User:Maschen/twistor theory

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Motivation

The motivation and one of the initial aims of twistor theory is to provide an adequate formalism for the union of quantum theory and general relativity. Twistors are essentially complex objects, like wavefunctions in quantum mechanics, as well as endowed with holomorphic and algebraic structure sufficient to encode space-time points. In this sense twistor space can be considered more primitive than the space-time itself and indeed provides a background against which space-time could be meaningfully quantized.

Stereographic projection

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Main article: Stereographic projection

In quantum mechanics a state of an object is described by a vector in Hilbert space. This is a linear superposition of some basis vectors, for example, in the case of an electron (or any spin-1/2 particle) it is some linear superposition of the spin "up" and "down" states. The two component vector (z,w) is called a spinor. The norm of this spinor caries no information - indeed, it is usually set to 1. Only the direction is physically significant which forces us to consider the projective (spin) space. In the electron example, spin state is therefore described by a point in CP¹, conveniently represented with Riemann sphere.

Properties of stereographic projection

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Stereographic projection is a shape (i.e. angle) preserving map. Hence circles on the sphere will be mapped to circles on the plane (except those containing the south pole, which are mapped to straight lines on the plane). This geometry is preserved by Möbius (fractional linear) transformation. In terms of spinors, this is the action of SL(2,C).

Relativity

Appearance of a moving sphere

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In relativity, an observer sees a t=const. section of its past null cone. This is a celestial (Riemann) 2-sphere that can be stereographically projected onto a complex plane. Möbius transformations of the plane correspond to Lorentz transformations of the celestial sphere.

This "spinorial" point of view has an interesting application in finding the apparent shape of a rapidly moving sphere.

Naively we would expect to see Lorentz contraction. However, no such contraction will be observed: every observer will see a circular outline (moving in some direction; of corse, they do not agree on the direction or the velocity of the sphere).

Lorentz transformation of sphere of vision

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The spherical outline on the t=const. section of observer's past light cone will remain spherical under Lorentz transformation since we know that Lorentz transformation on the celestial sphere is just Möbius transformation on the image of the sphere under stereographic projection. All these are shape preserving maps which proves out assertion.

Synge argument is another demonstration of the Lorentz-invariance of the circular outline on the celestial sphere. Here the blue timelike hyperplane intersects the observer's past lightcone and t=const. hyperplanes in a circle. It is clear that Lorentz transformation cannot change the shape of the circle.

Twistor theory

Twistors as spinors

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A space-time point can be represented as a Riemann sphere in terms of some section of its light cone. This is precisely how space-time points are represented in the projective twistor space.

The full twistor space is just a 4-dimensional complex vector space {(Z₀, Z₁, Z₂, Z₃)} = C⁴. More technical details are given below.

<<image: twistors as spinors for O(2,4)>>

Another way of defining twistors is as spinors for O(2,4) group (actually, the subgroup that preserves "time" and "space" orientations). Minkowski space M can be represented in R⁶ with metric signature (++−−−−) as a parabolic intersection of a hyperplane with the null cone of the origin. All generators of the cone intersect M except for the U = W generator. Adding this generator gives the compactified Minkowski space M^#. Projectively, M^# corresponds to a quadric.

Incidence

<<image: Klein correspondence>>

Twistors are essentially complex objects and in order to proceed we shall have to consider the complexification of the compactified Minkowski space. This is a quadric in the 5-dimensional complex projective space. On this quadric, there are two 3-parameter families of totally null 2-planes, one self-dual (SD, alpha-planes), the other anti-self-dual (ASD, beta-planes). Projective twistor space is the space of all alpha-planes. Through every point on the quadric there is a one-parameter family of alpha-planes which means that space-time points are represented as projective lines in the projective twistor space.

This correspondence between the lines in CP³ and points of a quadric in CP⁵ is known as the Klein correspondence. image: incidence relation

In terms of the coordinates (r0,r1,r2,r3) on CM, the complexified Minkowski space, and homogeneous coordinates (Z₀, Z₁, Z₂, Z₃) on CP³, the Klein correspondence can be espressed explicitly in terms of the incidence relation.

