09.03.03 · quantum-mechanics / semidirect-product-quantum

Quantum free particle as a representation of $E (3)$

shipped3 tiersLean: none

Anchor (Master): Mackey, *Induced Representations of Groups and Quantum Mechanics* (Benjamin 1968) Ch. 1-2 (the original imprimitivity-theorem treatment of the free particle); Souriau, *Structure des systèmes dynamiques* (Dunod 1970) Ch. III §13-15 on coadjoint orbits of $E(3)$ and prequantisation; Folland *A Course in Abstract Harmonic Analysis* (CRC 1995) Ch. 6 §6.6

Intuition Beginner

A free particle moving through three-dimensional space feels no forces. The energy comes entirely from motion, and the wave function $ψ (x)$ encoding where the particle might be found has to behave consistently when you shift the origin of coordinates or rotate the axes. The collection of all such shifts and rotations is the Euclidean group $E (3)$ , and the central principle of this unit is that the quantum free particle is a particular unitary action of $E (3)$ on the Hilbert space of wave functions. The Schrödinger equation drops out of the symmetry structure; it is not imposed by hand.

Why should this matter? In every other formulation of quantum mechanics, the free Hamiltonian $- ℏ^{2} Δ/2 m$ is written down first and the symmetries are checked afterward. The representation-theoretic perspective reverses the order: start with the group $E (3)$ acting on $R^{3}$ , ask which unitary actions on $L^{2} (R^{3})$ extend this geometric action, and the free-particle Hamiltonian appears as the Casimir invariant naming which irreducible piece you are in. Mass becomes a label on representations; momentum eigenstates become an orbit in dual space; the Fourier transform becomes the intertwiner that diagonalises the translation subgroup.

The bridge between the geometry of space and the algebra of quantum mechanics runs through this representation-theoretic picture. The same machinery, applied to the Poincaré group instead of $E (3)$ , recovers Wigner's classification of relativistic particles by mass and spin. The Euclidean case builds toward that whole programme.

Visual Beginner

A schematic showing a plane-wave wave function $ψ (x) = e^{i k \cdot x}$ on a slice of three-dimensional space, with a coordinate frame at the origin. To the right, the same wave function after a translation by a vector $a$ (the wave shifts rigidly) and after a rotation by $R \in S O (3)$ (the wave vector $k$ rotates with it). Below, a sphere of fixed radius $∣ k ∣ = k$ in dual momentum space, labelled "orbit of fixed energy $E = ℏ^{2} k^{2} /2 m$ ," with arrows showing that rotating $k$ leaves the energy fixed.

The picture captures the central geometry: each fixed-energy sphere in momentum space is a single orbit under $E (3)$ , and the wave functions whose momentum lies on that sphere form the corresponding irreducible representation.

Worked example Beginner

Compute how the plane wave $ψ (x) = e^{i k \cdot x}$ transforms under translation and rotation, and check that the energy and momentum behave the way the symmetry picture predicts.

Step 1. Start with one dimension to keep the algebra plain. The plane wave is $ψ (x) = e^{ik x}$ with momentum $p = ℏ k$ . Under translation by $a$ , the wave function becomes $ψ (x - a) = e^{ik (x - a)} = e^{- ik a} \cdot e^{ik x}$ . The translated wave function is the original times a phase factor $e^{- ik a}$ . The momentum value $ℏ k$ has not changed — only an overall phase.

Step 2. Now move to three dimensions. The plane wave is $ψ (x) = e^{i k \cdot x}$ . Translation by $a \in R^{3}$ gives $ψ (x - a) = e^{- i k \cdot a} \cdot ψ (x)$ . Same story: a phase factor depending on $a$ and $k$ . So plane waves are simultaneous eigenstates of all three translation generators.

Step 3. Rotate. If $R \in S O (3)$ is a rotation, the rotated plane wave is $ψ (R^{- 1} x) = e^{i k \cdot R^{- 1} x} = e^{i (R k) \cdot x}$ , where the second equality uses orthogonality of $R$ . The result is a plane wave with momentum $R k$ . Rotation moves the momentum vector while preserving its magnitude $∣ k ∣$ .

Step 4. Compute the energy. The free-particle energy is $E = ℏ^{2} ∣ k ∣^{2} / (2 m)$ . Translation preserves the energy because it does not touch $∣ k ∣$ . Rotation preserves the energy because $∣ R k ∣ = ∣ k ∣$ . Both pieces of $E (3)$ preserve the energy; this is exactly the statement that the free Hamiltonian commutes with the $E (3)$ action.

Step 5. Check the orbit picture. Fix a momentum magnitude $k_{0} > 0$ . The set ${k \in R^{3} : ∣ k ∣ = k_{0}}$ is a sphere of radius $k_{0}$ in momentum space. Under rotations, every point on this sphere maps to every other point — the sphere is a single rotation orbit. Translations multiply each plane wave by a phase but do not move $k$ off the sphere. So the sphere is a single $E (3)$ -orbit, and every plane wave with $∣ k ∣ = k_{0}$ belongs to the same irreducible piece of the representation.

What this tells us: the free particle of mass $m$ at fixed energy $E$ corresponds to one orbit in dual momentum space, namely the sphere of radius $2 m E /ℏ$ . The wave functions with momentum on this sphere form an irreducible piece of the $E (3)$ representation, and the energy is constant on this piece because $E (3)$ acts by symmetries of the free Hamiltonian.

Check your understanding Beginner

Formal definition Intermediate+

The Euclidean group in dimension $n$ is the semidirect product $$ E(n) = \mathbb{R}^n \rtimes O(n), $$ where $O (n)$ acts on $R^{n}$ by its defining representation. An element of $E (n)$ is a pair $(a, R)$ with $a \in R^{n}$ and $R \in O (n)$ , acting on $x \in R^{n}$ by $(a, R) \cdot x = R x + a$ . The group law is $$ (\mathbf{a}_1, R_1) \cdot (\mathbf{a}_2, R_2) = (\mathbf{a}_1 + R_1 \mathbf{a}_2,\ R_1 R_2), $$ which is the defining mixing of translations and rotations. The connected component is $SE (n) = R^{n} ⋊ S O (n)$ . This unit treats $n = 3$ and the connected component $SE (3)$ ; the same construction works for general $n$ . When clarity demands a distinction, $E (3)$ in what follows refers to $SE (3) = R^{3} ⋊ S O (3)$ .

The defining unitary representation of $E (3)$ on the Hilbert space $H = L^{2} (R^{3})$ is given by $$ \bigl(U(\mathbf{a}, R) \psi\bigr)(\mathbf{x}) = \psi\bigl(R^{-1}(\mathbf{x} - \mathbf{a})\bigr). $$ Translations act by $U (a, I) ψ (x) = ψ (x - a)$ , rotations by $U (0, R) ψ (x) = ψ (R^{- 1} x)$ . This is a strongly continuous unitary representation: unitarity follows from the change-of-variables formula together with $∣ det R ∣ = 1$ , and strong continuity follows from continuity of translation and rotation on $L^{2} (R^{3})$ (the latter requires a small approximation argument using the density of $C_{c} (R^{3})$ in $L^{2}$ ).

The Lie algebra $e (3)$ has basis ${P_{1}, P_{2}, P_{3}, J_{1}, J_{2}, J_{3}}$ where the $P_{j}$ generate translations and the $J_{k}$ generate rotations. The bracket relations are $$ [P_j, P_k] = 0, \qquad [J_j, J_k] = \varepsilon_{jkl} J_l, \qquad [J_j, P_k] = \varepsilon_{jkl} P_l, $$ where the $ε_{j k l}$ symbol is the totally antisymmetric tensor with $ε_{123} = 1$ . The translations form an abelian ideal; the rotations form the Lie subalgebra $so (3)$ ; and the rotations act on translations as on $R^{3}$ .

The infinitesimal generators on $H$ are realised by self-adjoint operators $$ P_j = -i \hbar \frac{\partial}{\partial x_j}, \qquad J_k = \varepsilon_{klm} x_l P_m, $$ with the Stone-theorem identifications $U (a, I) = exp (- i a \cdot P /ℏ)$ and $U (0, R (θ n)) = exp (- i θ n \cdot J /ℏ)$ for rotation by angle $θ$ about axis $n$ . The translations $P_{j}$ are self-adjoint as unbounded operators on the Schwartz subspace $S (R^{3}) \subset L^{2} (R^{3})$ , with self-adjoint extensions characterised by the spectral theorem applied to $- i \partial_{j}$ .

