From: mk_thisisit
Introduction to Twistor Theory
Twistor theory is described as one of the most revolutionary theories in physics, developed in the 1960s by Roger Penrose [01:30:52]. Today, it is experiencing its “Golden Age” [01:38:35]. Roger Penrose, a Nobel laureate in Physics, is considered the “father of twistor theory” [01:41:34]. His work on twistors is regarded as the most important new idea concerning the geometry of space-time after Einstein [01:08:44].
The theory is mathematical [02:38:20] and aims to describe the structure of space-time in an unusual way [02:41:34]. It proposes a different approach to space-time where the basic element is not a point, but a ray of light, which can be seen as the history of a photon [03:07:34]. This ray of light is a line in space-time, a “zero line” or “isotropically traveling at the speed of light” [03:20:34].
Historical Development and Core Concepts
Roger Penrose’s initial idea for twistor theory has a “complicated history” [03:37:34], originating from two main concepts [04:42:34]. During his time at Syracuse University in 1961-1962, he shared an office with Engelbert Singi, a German physicist [05:12:34]. Singi explained two crucial aspects of quantum field theory:
- Separation of fields into positive and negative frequencies: This complicated procedure involves dividing fields into frequencies and selecting only positive ones [05:46:34].
- Conformal invariance of Maxwell’s equations: Maxwell’s equations describe electricity, magnetism, and light, and are fundamental to understanding the phenomenon of light [06:16:34]. Conformal invariance means that these equations are independent of scale, so “large and small things are equal” [07:01:34]. This concept is visually represented in the art of M.C. Escher, where shapes retain their form while getting smaller towards the edge of a circle [07:09:34]. This property is particularly useful for objects with no mass, such as photons [08:06:34].
In twistor theory, the basic element, the twistor, is more complex than a simple light ray [08:55:34]. It takes into account the spin of the photon [08:57:34], which means the photon is not exactly localized but “spreads in a certain special way and twists around itself” [04:13:34]. This twisted configuration is the origin of the name “twistor” [04:23:34].
Spacetime in Twistor Theory
In twistor theory, the conventional notion of space-time as a four-dimensional space (three spatial dimensions and one time dimension) where a point represents a moment, is re-evaluated [02:43:34]. Instead, the points of space-time become a secondary construction [08:45:34], with the twistor as the basic element [08:48:34].
Complex Numbers and Frequencies
Complex numbers are fundamental to quantum mechanics [10:33:34]. The wave function of photons or any system is a system of complex numbers [10:40:34]. The problem of positive and negative frequencies in quantum field theory can be understood using complex numbers and the complex plane [11:49:34]. Functions that extend to one half of the complex plane are positive frequencies, and those that extend to the other half are negative frequencies [14:09:34].
The challenge lies in applying this concept to full four-dimensional space-time [14:34:34]. While one can make space-time complex, it results in an eight-dimensional space, which is not directly helpful [14:52:34]. The theory of twists, however, can elegantly describe positive and negative frequencies for massless objects [13:31:34].
Minkowski Geometry and Spacetime Signature
Spacetime has a specific geometry, referred to as Minkowski signature, which means it has one dimension of time and three dimensions of space [21:21:34]. This is distinct from purely mathematical four-dimensional spaces where all dimensions are spatial [21:46:34].
Minkowski geometry, introduced by Minkowski to understand special relativity [25:34:34], features one time dimension and three spatial dimensions [25:40:34]. Einstein initially disliked this concept but later realized it was crucial for developing his theory of general relativity, as it provided the framework needed to curve geometry and incorporate gravity [25:48:34]. This is known as pseudo-Riemannian geometry [24:29:34]. Twistor theory, in its physics-adapted version, is very naturally aligned with this specific geometry of space-time (one time, three space) [21:38:34].
Twistor Theory, Gravity, and Quantum Mechanics
A significant aspect of twistor theory is its potential to combine the theory of gravity and quantum theory [22:46:34]. Roger Penrose’s work on twistor theory aimed to describe positive and negative frequencies geometrically by dividing space into two halves [33:18:34].
