Twistor space explained
In mathematics and theoretical physics (especially twistor theory), twistor space is the complex vector space of solutions of the twistor equation
. It was described in the 1960s by
Roger Penrose and Malcolm MacCallum.
[1] According to
Andrew Hodges, twistor space is useful for conceptualizing the way photons travel through space, using four
complex numbers. He also posits that twistor space may aid in understanding the
asymmetry of the
weak nuclear force.
[2] Informal motivation
In the (translated) words of Jacques Hadamard: "the shortest path between two truths in the real domain passes through the complex domain." Therefore when studying four-dimensional space
it might be valuable to identify it with
However, since there is no canonical way of doing so, instead all
isomorphisms respecting orientation and metric between the two are considered. It turns out that
complex projective 3-space
parametrizes such isomorphisms together with complex coordinates. Thus one complex coordinate describes the identification and the other two describe a point in
. It turns out that
vector bundles with
self-dual connections on
(
instantons)
correspond bijectively to
holomorphic vector bundles on complex projective 3-space
.
Formal definition
For Minkowski space, denoted
, the solutions to the twistor equation are of the form
\OmegaA(x)=\omegaA-ixAA'\piA'
where
and
are two constant Weyl spinors and
is a point in Minkowski space. The
are the
Pauli matrices, with
the indexes on the matrices. This twistor space is a four-dimensional complex vector space, whose points are denoted by
, and with a hermitian form
\Sigma(Z)=\omegaA\bar\piA+\bar\omegaA'\piA'
which is invariant under the group SU(2,2) which is 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
This incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space, denoted
, which is isomorphic as a complex manifold to
.
Given a point
it is related to a line in projective twistor space where we can see the incidence relation as giving the linear embedding of a
parametrized by
.
The geometric relation between projective twistor space and complexified compactified Minkowski space is the same as the relation between lines and two-planes in twistor space; more precisely, twistor space is
It has associated to it the double fibration of flag manifolds
P\xleftarrow{\mu}F\xrightarrow{\nu}M
where
is the projective twistor space
and
is the compactified complexified Minkowski space
M=F2(T)=\operatorname{Gr}
\operatorname{Gr}2,4(C)
and the correspondence space between
and
is
In the above,
stands for
projective space,
a
Grassmannian, and
a
flag manifold. The double fibration gives rise to two
correspondences (see also
Penrose transform),
and
The compactified complexified Minkowski space
is embedded in
by the
Plücker embedding; the image is the
Klein quadric.
References
- Book: Ward . R.S. . Raymond O. Wells, Jr. . Wells . R.O. . Twistor Geometry and Field Theory . Cambridge University Press . 1991 . 0-521-42268-X .
- Book: Huggett . S.A. . Tod . K.P. . An introduction to twistor theory . Cambridge University Press . 1994 . 978-0-521-45689-0 .
Notes and References
- R. . Penrose . M.A.H. . MacCallum . Twistor theory: An approach to the quantisation of fields and space-time . Physics Reports . 6 . 4 . 241–315 . February 1973 . 10.1016/0370-1573(73)90008-2 . subscription .
- Book: Hodges, Andrew . One to Nine: The Inner Life of Numbers. 2010. Doubleday Canada. 978-0-385-67266-5. 142.