In my post on Wigner's theorem, I explained the famous result that any symmetry transformation on quantum states can be realized as a unitary or antiunitary operator on Hilbert space. But when we study symmetries of quantum systems, we usually have in mind not a single symmetry but a full** group** of symmetries; Wigner's theorem tells us nothing about how the operators corresponding to different symmetries in the same group should compose with one another.

Suppose, for example, that a quantum system transforms under the symmetry group $G$, and that the unitary or antiunitary operator corresponding to the element $g \in G$ is denoted by $\hat{U}_g$. Because any two unitary operators related by a phase are physically equivalent, it may not be the case that $\hat{U}_{g_1}$ and $\hat{U}_{g_2}$ compose to $\hat{U}_{g_1 g_2}$; instead, we will have a relationship like

$$\hat{U}_{g_1} \hat{U}_{g_2} = e^{i \phi(g_1, g_2)} \hat{U}_{g_1 g_2}.$$

At first glance, it seems like we just made a mistake in our definition of $\hat{U}_{g_1 g_2}$; since the Wigner operator corresponding to a symmetry is only defined up to a phase, we could freely redefine $\hat{U}_{g_1 g_2} \mapsto e^{- i \phi(g_1, g_2)} \hat{U}_{g_1 g_2}$ and obtain the expected composition law $\hat{U}_{g_1} \hat{U}_{g_2} = \hat{U}_{g_1g_2}.$ But remember that, because $G$ is a group, there may be many ways of making the group element $g_1 \cdot g_2$; for example, there may exist group elements $h_1, h_2, h_3$ with $g_1 g_2 = h_1 h_2 h_3$. If we wanted to make all of the unitary compositions "phase-free," we would need to enforce $\hat{U}_{g_1} \hat{U}_{g_2} = \hat{U}_{h_1} \hat{U}_{h_2} \hat{U}_{h_3},$ $\hat{U}_{g_1} \hat{U}_{g_2} = \hat{U}_{h_1} \hat{U}_{h_2 h_3},$ and so on. It turns out that it is not always possible to redefine phases in the Wigner unitaries so that *all* of these compositions are free of phases. (I give an explicit example where rephasing is not possible at the end of section 1.1 of this post.)

The formal statement that comes out of this observation is that there exist group actions of certain groups $G$ on the space of quantum states $\mathbb{P}(\mathcal{H})$ that cannot be realized as representations of $G$ on the Hilbert space $\mathcal{H}.$ Instead, it is necessary to construct a *central extension* group $E_G$ whose representation on $\mathcal{H}$ is consistent with the action of $G$ on $\mathbb{P}(\mathcal{H}).$ Classifying the central extensions of a given symmetry group (or, sometimes, a symmetry algebra) is an essential problem in quantum physics; for example, one can show that the algebra of infinitesimal conformal symmetries in 2D Euclidean space has a unique central extension in *the Virasoro algebra*, which explains why two-dimensional Euclidean conformal field theories all have Virasoro symmetry.

One very important theorem about central extensions, due to Bargmann in his 1954 paper, is that there are certain special symmetry groups with no nontrivial (smooth) central extensions. These are — and I'll explain what all of these terms mean later in this post — simply connected, finite-dimensional Lie groups with semisimple Lie algebras. With some extra work — mainly in dealing with the parenthetical "(smooth)" in the first sentence of this paragraph — one can show that any action of such a group on quantum states can always be represented on Hilbert space *without needing to worry about phases*. Probably the most famous application of this theorem is in the theory of spinors. The local isotropy group of a signature-$(p, q)$ pseudo-Riemannian manifold, denoted $\operatorname{O}(p, q)$, is finite-dimensional and has a semisimple Lie algebra. Its universal cover (or, more precisely, the universal cover of its identity component; more on that later) is finite-dimensional and simply connected with a semisimple Lie algebra and thus satisfies the conditions of Bargmann's theorem. As a result, for any quantum system whose space of states is acted on by the identity component of the symmetry group $\operatorname{O}(p, q)$, we can realize that action as a representation of the universal cover on Hilbert space. Spinors are basically just fields that transform in representations of this universal covering group, with some extra consistency conditions enforced.

In this post, I'll explain and motivate central extensions of groups, and central extensions of Lie algebras. I'll also give an explanation of Bargmann's theorem and sketch the proof, though I'll rely on some lemmas about Lie theory that I won't prove here. I suspect this post will eventually serve as preliminary reading for two future posts on (i) the Virasoro algebra in 2D conformal field theory and (ii) spinor fields on arbitrary manifolds, but I don't know for sure.

