From: mk_thisisit
Twistor theory, developed by Roger Penrose in the 1960s, is one of the most revolutionary theories in physics [00:01:30]. This theory views the structure of space-time in an unusual way [00:02:41]. While traditional understanding considers space-time as a four-dimensional space (three spatial dimensions and one time dimension) where a point is the basic unit, twistor theory posits that the fundamental element of space-time is a ray of light [00:02:46]. This can be conceptualized as the history of a photon, forming a line in space-time [00:03:17].
Core Concepts of Twistor Theory
A twistor is more complex than just a light ray; it also accounts for the momentum and angular momentum of the photon, including its rotation (helix and spin) [00:03:47]. When a photon has spin, it is not precisely localized, but rather spreads out and twists around itself, giving the theory its name [00:04:09].
The initial idea for twistor theory stems from two main concepts explained to Penrose by Engelbert Sching in 1961-1962 at the University of Syracuse [00:04:42].
Quantum Field Theory and Frequencies
One idea concerned quantum field theory and the separation of fields into positive and negative frequencies [00:05:37]. This complicated procedure, known as decomposition, involves dividing fields into frequencies and selecting only positive ones [00:05:56].
Conformal Invariance of Maxwell’s Equations
The second idea was the conformal invariance of Maxwell’s equations [00:06:14]. These equations describe electricity, magnetism, and light, and are significant for explaining the phenomenon of light [00:06:20]. Conformal invariance means that the equations are independent of scale; large and small things are treated equally [00:07:03]. This concept is particularly useful for describing massless objects like photons [00:08:06].
The Role of Complex Numbers
Complex numbers are fundamental to Quantum Mechanics, as the wave function of any system is described by complex numbers [00:10:30]. The problem of positive and negative frequencies can be understood using complex numbers, specifically within the complex plane [00:11:49]. Functions extending to one half of the complex plane represent positive frequencies, and those to the other half represent negative frequencies [00:14:09].
Twistor Theory and Space-Time Geometry
Twistor theory was originally developed to elegantly describe positive and negative frequencies for massless objects [00:13:31]. It is particularly adapted to space-time geometry with one dimension of time and three dimensions of space, known as Minkowski signature or pseudo-Riemannian geometry [00:21:21]. This specific geometry is crucial for Einstein’s theory of general relativity, which incorporates gravity by describing a curved Minkowski space [00:26:16].
According to a statement from Columbia University, Penrose’s work on twistors is considered “the most important new idea concerning space-time geometry after Einstein” [00:17:46].
Unification of Gravity and Quantum Theory
A significant goal for twistor theory is to unify gravity theory and quantum theory [00:22:46]. Professor Maciej Dunajski of the University of Cambridge, a collaborator of Penrose, indicated a breakthrough in twistor theory that could lead to a positive solution for combining these theories [00:22:15].
The Challenge of Gravitons
In Quantum Mechanics, particles like photons have both wave and particle properties [00:28:58]. The idea is that gravity, too, should have a particle form (gravitons) and a wave form (gravitational fields) [00:29:12]. However, gravitons have not yet been directly observed due to the extremely high energies required [00:29:22].
Penrose’s work on “nonlinear gravitons” was the first instance where curved twistor theory and curved twistor space were introduced [00:28:10]. This allows for incorporating the spin of the graviton [00:28:33]. However, a “fundamental confusion” in twistor theory arose because torsion (spin) and frequency mixed, meaning that early developments could only describe left-handed gravitons, not right-handed ones [00:30:04].
Boron Theory and Handedness
To address this, Penrose recently developed the idea of “borons” at a conference in Cambridge, which involves combining the standard twistor theory (inherently left-handed) with the dual twistor theory (right-handed) [00:31:41]. This combination is crucial for a proper quantum theory, as both helicities (left-handed and right-handed) are needed [00:36:49].
Weak interactions, as described by Chen-Ning Yang and Tsung-Dao Lee (Nobel laureates), are unique among physical theories because they are not left-right symmetric, unlike most of physics [00:37:33]. Penrose’s work aims to make twistor theory less torsion-dependent, allowing it to describe gravity, which is not a handed theory [00:38:27].
Quantum Mechanical Coupling
Combining these two “sides” of twistor theory introduces Quantum Mechanics. It’s analogous to position and momentum in Quantum Mechanics, which are canonically coupled [00:39:33]. In Quantum Mechanics, if a particle’s position is well-defined, its momentum is undefined, and vice-versa, a concept fundamental to Heisenberg’s work [00:39:55]. Similarly, twistors and dual twistors are canonically conjugated [00:41:21].
Future Implications and Speculation
Penrose believes that the “B-twistor theory,” which includes both twistor and dual twistor, could allow for the simultaneous consideration of left-handed and right-handed gravitons and facilitate the inclusion of general relativity [00:43:31]. While this is still speculation, it presents a premise for understanding the quantum aspects of general relativity [00:44:00].
Penrose once discussed his ideas about particle physics with Richard Feynman, who advised him to continue working on twistor theory rather than pursue the particle physics ideas at that time [00:44:34]. However, Penrose now believes there is a good chance that particle physics, particularly the strong force, could fit into the B-twistor theory [00:45:09]. This connection might involve a “triple product” construction related to quaternions and octonions, which are non-commutative algebraic structures [00:46:17].
While mathematics can be infinitely complicated, physics is interested only in certain aspects of it [00:52:27]. The structure of space-time and particle physics, with three spatial and one time dimension, suggests a finite complexity in nature [00:52:07].