If we restrict ourselves to real space-time points, the corresponding lines in CP³ lie in a 5-real-dimensional hypersurface PN.

<<image: spinor components>>

This transparency shows the same matrix equation of incidence relation written in terms of spinor components of a twistor in abstract-index notation. Indices A, A′ take values 0 and 1.

In complexified Minkowski space the incidence relation associates an alpha-plane with each projective twistor. image: momentum and angular momentum

The spinor components of a twistor have a natural interpretation in terms of the momentum and angular momentum of a zero rest mass particle. Technical details can be found in the Twistor Primer.

This identification also yields the spin in the form of the Pauli-Lubanski vector.

Quantization

<<image: quantization>>

The quantum mechanical commutation relations for momentum and angular momentum give simple commutation relation for twistors and dual twistors. Hence the quantisation rule for twistor theory: dual twistors are represented with derivative operators. Substituting into the expression for the spin, we observe that Euler homogeneity operator features in the formula. States with well defined spin s are therefore described by functions on twistor space which are homogeneous of degree −2(s + 1). image: massless field eqns

Field equations

Massless field of spin s is a spinor field with 2s indices, primed for s > 0 and unprimed for s < 0. We expect to be able to encode a spin s massless field with a homgeneity degree −2(s + 1) twistor function.

<<image: twistor representation>>

Indeed, this is achieved with a contour integral formula shown.

<<image: cohomology>>

An important observation can be made here that two different twistor functions encode the same field if their difference is a holomorphic function. Thus it is the equivalence classes of functions (strictly speaking, Cech cohomology representatives) that correspond to massless fields. For more information on Cech cohomology see Applications of Sheaf Cohomology in Twistor Theory.

Gravitation

<<image: linearised gravity>>

These are the massless fields equations and their solutions in the linearized case of gravitational interaction. As expected, homogeneity degree +2 and -6 functions encode the anti-self-dual and self-dual parts of the gravitational field.

Background on spinors

Roger Penrose was first led to the concept of twistors in his investigation of the structure of spacetime and it was he who first saw the wide range of applications for this new mathematical construct. The building blocks are spinors.

Notation:

• capital indices indicate the entries in a matrix, row vector or column vector.
• Latin and Greek indices take spacetime values.

Spinors

Let K^a be a vector in Minkowski spacetime of signature (+−−−). It can be represented in the form of a Hermitian matrix:

$K^{AA'}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}K^{0}+K^{3}&K^{1}+iK^{2}\\K^{1}-iK^{2}&K^{0}-K^{3}\end{pmatrix}}$

Twice the determinant is the square of the norm of the vector:

$2\det(K^{AA'})=g_{ab}K^{a}K^{b}=\|K\|^{2}$

with g_ab the components of the metric. If K^a is null future-pointing, there is a decomposition

$K^{AA'}=\kappa ^{A}{\bar {\kappa }}^{A'}\,,\quad \kappa ^{A}={\begin{pmatrix}\xi &\eta \end{pmatrix}}\,,\quad {\bar {\kappa }}^{A'}={\begin{pmatrix}{\bar {\xi }}\\{\bar {\eta }}\end{pmatrix}}$

where ξ, η are complex numbers, meaning κ^A is an element of C². The κ^A and κ^A′ are Hermitian conjugates of each other: κ^A is a spinor and κ^A′ is an element of the conjugate spinor space.

Thus the null cone can be parametrized by two complex numbers, the components of the spinor κ^A.

In the projective case, when we are interested only in the direction of κ^A, the space of parametrization reduces to $\mathbb {P} \mathbb {C} ^{1}$, the 1-dimensional complex projective space, which is homeomorphic to the celestial sphere S¹.

