Yes. Both universal covers and central extensions incurred during quantization come from the same fundamental concept:
Projective representations
If $\mathcal{H}$ is our Hilbert space of states, then distinct physical states are not vectors $\psi\in\mathcal{H}$, but rays, since multiplication by a complex number does not change the expectation values given by the rule
$$ \langle A\rangle_\psi = \frac{\langle \psi \vert A \vert \psi \rangle}{\langle \psi \vert \psi \rangle}$$
nor the transition probabilities
$$ P(\lvert \psi \rangle \to \lvert \phi \rangle) = \frac{\lvert \langle \psi \vert \phi \rangle\rvert^2}{\langle \phi \vert \phi \rangle\langle \psi \vert \psi \rangle}$$
The proper space to consider, where every element of the space is indeed a distinct physical state, is the projective Hilbert space
$$ \mathrm{P}\mathcal{H} := \mathcal{H} /\sim$$
$$ \lvert \psi \rangle \sim \lvert \phi \rangle :\Leftrightarrow \exists c\in\mathbb{C}: \lvert \psi \rangle = c\lvert\phi\rangle$$
which is just a fancy way to write that every complex ray has been shrunk to a point. By Wigner's theorem, every symmetry should have some, not necessarily unique, unitary representation $\rho : G \to \mathrm{U}(\mathcal{H})$. Since it has to descend to a well-defined ray transformation, the action of the symmetry is given by a group homomorphism into the projective unitary group $G \to \mathrm{PU}(\mathcal{H})$, which sits in an exact sequence
$$ 1 \to \mathrm{U}(1) \to \mathrm{U}(\mathcal{H}) \to \mathrm{PU}(\mathcal{H}) \to 1$$
where $\mathrm{U}(1)$ represents the "group of phases" that is divided out when passing to the projective space. It is already important to notice that this means $\mathrm{U}(\mathcal{H})$ is a central extension of $\mathrm{PU}(\mathcal{H})$ by $\mathrm{U}(1)$.
To classify all possible quantumly allowed representations of a symmetry group $G$, we need to understand the allowed Lie group homomorphisms $\sigma : G\to\mathrm{PU}(\mathcal{H})$. Since linear representations are nicer to work with than these weird projective things, we will look at
Classifying projective representations by unitary linear representations
For any $g\in G$, choose a representative $\Sigma(g)\in\mathrm{U}(\mathcal{H})$ for every $\sigma(g)\in\mathrm{PU}(\mathcal{H})$. This choice is highly non-unique, and is essentially responsible for how the central extension appears. Now, since for any $g,h\in G$ we have $\sigma(g)\sigma(h) = \sigma(gh)$, the choices of representatives must fulfill
$$ \Sigma(g)\Sigma(h) = C(g,h)\Sigma(gh)$$
for some $C : G\times G\to\mathrm{U}(1)$. Applying associativity to $\Sigma(g)\Sigma(h)\Sigma(k)$ gives the consistency requirement
$$ C(g,hk)C(h,k) = C(g,h)C(gh,k)\tag{1}$$
which is also called the cocycle identity. For any other choice $\Sigma'$, we must have
$$ \Sigma'(g) = f(g)\Sigma(g) $$
for some $f : G \to \mathrm{U}(1)$. $\Sigma'$ has an associated $C'$, and so we get
$$ C'(g,h)\Sigma'(gh) = \Sigma'(g)\Sigma'(h) = f(g)f(h)C(g,h)f(gh)^{-1}\Sigma'(gh)$$
which yields the consistency requirement
$$ C'(g,h)f(gh) = f(g)f(h)C(g,h)\tag{2}$$
Therefore, projective representations are classified giving the choice of unitary representatives $\Sigma$, but those that are related by $(2)$ give the same projective representation. Formally, the set
$$ H^2(G,\mathrm{U}(1)) := \{C : G\times G \to \mathrm{U}(1)\mid C \text{ fulfills } (1)\} / \sim$$
$$ C \sim C' :\Leftrightarrow \exists f : (2) \text{ holds }$$
classifies the projective representations of $G$. We want to use it to construct a unitary representation of something that classifies the projective representation:
Define the semi-direct product $G_C := G \ltimes_C \mathrm{U}(1)$ for any representative $C$ of an element in $H^2(G,\mathrm{U}(1)$ by endowing the Cartesion product $G \times \mathrm{U}(1)$ with the multiplication
$$ (g,\alpha)\cdot(h,\beta) := (gh,\alpha\beta C(g,h))$$
One may check that it is a central extension, i.e. the image of $\mathrm{U}(1)\to G \ltimes_C\mathrm{U}(1)$ is in the center of $G_C$, and
$$ 1 \to \mathrm{U}(1) \to G_C \to G \to 1$$
is exact. For any projective representation $\sigma$, fix $\Sigma,C$ and define the linear representation
$$ \sigma_C : G_C \to \mathrm{U}(\mathcal{H}), (g,\alpha) \mapsto \alpha\Sigma(g)$$
Conversely, every unitary representation $\rho$ of some $G_C$ gives a pair $\Sigma,C$ by $\Sigma(g) = \alpha^{-1}\rho(g,\alpha)$.
