Lie Groups

A Lie group is a group whose elements also form a smooth manifold, with multiplication and inversion depending smoothly on the group elements. Lie groups are the natural language of continuous symmetries: rotations, translations, phase transformations, time evolution, and many internal symmetry groups.

The slogan is:

A Lie group is a group with calculus.

This page introduces the smooth group object. Lie Algebras describes the infinitesimal generators near the identity.

Definition

A Lie group $G$ is both:

• a group;
• a smooth manifold;

such that the multiplication map

$$m:G\times G\to G, \qquad m(g,h)=gh,$$

and the inverse map

$$\iota:G\to G, \qquad \iota(g)=g^{-1},$$

are smooth maps.

The manifold condition lets group elements vary continuously and differentiably. The smoothness of multiplication and inversion says that composing and undoing transformations is compatible with that differentiable structure.

Why Lie Groups Matter in Quantum Mechanics

Quantum mechanics uses continuous transformations everywhere:

• spatial translations $T(a)$ depending on a displacement $a$;
• rotations $R(\hat{\mathbf n},\theta)$ depending on an axis and angle;
• phase transformations $e^{i\theta}$;
• time-evolution operators $U(t)$;
• spin rotations represented by $SU(2)$;
• families of unitary transformations generated by observables.

Continuous symmetry is powerful because it can be differentiated. Once a symmetry family is smooth near the identity, one can extract infinitesimal generators. In quantum mechanics, those generators are often observables: momentum, angular momentum, charge, or the Hamiltonian.

Lie Groups versus Discrete Groups

A finite group can be viewed as a zero-dimensional Lie group, but finite groups do not provide the usual infinitesimal-generator story. Most physics uses the term “Lie group” for groups with nontrivial continuous parameters.

For example:

• parity by itself is a two-element discrete group;
• translations of the line form a one-dimensional Lie group;
• three-dimensional rotations form a three-dimensional Lie group;
• $U(1)$ phase rotations form a one-dimensional compact Lie group.

Discrete and continuous symmetries are both important. Lie groups are the continuous side of the story.

Matrix Lie Groups

Many Lie groups in quantum mechanics are matrix groups. A matrix Lie group is a group of matrices that is also a smooth manifold inside a matrix space.

Important examples include:

$$GL(n,\mathbb C) = \{A\in M_n(\mathbb C):\det A\ne 0\},$$

the group of invertible complex matrices;

$$U(n) = \{U\in M_n(\mathbb C):U^\dagger U=I\},$$

the unitary group; and

$$SU(n) = \{U\in U(n):\det U=1\}.$$

Matrix Lie groups are convenient because multiplication, inversion, derivatives, and exponentials can be written using matrices. Not every Lie group must be introduced as a matrix group, but matrix groups cover many quantum-mechanical examples.

The Circle Group U(1)

The circle group is

$$U(1) = \{e^{i\theta}:\theta\in\mathbb R\}.$$

Its multiplication is

$$e^{i\theta}e^{i\phi} = e^{i(\theta+\phi)}.$$

The parameter $\theta$ is periodic:

$$\theta\sim \theta+2\pi.$$

Thus $U(1)$ is not the real line; it is a circle. This distinction is responsible for phase winding, quantized flux conditions, and many global phase effects.

In quantum mechanics, $U(1)$ appears as:

• phase multiplication of state representatives;
• internal phase symmetries connected to conserved charges;
• the structure group of complex line bundles;
• the group behind abelian gauge phases and Berry holonomy.

Translations

Translations of the line can be parameterized by $a\in\mathbb R$ with group law

$$a+b.$$

This Lie group is isomorphic to the additive group $(\mathbb R,+)$. On wavefunctions, translations are represented by unitary operators

$$T(a) = e^{-iaP/\hbar},$$

where $P$ is the momentum generator under the usual domain assumptions.

Translations in $d$ dimensions form $(\mathbb R^d,+)$. This group is abelian and noncompact.

Rotations and SO(3)

The group of proper rotations in three-dimensional Euclidean space is

$$SO(3) = \{R\in M_3(\mathbb R):R^TR=I,\ \det R=1\}.$$

It is a three-dimensional nonabelian Lie group. Its elements act on vectors by

$$\mathbf v\mapsto R\mathbf v.$$

Rotations about different axes generally do not commute. This noncommutativity is the finite-rotation version of the angular-momentum commutation relations.

The group $SO(3)$ is the correct group for ordinary spatial rotations of classical vectors. Quantum spinors require the closely related group $SU(2)$.

Spin and SU(2)

The group

$$SU(2) = \{U\in M_2(\mathbb C):U^\dagger U=I,\ \det U=1\}$$

is a three-dimensional compact Lie group. It is the double cover of $SO(3)$: two elements of $SU(2)$ correspond to the same ordinary rotation in $SO(3)$.

Spin-$1/2$ states transform naturally under $SU(2)$. A rotation by angle $\theta$ about unit vector $\hat{\mathbf n}$ is represented by

$$U(\hat{\mathbf n},\theta) = \exp \left( -\frac{i}{2}\theta\,\hat{\mathbf n}\cdot\boldsymbol\sigma \right).$$

The half-angle is not a trick of notation. It is the signature of the double-cover relation between $SU(2)$ and $SO(3)$.