I learned most of what I know about this subject from: (1) Chapter 13 of Wald's book on General Relativity, which defines spinor fields in four dimensions, and (2) Chapters 3 and 4 of Schottenloher's book on conformal field theory. Bargmann's 1954 paper is also an interesting read, but the proof of his theorem has been simplified considerably since its original formulation. The proof sketch I will give of Bargmann's theorem draws partially from these notes of Zinovy Reichstein, which give a nice direct proof of the fact that finite-dimensional semisimple Lie algebras have no nontrivial central extensions. (Traditionally, this statement is proven by showing that the set of inequivalent central extensions of a Lie algebra is in one-to-one correspondence with the second cohomology group of that Lie algebra, then invoking Whitehead's second lemma to show that the second cohomology of a semisimple Lie algebra vanishes. Reichstein's direct proof that there are no nontrivial central extensions actually gives an alternate proof of Whitehead's second lemma; I'll comment on this more in the section about central extensions of Lie algebras.)

**Prerequisites:** Group theory; basic familiarity with Lie groups and Lie algebras, including the definition of a Lie algebra representation. I'll make some side comments about cohomology and about continuous maps in the strong operator topology but they aren't essential to the main pedagogical flow. In section 2, I make some comments about the Lie algebra of an infinite-dimensional projective unitary group; the formalism needed to make sense of that idea is explained in detail in my post on Banach Lie groups and algebras, but if you're willing to accept me saying "it's the vector space of bounded antihermitian operators quotiented by multiples of the identity" on faith, then you can safely skip that background reading.

__Table of Contents__

- Projective Representations and Central Extensions of Groups
- Central Extensions of Lie Algebras
- Bargmann's Theorem

__1. Projective Representations and Central Extensions of Groups__

*projective space*$\mathbb{P}(\mathcal{H}).$ Since the only observables associated with states are the transition amplitudes

*group homomorphism*$\rho : G \rightarrow \operatorname{Aut}(\mathbb{P}(\mathcal{H})).$ Such a map is called a

*projective representation of $G$ on $\mathcal{H}$*. If $G$ has a topology, then we will usually assume that $\rho$ is continuous with respect to the topology on $G$. Making this assumption requires giving a topology to $\operatorname{Aut}(\mathbb{P}(\mathcal{H}))$; the natural choice is the

*strong operator topology*of pointwise convergence; we have $S_n \rightarrow S$ in $\operatorname{Aut}(\mathbb{P}(\mathcal{H}))$ if and only if, for each $[|\psi \rangle] \in \mathbb{P}(\mathcal{H})$, we have

*for a given symmetry group $G$, what is the full list of ways that $G$ can act on a quantum system?*This amounts to classifying all of the possible projective representations of $G$. In the next two subsections, we will explore two equivalent ways of classifying projective representations; the first in terms of cocycles, the second in terms of central extensions.

__1.1 Cocycles and Phases__

*equivalent*if there exists a function $\alpha : G \rightarrow \mathbb{R}$ so that

*any*phase function satisfying equation (1); see Bargmann section 3.) A phase function $e^{i \phi(g_1, g_2)}$ on $G$ satisfying (1) is sometimes called a

*2-cocycle*or just a

*cocycle*on $G$. I'll use this terminology when it's convenient, but the actual reasoning behind calling it a "cocycle" is irrelevant for the purposes of this post. I will only mention, for readers with a taste for homological algebra, that such a function is indeed a cocycle of the cochain complex of $G$'s group cohomology with respect to the trivial $G$-module on $\operatorname{U}(1)$; the phase functions of the form $e^{i \alpha(g_1 g_2) - i \alpha(g_1) - i \alpha(g_2)}$, used to define the equivalence relation on cocycles, are the coboundaries in the second chain group. In this sense, the set of allowed phase functions is exactly the second cohomology group of $G$ in the sense of group cohomology.

*do*exist nontrivial projective representations, i.e., nontrivial cocycle equivalence classes $[e^{i \phi(g_1, g_2)}]$. Let $\operatorname{W}$ be the Pauli group (I'm using $\operatorname{W}$ here for

*Wolfgang*Pauli, since I'll need P for projective space), i.e., a set of $2 \times 2$ unitary matrices $\{\hat{1}, \hat{X}, \hat{Y}, \hat{Z}\}$ satisfying $\hat{X}^2 = \hat{Y}^2 = \hat{Z}^2 = \hat{1}$ and $\hat{X}\hat{Y} = i \hat{Z} = - \hat{Y} \hat{X}, \hat{Y} \hat{Z} = i \hat{X} = - \hat{Z} \hat{Y}, \hat{Z} \hat{X} = i \hat{Y} = - \hat{X} \hat{Z}.$ This is a subgroup of the unitary group on $\mathbb{C}^2$, i.e. $\operatorname{W} \leq \operatorname{U}(\mathbb{C}^2)$, so if we quotient by phases, we obtain the