The free Hamiltonian of a particle of mass $m$ is $$ H_0 = \frac{1}{2m} \bigl(P_1^2 + P_2^2 + P_3^2\bigr) = -\frac{\hbar^2}{2m} \Delta, $$ the negative Laplacian divided by twice the mass. The Casimir-type expression $P^{2} := P_{1}^{2} + P_{2}^{2} + P_{3}^{2}$ is the Casimir of the translation subalgebra, and $H_{0} = P^{2} / (2 m)$ commutes with the entire $E (3)$ action because $P^{2}$ commutes with translations (translations commute among themselves) and is invariant under rotations (rotations act on $(P_{1}, P_{2}, P_{3})$ by orthogonal transformations, preserving the sum of squares).

A unitary representation of $E (3)$ on a Hilbert space $H^{'}$ is a strongly continuous homomorphism $U^{'} : E (3) \to U (H^{'})$ . Two representations are unitarily equivalent if an isometry $V : H^{'} \to H^{''}$ intertwines them: $V U^{'} (g) = U^{''} (g) V$ for every $g \in E (3)$ . A representation is irreducible if the only closed subspaces invariant under $U^{'} (E (3))$ are ${0}$ and $H^{'}$ .

Counterexamples to common slips

The Euclidean group is not a direct product. Rotations and translations satisfy $R \cdot a \neq = a \cdot R$ in $E (n)$ : rotating then translating by $a$ differs from translating by $a$ then rotating, because the second translation gets rotated to $R a$ first. The semidirect-product law $(a_{1}, R_{1}) (a_{2}, R_{2}) = (a_{1} + R_{1} a_{2}, R_{1} R_{2})$ encodes this.
The defining representation $U$ of $E (3)$ on $L^{2} (R^{3})$ is not irreducible. It decomposes into a direct integral over $∣ k ∣^{2} \in [0, \infty)$ , one irreducible piece per orbit of $S O (3)$ on dual momentum space. The Plancherel theorem makes this decomposition concrete via the Fourier transform.
For the zero-momentum orbit, the little group is the full $S O (3)$ (not $S O (2)$ ), and the corresponding representations are indexed by spin $j \in {0, \frac{1}{2}, 1, \dots}$ . The $j = 0$ piece is one-dimensional; the higher-spin pieces are the irreducible representations of $S O (3)$ realised on the unit cell. These zero-momentum irreps correspond to no physical free particle of definite momentum, but are admissible as $E (3)$ representations and play a role in the closure of the Plancherel measure at $∣ k ∣ = 0$ .
The "mass" parameter $m$ does not label the irreducible $E (3)$ representations directly. Mass only enters once one chooses a Hamiltonian $H_{0} = P^{2} / (2 m)$ ; the $E (3)$ representation theory itself is mass-blind. The orbit-and-stabiliser classification labels irreducibles by an orbit in $(R^{3})^{*}$ and an $S O (2)$ (or $S O (3)$ ) representation, with no $m$ -dependence.

Key theorem with proof Intermediate+

Theorem (Free-particle representation of $E (3)$ ; Hall Theorem 17.10). Let $H = L^{2} (R^{3})$ with the defining representation $U$ of $E (3) = R^{3} ⋊ S O (3)$ given by $(U (a, R) ψ) (x) = ψ (R^{- 1} (x - a))$ . The free Hamiltonian $H_{0} = - ℏ^{2} Δ/ (2 m)$ on $H$ commutes with $U (a, R)$ for every $(a, R) \in E (3)$ . The representation $U$ decomposes via the Fourier transform $\mathcal{F} : L^2(\mathbb{R}^3) \to L^2((\mathbb{R}^3)^)$ as a direct integral* $$ U \cong \int^{\oplus}{[0, \infty)} U{k_0}, d\mu(k_0), $$ where $U_{k_{0}}$ is the irreducible $E (3)$ -representation on $L^{2} (S_{k_{0}}^{2})$ with $S_{k_{0}}^{2}$ the momentum sphere of radius $k_{0}$ , and the measure $d μ (k_{0}) = k_{0}^{2} d k_{0}$ comes from the spherical decomposition $d^{3} k = k^{2} d k d Ω$ . The free Hamiltonian acts on $U_{k_{0}}$ as multiplication by the scalar $E (k_{0}) = ℏ^{2} k_{0}^{2} / (2 m)$ .

Proof. The argument has four steps. First, intertwine translations with multiplication via the Fourier transform. Second, identify the rotation action in momentum space. Third, decompose by orbits. Fourth, identify the free Hamiltonian as the Casimir.

Step 1: Fourier diagonalises translations. Let $F$ denote the unitary Fourier transform $F : L^{2} (R^{3}) \to L^{2} ((R^{3})^{*})$ defined on Schwartz functions by $$ (\mathcal{F} \psi)(\mathbf{k}) = \frac{1}{(2 \pi)^{3/2}} \int_{\mathbb{R}^3} \psi(\mathbf{x}), e^{-i \mathbf{k} \cdot \mathbf{x}}, d^3 x $$ and extended to $L^{2}$ by Plancherel's theorem. Translation by $a$ on the position side becomes multiplication by $e^{- i k \cdot a}$ on the momentum side: $F U (a, I) F^{- 1} = M_{e^{- i k \cdot a}}$ , where $M_{f}$ denotes multiplication by the function $f$ . This is the content of the convolution identity together with the fact that the Fourier transform of $ψ (x - a)$ is $e^{- i k \cdot a} (F ψ) (k)$ .

Step 2: Rotations act in momentum space by rotation. For $R \in S O (3)$ , $(F U (0, R) F^{- 1} ψ) (k) = ψ (R^{- 1} k)$ where $ψ = F ψ$ . The computation: $F [ψ (R^{- 1} \cdot)] (k) = (2 π)^{- 3/2} \int ψ (R^{- 1} x) e^{- i k \cdot x} d^{3} x = (2 π)^{- 3/2} \int ψ (y) e^{- i k \cdot R y} d^{3} y = ψ (R^{- 1} k)$ , using the change of variables $y = R^{- 1} x$ with unit Jacobian and the orthogonality identity $k \cdot R y = R^{T} k \cdot y = R^{- 1} k \cdot y$ .

So on the momentum side, $E (3)$ acts by $$ \bigl(\widetilde{U}(\mathbf{a}, R) \widehat{\psi}\bigr)(\mathbf{k}) = e^{-i \mathbf{k} \cdot \mathbf{a}}, \widehat{\psi}(R^{-1} \mathbf{k}), $$ where $U = F U F^{- 1}$ . Translations are multiplication operators; rotations are pullbacks.

Step 3: Decomposition by rotation orbits. The action of $S O (3)$ on $(R^{3})^{*} ≅ R^{3}$ has orbits the concentric spheres $S_{k_{0}}^{2} = {k : ∣ k ∣ = k_{0}}$ for $k_{0} > 0$ , together with the origin ${0}$ as a singleton orbit. The polar decomposition of momentum space is $R^{3} ∖ {0} ≅ (0, \infty) \times S^{2}$ , and Lebesgue measure factors as $d^{3} k = k^{2} d k d Ω$ . This gives a direct-integral decomposition $$ L^2((\mathbb{R}^3)^*, d^3 k) = \int^{\oplus}{(0, \infty)} L^2(S^2{k_0}, k_0^2, d\Omega), dk_0, $$ where each fibre $L^{2} (S_{k_{0}}^{2})$ is the Hilbert space of square-integrable functions on the radius- $k_{0}$ sphere.

The $E (3)$ action preserves this decomposition: translations multiply each fibre by a phase depending on the fibre point, and rotations preserve each sphere $S_{k_{0}}^{2}$ setwise. On the fibre over $k_{0}$ , the representation acts as $$ \bigl(U_{k_0}(\mathbf{a}, R) f\bigr)(\mathbf{k}) = e^{-i \mathbf{k} \cdot \mathbf{a}}, f(R^{-1} \mathbf{k}), \qquad \mathbf{k} \in S^2_{k_0}, $$ which is the representation of $E (3)$ on $L^{2} (S_{k_{0}}^{2})$ induced from the character $χ_{k_{0} \overset{e}{^}_{3}}$ of the translation subgroup at the reference point $k_{0} \overset{e}{^}_{3} \in S_{k_{0}}^{2}$ , with little group the stabiliser $S O (2)$ of $\overset{e}{^}_{3}$ under $S O (3)$ .

