“It has been generally believed that only the complex numbers could legitimately be used as the ground field in discussing quantum-mechanical operators. Over the complex field, Frobenius' theorem is of course not valid; the only division algebra over the complex field is formed by the complex numbers themselves. However, Frobenius' theorem is relevant precisely because the appropriate ground field for much of quantum mechanics is real rather than complex.”

— Freeman Dyson

[The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Jour. Math. Phys., 3, 1962, 1199–1215, https://books.google.com/books?id=nnyNUidX1OMC&pg=PA410] (p. 1200)

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "It has been generally believed that only the complex numbers could legitimately be used as the ground field in discussi…" by Freeman Dyson?

Freeman Dyson90

theoretical physicist and mathematician 1923

Related quotes

“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. Baez (1961) American mathematician and mathematical physicist

“Imaginary numbers are not imaginary and the theory of complex numbers is no more complex than the theory of real numbers. Complex numbers are as intuitive for an electronics engineer as -100 is for the average person with an overdrawn bank account.”

Mordechai Ben-Ari (1948) Israeli computer scientist

Source: Just a Theory: Exploring the Nature of Science (2005), Chapter 3, “Words Scientists Don’t Use: At Least Not the Way You Do” (p. 56)

“Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise is wishful thinking or "mathematics envy."”

Marvin Minsky (1927–2016) American cognitive scientist

Music, Mind, and Meaning (1981)

“Yet more complex are the environments we have called turbulent fields.”

Fred Emery (1925–1997) Australian psychologist

In these, dynamic processes, which create significant variances for the component organizations, arise from the field itself.
Source: The Causal Texture of Organizational Environments (1963), p. 30.

“An important point is that the p-adic field, or respectively the real or complex field, corresponding to a prime ideal, plays exactly the role, in arithmetic, that the field of power series in the neighborhood of a point plays in the theory of functions: that is why one calls it a local field.”

André Weil (1906–1998) French mathematician

as translated by Martin H. Krieger "A 1940 letter of André Weil on analogy in mathematics." http://www.ams.org/notices/200503/fea-weil.pdf Notices of the AMS 52, no. 3 (2005) pp. 334–341, quote on p. 340

“Theorems… record more complex patterns of thinking that once shown to be valid need not be repeated every time they are needed.”

Richard Hamming (1915–1998) American mathematician and information theorist

Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985)