*projective Pauli group*$\operatorname{PW} = \{1, X, Y, Z\} \leq \operatorname{Aut}(\mathbb{P}(\mathbb{C}^2)).$

*no way to rephase the Pauli operators*$\hat{X}, \hat{Y}, \hat{Z}$ to satisfy this phase-free multiplication law! In particular, the symmetry transformations $X$ and $Y$ commute, while the operators $\hat{X}$ and $\hat{Y}$ do not; there is no way to multiply $\hat{X}$ and $\hat{Y}$ by phases to make them commute. So the Pauli subgroup $\operatorname{PW}$ of the projective unitary group $\operatorname{PU}(\mathbb{C}^2)$ has a nontrivial projective representation on $\mathbb{C}^2$.

__1.2 Central Extensions__

*lift*this projective representation to a true representation, i.e., a group homomorphism $\rho : G \rightarrow \mathcal{I}(\mathcal{H})$, with $\mathcal{I}(\mathcal{H})$ the group of unitary and antiunitary operators on $\mathcal{H}$. (The symbol $\mathcal{I}$ here stands for "isometry.") This is not always possible, as evidenced by the construction of a nontrivial cocycle at the end of the previous subsection. What we can hope to do instead, is to

*extend*$G$, i.e., to find some

*larger*group $E_G$ with a representation on $\mathcal{H}$ that descends to the action of $G$ on $\mathbb{P}(\mathcal{H})$ under appropriate quotients. The group $E_G$ will be called a central extension of $G$ by $\operatorname{U}(1)$.

*extension*of $G$ by $\operatorname{U}(1)$? Well, $E_G$ contains a subset $\{(1, g)\}$ that is in bijection with $G$, and moreover there is a homomorphism $\pi : E_G \rightarrow G$ given by $(e^{i \alpha}, g) \mapsto g$ that maps $E_G$ onto $G$. $E_G$ also contains a sub

*group*$\{\{e^{i \alpha} e^{- i \phi(1, 1)}, 1)\}$ that is isomorphic to $\operatorname{U}(1)$ and that lies in the center of $E_G.$ Finally, the subgroup isomorphic to $\operatorname{U}(1)$ is exactly the kernel of $\pi$.

*central extension of $G$ by $\operatorname{U}(1)$.*We require a subgroup $\operatorname{U}(1) \leq E_G$ that lives in the center $Z(E_G)$, a surjective homomorphism $\pi : E_G \rightarrow G$, and the condition $\operatorname{ker}(\pi) = \operatorname{U}(1).$ Formally, this data can be written compactly in the form of a short exact sequence

*and*is a homomorphism; this means that the associated cocycle $\tau(g_1) \tau(g_2) \tau(g_1 g_2)^{-1}$ is trivial. Because we know from section 1.1 that there are cocycles not equivalent to trivial ones, we know there are central extensions for which no such $\tau$ can exist. A homomorphism $\tau : G \rightarrow E_G$ with $\pi \circ \tau = \operatorname{id}_G$ is called a

*splitting*; a central extension has a splitting if and only if it is equivalent to a trivial extension.

*for a given symmetry group $G$, what is the full list of ways that $G$ can act on a quantum system?*We classify all possible central extensions of $G$, then classify all possible representations of those central extensions. This sounds like a monumental task, but it isn't actually as hard as it seems; for many groups of interest, the list of central extensions is finite or trivial. We will return to that point in section 3; for now, though, let's turn to Lie algebras.

__2. Central Extensions of Lie Algebras__

*algebra*$\mathfrak{g}$? There are two ways to think about this assumption.

