Concept Map

This map shows how the main ideas in the volume fit together. It is a routing page: use it to decide which concept controls a calculation, which canonical page owns the details, and which mistakes to avoid.

An arrow in this page means “naturally leads to” or “is used to organize,” not strict logical equivalence. For example, symmetry can explain a degeneracy, but not every degeneracy is forced by an obvious symmetry.

At a Glance

Starting question	Symmetry data	Main outputs	First page
What transformations preserve the physics?	unitary or antiunitary operators	invariance, conserved transition probabilities, Hamiltonian constraints	Quantum Symmetries
What is conserved?	continuous symmetry and generator	constants of motion, good quantum numbers	Generators
How do rotations organize states?	angular momentum algebra	multiplets, spherical harmonics, spin, addition rules	Rotations in Three Dimensions
How do rotations organize operators?	adjoint action of rotations	tensor ranks, reduced matrix elements, selection rules	Tensor Operators and Selection Rules
Which matrix elements vanish?	state and operator transformation laws	selection rules and forbidden transitions	Selection Rules
How do inversion and time reversal constrain a model?	unitary or antiunitary action, transformed parameters, symmetry square	parity sectors, real structures, Kramers pairs	Discrete Symmetries
How can phase carry geometric or topological information?	eigenstate families, connections, curvature, and holonomy	Berry phases, Chern numbers, molecular and Hall geometry	Geometric Phases and Topology
What survives breaking, scale changes, or perturbations?	full Hamiltonian, residual group, limiting state, allowed deformation class	split multiplets, Goldstone modes, emergent and protected structure	Symmetry Breaking and Emergence
How is the formalism used in a concrete domain?	Hamiltonian, probe, scattering operator, channel, or parameter family	labels, blocks, selection rules, protected responses, and null tests	Symmetry in Applications
What changes when symmetry acts on relativistic local fields?	currents, particle and field representations, correlators, gauge redundancy, and the vacuum	charges, Ward identities, CPT, Goldstone modes, and anomaly constraints	Bridge to Quantum Field Theory
Where can I check a formula, convention, or worked problem?	basis, phase, normalization, and status metadata	identities, tables, diagnostics, solved practice, and reproducibility routes	Reference and Problem Lab
What happens to phase around a loop?	connection, curvature, and holonomy	Aharonov–Bohm phase, magnetic translation algebra, Berry phase	Gauge, Phase, and Magnetic Geometry

Transformation Chain

The basic symmetry chain is:

transformations
  -> quantum symmetry operators
  -> unitary or antiunitary action
  -> Hamiltonian invariance
  -> generators for continuous families
  -> commutators with H
  -> conserved quantities
  -> good quantum numbers
  -> degeneracies and selection rules

The starting point is a transformation of states and observables. A transformation becomes a symmetry of a Hamiltonian only after an invariance condition is checked. For a unitary symmetry this often appears as

UHU^\dagger=H.

For a continuous unitary family,

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

the Hermitian operator $G$ is the generator. If

[H,G]=0

and $G$ has no explicit time dependence, then $G$ is conserved. This is the practical bridge from transformations to constants of motion.

Use this chain when a problem asks whether momentum, angular momentum, parity, spin projection, or another good quantum number is protected by the Hamiltonian.

Rotation and Spin Chain

Rotations form the most important concrete example:

spatial rotations
  -> SO(3)
  -> projective quantum action
  -> SU(2) double cover
  -> angular momentum algebra
  -> orbital angular momentum and spin
  -> addition of angular momentum
  -> tensor operators and selection rules

Orbital angular momentum is generated by rotations of position space. It leads to spherical harmonics and central-potential labels:

L^2Y_\ell^m = \hbar^2\ell(\ell+1)Y_\ell^m, \qquad L_zY_\ell^m = \hbar mY_\ell^m.

Spin uses the same algebra but is not literally orbital motion. Spin- $1/2$ states are spinors, and their rotations are represented through $SU(2)$ :

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

When systems have more than one angular momentum, tensor products create a total rotation generator and coupled and uncoupled bases. Clebsch–Gordan coefficients are the change-of-basis amplitudes, and Wigner symbols organize recoupling when there are several angular momenta.

Use this chain when a problem contains $j,m$ , $\ell,m$ , spinors, spherical harmonics, magnetic sublevels, or multiplets.

Selection-Rule Chain

Selection rules are symmetry zeros:

state labels + operator transformation
  -> allowed representation products
  -> angular, parity, charge, or spin constraints
  -> matrix element zero or allowed amplitude

The typical matrix element is

\langle f|O|i\rangle.

If the states and operator transform incompatibly under a symmetry, the matrix element must vanish. For parity, a nonzero matrix element requires

\pi_f\pi_O\pi_i=1.

For an irreducible spherical tensor $T_q^{(k)}$ , rotational symmetry gives

m_f=m_i+q, \qquad \lvert j_i-k\rvert \le j_f \le j_i+k.

Use this chain before doing an integral. It tells you whether an integral is allowed to be nonzero at all.

Discrete-Symmetry Chain

Discrete symmetries do not require infinitesimal generators:

parity, time reversal, and related operations
  -> transformation of states and operators
  -> Hamiltonian constraints
  -> protected labels or degeneracies
  -> forbidden couplings or weakly allowed effects

Parity as Spatial Inversion introduces inversion as a spatial operation. The fuller Parity page treats parity as a discrete symmetry; parity sends ordinary polar vectors such as position to their negatives. Time Reversal is antiunitary and complex conjugates amplitudes in addition to reversing momenta and angular momenta; Antiunitary Time Reversal explains why that conjugation is forced. Time Reversal for Spinless Particles gives the $\Theta=K$ case with $\Theta^2=+I$ . Discrete Symmetries in Hamiltonians turns these sign rules into allowed and forbidden Hamiltonian terms. For spin- $1/2$ systems, time reversal squares to $-I$ , which underlies Kramers Degeneracy.

