Twistor theory

The twistor theory is an attempt to create a unified theory of gravitation and quantum field theory. The basic ideas of twistor theory go back to 1967 and were developed by the British mathematician and physicist Roger Penrose. The theory emerged from the studies of spin networks. An important feature of twistor theory is its mathematical character. She has to date not yielded fundamental explanations for measurable quantities. The twistor theory is still no established physical theory, but has been found in mathematics variety of applications.

• 6.1 helicity
• 6.2 Overview of the homogeneity of the particle families

The twistor theory and classical theories of gravity and quantum

In essence, the twistor theory attempts to merge the basic mathematical properties of the theory of relativity and quantum mechanics. In the case of relativity theory are the Minkowski space and its curvilinear generalization, called Riemannian manifolds with signature 1, both of which have four dimensions. In the case of quantum mechanics are the complex numbers, to which the non-local nature of quantum theory are due (eg Einstein - Podolsky-Rosen paradox). The twistor theory is characterized by many considerations of symmetry and mathematical elegance. In the twistor theory is now trying to analyze through a reinterpretation in the context of twistor geometry of the most fundamental aspects of relativity and quantum mechanics from a new perspective.

Basic intuition: The fundamental properties of twistor theory

The elementary properties of twistor theory are the twistors. Transforming a twistor from the twistor space in the Minkowski space, we obtain an ordinary beam of light, as he is known as a causal connection between two events in special relativity theory. It should be noted that not the events constitute the basic entities in the twistor theory, but their causal connection by light rays. Events are now perceived in the twistor theory as secondary constructs. Thus, there are, for example, events in special relativity theory at the head of two causality cone. In the twistor theory now and this must be reinterpreted and interprets an event as the intersection of a particular group of light rays. Transforming the band of light rays that lie on the causality cone, in the twistor space, we obtain the twistor picture a Riemann sphere in twistor space.

Mathematical foundations of twistor geometry

The idea of ​​twistor geometry is now to familiar objects and properties of special relativity and quantum mechanics to transfer into the twistor language and analyze with the existing in twistor space mathematical possibilities. The correspondence between twistor space and Minkowski space is described by the twistor equation: The twistor space of the underlying mathematical structure is a four-dimensional vector space over the field of complex numbers with the signature 0 The vectors of the twistor space is called twistors.

The twistor equation

Given a point in Minkowski space. In the standard basis of this point have 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 satisfied:

From this basic equation, all the other basics of twistor theory can be derived.

The complex conjugation and dual twistors

The complex conjugation of a Twistors can construct a dual twistor. The components of the dual Twistors in the standard representation are:.

The Hermitian scalar product

The complex conjugation of a Twistors can be in the twistor space is a Hermitian scalar product launch. This scalar product is a standard and has the signature. A twistor is then exactly consistent with a space-time point in Minkowski space if the norm of the Twistors disappears.

Twistors and the special theory of relativity

A twistor can be decomposed according to its spinor parts, both of 2- spinors are. The complex conjugation of Twistors results. The conformity of a Twistors and a space-time point can be written now as

Where the coordinates of in the following matrix notation are given:

The momentum of a massless particle can be expressed by the exterior product. Next, the angular momentum of the particle with respect to the coordinate zero point of the spinor shares can be calculated. From these quantities, the helicity of a particle can be calculated.

Twistors and quantum mechanics

Quantization in the twistor theory is given by a commutator relation:

A twistor wave function has the form. From a twistor wave function is required that it is independent of. This leads to the twistor wave functions for criterion that. Formally, this is equivalent to saying that the twistor wave function satisfies the Cauchy -Riemann condition, which in turn means that the twistor wave functions are holomorphic functions of.

Thus, the complex conjugate twistor variable acts as a differentiation:

Helicity

The symmetrized Helizitätsoperator is

The operator is called the homogeneity operator. It has the characteristic that its eigenvalues ​​exactly indicative of the degree of homogeneity of the function to which it is applied. Is now the helicity of a particle is known, we can conclude the degree of homogeneity calculate the Twistorfunktion must have to possess a corresponding particles:

Overview of the homogeneity of the particle families

Important terms related to the twistor theory

• Spinor
• Minkowski space
• Projective geometry
• Symmetry
• Complex Numbers
• Conjugation ( Mathematics )