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A Jordan algebra is an algebra (in the sense of: is a monoid) which is not associative but is commutative, subject to some further conditions which are modeled after the archetypical example: for $(A, \cdot)$ any associative algebra, equipping it with the symmetrized product
makes $(A, \circ)$ a Jordan algebra. It is this relation that originally motivated the notion in discussion of quantum mechanics, for the symmetrized product and hence the Jordan algebra structure of the algebra of observables of a quantum mechanical system is what remains when one ignores the otherwise all-important commutators and hence the Hamiltonian flows on observables.
But later Jordan algebras have been studied largely for their own sake.
More recently, the Bohr topos associated to a noncommutative algebra of observables was found to really see the underlying Jordan algebra structure. See at Bohr topos and poset of commutative subalgebras for more on this.
A Jordan algebra is a commutative nonassociative algebra $J$ satisfying the Jordan identity $(x y) (x x) = x (y (x x))$ for all $x,y$ in $J$. It follows that $J$ is power-associative, and the Jordan identity generalizes to
for natural numbers $m, n \geq 1$ (and, trivially, for $m, n \geq 0$ if there is an identity element).
If $k$ is a field whose characteristic is not $2$ (or is any commutative ring in which $2$ is invertible), then to any associative $k$-algebra $A$ with product $\cdot$, one associates a Jordan $k$-algebra with the same underlying vector space and whose Jordan product $\circ$ is given by
Such Jordan algebras are called special Jordan algebras; all others are called exceptional.
Jordan algebras had their origin in the study of the foundations of quantum theory. In 1932, Pascual Jordan tried to isolate some axioms that an ‘algebra of observables’ should satisfy (Jordan 32).
The unadorned phrase ‘algebra’ usually signals an associative algebra, but this is not the kind of algebra Jordan was led to. In both classical and quantum mechanics, observables are closed under addition and multiplication by real scalars. In classical mechanics we can also multiply observables, but in quantum mechanics this becomes problematic. After all, given two bounded self-adjoint linear operators on a complex Hilbert space, their product is self-adjoint if and only if they commute.
However, in quantum mechanics one can still raise an observable to a power and obtain another observable. From squaring and taking real linear combinations, one can construct a commutative product using the polarization identity:
This is sometimes called the anti-commutator (or more precisely, half the anti-commutator). Notice that it is analogous to the more famous commutator
and that both together recover the full algebra of observables in that
for all $x,y$. (A JLB-algebra is a Banach space equipped with the compatible structures of both a Jordan algebra and a Lie algebra, and these are equivalent to $C^*$-algebras in just this way, using $\circ$ and ${\bullet} \coloneqq \frac{1}{2} \mathrm{i} [{-},{-}]$ as the operations.)
From the point of view of deformation quantization of Poisson manifolds, one can read this as follows: the deformation quantization of a Poisson manifold $(X,\{-,-\})$ breaks up into two pieces:
the Poisson bracket $\{-,-\}$ on $C^\infty(X)$ deforms to the commutator;
the pointwise multiplication on $C^\infty(X)$ deforms into the Jordan algebra structure.
This perspective on deformation quantization making the role of Jordan algebras explicit is mentioned for instance in (Bates-Weinstein, p. 80).
The symmetrized product $(-) \circ (-)$ is not associative, in general, but it is power-associative: any way of parenthesizing a product of copies of the same observable $x$ gives the same result. This led Jordan to define what is now called a formally real Jordan algebra: a commutative and power-associative algebra $J$ satisfying
for all $n$. The last condition (as in any formally real algebra) gives $J$ a partial ordering: if we write $x \le y$ when the element $y - x$ is a sum of squares, it says that
So, in a formally real Jordan algebra we can reasonably talk about one observable being ‘greater’ than another.
In fact the Jordan identity $(x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x))$ is a consequence of the above definition of formally real Jordan algebra. So, every formally real Jordan algebra is a Jordan algebra (but not conversely).
For more on this see also at order-theoretic structure in quantum mechanics – Relation to non-commutative geometry.
In 1934, Jordan published a paper with von Neumann and Wigner classifying finite-dimensional formally real Jordan algebras (Jordan-vonNeumann-Wigner 34).