The “Fundamental Confusion” and Gravitons
Initially, the two halves of twistor space were related to positive and negative frequencies [27:20:34]. However, this evolved to relate more to right-handed or left-handed helicity (spin direction) of photons [27:41:34]. This led to what Penrose calls “the fundamental confusion in twistor theory” [28:05:34]: positive frequency and positive helicity occur together [31:04:34].
To integrate twistor theory with general relativity, Penrose developed the “curved twistor theory,” which introduced the concept of the “nonlinear graviton” [28:10:34]. A graviton is a hypothetical gravitational particle, analogous to a photon for the electromagnetic field [28:38:34]. When a curved twistor space is created, the spin of the graviton can be incorporated [28:31:34].
However, existing twistor theory solutions describe only “half” of the gravitons, specifically left-handed ones, due to the mixing of torsion and frequency [30:06:34]. To create a proper quantum theory of gravity, both left-handed and right-handed gravitons are needed [36:49:34].
Borons: Combining Twistor and Dual Twistor Space
To address this, Penrose recently developed the idea of “borons,” a structure that combines the normal twistor theory (which is “left-handed”) and the dual twistor theory (which is “right-handed”) [31:47:34]. This combination is crucial for introducing quantum mechanics [39:18:34].
In quantum mechanics, concepts like position and momentum are canonically conjugated variables, meaning they cannot be simultaneously defined with arbitrary precision (Heisenberg’s uncertainty principle) [39:37:34]. Similarly, twistor and dual twistor spaces are canonically conjugated variables [41:24:34]. By creating an algebra that includes both, it is hoped that the theory can relate to elementary particle physics [42:01:34].
Comparison with String Theory
Penrose expresses skepticism about string theory, primarily due to its high dimensionality [18:06:34]. He finds it problematic that string theory initially required 26 dimensions, and later 10 or 11, which he considers “far from the four dimensions that are needed for space-time” [18:22:34]. He believes it doesn’t make geometric sense or fit coherently into the geometry of space-time [18:54:34]. In contrast, twistor theory relates “very directly to space-time with one dimension of time and three dimensions of space” [19:39:34].
Connections to Other Mathematical Concepts
Twistor theory uses concepts like cohomology [20:22:34], a mathematical construct that plays beautifully with it, especially for describing fields like electromagnetism [20:37:34].
Quaternions and Octonions
The potential connection of twistor theory to particle physics, particularly the strong force, might involve mathematical structures like quaternions and octonions [42:28:34].
- Quaternions: Discovered by Hamilton, quaternions are an algebra where
ABis not equal toBA(non-commutative) [46:51:34]. They involve a “vector product” in three-dimensional space where the product is perpendicular to the plane of the two multiplied vectors [50:03:34]. - Octonions: These go a step further than quaternions [50:50:34]. They are eight-dimensional structures where the associative property (
A(BC) = (AB)C) does not always hold (A(BC)is not equal to(AB)C) [48:03:34]. Penrose explored whether split octonions (those with four pluses and four minuses in their signature) relate to twistor theory, concluding they connect to the boron theory [48:41:34].
Future Directions and Speculation
Penrose speculates that the “b-twistor theory,” which includes both twistor and dual twistor spaces, could allow for the simultaneous consideration of left-handed and right-handed phenomena [43:31:34], and might allow for the consideration of general relativity within the framework [43:42:34]. He believes there are good premises that it could eventually help understand the quantum aspects of general relativity [43:51:34].
While he once had ideas about fitting particle physics into twistor theory, Richard Feynman advised him against it [44:17:34]. However, Penrose now believes there’s a good chance that b-twistor theory could fit particle physics, particularly the strong interactions [45:09:34]. This is partly because the construction involving ‘cocoons’ (likely related to octonions in the boron theory) involves a “triple product” [49:40:34].
Penrose views twistor theory as his most developed theory, even more so than his work on black holes, which received a Nobel Prize due to observational evidence [35:59:34]. The “missing element” in twistor theory is its full connection with general relativity in appropriately curved space-time, which remains a goal for further development [36:25:34].
Ultimately, while mathematics can be infinitely complicated, physics is interested only in certain aspects of mathematics that exist in nature [52:27:34]. Abstract thinking, particularly in these fields, is seen as a “never-ending story” [01:17:34].