Stone-von Neumann theorem

In mathematics and in theoretical physics, the Stone-von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. The name is for Marshall Stone and John von Neumann.

Contents

1 Trying to represent the commutation relations 2 Weyl form of the canonical commutation relations 3 Another formulation 4 The Heisenberg group 5 Relation to the Fourier transform 6 Representations of finite Heisenberg groups 7 Generalizations 8 References

Trying to represent the commutation relations

In quantum mechanics, physical observables are represented mathematically by linear operators on Hilbert spaces. For a single particle moving on the real line R, there are two important observables: position and momentum. In the quantum-mechanical description of such a particle, the position operator Q and momentum operator P are respectively given by

<math> [Q \psi](x) = x \psi(x) \quad <math>

<math> [P \psi](x) = \frac{\hbar}{i}\psi'(x) <math>

on the domain V of infinitely differentiable functions of compact support on R. We assume <math> \hbar <math> is a fixed non-zero real number — in quantum theory <math> \hbar <math> is (up to a factor of 2π) Planck's constant, which is not dimensionless; it takes a small numerical value in terms of units in the macroscopic world. The operators P, Q satisfy the commutation relation

<math> Q P- P Q = \frac{\hbar}{i} \mathbf{1} <math>

Already in his classic volume, Hermann Weyl observed that this commutation law was impossible for linear operators P, Q acting on finite dimensional spaces (as is clear by applying the trace of a matrix), unless <math> \hbar <math> vanishes.

In the theory of quantization of classical mechanics, the question naturally arises whether it is possible to classify pairs of operators which satisfy the above commutation relations. The answer in general is no, without additional assumptions. To give a simple counterexample, consider the operators Q₊ and P₊ defined as operators in the same form as Q, P above, but acting on a different space, that is the space of infinitely differentiable functions of compact support on (0, ∞). The multiplication operator Q₊ is an essentially self-adjoint operator. It is also a non-negative operator, that is

<math> \langle Q_+ \psi | \psi \rangle \geq 0 \quad \forall \psi \in \operatorname{dom}(Q) <math>

so cannot possibly be equivalent to Q. Note that P₊ fails to be an essentially self adjoint operator on the given domain.

Weyl form of the canonical commutation relations

Instead of considering the operators P, Q, we will consider the pair of one-parameter groups of unitary operators e^{ia P} and e^{ib Q}; these operators are well-defined since P, Q are essentially self-adjoint on the domain V and so have unique self-adjoint extensions. Clearly e^{ib Q} is multiplication by the function e^{ib x}, while e^{ia P} is the operator of left translation by a, that is,

<math> [\operatorname{e}^{iaP} \psi](x) = \psi(x + \hbar a). <math>

Theorem. Let H be a separable Hilbert space and A, B self-adjoint operators on H. If

<math> \operatorname{e}^{ibA} \operatorname{e}^{iaB} = \operatorname{e}^{i \hbar a b}\operatorname{e}^{iaB} \operatorname{e}^{ibA}\quad <math>

then H is a finite or countably infinite Hilbert direct sum of Hilbert spaces {H_k}_k, each one invariant under both unitary groups e^{ib A} and e^{ia B}. Moreover, for each index k there is a unitary operator V_k : H_k → L²(R) such that

<math> V_k \operatorname{e}^{ibA} V_k^* = \operatorname{e}^{ibQ} <math>

<math> V_k \operatorname{e}^{iaB} V_k^* = \operatorname{e}^{iaP} <math>

Stated another way, any representation of the canonical commutation relations is a countable direct sum of isomorphic copies of e^{ia P} and e^{ib Q}.

This statement is usually referred to as the uniqueness of the Weyl form of the canonical commutation relations.

Another formulation

We can formulate this somewhat differently, noting that the unitary groups {e^{is P}} and {e^{it Q}} are jointly irreducible. This means that there are no closed subspaces other than {0} and L²(R) which are invariant under all the operators e^{is P} and e^{it Q}.

Theorem. Let H be a (non-trivial) separable Hilbert space A, B self-adjoint operators on H such that the Weyl commutation relations above hold and {e^{it A}} and {e^{is B}} are jointly irreducible. Then in the previous theorem the direct sum reduces to a single summand.

Historically this theorem was significant because it was a key step in proving that Heisenberg's matrix mechanics which presents quantum mechanical observables and dynamics in terms of infinite matrices, is unitarily equivalent to Schrödinger's wave mechanical formulation (see Schrödinger picture).

The Heisenberg group

The commutation relations for P, Q look very similar to the commutation relations that define the Lie algebra of general Heisenberg group H_n for n a positive integer. This is the Lie group of (n+2) × (n+2) square matrices of the form

<math> \operatorname{M}(a,b,c) = \begin{bmatrix} 1 & a & c \\ 0 & 1_n & b \\ 0 & 0 & 1 \end{bmatrix} <math>

In fact, using the Heisenberg group, we can formulate a far-reaching generalization of the Stone von Neumann theorem. Note that the center of H_n consists of matrices M(0, 0, c).