Since the determinant is the unique skew-symmetric form of maximal rank (up to a constant factor), we have the following decomposition:

$2\det(K^{AA'})=\varepsilon _{AB}\varepsilon _{A'B'}K^{AA'}K^{BB'}=g_{ab}K^{a}K^{b}$

$g_{ab}=\varepsilon _{AB}\varepsilon _{A'B'}$

where ε is the Levi-civita symbol in two dimensions, with non-zero entries ε₀₁ = −ε₁₀.

Now we can define raising and lowering of spinor indices via ε:

$\kappa _{A}=\kappa ^{B}\varepsilon _{BA}\,\quad \kappa _{A'}=\kappa ^{B'}\varepsilon _{B'A'}$

wherein the order of indices is important since ε is antisymmetric, even though the metric is symmetric. The norm is:

$\kappa _{A}\kappa ^{A}=\kappa _{A'}\kappa ^{A'}=0$

Spinor calculations are equivalent to tensor calculations in many ways. The main difference is that in the spinor case there are unprimed and primed (conjugate) spaces with their respective duals. The isomorphisms between the spaces and their duals are given by the skew-symmetric form so care must be taken to write the indices in the correct order.

Electromagnetism in spinor notation

We present here the basic results written in spinor notation. Every complex-valued antisymmetric second-order tensor F_ab can be written as:

$F_{ab}=\phi _{AB}\varepsilon _{A'B'}+\psi _{A'B'}\varepsilon _{AB}$

where ϕ_AB and ψ_AB are symmetric spinors. For real-valued F_ab:

$\psi _{A'B'}={\overline {\phi _{AB}}}={\bar {\phi }}_{A'B'}$

and the tensor dual to F is:

$\star F_{ab}=-i\phi _{AB}\varepsilon _{A'B'}+i\psi _{A'B'}\varepsilon _{AB}$

Maxwell's equations have the form:

$\nabla _{B'}^{A}\phi _{AB}=2\pi J_{BB'}$

$\phi _{AB}=\nabla _{A'(A}\Phi _{B)}^{A'}$

for some four-potential Φ_b and four-current J_b. The expression for the stress-energy tensor is also particularly simple:

$T_{ab}={\frac {1}{2\pi }}\phi _{AB}{\bar {\phi }}_{A'B'}$

Twistors

Approach to twistor by momentum, angular momentum, and spin

One of the easiest and most straightforward ways of defining twistors uses the transformation properties of linear and angular momentum of a particle under a shift of origin. Consider a change of origin from 0 to a point Q with coordinates q^a. With respect to the new origin, the momentum is independent of the origin,

$p^{a}(Q)=p^{a}(0)$

although the angular momentum is not:

$M^{ab}(Q)=M^{ab}(0)+(q^{a}p^{b}-q^{b}p^{a})$

The the Pauli-Lubanski spin vector, built out of the momentum, angular momentum, and four-dimensional Levi-Civita symbol:

$S_{a}(Q)={\frac {1}{2}}\varepsilon _{abcd}p^{b}M^{cd}=\star M_{ab}p^{b}$

is also independent of the origin:

$S_{a}(Q)={\frac {1}{2}}\varepsilon _{abcd}p^{b}M^{cd}=\star M_{ab}p^{b}$

If the momentum is future pointing and null, then from the above result we can decompose it into:

$p^{a}={\bar {\pi }}_{A}\pi _{A'}$

while for the angular momentum:

$M^{ab}={\bar {\mu }}_{AB}\varepsilon _{A'B'}+\mu _{A'B'}\varepsilon _{AB}$

with dual:

$\star M^{ab}=-i{\bar {\mu }}_{AB}\varepsilon _{A'B'}+i\mu _{A'B'}\varepsilon _{AB}$

and finally for the Pauli-Lubanski pseudovector:

$S_{a}=i{\bar {\mu }}_{AB}{\bar {\pi }}^{B}\pi _{A'}\varepsilon _{A'B'}-i\mu _{A'B'}\varepsilon _{AB}$

In nature we only observe massles particles with definite handedness, i.e. with

$S_{a}=sp_{a}$

then:

