Stone - Von Neumann Theorem

Home , Heisenberg group, Unitary representation

(April 22, 2015) Stone - von Neumann theorem

Paul Garrett [email protected] http://www.math.umn.edu/egarrett/

The bibliography indicates the origins of these issues in proving uniqueness, up to isomorphism, of the model for the commutation rules of quantum mechanics. As Mackey and Weil observed, some form of the argument succeeds in quite general circumstances. The argument here is an amalgam and adaptation of those in sources in the bibliography. The proof here applies nearly verbatim, with suitable adjustment of the notion of Schwartz space, to Heisenberg groups over other local ﬁelds in place of R. Fix a ﬁnite-dimensional R-vectorspace V with non-degenerate alternating form h, i. The corresponding Heisenberg group has Lie algebra h = V ⊕ R, with Lie bracket

0 0 0 [v + z, v + z ] = hv, v i ∈ R ≈ {0} ⊕ R ⊂ V ⊕ R Exponentiating to the Lie group H, the group operation is hv, v0i exp(v + z) · exp(v0 + z0) = exp v + v0 + z + z0 + 2

This is explained by literal matrix exponentiation: for example, with v = (x, y) ∈ R2,    xy  0 x z 1 x z + 2 exp(v + z) = exp  0 0 y  =  0 1 y  (with v = (x, y)) 0 0 0 0 0 1

We use Lie algebra coordinates on H, often suppressing the exponential on abelian subgroups, but reinstating an explicit exponential notation when necessary to avoid confusion.

[0.0.1] Theorem: (Stone-vonNeumann) For ﬁxed non-trivial unitary central character, up to isomorphism there is a unique irreducible unitary representation of the Heisenberg group with that central character. Further, any unitary representation with that central character is a multiple of that irreducible.

Proof: We construct the irreducible for given non-trivial central character ω as an L2 induced representation, prove its irreducibility, and then prove uniqueness. Given a unitary representation π of the Heisenberg group H, the associated Weyl transform [1] πϕ of a Schwartz function ϕ on the alternating space V is the action of π integrated over V ⊂ H: Z πϕ = ϕ(v) · π(v) dv (ϕ ∈ S V , unitary π) V

[1] Unitaries π with central character ω are acted-on by functions ϕ on H that are compactly supported modulo the center, and transform by the complex conjugate character ω under the center, by Z ϕ · x = f(h) π(h)(x) dx (for x ∈ π) H/Z

The convolution product on such functions is Z (ϕ ∗ ψ)(g) = ϕ(gh−1) ψ(h) dh H/Z

Over R, as opposed to Qp, one can also consider the reduced Heisenberg group H/ ker ω, which has the feature that Z/ ker ω has become compact. This device is unavailable for Qp.

1 Paul Garrett: Stone - von Neumann theorem (April 22, 2015)

Given a Lagrangian subspace W of V , W + Z is a maximal abelian subgroup of H. Extend a non-trivial central character ω trivially to W + Z by

ω(w + z) = ω(z)

2 H [0.0.2] Claim: For the L induced representation σ = IndW +Z ω the map ϕ → σϕ on S V extends to an isometric bijection from L2(V ) to Hilbert-Schmidt operators on σ.

Proof: We determine the kernel functions for operators σϕ. Let V = W ⊕ W 0 with complementary Lagrangian subspace W 0 to W . On non-abelian subgroups of H we may revert to explicit denotation of the exponential map. The relation ex+y = ex · ey · e−hx,yi/2 for x, y ∈ V gives an operator equality

Z Z Z Z −hx, yi σϕ = ϕ(x + y) σ(ex+y) dx dy = ϕ(x + y) σ(ex ey) ω dx dy W 0 W W 0 W 2

x H x w0 w0 x The action of e ∈ H on f in the usual model of IndW +Z ω is by right translation (e f)(e ) = f(e · e ). 0 0 0 The relation ew ex = ex ew e−hx,w i gives

0 0 0 0 0 (exf)(ew ) = f(ew ex) = f(ex ew e−hx,w i) = ω(−hx, w0i) · f(ew ) (for w0 ∈ W 0 and x ∈ W )

