Combining Ray Representations

Ray representations can combine differently on a direct sum and a tensor product. Direct sums require compatible phase multipliers if the group is to act consistently on every ray of the combined space. Tensor products multiply the multipliers, so two nontrivial projective classes can cancel. These elementary facts explain several representation-theoretic arguments that would otherwise be disguised as statements about forbidden superpositions.

Required background. Projective Representations and Multipliers defines the multiplier class. Direct Sums and Superselection distinguishes the full Hilbert-space ray description from restricted sector observables.

A ray action is a group action before phases are chosen

For each $g\in G$, a ray symmetry $S_g$ obeys $S_gS_h=S_{gh}$. Choosing unitary lifts gives

$$U_gU_h=\omega(g,h)U_{gh}.$$

The action on rays is independent of the chosen phase for each $U_g$. On a complex Hilbert space of dimension at least two, the phase uniqueness in Wigner’s Theorem means that an operator inducing the identity on every ray must be one scalar multiple of the identity.

This last phrase is crucial. An operator that multiplies different subspaces by different phases need not induce the identity on a ray with components in both subspaces.

For clarity, the constructions below use unitary lifts throughout. Tensoring or summing actions of different linearity types requires a separate convention and cannot be handled by ignoring antiunitarity.

Direct sums need a common multiplier

Suppose $U^{(1)}$ and $U^{(2)}$ act on nonzero spaces $\mathcal H_1,\mathcal H_2$ with multipliers $\omega_1,\omega_2$. Define

$$U_g=U_g^{(1)}\oplus U_g^{(2)}.$$

Then

$$U_gU_hU_{gh}^{-1} =\omega_1(g,h)I_1\oplus\omega_2(g,h)I_2.$$

This is scalar on the full direct sum exactly when $\omega_1(g,h)=\omega_2(g,h)$. If the multiplier classes agree, independent rephasings of the two component representations can arrange that equality. If the classes differ, no such simultaneous phase convention exists.

To see the ray obstruction directly, apply the displayed operator to $\psi_1\oplus\psi_2$ with both components nonzero. Unequal phases change their relative phase and hence change the full ray. Each component separately still has a perfectly consistent projective action.

Thus there are two different questions: whether the group acts on every combined ray, and whether it acts on the union of separate sector ray spaces. The latter permits different multipliers between sectors. Turning that possibility into a physical superselection claim additionally requires specifying the allowed observables.

Tensor products multiply rather than compare multipliers

For simultaneous subsystem actions, set

$$V_g=U_g^{(1)}\otimes U_g^{(2)}.$$

The Hilbert tensor product is well defined for these bounded unitaries, and

$$V_gV_h =\omega_1(g,h)\omega_2(g,h)V_{gh}.$$

The combined projective class is the product of the two classes. It need not equal either factor. In particular, inverse classes give an ordinary representation after a suitable phase choice.

The conjugate representation makes this cancellation explicit. On the conjugate Hilbert space $\overline{\mathcal H}$, the operators $\overline{U_g}$ have multiplier $\overline{\omega(g,h)}$. Therefore $U_g\otimes\overline{U_g}$ is an ordinary representation.

Under the standard identification with Hilbert–Schmidt operators, this representation acts as

$$X\longmapsto U_g XU_g^*.$$

The phases cancel in the observable transformation. This explains how a projective vector implementation can induce an exact group action on operators.

A finite example separates the two constructions

Use the projective action of $\mathbb Z_2\times\mathbb Z_2$ generated by Pauli $X$ and $Z$, whose scalar commutator is $-I$. Its direct sum with a one-dimensional trivial representation does not define a projective representation on all rays: the commutator becomes

$$(-I_2)\oplus I_1,$$

which is not scalar.

Two copies on a tensor product behave differently. The generators $X\otimes X$ and $Z\otimes Z$ commute because the two minus signs cancel, and both square to $I$. They therefore define an ordinary representation of the abelian group. Equivalently, the sign multiplier squares to one.

This is a structural calculation, independent of any claim about which superpositions a particular apparatus can prepare. For spatial rotations the related integer-versus-half-integer issue is treated in Spin as a Projective Representation.

Do not infer a universal superselection law from a mismatch

A mismatch says that a specified group cannot act projectively on the full direct sum with the proposed sector actions. One may restrict the state description, change the symmetry group to an extension, or enlarge the physical model. Which option is appropriate is additional physics.

The Galilei mass argument is an important example: different masses carry different multipliers for the unextended group. Its conclusion must retain that group choice and the assumptions about available observables. It is not a theorem forbidding every coherent comparison between quantities called masses.

Exercise: cancel a multiplier with a conjugate

Show directly that $\alpha_g(X)=U_gXU_g^*$ satisfies $\alpha_g\alpha_h=\alpha_{gh}$. Does this imply that the original multiplier is a coboundary?

Solution

Substituting $U_gU_h=\omega(g,h)U_{gh}$ introduces a phase on the left of $X$ and its complex conjugate on the right. They cancel, giving $\alpha_g\alpha_h(X)=\alpha_{gh}(X)$. This holds for every multiplier, including the Pauli example above, whose commutator obstruction proves it is nontrivial. Exact observable composition does not supply a globally consistent vector phase choice.

References

• V. Bargmann, “On Unitary Ray Representations of Continuous Groups”, Annals of Mathematics 59, 1–46, 1954 — ray representations and multiplier compatibility.
• D. Giulini, “Superselection Rules”, arXiv:0710.1516 [quant-ph], 2007, revised 2009 — the assumptions connecting symmetry representations and superselection.
• K. Landsman, Foundations of Quantum Theory: From Classical Concepts to Operator Algebras, Fundamental Theories of Physics 188, Springer, 2017, Chapter 5 — projective symmetries and operator actions.