Quantum free particle as a representation of E(3)

Intuition [Beginner]

A free quantum particle in three-dimensional space is described by a wave function $\psi (x,y,z)$ whose squared magnitude gives the probability density for finding the particle at a given location. The particle has no forces acting on it, so it moves in a straight line at constant speed.

Space itself has symmetries: you can rotate your coordinate axes, or shift your origin to a different point. The Euclidean group E(3) is the collection of all such rotations and translations. A central principle of quantum mechanics is that physical symmetries must act on wave functions in a way that preserves probabilities. This means E(3) acts on the space of wave functions by unitary transformations (linear maps that preserve the squared-magnitude integral).

The free particle provides the cleanest example of this principle. Because the particle feels no potential, the Hamiltonian (energy operator) commutes with all rotations and translations. The wave functions organise themselves into patterns dictated by the symmetry group: different momentum values label different orbits of the translation subgroup, and different angular momenta label how the wave function transforms under rotations.

Visual [Beginner]

A wave function $\psi (x)$ drawn as a complex-valued wave over a two-dimensional slice of space, with a set of coordinate axes. On the right, the same wave function after a translation (shifted to the right) and after a rotation by 90 degrees (rotated pattern). Below, a group diagram showing E(3) as a combination of SO(3) (rotations, drawn as a sphere) and R^3 (translations, drawn as arrows), with the semidirect product connecting them.

The key insight is that rotations and translations do not commute with each other: rotating then translating differs from translating then rotating, because the rotation changes the direction of the translation.

Worked example [Beginner]

The plane wave. Consider a free particle in one spatial dimension, so the wave function depends on a single coordinate $x$. A plane wave $\psi (x)=e^{ikx}$ has definite momentum $p=\hslash k$ (where $\hslash$ is the reduced Planck constant). Under a translation $x\mapsto x+a$, the wave function transforms as $\psi (x)\mapsto \psi (x-a)=e^{ik(x-a)}=e^{-ika}\psi (x)$. The translation multiplies the wave function by a phase factor $e^{-ika}$ depending only on the translation distance and the momentum.

Under a spatial reflection $x\mapsto -x$, the wave function becomes $\psi (-x)=e^{-ikx}$, which is a plane wave with momentum $-\hslash k$. So reflections reverse the direction of momentum. These two transformation rules show how the one-dimensional Euclidean group acts on plane-wave states: translations give phase factors, and reflections flip the momentum sign.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Euclidean group E(3)). The Euclidean group $\mathrm{E}(3)=\mathrm{SO}(3)⋉{\mathbb{R}}^3$ is the semidirect product of the rotation group $\mathrm{SO}(3)$ with the translation group ${\mathbb{R}}^3$, with multiplication $(R_1,v_1)(R_2,v_2)=(R_1{R}_2,v_1+R_1{v}_2)$. Its Lie algebra $\mathfrak{e}(3)=\mathfrak{so}(3)⋉{\mathbb{R}}^3$ has generators $L_i$ (angular momentum) and $P_i$ (linear momentum) satisfying $[L_i,L_j]={ϵ}_{ijk}{L}_k$, $[L_i,P_j]={ϵ}_{ijk}{P}_k$, $[P_i,P_j]=0$.

Definition (Free-particle representation). The Hilbert space of a spinless free particle is $\mathcal{H}=L^2({\mathbb{R}}^3)$. The unitary representation $U:\mathrm{E}(3)\to \mathrm{U}(\mathcal{H})$ is defined by $(U(R,v)\psi )(x)=\psi (R^{-1}(x-v))$. The corresponding Lie algebra representation gives the momentum operators $P_i=-i\hslash \partial /\partial x_i$ and angular momentum operators $L_i=-i\hslash {ϵ}_{ijk}{x}_j\partial /\partial x_k$.

The Casimir operator $P^2=P_1^2+P_2^2+P_3^2$ commutes with all generators of $\mathfrak{e}(3)$ and is proportional to the free Hamiltonian $H=P^2/(2m)$. The eigenvalue $p^2={\hslash }^2(k_1^2+k_2^2+k_3^2)$ labels the irreducible representations: each mass shell $\{k\in {\mathbb{R}}^3:|k{|}^2=k_0^2\}$ carries one irreducible of E(3).

