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At zero central parameter, distinct regular characters are inequivalent

Statement

On $V = R^{2}$ with $ω ((q, p), (q^{'}, p^{'})) = q p^{'} - p q^{'}$ , set $λ = 0$ . For each real-linear functional $ℓ : V \to R$ , the one-dimensional operators $W_{ℓ} (v) z = e^{i ℓ (v)} z$ form a regular irreducible Weyl system with central parameter zero. The choices $ℓ_{0} (q, p) = 0$ and $ℓ_{1} (q, p) = π q$ are inequivalent. Thus the nonzero central-parameter hypothesis in the finite-dimensional uniqueness theorem is essential.

Facts & Assumptions

Given: The standard symplectic plane, the parameter $λ = 0$ , and a real-linear functional $ℓ$ .

[F1]

A Weyl system satisfies $W (0) = I$ and $W (v) W (w) = e^{- i λ ω (v, w) / 2} W (v + w)$ ; regularity is strong continuity of each line $t \mapsto W (t v)$ (Regular Weyl systems and irreducibility).

[F2]

With $⟨ z, w ⟩ = z \overline{w}$ , $C$ is a one-dimensional complex Hilbert space: the inner-product axioms are immediate and its norm metric is complete by The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts. Hence it has no nonzero proper linear subspaces (Real and complex inner-product spaces and their induced length, Hilbert space).

[F3]

$e^{i π} = - 1$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

Proof

technique · direct

Given: The data in the Statement.

1.1F1givenalgebra

Since $ℓ$ is real-linear, $W_{ℓ} (0) = 1$ and $W_{ℓ} (v) W_{ℓ} (w) = e^{i (ℓ (v) + ℓ (w))} = e^{i ℓ (v + w)} = W_{ℓ} (v + w)$ . This is exactly the Weyl relation at $λ = 0$ , and each scalar has modulus one.

2.1F1F2step 1.1

For every $v$ , the map $t \mapsto W_{ℓ} (t v) = e^{i t ℓ (v)}$ is continuous, so the system is regular. Its Hilbert space is one-dimensional, hence [F2] shows it is irreducible.

3.1F3step 1.1algebra∎

If a unitary intertwines $W_{ℓ_{0}}$ with $W_{ℓ_{1}}$ , it is a nonzero scalar on $C$ , and the intertwining identity forces $e^{i ℓ_{0} (v)} = e^{i ℓ_{1} (v)}$ for every $v$ . At $(q, p) = (1, 0)$ these values are $1$ and $e^{i π} = - 1$ by [F3], so they differ. Thus the two regular irreducible systems are inequivalent.

Boundary cases

The example uses $λ = 0$ ; it does not contradict uniqueness for fixed $λ \neq 0$ .
The inequivalence claim concerns the displayed pair of characters only.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Roland Speicher, Mathematical Aspects of Quantum Mechanics, Lecture Notes (standard reference, not scraped)