scholarly journals Palatial twistor theory and the twistor googly problem

2015 ◽  
Vol 373 (2047) ◽  
pp. 20140237 ◽  
Author(s):  
Roger Penrose
Keyword(s):  
Holomorphic Functions ◽  
Heisenberg Algebra ◽  
Twistor Theory ◽  
Negative Helicity ◽  
Nonlinear Graviton ◽  
Twistor Description ◽  
Massless Fields

A key obstruction to the twistor programme has been its so-called ‘googly problem’, unresolved for nearly 40 years, which asks for a twistor description of right -handed interacting massless fields (positive helicity), using the same twistor conventions that give rise to left -handed fields (negative helicity) in the standard ‘nonlinear graviton’ and Ward constructions. An explicit proposal for resolving this obstruction— palatial twistor theory —is put forward (illustrated in the case of gravitation). This incorporates the concept of a non-commutative holomorphic quantized twistor ‘Heisenberg algebra’, extending the sheaves of holomorphic functions of conventional twistor theory to include the operators of twistor differentiation.

A POLAR DECOMPOSITION FOR HOLOMORPHIC FUNCTIONS ON A STRIP

2001 ◽  
Vol 33 (3) ◽  
pp. 309-319 ◽  
Author(s):  
KONRAD SCHMÜDGEN
Keyword(s):  
Holomorphic Function ◽  
Real Axis ◽  
Polar Decomposition ◽  
Holomorphic Functions ◽  
Heisenberg Algebra ◽  
Boundary Values ◽  
Operator Representation ◽  
The Real ◽  
Almost Everywhere

Let f be a holomorphic function on the strip {z ∈ [Copf ] : −α < Im z < α}, where α > 0, belonging to the class [Hscr ](α,−α;ε) defined below. It is shown that there exist holomorphic functions w1 on {z ∈ [Copf ] : 0 < Im z < 2α} and w2 on {z ∈ [Copf ] : −2α < Im z < 2α}, such that w1 and w2 have boundary values of modulus one on the real axis, and satisfy the relationsw1(z)=f(z-αi)w2(z-2αi) and w2(z+2αi)=f(z+αi)w1(z)for 0 < Im z < 2α, where f(z) := f(z). This leads to a ‘polar decomposition’ f(z) = uf(z + αi)gf(z) of the function f(z), where uf (z + αi) and gf(z) are holomorphic functions for −α < Im z < α, such that [mid ]uf(x)[mid ] = 1 and gf(x) [ges ] 0 almost everywhere on the real axis. As a byproduct, an operator representation of a q-deformed Heisenberg algebra is developed.

On the twistor description of massive fields

1981 ◽  
Vol 374 (1758) ◽  
pp. 431-445 ◽  
Keyword(s):  
Minkowski Space ◽  
Complex Manifold ◽  
Twistor Space ◽  
Spin Operator ◽  
Dirac Equations ◽  
Penrose Transform ◽  
Twistor Description ◽  
Integral Formulae ◽  
Massless Fields ◽  
Massive Fields

This paper forms a part of the twistor programme whereby constructions of physics on Minkowski space are transferred, it is hoped, to simpler constructions on Penrose’s twistor-space . We show how the Penrose transform may be used to describe solutions of the Dirac equations on Minkowski space in terms of certain cohomology classes on a related five-dimensional complex manifold. This is accomplished along the same lines as the corresponding representation of massless fields. It means that Penrose’s integral formulae for massive fields may be interpreted cohomologically. We also give a brief discussion of the spin operator in twistor space.

scholarly journals twistors and gauge fields

1986 ◽  
Vol 9 (2) ◽  
pp. 209-221
Author(s):  
A. G. Sergeev
Keyword(s):  
Gauge Fields ◽  
Twistor Theory ◽  
Yang Mills ◽  
Geometrical Properties ◽  
Minkowsky Space ◽  
Basic Ideas ◽  
Massless Fields

We describe briefly the basic ideas and results of the twistor theory. The main points: twistor representation of Minkowsky space, Penrose correspondence and its geometrical properties, twistor interpretation of linear massless fields, Yang-Mills fields (including instantons and monopoles) and Einstein-Hilbert equations.

scholarly journals Exotica or the failure of the strong cosmic censorship in four dimensions

