Twistor, Cohomology, Foundations of Physics

Twistor, Cohomology, Foundations of Physics

                                     Three Lectures by                     

 Tian Yu Cao (Boston University)

Abstract:

A twistorial framework for fundamental physics is outlined, in which (1) spacetime is derived from twistorial constructions; (2) physical agents are holomorphic functions defined on twistor space, understood as elements of Cech sheaf cohomology groups and appearing in spacetime as fields and particles, (2a) whose dynamics is encoded in the holomorphic structures and revealed through Penrose transforms and (2b) whose scattering processes are describable by twistor diagrams or correlators of twistor operators; (3) the nonlinearity of fundamental interactions (gravity and non-abelian gauge interactions) is the manifestation of nonlinear self-interactions of twistor agents through the deformation of twistor space effected by the agents.

A consistent framework of quantum gravity is attainable within the twistor formalism because (1) gravity can be most efficiently handled by the formalism and, most importantly, (2) the formalism can be proven intrinsically quantal in nature, from which the Planck constant and the whole quantum edifice built thereupon can be derived.

Many technical and conceptual details have to be clarified and fixed before the formalism can be turned into an effective and active research program, but the foundational pillars of the program provided by the formalism are rock solid. 

First Lecture: Foundational Framework for Physics.

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(1)  Einstein’s foundational thinking: A foundational framework, consisting of mathematical formalism and its physical interpretation guided by heuristic principles, defines what physics is and how it is evolving.

(2) Foundational frameworks of general relativity (local, relational and dynamic spacetime and its underpinning, continuous manifold) and the standard model (quantum field theory and its two pillars: Minkowski spacetime and the quantum postulate). 

(3)  No consistent framework for quantum gravity due to the irreconcilable conflict between the constitutive principles of general relativity (background independence) and quantum theory (discreteness).

(4)  Einstein’s encoding-structuralist view of spacetime.

(5)  The quantum postulate. Historically, (a) the notion of quantum was originated from a hypothesis, for understanding electrodynamic phenomena (black body radiation, photoelectric effect, specific heat, atomic spectrum, etc.), about the indivisibility of basic cells in phase space; (b) physical explanations of the notion were attempted but all failed, rendering it a non-disputable universally valid postulate. Conceptually, (a) the notion of quantum is definable and applicable only in the framework of phase space, or even richer ones, rather than the thinner one of spacetime; and (b) the root cause of duality and many other puzzling features in quantum theory may lie in the inappropriate attempts at making the essentially phase space phenomena intelligible in terms of spacetime notions.

Second Lecture: Twistor: Reformulation and New Openings

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(1) Spinor formalism guided by holomorphic principle.

1.  Klein representation and its further generalizations (complexification and compactification); the derivation of spacetime from twistor construction.
2.  Penrose transforms: the cohomological origin of spacetime fields (as solutions to wave equations)
3. Nonlinear graviton construction: the deformation of twistor space by cohomology elements explains the nonlinearities of gravity and nonabelian gauge fields.
4. Twistor diagrams replacing the Feynman diagrams for physical processes.

Third Lecture: Twistor: A New Foundational Framework for Physics.

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(1) Core ingredients of the twistor frame: holomorphicity principle, spinor-twistor formalism and its physical interpretation (describing physically primary massless, spin-half, self-interacting physical degrees of freedom).

(2) Why twistor formalism is intrinsically quantal in nature? (a) Spin, a new puzzlement: From the Einstein-de Haas experiments of 1915-16 to the Stern-Gerlach ones of 1921-22 to Pauli’s puzzlement of 1924. (b) The above developments entailed the existence of something later named spin, but was, contrary to widely spread misconception, completely independent from Planck’s quantum and Dirac’s relativistic theory of electron. (c) A symplectic manifold, such as the twistor space, with spinor structures for the description of spin has indivisible volume units. (d) The underlying discreteness of such symplectic-Poissonian manifolds entails and supports non-commutative algebras, which are continuous to the Poisson algebra (in terms of Poisson brackets) or as its deformations, with the observables defined on the manifold being turned to operators by the machinery similar to the machinery of geometric quantization.

(3) How to derive the Planck constant and all of its offspring? Conceptually it is almost trivial if we mobilize symplectomorphism, Liouville’s theorem and Darboux’s theorem although the technical works are extremely tedious! 

(4) The last ingredient for quantum gravity or the quantum twistor theory: how to define operator product expansions in the non-local twistor framework while all the existing formulations for OPEs, the string formulation included, are based on the notion of locality?

(5)  Recent developments in revealing the hidden structures of gluon/graviton scattering amplitudes, in terms of amplituhedron, Grassmanians and holography, and their relationship with notions originated from strings.

(6) Discussions.      

附注：由于本讲座只介绍主要概念和总体思路，不涉细节，如果有心学者（1）能事先熟悉下面所附文献，或将有助于对本讲座内容的理解和判断；（2）对本讲座中涉及的数学-物理-方法论-形而上学等方面的细节感兴趣，可与主办单位或主讲者联系，以便安排个别讨论或小型工作坊。

For an overview of twistor, see (2, 3, 28, 29, 42, 43, 48); for the basics of twistor, see (19, 24-27, 30, 31, 34, 44); for twistor and cohomology, see (12, 19, 31); for twistor diagram, see (15-18); for recent rapid developments, see (1, 4-11, 13, 20-23, 31-33, 35-40); for twistor and quantum, see (41, 45-47). More relevant references will be given in PPTs at lectures.

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