Step 4: Irreducibility and the Casimir. Each $U_{k_{0}}$ is irreducible. The Mackey machine for semidirect products with abelian normal factor (treated as a separate theorem in the Advanced results below) identifies the irreducible representations of $R^{3} ⋊ S O (3)$ with pairs $(O, σ)$ where $O$ is an $S O (3)$ -orbit on $(R^{3})^{*}$ and $σ$ is an irreducible representation of the little group fixing a chosen point of $O$ . For the orbit $S_{k_{0}}^{2}$ with $k_{0} > 0$ , the little group is $S O (2)$ , and its irreducible representations are the characters $σ_{ℓ} : R (θ) \mapsto e^{i ℓ θ}$ for $ℓ \in Z$ . The free-particle representation $U_{k_{0}}$ above corresponds to the identity little-group character $ℓ = 0$ (the spin-zero free particle); higher $ℓ$ values give the spin- $ℓ$ free particles, realised on $L^{2}$ sections of the line bundle $S^{2} \times_{S O (2)} C_{ℓ} \to S^{2}$ .

The free Hamiltonian $H_{0} = P^{2} / (2 m)$ in momentum space is multiplication by $ℏ^{2} ∣ k ∣^{2} / (2 m)$ . On the fibre over $k_{0}$ , this is the constant scalar $E (k_{0}) = ℏ^{2} k_{0}^{2} / (2 m)$ . So $H_{0}$ acts on the irreducible piece $U_{k_{0}}$ as the scalar $E (k_{0})$ , identifying $H_{0}$ as a Casimir of the translation subalgebra. Two representations $U_{k_{0}}$ and $U_{k_{0}^{'}}$ with $k_{0} \neq = k_{0}^{'}$ have distinct Casimir eigenvalues, hence are inequivalent. The direct-integral decomposition is the disintegration of the regular representation into mass- $m$ free-particle states organised by energy. $□$

Bridge. The free-particle representation builds toward the Wigner classification of relativistic particles and identifies the central idea that energy and momentum are quantum-mechanical labels precisely because they are the orbit and little-group invariants of a noncompact Lie group acting on a Hilbert space. The foundational reason this works is exactly that the Euclidean group is a semidirect product with abelian normal factor, which makes the Mackey machine applicable. This is exactly the structural fact that identifies free-particle states with $E (3)$ -irreducible representations: putting these together, the Hamiltonian is the Casimir, momentum eigenstates are the dual-space coordinates of the orbit, and the Fourier transform is the intertwiner that diagonalises the translation subgroup. The central insight is that the Schrödinger equation is the one-parameter subgroup of time evolution sitting inside a representation of an enlarged spacetime symmetry group — for the free particle, the Galilean group with $E (3)$ as its spatial subgroup — and the bridge is that asking which Hilbert-space representations carry the same orbit-stabiliser structure as classical phase space recovers all of free-particle quantum mechanics from group theory alone.

The same machinery appears again in 03.14.01 (the modern-geometry rendering of this construction with semidirect-product representation theory packaged differently) and builds toward the Poincaré-group analysis where mass and spin appear as orbit-and-little-group labels, generalises beyond $E (3)$ to every semidirect product with abelian normal factor, and is dual to the Stone-von Neumann uniqueness story for the Heisenberg group in the sense that both are imprimitivity-theorem instances. The bridge is the recognition that the free particle is one specific irreducible piece of the regular representation of $E (3)$ on its natural Hilbert space, and Mackey's classification accounts for every other piece as a different physical free system (zero momentum + spin, or higher-spin free particles).

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that the angular-momentum operators $J_{k} = ε_{k l m} x_{l} P_{m}$ satisfy the $so (3)$ commutation relations $[J_{j}, J_{k}] = i ℏ ε_{j k l} J_{l}$ on the Schwartz subspace of $L^{2} (R^{3})$ .

Hint

Use the canonical commutator $[x_{a}, P_{b}] = i ℏ δ_{ab}$ and expand $[J_{j}, J_{k}] = [ε_{j ab} x_{a} P_{b}, ε_{k c d} x_{c} P_{d}]$ using the Leibniz rule for commutators.

Answer

Compute $[J_{j}, J_{k}]$ on Schwartz functions. Expand using $[A B, C] = A [B, C] + [A, C] B$ : $$ [J_j, J_k] = \varepsilon_{jab} \varepsilon_{kcd}, [x_a P_b, x_c P_d]. $$ Using $[x_{a} P_{b}, x_{c} P_{d}] = x_{a} [P_{b}, x_{c}] P_{d} + x_{c} [x_{a}, P_{d}] P_{b} = - i ℏ (x_{a} δ_{b c} P_{d} - x_{c} δ_{a d} P_{b})$ . So $$ [J_j, J_k] = -i \hbar, \varepsilon_{jab} \varepsilon_{kcd} (x_a \delta_{bc} P_d - x_c \delta_{ad} P_b). $$ Contract the deltas: $ε_{j ab} ε_{k b d} = δ_{j k} δ_{a d} - δ_{j d} δ_{ak}$ (the $εε$ identity). The first term gives $- i ℏ (δ_{j k} δ_{a d} - δ_{j d} δ_{ak}) x_{a} P_{d} = - i ℏ (δ_{j k} x_{a} P_{a} - x_{k} P_{j})$ , and the symmetric second term contributes the cross piece. Combining and using $ε_{j k l} J_{l} = ε_{j k l} ε_{l mn} x_{m} P_{n} = (δ_{j m} δ_{k n} - δ_{j n} δ_{k m}) x_{m} P_{n} = x_{j} P_{k} - x_{k} P_{j}$ , one arrives at $[J_{j}, J_{k}] = i ℏ ε_{j k l} J_{l}$ . Rubric: full credit for the $εε$ contraction and the final identification with $ε_{j k l} J_{l}$ .

Exercise 4 (medium, short-answer).

Identify the stabiliser subgroup of $k_{0} = (0, 0, k_{0})$ with $k_{0} > 0$ under the action of $S O (3)$ on $R^{3}$ , and explain why this stabiliser is called the little group at $k_{0}$ .

Hint

A rotation fixes $k_{0}$ if and only if its rotation axis is parallel to $k_{0}$ . Rotations about the $z$ -axis form a one-parameter subgroup.

Answer

The stabiliser of $k_{0} = k_{0} \overset{e}{^}_{3}$ in $S O (3)$ is the subgroup of rotations about the $z$ -axis, isomorphic to $S O (2) ≅ U (1)$ . Indeed $R k_{0} = k_{0}$ requires $R \overset{e}{^}_{3} = \overset{e}{^}_{3}$ , which holds iff $R \in S O (2)_{z} = {R (θ \overset{e}{^}_{3}) : θ \in [0, 2 π)}$ . The orbit $S O (3) \cdot k_{0} = S_{k_{0}}^{2}$ is a single sphere, and the orbit-stabiliser theorem gives $S_{k_{0}}^{2} ≅ S O (3) / S O (2)$ . The stabiliser is called the little group in the Wigner-Mackey language because the irreducible representations of $E (3)$ on the orbit are induced from irreducible representations of this stabiliser; the stabiliser is the "little" piece of $S O (3)$ that remains after the orbit-defining direction has been singled out. Rubric: full credit for the identification $S O (2)_{z}$ , the orbit picture $S O (3) / S O (2) = S^{2}$ , and the induced-representation interpretation.

Exercise 5 (medium, symbolic).

Verify that $H_{0} = P^{2} / (2 m)$ commutes with both the translation generators $P_{j}$ and the angular-momentum generators $J_{k}$ on the Schwartz subspace.

Hint

For $[H_{0}, P_{j}]$ : use $[P_{a}^{2}, P_{j}] = 0$ since translations commute. For $[H_{0}, J_{k}]$ : expand $J_{k} = ε_{k l m} x_{l} P_{m}$ and use $[P_{a}^{2}, x_{l}] = - 2 i ℏ P_{a} δ_{a l}$ .

Answer

For $[H_{0}, P_{j}]$ : write $H_{0} = (P_{a}^{2}) / (2 m)$ summed on $a$ . Each $[P_{a}^{2}, P_{j}] = P_{a} [P_{a}, P_{j}] + [P_{a}, P_{j}] P_{a} = 0$ because $[P_{a}, P_{j}] = 0$ . So $[H_{0}, P_{j}] = 0$ .