- We imagine that we have some physical system that, at the classical level, is symmetric under a Lie group $G$. Is every element of $G$ a symmetry transformation of the quantization of that system? The answer appears to be no, in general; parity violation, for example, indicates that the Standard Model is not invariant under the full 4D Lorentz group $\operatorname{O}(1, 3),$ since the weak interaction is not symmetric under spatial reflections. But it seems reasonable enough to assume that there is a neighborhood of the identity in $G$ that is included among the quantum symmetries of the system — these symmetry transformations are
*so small*, goes the reasoning, that they couldn't possibly break down under quantization. Because any neighborhood of the identity in a Lie group generates the entire connected component $G_e$ containing the identity, it is generally assumed that a system classically symmetric under $G$ is quantumly symmetric under $G_e$. This in turn should induce an action of the Lie algebra $\mathfrak{g}$ on our quantum system in terms of "infinitesimal symmetries," and if we understand how those infinitesimal symmetries work then we can understand how the full symmetry group $G_e$ works. - We forget about Lie groups and just assume, by hand, that a particular Lie algebra $\mathfrak{g}$ is included among the infinitesimal symmetries of our quantum system. This is often done on physical grounds — for example, it is assumed that the algebra of two-dimensional conformal killing vector fields acts as an infinitesimal symmetry algebra on the Hilbert space of any 2D CFT, without thinking of it as the Lie algebra of any particular group. From this point of view, all we need to do is define what we mean by an "infinitesimal symmetry," and we can start working. At the end of the day, this isn't such a wild thing to do; doing physics is about making a set of assumptions about how nature is represented in mathematical structures, making predictions from those structures, and trying to verify those predictions experimentally. The assumption that 2D CFTs carry an infinitesimal symmetry action of the 2D conformal algebra, for example, holds up extremely well in experimental studies of critical spin chains. So it seems like a good assumption to make; do we really need to worry about justifying it post-facto?

$$\operatorname{Aut}(\mathbb{P}(\mathcal{H})) = \operatorname{PU}(\mathcal{H}) \cup \operatorname{PAU}(\mathcal{H}).$$

**Banach Lie group**with respect to a smooth structure induced by the operator norm. Its Lie algebra, $\mathfrak{pu}(\mathcal{H}),$ is the set of equivalence classes of bounded antihermitian operators on $\mathcal{H}$, where two operators are equivalent if their difference is a multiple of the identity. The Lie bracket on this algebra is given by the usual commutator of operators. We'll adopt the usual physics convention of multiplying every element of $\mathfrak{pu}(\mathcal{H})$ by $i$ so we can think of it as the space of bounded

*Hermitian*operators on $\mathcal{H}$ up to addition by multiples of the identity.

*unitary*representation $\rho : G_e \rightarrow \operatorname{PU}(\mathcal{H}).$ If we further assume that $\rho$ is differentiable

*as a map between Banach manifolds*, then its differential automatically furnishes us with a Lie algebra homomorphism

*definition*of a projective representation of the algebra $\mathfrak{g}$ on $\mathcal{H}.$

*cocycle*of $\mathfrak{g}$; as in the case of groups, this terminology comes from the fact that $\phi$ is a 2-cocycle in the Chevalley-Eilenberg cochain complex used to define the Lie algebra cohomology of $\mathfrak{g}.$ Because the translation $\hat{H}_{X} \mapsto \hat{H}_{X} + \alpha(X)$ for $\alpha$ linear does not affect the projective representation $\rho : \mathfrak{g} \rightarrow \mathfrak{pu}(\mathcal{H}),$ we will say that two cocycles $\phi$ and $\xi$ are equivalent if there exists some real, linear function $\alpha$ on $\mathfrak{g}$ satisfying

(As an aside, the condition $\phi(X_1, X_2) = \alpha([X_1, X_2])$ is exactly the condition for $\phi$ to be a 2-coboundary in the Chevalley-Eilenberg cochain complex.)

**extended**Lie algebra $\mathfrak{e}_{\mathfrak{g}}$ on the set $\mathbb{R} \oplus \mathfrak{g}$ with Lie bracket

**splitting**, and the existence of a splitting is equivalent to triviality of a central extension.

__3. Bargmann's Theorem__

**Bargmann's theorem**, which is that when $G$ is finite-dimensional and semisimple, $\widetilde{G}$ has no nontrivial central extensions into Lie groups. The second part of Bargmann's theorem, which I will comment on at the end of the post, says that every projective representation of $\widetilde{G}$ can be realized as a true representation of a central extension into a Lie group. The two parts of Bargmann's theorem together imply a beautiful result:

**every projective representation of a finite-dimensional, connected, semisimple $G$ is induced by a true representation of $\widetilde{G}.$**This is an extremely powerful result; it's at the origin of the theory of spinors, which are representations of the universal cover of the connected component of the Lorentz group.

*simple*if it has no nontrivial ideals. Recall that an

*ideal*is a Lie subalgebra $\mathfrak{h} \subset \mathfrak{g}$ that is "absorbing" under the Lie bracket: it satisfies

*only*ideals. A Lie algebra is called semisimple if it can be written as a direct sum of simple Lie algebras:

*linear*map, not necessarily a Lie algebra homomorphism, with

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