In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named after Marshall Stone and John von Neumann.^[1][2][3][4]

Representation issues of 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 ${\displaystyle \mathbb {R} }$, there are two important observables: position and momentum. In the Schrödinger representation quantum description of such a particle, the position operator x and momentum operator ${\displaystyle p}$ are respectively given by

$${\displaystyle {\begin{aligned}[][x\psi ](x_{0})&=x_{0}\psi (x_{0})\\[][p\psi ](x_{0})&=-i\hbar {\frac {\partial \psi }{\partial x}}(x_{0})\end{aligned}}}$$

on the domain ${\displaystyle V}$ of infinitely differentiable functions of compact support on ${\displaystyle \mathbb {R} }$. Assume ${\displaystyle \hbar }$ to be a fixed non-zero real number—in quantum theory ${\displaystyle \hbar }$ is the reduced Planck's constant, which carries units of action (energy times time).

The operators ${\displaystyle x}$, ${\displaystyle p}$ satisfy the canonical commutation relation Lie algebra,

$${\displaystyle [x,p]=xp-px=i\hbar .}$$

Already in his classic book,^[5] Hermann Weyl observed that this commutation law was impossible to satisfy for linear operators p, x acting on finite-dimensional spaces unless ℏ vanishes. This is apparent from taking the trace over both sides of the latter equation and using the relation Trace(AB) = Trace(BA); the left-hand side is zero, the right-hand side is non-zero. Further analysis shows that any two self-adjoint operators satisfying the above commutation relation cannot be both bounded (in fact, a theorem of Wielandt shows the relation cannot be satisfied by elements of any normed algebra^{[note 1]}). For notational convenience, the nonvanishing square root of ℏ may be absorbed into the normalization of p and x, so that, effectively, it is replaced by 1. We assume this normalization in what follows.

The idea of the Stone–von Neumann theorem is that any two irreducible representations of the canonical commutation relations are unitarily equivalent. Since, however, the operators involved are necessarily unbounded (as noted above), there are tricky domain issues that allow for counter-examples.^{[6]: Example 14.5} To obtain a rigorous result, one must require that the operators satisfy the exponentiated form of the canonical commutation relations, known as the Weyl relations. The exponentiated operators are bounded and unitary. Although, as noted below, these relations are formally equivalent to the standard canonical commutation relations, this equivalence is not rigorous, because (again) of the unbounded nature of the operators. (There is also a discrete analog of the Weyl relations, which can hold in a finite-dimensional space,^{[6]: Chapter 14, Exercise 5} namely Sylvester's clock and shift matrices in the finite Heisenberg group, discussed below.)

Uniqueness of representation

One would like to classify representations of the canonical commutation relation by two self-adjoint operators acting on separable Hilbert spaces, up to unitary equivalence. By Stone's theorem, there is a one-to-one correspondence between self-adjoint operators and (strongly continuous) one-parameter unitary groups.

Let Q and P be two self-adjoint operators satisfying the canonical commutation relation, [Q, P] = i, and s and t two real parameters. Introduce e^itQ and e^isP, the corresponding unitary groups given by functional calculus. (For the explicit operators x and p defined above, these are multiplication by e^itx and pullback by translation x → x + s.) A formal computation^{[6]: Section 14.2} (using a special case of the Baker–Campbell–Hausdorff formula) readily yields

$${\displaystyle e^{itQ}e^{isP}=e^{-ist}e^{isP}e^{itQ}.}$$

Conversely, given two one-parameter unitary groups U(t) and V(s) satisfying the braiding relation

formally differentiating at 0 shows that the two infinitesimal generators satisfy the above canonical commutation relation. This braiding formulation of the canonical commutation relations (CCR) for one-parameter unitary groups is called the Weyl form of the CCR.

It is important to note that the preceding derivation is purely formal. Since the operators involved are unbounded, technical issues prevent application of the Baker–Campbell–Hausdorff formula without additional domain assumptions. Indeed, there exist operators satisfying the canonical commutation relation but not the Weyl relations (E1).^{[6]: Example 14.5} Nevertheless, in "good" cases, we expect that operators satisfying the canonical commutation relation will also satisfy the Weyl relations.

