Symmetry Principles

Symmetry principles tell us which transformations preserve quantum predictions, how those transformations act on states and observables, and what follows when they also preserve a Hamiltonian. This chapter is the conceptual gateway from the core formalism to generators, conservation laws, angular momentum, spin, discrete symmetries, selection rules, and geometric phases.

The essential sequence is

Physical transformation
→ action on rays
→ unitary or antiunitary representative
→ action on states, observables, and dynamics
→ Hamiltonian invariance test
→ quantum numbers, constraints, multiplets, and selection rules

The sequence prevents a common shortcut: calling an available transformation a symmetry before checking what it preserves.

Canonical Boundaries

This page owns the map and problem-solving workflow. The detailed statements have separate canonical homes.

Object	Canonical home	Role here
pure states as rays	Rays and Global Phase	supplies the physical state space
abstract groups and representations	Groups and Representations	supplies the mathematical language
transition-probability definition	Quantum Symmetries	defines a quantum symmetry
unitary or antiunitary implementation	Wigner’s Theorem Preview	states the structural theorem
active versus passive conventions	Active and Passive Transformations	fixes signs and inverse actions
Hamiltonian filtering	Symmetry Constraints on Hamiltonians	gives the construction algorithm
symmetry-protected eigenspaces	Degeneracy and Multiplets	owns the representation-theoretic consequences
chapter route and diagnostic workflow	this page	connects the canonical pieces

Continuous generators and conservation laws begin with one-parameter unitary groups. Concrete translations, rotations, spin, parity, and time reversal belong to their later chapters.

The Core Definition

Pure physical states are rays rather than individual normalized vectors. A quantum symmetry is therefore first a transformation of rays that preserves transition probabilities. If rays represented by $|\psi\rangle$ and $|\phi\rangle$ are mapped to rays represented by $|\psi'\rangle$ and $|\phi'\rangle$, then

$$|\langle\phi'|\psi'\rangle|^2 = |\langle\phi|\psi\rangle|^2.$$

Wigner’s theorem states, under its standard assumptions, that such a ray transformation can be represented on Hilbert-space vectors by either a unitary or an antiunitary operator. The representative is not unique: multiplying it by an overall phase leaves its action on rays unchanged.

This theorem answers how prediction-preserving transformations can act. It does not say that every such transformation is a symmetry of a particular Hamiltonian. Dynamics must still be tested.

From Transformation to Dynamical Symmetry

Let $S$ represent a candidate transformation. An active state transformation is

$$|\psi\rangle \longmapsto S|\psi\rangle.$$

For a unitary transformation, an observable transforms by conjugation:

$$A \longmapsto SAS^{-1}.$$

A time-independent Hamiltonian is invariant when

$$SHS^{-1}=H.$$

For unitary $S$, this is equivalent to

$$[S,H]=0.$$

The conjugation equation remains the safer statement for antiunitary transformations because antiunitary maps are antilinear and ordinary commutator manipulations can hide complex conjugation.

Three statements must remain distinct:

1. $S$ is a valid transformation of quantum states.
2. $S$ is a symmetry of the Hamiltonian.
3. A particular state ray is invariant under $S$.

A parity-symmetric Hamiltonian, for example, can evolve a state that is not itself a parity eigenstate. Symmetry of the laws is not invariance of every allowed state.

The Seven-Step Workflow

Use this sequence before calculating matrix elements or diagonalizing a Hamiltonian.

1. Name the physical transformation

Specify the operation: translation, rotation, inversion, time reversal, particle exchange, an internal transformation, or a change of external parameters. State what physical objects it acts on.

2. Fix active or passive language

An active transformation changes the state or system while the descriptive frame is fixed. A passive transformation changes coordinates, basis, or frame while the physical state is held fixed. The two descriptions commonly use inverse matrices, so an unstated convention creates sign errors in generators.

3. Identify the state space

State whether the transformation acts on spatial wavefunctions, spinors, tensor-product states, a degenerate eigenspace, or another Hilbert space. A group without its representation space does not determine the physics.

4. Determine unitary or antiunitary action

Continuous symmetries connected to the identity are represented unitarily. Time reversal is the central antiunitary example. Do not infer the class from a classical picture alone; check its action on amplitudes and on $i$.