For $[H_{0}, J_{k}]$ : compute $[P_{a}^{2}, J_{k}] = [P_{a}^{2}, ε_{k l m} x_{l} P_{m}]$ . Using $[P_{a}^{2}, x_{l}] = - 2 i ℏ P_{a} δ_{a l}$ (from $[P_{a}, x_{l}] = - i ℏ δ_{a l}$ applied twice via Leibniz), we get $[P_{a}^{2}, x_{l} P_{m}] = - 2 i ℏ P_{a} δ_{a l} P_{m} = - 2 i ℏ δ_{a l} P_{a} P_{m}$ . Then $ε_{k l m} δ_{a l} P_{a} P_{m} = ε_{k am} P_{a} P_{m}$ . The symmetric tensor $P_{a} P_{m}$ contracted with the antisymmetric $ε_{k am}$ vanishes. So $[P_{a}^{2}, J_{k}] = 0$ summed on $a$ , giving $[H_{0}, J_{k}] = 0$ .

Together: $H_{0}$ commutes with every generator of $e (3)$ , hence with the entire $E (3)$ action by exponentiating. Rubric: full credit for both commutators with the $ε \cdot$ symmetric-tensor vanishing identified.

Exercise 7 (hard, short-answer).

State Mackey's theorem on irreducible unitary representations of $R^{n} ⋊ H$ (with $H$ acting on $R^{n}$ by an orthogonal action), and apply it to enumerate the irreducible unitary representations of $E (3) = R^{3} ⋊ S O (3)$ .

Hint

The Mackey machine for semidirect products with abelian normal factor associates each irreducible representation with a pair $(O, σ)$ where $O$ is an $H$ -orbit in the dual of $R^{n}$ and $σ$ is an irreducible representation of the stabiliser of any point of $O$ .

Answer

Mackey's theorem. Let $G = A ⋊ H$ be a semidirect product with $A$ a locally compact abelian group and $H$ acting on $A$ continuously. Let $A$ denote the Pontryagin dual of $A$ , with $H$ acting on $A$ via the dual action. Assume the action of $H$ on $A$ is regular (orbits are locally closed). Then the irreducible unitary representations of $G$ (up to unitary equivalence) are in bijection with pairs $(O, σ)$ where $O \subset A$ is an $H$ -orbit, and $σ$ is an irreducible unitary representation of the stabiliser $H_{χ}$ at any point $χ \in O$ (different choices of $χ$ give equivalent stabilisers, conjugate in $H$ ). The corresponding $G$ -representation is induced from the character-times- $σ$ representation of $A ⋊ H_{χ}$ via Mackey induction $Ind_{A ⋊ H_{χ}}^{G}$ .

Application to $E (3) = R^{3} ⋊ S O (3)$ . The dual $R^{3} ≅ R^{3}$ via $χ_{k} (a) = e^{i k \cdot a}$ , with $S O (3)$ acting by rotation. Orbits: $O_{0} = {0}$ (singleton) with stabiliser $S O (3)$ ; $O_{k_{0}} = S_{k_{0}}^{2}$ for $k_{0} > 0$ with stabiliser $S O (2)$ . Irreducible representations of $E (3)$ are then enumerated as:

$(O_{0}, V_{j})$ for each irrep $V_{j}$ of $S O (3)$ indexed by $j \in {0, 1, 2, \dots}$ : these are the spin- $j$ representations of $S O (3)$ extended by the action of translations (translations act by the identity since the character is the unit character at $0$ ). Finite-dimensional, dimension $2 j + 1$ .
$(O_{k_{0}}, σ_{ℓ})$ for each $k_{0} > 0$ and each character $σ_{ℓ} : S O (2) \to U (1)$ , $σ_{ℓ} (R (θ)) = e^{i ℓ θ}$ with $ℓ \in Z$ : these are the spin- $ℓ$ free-particle representations of $E (3)$ at momentum magnitude $k_{0}$ , realised on $L^{2}$ -sections of the homogeneous line bundle $S^{2} \times_{S O (2)} C_{ℓ} \to S_{k_{0}}^{2}$ . The spin-zero case $ℓ = 0$ is the scalar free particle on $L^{2} (S_{k_{0}}^{2})$ .

The orbit $O_{k_{0}}$ together with $σ_{0}$ is the irreducible piece $U_{k_{0}}$ appearing in the direct-integral decomposition of $L^{2} (R^{3})$ . Rubric: full credit for the orbit-and-stabiliser parametrisation, the identification of stabilisers ( $S O (3)$ at the origin, $S O (2)$ on each sphere), and the recognition that the regular $L^{2} (R^{3})$ representation is built from the $ℓ = 0$ pieces only.

Exercise 8 (hard, short-answer).

The free Hamiltonian $H_{0} = P^{2} / (2 m)$ has continuous spectrum $[0, \infty)$ with no eigenvalues in $L^{2} (R^{3})$ . Explain in representation-theoretic terms why $H_{0}$ has no proper eigenvectors, and identify what the "would-be" eigenstates are.

Hint

A proper eigenvector of $H_{0}$ would be supported on a single orbit $S_{k_{0}}^{2}$ in momentum space. Single orbits have zero Lebesgue measure in $R^{3}$ .

Answer

A vector $ψ \in L^{2} (R^{3})$ that is an eigenvector of $H_{0} =$ multiplication-by- $ℏ^{2} ∣ k ∣^{2} / (2 m)$ with eigenvalue $E_{0}$ would satisfy $(ℏ^{2} ∣ k ∣^{2} / (2 m) - E_{0}) ψ (k) = 0$ almost everywhere, hence $ψ$ is supported on the sphere $∣ k ∣^{2} = 2 m E_{0} / ℏ^{2}$ . This sphere is a measure-zero subset of $R^{3}$ , so the only $L^{2}$ representative is the zero vector.

The "would-be" eigenstates are the plane waves $e^{i k \cdot x}$ , which are not in $L^{2} (R^{3})$ but live in the larger space of tempered distributions (or in a rigged Hilbert space $S \subset L^{2} \subset S^{'}$ ). In the direct-integral decomposition $L^{2} (R^{3}) = \int^{\oplus} L^{2} (S_{k_{0}}^{2}) k_{0}^{2} d k_{0}$ , the spectral resolution of $H_{0}$ is the disintegration: each fibre $L^{2} (S_{k_{0}}^{2})$ is the "generalised $E_{0}$ -eigenspace" with $E_{0} = ℏ^{2} k_{0}^{2} / (2 m)$ , but no element of $L^{2} (R^{3})$ is concentrated on a single fibre.

Representation-theoretically: the free-particle Hamiltonian acts as a scalar on each irreducible piece $U_{k_{0}}$ , so the spectral resolution of $H_{0}$ is exactly the disintegration into irreducible $E (3)$ -pieces. The lack of proper eigenvectors reflects the non-discreteness of the orbit set ${S_{k_{0}}^{2} : k_{0} \geq 0}$ in $R^{3} / S O (3) = [0, \infty)$ , equivalently the absence of a normalisable invariant vector in any irreducible piece $L^{2} (S_{k_{0}}^{2})$ for the full $E (3)$ action. Rubric: full credit for the measure-zero argument, the rigged-Hilbert-space identification of plane waves, and the recognition that the lack of $H_{0}$ -eigenstates is the lack of $E (3)$ -invariant vectors in irreducible pieces.

Advanced results Master

Theorem (Mackey machine for semidirect products with abelian normal factor; Folland Theorem 6.42 / Mackey 1968 Ch. 1). Let $G = A ⋊ H$ be a second-countable locally compact group with $A$ abelian and $H$ acting on $A$ continuously by automorphisms. Suppose the dual action of $H$ on $A$ is regular. The irreducible unitary representations of $G$ are parametrised, up to unitary equivalence, by pairs $(O, [σ])$ where $O$ is an $H$ -orbit in $A$ and $[σ]$ is a unitary-equivalence class of irreducible representations of the stabiliser $H_{χ}$ for any $χ \in O$ . The representation $π_{O, σ}$ is the Mackey-induced representation $Ind_{A ⋊ H_{χ}}^{G} (χ \otimes σ)$ .

For $G = E (3) = R^{3} ⋊ S O (3)$ , the dual is $R^{3} ≅ R^{3}$ via $χ_{k} (a) = e^{i k \cdot a}$ , and the $S O (3)$ action is rotation. The orbit structure is ${0} \cup {S_{k_{0}}^{2} : k_{0} > 0}$ ; the stabiliser at $0$ is the full $S O (3)$ , and the stabiliser at $k_{0} = k_{0} \overset{e}{^}_{3}$ is $S O (2)_{z}$ . The irreducible unitary representations of $E (3)$ are then $({0}, V_{j})$ for $j \in {0, 1, 2, \dots}$ (the $S O (3)$ -irreps, dimension $2 j + 1$ , on which translations act by the identity) and $(S_{k_{0}}^{2}, σ_{ℓ})$ for $k_{0} > 0$ , $ℓ \in Z$ .

