User:Marsupilamov/Mathematics of the Schrödinger representation

The algebra of observables

TLDR

Any C*-algebra is a subalgebra of operators in a Hilbert space (Gelfand-Naimark theorem), states correspond to cyclic representations (Gelfand–Naimark–Segal construction) and they form a convex set, pure states correspond to irreducible representations, they are the extremities of the convex set. Pure states form the spectrum $\Phi _{A}$, and we have the Gelfand representation :

$A\longrightarrow C_{0}(\Phi _{A})$.

A commutative C*-algebra identifies with continuous functions vanishing at infinity on its spectrum $A\simeq C_{0}(\Phi _{A})$.

States

A state on a C*-algebra A is a positive linear functional f of norm 1, so that f(1) = 1.

Cyclic representations

A *-representation of a C*-algebra A on a Hilbert space H is a morphism of *-algebra

$\rho :A\longrightarrow {\mathcal {B}}(H)$

For a representation π of a C*-algebra A on a Hilbert space H, an element ξ is called a cyclic vector if its orbit is norm dense in H. Any non-zero vector of an irreducible representation is cyclic. For a ξ cyclic vector,

$x\mapsto \langle \xi ,\pi (x)\xi \rangle$

is a state of A, which determines the *-representation up to unitary isomorphism. Reciprocally, the representation determines the state up to a positive constant. The sum of the states is associated to a subrepresentation of the direct sum.

Theorem. (GNS construction) All states arise'this way, as cyclic vectors in a *-representation.

The GNS construction is at the heart of the proof of the Gelfand–Naimark theorem characterizing C*-algebras as algebras of operators. A C*-algebra has sufficiently many pure states (see below) so that the direct sum of corresponding irreducible GNS representations is faithful.

The direct sum of the corresponding GNS representations of all positive linear functionals is called the universal representation of A. Since every nondegenerate representation is a direct sum of cyclic representations, any other representation is a *-homomorphic image of π.

If π is the universal representation of a C*-algebra A, the closure of π(A) in the weak operator topology is called the enveloping von Neumann algebra of A. It can be identified with the double dual A**.

Irreducibility

Theorem. The set of states of a C*-algebra A with a unit element is a compact convex set under the weak-* topology. In general, (regardless of whether or not A has a unit element) the set of positive functionals of norm ≤ 1 is a compact convex set.

Both of these results follow immediately from the Banach–Alaoglu theorem.

In the unital commutative case, for the C*-algebra C(X) of continuous functions on some compact X, Riesz representation theorem says that the positive functionals of norm ≤ 1 are precisely the Borel positive measures on X with total mass ≤ 1. It follows from Krein–Milman theorem that the extremal states are the Dirac point-mass measures.

On the other hand, a representation of C(X) is irreducible if and only if it is one dimensional. Therefore the GNS representation of C(X) corresponding to a measure μ is irreducible if and only if μ is an extremal state. This is in fact true for C*-algebras in general.

Theorem. Let A be a C*-algebra. If π is a *-representation of A on the Hilbert space H with unit norm cyclic vector ξ, then π is irreducible if and only if the corresponding state f is an extreme point of the convex set of positive linear functionals on A of norm ≤ 1.

To prove this result one notes first that a representation is irreducible if and only if the commutant of π(A), denoted by π(A)', consists of scalar multiples of the identity.

Any positive linear functionals g on A dominated by f is of the form

$g(x^{*}x)=\langle \pi (x)\xi ,\pi (x)T_{g}\,\xi \rangle$

for some positive operator T_g in π(A)' with 0 ≤ T ≤ 1 in the operator order. This is a version of the Radon–Nikodym theorem.

For such g, one can write f as a sum of positive linear functionals: f = g + g' . So π is unitarily equivalent to a subrepresentation of π_g ⊕ π_g' . This shows that π is irreducible if and only if any such π_g is unitarily equivalent to π, i.e. g is a scalar multiple of f, which proves the theorem.

Extremal states are usually called pure states. Note that a state is a pure state if and only if it is extremal in the convex set of states.

The theorems above for C*-algebras are valid more generally in the context of B*-algebras with approximate identity.

The Gelfan'd Naimark theorem

The functor which associates to a locally compact Hausdorff topological space X, the commutative C*-algebra: C₀(X) of continuous complex-valued functions on X which vanish at infinity is an equivalence of categories.

Its inverse is given by the spectrum.

