Quantum Implementations, Projective Actions, and Central Extensions

A quantum symmetry acts on physical rays, not on arbitrarily phased state vectors. Consequently, after choosing one Hilbert-space implementer $U(g)$ for each group element, the implementers may compose only up to a phase. That is a consistent projective action. Its phase law is encoded by a two-cocycle, and the same action can be represented linearly after enlarging the symmetry group by a central extension.

This page develops that chain carefully: ray action $\rightarrow$ multiplier $\rightarrow$ cocycle class $\rightarrow$ central extension. It also separates three mechanisms that are often conflated: a global covering group, a central term in a Lie algebra, and an anomaly obstructing gauging. The main derivation assumes a ray action implemented by unitary operators on one Hilbert space or superselection sector. Antiunitary elements conjugate complex phases and therefore obey a twisted cocycle law; they are treated conceptually in Internal, Spacetime, Discrete, and Antiunitary Symmetries. Full group-cohomology classification is outside the present scope.

Required background. What Is a Symmetry of a QFT? supplies the operational definition of a symmetry action. Representations, Intertwiners, and Invariants supplies the representation and invariant-subspace language used below.

Helpful background. Commutators and Operator Exponentials reviews the generator and exponential manipulations used in the infinitesimal discussion.

From rays to a multiplier

A pure state is a ray: $\lvert\psi\rangle$ and $e^{i\chi}\lvert\psi\rangle$ represent the same state. Suppose the physical ray transformation associated with $g\in G$ is implemented by a unitary operator $U(g)$. The implementer is itself defined only up to an overall phase. Therefore the group law on rays permits

$$U(g)U(h)=\omega(g,h)U(gh), \qquad \omega(g,h)\in U(1).$$

After choosing $U(e)=\mathbf 1$, the phases can be normalized so that

$$\omega(e,g)=\omega(g,e)=1.$$

Associativity of operator multiplication gives a nontrivial condition. The two ways of multiplying three implementers are

$$\begin{aligned} \bigl(U(g)U(h)\bigr)U(k) &=\omega(g,h)\omega(gh,k)U(ghk),\\ U(g)\bigl(U(h)U(k)\bigr) &=\omega(h,k)\omega(g,hk)U(ghk). \end{aligned}$$

Hence

$$\boxed{ \omega(g,h)\omega(gh,k) = \omega(h,k)\omega(g,hk) }.$$

This is the multiplier, or two-cocycle, condition. It is not an optional extra rule: it is precisely what makes the projective multiplication associative. Weinberg derives the multiplier law, its associativity condition, and its algebraic and topological origins at Weinberg 1995, § 2.7, pp. 81–83.

If $G$ contains antiunitary elements, the leftmost implementer acts on a phase by complex conjugation. With $g\cdot z=z$ for unitary $g$ and $g\cdot z=z^*$ for antiunitary $g$, the right-hand side is replaced by $(g\cdot\omega(h,k))\omega(g,hk)$. The untwisted formulas below should not be applied to that case without this modification.

Rephasing and the invariant class

The numerical function $\omega$ depends on the arbitrary phases chosen for the implementers. Let

$$U'(g)=\beta(g)U(g), \qquad \beta(g)\in U(1).$$

For a topological or Lie group, both the cocycle and the rephasing must obey the relevant continuity or Borel regularity; an algebraic rephasing that destroys that structure is not a physical linearization.

Then

$$\omega'(g,h) = \frac{\beta(g)\beta(h)}{\beta(gh)} \omega(g,h).$$

Thus two multipliers related by this transformation describe the same ray action. The invariant datum is the equivalence class $[\omega]$, not the particular phase table. If some choice of $\beta$ makes $\omega'(g,h)=1$ for every pair, the projective action can be linearized on $G$. Otherwise it is genuinely projective on the stated space. The set of these multiplier classes is denoted $H^2(G,U(1))$: it consists of $U(1)$-valued two-cocycles modulo the rephasings just derived, in the relevant regularity category.

The qualification “on the stated space” matters. Different symmetry-invariant superselection sectors may carry different multiplier classes. If the group permutes sectors, only a sector stabilizer acts within one sector, so projectivity there is a statement about that stabilizer. A block-diagonal collection of implementers need not have one scalar multiplier on the direct sum. A central operator may act by different scalars in different irreducible sectors; it is scalar on a sector only under the appropriate irreducibility or superselection assumptions.

Lifting to a central extension

Every unitary multiplier above defines a group

$$\widehat G=U(1)\times_\omega G$$

with multiplication

$$(z,g)(w,h) = \bigl(zw\,\omega(g,h),gh\bigr).$$

The cocycle condition is exactly the associativity condition for this product. There is an exact sequence

$$1\longrightarrow U(1) \longrightarrow \widehat G \longrightarrow G \longrightarrow 1,$$

and the inserted $U(1)$ commutes with all of $\widehat G$, so the extension is central. The projective representation of $G$ becomes the ordinary representation

$$\widehat U(z,g)=z\,U(g)$$

of $\widehat G$. This replacement and its relation to global covering groups are developed at Weinberg 1995, § 2.7, pp. 81–90.