Theorem (Free-particle Hamiltonian as Casimir; Hall §17.4). On each irreducible representation $π_{S_{k_{0}}^{2}, σ_{ℓ}}$ of $E (3)$ , the operator $P^{2} = P_{1}^{2} + P_{2}^{2} + P_{3}^{2}$ acts as the scalar $ℏ^{2} k_{0}^{2}$ . The free Hamiltonian of a mass- $m$ particle is $H_{0} = P^{2} / (2 m)$ , hence acts on $π_{S_{k_{0}}^{2}, σ_{ℓ}}$ as the scalar $ℏ^{2} k_{0}^{2} / (2 m)$ . The operator $P^{2}$ is the Casimir element of the universal enveloping algebra $U (e (3))$ restricted to the abelian translation subalgebra, well-defined as a self-adjoint operator on smooth vectors of any unitary representation of $E (3)$ by Nelson's theorem.

The Casimir interpretation generalises: any quadratic invariant of $e (3)$ under the adjoint action of $S O (3)$ acts as a scalar on each irreducible representation. The translation Casimir $P^{2}$ labels the orbit; the mixed Casimir $J \cdot P$ vanishes on the spin-zero representations $σ_{0}$ but is a nonzero scalar on the higher-spin ones, with eigenvalue $ℏ^{2} k_{0} ℓ$ on $(S_{k_{0}}^{2}, σ_{ℓ})$ . Together $P^{2}$ and $J \cdot P$ label the spin- $ℓ$ free particle of momentum magnitude $k_{0}$ .

Theorem (Imprimitivity theorem; Mackey 1949, Proc. Nat. Acad. Sci. 35). Let $G$ be a locally compact group acting transitively on a homogeneous space $G / H$ . There is a bijective correspondence between unitary representations of $G$ together with a system of imprimitivity based on $G / H$ (a projection-valued measure on $G / H$ covariant under $G$ ) and unitary representations of $H$ . The correspondence is induction: a representation $σ$ of $H$ induces a representation $Ind_{H}^{G} (σ)$ on $L^{2} (G / H)$ -sections of a homogeneous bundle, and every $G$ -representation with the imprimitivity structure arises this way.

Applied to the free-particle representation: $G / H = E (3) / (R^{3} ⋊ S O (2)) = S^{2}$ is the momentum sphere, the system of imprimitivity is multiplication by indicator functions on subsets of $S^{2}$ , and the imprimitivity theorem identifies the free-particle representation as the unique (up to equivalence) $E (3)$ -representation with this system of imprimitivity and identity $S O (2)$ character. This is Mackey's original 1949 motivation for the imprimitivity theorem: to algebraicise the notion of "a particle localised at a position" via projection-valued measures on configuration space.

Theorem (Plancherel formula for $E (3)$ ; Stein-Wallach 1975 §3.4). The Plancherel measure of $E (3)$ is supported on the principal series of irreducibles ${(S_{k_{0}}^{2}, σ_{0}) : k_{0} > 0}$ (the scalar free-particle representations) plus a discrete contribution from the zero-momentum orbit. For $f \in C_{c} (E (3))$ , $$ |f|{L^2(E(3))}^2 = \int_0^\infty |\widehat{f}(k_0)|{HS}^2, k_0^2, dk_0 + \sum_{j \geq 0} (2j + 1), |\widehat{f}j|{HS}^2, $$ where $f (k_{0})$ is the operator-valued Fourier transform of $f$ at the orbit $(S_{k_{0}}^{2}, σ_{0})$ and the Hilbert-Schmidt norm is taken there. The measure $k_{0}^{2} d k_{0}$ is the Plancherel density for the principal-series component.

The Plancherel formula gives the precise spectral decomposition of the regular representation of $E (3)$ on $L^{2} (E (3))$ , refining the direct-integral decomposition of the position-space representation on $L^{2} (R^{3})$ . The free-particle representation appears as the spin-zero principal-series piece.

Theorem (Galilean extension and the projective phase). The Galilean group $Gal (3) = (R ⋊ E (3)) ⋊_{boost} R^{3}$ acts on free quantum systems, with $E (3)$ as the subgroup of spatial symmetries and the $R$ factor as time translation. The natural unitary action on $L^{2} (R^{3})$ is projective, with projective cocycle (the Bargmann cocycle) given by $$ \omega(g_1, g_2) = \exp\biggl(\frac{i}{\hbar} \biggl[\frac{m}{2}, |\mathbf{v}_1|^2, t_2 + m, \mathbf{v}_1 \cdot (R_1 \mathbf{a}_2 + \mathbf{v}_1 t_2)\biggr]\biggr) $$ for boost parameters $v_{i}$ and translations $(t_{i}, a_{i})$ . The central extension trivialising this cocycle is the Bargmann group $Gal (3)$ , and mass enters as the central charge: each projective irrep of $Gal (3)$ corresponds to a mass- $m$ ordinary irrep of $Gal (3)$ . The free-particle $E (3)$ -representation at fixed $k_{0}$ lifts to a mass- $m$ irrep of $Gal (3)$ with $k_{0}$ varying over the entire mass shell as time evolves.

This is Bargmann's 1954 Ann. Math. result: there is no nonprojective unitary action of $Gal (3)$ on a quantum free particle of nonzero mass; mass is a cohomology class. The $E (3)$ analysis is the spatial slice of this story, and the Schrödinger equation $i ℏ \partial_{t} ψ = H_{0} ψ$ is the time-translation subgroup of the projective $Gal (3)$ representation.

Theorem (Coadjoint-orbit picture; Souriau 1970). The coadjoint orbits of $E (3)$ on $\mathfrak{e}(3)^ $a r e p a r am e t r i se d b y t h eor bi t in v a r ian t s$ |\mathbf{p}| $(m o m e n t u mma g ni t u d e) an d$ \mathbf{p} \cdot \mathbf{j} $(h e l i c i t y, w h er e$ \mathbf{j} $i s t h e an g u l a r - m o m e n t u m v ec t or) . F or$ |\mathbf{p}| > 0 $, t h eco a d j o in t or bi t i s t h eco t an g e n t b u n d l e$ T^* S^2_{|\mathbf{p}|} $w i t ha sy m pl ec t i c f or m co mbinin g t h e L i o uv i l l e f or m o n$ T^* S^2 $an d t h e h e l i c i t y co n t r ib u t i o n$ \mathbf{p} \cdot \mathbf{j} = \ell $a s ama g n e t i c - m o n o p o l e t er m . G eo m e t r i c q u an t i s a t i o n o f t hi sco a d j o in t or bi t, in t h eK o n s t an t - S o u r ia u se n se, r eco v er s t h e i r r e d u c ib l er e p r ese n t a t i o n$ (S^2_{|\mathbf{p}|}, \sigma_\ell)$ exactly.*

The coadjoint-orbit picture identifies classical and quantum free particles via the Kirillov-Konstant-Souriau orbit method. Mass-zero limit cases (such as the helicity representations of the Poincaré group used for photons) and their Euclidean analogues all fit this framework. Souriau's 1970 monograph treats both the classical orbit geometry and the prequantum bundle that lifts it to the quantum representation.

Theorem (Wigner-style classification refinement; Wigner 1939 adapted). Every projective unitary representation of $E (3)$ comes from a unitary representation of $E (3)$ itself, since $H^{2} (E (3), U (1)) = 0$ . By contrast, the Galilean group $Gal (3)$ has nonvanishing $H^{2}$ generated by the mass cocycle, and the Poincaré group $P = R^{1, 3} ⋊ O (1, 3)$ has $H^{2} = 0$ but is itself the unique central extension of the Galilean group in the relativistic limit.

This cohomological refinement was identified for the Galilean case by Inönü-Wigner (1953, Proc. Nat. Acad. Sci. 39) and for the Poincaré case in Wigner's original 1939 paper. The Euclidean $E (3)$ is cohomologically simpler than its Galilean and Poincaré extensions; this is why the spatial part of the free-particle theory is so clean and why the mass parameter enters only through the time-evolution extension.

Synthesis. The free particle in three-dimensional space is the irreducible unitary representation of $E (3) = R^{3} ⋊ S O (3)$ at momentum magnitude $k_{0} > 0$ with spin-zero little-group character. The foundational reason this is the right framework is exactly that the Euclidean group is a semidirect product with abelian normal factor, making the Mackey machine applicable, and the orbit-stabiliser decomposition is exactly what produces momentum eigenstates and angular-momentum quantisation in one breath. Putting these together, the free Hamiltonian is the Casimir of the translation subalgebra; the Fourier transform is the intertwiner that diagonalises this Casimir; the Schrödinger equation is the time-translation one-parameter subgroup inside the projective Galilean extension; and the absence of $L^{2}$ -normalisable energy eigenstates is identified with the continuous-spectrum nature of the orbit set $[0, \infty)$ . The bridge is that quantum free-particle physics is exactly the representation theory of $E (3)$ in the spin-zero principal series, with mass entering as the central charge of the Bargmann extension to the Galilean group rather than as an $E (3)$ representation label.

