))
f(x)dx=∑
i∈I f(xi)
|g0(xi)| generalizes the scaling property δ(λx) = |λ1|δ(x) for λ∈R\{0}to the composition with a function.
The delta function is similarly dened in higher dimensions as δn:C0∞(Rn)→R with δn(f) =f(0), formally written as δn(x) = δ(x1). . . δ(xn).
Using the basic properties, we can deduce special rules which we use in the fol-lowing calculations. We will need the area of the (n−1)−sphere Sn−1 ⊂ Rn for n∈N, n≥2, which is given by
vol(Sn−1) = 2πn/2
Γ(n/2), (8.1.2)
whereΓ denotes the Gamma function.
88
8.2 Correlation Matrices of Rank One 89 Lemma 8.2. Forn ∈N, n≥2, letk.k denote the standard Euclidean norm on Rn.
Then ∫
Rn
δ(
kxk2 −1)
dx = 1
2 vol(Sn−1). (8.1.3) Proof. Since the integrand is radial symmetric, one obtains
∫
Rn
δ(
kxk2−1)
dx= vol(Sn−1)
∫ ∞
0
dr rn−1δ(r2−1), where using formula (8.1.1) one calculates ∫∞
0 dr rn−1δ(r2−1) = 12.
In the following calculations, additionally we use the Beta function which is given by
B(x, y) =
∫ 1 0
dt tx−1(1−t)y−1 = Γ(x)Γ(y)
Γ(x+y) for x, y >0. (8.1.4)
8.2 Correlation Matrices of Rank One
We start the analysis with considering the caseβ = 0. In this case, F is the family of hermitian f ×f-matrices of rank one with non-vanishing eigenvalue α > 0. An element xin F can be represented as
x=α|u)(u| for u∈Cf with kuk= 1, (8.2.5) where again (.|.) denotes the standard Euclidean scalar product on Cf and k.k the induced norm. The functions L and D on F×F dened in (2.4.30) coincide, and simplify to
D(x, y) =L(x, y) = 1 2 Tr(
xy)2
. (8.2.6)
Since the functionD is always non-negative, the causal structure dened in Deni-tion 6.3 can be lightly stated: If x, y ∈ F are represented via (8.2.5) by vectors ux and uy in Cf, the light-cones are I(x) = {y ∈ F : (ux|uy) 6= 0}, J(x) = F and K(x) = ∅.
With regard to Denition 6.5, the invariant normalized volume measure on the complex sphere is a homogenizer. Denoting byduthe Lebesgue measure on Cf, the homogenizerµ∈Mcan be written as
dµ= 2
vol(S2f−1) δ(kuk2−1)du, (8.2.7) where formula (8.1.3) proves the normalization. The corresponding integral oper-ators Lµ and Dµ obviously coincide. The operator Lµ has already been analyzed in [15, Lemma 1.10], containing the proof of the following Lemma:
Lemma 8.3. The operator Lµ =Dµ is positive semi-denite with rk Lµ ≤f4. This spectral property immediately reveals the solution of the causal variational principle:
90 8 Causal Variational Principles on Flag Manifolds Proposition 8.4. Let F be the set of hermitian f ×f-matrices of rank one with non-trivial eigenvalueα >0, letD be the function onF×Fdened in (8.2.6). Then the minimal action of the corresponding variational principle (6.1.5) is
Smin =S[µ] = α4
f(f+ 1), (8.2.8)
where µ denotes the homogenizer. Each minimizer is generically timelike.
Proof. According to Lemma 8.3, we can apply Proposition 6.6, yielding that the minimal action is given by
Smin =S[µ] =`µ(x) for all x∈F.
Representing elements in F as in (8.2.5) and using formula (8.2.7) for the homoge-nizer, we obtain for x=α|e1)(e1|
Thus the integral on the right hand side of equation (8.2.9) simplies to Smin =α4 vol(S2f−3)(2π)
Using the generalized scaling property (8.1.1) applied on the u2-integral and for-mula (8.1.2) for the area of the sphere, one calculates
Smin = α4 Γ(f) Substitutingu21 =s, the last expression can be transformed to
Smin = α4 Γ(f)
where in the last step we used the denition of the Beta function (8.1.4). Repre-senting the Beta function via the Γ-function, we nally obtain
Smin = α4 Γ(f) 2 Γ(f −1)
Γ(3)Γ(f−1)
Γ(f + 2) = α4 f(f+ 1).
Since the functions L and D coincide, the eigenvalue ν0 of the operator Dµ corre-sponding to the constant function1F is given byν0 =`µ(x) = Smin . Proposition 6.15 yields that each minimizer is generically timelike.
We remark that the value of the minimal action ts with the numerical values of Table 4.1 for α = f. We conclude that the numerically calculated solutions dene generically timelike minimizers, supported at only a nite number of points with equal weighting factors. The additional freedom of the variational principle in Section 4.3 in the choice of the non-vanishing eigenvalue did not yield a smaller action.