5. Transform states, observables, and parameters

Write all three actions explicitly. External electric or magnetic fields may transform too. Holding a parameter fixed when the physical transformation should reverse it can turn a covariance statement into a false symmetry claim.

6. Test the Hamiltonian

Evaluate $SHS^{-1}$ and compare it with $H$. For a parameterized family, distinguish

$$SH(\lambda)S^{-1} = H(\lambda)$$

from covariance between different parameter values,

$$SH(\lambda)S^{-1} = H(\lambda').$$

Only the first is invariance at fixed parameters.

7. Extract consequences carefully

Depending on the symmetry, consequences can include:

• invariant subspaces and block-diagonal Hamiltonians;
• conserved generators for continuous symmetries;
• good quantum numbers;
• relations among matrix elements;
• degeneracy within irreducible multiplets;
• forbidden Hamiltonian terms;
• selection rules;
• Kramers pairing for suitable antiunitary symmetry;
• superselection structure when coherent mixing is excluded.

No single consequence follows from every symmetry. The representation and the Hamiltonian determine which conclusions are justified.

Continuous Symmetries

A differentiable one-parameter unitary family can be written locally as

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

where $G$ is Hermitian. If

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

for all $\alpha$ near zero, differentiating at $\alpha=0$ gives

$$[G,H]=0.$$

For a time-independent $G$, closed-system Schrödinger evolution then gives

$$\frac{d}{dt}\langle G\rangle=0.$$

Translations lead to momentum generators, rotations to angular momentum, and time translations to the Hamiltonian. One-parameter unitary groups develop this chain, including its assumptions and limitations.

Groups, Representations, and Multiplets

The symmetry group records how transformations compose. A representation records how they act on a chosen vector space. The same abstract group can act differently on scalar wavefunctions, vectors, spinors, operators, and many-particle states.

If a Hamiltonian commutes with every representative $U(g)$ of a group $G$, then each energy eigenspace is invariant under the group action:

$$\begin{aligned} H|\psi\rangle &= E|\psi\rangle, \\ H U(g)|\psi\rangle &= U(g)H|\psi\rangle \\ &=E U(g)|\psi\rangle. \end{aligned}$$

The eigenspace can therefore be decomposed into representation multiplets. When an irreducible representation has dimension greater than one and appears once, symmetry forces equal energy within that multiplet. Important cautions remain:

• one-dimensional irreducible representations do not force degeneracy;
• several copies of the same representation can mix;
• accidental degeneracy can occur without being required by the stated symmetry;
• an additional hidden symmetry can explain a degeneracy missed by the first group;
• a symmetry-breaking perturbation can split a multiplet according to the remaining subgroup.

Use Symmetry Groups and Representations for the working representation language and Degeneracy and Multiplets for the spectral consequences.

Projective Action and Rays

Quantum states are rays, so composition may be exact on rays while vector representatives compose only up to phase:

$$U(g_1)U(g_2) = e^{i\omega(g_1,g_2)} U(g_1g_2).$$

This is a projective representation. The phase does not alter an individual ray, but its consistency encodes real structure. Spin-$1/2$ is the central example: physical rotations form $SO(3)$, while spinors carry ordinary representations of its double cover $SU(2)$.

For a spin-$1/2$ rotation,

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

so

$$U(\hat{\mathbf n},2\pi)=-I.$$

The vector changes sign, but the ray does not. Projective Representations owns the general explanation; later spin pages develop the concrete rotations.

Unitary and Antiunitary Branches

Feature	Unitary representative	Antiunitary representative
scalar action	linear	antilinear
inner product	preserved	complex conjugated
action on $i$	$i\mapsto i$	$i\mapsto-i$
connected continuous family	standard case	not connected continuously to identity
typical example	translation, rotation, parity	time reversal
Hamiltonian test	$UHU^{-1}=H$	$\Theta H\Theta^{-1}=H$

Antiunitarity is not an optional convention for time reversal. Reversing momenta while preserving the form of Schrödinger evolution requires complex conjugation of amplitudes. The details and basis dependence of the conjugation operator belong in Antiunitary Symmetries and the Discrete Symmetries chapter.

