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Stone–von Neumann theorem

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Template:Short description 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.<ref>Template:Citation</ref><ref>Template:Citation</ref><ref>Template:Citation</ref><ref>Template:Citation</ref>

Representation issues of the commutation relationsEdit

In quantum mechanics, physical observables are represented mathematically by linear operators on Hilbert spaces.

For a single particle moving on the real line <math>\mathbb{R}</math>, there are two important observables: position and momentum. In the Schrödinger representation quantum description of such a particle, the position operator Template:Mvar and momentum operator <math>p</math> are respectively given by <math display="block">\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}</math> on the domain <math>V</math> of infinitely differentiable functions of compact support on <math>\mathbb{R}</math>. Assume <math>\hbar</math> to be a fixed non-zero real number—in quantum theory <math>\hbar</math> is the reduced Planck constant, which carries units of action (energy times time).

The operators <math>x</math>, <math>p</math> satisfy the canonical commutation relation Lie algebra, <math display="block"> [x,p] = x p - p x = i \hbar.</math>

Already in his classic book,<ref>Weyl, H. (1927), "Quantenmechanik und Gruppentheorie", Zeitschrift für Physik, 46 (1927) pp. 1–46, {{#invoke:doi|main}}; Weyl, H., The Theory of Groups and Quantum Mechanics, Dover Publications, 1950, Template:Isbn.</ref> Hermann Weyl observed that this commutation law was impossible to satisfy for linear operators Template:Mvar, Template:Mvar acting on finite-dimensional spaces unless Template:Math vanishes. This is apparent from taking the trace over both sides of the latter equation and using the relation Template:Math; 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<ref group=note>Template:Math, hence Template:Math, so that, Template:Math.</ref>). For notational convenience, the nonvanishing square root of Template:Math may be absorbed into the normalization of Template:Mvar and Template:Mvar, 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.<ref name="Hall 2013">Template:Citation</ref>Template:Rp 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,Template:R namely Sylvester's clock and shift matrices in the finite Heisenberg group, discussed below.)

Uniqueness of representationEdit

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 Template:Mvar and Template:Mvar be two self-adjoint operators satisfying the canonical commutation relation, Template:Math, and Template:Mvar and Template:Mvar two real parameters. Introduce Template:Math and Template:Math, the corresponding unitary groups given by functional calculus. (For the explicit operators Template:Math and Template:Math defined above, these are multiplication by Template:Math and pullback by translation Template:Math.) A formal computationTemplate:R (using a special case of the Baker–Campbell–Hausdorff formula) readily yields <math display="block">e^{itQ} e^{isP} = e^{-i st} e^{isP} e^{itQ} .</math>

Conversely, given two one-parameter unitary groups Template:Math and Template:Math satisfying the braiding relation Template:Equation box 1 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 (Template:EquationNote).Template:R 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 Template:Math and Template:Math 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.Template:R In other words, for any two such Template:Math and Template:Math acting jointly irreducibly on a Hilbert space Template:Mvar, there is a unitary operator Template:Math so that <math display="block">W^*U(t)W = e^{itx} \quad \text{and} \quad W^*V(s)W = e^{isp},</math> where Template:Mvar and Template:Mvar are the explicit position and momentum operators from earlier. When Template:Mvar is Template:Mvar in this equation, so, then, in the Template:Mvar-representation, it is evident that Template:Mvar is unitarily equivalent to Template:Math, and the spectrum of Template:Mvar must range along the entire real line. The analog argument holds for Template:Mvar.

There is also a straightforward extension of the Stone–von Neumann theorem to Template:Mvar degrees of freedom.Template:R 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), <math display="block"> [U(t)\psi ] (x)=e^{itx} \psi(x), \qquad [V(s)\psi ](x)= \psi(x+s) .</math> Template:See also

Representation theory formulationEdit

In terms of representation theory, the Stone–von Neumann theorem classifies certain unitary representations of the Heisenberg group. This is discussed in more detail in the Heisenberg group section, below.

Informally stated, with certain technical assumptions, every representation of the Heisenberg group Template:Math is equivalent to the position operators and momentum operators on Template:Math. Alternatively, that they are all equivalent to the Weyl algebra (or CCR algebra) on a symplectic space of dimension Template:Math.