$\mu _{A'B'}\pi ^{B'}\pi ^{A'}=0$

and

$\mu _{A'B'}=\alpha _{(A'}\beta _{B')}={\frac {1}{2}}(\alpha _{A'}\beta ^{B'}+\alpha _{B'}\beta ^{A'})$

where the round brackets ( ) denote index symmetrization, and either α_A′ or β_B′ are proportional to π_a′. Always:

$\pi _{A}\pi ^{A}=0$

The same arguments apply to μ_AB. Introducing ω_A by the equation:

${\bar {\mu }}_{AB}=i\omega _{(A}{\bar {\pi }}_{B)}$

These form the components of a twistor:

$Z^{\alpha }=(\omega ^{A},\pi _{A'})$

with conjugate:

${\bar {Z}}_{\alpha }=({\bar {\pi }}_{A},{\bar {\omega }}^{A'})$

The helicity of a massless particle is given by half of the square of twistor norm:

$s={\frac {1}{2}}(\omega ^{A}{\bar {\pi }}_{A}+{\bar {\omega }}^{A'}\pi _{A'})={\frac {1}{2}}Z^{\alpha }{\bar {Z}}_{\alpha }$

Quantization

The canonical commutation rules for Minkowski spacetime are those of the Poincaré group:

$[p_{a},p_{b}]=0$

$[q^{a},q^{b}]=0$

$[p_{a},q^{b}]=i\hbar g_{a}^{b}$

induce the following commutation relations on the twistor space:

$[Z^{\alpha },Z^{\beta }]=0$

$[{\bar {Z}}_{\alpha },{\bar {Z}}_{\beta }]=0$

$[Z^{\alpha },{\bar {Z}}_{\beta }]=\hbar \delta _{\beta }^{\alpha }$

One way of quantizing the theory is to use the following substitution:

$Z^{\alpha }\rightarrow Z^{\alpha }$

${\bar {Z}}_{\alpha }\rightarrow -\hbar {\frac {\partial }{\partial Z^{\alpha }}}$

The spin operator can be easily derived in the non-commutative case following the same procedure. The result is the symmetrized form:

$s={\frac {1}{4}}(Z^{\alpha }{\bar {Z}}_{\alpha }+{\bar {Z}}_{\alpha }Z^{\alpha })=-{\frac {\hbar }{2}}\left(Z^{\alpha }{\frac {\partial }{\partial Z^{\alpha }}}+2\right)$

Therefore, if we want a twistor function f(Z^α) to be an eigenstate of the spin operator with eigenvalue ħs, the function f(Z^α) must be homogeneous of degree −2(s + 1).

Klein Correspondence

In this section we outline some basic twistor geometry. Let M be complexified and compactified Minkowski space. We can think of it as the Grassmannian manifold $G_{2,4}\mathbb {C}$.

Incidence

The basic concept is that of incidence. The twistor $Z^{\alpha }=(\omega ^{A},\pi _{A'})$ is incident with a spacetime point x^a if and only if:

$\omega ^{A}=ix^{AA'}\pi _{A'}$

The dual twistor:

$W_{\alpha }=(\mu _{A},\nu ^{A'})$

is incident with a spacetime point x^a if and only if:

$\nu ^{A'}=-ix^{AA'}\mu _{A}$

The twistor is incident with its dual if

$Z^{\alpha }W_{\alpha }=0$

In the projective twistor space $\mathbb {P} \mathbb {T}$, Z^α defines a point as an equivalence class of all twistors in $\mathbb {T}$ proportional to Z^α. The set of all spacetime points incident with it forms a totally null complex 2-plane in M (α-plane). If we denote the incidence relation by $\leftrightsquigarrow$, then

$x^{a}\leftrightsquigarrow Z^{\alpha }\rightarrow x^{a}+\lambda ^{A}\pi ^{A'}\leftrightsquigarrow Z^{\alpha }$

for all λ^A. Any two vectors in the α-plane are orthogonal to each other. Since Z^α and λZ^α have the same incidence properties for nonzero λ, , it is natural to study incidence on the projective twistor space PT.

