From: mk_thisisit
Complex numbers are fundamental to understanding quantum mechanics and the structure of space-time in certain theoretical frameworks [00:00:03]. They are crucial for describing various physical phenomena, particularly those involving wave functions and fields [00:10:33].
What Are Complex Numbers?
A complex number is defined by the inclusion of the square root of minus one (i) [00:10:14]. Unlike ordinary mathematics that primarily deals with real numbers, complex numbers combine a real number with an imaginary number [00:11:07]. While often called “imaginary,” these numbers are considered just as “real” in mathematics, though they cannot be directly observed or measured in the same way as real numbers [00:11:16]. They create a “very beautiful algebra and mathematical analysis” that extends the ordinary analysis of real numbers [00:11:34].
Complex numbers are typically represented on a complex plane, also known as the Wessel plane or Gauss plane, where real numbers lie on one axis and imaginary numbers on another [00:11:58]. Each point on this plane represents a unique complex number [00:12:40].
Role in Quantum Mechanics
The entire structure of quantum mechanics depends on complex numbers [00:10:38].
- Wave Functions: The wave function of photons or any quantum system is described as a system of complex numbers [00:10:40].
- Schrödinger Equation: The Singer (Schrödinger) equation, which dictates the evolution of the wave function, is itself a complex equation [00:10:47]. To understand the physics of quantum mechanics, it is necessary to understand complex numbers [00:10:55].
Complex Numbers and Frequencies
Complex numbers are crucial for understanding the concept of positive and negative frequencies in quantum field theory [00:05:40], particularly for phenomena like light described by Maxwell’s equations [00:06:16]. This separation into positive and negative frequencies is a complicated procedure, often involving decomposition [00:05:56].
In the context of the complex plane, functions that extend to one half of the plane represent positive frequencies, while those extending to the other half represent negative frequencies [00:14:06]. This provides a clear geometric separation of these frequencies [00:14:22].
Complex Numbers and Twistor Theory
Twistor theory, developed by Roger Penrose in the 1960s, is one of the most revolutionary theories in physics [00:01:30]. It views the basic element of space-time not as a point, but as a ray of light [00:03:10]. This theory heavily relies on the mathematics of complex numbers [00:12:47].
Conformal Invariance
A key property that led to twistor theory was the conformal invariance of Maxwell’s equations [00:06:14]. Conformal invariance means that equations remain consistent regardless of scale, where large and small things are treated equally [00:07:03]. This concept is particularly useful for describing massless objects like photons, which fit perfectly into a conformal approach [00:08:06]. Twistor theory aims to describe positive and negative frequencies in a conformally invariant way, unlike standard methods that depend on the geometry’s metric [00:09:22].
Space-time and Twistors
While space-time can be made complex, it doesn’t significantly simplify the problem, as it results in an eight-line space [00:00:03]. In twistor theory, the points of space-time are considered secondary constructs, with the basic element being the twistor itself [00:08:45]. Twistors are more complex than simple light rays, incorporating aspects like the photon’s momentum, angular momentum, and spin [00:03:43]. A photon with spin is not entirely localized; it “spreads out” and creates a “twisted swistow configuration” that gives the theory its name [00:04:13].
Twistor theory, in its foundational form, deals with massless objects and is highly adapted to space-time with one time dimension and three spatial dimensions (Minkowski signature) [00:13:31], [00:21:21].
Fundamental Confusion and Dual Twistors
Initially, the division in twistor space was related to positive and negative frequencies [00:27:23]. However, it later became more connected to the handedness (right-handed or left-handed spin) of photons [00:27:41]. This led to what Penrose calls a “fundamental confusion in twistor theory,” where positive frequency and positive helicity occur together [00:28:02], [00:31:04].
To address this, the concept of a “dual twistor” was introduced [00:31:17]. Dual twistors are similar to twistors but are “reversed,” meaning, for example, a positive frequency might correspond to a negative helicity [00:31:29].
Borons and Quantum Gravity
A recent development involves combining these two concepts—twistors and dual twistors—into a structure called “borons” [00:31:41], [00:31:49]. This combination is crucial for introducing quantum mechanics into the framework [00:39:18].
The process is analogous to the canonical coupling of position and momentum in quantum mechanics, a concept central to Heisenberg’s uncertainty principle [00:39:37]. In classical physics, position and momentum operators commute (AB = BA), but in quantum mechanics, they do not (AB ≠ BA) [00:40:30]. The difference (AB - BA) yields something fundamental [00:40:45]. Similarly, twistors and dual twistors are canonically conjugated variables [00:41:21]. By creating an algebra that includes both, it allows for the consideration of both left-handed and right-handed helicities simultaneously, which is necessary for a proper quantum theory of gravity [00:36:52], [00:43:36].
This unified approach holds promise for potentially relating to fundamental forces in particle physics, particularly the strong force, and for integrating general relativity into a quantum framework [00:42:24], [00:43:42].