Group Actions

A group action is a rule telling how each element of a group transforms points of some space. The group itself gives the abstract composition law; the action tells what the transformations actually do.

This distinction is essential in quantum mechanics. The same abstract group can act on positions, momenta, spin states, wavefunctions, rays, observables, labels of identical particles, or parameters in a Hamiltonian. The physical meaning depends on the action.

Definition

Let $G$ be a group and let $X$ be a set. A left action of $G$ on $X$ is a map

$$G\times X\to X, \qquad (g,x)\mapsto g\cdot x,$$

such that

$$e\cdot x=x$$

for every $x\in X$, and

$$(gh)\cdot x = g\cdot(h\cdot x)$$

for every $g,h\in G$ and $x\in X$.

The first condition says that the identity group element does nothing. The second says that group multiplication matches composition of transformations. If $gh$ means “do $h$ first, then $g$”, the formula above is exactly the usual convention for composing functions.

Right Actions

Some subjects use right actions, written

$$x\cdot g.$$

Then the compatibility condition is

$$(x\cdot g)\cdot h = x\cdot(gh).$$

Left and right conventions are both valid, but they must not be mixed casually. Quantum mechanics usually writes operators acting on state vectors from the left, while some geometric and gauge-theory conventions naturally use right actions on frames or principal bundles.

Why Actions Matter

The abstract group $\mathbb Z_2$ has two elements. By itself, it does not say what physical transformation is being performed. A $\mathbb Z_2$ action could be:

• parity acting on position by $x\mapsto -x$;
• spin flip acting on a two-level system;
• exchange of two identical particle labels;
• multiplication of a field by $-1$;
• time reversal squared in a system where the square is the identity on rays.

The group law is the same in each case. The action is different.

Examples on Sets and Spaces

Translations of the line form a group under addition. They act on points $x\in\mathbb R$ by

$$a\cdot x = x+a.$$

The action condition is

$$(a+b)\cdot x = x+a+b = a\cdot(b\cdot x)$$

after choosing the convention for the order of addition.

Rotations in $SO(3)$ act on vectors in $\mathbb R^3$ by matrix multiplication:

$$R\cdot \mathbf v = R\mathbf v.$$

The group law is matrix multiplication, and the action condition is

$$(R_1R_2)\mathbf v = R_1(R_2\mathbf v).$$

The symmetric group $S_n$ acts on the set of labels $\{1,\ldots,n\}$ by permutation:

$$\sigma\cdot i = \sigma(i).$$

It can also act on $n$-tuples by permuting entries:

$$\sigma\cdot(x_1,\ldots,x_n) = (x_{\sigma^{-1}(1)},\ldots,x_{\sigma^{-1}(n)}),$$

where the inverse appears so that the convention gives a left action on functions or components. This is a common place where composition-order mistakes enter.

Linear Actions

If $X$ is a vector space $V$, an action is linear when each map

$$v\mapsto g\cdot v$$

is a linear map $V\to V$. A linear group action is essentially a representation:

$$g\mapsto R(g), \qquad R(gh)=R(g)R(h),$$

where $R(g)$ is a linear operator on $V$.

For quantum mechanics, linear and unitary actions are central because state spaces are complex vector spaces with inner products. But not every action is a linear action. Rotations can act on ordinary points of space, permutations can act on labels, and a group can act on a set of Hamiltonians without that set being a vector space in any useful way.

Orbits

The orbit of $x\in X$ is the set of all points reachable from $x$ by the group action:

$$G\cdot x = \{g\cdot x:g\in G\}.$$

Orbits organize a space into symmetry-related pieces. If two points lie in the same orbit, the group can transform one into the other.

Examples:

• all points on a sphere of fixed radius form an orbit under $SO(3)$;
• all translated points on the real line lie in one orbit under the full translation group;
• all basis labels lie in one orbit under the full permutation group $S_n$;
• a spin multiplet is an orbit-like family under rotations, though quantum rays and representation theory add extra structure.

In quantum mechanics, orbits are useful for recognizing degeneracy patterns and equivalent configurations. If a Hamiltonian is invariant under a group action, states related by the symmetry often have related energies or matrix elements.

Stabilizers

The stabilizer of $x\in X$ is the subgroup of elements that leave $x$ fixed:

$$G_x = \{g\in G:g\cdot x=x\}.$$

The stabilizer is also called the isotropy subgroup or little group.

For a point on a sphere, the stabilizer inside $SO(3)$ consists of rotations about the axis through that point. For a generic vector in $\mathbb R^3$, this stabilizer is an $SO(2)$ subgroup.

For finite groups, orbit and stabilizer sizes satisfy

$$\lvert G\cdot x\rvert = \frac{\lvert G\rvert}{\lvert G_x\rvert}.$$

This formula says that points with more symmetry have smaller orbits. In physics language, a highly symmetric configuration has a large stabilizer; a generic perturbation often reduces the stabilizer.

Invariant Functions

A function $f:X\to Y$ is invariant under a group action when

$$f(g\cdot x)=f(x)$$

for every $g\in G$ and $x\in X$.

For rotations, the radius

$$r^2=x^2+y^2+z^2$$

is invariant. For translations, differences of positions are invariant under simultaneous translation of all positions. For permutations, symmetric polynomials are invariant under relabeling.

Hamiltonian symmetry is an invariant statement. In a coordinate-space model, one might have

$$H(g\cdot x)=H(x).$$

In Hilbert-space language, a unitary symmetry is expressed as

$$U(g)HU(g)^{-1}=H.$$

The notation differs, but the idea is the same: the relevant object does not change under the action.