Therefore, projective representations are in bijection to linear representations of central extensions.
On the level of the Lie algebras, we have $\mathfrak{u}(\mathcal{H}) = \mathfrak{pu}(\mathcal{H})\oplus\mathbb{R}$, where the basis element $\mathrm{i}$ of $\mathbb{R}$ generates multiples of the identity $\mathrm{e}^{\mathrm{i}\phi}\mathrm{Id}$. We omit the $\mathrm{Id}$ in the following, whenever a real number is added to an element of the Lie algebra, it is implied to be multiplied by it.
Repeating the arguments above for the Lie algebras, we get that the projective representation $\sigma : G \to \mathrm{PU}(\mathcal{H})$ induces a representation of the Lie algebra $\phi : \mathfrak{g}\to\mathfrak{pu}(\mathcal{H})$. A choice of representatives $\Phi$ in $\mathfrak{u}(H)$ classifies such a projective representation together with an element $\theta$ in
$$ H^2(\mathfrak{g},\mathbb{R}) := \{\theta : \mathfrak{g}\times\mathfrak{g} \to \mathbb{R}\mid \text{ fulfills } (1') \text{ and } \theta(u,v) = -\theta(v,u)\} / \sim$$
$$ \theta \sim \theta' :\Leftrightarrow \exists (b : \mathfrak{g}\to\mathbb{R}) :\theta'(u,v) = \theta(u,v) + b([u,v])$$
with consistency condition
$$ \theta([u,v],w) + \theta ([w,u],v) + \theta([v,w],u) = 0 \tag{1'}$$
that $\theta$ respects the Jacobi identity, essentially.
Thus, a projective representation of $\mathfrak{g}$ is classified by $\Phi$ together with a $\theta\in H^2(\mathfrak{g},\mathbb{R})$. Here, the central extension is defined by $\mathfrak{g}_\theta := \mathfrak{g}\oplus\mathbb{R}$ with Lie bracket
$$ [u\oplus y,v\oplus z] = [u,v]\oplus\theta(u,v)$$
and we get a linear representation of it into $\mathfrak{u}(\mathcal{H})$ by
$$ \phi_\theta(u\oplus z) := \Phi(u) + a$$
Again, we obtain a bijection between projective representations of $\mathfrak{g}$ and those of its central extensions $\mathfrak{g}_\theta$.
Universal covers, central charges
We are finally in the position to decide which representations of $G$ we must allow quantumly. We distinguish three cases:
There are no non-trivial central extensions of either $\mathfrak{g}$ or $G$. In this case, all projective representations of $G$ are already given by the linear representations of $G$. This is the case for e.g. $\mathrm{SU}(n)$.
There are no non-trivial central extensions of $\mathfrak{g}$, but there are discrete central extensions of $G$ by $\mathbb{Z}_n$ instead of $\mathrm{U}(1)$. Those evidently also descend to projective representations of $G$. Central extensions of Lie groups by discrete groups are just covering groups of them, because the universal cover $\overline{G}$ gives the group $G$ as the quotient $\overline{G}/\Gamma$ by a discrete central subgroup $\Gamma$ isomorphic to the fundamental group of the covered group. Thus we get that all projective representations of $G$ are given by linear representations of the universal cover. No central charges occur. This is the case for e.g. $\mathrm{SO}(n)$.
There are non-trivial central extensions of $\mathfrak{g}$, and consequently also of $G$. If the element $\theta\in H^2(\mathfrak{g},\mathbb{R})$ is not zero, there is a central charge - the generator of the $\oplus\mathbb{R}$ in $\mathfrak{g}_\theta$, or equivalently the conserved charge belonging to the central subgroup $\mathrm{U}(1)\subset G_C$. This happens for the Witt algebra, where inequivalent $\theta(L_m,L_n) = \frac{c}{12}(m^3 - m)\delta_{m,-n}$ are classified by real numbers $c\in \mathbb{R}$.
This post imported from StackExchange Physics at 2015-11-01 18:02 (UTC), posted by SE-user ACuriousMind
Q&A (4872)
