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Individual finite symplectic implementers

Statement

Assume the Axiom of Choice. On $L^{2} (R^{r})$ , for $r \geq 1$ and $ℏ > 0$ , define the Weyl unitaries in grouped label order $(q, p) \in R^{r} \oplus R^{r}$ by $[W (q, p) f] (x) = e^{i \sqrt{ℏ} q \cdot (x + \sqrt{ℏ} p / 2)} f (x + \sqrt{ℏ} p) .$ For $J_{0} = (\begin{array}{cc} 0 & I \\ - I & 0 \end{array})$ , every $T \in Sp (2 r, R)$ has a unitary $M_{T}$ such that $M_{T}^{*} W (z) M_{T} = W (T z) (z \in R^{2 r}) .$ Any two such unitaries differ by a scalar of modulus one. The covariance identity holds on all of $L^{2}$ ; the corresponding unbounded quadrature identities hold on the common transported Schwartz domain. This is an individual existence and phase-uniqueness result; it does not identify the global topology or kernel of a continuously chosen implementer family.

Facts & Assumptions

Given: AC (and hence Countable Choice), finite $r \geq 1$ , $ℏ > 0$ , the displayed Weyl system, and $T \in Sp (2 r, R)$ .

[A1]

AC supplies Countable Choice by applying a choice function to the family of values of any sequence of nonempty sets. This supplies the choice hypothesis of the regular-irreducibility and Fourier inputs below; AC also permits the spectral-calculus input [F4]. The proof constructs implementers directly and does not use the Stone–von Neumann theorem (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The displayed $W$ are unitary and form a strongly continuous irreducible Weyl system. Under $(q_{0}, p_{0}) = (- \sqrt{ℏ} p, q / \sqrt{ℏ})$ they are the Schrödinger system with fixed nonzero central parameter $ℏ$ ; the root lemma proves regularity and irreducibility under Countable Choice (The Schrödinger Weyl system is regular and irreducible).

[F2]

Every real symplectic matrix factors into symmetric upper and lower shears, invertible coordinate changes and partial Fourier rotations (Finite symplectic generator factorization).

[F3]

The imported $2 π$ -frequency transform $F_{2 π}$ is unitary on $L^{2}$ and a topological automorphism of Schwartz space (Fourier transform on complex L1 classes, Plancherel theorem, Fourier transform is a topological automorphism of Schwartz space). The angular-frequency unitary is exactly $F_{a n g} f (k) = (2 π)^{- r / 2} (F_{2 π} f) (k / (2 π)) = (2 π)^{- r / 2} \int e^{- i k \cdot x} f (x) d x .$ The real dilation is an $L^{2}$ unitary by change of variables and a Schwartz automorphism: for multi-indices $α, β \in N_{0}^{r}$ , define $p_{α, β} (g) = \sup_{k} ∣ k^{α} \partial^{β} g (k) ∣$ ; then $p_{α, β} ⁣ ((2 π)^{- r / 2} g (\cdot / (2 π))) = (2 π)^{∣ α ∣ - ∣ β ∣ - r / 2} p_{α, β} (g),$ and its inverse has the corresponding finite reciprocal factors. Thus $F_{a n g}$ has both properties. Integration by parts in its kernel gives $F_{a n g} (- i \partial_{x_{j}} f) = k_{j} F_{a n g} f$ for every $f \in S$ .

[F4]

For a bounded normal operator on a nonzero complex Hilbert space, its spectral projection valued measure represents its bounded Borel functional calculus; a projection commuting with the operator's generated unitaries is therefore available from the spectral calculus of a commuting self-adjoint operator (Spectral theorem for bounded normal operators pvm form).

[F5]

If $T : U \to V$ is a $C^{1}$ diffeomorphism of open subsets of $R^{r}$ and $g \in L^{1} (V)$ , then $\int_{V} g (y) d y = \int_{U} g (T (x)) ∣ \det D T (x) ∣ d x$ ; for $T (x) = A x$ and $g = ∣ f ∣^{2}$ , this gives $∥ D_{A} f ∥_{2} = ∥ f ∥_{2}$ (A C^1 diffeomorphism satisfies the change-of-variables formula for L^1 functions).

