扭量理论1967

In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a unification of general relativity (space-time geometry) and quantum mechanics (quantum theory) and has subsequently evolved into a widely studied branch of theoretical and mathematical physics. Penrose's idea was that twistor space should be the basic arena for physics from which space-time itself should emerge. It has led to powerful mathematical tools that have applications to differential and integral geometry, nonlinear differential equations and representation theory, and in physics to general relativity, quantum field theory, and the theory of scattering amplitudes.

Twistor theory arose in the context of the rapidly expanding mathematical developments in Einstein's theory of general relativity in the late 1950s and in the 1960s and carries a number of influences from that period. In particular, Roger Penrose has credited Ivor Robinson as an important early influence in the development of twistor theory, through his construction of so-called Robinson congruences.

Overview

Projective twistor space PT is projective 3-space CP ³, the simplest 3-dimensional compact algebraic variety. It has a physical interpretation as the space of massless particles with spin. It is the projectivisation of a 4-dimensional complex vector space, non-projective twistor space T, with a Hermitian form of signature (2, 2) and a holomorphic volume form. This can be most naturally understood as the space of chiral (Weyl) spinors for the conformal group SO(4,2)/Z2 of Minkowski space; it is the fundamental representation of the spin group SU(2,2) of the conformal group. This definition can be extended to arbitrary dimensions except that beyond dimension four, one defines projective twistor space to be the space of projective pure spinors for the conformal group.

In its original form, twistor theory encodes physical fields on Minkowski space in terms of complex analytic objects on twistor space via the Penrose transform. This is especially natural for massless fields of arbitrary spin. In the first instance these are obtained via contour integral formulae in terms of free holomorphic functions on regions in twistor space.

Twistor correspondence

Denote Minkowski space by ⁠ M ⁠, with coordinates x^(a)=(t,x,y,z) and Lorentzian metric η _(ab) signature (1,3). Introduce 2-component spinor indices A=0,1; A'=0',1', and set （公式略）

Non-projective twistor space T is a four-dimensional complex vector space with coordinates denoted by Z^(α)=(ω ^(A), π _(A')) where ω ^(A) and π _(A') are two constant Weyl spinors. The hermitian form can be expressed by defining a complex conjugation from T to its dual T ^(*) Z_(α)=(π _(A), ω ^(A')) so that the Hermitian form can be expressed as Z^(α)Z_(α)=ω ^(A)π _(A)+ω ^(A')π _(A').

This together with the holomorphic volume form, ε _(α β γ δ)Z^(α)dZ^(β)∧ dZ^(γ)∧ dZ^(δ) is invariant under the group SU(2,2), a quadruple cover of the conformal group C(1,3) of compactified Minkowski spacetime.

Points in Minkowski space are related to subspaces of twistor space through the incidence relation ω ^(A)=ix^(AA')π _(A').

The incidence relation is preserved under an overall complex rescaling of the twistor. This scaling assigns a flagpole extent, flag-plane direction, and a spinorial sign to the ray ⁠ Z ∈ PT ⁠.

Variations

Supertwistors

Supertwistors are a supersymmetric extension of twistors introduced by Alan Ferber in 1978. Non-projective twistor space is extended by fermionic coordinates where N is the number of supersymmetries so that a twistor is now given by (ω ^(A), π _(A'), η ⁱ),i=1,…,N with η ⁱ anticommuting. The super conformal group SU(2,2 | N) naturally acts on this space and a supersymmetric version of the Penrose transform takes cohomology classes on supertwistor space to massless supersymmetric multiplets on super Minkowski space. The N

= 4 case provides the target for Penrose's original twistor string and the N=8 case is that for Skinner's supergravity generalisation.

Higher dimensional generalization of the Klein correspondence

A higher dimensional generalization of the Klein correspondence underlying twistor theory, applicable to isotropic subspaces of conformally compactified (complexified) Minkowski space and its super-space extensions, was developed by J. Harnad and S. Shnider.

Hyperkähler manifolds

Hyperkähler manifolds of dimension 4k also admit a twistor correspondence with a twistor space of complex dimension 2k+1.