The problem thus becomes classifying two jointly irreducible one-parameter unitary groups U(t) and V(s) which satisfy the Weyl relation on separable Hilbert spaces. The answer is the content of the Stone–von Neumann theorem: all such pairs of one-parameter unitary groups are unitarily equivalent.^{[6]: Theorem 14.8} In other words, for any two such U(t) and V(s) acting jointly irreducibly on a Hilbert space H, there is a unitary operator W : L²(R) → H so that

$${\displaystyle W^{*}U(t)W=e^{itx}\quad {\text{and}}\quad W^{*}V(s)W=e^{isp},}$$

where p and x are the explicit position and momentum operators from earlier. When W is U in this equation, so, then, in the x-representation, it is evident that P is unitarily equivalent to e^−itQ P e^itQ = P + t, and the spectrum of P must range along the entire real line. The analog argument holds for Q.

There is also a straightforward extension of the Stone–von Neumann theorem to n degrees of freedom.^{[6]: Theorem 14.8} Historically, this result 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),

$${\displaystyle [U(t)\psi ](x)=e^{itx}\psi (x),\qquad [V(s)\psi ](x)=\psi (x+s).}$$

Reformulation via Fourier transform

Let G be a locally compact abelian group and G^{^} be the Pontryagin dual of G. The Fourier–Plancherel transform defined by

$${\displaystyle f\mapsto {\hat {f}}(\gamma )=\int _{G}{\overline {\gamma (t)}}f(t)d\mu (t)}$$

extends to a C*-isomorphism from the group C*-algebra C*(G) of G and C₀(G^{^}), i.e. the spectrum of C*(G) is precisely G^{^}. When G is the real line R, this is Stone's theorem characterizing one-parameter unitary groups. The theorem of Stone–von Neumann can also be restated using similar language.

The group G acts on the C*-algebra C₀(G) by right translation ρ: for s in G and f in C₀(G),

$${\displaystyle (s\cdot f)(t)=f(t+s).}$$

Under the isomorphism given above, this action becomes the natural action of G on C*(G^{^}):

$${\displaystyle {\widehat {(s\cdot f)}}(\gamma )=\gamma (s){\hat {f}}(\gamma ).}$$

So a covariant representation corresponding to the C*-crossed product

$${\displaystyle C^{*}\left({\hat {G}}\right)\rtimes _{\hat {\rho }}G}$$

is a unitary representation U(s) of G and V(γ) of G^{^} such that

$${\displaystyle U(s)V(\gamma )U^{*}(s)=\gamma (s)V(\gamma ).}$$

It is a general fact that covariant representations are in one-to-one correspondence with *-representation of the corresponding crossed product. On the other hand, all irreducible representations of

$${\displaystyle C_{0}(G)\rtimes _{\rho }G}$$

are unitarily equivalent to the ${\displaystyle {\mathcal {K}}\left(L^{2}(G)\right)}$, the compact operators on L²(G)). Therefore, all pairs {U(s), V(γ)} are unitarily equivalent. Specializing to the case where G = R yields the Stone–von Neumann theorem.

The Heisenberg group

The above canonical commutation relations for P, Q are identical to the commutation relations that specify the Lie algebra of the general Heisenberg group H_2n+1 for n a positive integer. This is the Lie group of (n + 2) × (n + 2) square matrices of the form

$${\displaystyle \mathrm {M} (a,b,c)={\begin{bmatrix}1&a&c\\0&1_{n}&b\\0&0&1\end{bmatrix}}.}$$

In fact, using the Heisenberg group, one can reformulate the Stone von Neumann theorem in the language of representation theory.

Note that the center of H_2n+1 consists of matrices M(0, 0, c). However, this center is not the identity operator in Heisenberg's original CCRs. The Heisenberg group Lie algebra generators, e.g. for n = 1, are

$${\displaystyle {\begin{aligned}P&={\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix}},&Q&={\begin{bmatrix}0&0&0\\0&0&1\\0&0&0\end{bmatrix}},&z&={\begin{bmatrix}0&0&1\\0&0&0\\0&0&0\end{bmatrix}},\end{aligned}}}$$

and the central generator z = log M(0, 0, 1) = exp(z) − 1 is not the identity.

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

$${\displaystyle \left[U_{h}(\mathrm {M} (a,b,c))\right]\psi (x)=e^{i(b\cdot x+hc)}\psi (x+ha).}$$

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 ha 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; this claim, nevertheless, follows easily from the Stone–von Neumann theorem as stated above. We will sketch below 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.

One representation of the Heisenberg group which is important in number theory and the theory of modular forms is the theta representation, so named because the Jacobi theta function is invariant under the action of the discrete subgroup of the Heisenberg group.