The central insight that emerges from this assembly is that the four classical labels of a free particle — energy, momentum, angular momentum, mass — are not four independent inputs but four projections of a single representation-theoretic object: $∣ p ∣$ labels the orbit, $\hat{p}$ coordinatises a point on the orbit, the little-group character $ℓ$ labels the spin, and $m$ is the cocycle class of the Galilean extension. This pattern recurs in the relativistic theory: replacing $E (3)$ by the Poincaré group $R^{1, 3} ⋊ O (1, 3)$ produces the Wigner classification of elementary particles by mass and spin, with the orbits in momentum space now the mass-shell hyperboloids and the little groups one of $S O (3)$ (massive), $E (2)$ (massless), $S O (2, 1)$ (tachyonic), or the identity (zero-momentum). The bridge from the Euclidean to the Poincaré story is the recognition that the same Mackey-machine technology, adapted to the new orbit-and-stabiliser structure of $R^{1, 3}$ under $O (1, 3)$ , classifies all relativistic free particles. Geometric quantisation via the coadjoint-orbit method (Konstant-Souriau-Kirillov) gives the parallel construction at the classical-quantum interface; the imprimitivity theorem (Mackey 1949) gives the algebraic statement that the system of position-projection measures on configuration space uniquely determines the quantisation; and the rigged-Hilbert-space framework (Gel'fand-Vilenkin 1964) supplies the analytic setting in which plane waves are honest distributional eigenstates of the position-translation generators. All of these are different presentations of the one representation-theoretic identification: the free particle is the $E (3)$ -irrep at fixed $k_{0}$ , and every other feature of free-particle quantum mechanics is a property of this irrep.

Full proof set Master

Proposition (Self-adjointness of $H_{0}$ ). The free Hamiltonian $H_{0} = - ℏ^{2} Δ/ (2 m)$ on the domain $S (R^{3}) \subset L^{2} (R^{3})$ is essentially self-adjoint, and its self-adjoint closure has domain $H^{2} (R^{3})$ , the Sobolev space of order two.

Proof. The Fourier transform $F : L^{2} (R^{3}) \to L^{2} (R^{3})$ is unitary, and conjugating $- Δ$ by $F$ gives multiplication by $∣ k ∣^{2}$ . The multiplication operator $M_{∣ k ∣^{2}}$ on its maximal domain $Dom (M_{∣ k ∣^{2}}) = {f \in L^{2} : ∣ k ∣^{2} f \in L^{2}}$ is self-adjoint, since multiplication by a real-valued measurable function on an $L^{2}$ -space is always self-adjoint on its maximal domain. The unitary conjugate $F^{- 1} M_{∣ k ∣^{2}} F = - Δ$ is therefore self-adjoint on the corresponding conjugated domain, which is the Sobolev space $H^{2} (R^{3}) = {ψ \in L^{2} : Δ ψ \in L^{2} in the distributional sense}$ . Schwartz functions are dense in $H^{2}$ , hence the restriction of $- Δ$ to $S (R^{3})$ is essentially self-adjoint with the same closure. Dividing by $2 m / ℏ^{2}$ does not affect self-adjointness. $□$

Proposition (Stone's theorem applied to translations). The one-parameter family $a \mapsto U (a, I)$ of translations on $L^{2} (R^{3})$ is strongly continuous, and its infinitesimal generators are $P_{j} = - i ℏ \partial_{j}$ for $j = 1, 2, 3$ , with $U (a, I) = exp (- i a \cdot P /ℏ)$ .

Proof. Strong continuity: for $ψ \in L^{2} (R^{3})$ , $∥ U (a, I) ψ - ψ ∥_{2} \to 0$ as $a \to 0$ by the standard $L^{2}$ -continuity-of-translation argument (verify on $C_{c}$ first by uniform continuity, extend to $L^{2}$ by density and the unitarity of translation).

Generator: for $ψ \in S (R^{3})$ and a unit vector $\overset{e}{^}_{j}$ , $\frac{d}{d t}_{t = 0} U (t \overset{e}{^}_{j}, I) ψ = - \partial_{j} ψ = (- i ℏ)^{- 1} \cdot (- i ℏ) \partial_{j} ψ = (- i /ℏ) P_{j} ψ$ . Multiplying by $i ℏ$ identifies the Stone generator: $i ℏ \frac{d}{d t} U (t \overset{e}{^}_{j}, I) ψ = P_{j} ψ$ . By Stone's theorem applied to the strongly continuous one-parameter group $t \mapsto U (t \overset{e}{^}_{j}, I)$ , the generator $P_{j}$ has a unique self-adjoint extension and $U (t \overset{e}{^}_{j}, I) = exp (- i t P_{j} /ℏ)$ . Combining the three commuting generators gives $U (a, I) = exp (- i a \cdot P /ℏ)$ on $S$ , extended to $L^{2}$ by closure. $□$

Proposition (Plancherel and the direct-integral structure). The unitary Fourier transform $F : L^{2} (R^{3}) \to L^{2} (R^{3})$ intertwines the $E (3)$ action with $U (a, R) ψ (k) = e^{- i k \cdot a} ψ (R^{- 1} k)$ , and the polar decomposition $R^{3} = (0, \infty) \times S^{2} \cup {0}$ gives the direct-integral decomposition $L^{2} (R^{3}) = \int_{(0, \infty)}^{\oplus} L^{2} (S^{2}, k_{0}^{2} d Ω) d k_{0}$ .

Proof. The intertwining $F U (a, R) F^{- 1} = U (a, R)$ was computed in the Intermediate-tier proof. For the direct-integral statement: any $ψ \in L^{2} (R^{3})$ has $\int_{R^{3}} ∣ ψ ∣^{2} d^{3} k = \int_{0}^{\infty} \int_{S^{2}} ∣ ψ (k_{0} ω) ∣^{2} k_{0}^{2} d Ω (ω) d k_{0}$ by spherical change of variables, where $ω \in S^{2}$ is a unit vector. This identifies the Hilbert-space isomorphism with the direct integral $\int^{\oplus} L^{2} (S^{2}, k_{0}^{2} d Ω) d k_{0}$ . The fibre at $k_{0}$ is $L^{2} (S^{2})$ as an abstract Hilbert space, with the inner product $⟨ f, g ⟩_{k_{0}} = \int_{S^{2}} \overline{f (ω)} g (ω) k_{0}^{2} d Ω (ω) = k_{0}^{2} ⟨ f, g ⟩_{L^{2} (S^{2})}$ , so up to the conventional weight $k_{0}^{2}$ each fibre is the standard $L^{2} (S^{2})$ . $□$

Proposition (Irreducibility of the fibre representation). For each $k_{0} > 0$ , the representation $U_{k_{0}}$ of $E (3)$ on $L^{2} (S_{k_{0}}^{2})$ defined by $(U_{k_{0}} (a, R) f) (k) = e^{- i k \cdot a} f (R^{- 1} k)$ is irreducible.

Proof. Suppose $W \subseteq L^{2} (S_{k_{0}}^{2})$ is a closed $E (3)$ -invariant subspace. We will show $W = {0}$ or $W = L^{2} (S_{k_{0}}^{2})$ .

The translations ${U_{k_{0}} (a, I) : a \in R^{3}}$ act by multiplication operators $M_{e^{- i k \cdot a}}$ on $L^{2} (S_{k_{0}}^{2})$ . The von Neumann algebra they generate is the algebra of essentially bounded multiplication operators ${M_{g} : g \in L^{\infty} (S_{k_{0}}^{2})}$ (since the family ${e^{- i k \cdot a}}_{a \in R^{3}}$ separates points on $S_{k_{0}}^{2}$ — different points $k, k^{'}$ on the sphere give different characters $a \mapsto e^{- i k \cdot a}$ , and the Stone-Weierstrass theorem applied to the algebra they generate identifies it with $C (S_{k_{0}}^{2})$ , weak- $*$ -dense in $L^{\infty}$ ). So a translation-invariant closed subspace $W$ is preserved by all multiplication operators $M_{g}$ , $g \in L^{\infty} (S_{k_{0}}^{2})$ , which forces $W = L^{2} (E)$ for some measurable $E \subseteq S_{k_{0}}^{2}$ (with characteristic function $1_{E}$ acting as the orthogonal projection onto $W$ ).

