Exceptional quantum geometry and particle physics

In [3] some constructions of exceptional Jordan algebras due to H. Freudenthal, T. A. Springer, and J. Tits were carried over to quadratic Jordan algebras (as in [4]) of arbitrary characteristic. The question was left open whether the Tits Constructions yielded all exceptional finite-dimensional central simple algebras in characteristic 2 (it was known that they do for characteristics #2). In this paper we settle this question in the affirmative. This result completes the structure theory for finite-dimensional quadratic Jordan algebras. The Tits Constructions as given in [3] involve the construction of a norm form. To prove that all the exceptional algebras arise from these constructions we need to show that all such algebras have a suitable norm form. This necessitates a slight detour in ??1 and 2 to verify that Jordan algebras in characteristic 2 have generic norms with the same properties as in the other characteristics. In ?3 we define the centroid for quadratic Jordan algebras and the corresponding notion of central simple algebras. In the next section we establish certain conditions under which a central simple Jordan algebra remains simple upon extension of the base field. In ?5 we apply these results to show that every exceptional finite-dimensional central simple algebra is a form of the 27-dimensional exceptional algebra ?((i3) (i.e. becomes ( upon suitable extension of the base field). In the final section we show that the Tits Constructions yield precisely all the exceptional finite-dimensional central simple Jordan algebras. Our proofs will be valid for all characteristics.

It is well known that the Lie algebra of derivations of the exceptional Jordan algebra M 3 8 over an algebraically closed field of characteristic 0 or over the field of reals is the exceptional Lie algebra F 4 of Killing-Cartan and that the Lie algebra of linear transformations in M 3 8 leaving a certain cubic norm form defined on M 3 8 invariant is the exceptional Lie algebra E 6 (Chevalley-Schafer [7], Freudenthal [11]). The program envisaged in the present series of papers is the extension of these results and their group analogues to arbitrary central simple Jordan algebras. The present paper gives the definition of the groups and Lie algebras in the general case and gives their determination for special Jordan algebras with some restrictions on the base field. In a second paper we plan to consider the groups of automorphisms of arbitrary reduced exceptional simple Jordan algebras and in a third paper we hope to study the generalized groups of types E 6 which arise from these Jordan algebras.

The set of real functions is countable since the functions must be computable, i.e. there must be an algorithm for computing them. But the set of algorithms is countable. Uncomputable functions are useless, they do not exist in applied mathematics. The set of computable real numbers is also countable. Uncomputable numbers are useless. The definition of algebra of computable real functions is given and a classification of subalgebras with one-element bases is constructed. This classification is a classification of functions too. Algebras with multielement bases are fictitious, they are useless for classification of functions. All infinite sequences of inclusions of subalgebras are constructed.

In [2]2 we classified the isometric mappings of one C*-algebra (uniformly closed, self-adjoint operator algebra) onto another. It was remarked in that paper that the results obtained were a non-commutative extension of results of Banach [1] and Stone [7]. While this was true in spirit, we were well aware that it was not accurate to the letter. Banach and Stone deal with the algebra of real continuous functions on a compact-Hausdorff space, and our results concerning C*-algebras are actually the non-commutative analogue of results concerning the complex function algebra. The strict non-commutative analogue of the real function algebra is the Jordan algebra of self-adjoint elements in a C*-algebra (Jordan C*-algebra). The complex and real theorems follow very easily from one another in the commutative case, so that one might justifiably consider the C*-algebra theorem an extension of both of the function algebra theorems. Despite such trifling considerations, two questions still remain: what are the isometries of one C*-algebra onto another, and what are the isometries of one Jordan C*-algebra onto another? At the time [21 was written, the C*-algebra seemed the more natural object to consider. In view of the results obtained, answering the Jordan C*-algebra questions appeared to be an unnecessary decoration to the theory. We felt that the Jordan C*-algebra results could be obtained from the C*-algebra results in the same way that the real function algebra theorem follows from the complex function algebra theorem (viz., by showing that the complexified linear map is everywhere isometric). Subsequent investigations have changed our attitude in this matter. An important application of these considerations requires a Jordan C*-algebra theorem for one thing, and our attempts to derive this theorem directly from the C*-algebra theorem failed for another. The result in question is contained in Theorem 2 of ?2 and states (in normalized form) that an isometry between two Jordan C*-algebras which carries the identity into the identity is a C* (Jordan) -isomorphism. This theorem was eventually proved with the aid of a Generalized Schwarz Inequality (cf. Theorem 1 of ?2). In effect, an alternative ending has been given to the proof of [Theorem 7; 2]. This ending is by no means simpler or shorter than the one given in [2] (though it is, perhaps, less contrived), but it is flexible enough to allow us to draw the desired Jordan C*-algebra conclusion. The critical application of these results is contained in Corollary 3. A discussion accompanies Corollaries 3 and 4, but a few additional remarks are in order.

Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan - von Neumann - Wigner classification of finite dimensional Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra $\frak{J}$. The automorphism group $\rm{F}_4$ of $\frak{J}$ and its maximal Borel-de~Siebenthal subgroups $\frac{\rm SU(3)\times \rm SU(3)}{\mathbb{Z}_3}$ and ${\rm Spin}(9)$ are studied in some detail with an eye to possible applications to the fundamental fermions in the Standard Model of particle physics.