Key theorem with proof [Intermediate+]

Theorem (Irreducible decomposition). The free-particle representation of E(3) on $L^2({\mathbb{R}}^3)$ decomposes as a direct integral of irreducible representations labelled by the positive real number $|k|$. Each irreducible ${\mathcal{H}}_k$ consists of wave functions supported on the momentum sphere of radius $|k|$ in Fourier space, on which SO(3) acts by rotation.

Proof sketch. Pass to the momentum-space (Fourier) representation: $\hat{\psi }(k)=(2\pi \hslash {)}^{-3/2}\int e^{-ik\cdot x/\hslash }\psi (x)d^{3}x$. In momentum space, the representation of E(3) acts by $(U(R,v)\hat{\psi })(k)=e^{-ik\cdot v/\hslash }\hat{\psi }(R^{-1}k)$. The Casimir $P^2$ acts as multiplication by ${\hslash }^2|k{|}^2$, so the spectral decomposition of $P^2$ gives the direct integral over mass shells ${\Sigma }_{|k|}=\{k\in {\mathbb{R}}^3:|k|=|k|\}$. On each shell, the translations act by phases $e^{-ik\cdot v/\hslash }$ and SO(3) acts by rotating the argument, yielding an irreducible representation induced from the stabiliser of a point on the sphere. $\square$

Bridge. This decomposition realises Mackey's theory of induced representations in the concrete setting of quantum mechanics ^{[Mackey 1968]}; the mass-shell decomposition mirrors how the orbits of a group action classify irreducible representations, just as the Delaunay decomposition in lattice theory classifies translationally invariant structures. The Fourier transform to momentum space is the intertwining operator that diagonalises the translation subgroup, analogous to how the Fourier transform on finite groups diagonalises the regular representation in 07.01.05, and the mass Casimir plays the same labelling role that the quadratic Casimir plays in the classification of irreducible representations of compact Lie groups in 07.01.12.

Exercises [Intermediate+]

Advanced results [Master]

Mackey's imprimitivity theorem. The free-particle representation is an induced representation $U={\mathrm{Ind}}_{\mathrm{Stab}(k_0)}^{\mathrm{E}(3)}({\chi }_{k_0})$ where ${\chi }_{k_0}$ is the character of the stabiliser subgroup of a fixed momentum vector $k_0$. Mackey's imprimitivity theorem gives a bijection between irreducible unitary representations of $\mathrm{E}(3)$ and pairs $(\mathcal{O},\sigma )$ where $\mathcal{O}$ is an orbit of $\mathrm{SO}(3)$ on ${\mathbb{R}}^3$ and $\sigma$ is an irreducible of the stabiliser. For the free particle, $\mathcal{O}=S_{|k|}^2$ (a sphere) and the stabiliser is $\mathrm{SO}(2)$ (rotations about the momentum axis).

Souriau's symplectic viewpoint. Souriau's 1970 formulation identifies the free-particle coadjoint orbit $m\mathcal{O}\subset \mathfrak{e}(3{)}^{\ast }$ with the classical phase space, equipped with the Kirillov-Kostant-Souriau symplectic form. Quantisation maps this orbit to the Hilbert space ${\mathcal{H}}_k$, and the KKS form becomes the symplectic structure underlying the Schrodinger equation.

Synthesis. The free-particle representation of E(3) sits at the crossroads of representation theory, symplectic geometry, and quantum physics; Mackey's orbit method ^{[Mackey 1968]} provides the classification framework where the coadjoint orbit $\mathcal{O}\subset \mathfrak{e}(3{)}^{\ast }$ determines the irreducible representation, paralleling how the orbit of $\mathrm{SO}(3)$ on ${\mathbb{R}}^3$ labels the mass shell in the physical picture, while Souriau's symplectic construction ^{[Souriau 1970]} unifies the classical and quantum descriptions by identifying the coadjoint orbit as the phase space itself, just as the cotangent bundle $T^{\ast }Q$ serves as the universal phase space in the Hamiltonian mechanics of 09.05.01 pending. The Casimir value $|k|$ is the symplectic invariant distinguishing different orbits, playing the same structural role as the energy level in the quantum oscillator of 09.04.01 pending, and the direct integral decomposition over mass shells is the spectral-theoretic counterpart of the orbit decomposition in the adjoint representation.

