Wigner’s symmetry representation theorem

Bargmann 5 — checking (anti)linearity is universal

Chevalier, Proposition 2, step 4: \(A\) is linear or antilinear on \([\varphi_0]^\perp\) but not both. Using this, \(A\) is extended to all of \(H\).

We know that \(A\) is linear or antilinear on each ray. If \(\psi_1,\psi_2\) are orthogonal in \([\varphi_0]^\perp\), and \(A\) is linear on \([\psi_1]\) and antilinear on \([\psi_2]\), we have \(\Vert A(\psi_1+\psi_2)\Vert^2=\Vert A(\psi_1)\Vert^2+\Vert A(\psi_2)\Vert^2\) by additivity. But also \[\begin{aligned} \Vert A(\psi_1+\psi_2)\Vert^2&=\langle A(\psi_1+\psi_2)|A(\psi_1+\psi_2)\rangle\\&=|\mathrm{i} \langle A(\psi_1+\psi_2)|A(\psi_1+\psi_2)\rangle|\\&=|\langle A(\psi_1+\psi_2)|\mathrm{i}A(\psi_1)+\mathrm{i}A(\psi_2)\rangle|\\&=|\langle A(\psi_1+\psi_2)|A(\mathrm{i}\psi_1)-A(\mathrm{i}\psi_2)\rangle|\\&=|\Vert A(\psi_1)\Vert^2-\Vert A(\psi_2)\Vert^2|\end{aligned}\] The key part of this sequence is the use of the linearity/antilinearity in the respective rays, \(|\langle A(\psi_1+\psi_2)|\mathrm{i}A(\psi_1)+\mathrm{i}A(\psi_2)\rangle|=|\langle A(\psi_1+\psi_2)|A(\mathrm{i}\psi_1)-A(\mathrm{i}\psi_2)\rangle|\). The upshot is that \(\Vert A(\psi_1)\Vert^2+\Vert A(\psi_2)\Vert^2=|\Vert A(\psi_1)\Vert^2-\Vert A(\psi_2)\Vert^2|\), which implies that one of the ‘rays’ is in fact zero, i.e. not really a ray. Thus orthogonal rays all have the same linearity or antilinearity character.

The action of \(A\) on any vector \(\chi=\alpha_1\psi_1+\alpha_2\psi_2\) in the subspace spanned by \(\psi_1,\psi_2\) maintains their (anti)linear character: \(A(\alpha\chi)=\alpha'\alpha_1' A(\psi_1)+\alpha'\alpha_2' A(\psi_2)=\alpha'A(\chi)\), where the prime operation is the appropriate identity or conjugation for the linear/antilinear character of \(\psi_1,\psi_2\).

We can extend the definition of \(A\) to the whole of \(H\): \(A(\alpha\varphi_0+\varphi)=\alpha'A(\varphi_0)+A(\varphi)\), where \(\varphi \in [\varphi_0]^\perp\).