This result has a number of important consequences. It may be seen to imply that no simple exceptional finite dimensional Jordan algebra of characteristic not two is a homomorphic image of a special(2) Jordan algebra. But there is an exceptional simple Jordan algebra S& (of dimension 27) over a field a of characteristic not two which is generated by three of its elements. Then ID is a homomorphic image of the free Jordan algebra '3 on three generators. It follows that '3 is not special, and we also know that '3 is not isomorphic to the free special Jordan algebra 3 [x, y, z, 1 ] consisting of all reversible polynomials(') in the free associative algebra a [x, y, , 1Z] of all polynomials on the three generators x, y, z. 2. Elementary properties of 60 and certain subalgebras. We begin with a brief description of the algebra eo of our theorem. Let Z be an algebra with a unity element e over a field 0 of characteristic not two. Suppose that Z has an involution T over a; that is, an antiautomorphism

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.

A well known theorem, due to Wedderburn, states that if $$ \mathfrak{B} $$ is an arbitrary associative algebra with an identity 1 and $$ \mathfrak{A} $$ is a finite dimensional central simple subalgebra containing 1, then $$ \mathfrak{B} $$ is the Kronecker product $$ \mathfrak{B} = \mathfrak{A} \otimes \mathfrak{U} $$ where $$ \mathfrak{U} $$ is the subalgebra of elements which commute with every $$ a\,\varepsilon \,\mathfrak{A} $$ . Recently Kaplansky indicated that the same result holds for $$ \mathfrak{B} $$ any alternative algebra and $$ \mathfrak{A} $$ an algebra of Cayley numbers.2 It is the purpose of the present note to give a new proof of this result and to show that an analogous result holds for $$ \mathfrak{B} $$ any Jordan algebra and $$ \mathfrak{A} $$ an exceptional simple Jordan algebra of finite dimensions.

We study exceptional Jordan algebras and related exceptional group schemes over commutative rings from a geometric point of view, using appropriate torsors to parametrize and explain classical and new constructions, and proving that over rings, they give rise to nonisomorphic structures.We begin by showing that isotopes of Albert algebras are obtained as twists by a certain $\mathrm F_4$ -torsor with total space a group of type $\mathrm E_6$ and, using this, that Albert algebras over rings in general admit nonisomorphic isotopes even in the split case, as opposed to the situation over fields. We then consider certain $\mathrm D_4$ -torsors constructed from reduced Albert algebras, and show how these give rise to a class of generalised reduced Albert algebras constructed from compositions of quadratic forms. Showing that this torsor is nontrivial, we conclude that the Albert algebra does not uniquely determine the underlying composition, even in the split case. In a similar vein, we show that a given reduced Albert algebra can admit two coordinate algebras which are nonisomorphic and have nonisometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.

Let $\mathcal{J}^1$ be the real form of a complex simple Jordan algebra such that the automorphism group is $\mathrm{F}_{4(-20)}$. By using some orbit types of $\mathrm{F}_{4(-20)}$ on $\mathcal{J}^1$, for $\mathrm{F}_{4(-20)}$, explicitly, we give the Iwasawa decomposition, the Oshima--Sekiguchi's $K_{\epsilon}-$Iwasawa decomposition, the Matsuki decomposition, and the Bruhat and Gauss decompositions.

We give an invariant classification of orbits of the fundamental representations of the exceptional groups E7(7) and E6(6) which classify BPS states in string and M theories toroidally compactified to d=4 and 5. The exceptional Jordan algebra and the exceptional Freudenthal triple system and their cubic and quartic invariants play a major role in this classification. The cubic and quartic invariants correspond to the black hole entropy in d=5 and d=4, respectively. The classification of BPS states preserving different numbers of supersymmetries is in close parallel to the classification of the little groups and the orbits of timelike, lightlike and spacelike vectors in Minkowski space. The orbits of BPS black holes in N=2 Maxwell–Einstein supergravity theories in d=4 and d=5 with symmetric space geometries are also classified including the exceptional N=2 theory, which has E7(-25) and E6(-26) as its symmetry in the respective dimensions.

While most theorists are tied to the mast of four dimensions, some have found it irresistible to speculate about eleven dimensions, the domain of M-theory. We outline a program which starts from the light-cone description of supergravity, and tracks its divergences to suggest the existence of an infinite component theory which in the light-cone relies on the coset $F_4/SO(9)$, long known to be linked to the Exceptional Jordan Algebra.

Quantum theory’s Hilbert space apparatus in its finite-dimensional version is nearly reconstructed from four simple and quantum-mechanically motivated postulates for a quantum logic. The recon- struction process is not complete, since it excludes the two-dimensional Hilbert space and still includes the exceptional Jordan algebras, which are not part of the Hilbert space apparatus. Options for physically meaningful potential generalizations of the apparatus are discussed.

The problem of speciality for Bernstein Jordan algebras is considered. Two examples of exceptional Bernstein Jordan algebras with special nucleus are constructed. The examples correspond to two different types of exceptional Jordan algebras: the first one is i-exceptional, and the second one is i-special.

String theories and the Jordan algebras