theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...

, twistor theory was proposed by

Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus fello ...

in 1967 as a possible path to

quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...

and has evolved into a branch of

theoretical A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...

and

mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...

. Penrose proposed that twistor space should be the basic arena for physics from which space-time itself should emerge. It leads to a powerful set of mathematical tools that have applications to differential and

integral geometry In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformati ...

nonlinear differential equation In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...

s and

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

and in physics to

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

and

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...

, in particular to

scattering amplitude In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.Einstein–Cartan–Sciama–Kibble theory.

Overview

Mathematically, projective twistor space

\mathbb

is a

3-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

, complex projective 3-space

\mathbb^3

. It has the physical interpretation of the space of

massless particle In particle physics, a massless particle is an elementary particle whose invariant mass is zero. There are two known gauge boson massless particles: the photon (carrier of electromagnetism) and the gluon (carrier of the strong force). However, g ...

s with spin. It is the

projectivisation In mathematics, projectivization is a procedure which associates with a non-zero vector space ''V'' a projective space (V), whose elements are one-dimensional Linear subspace, subspaces of ''V''. More generally, any subset ''S'' of ''V'' closed un ...

of a 4-dimensional

complex vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

, non-projective twistor space

\mathbb

with a

Hermitian form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...

signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...

(2,2) and a

holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...

volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the ...

. This can be most naturally understood as the space of

chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...

(

Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is ass ...

)

spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...

s for the

conformal group In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space. Se ...

SO(4,2)/\mathbb_2

Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...

; it is the

fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defini ...

of the

spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a L ...

SU(2,2)

of the conformal group. This definition can be extended to arbitrary dimensions except that beyond dimension four, one defines projective twistor space to be the space of projective pure spinors for the conformal group. In its original form, twistor theory encodes

physical field In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. For example, on a weather map, the surface temperature is described by assigning a number to each point o ...

s on Minkowski space into

complex analytic Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...

objects on twistor space via the

Penrose transform In theoretical physics, the Penrose transform, introduced by , is a complex analogue of the Radon transform that relates massless fields on spacetime to sheaf cohomology, cohomology of sheaf (mathematics), sheaves on complex projective space. The p ...

. This is especially natural for massless fields of arbitrary spin. In the first instance these are obtained via

contour integral In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. ...

formulae in terms of free holomorphic functions on regions in twistor space. The holomorphic twistor functions that give rise to solutions to the massless field equations can be more deeply understood as

Čech Čech (feminine Čechová) is a Czech surname meaning Czech. It was used to distinguish an inhabitant of Bohemia from Slovaks, Moravians and other ethnic groups. Notable people with the surname include: * Dana Čechová (born 1983), Czech tabl ...

representatives of analytic cohomology classes on regions in

\mathbb

. These correspondences have been extended to certain nonlinear fields, including

self-dual In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involutio ...

gravity in Penrose's

nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...

graviton In theories of quantum gravity, the graviton is the hypothetical quantum of gravity, an elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathem ...

construction and self-dual Yang–Mills fields in the so-called Ward construction; the former gives rise to deformations of the underlying complex structure of regions in

\mathbb

, and the latter to certain holomorphic vector bundles over regions in

\mathbb

. These constructions have had wide applications, including inter alia the theory of

integrable systems In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...

. The self-duality condition is a major limitation for incorporating the full nonlinearities of physical theories, although it does suffice for Yang–Mills–Higgs monopoles and

instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. Mo ...

s (see

ADHM construction In mathematical physics and gauge theory, the ADHM construction or monad construction is the construction of all instantons using methods of linear algebra by Michael Atiyah, Vladimir Drinfeld, Nigel Hitchin, Yuri I. Manin in their paper "Constru ...

). An early attempt to overcome this restriction was the introduction of ambitwistors by

Edward Witten Edward Witten (born August 26, 1951) is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, q ...

and by Isenberg, Yasskin & Green. Ambitwistor space is the space of complexified light rays or massless particles and can be regarded as a complexification or cotangent bundle of the original twistor description. These apply to general fields but the field equations are no longer so simply expressed. Twistorial formulae for

interactions Interaction is action that occurs between two or more objects, with broad use in philosophy and the sciences. It may refer to: Science * Interaction hypothesis, a theory of second language acquisition * Interaction (statistics) * Interactions o ...

beyond the self-dual sector first arose from Witten's twistor string theory. This is a quantum theory of holomorphic maps of a

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

into twistor space. It gave rise to the remarkably compact RSV (Roiban, Spradlin & Volovich) formulae for tree-level S-matrices of Yang–Mills theories, but its gravity degrees of freedom gave rise to a version of conformal

supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...

limiting its applicability;

conformal gravity Conformal gravity refers to gravity theories that are invariant under conformal transformations in the Riemannian geometry sense; more accurately, they are invariant under Weyl transformations g_\rightarrow\Omega^2(x)g_ where g_ is the metric te ...

is an unphysical theory containing

ghosts A ghost is the soul or spirit of a dead person or animal that is believed to be able to appear to the living. In ghostlore, descriptions of ghosts vary widely from an invisible presence to translucent or barely visible wispy shapes, to rea ...

