Editing Singlet state (section)

== Mathematical representations ==
The ability of [[positronium]] to form both singlet and triplet states is described mathematically by saying that the [[tensor product|product]] of two doublet representations (meaning the electron and positron, which are both spin-1/2 doublets) can be decomposed into the sum of an [[Adjoint representation of a Lie group|adjoint representation]] (the triplet or spin&nbsp;1 state) and a [[trivial representation]] (the singlet or spin&nbsp;0 state). While the particle interpretation of the positronium triplet and singlet states is arguably more intuitive, the mathematical description enables precise calculations of quantum states and probabilities.

This greater mathematical precision for example makes it possible to assess how singlets and doublets behave under rotation operations. Since a spin-1/2 electron transforms as a doublet under rotation, its experimental response to rotation can be predicted by using the [[fundamental representation]] of that doublet, specifically the [[Lie group]] [[SU(2)]].<ref>{{cite book |author-link=J. J. Sakurai |first=J.J. |last=Sakurai |title=Modern Quantum Mechanics |publisher=Addison Wesley |year=1985}}</ref> Applying the operator <math>\vec{S}^2</math> to the spin state of the electron thus will always result in <math display="inline">\hbar^2 \left(\frac{1}{2}\right) \left(\frac{1}{2} + 1\right) = \left(\frac{3}{4}\right) \hbar^2</math>, or spin-1/2, since the spin-up and spin-down states are both [[eigenstate]]s of the operator with the same eigenvalue.

Similarly, for a system of two electrons, it is possible to measure the total spin by applying <math>\left(\vec{S}_1 + \vec{S}_2\right)^2</math>, where <math>\vec{S}_1</math> acts on electron&nbsp;1 and <math>\vec{S}_2</math> acts on electron&nbsp;2. Since this system has two possible spins, it also has two possible eigenvalues and corresponding eigenstates for the total spin operator, corresponding to the spin&nbsp;0 and spin&nbsp;1 states.