Identify the representation space of σ with L2(W 0) via exp(W + Z)\H ≈ exp(W 0) ≈ W 0. For f ∈ L2(W 0) and w0 ∈ W 0, Z Z −hx, yi (σϕ)f(w0) = ω ϕ(x + y)(exeyf)(w0) dx dy 2 W 0 W From (exeyf)(w0) = ω(−hx, w0i) · (eyf)(w0) = ω(−hx, w0i) · f(y + w0) rearrange to see the kernel function: Z Z −hx, yi (σϕ)f(w0) = ω ω(−hx, w0i) ϕ(x + y) f(y + w0) dx dy 2 W 0 W Z Z −hx, y + 2w0i Z Z −hx, y + w0i = ω ϕ(x + y) f(y + w0) dx dy = ω ϕ(x + y − w0) f(y) dx dy 2 2 W 0 W W 0 W Z Z 0 0 −hx, y + w i 0 = Kϕ(·, y) f(y) dy (with kernel Kϕ(w , y) = ω ϕ(x + y − w ) dx) W 0 W 2

The integral in x ∈ W is a Fourier transform FW along W ⊂ V , with respect to the character x → ω(hx, yi/2), making the Fourier transform a Schwartz function on W 0. Indeed, this (partial) Fourier transform is a homeomorphism S V → S V , so

0 0 y + w 0 0 0 Kϕ(w , y) = FW ϕ , y − w ∈ S (W × W ) 2

Further, the partial Fourier transform is an isometry in the corresponding L2 metrics, so extends by continuity from S V → S V to an isometry L2(W × W 0) → L2(W 0 × W 0). Thus, ϕ ∈ L2(W × W 0) gives a Hilbert- Schmidt operator on L2(W 0). Conversely, every Hilbert-Schmidt operator is given by such a kernel, which arises from some ϕ ∈ L2(W × W 0). ///

[0.0.3] Corollary: Finite-rank operators are in the image of L2(V ) under σ. ///

[0.0.4] Corollary: The only continuous linear operators on σ commuting with σh for all h ∈ H are scalars.

2 Paul Garrett: Stone - von Neumann theorem (April 22, 2015)

Proof: Such T commutes with all integral operators σϕ for ϕ ∈ S V , therefore with all operators arising from ϕ ∈ L2(V ), therefore with all Hilbert-Schmidt operators, including all ﬁnite-rank operators. Thus, for any vector e ∈ V , the rank-one orthogonal projector P to C · e commutes with T , and P ◦ T = T ◦ P implies that T stabilizes the line C·e. Necessarily T acts by a scalar λ on any line it stabilizes. Likewise, for another vector e0, T acts by a scalar λ0 on C · e0, and by a scalar on the line C · (e + e0), as well. Thus, λ = λ0, and T is a scalar. ///

[0.0.5] Corollary: The representation σ is irreducible.

Proof: For a closed stable subspace π, the orthogonal projector P to π commutes with H. By the previous corollary, P is scalar, so is 0 or 1. ///

Let ∗ be convolution of (Z, ω)-equivariant smooth compactly-supported functions on H: Z (f ∗ g)(x) = f(xy−1) g(y) dy H/Z

∞ ∞ Transport this convolution to Cc (V ), via the identiﬁcation of ϕ ∈ Cc (V ) with (Z, ω)-equivariant functions ϕ on H by e v ϕe(e · z) = ω(z) · ϕ(v) (for v ∈ V and z ∈ Z) ∞ Thus, for f, g ∈ Cc (V ), Z Z Z v −y y v−y −hv,yi y hv, yi (f ∗ g)(v) = f(e · e ) g(e ) dy = f(e · e 2 ) g(e ) dy = ω f(v − y) g(y) dy V H/Z V 2

Unsurprisingly, this convolution is not the convolution on the additive group V . This convolution extends to Schwartz functions S V on V . Under convolution, the Schwartz functions ϕ on V give an algebra A of operators σϕ on the irreducible σ, compatible with composition of operators: σϕ ◦ σϕ0 = σ(ϕ ∗ ϕ0). For any unitary π of H with central character ω, ϕ → πϕ maps S V to an algebra Aπ of operators on π, and respects Hilbert space adjoints.

0 For f ∈ S W with |f|L2(W 0) = 1, the orthogonal projector P to the line C · f is Hilbert-Schmidt, so is in V 2 the image of the Weyl transform. That is, P = σϕ for σ = IndW +Z ω and some ϕ ∈ L (V ), with

ϕ ∗ ϕ = ϕ ϕ∨ = ϕ σϕ ◦ σh ◦ σϕ = hσhf, fi · σϕ (for h ∈ H)

In fact, ϕ ∈ S V , because f ∈ S V , so the kernel function of P is K(u, v) = f(u) f (v), and we have the relation y + w0 f(w0) f (y) = K(w0, y) = K (w0, y) = F ϕ , y − w0 ϕ W 2 The (partial) Fourier transform maps Schwartz functions to Schwartz functions. Thus, for any unitary π with central character ω, the integral operator πϕ makes sense.

