Projective Representations

Quantum symmetries act on physical rays. Because a ray is unchanged by an overall phase, a symmetry group may be represented on Hilbert-space vectors up to phases. Such a representation is called projective. It should be compared with an ordinary representation, where the group multiplication law is implemented exactly by linear operators.

For the pure-state space of rays itself, see Projective Hilbert Space.

Rays Allow Phase Freedom

If two normalized vectors differ by a global phase,

$$\lvert\psi'\rangle = e^{i\alpha}\lvert\psi\rangle,$$

they represent the same pure state. Therefore a symmetry operation only has to act consistently on rays, not on a specific phase choice for every vector.

This is the origin of projective representations in quantum mechanics.

Projective Composition Law

We first restrict to unitary implementations. For a group $G$, an ordinary unitary representation satisfies

$$U(g)U(h)=U(gh).$$

A projective representation allows a phase:

$$U(g)U(h) = e^{i\alpha(g,h)}U(gh).$$

The multiplier $e^{i\alpha(g,h)}$ is independent of the vector being transformed. It does not change a ray, but not every multiplier can be removed by redefining the phases of the individual operators.

Associativity imposes a condition. Comparing $(U(g)U(h))U(k)$ with $U(g)(U(h)U(k))$ gives

$$\alpha(g,h)+\alpha(gh,k) =\alpha(h,k)+\alpha(g,hk)\pmod{2\pi}.$$

Thus the phase is constrained, not arbitrary; this is the cocycle condition.

Spin and Rotations

The most important example in elementary quantum mechanics is spin. Ordinary spatial rotations form $SO(3)$, but spin-$1/2$ states transform naturally under $SU(2)$, the double cover of $SO(3)$. The group-level comparison is developed in SU(2) versus SO(3).

For the distinction between a harmless global sign on one ray and a measurable relative sign in interference, see Spinors and 2π Rotations.

A spin-$1/2$ rotation is represented by

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

Following the continuous lifted rotation from $0$ to $2\pi$ gives

$$U(2\pi)=-I.$$

This changes the vector but not the ray. A $4\pi$ rotation returns the vector itself. A single-valued projective choice of $U(R)$ on $SO(3)$ cannot be continuous everywhere; the continuous spinor representation instead lives on $SU(2)$.

Physical Meaning of the Phase

The statement “global phase is unobservable” is true for a single isolated state. It does not mean phases in symmetry composition are irrelevant. Projective phases determine which Hilbert-space representations are possible and how states transform under combined operations.

They are especially important in:

• spinor representations of rotations,
• magnetic translations,
• Galilean boosts and mass as a central charge.

For example, Galilean Boosts shows how mass enters the multiplier. The rotation sign is analyzed experimentally in Spinors and 2π Rotations.

Rephasing and Convention Dependence

Changing representatives by phases,

$$U(g)\longmapsto e^{i\beta(g)}U(g),$$

changes the multiplier according to

$$\alpha'(g,h)=\alpha(g,h)+\beta(g)+\beta(h)-\beta(gh) \pmod{2\pi}.$$

It leaves the ray action unchanged. A trivial multiplier is one that can be removed by such a choice, with the required continuity conditions respected. A nontrivial multiplier is a property of the representation, not a poor phase convention.

If the group includes antiunitary elements, these formulas need a modification: $U(g)$ conjugates a scalar phase when $g$ is antiunitary. Writing $s(g)=+1$ for unitary and $s(g)=-1$ for antiunitary implementations replaces $\alpha(h,k)$ by $s(g)\alpha(h,k)$ in the cocycle equation and $\beta(h)$ by $s(g)\beta(h)$ in the rephasing formula.

Exercises

1. Show that $U(2\pi)=-I$ for a spin-$1/2$ rotation about the $z$ axis.

Solution

For spin-$1/2$,

$$U(\hat z,\theta) = \exp\left(-\frac{i}{2}\theta\sigma_z\right).$$

For $\theta=2\pi$,

$$U(\hat z,2\pi)=e^{-i\pi\sigma_z}.$$

Since $\sigma_z$ has eigenvalues $\pm1$, both exponential eigenvalues are $-1$. Thus $U(\hat z,2\pi)=-I$.

2. Why does a projective phase in $U(g)U(h)=e^{i\alpha(g,h)}U(gh)$ not change transition probabilities?

Solution

The phase multiplies the transformed vector by an overall unit complex number. Transition probabilities use absolute squares of inner products, so the phase cancels with its complex conjugate.

3. Derive the rephasing formula by multiplying $U'(g)=e^{i\beta(g)}U(g)$ and $U'(h)$. Why is the sign of $\beta(h)$ different if $U(g)$ is antiunitary?

Solution

For unitary $U(g)$,

$$U'(g)U'(h)=e^{i[\beta(g)+\beta(h)+\alpha(g,h)-\beta(gh)]}U'(gh).$$

For antiunitary $U(g)$, passing $e^{i\beta(h)}$ through it gives $e^{-i\beta(h)}$. That is the origin of the factor $s(g)$. It cannot be dropped by treating an antiunitary map as an ordinary matrix.

References

• V. Bargmann, “On unitary ray representations of continuous groups,” Annals of Mathematics 59, 1–46 (1954).
• B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed., Springer, 2015.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, chapters 3–4.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.