They began by defining an ideal of a formally real Jordan algebra $J$ to be a linear subspace $S \subseteq J$ such that $x \in S$ implies $x \circ y \in S$ for all $y \in J$. Next they defined $J$ to be simple when its only ideals were $\{0\}$ and $J$ itself. Then they proved that any finite-dimensional formally Jordan algebra is a direct sum of simple ones.
This reduced the classification problem to the task of classifying simple finite-dimensional formally real Jordan algebras. There are four families of these, and one exception:
The $n \times n$ self-adjoint real matrices, $\mathfrak{h}_n(\mathbb{R})$, with the product $a \circ b = {1\over 2}(a b + b a)$.
The $n \times n$ self-adjoint complex matrices, $\mathfrak{h}_n(\mathbb{C})$, with the product $a \circ b = {1\over 2}(a b + b a)$.
The $n \times n$ self-adjoint quaternionic matrices, $\mathfrak{h}_n(\mathbb{H})$, with the product $a \circ b = {1\over 2}(a b + b a)$.
The $n \times n$ self-adjoint octonionic matrices, $\mathfrak{h}_n(\mathbb{O})$, with the product $a \circ b = {1\over 2}(a b + b a)$, where $n \le 3$.
The space $\mathbb{R}^n \oplus \mathbb{R}$ with the product $(x,t) \circ (x', t') = (t x' + t' x, x \cdot x' + t t').$
Here we say a square matrix with entries in the $\ast$-algebra $A$ is hermitian if it equals its conjugate transpose. (Note that $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ are all $\ast$-algebras.)
Because the octonions are an alternative algebra but not associative, we cannot go beyond $3 \times 3$ matrices and still get a Jordan algebra. The $1 \times 1$ self-adjoint octonionic matrices are just the real numbers, and the $2 \times 2$ ones are isomorphic to the spin factor $\mathbb{R}^9 \oplus \mathbb{R}$. The $3 \times 3$ self-adjoint octonionic matrices form the Albert algebra.
Jordan algebras in the fifth family are called spin factors. This family has some overlaps with the others. Most notably:
The Jordan algebra of $2 \times 2$ self-adjoint real matrices is isomorphic to the spin factor $\mathbb{R}^2 \oplus \mathbb{R}$.
The Jordan algebra of $2 \times 2$ self-adjoint complex matrices is isomorphic to the spin factor $\mathbb{R}^3 \oplus \mathbb{R}$.
The Jordan algebra of $2 \times 2$ self-adjoint quaternionic matrices is isomorphic to the spin factor $\mathbb{R}^5 \oplus \mathbb{R}$.
The Jordan algebra of $2 \times 2$ self-adjoint octonionic matrices is isomorphic to the spin factor $\mathbb{R}^9 \oplus \mathbb{R}$.
Because the spin factor $\mathbb{R}^n \oplus \mathbb{R}$ can be identified with $(n+1)$-dimensional Minkowski space, this sets up a relation between the real numbers, complex numbers, quaternions and octonions and Minkowski space in 3,4,6 and 10 dimensions — a pattern which becomes important in string theory. For more details, see division algebras and supersymmetry.
John Baez, The octonions, Sec. 3.3: $\mathbb{O}P^1$ and Lorentzian geometry. (web)
John Baez and John Huerta, Division algebras and supersymmetry I. (arXiv)
In 1983, Zelmanov drastically generalized the result of Jordan, von Neumann and Wigner by classifying all simple Jordan algebras, including infinite-dimensional ones (Zelmanov 83).
Among the exceptional Jordan algebras over the real numbers, there is a remarkable $27$-dimensional example: the Albert algebra $\mathbb{Al}$ of self-adjoint $3\times 3$ matrices over the octonions with the same formula as above for the product in terms of matrix product. Notice that the octonions and their matrices do not form associative algebras, but only alternative algebras, so the Jordan identity for the Albert algebra is not automatic (it does not hold for all alternative algebras) but is a consequence of more special circumstances.
