Fedja Hadrovich

Twistor Primer

Introduction

In the past 30 years a lot of work has been done on developing twistor theory. Its creator, 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. Yet 30 years later, twistors remain relatively unknown even in the mathematical physics community. The reason for this may be the air of mystery that seems to surround the subject even though it provides a very elegant formalism for both general relativity and quantum theory. These notes are based on a graduate lecture course given by R. Penrose in Mathematical Institute, Oxford, in 1997 and should give a brief introduction to the basic definitions. Let us begin with the building blocks: spinors.


Spinors

Let tex2html_wrap_inline365 be a vector in a (+--) spacetime. We choose to represent it in the form of a Hermitian matrix:

eqnarray58

Note that tex2html_wrap_inline369 . If tex2html_wrap_inline365 is null future-pointing, there is a decomposition

eqnarray68

i.e. tex2html_wrap_inline373 . Now tex2html_wrap_inline375 is a spinor and tex2html_wrap_inline377 is an element of the conjugate spin space.

The notation may be unusual, but all we are doing is elementary algebra. We have shown that we can parametrise the null cone by 2 complex numbers, the components of the spinor tex2html_wrap_inline379 . In the projective case, when we are interested only in the direction of tex2html_wrap_inline365 , the space of parametrisation reduces to tex2html_wrap_inline383 , the 1-dimensional complex projective space, which is homeomorphic to tex2html_wrap_inline385 , the celestial sphere.

Since tex2html_wrap_inline387 is (up to a constant factor) the unique skew-symmetric form of maximal rank, we have the following decomposition:

eqnarray88

eqnarray96

where tex2html_wrap_inline389 , with all diagonal entries zero.

Now we can define raising and lowering of spinor indices (nb. order of indices is important since tex2html_wrap_inline391 s are skew-symmetric!):

eqnarray105

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.

Note that tex2html_wrap_inline393 since tex2html_wrap_inline391 s are skew.

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Electromagnetism in spinor notation

We present here the basic results written in spinor notation. You can prove all of them as an exercise.

Every skew tex2html_wrap_inline397 can be written as

eqnarray115

where tex2html_wrap_inline399 and tex2html_wrap_inline401 are symmetric. If tex2html_wrap_inline397 is real then tex2html_wrap_inline405 . Dualisation is particularly easy:

eqnarray124

Maxwell's equations have the form:

eqnarray129

for some 4-potential tex2html_wrap_inline407 and 4-current tex2html_wrap_inline409 . The expression for the stress-energy tensor is also particularly simple:

eqnarray136


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Twistors

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 tex2html_wrap_inline415 . With respect to the new origin,

eqnarray143

We define the Pauli-Lubanski spin vector:

eqnarray147

It is easy to show that tex2html_wrap_inline417 . Now let tex2html_wrap_inline419 be future null, so that we can write tex2html_wrap_inline421 . Since tex2html_wrap_inline423 is skew it can be decomposed as

eqnarray155

The dual is then easy to write:

eqnarray160

and

eqnarray165

In nature we only observe massles particles with definite handedness, i.e. with tex2html_wrap_inline425 .

eqnarray169

follows immediately and hence tex2html_wrap_inline427 , where either tex2html_wrap_inline429 or tex2html_wrap_inline431 are proportional to tex2html_wrap_inline433 (note that tex2html_wrap_inline435 always) and () denotes index symmetrisation. The same arguments apply to tex2html_wrap_inline439 . We can now define tex2html_wrap_inline441 with

eqnarray180

At last, we can say, tex2html_wrap_inline443 is a twistor.

Now we have

eqnarray186

if we define tex2html_wrap_inline445 .

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Quantisation

Canonical commutation rules for Minkowski spacetime

eqnarray200

induce the following commutation relations on the twistor space:

eqnarray202

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

eqnarray207

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

eqnarray211

Therefore, if we want a twistor function tex2html_wrap_inline447 to be an eigenstate of spin operator with eigenvalue tex2html_wrap_inline449 , tex2html_wrap_inline447 must be homogeneous of degree -2s-2.


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Klein Correspondence

In this section we outline some basic twistor geometry. Let tex2html_wrap_inline455 be complexified compactified Minkowski space. (We can think of it as the Grassmannian manifold tex2html_wrap_inline457 .) The basic concept is that of incidence.

  • Twistor tex2html_wrap_inline443 is incident with a spacetime point tex2html_wrap_inline461 iff

    eqnarray223

  • Dual twistor tex2html_wrap_inline463 is incident with a spacetime point tex2html_wrap_inline461 iff

    eqnarray228

  • tex2html_wrap_inline467 is incident with tex2html_wrap_inline469 iff tex2html_wrap_inline471 .

In the projective twistor space tex2html_wrap_inline473 , tex2html_wrap_inline443 defines a point as an equivalence class of all twistors in tex2html_wrap_inline477 proportinal to tex2html_wrap_inline467 . The set of all spacetime points incident with it forms a totally null complex 2-plane in tex2html_wrap_inline455 ( tex2html_wrap_inline483 -plane). If we denote the incidence relation with tex2html_wrap_inline485 , then

eqnarray234

for all tex2html_wrap_inline487 . Any two vectors in the tex2html_wrap_inline483 -plane are orthogonal to each other. Since tex2html_wrap_inline467 and tex2html_wrap_inline493 have the same incidence properties for tex2html_wrap_inline495 , it is natural to study incidence on the projective twistor space tex2html_wrap_inline473 .

Dual twistors now correspond to planes in tex2html_wrap_inline473 and are incident on tex2html_wrap_inline501 -planes in tex2html_wrap_inline455 . These are also totally null complex 2-planes in tex2html_wrap_inline455 .

Finally, it can be shown that tex2html_wrap_inline507 iff 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:

tabular237


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Classical fields

Let tex2html_wrap_inline545 be a spin n/2 field and tex2html_wrap_inline549 a spin -n/2 field on tex2html_wrap_inline455 , where n is the number of spinor indices on tex2html_wrap_inline557 s.

Zero mass field equations are given by

eqnarray245

or tex2html_wrap_inline559 for zero spin case.

Solutions can then be written in the form:

eqnarray251

As an exercise, show that for

eqnarray269


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