The Gelfand representation

Any Banach algebra A is represented in its spectrum $\Phi _{A}$: the space of characters (non-zero algebra homomorphism $\phi :A\to {\mathbb {C} }$) is called a character of $A$; the set of all characters of A is denoted by $\Phi _{A}$, equipped with the relative weak-* topology. From the Banach-Alaoglu theorem, the spectrum $\Phi _{A}$ is locally compact and Hausdorff, and compact and only if the algebra A is unital.

Given $a\in A$, one defines the function ${\widehat {a}}:\Phi _{A}\to {\mathbb {C} }$ by ${\widehat {a}}(\phi )=\phi (a)$. The definition of $\Phi _{A}$ and the topology on it ensure that ${\widehat {a}}$ is continuous and vanishes at infinity, and that the map $a\mapsto {\widehat {a}}$ defines a norm-decreasing, unit-preserving algebra homomorphism from A to $C_{0}(\Phi _{A})$. This homomorphism is the Gelfand representation of A, and ${\widehat {a}}$ is the Gelfand transform of the element $a$. In general the representation is neither injective nor surjective.

In the case where A has an identity element, there is a bijection between $\Phi _{A}$ and the set of maximal proper ideals in A (this relies on the Gelfand-Mazur theorem). As a consequence, the kernel of the Gelfand representation $A\to C_{0}(\Phi _{A})$ may be identified with the Jacobson radical of A. Thus the Gelfand representation is injective if and only if A is (Jacobson) semisimple.

The Weyl algebra and the Heisenberg group

The Heisenberg group

The general abstraction of a Heisenberg group is constructed from any symplectic vector space. The Heisenberg group H(V) on (V,ω) (or simply V for brevity) is the set V×R endowed with the group law

$(v,t)\cdot (v',t')=\left(v+v',t+t'+{\frac {1}{2}}\omega (v,v')\right).$.

The Heisenberg group is a central extension of the additive group V. Thus there is an exact sequence

$0\to \mathbb {R} \to H(V)\to V\to 0.$

For any polarization (or pair $L,L'$ of transverse Lagrangian subspaces) of V, the Heisenberg group admits a canonical faithful representation in $\mathbf {k} \oplus L\oplus \mathbf {k}$, given by

$(q,p,t)\longrightarrow {\begin{bmatrix}1&p&t\\0&I_{L}&q\\0&0&1\end{bmatrix}}$,

The Weyl algebra

The Weyl algebra W(V) of the alternate 2-form $\omega$ on V is the quotient of the tensor algebra of V, $T(V)$ by the ideal generated by tensor of the form:

$(v\otimes w-w\otimes v-\omega (v,w),$ for $v,w\in V.$

In other words, $W(V)$ is the algebra generated by V subject only to the relation $vw-wv=\omega (v,w)$.

By the Poincaré-Birkhoff-Witt theorem, the Weyl algebra identifies with the enveloping algebra of the Lie algebra tangent to the Heisenberg group.

The Weyl algebra as a quantization of the symmetric algebra

We assume the characteristic of k to be 0.

Like for Clifford algebras, the Weyl algebra receives a linear isomorphism from the symmetric algebra of V, and is thus filtrated. The associated graduated algebra is the symmetric algebra.

In that sense, the Weyl algebra is a quantization of the symmetric algebra, which is essentially the same as the Moyal quantization (if for the latter, one restricts to polynomial functions), but the former is in terms of generators and relations (considered to be differential operators) and the latter is in terms of a deformed multiplication.

The Wigner-Moyal transform

Consider the sesquilinear form

$W:{\overline {L^{2}(L)}}\otimes L^{2}(L)\longrightarrow L^{2}(L\oplus L^{\vee })$

given by

$W_{\phi ,\psi }(x,p)={\frac {1}{(\pi \hbar )^{n}}}\int _{L}\langle e^{-ip(x+y)/\hbar }\phi (x+y)|\psi (x-y)e^{-ip(x-y)/\hbar }\rangle \,dy\ .$

For normed &:phi, the map is unitary in :psi.

The Schrödinger representation

Assume now that V is a hermitian vector space. Then for any Lagrangian subspace L, the Heisenberg group has a unitary representation in the Hilbert space

${\mathcal {H}}(L)=L^{2}(L)$, where position vectors x of L act by translation on $\psi \mapsto \psi (x-x_{0})$ and momentum vectors p orthogonal to L act by