For a Lie or topological symmetry, this formula first specifies the underlying algebraic group. The extension must also be supplied with the compatible topology or measurable structure. A global continuous section need not exist, so a Borel or patchwise cocycle—or a direct covering-group description—may be the appropriate realization.

This construction does not say that every useful extension is literally a covering group. A connected Lie group also has a universal cover whose kernel is discrete and central. For example,

$$1\longrightarrow\mathbb Z_2 \longrightarrow SU(2) \longrightarrow SO(3) \longrightarrow1.$$

Passing to a cover removes projectivity caused by the global topology of the group. A general $U(1)$ central extension can instead add a continuous central direction. Both replace a projective action by a linear one, but their topology and infinitesimal algebras need not be the same.

Infinitesimal central terms

For a smooth action near the identity, let $Q_a$ be Hermitian generators. The most general scalar central term on the chosen sector has the form

$$[Q_a,Q_b] = i f_{ab}{}^c Q_c +i\kappa_{ab}\mathbf 1, \qquad \kappa_{ab}=-\kappa_{ba}\in\mathbb R.$$

The Jacobi identity requires

$$f_{ab}{}^d\kappa_{dc} +f_{bc}{}^d\kappa_{da} +f_{ca}{}^d\kappa_{db} =0.$$

Generator origins are not unique. Under

$$Q'_a=Q_a+\lambda_a\mathbf 1, \qquad \lambda_a\in\mathbb R,$$

the central coefficient changes to

$$\kappa'_{ab} = \kappa_{ab}-f_{ab}{}^c\lambda_c.$$

A term of this removable form is therefore not invariant information. More generally, a central generator commutes with every represented symmetry generator; it need not be a multiple of the identity on a reducible Hilbert space.

Group and Lie-algebra extensions must be kept distinct:

• differentiating a smooth multiplier can produce $\kappa_{ab}$;
• a projective action can be purely topological even when its Lie algebra has no central term;
• a Lie-algebra cocycle need not exponentiate to the chosen global group without the required topology and integrality conditions.

The infinitesimal cocycle and its removable shifts are developed at Weinberg 1995, § 2.7, pp. 83–84; the global covering-group criterion and Lorentz example follow at Weinberg 1995, § 2.7, pp. 87–90.

Spin: projective rotations and Lorentz transformations

A path of spatial rotations from angle zero to $2\pi$ closes at the identity of $SO(3)$. Its lift to $SU(2)$ that starts at $\mathbf 1$ ends at $-\mathbf 1$, and the spin-$\tfrac12$ representation maps that endpoint to a minus sign on state vectors. The transported ray is unchanged. Spinors therefore carry projective $SO(3)$ actions and ordinary $SU(2)$ representations.

Relativistically, the corresponding connected statement is

$$SO^+(1,3) \simeq SL(2,\mathbb C)/\mathbb Z_2.$$

Spinor fields transform linearly under $SL(2,\mathbb C)$, the universal cover of the proper orthochronous Lorentz group, rather than as ordinary single-valued representations of $SO^+(1,3)$. The ray interpretation and covering-group resolution are explained at Schwartz 2014, § 10.5.1, pp. 176–177.

There is a sector qualification. On a fixed fermion-parity sector, the nontrivial kernel element acts as a scalar. On a state space containing both bosonic and fermionic sectors it acts as $(-1)^F$, not as one common scalar. Statements about a projective Lorentz action must therefore specify which rays and superselection structure are being represented. The formula above also concerns the connected Lorentz group, not the disconnected full group $O(1,3)$.

Galilean mass: an algebraic extension

The Galilei group provides a different mechanism. Its quantum implementation admits the central generator $M$ with

$$[K_i,P_j]=i\delta_{ij}M,$$

where $K_i$ generates boosts and $P_j$ translations. In a fixed-mass sector, $M=m\mathbf 1$, so boosts and translations commute only projectively when viewed as transformations of rays. Unlike the spin example, this extension is already visible infinitesimally. Weinberg records this mass central extension in the bounded discussion at Weinberg 1995, § 2.7, p. 86.

This nonrelativistic example does not imply that arbitrary relativistic internal symmetries admit mass-like central terms. Allowed extensions depend on the symmetry algebra, global group, dimension, locality assumptions, and the sector under study.

A bounded finite-group QFT application

Projective actions can also occur on degrees of freedom attached to defects, boundaries, or twisted sectors even when the bulk theory has an ordinary finite symmetry. In the two-dimensional finite-group example of Gaiotto, Kapustin, Seiberg, and Willett, a discrete-torsion cocycle forces suitable boundary degrees of freedom to transform projectively and changes the twisted-sector operator data Gaiotto et al. 2015, § 2, pp. 8–10, arXiv PDF.

This is a controlled application, not a proof that every twisted sector has a projective action or that every central extension is discrete torsion. Which subgroup acts in a twisted sector and which cocycle it sees depend on the finite group, the twist, the spacetime construction, and the chosen topological weighting.