More formally, there is a unique (up to scale) non-trivial central strongly continuous unitary representation.

This was later generalized by Mackey theory – and was the motivation for the introduction of the Heisenberg group in quantum physics.

In detail:

In all cases, if one has a representation Template:Math, where Template:Math is an algebraTemplate:Clarify and the center maps to zero, then one simply has a representation of the corresponding abelian group or algebra, which is Fourier theory.Template:Clarify

If the center does not map to zero, one has a more interesting theory, particularly if one restricts oneself to central representations.

Concretely, by a central representation one means a representation such that the center of the Heisenberg group maps into the center of the algebra: for example, if one is studying matrix representations or representations by operators on a Hilbert space, then the center of the matrix algebra or the operator algebra is the scalar matrices. Thus the representation of the center of the Heisenberg group is determined by a scale value, called the quantization value (in physics terms, the Planck constant), and if this goes to zero, one gets a representation of the abelian group (in physics terms, this is the classical limit).

More formally, the group algebra of the Heisenberg group over its field of scalars K, written Template:Math, has center Template:Math, so rather than simply thinking of the group algebra as an algebra over the field Template:Mvar, one may think of it as an algebra over the commutative algebra Template:Math. As the center of a matrix algebra or operator algebra is the scalar matrices, a Template:Math-structure on the matrix algebra is a choice of scalar matrix – a choice of scale. Given such a choice of scale, a central representation of the Heisenberg group is a map of Template:Math-algebras Template:Math, which is the formal way of saying that it sends the center to a chosen scale.

Then the Stone–von Neumann theorem is that, given the standard quantum mechanical scale (effectively, the value of ħ), every strongly continuous unitary representation is unitarily equivalent to the standard representation with position and momentum.

Reformulation via Fourier transformEdit

Let Template:Mvar be a locally compact abelian group and Template:Math be the Pontryagin dual of Template:Mvar. The Fourier–Plancherel transform defined by <math display="block">f \mapsto {\hat f}(\gamma) = \int_G \overline{\gamma(t)} f(t) d \mu (t)</math> extends to a C*-isomorphism from the group C*-algebra Template:Math of Template:Mvar and Template:Math, i.e. the spectrum of Template:Math is precisely Template:Math. When Template:Mvar is the real line Template:Math, 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 Template:Mvar acts on the Template:Mvar*-algebra Template:Math by right translation Template:Mvar: for Template:Mvar in Template:Mvar and Template:Mvar in Template:Math, <math display="block">(s \cdot f)(t) = f(t + s).</math>

Under the isomorphism given above, this action becomes the natural action of Template:Mvar on Template:Math: <math display="block"> \widehat{ (s \cdot f) }(\gamma) = \gamma(s) \hat{f} (\gamma).</math>

So a covariant representation corresponding to the Template:Mvar*-crossed product <math display="block">C^*\left( \hat{G} \right) \rtimes_{\hat{\rho}} G </math> is a unitary representation Template:Math of Template:Mvar and Template:Math of Template:Math such that <math display="block">U(s) V(\gamma) U^*(s) = \gamma(s) V(\gamma).</math>

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 <math display="block">C_0(G) \rtimes_\rho G </math> are unitarily equivalent to the <math>{\mathcal K}\left(L^2(G)\right)</math>, the compact operators on Template:Math. Therefore, all pairs Template:Math are unitarily equivalent. Specializing to the case where Template:Math yields the Stone–von Neumann theorem.

Heisenberg groupEdit

The above canonical commutation relations for Template:Mvar, Template:Mvar are identical to the commutation relations that specify the Lie algebra of the general Heisenberg group Template:Math for Template:Mvar a positive integer. This is the Lie group of Template:Math square matrices of the form <math display="block"> \mathrm{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, one can reformulate the Stone von Neumann theorem in the language of representation theory.

Note that the center of Template:Math consists of matrices Template:Math. However, this center is not the identity operator in Heisenberg's original CCRs. The Heisenberg group Lie algebra generators, e.g. for Template:Math, are <math display="block">\begin{align} 

 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{align}</math> and the central generator Template:Math is not the identity.

