相关论文: A comment on the magical realizations of W_3
A celebrated result of Koecher and Vinberg asserts the one-one correspondence between the finite dimensional formally real Jordan algebras and Euclidean symmetric cones. We extend this result to the infinite dimensional setting.
The exceptional Jordan algebra is the algebra of $3\times 3$ Hermitian matrices with octonionic entries. It is the only one from Jordan's algebraic formulation of quantum mechanics which is not equivalent to the conventional formulation of…
A representation of finite-dimensional probabilistic models in terms of formally real Jordan algebras is obtained, in a strikingly easy way, from simple assumptions. This provides a framework in which real, complex and quaternionic quantum…
It is shown that the Sugawara-type construction for W3-algebra associated with the four magical Jordan algebras leads to the anomalous theory of W3-gravity.
While canonical quantization solves many problems there are some problems where it fails. A close examination of the classical/quantum connection leads to a new connection that permits quantum and classical realms to coexist, as is the case…
We study finite-dimensional commutative algebras, which satisfy the Jacobi identity. Such algebras are Jordan algebras. We describe some of their properties and give a classification in dimensions $n<7$ over algebraically closed fields of…
This paper fails to derive quantum mechanics from a few simple postulates. But it gets very close --- and it does so without much exertion. More exactly, I obtain a representation of finite-dimensional probabilistic systems in terms of…
There exists the problem to construct a quantum algebra of observables in lightcone QCD beyond the perturbative regime. It has recently established that the boundary gauge fields are crucial for a consistent construction of the classical…
In this paper, we classify Jordan superalgebras of dimension up to three over an algebraically closed field of characteristic different of two. Our main motivation to obtain such classification comes out from the intention to give an answer…
In this article we begin the study of representations of simple finite-dimensional noncommutative Jordan superalgebras. In the case of degree $\geq 3$ we show that any finite-dimensional representation is completely reducible and, depending…
We define and compute explicitly the classical limit of the realizations of $W_n$ appearing as hamiltonian structures of generalized KdV hierarchies. The classical limit is obtained by taking the commutative limit of the ring of…
We describe a model where breaking of W3 symmetry will lead to the emergence of time and subsequently of space. Surprisingly the simplest such models which lead to higher dimensional spacetimes are based on the four "magical" Jordan…
We show how our Universe can emerge from a symmetry breaking of a multicomponent $W_3$ algebra, where the components in addition form a Jordan algebra. We discuss how symmetry breaking related to the Jordan algebras $H_3(C)$ and $H_3(O)$…
In 2003 Peter Cameron introduced the concept of a Jordan scheme and asked whether there exist Jordan schemes which are not symmetrisations of coherent configurations (proper Jordan schemes). The question was answered affirmatively by the…
We propose a model for the universe based on Jordan algebras. The action consists of cubic terms with coefficients being the structure constants of a Jordan algebra. Coupling constants only enter the theory via symmetry breaking which also…
Three elementary canonical transformations are shown both to have quantum implementations as finite transformations and to generate, classically and infinitesimally, the full canonical algebra. A general canonical transformation can, in…
We construct canonical realizations of the $\mathfrak{bms}_3$ algebra as symmetry algebras of a free Klein-Gordon (KG) field in $2+1$ dimensions, for both the massive and massless case. We consider two types of realizations, one on-shell,…
This paper extends the tools of C*-algebraic strict quantization toward analyzing the classical limits of unbounded quantities in quantum theories. We introduce the approach first in the simple case of finite systems. Then we apply this…
Jordan, Wigner and von Neumann classified the possible algebras of quantum mechanical observables, and found they fell into 4 "ordinary" families, plus one remarkable outlier: the exceptional Jordan algebra. We point out an intriguing…
Realizations of algebras in terms of canonical or bosonic variables can often be used to simplify calculations and to exhibit underlying properties. There is a long history of using such methods in order to study symmetry groups related to…