, but its interactions are combined with those of Yang–Mills theory in loop amplitudes calculated via twistor string theory. Despite its shortcomings, twistor string theory led to rapid developments in the study of scattering amplitudes. One was the so-called MHV formalism loosely based on disconnected strings, but was given a more basic foundation in terms of a twistor action for full Yang–Mills theory in twistor space. Another key development was the introduction of BCFW recursion. This has a natural formulation in twistor space that in turn led to remarkable formulations of scattering amplitudes in terms of Grassmann integral formulae and

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

s. These ideas have evolved more recently into the positive

Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the ...

and amplituhedron. Twistor string theory was extended first by generalising the RSV Yang–Mills amplitude formula, and then by finding the underlying

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...

. The extension to gravity was given by Cachazo & Skinner, and formulated as a twistor string theory for maximal supergravity by David Skinner. Analogous formulae were then found in all dimensions by Cachazo, He & Yuan for Yang–Mills theory and gravity and subsequently for a variety of other theories. They were then understood as string theories in ambitwistor space by Mason & Skinner in a general framework that includes the original twistor string and extends to give a number of new models and formulae. As string theories they have the same

critical dimension In the renormalization group analysis of phase transitions in physics, a critical dimension is the dimensionality of space at which the character of the phase transition changes. Below the lower critical dimension there is no phase transition. ...

s as conventional string theory; for example the type II supersymmetric versions are critical in ten dimensions and are equivalent to the full field theory of type II supergravities in ten dimensions (this is distinct from conventional string theories that also have a further infinite hierarchy of massive higher spin states that provide an ultraviolet completion). They extend to give formulae for loop amplitudes and can be defined on curved backgrounds.

The twistor correspondence

Denote

M

, with coordinates

x^a = (t, x, y, z)

and Lorentzian metric

\eta_

signature

(1, 3)

. Introduce 2-component spinor indices

A = 0, 1;\; A' = 0', 1',

and set :

x^ = \frac\begin t - z & x + iy \\ x - iy & t + z \end.

Non-projective twistor space

\mathbb

is a four-dimensional complex vector space with coordinates denoted by

Z^ = \left(\omega^,\, \pi_\right)

where

\omega^A

and

\pi_

are two constant

Weyl spinor In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...

s. The hermitian form can be expressed by defining a complex conjugation from

\mathbb

to its dual

\mathbb^*

\bar Z_\alpha = \left(\bar\pi_A,\, \bar \omega^\right)

so that the Hermitian form can be expressed as :

Z^\alpha \bar Z_\alpha = \omega^\bar\pi_ + \bar\omega^\pi_.

This together with the holomorphic volume form,

\varepsilon_ Z^\alpha dZ^\beta \wedge dZ^\gamma \wedge dZ^\delta

is invariant under the group SU(2,2), a quadruple cover of the conformal group C(1,3) of compactified Minkowski spacetime. Points in Minkowski space are related to subspaces of twistor space through the incidence relation :

\omega^ = ix^\pi_.

The incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space

\mathbb,

which is isomorphic as a complex manifold to

\mathbb^3

. A point

x\in M

thereby determines a line

\mathbb^1

\mathbb

parametrised by

\pi_.

A twistor

Z^\alpha

is easiest understood in space-time for complex values of the coordinates where it defines a totally null two-plane that is self-dual. Take

x

to be real, then if

Z^\alpha \bar Z_\alpha

vanishes, then

x

lies on a light ray, whereas if

Z^\alpha \bar Z_\alpha

is non-vanishing, there are no solutions, and indeed then

Z^

corresponds to a massless particle with spin that are not localised in real space-time.