Theorem. For each non-zero real number h there is an irreducible representation U_h acting on on the Hilbert space L²(Rⁿ) by

<math> [U_h(\operatorname{M}(a,b,c))]\psi(x) = e^{i (b \cdot x + h c)} \psi(x+h a) <math>

All these representations are unitarily inequivalent and any irreducible representation which is not trivial on the center of H_n is unitarily equivalent to exactly one of these.

Note that U_h is a unitary operator because it is the composition of two operators which are easily seen to be unitary: the translation to the left by h a and multiplication by a function of absolute value 1. To show U_h is multiplicative is a straightforward calculation. The hard part of the theorem is showing the uniqueness which is beyond the scope of the article. However, below we sketch a proof of the corresponding Stone-von Neumann theorem for certain finite Heisenberg groups.

In particular, irreducible representations π, π' of the Heisenberg group H_n which are non-trivial on the center of H_n are unitarily equivalent if and only if π(z) = π'(z) for any z in the center of H_n.

A representation of the Heisenberg group can also be given on a Hilbert space of functions closely related to theta functions.

Relation to the Fourier transform

For any non-zero h, the mapping

<math> \alpha_h: \operatorname{M}(a,b,c) \rightarrow \operatorname{M}(-h^{-1} b,h a, c -a b) <math>

is an automorphism of H_n which is the identity on the center of H_n. In particular, the representations U_h and U_h α are unitarily equivalent. This means that there is a unitary operator W on L²(Rⁿ) such that for any g in H_n,

<math> W U_h(g) W^* = U_h \alpha (g) \quad <math>

Moreover, by irreducibility of the representations U_h, it follows that up to a scalar, such an operator W is unique (cf. Schur's lemma).

Theorem. The operator W is, up to a scalar multiple, the Fourier transform on L²(Rⁿ).

This means that (ignoring the factor of (2 π)^n/2 in the definition of the Fourier transform)

<math> \int_{\mathbb{R}^n} e^{-i x \cdot p} e^{i (b \cdot x + h c)}\psi (x+h a) \ dx = e^{ i (h a \cdot p + h (c - b \cdot a))} \int_{\mathbb{R}^n} e^{-i y \cdot ( p - b)} \psi(y) \ dy <math>

The previous theorem can actually be used to prove the unitary nature of the Fourier transform, also known as the Plancherel theorem. Moroever, note that

<math> (\alpha_h)^2 \operatorname{M}(a,b,c) =\operatorname{M}(- a, -b, c) <math>

Theorem. The operator W₁ such that

<math> W_1 U_h W_1^* = U_h \alpha^2 (g) \quad <math>

is the reflection operator

<math> [W_1 \psi](x) = \psi(-x).\quad <math>

From this fact the Fourier inversion formula easily follows.

Representations of finite Heisenberg groups

The Heisenberg group H_n(K) is defined for any commutative ring K. In this section let us specialize to the field K = Z/p Z for p a prime. This field has the property that there is an imbedding ω of K as an additive group into the circle group T. Note that H_n(K) is finite with cardinality |K|^{2 n+1}. For finite Heisenberg group H_n(K) one can give a simple proof of the Stone-von Neumann theorem using simple properties of character functions of representations. These properties follow from the orthogonality relations for characters of representations of finite groups.

For any non-zero h in K define the representation U_h on the finite-dimensional inner product space l²(Kⁿ) by

<math> [U_h \operatorname{M}(a,b,c) \psi](x) = \omega(b \cdot x + h c) \psi(x+ h a) <math>

Theorem. For a fixed non-zero h, the character function χ of U_h is given by:

<math> \chi (\operatorname{M}(a,b,c)) = \left\{ \begin{matrix} |\mathbf{K}|^n \ \omega( h c) & \mbox{ if } a = b = 0 \\ 0 & \mbox{ otherwise} \end{matrix} \right. <math>

It follows that

<math> \frac{1}{|\operatorname{H}_n(\mathbf{K})|} \sum_{g \in \operatorname{H}_n(\mathbf{K})} |\chi(g)|^2 = \frac{1}{|\mathbf{K}|^{2 n+1}} |\mathbf{K}|^{2 n} |\mathbf{K}| = 1 <math>

By the orthogonality relations for characters of representations of finite groups this fact implies the corresponding Stone-von Neumann theorem for Heisenberg groups H_n(Z/p Z), particularly:

• Irreducibility of U_h
• Pairwise inequivalence of all the representations U_h.

Generalizations

The Stone-von Neumann theorem admits numerous generalizations. Much of the early work of George Mackey was directed at obtaining a formulation of the theory of induced representations developed originally by Frobenius for finite groups to the context of unitary representations of locally compact topological groups.

See also CCR algebra

References

• H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications, 1950
• A. Kirillov, Elements de la Theorie des Representations, Editions MIR, 1974