Generators

A generator is the observable that appears in the infinitesimal form of a continuous unitary transformation, usually a one-parameter unitary group. The smooth symmetry groups behind such transformations are Lie groups, and their infinitesimal commutator structure is a Lie algebra. Translations are generated by momentum, rotations by angular momentum, and time translations by the Hamiltonian.

The classical counterpart is an infinitesimal canonical transformation generated by a phase-space function through the Poisson bracket.

The standard form is

$$U(\alpha) = e^{-i\alpha G/\hbar},$$

where $\alpha$ is the transformation parameter and $G$ is the generator.

Generator as a Derivative

For a continuous unitary family with $U(0)=I$,

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

Therefore

$$G = i\hbar \left.\frac{dU}{d\alpha}\right|_{\alpha=0}.$$

This formula is the practical meaning of “generator”: $G$ is the coefficient of the infinitesimal transformation.

Self-Adjointness and Observables

If $U(\alpha)$ is unitary for real $\alpha$, the generator is self-adjoint under the usual domain assumptions. Physically, this is why generators are observables.

For small $\alpha$,

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

Unitarity to first order requires $G^\dagger=G$.

Infinitesimal State Transformation

An active transformation of the state is

$$\lvert\psi\rangle \longmapsto U(\alpha)\lvert\psi\rangle.$$

For small $\alpha$,

$$\delta\lvert\psi\rangle = -\frac{i}{\hbar}\alpha G\lvert\psi\rangle.$$

This is the state-space version of an infinitesimal symmetry transformation.

Infinitesimal Operator Transformation

With the active convention used in this volume,

$$A\longmapsto U(\alpha)AU^\dagger(\alpha).$$

To first order,

$$\delta A = -\frac{i}{\hbar}\alpha [G,A].$$

If instead one uses the passive or Heisenberg-style convention $A\mapsto U^\dagger A U$, the sign flips:

$$\delta A = \frac{i}{\hbar}\alpha [G,A].$$

Many sign disagreements in books come from switching between these two conventions without saying so.

Translation Generator

For spatial translations along $x$,

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

The generator $P$ is the momentum operator. In the position representation on the line,

$$P=-i\hbar\frac{d}{dx}.$$

This is not a separate postulate here; it follows from requiring translations to shift wavefunctions.

Rotation Generator

For a rotation by angle $\theta$ about unit vector $\hat{\mathbf n}$,

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

The generator is $\hat{\mathbf n}\cdot\mathbf J$. The components of $\mathbf J$ satisfy the angular momentum algebra

$$[J_i,J_j] = i\hbar\sum_k\epsilon_{ijk}J_k.$$

Time-Translation Generator

For time-independent closed-system evolution,

$$U(t)=e^{-iHt/\hbar}.$$

The generator is the Hamiltonian $H$. This is why the Hamiltonian is more than an energy function: it is the operator that generates time evolution.

Sign and Parameter Conventions

The parameter $\alpha$ may have dimensions. For translations, $\alpha=a$ has dimensions of length and $G=P$ has dimensions of momentum. For rotations, $\alpha=\theta$ is dimensionless and $G=\hat{\mathbf n}\cdot\mathbf J$ has dimensions of angular momentum. The combination $\alpha G/\hbar$ must be dimensionless.

Common Mistakes

• Forgetting the factor of $\hbar$ in the exponential.
• Mixing active and passive transformation signs.
• Assuming all Hermitian-looking formal differential operators are self-adjoint on the intended domain.
• Treating the Hamiltonian only as energy rather than as a generator.
• Writing a finite transformation without checking the domain or boundary conditions of its generator.

Cross-Links

• Unitary Symmetries
• One-Parameter Unitary Groups
• Active and Passive Transformations
• Canonical Transformations
• Lie Groups
• Lie Algebras
• Infinitesimal Transformations
• Commutators and Conservation Laws
• Quantum Noether Principle
• From Quantum Generators to Noether Currents
• Translations and Momentum
• Momentum Operator as Generator
• Hamiltonians as Generators
• Angular Momentum Operators
• Angular Momentum Algebra
• Momentum Operator

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.

Exercises

1. Starting from $U(\alpha)=I-i\alpha G/\hbar+O(\alpha^2)$, show that first-order unitarity implies $G^\dagger=G$.

Solution

Compute

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

This equals

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

For this to equal $I$ to first order for arbitrary $\alpha$, one needs $G^\dagger=G$.

2. What is the generator of a rotation by angle $\theta$ about the $z$ axis?

Solution

With the convention

$$U(\hat z,\theta) = e^{-i\theta J_z/\hbar},$$

the generator is $J_z$.