Equivariant Maps

Invariant maps are not the only useful maps. A map $F:X\to Y$ between two $G$-spaces is equivariant if

$$F(g\cdot x) = g\cdot F(x).$$

Equivariance means that applying the group action before or after the map gives compatible results. In physics, tensor operators, vector fields, spinor maps, and selection-rule machinery often rely on equivariance rather than strict invariance.

For example, a vector-valued quantity may rotate when the system is rotated. It is not invariant, but it transforms covariantly with the action.

Actions on Quantum States

A quantum symmetry acts first on physical states, which are rays. After choosing Hilbert-space representatives, the action is commonly implemented by unitary or antiunitary operators. For an ordinary unitary representation,

$$g\cdot\lvert\psi\rangle = U(g)\lvert\psi\rangle.$$

The action condition becomes

$$U(gh)\lvert\psi\rangle = U(g)U(h)\lvert\psi\rangle.$$

For ray states, phases can enter. A physical action on rays may lift to a projective action on vectors, where multiplication holds only up to phase. That issue belongs to projective representations; the group-action concept explains what is being acted on.

Actions on Observables

If states transform by a unitary $U(g)$, observables transform by conjugation:

$$A \mapsto U(g)AU(g)^{-1}.$$

This is also a group action because applying $h$ and then $g$ gives conjugation by $U(g)U(h)$.

A Hamiltonian is invariant under the symmetry when

$$U(g)HU(g)^{-1}=H$$

for every relevant $g$. An observable that transforms into itself is invariant. An observable that transforms into a component of a vector or tensor operator is not invariant, but it may still transform equivariantly.

Faithful, Free, and Transitive Actions

An action is faithful if different group elements produce different transformations of $X$. Equivalently, only the identity acts trivially on every point.

An action is free if no nonidentity element fixes any point:

$$g\cdot x=x \quad\Longrightarrow\quad g=e.$$

An action is transitive if every point can be reached from every other point:

$$\forall x,y\in X, \quad \exists g\in G \quad \text{such that} \quad g\cdot x=y.$$

These words are useful because physical examples vary. The rotation action of $SO(3)$ on a sphere is transitive but not free, because rotations about the axis through a point fix that point. A trivial action, where every group element fixes every point, is not faithful unless the group has only the identity.

Common Mistakes

• Confusing a group with an action of that group.
• Forgetting to check the action condition $(gh)\cdot x=g\cdot(h\cdot x)$.
• Mixing left-action and right-action conventions.
• Assuming every group action is faithful.
• Calling a point invariant when only its orbit or a function of it is invariant.
• Confusing a stabilizer subgroup with the whole symmetry group.
• Treating every vector-valued transforming object as invariant instead of equivariant.
• Ignoring phases when a group action on rays is lifted to Hilbert-space vectors.

Cross-Links

• Groups
• Representations
• Symmetric Group
• Unitary Representations
• Antiunitary Symmetries, First Look
• Lie Groups
• SO(3)
• Sets, Functions, and Maps
• Vector Spaces and Dual Spaces
• Matrices as Linear Maps
• Unitary Operators
• Quantum Symmetries
• Unitary Symmetries
• Projective Representations
• Symmetry Constraints on Hamiltonians
• Why Symmetry Matters
• Indistinguishability

References

• M. Artin, Algebra, 2nd ed., Pearson, 2011.
• D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd ed., Wiley, 2004.
• B. C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

1. Let $\mathbb Z$ act on $\mathbb R$ by $n\cdot x=x+n$. Verify the two action axioms.

Solution

The identity of $\mathbb Z$ under addition is $0$, and

$$0\cdot x=x+0=x.$$

For $m,n\in\mathbb Z$,

$$(m+n)\cdot x = x+m+n,$$

while

$$m\cdot(n\cdot x) = m\cdot(x+n) = x+n+m.$$

Since addition is commutative here, these agree. Thus the rule defines an action.

2. For the action of $SO(3)$ on $\mathbb R^3$, what is the stabilizer of the vector $(0,0,1)$?

Solution

The stabilizer consists of rotations that leave the $z$ axis fixed and preserve the vector $(0,0,1)$. These are rotations about the $z$ axis, forming a subgroup isomorphic to $SO(2)$.

3. Let $S_3$ act on the set $\{1,2,3\}$ by permutation. What is the orbit of $1$ and what is its stabilizer?

Solution

The orbit of $1$ is all of $\{1,2,3\}$, since some permutation sends $1$ to any chosen element. The stabilizer consists of permutations that keep $1$ fixed; they may either keep or exchange $2$ and $3$. Thus the stabilizer has two elements and is isomorphic to $S_2$.

4. Suppose $U(g)HU(g)^{-1}=H$ for every $g\in G$. Show that if $H\lvert\psi\rangle=E\lvert\psi\rangle$, then $U(g)\lvert\psi\rangle$ is also an eigenstate with eigenvalue $E$.

Solution

From $U(g)HU(g)^{-1}=H$, one gets

$$HU(g) = U(g)H.$$

Therefore

$$H\,U(g)\lvert\psi\rangle = U(g)H\lvert\psi\rangle = E\,U(g)\lvert\psi\rangle.$$

So the transformed state is again an eigenstate with the same eigenvalue.

5. Give an example of a quantity that is equivariant rather than invariant.

Solution

A position vector in three-dimensional space is equivariant under rotations: it changes as $\mathbf r\mapsto R\mathbf r$. It is not invariant as a vector, but its length squared $\mathbf r\cdot\mathbf r$ is invariant. The vector transforms compatibly with the group action, while the scalar length does not change.