Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way".... This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics.
John C. BaezRelated topics
appear classification complex light number quantum real reversal symmetry three treat using way HilbertRelated quotes
At the beginning of this century a self-destructive democratic principle was advanced in mathematics (especially by Hilbert), according to which all axiom systems have equal right to be analyzed, and the value of a mathematical achievement is determined, not by its significance and usefulness as in other sciences, but by its difficulty alone, as in mountaineering. This principle quickly led mathematicians to break from physics and to separate from all other sciences. In the eyes of all normal people, they were transformed into a sinister priestly caste... Bizarre questions like Fermat's problem or problems on sums of prime numbers were elevated to supposedly central problems of mathematics.
Vladimir Arnold
It is essential to realize at this point that the formalism to be developed, although we cast it initially primarily in the framework of natural systems, is in fact applicable to any situation in which a class of objects is associated with real numbers, or in fact classified or indexed by any set whatever. It is thus applicable to any situation in which classification, or recognition, or discrimination is involved; indeed, one of the aims of our formalism is to point up the essential equivalence of the measurement problem in physics with all types of recognition or classification mechanisms based on observable properties of the objects being recognized or classified.
Robert Rosen
First you must have faith in an eternal world independent of you; then you must have faith in your ability to perceive it, and finally you must try to explain it by means of concepts or mathematical constructions. But don't always accept traditional concepts without reexamining them. Even overthrow my relativity theory, if you find a better one.... You must believe that the world was created as a unified whole which is comprehensible to man. Of course, it's going to take an infinitely long time to investigate this unified creation. But to me that is the highest and most sacred duty-unifying physics. Simplicity is the criterion of the universe.
Albert Einstein
Given my interest in the concept of time, it was only natural that my attention was focused on the second principle, as I felt from the start that it would introduce a new, unexpected element into the description of physical world evolution. No doubt it was the same impression illustrious physicists such as Boltzmann and Planck would have felt before me. A huge part of my scientific career would then be devoted to the elucidation of macroscopic as well as microscopic aspects of the second principle, in order to extend its validity to new situations, and to the other fundamental approaches of theoretical physics, such as classical and quantum dynamics.
Ilya Prigogine
Considering the pervasive importance of quantum mechanics in modern physics, it is odd how rarely one hears of efforts to test quantum mechanics experimentally with high precision....The trouble is that it is very difficult to find any logically consistent generalization of quantum mechanics. One obvious target for generalization is the linearity of quantum mechanics, but if we arbitrarily add nonlinear terms to the Schrodinger equation, how do we know that the theory we obtain will have a sensible physical interpretation? At least in part, it is the dearth of generalized versions of quantum mechanics that has made it so hard to plan experimental tests of quantum mechanics.
Steven Weinberg