The formalism of Jordan algebras seems rather removed from the actual practice of physics, because in quantum theory we hardly ever take two observables $a$ and $b$ and form their Jordan product ${1\over 2}(a b + b a)$. As hinted in the previous section, it is better to think of this operation as derived from the process of squaring an observable, which is something we actually do. But still, one must ask: can we see the classification of finite-dimensional formally real Jordan algebras, and thus the special role of normed division algebras, as arising from some axioms more closely tied to quantum theory as physicists usually practice it?
One answer involves the Koecher–Vinberg classification of self-dual homogeneous convex cones. Consider first the case of ordinary quantum theory. If a quantum system has the Hilbert space $\mathbb{C}^n$, observables are described by self-adjoint $n \times n$ complex matrices: elements of the Jordan algebra $\mathfrak{h}_n(\mathbb{C})$. But matrices of this form that are nonnegative and have trace 1 also play another role. They are called density matrices, and they describe states of our quantum system: not just pure states, but also more general mixed states. The idea is that any density matrix $\rho \in \mathfrak{h}_n(\mathbb{C})$ allows us to define expectation values of observables $a \in \mathfrak{h}_n(\mathbb{C})$ via
The map sending observables to their expectation values is real-linear. The fact that $\rho$ is nonnegative is equivalent to
and the fact that $\rho$ has trace 1 is equivalent to
All of this generalizes to an arbitrary finite-dimensional formally real Jordan algebra $J$. Any such algebra automatically has an identity element. This lets us define a state on $J$ to be a linear functional $\langle \cdot \rangle : J \to \mathbb{R}$ that is
nonnegative:
and normalized:
But in fact, there is a bijective correspondence between linear functionals on $J$ and elements of $J$. The reason is that every finite-dimensional Jordan algebra has a trace
defined so that $\mathrm{tr}(a)$ is the trace of the linear operator ‘multiplication by $a$’. Such a Jordan algebra is then formally real if and only if
is a real-valued inner product. So, when $J$ is a finite-dimensional formally real Jordan algebra, any linear functional $\langle \cdot \rangle : J \to \mathbb{R}$ can be written as
for a unique element $\rho \in J$. Conversely, every element $\rho \in J$ gives a linear functional by this formula. While not obvious, it is true that the linear functional $\langle \cdot \rangle$ is nonnegative if and only if $\rho \ge 0$ in terms of the ordering on $J$. More obviously, $\langle \cdot \rangle$ is normalized if and only if $\mathrm{tr}(\rho) = 1$. So, states can be identified with certain special observables: namely, those observables $\rho \in J$ with $\rho \ge 0$ and $\mathrm{tr}(\rho) = 1$.
These ideas help motivate an important theorem of Koecher and Vinberg. The idea is to axiomatize the situation we we have just described, in a way that does not mention the Jordan product in $J$, but instead emphasizes:
the isomorphism between $J$ and its dual space
the fact that ‘positive’ elements of $J$ form a cone.
To find appropriate axioms, suppose $J$ is a finite-dimensional formally real Jordan algebra. Then seven facts are always true.
The set of positive observables
is a cone: that is, $a \in C$ implies that every positive multiple of $a$ is also in $C$.
This cone is convex:
if $a,b \in C$ then any linear combination $x a + (1-x) b$ with $0 \le x \le 1$ also lies in $C$.
It is an open set.
It is regular, meaning that if $a$ and $-a$ are both in the closure $\overline{C}$, then $a = 0$.
This condition may seem obscure, but if we note that
we see that $C$ being regular simply means
a perfectly plausible assumption.
Recall that $J$ has an inner product; this is what lets us identify linear functionals on $J$ with elements of $J$. This also lets us define the dual cone
which one can check is indeed a cone.
The fifth fact about $C$ is that it is self-dual, meaning $C = C^*$.
This formalizes the fact that states may be identified with special observables.
$C$ is homogeneous: given any two points $a,b \in C$, there is a real-linear transformation $T : A \to A$ mapping $C$ to itself in a bijective way, with the property that $T a = b$. This says that cone $C$ is highly symmetrical: no point of $C$ is any ‘better’ than any other, at least if we only consider the linear structure of the space $A$, ignoring the Jordan product and the trace.
From another viewpoint, however, there is a very special point of $C$, namely the identity $1$ of our Jordan algebra. And this brings us to our seventh and final fact: the cone $C$ is pointed meaning that it is equipped with a distinguished element (in this case $1 \in C$).
