From: mk_thisisit
Twistor theory, described by Roger Penrose, its developer, as “one of the most revolutionary theories in physics” [01:30:40], explores the geometry of space-time in a unique way. It is a mathematical theory that provides a different perspective on reality [02:24:00].
Origins and Development
Twistor theory was developed in the 1960s by Roger Penrose [01:32:00]. Penrose obtained his doctorate in Cambridge and conducted research in Princeton before spending six months at the University of Syracuse in 1961-1962 [04:52:00]. While at Syracuse, he shared an office with Engelbert Singi, who explained two important aspects of quantum field theory that were crucial to the development of twistor theory [05:37:00]:
- Separation of fields into positive and negative frequencies: This complex procedure involves dividing fields into frequencies and selecting only positive ones [05:46:00].
- Conformal invariance of Maxwell’s equations: Maxwell’s equations describe electricity, magnetism, and light, and their conformal invariance means they are “regardless of scale” [06:16:00] [07:03:00]. This concept, similar to M.C. Escher’s paintings where shapes retain their form while getting smaller towards the edge [07:09:00], was very important as it allowed for the study of radiation and fields at infinity [07:43:00]. Conformal invariance is particularly useful for massless objects like photons [08:06:00].
The initial idea for twistor theory was to geometrically represent positive and negative frequencies using conformal invariance, a problem that standard methods could not solve [09:16:00] [09:44:00].
Core Concepts
Space-Time and Light Rays
Unlike traditional physics where a point in four-dimensional space-time (three spatial, one temporal) is the basic unit [02:46:00], twistor theory posits that the basic element is a ray of light [03:13:00]. This ray can be seen as the history of a photon, forming a “zero line” or “isotropically traveling at the speed of light” line in space-time [03:17:00].
What is a Twistor?
A twistor is more than just the trajectory of a photon [03:38:00]. The concept includes other aspects such as a photon’s:
- Momentum [03:47:00]
- Angular momentum [03:47:00]
- Rotation (helix and spin) [03:56:00]
When a photon’s spin direction (clockwise or counterclockwise) is considered, it is not perfectly localized but spreads and twists around itself [04:09:00]. This “twisted configuration” gave the theory its name [04:23:00].
The Role of Complex Numbers
Complex numbers are “absolutely fundamental” and “crucial for Quantum Mechanics” [10:33:00]. They combine a real number with an imaginary number (involving the square root of -1) [11:07:00]. The problem of positive and negative frequencies can be understood well within the context of complex numbers, visualized on a complex plane [11:49:00]. Twistor theory is deeply rooted in the mathematics of complex numbers [12:47:00].
Twistor Space and Helicity
Originally, twistor theory primarily dealt with massless objects [13:37:00]. It allowed for an elegant description of positive and negative frequencies [13:40:00]. While simple in one dimension (functions extending to one half of the complex plane are positive frequencies, the other half are negative) [14:02:00], applying this to four-dimensional space-time is challenging [14:34:00].
Penrose observed a “fundamental confusion” [02:05:00] in twistor theory: the concept of positive and negative frequencies became intertwined with right-handed and left-handed helicity (spin) of the photon [02:39:00]. Initially, twistor space was related to positive/negative frequencies, but later shifted to right-handed/left-handed helicity [02:52:00].
To resolve this, the concept of dual twistors was introduced [03:17:00]. These are similar to twistors but are “reversed” regarding frequency/helicity association [03:29:00]. The combination of twistors and dual twistors forms a new structure called borons [03:49:00] [03:17:00].
Connection to Geometry
Twistor theory is specifically adapted to space-time with a Minkowski signature, meaning one dimension of time and three dimensions of space (pseudo-Ranowski geometry) [02:11:00] [02:36:00]. This differs from mathematicians’ preference for four spatial dimensions (proper Ranowski geometry) [02:48:00]. Minkowski geometry, which Einstein initially disliked but later found key to his work, describes special relativity based on this 1-time, 3-space signature [02:50:00].