Use this chain when a Hamiltonian contains external fields, magnetic moments, spin–orbit terms, or operators whose behavior under inversion or time reversal matters.

Phase and Geometry Chain

Phase geometry is the global side of the map:

phase freedom of state vectors
  -> local gauge choices
  -> connection
  -> curvature
  -> holonomy around loops
  -> topological quantum numbers

The physical state is a ray, so a local phase choice is a gauge convention:

\lvert n(R)\rangle \mapsto e^{i\chi(R)} \lvert n(R)\rangle.

The Berry connection

A_n=i\langle n|dn\rangle

records local phase comparison, while its curvature

F_n=dA_n

records local geometric phase flux. A closed loop can produce a Berry phase

\gamma_n[C] = \oint_C A_n.

The Aharonov–Bohm effect is a related holonomy phenomenon in real configuration space rather than adiabatic parameter space. Chern numbers appear when Berry curvature is integrated over a closed two-dimensional parameter space with an isolated eigenstate or band.

Use this chain when a problem involves cyclic adiabatic evolution, flux-threaded loops, gauge choices, conical intersections, or band topology.

Concept Separations

Do not confuse	Difference
Transformation and symmetry	A transformation is available; it is a symmetry only if it preserves the Hamiltonian or relevant dynamics.
Generator and conserved quantity	A generator is conserved only when it commutes with the Hamiltonian and has no explicit time dependence.
Orbital angular momentum and spin	They share the algebra, but orbital angular momentum acts on spatial wavefunctions while spin is intrinsic.
$SO(3)$ and $SU(2)$	$SU(2)$ double-covers $SO(3)$ and acts naturally on spinors.
Selection rule and small rate	A selection rule can force a matrix element to vanish; an allowed matrix element may still be dynamically small.
Berry connection and Berry phase	The connection is gauge dependent; closed-loop phase factors and curvature are the gauge-invariant data.
Aharonov–Bohm and Berry phase	Both are holonomies, but the base space and physical connection are different.

Where to Go Next

If you are trying to…	Open
choose a guided reading order through the volume	Learning Path
decide whether a transformation is a symmetry	Symmetry Constraints on Hamiltonians
derive a conserved quantity from a continuous symmetry	Commutators and Conservation Laws
orient spatial rotations before the full algebra	Rotations Preview
work through active, passive, and wavefunction rotations	Rotations in Three Dimensions
understand why SO(3) and SU(2) both appear	SO(3) and SU(2) Preview
identify the operator that generates rotations	Angular Momentum Operators
compute orbital angular momentum in wavefunctions	Position-Space Representation
separate radial and angular variables	Spherical Coordinates
understand a pure angular-motion model	Rigid Rotor
connect orbital angular momentum to magnetic fields	Magnetic Moments from Orbital Motion
place spin inside the angular momentum algebra	Spin as Intrinsic Angular Momentum
compute spin-component outcome probabilities	Spin Measurements
understand the spinor sign after a full turn	Spinors and 2π Rotations
move beyond spin- $1/2$ multiplets	Higher Spin Systems
track spin magnetic moments and g factors	Magnetic Moments and g-Factors
start angular-momentum addition	Addition of Angular Momentum
define the total rotation generator	Total Angular Momentum
understand spatial inversion before discrete symmetries	Parity as Spatial Inversion
compute angular momentum ladder actions	Ladder Operators
understand spin- $1/2$ rotations	Spin Rotations
add two angular momenta	Clebsch–Gordan Coefficients
read coefficient tables safely	Clebsch–Gordan Tables and Conventions
add orbital and spin angular momentum	Addition of Orbital and Spin Angular Momentum
choose LS, jj, or hyperfine labels	Angular Momentum Coupling Schemes
connect singlet/triplet symmetry to identical particles	Identical Particles and Exchange Symmetry Preview
classify operators under rotations	Scalar, Vector, and Tensor Operators
test operator transformation laws infinitesimally	Commutators with Angular Momentum
apply even/odd matrix-element rules	Parity Selection Rules
compare electric and magnetic multipole rules	Multipole Operators
apply selection rules to atomic lines	Applications to Atomic Spectra
apply selection rules to molecular rotation	Applications to Molecular Rotations
apply an angular selection rule	Wigner–Eckart Theorem
track parity or time reversal	Parity and Time Reversal
compute a geometric phase	Berry Phase Problems
check conventions, formulas, worked practice, or computational status	Reference and Problem Lab

Common Mistakes

Following arrows backward as if every consequence uniquely identifies a cause.
Treating a conserved label as protected before checking the Hamiltonian.
Using spin pictures that hide the Hilbert-space representation.
Applying orbital selection rules after spin–orbit coupling has changed the good labels.
Treating gauge-dependent connections as directly observable.
Calling a visual mnemonic a derivation.

Cross-Links

References

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.
A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
A. Shapere and F. Wilczek, eds., Geometric Phases in Physics, World Scientific, 1989.

Exercises

A Hamiltonian commutes with $J_z$ but not with $J^2$ . Which part of the map should you use first, and what label is safer to keep?

Solution

Use the transformation-to-generator chain for continuous symmetries. If $[H,J_z]=0$ , the magnetic label $m$ associated with the chosen axis may remain good. If $[H,J^2]\ne0$ , the full rotational multiplet label $j$ is not generally protected.

Why does a Berry connection being gauge dependent not make the Berry phase useless?

Solution

The local connection changes under a phase convention choice, but its closed-loop phase factor is invariant modulo $2\pi$ when the gauge is single-valued around the loop. The curvature is also gauge invariant in the abelian nondegenerate case.