Full proof set [Master]

Proposition (Irreducibility on a mass shell). For each $k_0>0$, the representation of E(3) on $L^2(S_{k_0}^2)$ (square-integrable functions on the sphere of radius $k_0$) is irreducible.

Proof. Suppose $W\subseteq L^2(S_{k_0}^2)$ is a closed invariant subspace. The action of translations gives $(U(0,v)f)(k)=e^{-ik\cdot v/\hslash }f(k)$. Since the exponential functions $\{e^{-ik\cdot v/\hslash }{\}}_{v\in {\mathbb{R}}^3}$ span a dense subspace of $C(S_{k_0}^2)$ by the Stone-Weierstrass theorem restricted to the sphere, invariance under all translations forces $W$ to contain all functions whose restriction to the sphere lies in the span of these exponentials. The SO(3) action then rotates these functions to all angular directions. Any nonzero $f\in W$ generates all of $L^2(S_{k_0}^2)$ under the combined E(3) action, so $W=L^2(S_{k_0}^2)$. $\square$

Connections [Master]

• The orbit method applied to E(3) is a blueprint for the orbit method for the Poincare group, which classifies relativistic particles by mass and spin; the Souriau moment map framework developed here generalises directly to the symplectic G-spaces of 05.03.02.

• The Fourier transform diagonalising the translation action on ${\mathbb{R}}^3$ is the same tool used to diagonalise the translation action on ${\mathbb{R}}^n$ in the representation theory of abelian groups in 07.01.09, where the non-abelian Fourier transform decomposes group algebras.

• The direct integral decomposition of $L^2({\mathbb{R}}^3)$ into irreducible E(3)-modules labelled by $|k|$ parallels the spectral decomposition of the hydrogen atom Hamiltonian in 09.04.01 pending, where the energy levels label the irreducible representations of the rotation group appearing in the bound-state wave functions.

Bibliography [Master]

@book{hall2013,
  author = {Hall, Brian C.},
  title = {Quantum Theory for Mathematicians},
  publisher = {Springer},
  year = {2013}
}

@book{souriau1970,
  author = {Souriau, Jean-Marie},
  title = {Structure des Syst\`emes Dynamiques},
  publisher = {Dunod},
  year = {1970}
}

@book{mackey1968,
  author = {Mackey, George W.},
  title = {Induced Representations of Groups and Quantum Mechanics},
  publisher = {Benjamin},
  year = {1968}
}

@book{taylor2011,
  author = {Taylor, Michael E.},
  title = {Partial Differential Equations I},
  publisher = {Springer},
  year = {2011}
}

Historical & philosophical context [Master]

The identification of quantum states with unitary group representations was systematised by Wigner in his 1939 paper on the unitary representations of the inhomogeneous Lorentz group ^{[Hall 2013]}. Wigner showed that relativistic particles are classified by the irreducible representations of the Poincare group, with mass and spin as the labelling Casimir invariants. The nonrelativistic free-particle case (E(3) instead of the Poincare group) is the pedagogically cleaner precursor that already exhibits the key structural features: a non-compact group, continuous spectrum, and the orbit method.

Mackey's work in the 1950s and 1960s ^{[Mackey 1968]} placed Wigner's physical insight into the rigorous mathematical framework of induced representations and the imprimitivity theorem. Souriau's 1970 monograph ^{[Souriau 1970]} reinterpreted the same classification through symplectic geometry, identifying the coadjoint orbit as the classical phase space and quantisation as the passage from orbit to representation.

Philosophically, the free-particle representation embodies the thesis that the symmetries of a physical system determine its quantum structure. The Euclidean group constrains what kinds of wave functions are possible, and the mass Casimir selects the physical sector. This "group-first" approach, pioneered by Wigner and developed by Mackey, Kirillov, and Souriau, became the dominant paradigm in mathematical physics and representation theory.