[0.0.6] Claim: With ﬁxed ϕ as above, the images (πh ◦ πϕ)(x) for h ∈ H and x ∈ π are dense in the Hilbert space π.

Proof: Let y be orthogonal to all the indicated images. For h = eu in H, suppressing π, Z Z Z 0 = hh · ϕ · h−1x, yi = ϕ(v) heu · ev · e−ux, yi dv = ϕ(v) hev+hu,vix, yi dv = ϕ(v) ωhu, vi hev · x, yi dv V V V

The function v → ϕ(v) · hev · x, yi is continuous and in L2(V ), because ϕ is Schwartz, and because

|hev · x, yi| = |hσ(ev)(x), yi| ≤ |σ(ev)x| · |y| = |x| · |y| (because σ is unitary)

3 Paul Garrett: Stone - von Neumann theorem (April 22, 2015)

The vanishing Z 0 = ϕ(v) · ωhu, vi · hev · x, yi dv V is vanishing of the Fourier transform of v → ϕ(v) · hev · x, yi. Since ϕ ∈ S V , its Fourier transform is also in S V , so is 0 pointwise, giving the claimed density. ///

From ϕ ∗ ϕ = ϕ and ϕ∨ = ϕ, the image πϕ of ϕ under any unitary representation π is a projector to the 0 0 subspace X = (πϕ)(Y ) where Y is the representation space of π. As X is the kernel of 1Y −πϕ, it is closed. With X the representation space of σ, we want to deﬁne an H-isomorphism j : X ⊗ X0 → Y by j(σh)f ⊗ x0 = πh(x0) (for x0 ∈ X0 = (πϕ)Y , and h ∈ H))

The inner product on an algebraic tensor product C = A ⊗ B of inner product spaces is given by

0 0 0 0 ha ⊗ b, a ⊗ b iC = ha, a iA · hb, b iB

0 [0.0.7] Claim: For h1, h2 ∈ H and x1, x2 ∈ X , D E D E D E (πh1 ◦ πϕ)(x1), (πh2 ◦ πϕ)(x2) = (σh1)(f), (σh2)(f) · (πh1)(x1), (πh2)(x2) Y X X0

Proof: Suppress σ and π. Moving all the operators to one side, using the unitariness of π, and ϕ∨ = ϕ, −1 hh1 · ϕ · x1, h2 · ϕ · x2iY = hϕ · h2 h1 · ϕ · x1, x2iY

Using ϕ · h · ϕ = hh · f, fiX · ϕ, −1 hϕ · h2 h1 · ϕ · x1, x2iY = hh · f, fiX · hϕ · x1, x2iY Since ϕ ∗ ϕ = ϕ, and since π is unitary,

hϕ · x1, x2iY = hϕ · ϕ · x1, x2iY = hϕ · x1, ϕ · x2iY giving the asserted equality. /// P Since σ is irreducible, the collection of all ﬁnite complex-linear combinations i ci σhi · f with hi ∈ H is dense in the representation space X of σ. Try to deﬁne

0 X 0 X 0 j : X ⊗ X −→ Y by j (σhi)f ⊗ xi = (πhi)(xi) i i P 0 This is well-deﬁned: the inner product formula just proven shows that, for i(σhi)f ⊗ xi = 0, 2 X 0 X D 0 0 E X D E D 0 0 E (πhi)(xi) = (πhi)(xi), (πhj)(xj) = (σhi)f, (σhj)f · (πhi)(xi), (πhj)(xj) Y X0 i ij ij

2 X 0 = (σhi)f ⊗ xi = 0 X⊗X0 i Thus, the map is an isometry on a dense subspace of a Hilbert space, with dense image in another Hilbert space. Extend the map by continuity (isometry). The extension is surjective, hence an isometric isomorphism X ⊗ X0 → Y , and −1 j ◦ (σh ⊗ 1X0 ) ◦ j = πh (for all h ∈ H) so π is a multiple of σ in this sense. In particular, for π irreducible, necessarily X0 ≈ {0}, and π ≈ σ. ///

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