Twistor space ( $T ≅ C^{4}$ )

In flat twistor theory, a twistor is a pair

Z = (ω^{A}, π_{A^{'}})

of two-component spinors. The twistor incidence relation with a spacetime point is

ω^{A} = i x^{A A^{'}} π_{A^{'}}

. The nonzero vector

Z

and its projective class play different roles: the latter belongs to projective twistor space.

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Euclidean reality structure on twistor space ( $J^{2} = - 1, J (W) = W$ )

 0  0

Choose an antilinear quaternionic structure

J

C^{4}

. It induces a fixed-point-free involution on projective twistor space. A real Euclidean spacetime point is a

J

-invariant complex two-plane, equivalently a quaternionic line in

H^{2}

. These form

HP^{1} ≅ S^{4}

, the conformal compactification of Euclidean space. Every projective twistor lies on the unique invariant line

P span_{C} (Z, J Z)

, giving a fibration

CP^{3} \to S^{4}

with fiber

CP^{1}

Twistor Hermitian form ( $h (Z, W) = ω^{A} \overline{ρ}_{A} + π_{A^{'}} \overline{η}^{A^{'}}$ )

 0  0

For

Z = (ω, π)

and

W = (η, ρ)

the displayed Hermitian form has signature

(2, 2)

. It is zero on every pair of vectors in

W_{x} = {(i x π, π)}

when

x

is Hermitian, that is, Lorentzian real. Conversely a transverse totally isotropic two-plane is such a graph with

x

Hermitian. This gives the Lorentzian reality structure on twistor space.

Lorentzian reality structure on twistor space

 0  0

Real compactified Minkowski points are the two-dimensional totally isotropic subspaces for the signature-

(2, 2)

twistor Hermitian form. In a graph chart this is the usual Hermitian reality condition on

x^{A A^{'}}

. Their twistor lines lie in projective null twistor space. On the Klein quadric the same real form is the projectivized null cone of a real quadratic form of signature

(2, 4)

Null twistor ( $h (Z, Z) = 0$ )

 0  0

A nonzero twistor null for the twistor Hermitian form. Its projective class determines a real null geodesic in compactified Minkowski space. For nonzero

π

, its affine real solutions of the twistor incidence relation have the form

x = x_{0} + r \overline{π}^{A} π^{A^{'}}

r \in R

. Twistors with

π = 0

encode generators at infinity rather than finite affine solutions.

Projective null twistor space ( $PN = {[Z] \in CP^{3} : h (Z, Z) = 0}$ )

 0  0

The five-real-dimensional space of projective null twistors. A projective line lies entirely in it precisely when its corresponding two-plane is totally isotropic for the twistor Hermitian form. Such lines represent real points of Minkowski conformal compactification. Two independent null twistors lie on one such line exactly when

h (X, Y) = 0

Twistor incidence relation ( $ω^{A} = i x^{A A^{'}} π_{A^{'}}$ )

 0  0

For a fixed affine spacetime point

x

, this linear relation cuts out a two-dimensional subspace

W_{x} = {(i x π, π)}

of twistor space. Projectivizing gives its twistor line. For a fixed projective twistor with

π \neq = 0

, its solutions are

x = x_{0} + u^{A} π^{A^{'}}

, a totally null affine alpha-plane. The compact description replaces affine solutions by the two-planes containing the fixed twistor line in the vector space.

Projective twistor space ( $PT ≅ CP^{3}$ )

 0  0

The projective space of one-dimensional complex subspaces of twistor space. A point corresponds to an alpha-plane in complexified compactified Minkowski space; a projective line corresponds to a spacetime point.

Affine projective twistor space ( $P T_{aff} = CP^{3} ∖ CP_{\infty}^{1}$ )

 0  0

For affine complexified Minkowski space, the relevant twistor domain is

P T_{aff} = {[ω^{A}, π_{A^{'}}] \in CP^{3} : π \neq = 0} = CP^{3} ∖ CP_{\infty}^{1} ≅ Tot (O (1) \oplus O (1)) .

(1)

Incidence can be written

ω^{A} = x^{A A^{'}} π_{A^{'}}

, absorbing the conventional factor

i

into the definition of

x

. Each space-time point gives a twistor line

L_{x} ≅ CP^{1}

. On charts

π_{0^{'}} \neq = 0

and

π_{1^{'}} \neq = 0

, the fibre coordinates transform by

ω / π_{1^{'}} = (ω / π_{0^{'}}) / λ

, with

λ = π_{1^{'}} / π_{0^{'}}

, giving the two

O (1)

summands. The full projective twistor space corresponds to complexified compactified Minkowski space.

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Twistor space by

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Twistor space is a mathematical construction that arises in the context of theoretical physics, particularly in the study of certain fundamental aspects of spacetime and quantum field theory. Introduced by Roger Penrose in the 1960s, twistor theory provides a framework for understanding the relationships between geometrical and physical entities in a novel way, combining aspects of geometry with concepts in physics.

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Twistor space ( $T ≅ C^{4}$ )

Euclidean reality structure on twistor space ( $J^{2} = - 1, J (W) = W$ )

Twistor Hermitian form ( $h (Z, W) = ω^{A} \overline{ρ}_{A} + π_{A^{'}} \overline{η}^{A^{'}}$ )

Lorentzian reality structure on twistor space

Null twistor ( $h (Z, Z) = 0$ )

Projective null twistor space ( $PN = {[Z] \in CP^{3} : h (Z, Z) = 0}$ )

Twistor incidence relation ( $ω^{A} = i x^{A A^{'}} π_{A^{'}}$ )

Projective twistor space ( $PT ≅ CP^{3}$ )

Affine projective twistor space ( $P T_{aff} = CP^{3} ∖ CP_{\infty}^{1}$ )

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