Proof

1.1givenA1F1F5algebra

The formula for $W$ is a unitary composition of translation and a modulus-one multiplier; its inverse is $W (- q, - p)$ . Direct substitution gives $W (q, p) W (q^{'}, p^{'}) = e^{- i ℏ (q \cdot p^{'} - p \cdot q^{'}) / 2} W (q + q^{'}, p + p^{'}) .$ For a real symmetric $X$ , the chirp $[C_{X} f] (x) = e^{i x^{T} X x / 2} f (x)$ is unitary, and substitution in the Weyl formula gives $C_{X}^{*} W (q, p) C_{X} = W (q + X p, p)$ , the upper shear. For invertible real $A$ , $[D_{A} f] (x) = ∣ \det A ∣^{1 / 2} f (A x)$ is unitary by [F5] applied to the diffeomorphism $x \mapsto A x$ and the integrable function $∣ f ∣^{2}$ ; direct substitution gives $D_{A}^{*} W (q, p) D_{A} = W (A^{- T} q, A p)$ , an invertible coordinate change.

2.1F2F3step 1.1algebra

Let $F_{a n g}$ be the exact angular-frequency unitary of [F3]. Substitution into the Fourier integral on Schwartz functions gives $[F_{a n g} W (q, p) f] (k) = e^{i \sqrt{ℏ} p \cdot (k - \sqrt{ℏ} q / 2)} [F_{a n g} f] (k - \sqrt{ℏ} q),$ which is exactly $[W (p, - q) F_{a n g} f] (k)$ . Both sides are bounded operators, and Schwartz functions are dense, so this identity holds on all of $L^{2}$ . Applying it to selected coordinates gives partial Fourier rotations. Conjugating an upper shear by the full or partial Fourier rotation gives every lower shear, since the matrix identity is $(\begin{array}{cc} 0 & - I \\ I & 0 \end{array}) U_{X} (\begin{array}{cc} 0 & I \\ - I & 0 \end{array}) = L_{- X}$ . Thus each generator and inverse in [F2] has a unitary Weyl implementer.

3.1F2step 1.1step 2.1algebra

Each generator unitary maps Schwartz space onto itself: this follows from the product rule for the smooth quadratic chirp, the chain rule for an invertible linear coordinate change, and Fourier invariance of Schwartz space. On any connected parameter neighborhood on which $A$ remains invertible, the chirp and coordinate formulas vary continuously in every Schwartz seminorm; the Fourier factor is fixed. If $G_{1}^{*} W (z) G_{1} = W (T_{1} z)$ and $G_{2}^{*} W (z) G_{2} = W (T_{2} z)$ , then $(G_{1} G_{2})^{*} W (z) (G_{1} G_{2}) = W (T_{2} T_{1} z)$ . Therefore, for any finite factorization $T = T_{k} \dots T_{1}$ supplied by [F2], multiplying the corresponding generator unitaries in the order $G_{1} \dots G_{k}$ gives a unitary $M_{T}$ with $M_{T}^{*} W (z) M_{T} = W (T z)$ for every label $z$ . This proves existence and does not invoke a general metaplectic existence theorem.

4.1F1F4step 3.1algebra∎

Suppose $M_{1}, M_{2}$ implement the same $T$ . Then $X = M_{2} M_{1}^{*}$ commutes with every $W (z)$ by applying their two covariance identities. Since the Weyl system is irreducible by [F1], every spectral projection of either bounded self-adjoint operator $Re X = (X + X^{*}) / 2$ and $Im X = (X - X^{*}) / (2 i)$ commutes with every $W (z)$ and has invariant range. Irreducibility forces each such projection to be $0$ or $I$ . The spectral measure in [F4] then implies each self-adjoint part is scalar: if a bounded self-adjoint operator had two distinct spectral values, disjoint neighborhoods of those values would yield a nonzero proper spectral projection. Hence $X = λ I$ for some $λ \in C$ ; since $X$ is unitary, $∣ λ ∣ = 1$ . Thus $M_{1}, M_{2}$ differ only by a phase.

Individual finite symplectic implementers

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources