Twistor theory

The twistor theory is an attempt to create a unified theory of gravitation and quantum field theory to create. The basic ideas of the Twistor theory date back to 1967 and were developed by the British mathematician and physicist Roger Penrose . The theory emerged from investigations into spin networks . The Twistor theory is not an established physical theory to this day, but has found diverse applications in mathematics.

The Twistor Theory and Classical Theories of Gravity and Quantum

In essence, the Twistor Theory tries to bring together the basic mathematical properties of relativity and quantum mechanics . In the case of the theory of relativity, these are the Minkowski space and its curvilinear generalization, so-called Riemannian manifolds with the signature 1, both of which have four dimensions . In the case of quantum mechanics, these are the complex numbers to which the nonlocal properties of quantum theory can be traced (e.g. Einstein-Podolsky-Rosen paradox ). The Twistor theory is characterized by many considerations of symmetry and mathematical elegance. In the twistor theory, an attempt is now made to analyze the most fundamental aspects of relativity and quantum mechanics from a new perspective through a reinterpretation within the framework of the twistor geometry.

Elementary Intuition: The Fundamental Objects of Twistor Theory

The elementary objects of the twistor theory are the twistors. If you transform a twistor from the twistor space into the Minkowski space, you get an ordinary ray of light, as we know it as a causal connection between two events in the special theory of relativity. It should be noted that in the Twistor theory, it is not the events that represent the elementary entities , but their causal connection through light rays. Events are seen as secondary constructs in the Twistor theory. So are z. B. Events in special relativity at the apex of two causality cones. In the Twistor theory, this fact is now reinterpreted and an event is interpreted as the intersection of a group of special light rays. If one transforms the family of light rays, which lie on the causality cone, into the twistor space, one obtains a Riemann sphere in the twistor space in the twistor image.

Mathematical basics of twistor geometry

The idea of ​​Twistor geometry consists in transferring well-known objects and properties of the special theory of relativity and quantum mechanics into the Twistor language and analyzing them with the mathematical possibilities that exist in Twistor space. The correspondence between Twistor space and Minkowski space is described by the Twistor equation: The mathematical structure on which the Twistor space is based is a four-dimensional vector space over the body of complex numbers with the signature 0. The vectors of the Twistor space are called Twistors.

The Twistor Equation

Given a point in the Minkowski space . In the standard base, this point has the coordinates . The twistor space is now a four-dimensional complex vector space . In standard coordinates, an element of this space has four complex coordinates . The twistor coincides with the space-time point if the following relation is fulfilled: ${\displaystyle R}$ ${\displaystyle \mathbb {M} }$ ${\displaystyle (t,x,y,z)}$ ${\displaystyle (Z^{0},Z^{1},Z^{2},Z^{3})}$ ${\displaystyle \mathbf {T} }$ ${\displaystyle R}$

${\displaystyle {\begin{bmatrix}Z^{0}\\Z^{1}\\\end{bmatrix}}={\frac {i}{\sqrt {2}}}{\begin{bmatrix}t+z&x+iy\\x-iy&t-z\\\end{bmatrix}}{\begin{bmatrix}Z^{2}\\Z^{3}\\\end{bmatrix}}}$

All further principles of the Twistor theory can be derived from this basic equation.

The complex conjugation and dual twistors

A dual twistor can be constructed through the complex conjugation of a twistor . The components of the dual Twistors in the standard representation are: . ${\displaystyle T^{*}}$ ${\displaystyle (Z_{0}^{*},Z_{1}^{*},Z_{2}^{*},Z_{3}^{*})=({\overline {Z^{2}}},{\overline {Z^{3}}},{\overline {Z^{0}}},{\overline {Z^{1}}})}$

The Hermitian scalar product

A Hermitian scalar product can be introduced into the twistor space through the complex conjugation of a twistor. This dot product induces a norm and has the signature . A twistor coincides with a space-time point in Minkowski space if the norm of the twistor disappears. ${\displaystyle (+,+,-,-)}$ ${\displaystyle {\overline {T}}_{\alpha }T^{\alpha }=0}$

Twistors and the special theory of relativity

A twistor can be broken down into its spinorial parts, both of which are 2- spinors . The complex conjugation of the twistor gives . The correspondence of a twistor and a space-time point can now be written as ${\displaystyle \mathbf {T} }$ ${\displaystyle \mathbf {T} =(\mathbf {\omega } ,\mathbf {\pi } )}$ ${\displaystyle \mathbf {\omega } ,\mathbf {\pi } }$ ${\displaystyle \mathbf {T} }$ ${\displaystyle {\overline {\mathbf {T} }}=({\overline {\mathbf {\pi } }},{\overline {\mathbf {\omega } }})}$ ${\displaystyle \mathbf {T} }$ ${\displaystyle R}$

