Models of the Universe based on Jordan algebras

与晚期宇宙学模型对偶的弹跳与坍缩宇宙

Bouncing and collapsing universes dual to late-time cosmological models

本文利用 Jordan frame-Einstein frame 对应关系，证明晚期物理宇宙的演化（如 $Λ$CDM 和 quintessence 模型）可对偶描述为 Brans-Dicke 理论下的弹跳或坍缩宇宙，并研究了共形映射的稳定性。

On a Jordan-algebraic formulation of quantum mechanics: Hilbert space construction

In this note I discuss some aspects of a formulation of quantum mechanics based entirely on the Jordan algebra of observables. After reviewing some facts of the formulation in the \CS -approach I present a Jordan-algebraic Hilbert space construction (inspired by the usual GNS-construction), thereby obtaining a real Hilbert space and a (Jordan-) representation of the algebra of observables on this space. Taking the usual case as a guideline I subsequently derive a Schrödinger equation on this Hilbert space.

Differential calculus on Jordan algebra and Jordan modules

Having in mind applications to particle physics we develop the differential calculus over Jordan algebras and the theory of connections on Jordan modules. In particular we focus on differential calculus over the exceptional Jordan algebra and provide a complete characterization of the theory of connections for free Jordan modules.

Jordan Algebras and Extremal Black Holes

We review various properties of the exceptional Euclidean Jordan algebra of degree three. Euclidean Jordan algebras of degree three and their corresponding Freudenthal triple systems were recently shown to be intimately related to extremal black holes in N=2, d=4 homogeneous supergravities. Using a novel type of eigenvalue problem with eigenmatrix solutions, we elucidate the rich matrix geometry underlying the exceptional N=2, d=4 homogeneous supergravity and explore the relations to extremal black holes.

Asymptotic commutativity in Jordan algebras

We estimate commutators of quadratic operators $Q_a$ in Jordan algebras. These estimates can be used to construct the scattering theory in quantum fields theories formulated in terms of Jordan algebras.

Exceptional Lie Algebras, SU(3) and Jordan Pairs

A simple unifying view of the exceptional Lie algebras is presented. The underlying Jordan pair content and role are exhibited. Each algebra contains three Jordan pairs sharing the same Lie algebra of automorphisms and the same external su(3) symmetry. Eventual physical applications and implications of the theory are outlined.

Scattering in algebraic approach to quantum theory. Jordan algebras

Using geometric approach we formulate quantum theory in terms of Jordan algebras. We analyze the notion of (quasi)particle (=elementary excitation of translation-invariant stationary state) and the scattering of (quasi)particles in this framework.

Reduction of Lie-Jordan Banach algebras and quantum states

A theory of reduction of Lie-Jordan Banach algebras with respect to either a Jordan ideal or a Lie-Jordan subalgebra is presented. This theory is compared with the standard reduction of C*-algebras of observables of a quantum system in the presence of quantum constraints. It is shown that the later corresponds to the particular instance of the reduction of Lie-Jordan Banach algebras with respect to a Lie-Jordan subalgebra as described in this paper. The space of states of the reduced Lie-Jordan Banach algebras is described in terms of equivalence classes of extensions to the full algebra and t…

Exceptional Lie Algebras, SU(3) and Jordan Pairs Part 2: Zorn-type Representations

A representation of the exceptional Lie algebras is presented. It reflects a simple unifying view and it is realized in terms of Zorn-type matrices. The role of the underlying Jordan pair and Jordan algebra content is crucial in the development of the structure. Each algebra contains three Jordan pairs sharing the same Lie algebra of automorphisms and the same external su(3) symmetry. Applications in physics are outlined.

Jordan C*-Algebras and Supergravity

It is known that black hole charge vectors of N=8 and magic N=2 supergravity in four and five dimensions can be represented as elements of Jordan algebras of degree three over the octonions and split-octonions and their Freudenthal triple systems. We show both such Jordan algebras are contained in the exceptional Jordan C*-algebra and construct its corresponding Freudenthal triple system and single variable extension. The transformation groups for these structures give rise to the complex forms of the U-duality groups for N=8 and magic N=2 supergravities in three, four and five dimensions.

Jordan Algebraic Interpretation of Maximal Parabolic Subalgebras : Exceptional Lie Algebras

With this paper we start a programme aiming at connecting two vast scientific areas: Jordan algebras and representation theory. Within representation theory, we focus on non-compact, real forms of semisimple Lie algebras and groups as well as on the modern theory of their induced representations, in which a central role is played by the parabolic subalgebras and subgroups. The aim of the present paper and its sequels is to present a Jordan algebraic interpretations of maximal parabolic subalgebras. In this first paper, we confine ourselves to maximal parabolic subalgebras of the non-compact re…