Relation to the Fourier transform

For any non-zero h, the mapping

$${\displaystyle \alpha _{h}:\mathrm {M} (a,b,c)\to \mathrm {M} \left(-h^{-1}b,ha,c-a\cdot b\right)}$$

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

$${\displaystyle WU_{h}(g)W^{*}=U_{h}\alpha (g).}$$

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). Since W is unitary, this scalar multiple is uniquely determined and hence such an operator W is unique.

Theorem —  The operator W is the Fourier transform on L²(Rⁿ).

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

$${\displaystyle \int _{\mathbf {R} ^{n}}e^{-ix\cdot p}e^{i(b\cdot x+hc)}\psi (x+ha)\ dx=e^{i(ha\cdot p+h(c-b\cdot a))}\int _{\mathbf {R} ^{n}}e^{-iy\cdot (p-b)}\psi (y)\ dy.}$$

This theorem has the immediate implication that the Fourier transform is unitary, also known as the Plancherel theorem. Moreover,

$${\displaystyle (\alpha _{h})^{2}\mathrm {M} (a,b,c)=\mathrm {M} (-a,-b,c).}$$

Theorem —  The operator W₁ such that

$${\displaystyle W_{1}U_{h}W_{1}^{*}=U_{h}\alpha ^{2}(g)}$$

is the reflection operator

$${\displaystyle [W_{1}\psi ](x)=\psi (-x).}$$

From this fact the Fourier inversion formula easily follows.

Example: The Segal–Bargmann space

The Segal–Bargmann space is the space of holomorphic functions on Cⁿ that are square-integrable with respect to a Gaussian measure. Fock observed in 1920s that the operators

$${\displaystyle a_{j}={\frac {\partial }{\partial z_{j}}},\qquad a_{j}^{*}=z_{j},}$$

acting on holomorphic functions, satisfy the same commutation relations as the usual annihilation and creation operators, namely,

$${\displaystyle \left[a_{j},a_{k}^{*}\right]=\delta _{j,k}.}$$

In 1961, Bargmann showed that a^∗
_j is actually the adjoint of a_j with respect to the inner product coming from the Gaussian measure. By taking appropriate linear combinations of a_j and a^∗
_j, one can then obtain "position" and "momentum" operators satisfying the canonical commutation relations. It is not hard to show that the exponentials of these operators satisfy the Weyl relations and that the exponentiated operators act irreducibly.^{[6]: Section 14.4} The Stone–von Neumann theorem therefore applies and implies the existence of a unitary map from L²(Rⁿ) to the Segal–Bargmann space that intertwines the usual annihilation and creation operators with the operators a_j and a^∗
_j. This unitary map is the Segal–Bargmann transform.

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/pZ for p a prime. This field has the property that there is an embedding ω of K as an additive group into the circle group T. Note that H_n(K) is finite with cardinality |K|^2n + 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

$${\displaystyle \left[U_{h}\mathrm {M} (a,b,c)\psi \right](x)=\omega (b\cdot x+hc)\psi (x+ha).}$$

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

$${\displaystyle \chi (\mathrm {M} (a,b,c))={\begin{cases}|K|^{n}\,\omega (hc)&{\text{if }}a=b=0\\0&{\text{otherwise}}\end{cases}}}$$

It follows that

$${\displaystyle {\frac {1}{\left|H_{n}(\mathbf {K} )\right|}}\sum _{g\in H_{n}(K)}|\chi (g)|^{2}={\frac {1}{|K|^{2n+1}}}|K|^{2n}|K|=1.}$$

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/pZ), particularly:

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

Actually, all irreducible representations of H_n(K) on which the center acts nontrivially arise in this way.^{[6]: Chapter 14, Exercise 5}

Generalizations

The Stone–von Neumann theorem admits numerous generalizations. Much of the early work of George Mackey was directed at obtaining a formulation^[7] 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

Notes

References

• Kirillov, A. A. (1976), Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, vol. 220, Berlin, New York: Springer-Verlag, ISBN 978-0-387-07476-4, MR 0407202
• Rosenberg, Jonathan (2004) "A Selective History of the Stone–von Neumann Theorem" Contemporary Mathematics 365. American Mathematical Society.
• Summers, Stephen J. (2001). "On the Stone–von Neumann Uniqueness Theorem and Its Ramifications." In John von Neumann and the foundations of quantum physics, pp. 135-152. Springer, Dordrecht, 2001, online.