Adding rotation invariance: $U_{k_{0}} (0, R) L^{2} (E) = L^{2} (R \cdot E)$ , so rotation-invariance of $W$ forces $E$ to be $S O (3)$ -invariant as a subset of $S_{k_{0}}^{2}$ . But $S O (3)$ acts transitively on $S_{k_{0}}^{2}$ , so the only $S O (3)$ -invariant measurable subsets are $\emptyset$ and $S_{k_{0}}^{2}$ (up to measure-zero correction). Hence $W = {0}$ or $W = L^{2} (S_{k_{0}}^{2})$ , and $U_{k_{0}}$ is irreducible. $□$

Proposition (Mackey machine, sketch of proof). Following Folland §6.5-6.6, the proof of the Mackey machine for $G = A ⋊ H$ has three steps: (i) any irreducible representation $π$ of $G$ restricts to a representation of $A$ , whose spectral decomposition gives a projection-valued measure on $A$ covariant under the $H$ -action — equivalently a system of imprimitivity based on a single orbit $O \subset A$ ; (ii) the imprimitivity theorem identifies such systems of imprimitivity with representations of the stabiliser $H_{χ}$ at any $χ \in O$ , via Mackey induction; (iii) different choices of $χ$ give conjugate stabilisers and equivalent induced representations, so the data $(O, [σ])$ uniquely determines $π$ up to equivalence.

For $G = E (3)$ : step (i) gives the orbit set ${S_{k_{0}}^{2}} \cup {0}$ as the parameter space for $O$ . Step (ii) identifies the induced representations as $L^{2}$ -sections of the homogeneous line bundle $S^{2} \times_{S O (2)} C_{ℓ} \to S_{k_{0}}^{2}$ for $(O, σ_{ℓ})$ on the sphere orbits, and as $V_{j}$ for the singleton orbit. Step (iii) is automatic since $S O (2) = S O (2)$ and $S O (3) = S O (3)$ as the only conjugacy classes at issue. The full proof of the imprimitivity theorem itself is in Folland §6.5 or Mackey 1968 Ch. 1; we cite it without reproducing. $□$

Proposition (Casimir characterisation). On the irreducible representation $(S_{k_{0}}^{2}, σ_{ℓ})$ of $E (3)$ , the translation Casimir $P^{2}$ acts as $ℏ^{2} k_{0}^{2} \cdot id$ and the mixed Casimir $J \cdot P$ acts as $ℏ^{2} k_{0} ℓ \cdot id$ .

Proof. In the Fourier picture, $P^{2} = ℏ^{2} ∣ k ∣^{2}$ as a multiplication operator. On the fibre $L^{2} (S_{k_{0}}^{2})$ , $∣ k ∣^{2} = k_{0}^{2}$ is constant, so $P^{2} = ℏ^{2} k_{0}^{2} \cdot id$ .

For the mixed Casimir $J \cdot P = \sum_{j} J_{j} P_{j}$ : the operator commutes with all of $e (3)$ (a direct check using $[J_{i}, J_{j} P_{j}] + cyc = 0$ and similarly for $P_{i}$ , using the $e (3)$ brackets). By Schur's lemma on the irreducible $(O, σ_{ℓ})$ , the operator acts as a scalar. Computing on the reference point $k_{0} = k_{0} \overset{e}{^}_{3} \in S_{k_{0}}^{2}$ in a local trivialisation of the homogeneous line bundle: $J \cdot P$ at $k_{0}$ reduces to $ℏ k_{0} J_{3}$ (the $P_{3}$ -component is $ℏ k_{0}$ and the others vanish on $k_{0}$ ), and $J_{3}$ acts on the $S O (2)$ -character $σ_{ℓ}$ by multiplication by $ℏ ℓ$ . So $J \cdot P$ acts as $ℏ^{2} k_{0} ℓ$ on $(S_{k_{0}}^{2}, σ_{ℓ})$ . $□$

Proposition (Generalised eigenstates and rigged Hilbert space). For each $k \in R^{3}$ , the plane wave $ϕ_{k} (x) = (2 π)^{- 3/2} e^{i k \cdot x}$ is a tempered distribution that is a generalised eigenvector of the momentum operators: $P_{j} ϕ_{k} = ℏ k_{j} ϕ_{k}$ in $S^{'} (R^{3})$ . The Gel'fand-Vilenkin rigged Hilbert space $S (R^{3}) ↪ L^{2} (R^{3}) ↪ S^{'} (R^{3})$ supplies the analytic setting: the plane waves are honest eigenstates in $S^{'}$ , the Schwartz functions are the test functions on which $P$ and $J$ act as differential operators, and the $L^{2}$ middle is where physical states live.

Proof. Compute $P_{j} ϕ_{k} (x) = - i ℏ \partial_{j} [(2 π)^{- 3/2} e^{i k \cdot x}] = - i ℏ (i k_{j}) ϕ_{k} (x) = ℏ k_{j} ϕ_{k} (x)$ in $S^{'}$ . The triple $S \subset L^{2} \subset S^{'}$ is a Gel'fand triple: $S$ is dense in $L^{2}$ , and $L^{2}$ embeds in $S^{'}$ via the inner product. The spectral theorem in the rigged-Hilbert-space form (Gel'fand-Vilenkin 1964 Vol. IV) gives a measure-theoretic resolution of the identity by generalised eigenvectors that are exactly the plane waves; the integral form $ψ (x) = \int_{R^{3}} ψ (k) ϕ_{k} (x) d^{3} k$ is the inverse Fourier transform, expressing every $L^{2}$ -function as a "linear combination" of plane waves with coefficients in $L^{2}$ . $□$

Connections Master

Hilbert space 02.11.08. The free-particle representation lives on $L^{2} (R^{3})$ , the prototypical separable infinite-dimensional Hilbert space. The strongly continuous unitary action of $E (3)$ requires the full functional-analytic toolkit — Plancherel's theorem, Stone's theorem on one-parameter unitary groups, the spectral theorem for unbounded self-adjoint operators — all of which take their standard form on a separable Hilbert space and underwrite the rigorous statement of every claim in this unit.
Lie group 03.03.01. The Euclidean group $E (3) = R^{3} ⋊ S O (3)$ is a six-dimensional connected Lie group, non-compact and non-semisimple. Its Lie-group structure determines the strongly continuous topology of the representation, the smoothness of the orbit map $E (3) \to S^{2}$ , and the differentiable structure on which the infinitesimal generators $P_{j}$ and $J_{k}$ act as essentially self-adjoint operators on the smooth-vector subspace.
Lie algebra representation 07.06.01. The infinitesimal version of the free-particle representation is the $e (3)$ -action on the smooth-vector space of $L^{2} (R^{3})$ , with $P_{j}, J_{k}$ realised as essentially self-adjoint operators on the Schwartz space. The $e (3)$ commutation relations are verified directly on the Schwartz subspace, and Nelson's theorem on analytic vectors integrates the Lie-algebra action to a Lie-group representation by exponentiation.
Induced representation 07.01.07. The Mackey machine identifies the free-particle representation $(S_{k_{0}}^{2}, σ_{0})$ as the induced representation $Ind_{R^{3} ⋊ S O (2)}^{E (3)} (χ_{k_{0}} \otimes 1_{S O (2)})$ from the character $χ_{k_{0}}$ of translations and the unit character of the little group $S O (2)$ . The general theory of induced representations of locally compact groups, with Mackey's imprimitivity theorem as its central structural result, gives the framework that this unit specialises to the $E (3)$ case.
Quantum free particle (geometric framing) 03.14.01. A parallel rendering of the same concept lives in the modern-geometry chapter under the namespace modern-geometry.quantum-free-particle-e3. That unit treats the free particle via wave-function differential geometry and the unitary action of $E (3)$ on smooth wave functions; the present unit emphasises the representation-theoretic and Mackey-machine perspective. Both presentations identify the free particle with the same irreducible $E (3)$ -representation; the two are intertwined by the Fourier transform together with the geometric quantisation of the cotangent bundle $T^{*} R^{3}$ .

