Podcast Episode Details
Back to Podcast Episodes
Jordan Algebra
https://g.co/gemini/share/0ffb256c7fb3
The genesis of Jordan algebras is a compelling chapter in the history of science, rooted not in the abstract explorations of pure mathematics but in the foundational crises of early 20th-century physics. In the early 1930s, the newly formulated theory of quantum mechanics was a spectacular success, yet its mathematical underpinnings were still being solidified. The German physicist Pascual Jordan, a key architect of quantum field theory , sought to place the theory on a more rigorous and conceptually fundamental algebraic footing. Collaborating with the eminent mathematician John von Neumann and physicist Eugene Wigner, Jordan embarked on a program to axiomatize the algebra of physical observables. This investigation led them to abandon one of the most sacred tenets of algebra—the associative law of multiplication—and in doing so, they uncovered a new and profound mathematical structure.
The central problem stemmed from the representation of physical observables in quantum theory. In the established Hilbert space formalism, quantities that can be measured, such as energy, position, or spin, are represented by self-adjoint (or Hermitian) operators. The set of all such operators on a given Hilbert space forms a real vector space: they can be added together and scaled by real numbers, and the result remains a self-adjoint operator. However, a critical inconsistency arises when one considers multiplication. The standard associative product of two Hermitian operators, A and B, is not, in general, Hermitian itself. The product AB is Hermitian if and only if A and B commute, a condition that corresponds physically to the simultaneous observability of the two quantities. This meant that the collection of physical observables was not closed under the standard algebraic product, posing a significant obstacle to formulating a self-contained "algebra of observables".
Jordan's crucial insight was to recognize that the problem lay not with the operators themselves, but with the choice of multiplication. He proposed a new product, now known as the Jordan product, defined by symmetrizing the associative matrix product :
Published on 2 months, 1 week ago