Revisiting Twistors

In twistor theory, α-planes are the building blocks of classical field theory in complexified compactified Minkowski space-time. The α-planes are totally null two-surfaces S in that, if p is any point on S, and if v and w are any two null tangent vectors at p ∈ S, the complexified Minkowski metric η satisfies the identity η(v,w) = v_aw^a = 0. By definition, their null tangent vectors have the two-component spinor form λ^Aπ^A′, where λ^A is varying and π^A′ is fixed. Therefore, the induced metric vanishes identically since η(v,w) = λ^Aπ^A′ μ_Aπ_A^′ = 0 = η(v, v) = λ^Aπ^A′ λ_Aπ_A^′ . One thus obtains a conformally invariant characterization of flat space-times. This definition can be generalized to complex or real Riemannian space-times with non-vanishing curvature, provided the Weyl curvature is anti-self-dual. One then finds that the curved metric g is such that g(v,w) = 0 on S, and the spinor field π_A^′ is covariantly constant on S. The corresponding holomorphic two-surfaces are called α-surfaces, and they form a three-complex-dimensional family. Twistor space is the space of all α-surfaces, and depends only on the conformal structure of complex space-time.

Projective twistor space PT is isomorphic to complex projective space CP³. The correspondence between flat space-time and twistor space shows that complex α-planes correspond to points in PT, and real null geodesics to points in PN, i.e. the space of null twistors. Moreover, a complex space-time point corresponds to a sphere in PT, and a real space-time point to a sphere in PN. Remarkably, the points x and y are null-separated iff the corresponding spheres in PT intersect. This is the twistor description of the light-cone structure of Minkowski space-time.

A conformally invariant isomorphism exists between the complex vector space of holomorphic solutions of  ◻φ = 0 on the forward tube of flat space-time, and the complex vector space of arbitrary complex-analytic functions of three variables, not subject to any differential equation. Moreover, when curvature is non-vanishing, there is a one-to-one correspondence between complex space-times with anti-self-dual Weyl curvature and scalar curvature R = 24Λ, and sufficiently small deformations of flat projective twistor space PT which preserve a one-form τ homogeneous of degree 2 and a three-form ρ homogeneous of degree 4, with τ ∧ dτ = 2Λρ. Thus, to solve the anti-self-dual Einstein equations, one has to study a geometric problem, i.e. finding the holomorphic curves in deformed projective twistor space.

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Stationarity or Homogeneity of Random Fields

Let (Ω, F, P) be a probability space on which all random objects will be defined. A filtration {F_t : t ≥ 0} of σ-algebras, is fixed and defines the information available at each time t.

Random field: A real-valued random field is a family of random variables Z(x) indexed by x ∈ R^d together with a collection of distribution functions of the form F_x1,…,_xn which satisfy

F_x1,…,_xn(b₁,…,b_n) = P[Z(x₁) ≤ b₁,…,Z(x_n) ≤ b_n], b₁,…,b_n ∈ R

The mean function of Z is m(x) = E[Z(x)] whereas the covariance function and the correlation function are respectively defined as

R(x, y) = E[Z(x)Z(y)] − m(x)m(y)

c(x, y) = R(x, x)/√(R(x, x)R(y, y))

Notice that the covariance function of a random field Z is a non-negative definite function on R^d × R^d, that is if x₁, . . . , x_k is any collection of points in R^d, and ξ₁, . . . , ξ_k are arbitrary real constants, then

∑_l=1^k∑_j=1^k ξ_lξ_j R(x_l, x_j) = ∑_l=1^k∑_j=1^k ξ_lξ_j E(Z(x_l) Z(x_j)) = E (∑_j=1^k ξ_j Z(x_j))² ≥ 0

Without loss of generality, we assumed m = 0. The property of non-negative definiteness characterizes covariance functions. Hence, given any function m : R^d → R and a non-negative definite function R : R^d × R^d → R, it is always possible to construct a random field for which m and R are the mean and covariance function, respectively.

Bochner’s Theorem: A continuous function R from R^d to the complex plane is non-negative definite if and only if it is the Fourier-Stieltjes transform of a measure F on R^d, that is the representation

R(x) = ∫_R^d e^ix.λ dF(λ)

holds for x ∈ R^d. Here, x.λ denotes the scalar product ∑_k=1^d x_kλ_k and F is a bounded,  real-valued function satisfying ∫_A dF(λ) ≥ 0 ∀ measurable A ⊂ R^d

The cross covariance function is defined as R₁₂(x, y) = E[Z₁(x)Z₂(y)] − m₁(x)m₂(y)

, where m₁ and m₂ are the respective mean functions. Obviously, R₁₂(x, y) = R₂₁(y, x). A family of processes Z_ι with ι belonging to some index set I can be considered as a process in the product space (R^d, I).