Three Diagnostic Examples

Parity in one dimension

Parity acts on a wavefunction as

$$(P\psi)(x)=\psi(-x),$$

with

$$P\hat xP^{-1}=-\hat x, \qquad P\hat pP^{-1}=-\hat p.$$

For

$$H = \frac{\hat p^2}{2m} +V(\hat x),$$

parity is a symmetry exactly when $V(-x)=V(x)$. The Hilbert space then separates into even and odd sectors, but a general superposition of those sectors is not itself parity invariant.

Spinless time reversal

In the position basis, spinless time reversal can be represented by complex conjugation $\Theta=K$. It satisfies

$$\Theta i\Theta^{-1}=-i, \qquad \Theta\hat p\Theta^{-1}=-\hat p.$$

A real scalar potential with no magnetic field gives a time-reversal-invariant Hamiltonian. The example shows why an antiunitary operation cannot be treated as an ordinary unitary matrix with a different label.

A constrained two-level Hamiltonian

A general Hermitian two-level Hamiltonian is

$$H = a_0 I +a_x\sigma_x +a_y\sigma_y +a_z\sigma_z,$$

with real coefficients. Suppose $P=\sigma_z$ is imposed as a unitary symmetry. Since

$$\begin{aligned} P\sigma_xP^{-1}&=-\sigma_x, \\ P\sigma_yP^{-1}&=-\sigma_y, \\ P\sigma_zP^{-1}&=\sigma_z, \end{aligned}$$

Hamiltonian invariance requires $a_x=a_y=0$. Symmetry has filtered the allowed operator terms before diagonalization.

Page Map

Page	Central question	Main output
Quantum Symmetries	What does a quantum symmetry preserve?	transition-probability definition
Active and Passive Transformations	Is the system changing or only its description?	inverse and generator-sign conventions
States, Observables, and Hamiltonians	Which object is transformed or invariant?	four-way diagnostic table
Wigner’s Theorem Preview	Why unitary or antiunitary?	theorem statement, assumptions, and proof architecture
Unitary Symmetries	How do ordinary quantum symmetries act?	conjugation and invariant inner products
Antiunitary Symmetries	What changes under antilinear symmetry?	complex conjugation and time-reversal structure
Projective Representations	Why can composition close only up to phase?	ray representations and spinor preview
Symmetry Groups and Representations	How does an abstract group act on Hilbert space?	irreducible sectors and representation labels
Symmetry Constraints on Hamiltonians	Which operator terms are allowed?	Hamiltonian construction algorithm
Degeneracy and Multiplets	When does symmetry organize equal energies?	invariant eigenspaces and multiplets
Superselection Sectors Preview	When are coherent superpositions operationally excluded?	sector decomposition and scope cautions

Reading Paths

First working pass

Read Quantum Symmetries, Active and Passive Transformations, States, Observables, and Hamiltonians, Unitary Symmetries, and Symmetry Constraints on Hamiltonians. Then continue to generators and concrete spatial symmetries.

Graduate structural pass

Add Wigner’s Theorem Preview, Antiunitary Symmetries, Projective Representations, Symmetry Groups and Representations, and Superselection Sectors Preview.

Problem-solving pass

Start from Symmetry Constraints on Hamiltonians when constructing a model. Use Degeneracy and Multiplets when interpreting a spectrum. Use Active and Passive Transformations whenever translation or rotation signs are in doubt.

Distinctions to Keep Visible

Do not confuse	Correct separation
transformation and symmetry	a transformation becomes a dynamical symmetry only after an invariance test
covariance and invariance	covariance may map $H(\lambda)$ to $H(\lambda')$; invariance returns the same fixed Hamiltonian
symmetric Hamiltonian and symmetric state	the dynamics can be invariant while a particular state is not
basis change and active operation	one redescribes the same physics; the other changes the physical state or apparatus
group and representation	the group gives composition; the representation gives its action on a chosen space
ordinary and projective representation	vector representatives may compose up to phase because states are rays
multiplet and accidental degeneracy	a multiplet transforms irreducibly; equal energies can also occur for other reasons
conserved label and superselection sector	conservation restricts dynamics; superselection additionally excludes observable coherence between sectors

Common Mistakes

• Calling every coordinate transformation a physical symmetry.
• Checking how states transform but not how external fields or Hamiltonian parameters transform.
• Replacing $SHS^{-1}=H$ by $[S,H]=0$ without noticing that $S$ is antiunitary.
• Assuming a symmetry forces every energy level to be degenerate.
• Treating a degenerate numerical eigenspace as a unique basis of symmetry eigenvectors.
• Reading a passive rotation formula with the active sign convention.
• Ignoring the phase freedom of symmetry representatives on rays.
• Treating time reversal as “run the movie backward” without its action on complex amplitudes.
• Inferring a superselection rule from ordinary energy conservation alone.