Dual twistors now correspond to planes in PT and are incident on β-planes in M. These are also totally null complex 2-planes in M. Finally, it can be shown that

$Z^{\alpha }\leftrightsquigarrow W_{\alpha }$

if and only if there is a spacetime point incident on both of them. Geometric correspondence defined by the incidence relation (Klein correspondence) is summarised in the following table:

Classical fields

Let $\phi _{ABC\ldots }=\phi _{(ABC\ldots )}$ and $\phi _{A'B'C'\ldots }=\phi _{(A'B'C'\ldots )}$ be two symmetric spin-(n/2) fields in M, where n is the number of spinor indices. Massless field of spin s is a spinor field with 2s indices, primed for s > 0 and unprimed for s < 0.

The field equation for a massless positive helicity field is given by:

$\nabla ^{AA'}\phi _{ABC\ldots }=0\,,\quad \nabla ^{AA'}\phi _{A'B'C'\ldots }=0$

while for a massless negative helicity field:

$\nabla ^{AA'}\phi _{ABC\ldots }=0\,,\quad \nabla ^{AA'}\phi _{A'B'C'\ldots }=0$

as for the massless spinless case we have the scalar wave equation:

$\Box \phi =0$

We expect to be able to encode a spin s massless field with a homgeneity degree −2(s + 1) twistor function. Indeed, this is achieved with contour integral formulae. The solutions can then be written in the form:

$\phi _{ABC\ldots }(R^{a})={\frac {1}{2\pi i}}\oint _{Z^{\alpha }\leftrightsquigarrow R^{a}}{\frac {\partial }{\partial \omega ^{A}}}{\frac {\partial }{\partial \omega ^{B}}}{\frac {\partial }{\partial \omega ^{C}}}\cdots f(Z^{\alpha })\pi _{p'}d\pi ^{p'}$

$\phi _{A'B'C'\ldots }(R^{a})={\frac {1}{2\pi i}}\oint _{Z^{\alpha }\leftrightsquigarrow R^{a}}\pi _{A'}\pi _{B'}\pi _{C'}\cdots f(Z^{\alpha })\pi _{p'}d\pi ^{p'}$

An important observation can be made here that two different twistor functions encode the same field if their difference is a holomorphic function. Thus it is the equivalence classes of functions (strictly speaking, Cech cohomology representatives) that correspond to massless fields. For more information on Cech cohomology see Applications of Sheaf Cohomology in Twistor Theory.

Consider the cases:

$f(Z^{\alpha })={\frac {1}{A_{\alpha }Z^{\alpha }B_{\alpha }Z^{\alpha }}}$

$\phi (r^{a})={\frac {2}{A_{C}B^{C}}}{\frac {1}{(r^{a}-q^{a})(r_{a}-q_{a})}}$

$q^{e}={\frac {i}{A_{C}B^{C}}}(A^{E'}B^{E}-A^{E}B^{E'})$

References

• R. Penrose (2005). "33". The Road to Reality. Vintage books. ISBN 978-00994-40680.

• S. A. Huggett, K. P. Tod (1994). An Introduction to Twistor Theory. London Mathematical Society Student Texts. Vol. 4. Cambridge University Press. ISBN 0-521-456-894.

• L. J. Mason, N. Michael, J. Woodhouse (1996). Integrability, Self-duality, and Twistor Theory. London Mathematical Society monographs, London Mathematical Society Oxford science publications. Vol. 15. Oxford University Press. ISBN 0-198-534-981.{{cite book}}: CS1 maint: multiple names: authors list (link)

• R. S. Ward, Raymond O'Neil Wells (1991). Twistor Geometry and Field Theory. Cambridge Monographs on Mathematical Physics. Cambridge University Press. ISBN 0-5214-2268-X. ISSN 0269-8242.

External links

• R. Penrose, F. Hadrovich, Twistor theory
• F. Hadrovich, Twistor Primer