Revisiting Roger Penrose's Twistor Theory

 

                  One possible twistor configuration conceived by Roger Penrose

 Though he's now remembered best for his Physics Nobel Prize (one half share) for work on stellar -collapsed black holes, e.g.  

• Roger Penrose Takes 1/2 Nobel Prize In Physics - ...

Roger Penrose has also done a vast amount of research in abstract theoretical physics, which includes twistors, conceived originally as an approach to quantum theory.  In exploring the twin concepts of twistors and twistor space, Penrose conjectured that space is not really empty but composed of fundamental units called twistors - out of which everything else is constituted. The primary proposition is that it is futile to try to understand the sundry objects in the universe unless the nature of empty space is first comprehended.

To approach twistors Penrose and his collaborators used complex numbers, e.g. of the form a + bi,  where i is an imaginary number, i.e  i= Ö -1.     For more on complex numbers see the previous post:

• What are Complex Numbers?
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To  depict the twistor space of T for M, Penrose invokes a complex 4-dimensional vector space for which standard complex coordinates (Z ^o,, Z ¹,  Z ²,  Z ³)  are used, i.e.:  

(Z ^o)

(Z ¹)

=
 (t + z  ..... x + iy)  ( Z ²)

 (x – iy   ..... t - z)  ( Z ³)

And we say a twistor Z is incident with a spacetime point R.  In his book, 'The Road To Reality' pp. 974-75, Penrose describes  the twistor representation process as follows:

" Z ^a  will sometimes be used to represent the twistor Z, where the components of Z in a standard frame would be (Z ^o,, Z ¹,  Z ²,  Z ³).  Each twistor Z, or  Z ^a, (an element of T) has a complex conjugate Z*,  which is a dual twistor (element of the dual twistor space T*).  In index form, Z* is written  Z*^a   , with a lower index, and its components (in the standard frame) would be: 

(Z*_{0  ,}  Z*₁, Z*₂, Z*₃)  =  (Z*₂, Z*_{3 ,} Z*_{0  ,}  Z*₁ )

He adds (ibid.):

"This notation is probably a little confusing.  The four quantities (complex numbers) on the left are simply the four components of the dual twistor Z*.  The four on the right are the respective complex conjugates of the complex numbers: (Z ²,  Z ³, Z^o,, Z ¹).  Thus, the component Z*₀   of Z* is the complex conjugate of the component  Z ²  of Z etc.  Note the interchange of the first two with the second two when forming the complex conjugation.  Since Z* is a dual twistor, we can form its (Hermitian) scalar product  with the original twistor Z to obtain the (squared) twistor norm:

Z ·  Z*  = Z*₀  ·  Z ^a 

⁼ 

 Z 0*  Z 0  +  Z ¹* Z ¹    +   Z ²* Z ²    +   Z ³* Z ³

⁼        ½  (‖ Z ^o + Z ² ‖ ² + ‖ Z ¹ + Z ³ ‖ ²  -   ‖ Z ⁰ - Z ² ‖ ²   -   ‖ Z ¹ - Z ³ ‖ ² )

where this last formula shows the Hermitian expression Z*₀ Z ²     has signature (+ + -  - )."

^{Penrose then points out that:}  

^{"Much of twistor space is most easily expressed in terms of   PT}  rather than  T . 

So that:  "The numbers   now provide homogenous coordinates for PT, so that the three independent ratios:

      Z^o, : Z ¹  : Z ² :  Z ³

serve to label points of PT.  The null projective twistors constitute the space PN which is the 5-real-dimensional subspace of the 6-real dimensional space PT  for which the twistor norm vanishes, i.e: 

Z*₀  ·  Z ^a   =   0 "

Penrose's twistor conjecture is not an easy concept but if one understands and uses complex spaces in conjunction with linear algebra the subject is rendered less abstruse.  Basically, the existence of the "dual twistor space"  T*   means one ends up dealing with an 8-dimensional space with 4 real dimensions and 4 complex dimensions.  Physically,  the twistor researchers  allocate three of these dimensions  for specifying position, two for angular directions, and one each for spin, energy and polarization of associated light rays.   

The visual  representation shown is actually a superposition of  two different twistor spaces - each appearing as twisted 'donut' (torus) spaces.   The extent of the respective twistor space depends on the energy, with small, highly localized space corresponding to  high energy and the larger donut shape corresponding to lower energy.

A fundamental perspective on twistors is that they is that they show space is not continuous but comprised of discrete units. Hence, space has a texture and one can think of it being quantized.   

For their part, twistor theoreticians have demonstrated mathematically that their creations are about more than geometry.  For example, various combinations can give rise to different particles.  So 2-twistor particles emerge as electrons, muons and other leptons.    

In his co-authored book (with Stephen Hawking), 'The Nature of Space and Time' ,   Penrose writes (p. 109):   "The basic idea of twistor theory is to exploit this link between quantum mechanics and spacetime structure,  as manifested  in the Riemann sphere - by extending this idea to  the whole of spacetime."

The most radical proposition of Penrose (p. 110) is that "light rays are to be regarded as more fundamental than spacetime points."  

In effect, spacetime becomes a  "secondary concept"  while the space of light rays (twistor space) becomes "the more fundamental concept."   Then a point in spacetime is represented by the set of light rays passing through it.  Thus, "a point in spacetime becomes a Riemann sphere in  twistor space."

A superb visual (and oral) introduction to twistor theory is presented by Penrose in a 2015  lecture below:

Roger Penrose. Twistor theory. - YouTube