**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 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:

**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.:**

^{o})

^{1})

(t + z ..... x + iy) ( Z

^{2})

^{3})

__' pp. 974-75, Penrose describes the twistor representation process as follows:__

*The Road To Reality*^{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:

_{0 , }Z*

_{1}, Z*

_{2}, Z*

_{3}) = (Z*

_{2}, Z*

_{3 , }Z*

_{0 , }Z*

_{1 })

*ibid*.):

^{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:

_{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 })

_{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 .

^{ }Z

^{o}

^{, }: Z

^{1 }

^{ }:

^{ }Z

^{2}

^{ }: Z

^{3}

_{0 }·

_{ }Z

^{a }= 0 "

**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 Nature__*' , Penrose writes (p. 109): "T*

__of Space and Time__*he 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*."

*light rays are to be regarded as more fundamental than spacetime points.*"

*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*."

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