Exercises

1. Symmetric Hamiltonian, nonsymmetric state

Let $[H,P]=0$ with $P^2=I$, and let $|e\rangle$ and $|o\rangle$ have parity $+1$ and $-1$. Is the state

$$|\psi\rangle = \frac{|e\rangle+|o\rangle}{\sqrt2}$$

parity invariant?

Solution

Applying parity gives

$$P|\psi\rangle = \frac{|e\rangle-|o\rangle}{\sqrt2}.$$

This is not the original state times one overall phase, so the ray is not parity invariant. The Hamiltonian can nevertheless be parity symmetric. Because $[H,P]=0$, the even and odd components evolve within their respective sectors.

2. Derive the generator condition

Let $U(\alpha)=e^{-i\alpha G/\hbar}$ and suppose $U(\alpha)HU(\alpha)^{-1}=H$ for all sufficiently small $\alpha$. Derive the infinitesimal condition.

Solution

Differentiate at $\alpha=0$:

$$\begin{aligned} 0 &= \left. \frac{d}{d\alpha} \left( UHU^{-1} \right) \right|_{\alpha=0} \\ &= -\frac{i}{\hbar}GH + \frac{i}{\hbar}HG \\ &= -\frac{i}{\hbar}[G,H]. \end{aligned}$$

Therefore $[G,H]=0$. The derivation assumes a differentiable unitary family and a common domain on which the operator manipulations are valid.

3. Classify spinless time reversal

Why can the spinless operation $\Theta=K$ preserve transition probabilities even though it is not unitary?

Solution

Complex conjugation is antiunitary. It reverses scalar multiplication,

$$K(c|\psi\rangle) = c^*K|\psi\rangle,$$

and conjugates inner products:

$$\langle K\phi|K\psi\rangle = \langle\phi|\psi\rangle^*.$$

Taking the absolute square removes that conjugation, so transition probabilities are preserved. Wigner’s theorem allows both unitary and antiunitary representatives for precisely this reason.

4. Filter a two-level Hamiltonian

For

$$H=a_0I+a_x\sigma_x+a_y\sigma_y+a_z\sigma_z,$$

impose invariance under $P=\sigma_x$. Which coefficients must vanish?

Solution

Conjugation gives

$$\begin{aligned} \sigma_x\sigma_x\sigma_x&=\sigma_x, \\ \sigma_x\sigma_y\sigma_x&=-\sigma_y, \\ \sigma_x\sigma_z\sigma_x&=-\sigma_z. \end{aligned}$$

Thus

$$PHP^{-1} = a_0I+a_x\sigma_x-a_y\sigma_y-a_z\sigma_z.$$

Equality with $H$ requires $a_y=a_z=0$. The coefficients $a_0$ and $a_x$ remain allowed.

5. Interpret a full spinor rotation

A spin-$1/2$ vector changes sign under a $2\pi$ rotation. Why does this not mean the isolated physical state has changed?

Solution

The rotation gives

$$|\psi\rangle \longmapsto -|\psi\rangle.$$

The two vectors represent the same ray, so every expectation value and transition probability involving the isolated state is unchanged. The sign can still become observable as a relative phase when one branch of a coherent interferometric superposition is rotated and another is not. Projective representation theory keeps these two statements consistent.

Cross-Links

• Symmetry, Angular Momentum, and Spin
• Why Symmetry Matters
• Concept Map
• Learning Path
• Notation and Conventions
• Continuous Symmetries and Conservation Laws
• One-Parameter Unitary Groups
• Translations and Momentum
• Discrete Symmetries
• Parity
• Groups
• Representations
• Rays and Global Phase
• Commutators

References

1. E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press (1959).
2. J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press (2020).
3. R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer (1994), doi:10.1007/978-1-4757-0576-8.
4. S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press (1995), Chapter 2.
5. H. F. Jones, Groups, Representations and Physics, 2nd ed., CRC Press (1998).
6. M. Tinkham, Group Theory and Quantum Mechanics, Dover (2003).