Variations

Supertwistors

Supertwistors are a

supersymmetric In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...

extension of twistors introduced by Alan Ferber in 1978. Non-projective twistor space is extended by

fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...

ic coordinates where

\mathcal

is the number of supersymmetries so that a twistor is now given by

\left(\omega^A,\, \pi_,\, \eta^i\right), i = 1, \ldots, \mathcal

with

\eta^i

anticommuting. The super conformal group

SU(2,2, \mathcal)

naturally acts on this space and a supersymmetric version of the Penrose transform takes cohomology classes on supertwistor space to massless supersymmetric multiplets on super Minkowski space. The

\mathcal = 4

case provides the target for Penrose's original twistor string and the

\mathcal = 8

case is that for Skinner's supergravity generalisation.

Hyperkähler manifolds

Hyperkähler manifold In differential geometry, a hyperkähler manifold is a Riemannian manifold (M, g) endowed with three integrable almost complex structures I, J, K that are Kähler with respect to the Riemannian metric g and satisfy the quaternionic relations I^2 ...

s of dimension

4k

also admit a twistor correspondence with a twistor space of complex dimension

2k+1

Palatial twistor theory

The nonlinear graviton construction encodes only anti-self-dual, i.e., left-handed fields. A first step towards the problem of modifying twistor space so as to encode a general gravitational field is the encoding of

right-handed In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subjecti ...

fields. Infinitesimally, these are encoded in twistor functions or

cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...

classes of

homogeneity Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the Uniformity (chemistry), uniformity of a Chemical substance, substance or organism. A material or image that is homogeneous is uniform in compos ...

−6. The task of using such twistor functions in a fully nonlinear way so as to obtain a ''

'' nonlinear graviton has been referred to as the (gravitational) googly problem (the word "

googly In the game of cricket, a googly refers to a type of delivery bowled by a right-arm leg spin bowler. It is different from the normal delivery for a leg-spin bowler in that it is turning the other way. The googly is ''not'' a variation of the ...

" is a term used in the game of

cricket Cricket is a bat-and-ball game played between two teams of eleven players on a field at the centre of which is a pitch with a wicket at each end, each comprising two bails balanced on three stumps. The batting side scores runs by striki ...

for a ball bowled with right-handed helicity using the apparent action that would normally give rise to left-handed helicity).Penrose 2004, p. 1000. The most recent proposal in this direction by Penrose in 2015 was based on

noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some ge ...

on twistor space and referred to as palatial twistor theory. The theory is named after

Buckingham Palace Buckingham Palace () is a London royal residence and the administrative headquarters of the monarch of the United Kingdom. Located in the City of Westminster, the palace is often at the centre of state occasions and royal hospitality. It ...

, where

Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded th ...

suggested to Penrose the use of a type of "

noncommutative algebra In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist ''a'' and ''b'' in the ring such that ''ab'' and ''ba'' are different. Equivalently, a ''noncommutative ring'' is a ring that is not ...

", an important component of the theory (the underlying twistor structure in palatial twistor theory was modeled not on the twistor space but on the non-commutative

twistor quantum algebra)."Michael Atiyah's Imaginative State of Mind"
– ''

Quanta Magazine ''Quanta Magazine'' is an editorially independent online publication of the Simons Foundation covering developments in physics, mathematics, biology and computer science. ''Undark Magazine'' described ''Quanta Magazine'' as "highly regarded for ...

''.

Notes

References

(2004), ''

The Road to Reality ''The Road to Reality: A Complete Guide to the Laws of the Universe'' is a book on modern physics by the British mathematical physicist Roger Penrose, published in 2004. It covers the basics of the Standard Model of particle physics, discussing ...

'', Alfred A. Knopf, ch. 33, pp. 958–1009. * Roger Penrose and

Wolfgang Rindler Wolfgang Rindler (18 May 1924 – 8 February 2019) was a physicist working in the field of general relativity where he is known for introducing the term "event horizon", Rindler coordinates, and (in collaboration with Roger Penrose) for the use of ...

(1984), ''Spinors and Space-Time; vol. 1, Two-Spinor Calculus and Relativitic Fields'', Cambridge University Press, Cambridge. * Roger Penrose and Wolfgang Rindler (1986), ''Spinors and Space-Time; vol. 2, Spinor and Twistor Methods in Space-Time Geometry'', Cambridge University Press, Cambridge.

Andrew Hodges Andrew Philip Hodges (; born 1949) is a British mathematician, author and emeritus senior research fellow at Wadham College, Oxford. Education Hodges was born in London in 1949 and educated at Birkbeck, University of London where he was award ...

Summary of recent developments.
* Huggett, Stephen (2005),
The Elements of Twistor Theory.
* Mason, L. J.,
The twistor programme and twistor strings: From twistor strings to quantum gravity?
* * Sparling, George (1999),
On Time Asymmetry.
*

* Universe Review:

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