Template:Math theorem

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

Note that Template:Math 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 Template:Math and multiplication by a function of absolute value 1. To show Template:Math 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 Template:Mvar, Template:Mvar of the Heisenberg group Template:Math which are non-trivial on the center of Template:Math are unitarily equivalent if and only if Template:Math for any Template:Mvar in the center of Template:Math.

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 transformEdit

For any non-zero Template:Mvar, the mapping <math display="block"> \alpha_h: \mathrm{M}(a,b,c) \to \mathrm{M} \left( -h^{-1} b,h a, c -a\cdot b \right) </math> is an automorphism of Template:Math which is the identity on the center of Template:Math. In particular, the representations Template:Math and Template:Math are unitarily equivalent. This means that there is a unitary operator Template:Mvar on Template:Math such that, for any Template:Mvar in Template:Math, <math display="block"> W U_h(g) W^* = U_h \alpha (g).</math>

Moreover, by irreducibility of the representations Template:Math, it follows that up to a scalar, such an operator Template:Mvar is unique (cf. Schur's lemma). Since Template:Mvar is unitary, this scalar multiple is uniquely determined and hence such an operator Template:Mvar is unique.

Template:Math theorem

This means that, ignoring the factor of Template:Math in the definition of the Fourier transform, <math display="block"> \int_{\mathbf{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_{\mathbf{R}^n} e^{-i y \cdot ( p - b)} \psi(y) \ dy.</math>

This theorem has the immediate implication that the Fourier transform is unitary, also known as the Plancherel theorem. Moreover, <math display="block"> (\alpha_h)^2 \mathrm{M}(a,b,c) =\mathrm{M}(- a, -b, c). </math>

Template:Math theorem

From this fact the Fourier inversion formula easily follows.

Example: Segal–Bargmann spaceEdit

The Segal–Bargmann space is the space of holomorphic functions on Template:Math that are square-integrable with respect to a Gaussian measure. Fock observed in 1920s that the operators <math display="block"> a_j = \frac{\partial}{\partial z_j}, \qquad a_j^* = z_j, </math> acting on holomorphic functions, satisfy the same commutation relations as the usual annihilation and creation operators, namely, <math display="block"> \left [a_j,a_k^* \right ] = \delta_{j,k}. </math>

In 1961, Bargmann showed that Template:Math is actually the adjoint of Template:Math with respect to the inner product coming from the Gaussian measure. By taking appropriate linear combinations of Template:Math and Template:Math, 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.Template:R The Stone–von Neumann theorem therefore applies and implies the existence of a unitary map from Template:Math to the Segal–Bargmann space that intertwines the usual annihilation and creation operators with the operators Template:Math and Template:Math. This unitary map is the Segal–Bargmann transform.

Representations of finite Heisenberg groupsEdit

The Heisenberg group Template:Math is defined for any commutative ring Template:Mvar. In this section let us specialize to the field Template:Math for Template:Mvar a prime. This field has the property that there is an embedding Template:Mvar of Template:Mvar as an additive group into the circle group Template:Math. Note that Template:Math is finite with cardinality Template:Math. For finite Heisenberg group Template:Math 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 Template:Mvar in Template:Mvar define the representation Template:Math on the finite-dimensional inner product space Template:Math by <math display="block">\left[U_h \mathrm{M}(a, b, c) \psi\right](x) = \omega(b \cdot x + h c) \psi(x + ha). </math>

Template:Math theorem

It follows that <math display="block"> \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. </math>

By the orthogonality relations for characters of representations of finite groups this fact implies the corresponding Stone–von Neumann theorem for Heisenberg groups Template:Math, particularly:

Actually, all irreducible representations of Template:Math on which the center acts nontrivially arise in this way.Template:R

GeneralizationsEdit

The Stone–von Neumann theorem admits numerous generalizations. Much of the early work of George Mackey was directed at obtaining a formulation<ref>Mackey, G. W. (1976). The Theory of Unitary Group Representations, The University of Chicago Press, 1976.</ref> 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 alsoEdit

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NotesEdit

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ReferencesEdit

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