Smooth Actions

A Lie group action on a smooth manifold $M$ is a group action

$$G\times M\to M, \qquad (g,p)\mapsto g\cdot p,$$

that is also smooth as a map of manifolds.

Smooth actions allow infinitesimal analysis. For a one-parameter family $g(t)$ with $g(0)=e$, the derivative of $g(t)\cdot p$ at $t=0$ gives an infinitesimal vector field on $M$. In quantum mechanics, the analogous derivative of a unitary family gives a generator.

Tangent Space at the Identity

The tangent space at the identity element,

$$T_eG,$$

contains the infinitesimal directions in the group. For a Lie group, this tangent space carries extra algebraic structure: the Lie algebra.

For a matrix Lie group, an infinitesimal element can be pictured as a matrix $X$ such that

$$g(t) = I+tX+O(t^2)$$

lies in the group to first order. The commutator of such infinitesimal matrices encodes the noncommutativity of the group near the identity.

This page only previews that structure. Lie Algebras owns the systematic discussion of generators, commutators, and structure constants.

Exponential Map Preview

For matrix Lie groups, the exponential map sends an infinitesimal generator $X$ to a group element:

$$\exp X = I+X+\frac{1}{2!}X^2+\frac{1}{3!}X^3+\cdots.$$

A one-parameter subgroup has the form

$$g(t)=\exp(tX).$$

In quantum mechanics, unitary one-parameter groups are often written

$$U(\alpha) = \exp \left( -\frac{i}{\hbar}\alpha G \right),$$

where $G$ is a Hermitian generator in the physics convention. This formula is a bridge between Lie groups and observables.

The exponential map is not always globally one-to-one or onto in every Lie group. It is, however, the central local tool near the identity for the matrix groups used most often in quantum mechanics.

Compact and Noncompact Examples

Compactness is a global topological property of the group manifold.

Examples:

• $U(1)$ is compact; it is a circle.
• $SU(2)$ is compact; as a manifold it is a three-sphere.
• $SO(3)$ is compact.
• $(\mathbb R,+)$ is noncompact.
• $GL(n,\mathbb C)$ is noncompact.

Compact groups have especially well-behaved unitary representation theory. Many angular-momentum tools rely on compactness of rotation groups and their covers.

Common Mistakes

• Treating every continuous parameterization as a global coordinate system on the group.
• Forgetting that $U(1)$ is a circle, not the real line.
• Confusing $SO(3)$ with $SU(2)$.
• Assuming continuous groups are automatically abelian.
• Mixing the Hermitian physics convention for generators with the anti-Hermitian mathematics convention.
• Ignoring the difference between a Lie group and its Lie algebra.
• Assuming the exponential map is globally one-to-one.
• Treating a finite discrete group as if it supplied infinitesimal generators.

Cross-Links

• Groups
• Group Actions
• Representations
• Unitary Representations
• Lie Algebras
• Manifolds, First Look
• Matrix Functions and Exponentials
• Unitary Operators
• U(1) Bundles and Quantum Phase
• SO(3)
• SU(2)
• SU(2) versus SO(3)
• Heisenberg Group
• Angular Momentum Algebra
• One-Parameter Unitary Groups
• Generators
• Unitary Symmetries
• Spin Rotations

References

• B. C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• H. Georgi, Lie Algebras in Particle Physics, 2nd ed., Westview Press, 1999.
• J. F. Cornwell, Group Theory in Physics, Vol. 1, Academic Press, 1984.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

1. Show that $U(1)$ is closed under multiplication and inversion.

Solution

If $z=e^{i\theta}$ and $w=e^{i\phi}$, then

$$zw = e^{i(\theta+\phi)} \in U(1).$$

The inverse is

$$z^{-1} = e^{-i\theta},$$

which is also in $U(1)$. Thus $U(1)$ is closed under multiplication and inversion.

2. Why is the additive group $(\mathbb R,+)$ a Lie group but not compact?

Solution

The real line is a smooth one-dimensional manifold, addition and inversion $a\mapsto -a$ are smooth, and the group axioms hold. It is not compact because the real line is unbounded and cannot be covered by finitely many bounded coordinate intervals in the way a compact manifold can.

3. Explain why $SO(3)$ is nonabelian.

Solution

Rotations about different axes generally do not commute. For example, rotating first about the $x$ axis and then about the $y$ axis usually gives a different final orientation than doing the two rotations in the opposite order. Therefore the group multiplication in $SO(3)$ is noncommutative.

4. For a one-parameter unitary family $U(\alpha)=\exp(-i\alpha G/\hbar)$, compute the first-order expansion near $\alpha=0$.

Solution

Using the exponential series,

$$U(\alpha) = I-\frac{i}{\hbar}\alpha G+O(\alpha^2).$$

Thus $G$ is the coefficient of the infinitesimal transformation in the physics Hermitian-generator convention.

5. Why does a finite symmetry group usually not lead to a conserved generator by differentiation?

Solution

A finite group has no nontrivial continuous parameter through the identity. Without a smooth one-parameter family, there is no derivative at the identity that produces an infinitesimal generator. Finite symmetries can still constrain spectra and matrix elements, but their consequences are not obtained by differentiating a continuous family.