2015 ◽  
Vol 12 (10) ◽  
pp. 1550121 ◽  
Author(s):  
Gábor Etesi
Keyword(s):  
Cosmological Constant ◽  
Cosmic Censorship ◽  
Twistor Theory ◽  
Wick Rotation ◽  
Four Dimensions ◽  
Precise Formulation ◽  
Ricci Flat ◽  
Lorentzian Metric ◽  
Strong Cosmic Censorship ◽  
Nonlinear Graviton

In this paper, a generic counterexample to the strong cosmic censor conjecture is exhibited. More precisely — taking into account that the conjecture lacks any precise formulation yet — first we make sense of what one would mean by a "generic counterexample" by introducing the mathematically unambigous and logically stronger concept of a "robust counterexample". Then making use of Penrose' nonlinear graviton construction (i.e. twistor theory) and a Wick rotation trick we construct a smooth Ricci-flat but not flat Lorentzian metric on the largest member of the Gompf — Taubes uncountable radial family of large exotic ℝ4's. We observe that this solution of the Lorentzian vacuum Einstein's equations with vanishing cosmological constant provides us with a sort of counterexample which is weaker than a "robust counterexample" but still reasonable to consider as a "generic counterexample". It is interesting that this kind of counterexample exists only in four dimensions.

A cohomological description of massive fields

1981 ◽  
Vol 378 (1772) ◽  
pp. 141-154 ◽  
Keyword(s):  
Differential Equations ◽  
Cohomology Group ◽  
Rest Mass ◽  
Holomorphic Functions ◽  
Basic Properties ◽  
Free Fields ◽  
Time Required ◽  
Twistor Description ◽  
Positive Frequency ◽  
Massive Fields

In this paper we present a generalization of the theory of the P -transform that encompasses the n -twistor description of massive fields. Attention is devoted to the two-twistor description, for which it is shown that the cohomology group H 2 ( P + 3 x P + 3 , O m, s ( — ξ — 2, — ŋ — 2)) is naturally isomorphic to the space of positive-frequency free fields of mass m and spin s , provided s — 1/2 | ξ — ŋ | is a non-negative integer, and vanishes otherwise. The sheaf O m, s ( — ξ — 2, — ŋ — 2) is a subsheaf of the standard sheaf of twisted holomorphic functions O ( — ξ — 2, — ŋ — 2) on P + 3 x P + 3 and satisfies a pair of differential equations determining the mass and the spin. In establishing these results extensive use is made of a certain class of two-point fields on space-time, required to be of positive frequency and of zero rest mass in each variable separately, and also subject to a condition of definite total mass and total spin. Such fields are of considerable interest in their own right, for example in connection with the theory of twistor diagrams, and in this paper we formulate a number of their basic properties.

On the twistor description of sourced fields

1989 ◽  
Vol 422 (1863) ◽  
pp. 367-385 ◽  
Keyword(s):  
Angular Momentum ◽  
World Line ◽  
Large Family ◽  
Conserved Quantities ◽  
Cohomology Groups ◽  
Relative Cohomology ◽  
Penrose Transform ◽  
Left Handed ◽  
Twistor Description ◽  
Massless Fields

Massless fields with source on an analytic world-line are double-valued, and it was shown by Bailey (1985) that a large family of such fields have a twistor description in terms of relative cohomology groups. In this paper it is proved that all right-handed massless fields are obtained in this way, and that if the sheaves O ( n - 2) are quotiented by the polynomials, then the relative cohomology of the resulting sheaves describes all left-handed sourced massless fields. The proof for right-handed fields uses techniques developed by Singer (1987, 1988) for applying the Penrose transform to situations in which the ‘pull-back mechanism’ is non-trivial. For the left-handed fields it is necessary to use some additional arguments involving the conserved quantities (e. g. momentum and angular momentum for spin 2) of these fields; it is shown that the conserved quantities are the obstructions to a twistor description of left-handed fields in terms of the cohomology of O ( n - 2).

An Introduction to Twistor Theory

1994 ◽  
Author(s):  
S. A. Huggett ◽  
K. P. Tod
Keyword(s):  
Twistor Theory

Bialgebra structures on the Heisenberg algebra

1989 ◽  
Vol 75 (1) ◽  
pp. 315-321
Author(s):  
Michel Cahen ◽  
Christian Ohn
Keyword(s):  
Heisenberg Algebra
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