${\displaystyle \mathbf {\omega } =i\mathbf {r} \mathbf {\pi } }$,

where the coordinates of are given in the following matrix notation: ${\displaystyle R}$

${\displaystyle \mathbf {r} ={\begin{bmatrix}r^{00'}&r^{01'}\\r^{10'}&r^{11'}\\\end{bmatrix}}={\frac {1}{\sqrt {2}}}{\begin{bmatrix}t+z&x+iy\\x-iy&t-z\\\end{bmatrix}}}$

The momentum of a massless particle can be expressed by the external product . Furthermore, the angular momentum of the particle with respect to the coordinate zero point can be calculated from the spinoral components. The helicity of a particle can also be calculated from these quantities . ${\displaystyle {\overline {\mathbf {\pi } }}\mathbf {\pi } }$

Twistors and Quantum Mechanics

The quantization in the Twistor theory is given by a commutator relation:

${\displaystyle \mathbf {T} ^{\alpha }{\overline {\mathbf {T} }}_{\beta }-{\overline {\mathbf {T} }}_{\beta }\mathbf {T} ^{\alpha }=\hbar \delta _{\beta }^{\alpha }}$

A twistor wave function has the shape . A twistor wave function is required to be independent of . For the twistor wave functions, this leads to the criterion that . Formally, this is synonymous with the fact that the twistor wave function fulfills the Cauchy-Riemann condition , which in turn means that the twistor wave functions are holomorphic functions of . ${\displaystyle f(\mathbf {T} ^{\alpha })}$ ${\displaystyle {\overline {\mathbf {T} }}_{\alpha }}$ ${\displaystyle {\frac {\partial }{\partial {\overline {\mathbf {T} }}_{\alpha }}}f(\mathbf {T} ^{\alpha })=0}$ ${\displaystyle \mathbf {T^{\alpha }} }$

The complex conjugate twistor variable thus functions as a differentiation: ${\displaystyle {\overline {\mathbf {T} }}_{\alpha }}$

${\displaystyle {\overline {\mathbf {T} }}_{\alpha }\mapsto -\hbar {\frac {\partial }{\partial \mathbf {T} ^{\alpha }}}}$

Helicity

The symmetrized helicity operator is

${\displaystyle s={\frac {1}{4}}(\mathbf {T} ^{\alpha }{\overline {\mathbf {T} }}_{\alpha }+{\overline {\mathbf {T} }}_{\alpha }\mathbf {T} ^{\alpha })\mapsto -{\frac {1}{2}}\hbar \left(2+\mathbf {T} ^{\alpha }{\frac {\partial }{\partial \mathbf {T} ^{\alpha }}}\right)}$

The operator is the so-called homogeneity operator . It has the property that its eigenvalues precisely indicate the degree of homogeneity of the function to which it is applied. If the helicity of a particle is known, the degree of homogeneity that a twistor function must have in order to have a corresponding particle can be calculated from this : ${\displaystyle \mathbf {T} ^{\alpha }{\frac {\partial }{\partial \mathbf {T} ^{\alpha }}}}$ ${\displaystyle H}$

${\displaystyle H=-2S-2}$

Overview of the homogeneity of the particle families

Particle type	Helicity ${\displaystyle S}$	Homogeneity of the particle	Homogeneity of the antiparticle
photon	${\displaystyle \pm 1}$	${\displaystyle 0}$	${\displaystyle -4}$
Neutrino (massless)	${\displaystyle \pm {\tfrac {1}{2}}}$	${\displaystyle -1}$	${\displaystyle -3}$
Scalar particles	${\displaystyle 0}$	${\displaystyle -2}$	${\displaystyle -2}$
Graviton	${\displaystyle \pm 2}$	${\displaystyle 2}$	${\displaystyle -6}$

Important terms related to the twistor theory

• Spinor
• Minkowski room
• Projective geometry
• symmetry
• Complex numbers
• Conjugation (math)

literature

• R. Penrose, W. Rindler : Spinors and space-time. Volume 1: Two-Spinor Calculus and Relativistic Fields. In: Cambridge Monographs on Mathematical Physics. Cambridge University Press, ISBN 0-521-33707-0 .
• R. Penrose, W. Rindler: Spinors and space-time. Volume 2: Spinor and Twistor Methods in Space Time Geometry. In: Cambridge Monographs on Mathematical Physics. Cambridge University Press, ISBN 0-521-34786-6 .
• RS Ward, Raymond O. Wells Jr .: Twistor Geometry and Field Theory . Cambridge University Press, ISBN 0-521-26890-7 .
• Maciej Dunajski: Solitons, instantons, and twistors. Oxford Univ. Press, Oxford 2010, ISBN 978-0-19-857062-2 .

Web links

• Penrose about the origins of twistor theory (English)