Implications and Applications
Quantum Gravity and Gravitons
A significant application of twistor theory is in the realm of quantum gravity [02:33:00]. When developing a curved twistor theory to fit general relativity, Penrose created the concept of the nonlinear graviton [02:19:00]. Gravitons are hypothesized gravitational particles, analogous to photons for the electromagnetic field [02:41:00]. The theory describes gravitons consistent with general relativity, meaning they are curved, not flat [02:55:00].
The “fundamental confusion” between torsion (spin) and frequency means that the initial formulation of twistor theory only described one half of the gravitational structure, specifically left-handed gravitons [03:06:00] [03:42:00]. To create a proper quantum theory of gravity, both left-handed and right-handed helicities are needed [03:52:00] [03:52:00].
The idea of combining the normal (left-handed) twistor theory with the dual (right-handed) twistor theory into “borons” introduces quantum mechanics, similar to how position and momentum are canonically coupled in quantum mechanics via Heisenberg’s uncertainty principle [03:58:00]. This approach aims to unify gravity theory and quantum theory [02:35:00].
Particle Physics
Penrose speculates that the “b-twistor theory” (which includes both twistors and dual twistors) could relate to the strong force, which binds particles together [04:24:00] [04:28:00]. This connection involves concepts like quaternions and octonions. Quaternions, discovered by Hamilton, involve an algebra where AB does not equal BA [04:49:00]. Octonions go a step further with an even more complicated structure where associativity does not hold (ABC is not equal to A(BC)) [04:52:00]. Penrose has explored whether specific types of octonions (with four pluses and four minuses) are related to twistor theory, concluding they have connections to the boron theory [04:57:00].
Comparison to Other Theories
Penrose views twistor theory as the “most important new idea concerning space-time geometry after Einstein” [01:49:00]. He expresses skepticism about string theory due to its reliance on higher dimensions (initially 26, then 10, now 11) [01:08:00]:
“In my opinion the main problem with string theory is that it doesn’t focus very carefully on the dimensionality of spacetime at first the idea of strings really appealed to me when I heard about it but then it turned out that it only works in 26 dimensions for me that’s the end in my opinion it’s the wrong number of dimensions later they reduced it to 10 dimensions which seemed like an improvement but it’s still far from the four dimensions that are needed for spacetime so don’t get excited about string theory when I found out that we have to consider it in 10 or 26 dimensions at the same time it was even more confusing to me it didn’t make geometric sense now there are ways of dealing with this string theory but they’re not very natural and they don’t really fit into the geometry of spacetime in a coherent way so after the dimensionality became too big I gave up on it although the ideas were initially attractive to me I gave up when the idea of 10 and 26 dimensions at the same time came up and now 11 all that seems to me Not focusing enough on the reality that we live in that we live in has four three spatial dimensions and one time dimension it is four dimensional without any particular sign you see them and TR this is also important because the twistor theory relates very directly to space-time with one dimension of time and three dimensions of space…” [01:08:00]
In contrast to string theory, twistor theory directly relates to the four-dimensional space-time of our reality (three spatial dimensions and one time dimension) [01:52:00].
Current Status and Future Directions
Twistor theory is considered a highly developed field [03:02:00]. While the connection to general relativity in appropriately curved space-time is still developing [03:33:00], Penrose believes there are “good premises” for achieving this with the b-twistor theory [04:49:00]. This would allow for the consideration of both left-handed and right-handed gravitons simultaneously, potentially leading to a more complete quantum theory of gravity [04:36:00].
The theory also continues to be developed by mathematicians, sometimes in directions that diverge from Penrose’s original physical motivations, such as exploring four spatial dimensions instead of one time and three space [02:44:00]. Key mathematical constructs, like “cohomology,” are essential for describing fields such as electromagnetism within twistor theory [02:25:00].
Penrose acknowledges that much of the future application of twistor theory to particle physics and quantum aspects of general relativity remains speculative [04:22:00] [04:06:00]. However, he maintains that the insights offered by twistor theory into the fundamental structure of space-time and its potential for unification are profound [04:24:00].