In short: when $J$ is a finite-dimensional formally real Jordan algebra, $C$ is a pointed homogeneous self-dual regular open convex cone. All the elements $a \in J$ are positive observables, but certain special ones, namely those with $\langle a, 1 \rangle = 1$, can also be viewed as states.
In fact, there is a category of pointed homogeneous self-dual regular open convex cones, where:
An object is a finite-dimensional real inner product space $V$ equipped with a pointed homogeneous self-dual regular open convex cone $C \subset V$.
A morphism from one object, say $(V,C)$, to another, say $(V',C')$, is a linear map $T : V \to V'$ preserving the inner product and mapping $C$ into $C'$.
Now for the payoff. The work of Koecher and Vinberg, nicely explained in Koecher’s Minnesota notes (Koecher) shows that:
The category of pointed homogeneous self-dual regular open convex cones is equivalent to the category of finite-dimensional formally real Jordan algebras.
This means that the theorem of Jordan, von Neumann and Wigner also classifies the pointed homogeneous self-dual regular convex cones!
Every pointed homogeneous self-dual regular open convex cone is isomorphic to a direct sum of those on this list:
the cone of positive elements in $\mathfrak{h}_n(\mathbb{R})$,
the cone of positive elements in $\mathfrak{h}_n(\mathbb{C})$,
the cone of positive elements in $\mathfrak{h}_n(\mathbb{H})$,
the cone of positive elements in $\mathfrak{h}_3(\mathbb{O})$,
the future lightcone in $\mathbb{R}^n \oplus \mathbb{R}$.
Some of this deserves a bit of explanation. For $\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H}$, an element $T \in \mathfrak{h}_n(\mathbb{K})$ is positive if and only if the corresponding operator $T : \mathbb{K}^n \to \mathbb{K}^n$ has
for all nonzero $v \in \mathbb{K}^n$. A similar trick works for defining positive elements of $\mathfrak{h}_3(\mathbb{O})$, but we do not need the details here. We say an element $(x,t) \in \mathbb{R}^n \oplus \mathbb{R}$ lies in the future lightcone if $t \gt 0$ and $t^2 - x \cdot x \gt 0$. This of course fits in nicely with the idea that the spin factors are connected to Minkowski spacetimes. Finally, there is an obvious notion of direct sum for Euclidean spaces with cones, where the direct sum of $(V,C)$ and $(V',C')$ is $V \oplus V'$ equipped with the cone
In short: finite-dimensional formally real Jordan algebras arise fairly naturally as observables starting from a formalism where nonnegative observables form a cone, as long as we insist on some properties of this cone.
For every associative algebra there is its semilattice of commutative subalgebras $ComSub(A)$. At least for $A, B$ von Neumann algebras without type $I_2$ von Neumann algebra factor-subfactors, the isomorphisms $ComSub(A) \to ComSub(B)$ correspond to isomorphisms between the corresponding Jordan algebras $A_J \to B_J$.
For more details see semilattice of commutative subalgebras.
The original articles include
Pascual Jordan, Über eine Klasse nichtassociativer hyperkomplexer Algebren, Nachr. Ges. Wiss. Göttingen (1932), 569–575.
Pascual Jordan, John von Neumann and Eugene Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934), 29–64.
E. I. Zelmanov, On prime Jordan algebras. II, Sibirsk Mat. J. 24 (1983), 89-104.
Introductions and surveys include
Harald Hanche-Olsen and Erling Stormer: Jordan Operator Algebras. (web)
Kevin McCrimmon, Jordan algebras and their applications, Bull. Amer. Math. Soc. 84 (1978), 612–627. (AMS website) and (Project Euclid website).
Max Koecher, The Minnesota Notes on Jordan Algebras and Their Applications, eds. Aloys Krieg and Sebastican Walcher, Lecture Notes in Mathematics 1710, Springer, Berlin, 1999. doi:10.1007/BFb0096285 (paywalled)
Paul Townsend, The Jordan formulation of Quantum Mechanics: a review (arXiv:1612.09228)
Remarks on Jordan algebras as algebras of observables in quantum physics are for instance in