# Physics Notes by Jakob Schwichtenberg

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group_theory:notions:central_extension
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# Central Extension

Central extensions are a standard trick to convert projective representations of some group into true representations of another group.

This is necessary, because when we only consider the “naive” normal representations of a group like the Lorentz group, we miss an important representation (the spin $\frac{1}{2}$) representation). Thus, we can either use a less restrictive definition of a representation, i.e. use projective representations instead of true representations, or we could simply work with true representations of the central extension of the given group.

For example, the projective representations of $SO(3,1)$ correspond to regular representations of $SL(2,\mathbb{C})$.

The central extension $\hat G$ of a given group $G$ by an abelian group $A$ is defined as a group such that $A$ is a subgroup of the center of $\hat G$ and that the quotient $\hat G/A = G$.

See page 178 in Moonshine beyond the Monster by Terry Gannon

The trick, that we use a larger group instead of a central extension, does not always work!

“In most cases physicists have been succesfull in hiding these central extensions by using larger symmetry groups […] However, there are two important exceptions for which this trick does not work: for the translation group R2n of translations in both position and momentum it is not possible to hide the phase factors: one obtains the Heisenberg group. The same problem occurs for the Galilei group: it is not a symmetry group of the (non relativistic) Schrödinger equation, but its central extension, the Bargmann group, is.

## Important Examples

• The classical Galilean group needs to be extended by the introduction of a central charge, called mass, and this yields the Bargmann group. (This is shown very nicely in QUANTIZATION ON A LIE GROUP: HIGHER-ORDER POLARIZATIONS by V. Aldaya, J. Guerrero and G. Marmo).
• The standard spatial rotation group $SO(3)$ needs to be extended by $\mathbb{Z}_2$, which yields $SU(2)$, because otherwise we are not able to describe spin $\frac{1}{2}$ particles.
• The algebra of fermionic non-Abelian charge densitites needs to be extended to the Mickelsson-Faddeev algebra (See this answer)