A central concept in the study of random fields is that of homogeneity or stationarity. A random field is homogeneous or (second-order) stationary if E[Z(x)²] is finite ∀ x and

• m(x) ≡ m is independent of x ∈ R^d

• R(x, y) solely depends on the difference x − y

Thus we may consider R(h) = Cov(Z(x), Z(x+h)) = E[Z(x) Z(x+h)] − m², h ∈ R^d,

and denote R the covariance function of Z. In this case, the following correspondence exists between the covariance and correlation function, respectively:

c(h) = R(h)/R(o)

i.e. c(h) ∝ R(h). For this reason, the attention is confined to either c or R. Two stationary random fields Z₁, Z₂ are stationarily correlated if their cross covariance function R₁₂(x, y) depends on the difference x − y only. The two random fields are uncorrelated if R₁₂ vanishes identically.

An interesting special class of homogeneous random fields that often arise in practice is the class of isotropic fields. These are characterized by the property that the covariance function R depends only on the length ∥h∥ of the vector h:

R(h) = R(∥h∥) .

In many applications, random fields are considered as functions of “time” and “space”. In this case, the parameter set is most conveniently written as (t,x) with t ∈ R₊ and x ∈ R^d. Such processes are often homogeneous in (t, x) and isotropic in x in the sense that

E[Z(t, x)Z(t + h, x + y)] = R(h, ∥y∥) ,

where R is a function from R² into R. In such a situation, the covariance function can be written as

R(t, ∥x∥) = ∫_R ∫_λ=0^∞ e^itu H_d (λ ∥x∥) dG(u, λ),

where

H_d(r) = (2/r)^{(d – 2)/2} Γ(d/2) J_{(d – 2)/2} (r)

and J_m is the Bessel function of the first kind of order m and G is a multiple of a distribution function on the half plane {(λ,u)|λ ≥ 0,u ∈ R}.

Category-Theoretic Sinks

The concept dual to that of source is called sink. Whereas the concepts of sources and sinks are dual to each other, frequently sources occur more naturally than sinks.

A sink is a pair ((f_i)_i∈I, A), sometimes denoted by (f_i,A)_I or (A_i →^f_i A)_I consisting of an object A (the codomain of the sink) and a family of morphisms f_i : A_i → A indexed by some class I. The family (A_i)_i∈I is called the domain of the sink. Composition of sinks is defined in the (obvious) way dual to that of composition of sources.

In Set, a sink (A_i →^f_i A)_I is an epi-sink if and only if it is jointly surjective, i.e., iff A = ∪_i∈I f_i[A_i]. In every construct, all jointly surjective sinks are epi-sinks. The converse implication holds, e.g., in Vec, Pos, Top, and Σ-Seq. A category A is thin if and only if every sink in A is an epi-sink.

Every epi-sink (= jointly surjective sink) is an extremal epi-sink in Set, Vec, and Ab. In Top an epi-sink (A_i →^f_i A) is extremal if and only if A carries the final topology f_i with respect to (f_i)_i∈I. In Pos an epi-sink (A_i →^f_i A)_I is extremal iff the ordering of A is the transitive closure of the relation consisting of all pairs (f_i(x), f_i(y)) with i ∈ I and x ≤ y in A_i. In Σ-Seq an epi-sink (A_i →^f_i A)_I is extremal iff each final state of A has the form f_i(q) for some i ∈ I and some final state q of A_i.

Every separator is extremal in Set, Vec, and Ab. In Pos the separators are precisely the nonempty posets, whereas the extremal separators are precisely the non-discrete posets. Top has no extremal separator.

An empty sink (∅, A) in a concrete category is final iff A is discrete.

A singleton sink A →^f B in a concrete category is final iff f is a final morphism.

A sink ((X_i, τ_i) →^f_i (X, τ))_I in Top is final iff τ is the final topology with respect to the maps (f_i)I, i.e. , τ = {U ⊆ X|∀ i ∈ I, f⁻¹[U] ∈ τ_i}.

A sink ((X_i, ≤_i) →^f_i (X, ≤))_I in Pos is final iff ρ is the transitive closure of the relation {(x, x)|x ∈ X} ∪ ∪_I {(f_i(x), f_i(y))|x ≤_i y}.

A sink ((X_i, α_i) →^f_i (X, α))_I in Spa(T) is final iff α = ∪_I Tf_i[α_i].