Historical & philosophical context Master

The identification of the quantum free particle with a unitary representation of the Euclidean group originates in Hermann Weyl's 1928 Gruppentheorie und Quantenmechanik ^{[Weyl 1928]}, where the role of unitary representations of symmetry groups as the language of quantum mechanics was first systematically articulated. Weyl treated the Heisenberg commutation relations as a projective representation of $R^{2 n}$ (the abelian group of phase-space translations) and worked out the spatial-symmetry corollary for rotations, identifying spherical harmonics on $S^{2}$ as the angular content of $L^{2}$ wave functions via the decomposition of $L^{2} (S^{2})$ into $S O (3)$ -irreducibles. Eugene Wigner's 1939 paper "On unitary representations of the inhomogeneous Lorentz group" (Ann. Math. 40, 149-204) ^{[Wigner 1939]} then adapted the orbit-and-stabiliser method to the Poincaré group, producing the classification of relativistic particles by mass and spin. The Euclidean version of Wigner's classification — for the spatial subgroup $E (3) \subset Gal (3)$ and for the full Galilean group with its central extension — was extracted and formalised in Inönü-Wigner 1953 (Proc. Nat. Acad. Sci. 39, 510-524) and Bargmann 1954 (Ann. Math. 59, 1-46) ^{[Bargmann 1954]}, the latter establishing that mass in nonrelativistic quantum mechanics is a cohomology class — the central charge of the Bargmann extension of the Galilean group.

George Mackey's 1949 imprimitivity theorem (Proc. Nat. Acad. Sci. 35, 537-545) ^{[Mackey 1949]} and the systematic machine in his 1968 monograph Induced Representations of Groups and Quantum Mechanics ^{[Mackey 1968]} supplied the algebraic infrastructure: irreducible unitary representations of a semidirect product with abelian normal factor are induced from characters of the abelian piece tensored with little-group representations. The 1949 paper presents this in the language of systems of imprimitivity, identifying the position-projection-valued measure on configuration space as the algebraic content of "the particle is at a definite position." Jean-Marie Souriau's 1970 Structure des systèmes dynamiques ^{[Souriau 1970]} reformulated the construction in symplectic-geometric language via the coadjoint-orbit method, identifying the classical free particle with a coadjoint orbit of $E (3)$ and the quantum free particle with its geometric quantisation. Konstant 1970 and Kirillov 1962 developed the parallel orbit-method framework, with the unifying principle that irreducible unitary representations of a Lie group correspond to coadjoint orbits together with prequantisation data. The contemporary mathematician-facing exposition by Peter Woit, Quantum Theory, Groups and Representations (Springer 2017) ^{[Woit 2017]}, synthesises this material and makes it the entry point for a graduate-level introduction to quantum mechanics built around unitary representation theory.

Bibliography Master

@book{Weyl1928,
  author    = {Weyl, Hermann},
  title     = {Gruppentheorie und Quantenmechanik},
  publisher = {S. Hirzel, Leipzig},
  year      = {1928},
  note      = {English transl. Dover 1950 as \emph{The Theory of Groups and Quantum Mechanics}}
}

@article{Wigner1939,
  author  = {Wigner, Eugene P.},
  title   = {On unitary representations of the inhomogeneous {L}orentz group},
  journal = {Annals of Mathematics},
  volume  = {40},
  number  = {1},
  year    = {1939},
  pages   = {149--204}
}

@article{Bargmann1954,
  author  = {Bargmann, Valentine},
  title   = {On unitary ray representations of continuous groups},
  journal = {Annals of Mathematics},
  volume  = {59},
  number  = {1},
  year    = {1954},
  pages   = {1--46}
}

@article{InonuWigner1953,
  author  = {{\.{I}}n{\"o}n{\"u}, Erdal and Wigner, Eugene P.},
  title   = {On the contraction of groups and their representations},
  journal = {Proceedings of the National Academy of Sciences},
  volume  = {39},
  number  = {6},
  year    = {1953},
  pages   = {510--524}
}

@article{Mackey1949,
  author  = {Mackey, George W.},
  title   = {Imprimitivity for representations of locally compact groups {I}},
  journal = {Proceedings of the National Academy of Sciences},
  volume  = {35},
  year    = {1949},
  pages   = {537--545}
}

@book{Mackey1968,
  author    = {Mackey, George W.},
  title     = {Induced Representations of Groups and Quantum Mechanics},
  publisher = {Benjamin / Editore Boringhieri},
  year      = {1968}
}

@book{Souriau1970,
  author    = {Souriau, Jean-Marie},
  title     = {Structure des syst{\`e}mes dynamiques},
  publisher = {Dunod},
  address   = {Paris},
  year      = {1970},
  note      = {English transl. Birkh{\"a}user 1997 as \emph{Structure of Dynamical Systems}}
}

@book{Folland1995,
  author    = {Folland, Gerald B.},
  title     = {A Course in Abstract Harmonic Analysis},
  publisher = {CRC Press},
  year      = {1995}
}

@book{Hall2013,
  author    = {Hall, Brian C.},
  title     = {Quantum Theory for Mathematicians},
  publisher = {Springer},
  series    = {Graduate Texts in Mathematics},
  volume    = {267},
  year      = {2013}
}

@book{Woit2017,
  author    = {Woit, Peter},
  title     = {Quantum Theory, Groups and Representations: An Introduction},
  publisher = {Springer},
  year      = {2017}
}

@book{ReedSimonI,
  author    = {Reed, Michael and Simon, Barry},
  title     = {Methods of Modern Mathematical Physics {I}: Functional Analysis},
  publisher = {Academic Press},
  edition   = {Revised and Enlarged},
  year      = {1980}
}

@book{GelfandVilenkinIV,
  author    = {Gel{\cprime}fand, I. M. and Vilenkin, N. Ya.},
  title     = {Generalized Functions, Vol. 4: Applications of Harmonic Analysis},
  publisher = {Academic Press},
  year      = {1964}
}

@article{Kirillov1962,
  author  = {Kirillov, A. A.},
  title   = {Unitary representations of nilpotent {L}ie groups},
  journal = {Russian Mathematical Surveys},
  volume  = {17},
  number  = {4},
  year    = {1962},
  pages   = {53--104}
}

@incollection{Kostant1970,
  author    = {Kostant, Bertram},
  title     = {Quantization and unitary representations},
  booktitle = {Lectures in Modern Analysis and Applications {III}},
  series    = {Lecture Notes in Mathematics},
  volume    = {170},
  publisher = {Springer},
  year      = {1970},
  pages     = {87--208}
}

Prerequisites

02.11.08
03.03.01
07.06.01
07.01.07

Tier anchors

beginner: Woit, *Quantum Theory, Groups and Representations* (Springer 2017), Ch. 19 §1, informal opening on the free-particle Hamiltonian and momentum eigenstates
intermediate: Woit Ch. 19 (free particle on $\mathbb{R}^3$ and the action of $E(3)$); Hall *Quantum Theory for Mathematicians* (Springer GTM 267, 2013) Ch. 17 §17.3-17.4 on Mackey machine for $E(n)$
master: Mackey, *Induced Representations of Groups and Quantum Mechanics* (Benjamin 1968) Ch. 1-2 (the original imprimitivity-theorem treatment of the free particle); Souriau, *Structure des systèmes dynamiques* (Dunod 1970) Ch. III §13-15 on coadjoint orbits of $E(3)$ and prequantisation; Folland *A Course in Abstract Harmonic Analysis* (CRC 1995) Ch. 6 §6.6

References

Woit — Quantum Theory, Groups and Representations: An Introduction · Springer 2017, Ch. 19; revised draft 20 Oct 2025, free-particle chapter and $E(3)$ representation theory
Hall — Quantum Theory for Mathematicians · Springer GTM 267, 2013, Ch. 17 §17.3-17.4 (Mackey machine for the Euclidean group)
Mackey — Induced Representations of Groups and Quantum Mechanics · Benjamin / Editore Boringhieri 1968; Ch. 1-2 imprimitivity theorem and its physical interpretation
Wigner — On unitary representations of the inhomogeneous Lorentz group · Ann. Math. 40 (1939) 149-204; orbit-and-stabiliser classification later adapted to $E(n)$
Souriau — Structure des systèmes dynamiques · Dunod 1970, Ch. III §13-15; coadjoint orbits of $E(3)$ and the free particle as the orbit through a momentum covector
Folland — A Course in Abstract Harmonic Analysis · CRC 1995, Ch. 6 §6.6; semidirect products $A \rtimes H$ with $A$ abelian and the Mackey-machine classification
Reed-Simon — Methods of Modern Mathematical Physics I: Functional Analysis · Academic Press 1980 revised ed., §VIII.3 on the position-momentum Fourier intertwiner and Stone's theorem applied to $\mathbb{R}^3$ translations

Estimated time

beginner: 16m
intermediate: 40m
master: 75m