Threaded scalar: an ordinary action and a negative test

For the charge-one complex scalar used throughout this volume, the standard integer-charge Hilbert space has

$$U(\alpha)=e^{-i\alpha Q}, \qquad U(\alpha)U(\beta)=U(\alpha+\beta), \qquad U(2\pi)=\mathbf 1.$$

This is an ordinary $U(1)$ representation. If

$$\Delta\mathcal L =h\phi^N+h^*(\phi^\dagger)^N, \qquad N\geq2,$$

and fixed nonzero $h$ reduces this $U(1)$ factor to its residual $\mathbb Z_N$ subgroup, a generator is

$$R=U(2\pi/N), \qquad R^N=\mathbf 1.$$

The residual action is again ordinary. More generally, if a chosen unitary implementer $V$ of a cyclic generator satisfies only

$$V^N=e^{i\gamma}\mathbf 1,$$

then

$$V'=e^{-i\gamma/N}V$$

obeys $(V')^N=\mathbf 1$. Thus this phase alone does not give an intrinsic projective class for a unitary cyclic group: in this setting $H^2(\mathbb Z_N,U(1))=0$. Nontrivial projectivity requires a different acting group or stabilizer with a nonzero multiplier class, or a twisted coefficient system such as one induced by antiunitary elements. Boundary or twisted-sector data can select such an action, but cannot make a lone unitary $\mathbb Z_N$ intrinsically projective. None of it follows from the term $h\phi^N$.

Transforming $h$ spurionically is also unrelated to projectivity. It moves in a family of source-dependent theories, whereas a multiplier compares Hilbert-space implementers of transformations within the fixed theory.

Central extensions, contact terms, and anomalies

Three superficially similar appearances of “extra terms” answer different questions.

Central extension. The symmetry acts consistently, but its implementers obey an extended group or algebra. Central means commuting with the represented symmetry generators; scalarity is a sector-dependent specialization.

Schwinger or contact term. A local current commutator may contain a distributional term. It defines a central extension of the integrated charge algebra only if it survives the spatial integration, boundary conditions, and operator prescription in precisely that form. See Contact Terms, Equal-Time Commutators, and Schwinger Terms and, when boundaries matter, Charge Algebras, Central Terms, and Corners.

Anomaly. A ’t Hooft anomaly does not prevent the global symmetry from acting consistently. It obstructs coupling that symmetry gauge-invariantly to arbitrary background fields, and therefore obstructs gauging it in the same dimension unless additional cancellation or inflow data are supplied. A perfectly consistent projective action on rays is not automatically anomalous. Conversely, an anomaly can exist without a projective representation of the ordinary zero-form symmetry on the bulk Hilbert space. The precise obstruction is developed in What Is an Anomaly?.

Common pitfalls

Treating a chosen multiplier as observable. Rephase the implementers before assigning meaning to $\omega(g,h)$. Only its equivalence class can be invariant.

Calling every phase a central charge. A central charge is a central generator or its sector eigenvalue. A removable implementation phase carries no such information.

Identifying every extension with a cover. The spin cover has a discrete kernel and no new local Lie-algebra generator. The Galilei mass extension is visible in the algebra.

Using untwisted formulas for antiunitary elements. Antiunitary implementers conjugate phases, so both the cocycle and rephasing laws are twisted.

Combining sectors without checking the center. An operator that is scalar in each sector can have different eigenvalues between sectors and therefore fail to be one scalar multiplier on their direct sum.

Equating projectivity with an anomaly. Projectivity describes a consistent ray action. An anomaly is a separate obstruction whose diagnosis requires background-field or gauging data.

Check your understanding

Let $V$ implement the generator of a unitary $\mathbb Z_N$ action on one invariant sector, and suppose $V^N=e^{i\gamma}\mathbf 1$. Show that this equation by itself does not define an intrinsic projective action. What additional kind of symmetry data could evade this cyclic rephasing argument?

Check

Choose $V'=e^{-i\gamma/N}V$. Then

$$(V')^N=e^{-i\gamma}V^N=\mathbf 1,$$

so the phase is removed and the cyclic action is linearized. A genuinely nontrivial multiplier can instead occur for an acting group or sector stabilizer with nonzero $H^2(G,U(1))$. Antiunitary elements provide another possibility because their coefficient action complex-conjugates phases and changes the coboundary equation. Neither possibility is generated by the single phase in $V^N$.

What to carry forward

A projective symmetry is an associative action on rays. Rephasing changes the multiplier but not its class; a central extension converts that projective action into an ordinary representation. Group topology, Lie-algebra central terms, local Schwinger terms, and anomalies can interact, but none should be identified with another without the relevant global, operator, and sector checks.

The anomaly chapter owns obstructions to gauging, the boundary chapter owns surface-charge central terms, and the supersymmetry volume owns supersymmetry central charges and their dynamical consequences.

References

• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 02 (2015): 172. DOI. Open PDF
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI