Editing Projective representation (section)

==Projective representations of Lie groups==
{{see also|Spinor|Spin-½}}
Studying projective representations of [[Lie group]]s leads one to consider true representations of their central extensions (see {{Slink|Group extension|Lie groups}}). In many cases of interest it suffices to consider representations of [[covering group]]s. Specifically, suppose <math>\hat G</math> is a connected cover of a connected Lie group <math>G</math>, so that <math>G\cong \hat G/N</math> for a discrete central subgroup <math>N</math> of <math>\hat G</math>. (Note that <math>\hat G</math> is a special sort of central extension of <math>G</math>.) Suppose also that <math>\Pi</math> is an irreducible unitary representation of <math>\hat G</math> (possibly infinite dimensional). Then by [[Schur's lemma]], the central subgroup <math>N</math> will act by scalar multiples of the identity. Thus, at the projective level, <math>\Pi</math> will descend to <math>G</math>. That is to say, for each <math>g\in G</math>, we can choose a preimage <math>\hat g</math> of <math>g</math> in <math>\hat G</math>, and define a projective representation <math>\rho</math> of <math>G</math> by setting
:<math>\rho(g) = \left[\Pi\left(\hat g\right)\right]</math>,
where <math>[A]</math> denotes the image in <math>\mathrm{PGL}(V)</math> of an operator <math>A\in\mathrm{GL}(V)</math>. Since <math>N</math> is contained in the center of <math>\hat G</math> and the center of <math>\hat G</math> [[Schur's lemma|acts as scalars]], the value of <math>\left[\Pi\left(\hat g\right)\right]</math> does not depend on the choice of <math>\hat g</math>.

The preceding construction is an important source of examples of projective representations. Bargmann's theorem (discussed below) gives a criterion under which ''every'' irreducible projective unitary representation of <math>G</math> arises in this way.

===Projective representations of SO(3)===
A physically important example of the above construction comes from the case of the [[rotation group SO(3)]], whose [[Rotation group SO(3)#Connection between SO(3) and SU(2)|universal cover is SU(2)]]. According to the [[representation theory of SU(2)]], there is exactly one irreducible representation of SU(2) in each dimension. When the dimension is odd (the "integer spin" case), the representation descends to an ordinary representation of SO(3).<ref>{{harvnb|Hall|2015}} Section 4.7</ref> When the dimension is even (the "fractional spin" case), the representation does not descend to an ordinary representation of SO(3) but does (by the result discussed above) descend to a projective representation of SO(3). Such projective representations of SO(3) (the ones that do not come from ordinary representations) are referred to as "spinorial representations", whose elements (vectors) are called [[spinors]].

By an argument discussed below, every finite-dimensional, irreducible ''projective'' representation of SO(3) comes from a finite-dimensional, irreducible ''ordinary'' representation of SU(2).

===Examples of covers, leading to projective representations===
Notable cases of covering groups giving interesting projective representations:

* The [[special orthogonal group]] SO(''n'', ''F'') is doubly covered by the [[Spin group]] Spin(''n'', ''F'').  
*In particular, the [[rotation group SO(3)|group SO(3)]] (the rotation group in 3 dimensions) is doubly covered by [[special unitary group|SU(2)]].  This has important applications in quantum mechanics, as the [[representation theory of SU(2)|study of representations of SU(2)]] leads to a nonrelativistic (low-velocity) theory of [[Spin (physics)|spin]].
* The group [[Lorentz group|SO<sup>+</sup>(3;1)]], isomorphic to the [[Möbius group]], is likewise doubly covered by [[special linear group|SL<sub>2</sub>]]('''C'''). Both are supergroups of aforementioned SO(3) and SU(2) respectively and form a [[special relativity|relativistic]] spin theory.
*The universal cover of the [[Poincaré group]] is a double cover (the [[semidirect product]] of SL<sub>2</sub>('''C''') with '''R'''<sup>4</sup>). The irreducible unitary representations of this cover give rise to projective representations of the Poincaré group, as in [[Wigner's classification]]. Passing to the cover is essential, in order to include the fractional spin case.
* The [[orthogonal group]] O(''n'') is double covered by the [[Pin group]] Pin<sub>±</sub>(''n'').
* The [[symplectic group]] Sp(2''n'')=Sp(2''n'', '''R''') (not to be confused with the compact real form of the symplectic group, sometimes also denoted by Sp(''m'')) is double covered by the [[metaplectic group]] Mp(2''n''). An important projective representation of Sp(2''n'') comes from the [[Oscillator representation|metaplectic representation]] of Mp(2''n'').

===Finite-dimensional projective unitary representations===
In quantum physics, [[Symmetry in quantum mechanics|symmetry]] of a physical system is typically implemented by means of a projective unitary representation <math>\rho</math> of a Lie group <math>G</math> on the quantum [[Hilbert space]], that is, a continuous homomorphism
:<math>\rho: G\rightarrow\mathrm{PU}(\mathcal H),</math>
where <math>\mathrm{PU}(\mathcal H)</math> is the quotient of the unitary group <math>\mathrm{U}(\mathcal H)</math> by the operators of the form <math>cI,\,|c| = 1</math>. The reason for taking the quotient is that physically, two vectors in the Hilbert space that are proportional represent the same physical state. [That is to say, the space of (pure) states is the [[Complex projective space|set of equivalence classes of unit vectors]], where two unit vectors are considered equivalent if they are proportional.] Thus, a unitary operator that is a multiple of the identity actually acts as the identity on the level of physical states.

A finite-dimensional projective representation of <math>G</math> then gives rise to a projective unitary representation <math>\rho_*</math> of the Lie algebra <math>\mathfrak g</math> of <math>G</math>. In the finite-dimensional case, it is always possible to "de-projectivize" the Lie-algebra representation <math>\rho_*</math> simply by choosing a representative for each <math>\rho_*(X)</math> having trace zero.<ref>{{harvnb|Hall|2013}} Proposition 16.46</ref> In light of the [[Lie group–Lie algebra correspondence#The correspondence|homomorphisms theorem]], it is then possible to de-projectivize <math>\rho</math> itself, but at the expense of passing to the universal cover <math>\tilde G</math> of <math>G</math>.<ref>{{harvnb|Hall|2013}} Theorem 16.47</ref> That is to say, every finite-dimensional projective unitary representation of <math>G</math> arises from an ordinary unitary representation of <math>\tilde G</math> by the procedure mentioned at the beginning of this section.

Specifically, since the Lie-algebra representation was de-projectivized by choosing a trace-zero representative, every finite-dimensional projective unitary representation of <math>G</math> arises from a ''determinant-one'' ordinary unitary representation of <math>\tilde G</math> (i.e., one in which each element of <math>\tilde G</math> acts as an operator with determinant one). If <math>\mathfrak g</math> is semisimple, then every element of <math>\mathfrak g</math> is a linear combination of commutators, in which case ''every'' representation of <math>\mathfrak g</math> is by operators with trace zero. In the semisimple case, then, the associated linear representation of <math>\tilde G</math> is unique.

Conversely, if <math>\rho</math> is an ''irreducible'' unitary representation of the universal cover <math>\tilde G</math> of <math>G</math>, then by [[Schur's lemma]], the center of <math>\tilde G</math> acts as scalar multiples of the identity. Thus, at the projective level, <math>\rho</math> descends to a projective representation of the original group <math>G</math>. Thus, there is a natural one-to-one correspondence between the irreducible projective representations of <math>G</math> and the irreducible, determinant-one ordinary representations of <math>\tilde G</math>. (In the semisimple case, the qualifier "determinant-one" may be omitted, because in that case, every representation of <math>\tilde G</math> is automatically determinant one.)

An important example is the case of [[Rotation group SO(3)|SO(3)]], whose universal cover is [[Rotation group SO(3)#Connection between SO(3) and SU(2)|SU(2)]]. Now, the Lie algebra <math>\mathrm{su}(2)</math> is semisimple. Furthermore, since SU(2) is a [[compact group]], every finite-dimensional representation of it admits an inner product with respect to which the representation is unitary.<ref>{{harvnb|Hall|2015}} proof of Theorem 4.28</ref> Thus, the irreducible ''projective'' representations of SO(3) are in one-to-one correspondence with the irreducible ''ordinary'' representations of SU(2).

===Infinite-dimensional projective unitary representations: the Heisenberg case===
The results of the previous subsection do not hold in the infinite-dimensional case, simply because the trace of <math>\rho_*(X)</math> is typically not well defined. Indeed, the result fails: Consider, for example, the translations in position space and in momentum space for a quantum particle moving in <math>\mathbb R^n</math>, acting on the Hilbert space <math>L^2(\mathbb R^n)</math>.<ref>{{harvnb|Hall|2013}} Example 16.56</ref> These operators are defined as follows:
:<math>\begin{align}
  (T_a f)(x) &= f(x - a) \\
  (S_a f)(x) &= e^{iax}f(x),
\end{align}</math>

for all <math>a\in\mathbb R^n</math>. These operators are simply continuous versions of the operators <math>T_a</math> and <math>S_a</math> described in the "First example" section above. As in that section, we can then define a ''projective'' unitary representation <math>\rho</math> of <math>\mathbb R^{2n}</math>:
:<math>\rho(a, b) = [T_a S_b],</math>
because the operators commute up to a [[phase factor]]. But no choice of the phase factors will lead to an ordinary unitary representation, since translations in position do not commute with translations in momentum (and multiplying by a nonzero constant will not change this). These operators do, however, come from an ordinary unitary representation of the [[Heisenberg group]], which is a one-dimensional central extension of <math>\mathbb R^{2n}</math>.<ref>{{harvnb|Hall|2013}} Exercise 6 in Chapter 14</ref> (See also the [[Stone–von Neumann theorem]].)

===Infinite-dimensional projective unitary representations: Bargmann's theorem===
On the other hand, [[Valentine Bargmann|Bargmann's]] theorem states that if the second [[Lie algebra cohomology]] group <math>H^2(\mathfrak g; \mathbb R)</math> of <math>\mathfrak g</math> is trivial, then every projective unitary representation of <math>G</math> can be de-projectivized after passing to the universal cover.<ref>{{harvnb|Bargmann|1954}}</ref><ref>{{harvnb|Simms|1971}}</ref> More precisely, suppose we begin with a projective unitary representation <math>\rho</math> of a Lie group <math>G</math>. Then the theorem states that <math>\rho</math> can be lifted to an ordinary unitary representation <math>\hat\rho</math> of the universal cover <math>\hat G</math> of <math>G</math>. This means that <math>\hat\rho</math> maps each element of the kernel of the covering map to a scalar multiple of the identity—so that at the projective level, <math>\hat\rho</math> descends to <math>G</math>—and that the associated projective representation of <math>G</math> is equal to <math>\rho</math>.

The theorem does not apply to the group <math>\mathbb R^{2n}</math>—as the previous example shows—because the second cohomology group of the associated commutative Lie algebra is nontrivial. Examples where the result does apply include semisimple groups (e.g., [[Representation theory of SL2(R)|SL(2,R)]]) and the [[Poincaré group]]. This last result is important for [[Wigner's classification]] of the projective unitary representations of the Poincaré group.

The proof of Bargmann's theorem goes by considering a [[central extension (mathematics)|central extension]] <math>H</math> of <math>G</math>, constructed similarly to the section above on linear representations and projective representations, as a subgroup of the direct product group <math>G\times U(\mathcal H)</math>, where <math>\mathcal H</math> is the Hilbert space on which <math>\rho</math> acts and <math>U(\mathcal H)</math> is the group of unitary operators on <math>\mathcal H</math>. The group <math>H</math> is defined as
:<math>H = \{(g, U) \mid \pi(U) = \rho(g)\}.</math>
As in the earlier section, the map <math>\phi: H \rightarrow G</math> given by <math>\phi(g, U) = g</math> is a surjective homomorphism whose kernel is <math>\{(e, cI) \mid |c| = 1\},</math> so that <math>H</math> is a central extension of <math>G</math>. Again as in the earlier section, we can then define a linear representation <math>\sigma</math> of <math>H</math> by setting <math>\sigma(g, U) = U</math>. Then <math>\sigma</math> is a lift of <math>\rho</math> in the sense that <math>\rho\circ\phi = \pi\circ\sigma</math>, where <math>\pi</math> is the quotient map from <math>U(\mathcal H)</math> to <math>PU(\mathcal H)</math>.

A key technical point is to show that <math>H</math> is a ''Lie'' group. (This claim is not so obvious, because if <math>\mathcal H</math> is infinite dimensional, the group <math>G\times U(\mathcal H)</math> is an infinite-dimensional [[topological group]].) Once this result is established, we see that <math>H</math> is a one-dimensional Lie group central extension of <math>G</math>, so that the Lie algebra <math>\mathfrak h</math> of <math>H</math> is also a one-dimensional central extension of <math>\mathfrak g</math> (note here that the adjective "one-dimensional" does not refer to <math>H</math> and <math>\mathfrak{h}</math>, but rather to the kernel of the projection map from those objects onto <math>G</math> and <math>\mathfrak{g}</math> respectively). But the cohomology group <math>H^2(\mathfrak g; \mathbb R)</math> [[Lie algebra cohomology#Cohomology in small dimensions|may be identified]] with the space of one-dimensional (again, in the aforementioned sense) central extensions of <math>\mathfrak g</math>; if <math>H^2(\mathfrak g; \mathbb R)</math> is trivial then every one-dimensional central extension of <math>\mathfrak g</math> is trivial. In that case, <math>\mathfrak h</math> is just the direct sum of <math>\mathfrak g</math> with a copy of the real line. It follows that the universal cover <math>\tilde H</math> of <math>H</math> must be just a direct product of the universal cover of <math>G</math> with a copy of the real line. We can then lift <math>\sigma</math> from <math>H</math> to <math>\tilde H</math> (by composing with the covering map) and finally restrict this lift to the universal cover <math>\tilde G</math> of <math>G</math>.