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I'm working through the proof of Wigner's theorem in Weinberg's The Quantum Theory of Fields Volume 1 Chapter 2 Appendix A but have hit a snag. In the footnote on page 94 Weinberg says

If $A_m^{*}A_n$ is real, then choose all $C$'s to vanish except for $C_k$, $C_l$, $C_m$ and $C_n$, and choose these four coefficients all to have different phases.

From the language he is using in the footnote as a whole, I would interpret Weinberg as meaning that if $A_k^*A_l$ is complex and $A_m^*A_n$ is real then, by simply choosing the $C$'s as specified, the equation
$$\sum_{kl}\Im(C_k^*C_l)\Im(A_k^*A_l)\ne0\tag{2.A.17} $$ will be automatically satisfied.

But surely that is not true. If I choose $$A_k = 1 + 3i,\ A_l = 5 + 7i,\ A_m = 9 + 11i,\ A_n = 9 + 11i$$
and $$C_k = 5 + 6i,\ C_l = 6 + 9i,\ C_m = 9 + 12i,\ C_n = 12 + 15i$$ with all other coefficients of $A$ and $C$ being zero then the conditions on $A$ are met and the four non-zero coefficients of $C$ have different phases but equation 2.A.17 is not satisfied.

What am I missing?

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1 Answer 1

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You are right that Weinberg's footnote cannot be interpreted in the sense that any choice of $C$'s that matches the cited conditions is guaranteed to satisfy Equation (2.A.17) -- you constructed a valid counterexample to that reading.

The footnote should rather be understood as a brief sketch of how a suitable state $\sum_k C_k\Psi_k$ can be found, focussing mainly on the number of $C$'s that need to be non-vaninishing and leaving parts of the technical details for the reader. Maybe some more context helps to clearify the intended meaning: in the sentence just before the one you quoted, Weinberg states that

If $A_m^*A_n$ is complex, then choose all $C$'s to vanish apart from $C_m$ and $C_n$, and choose these coefficients to have different phase.

If $A_m^*A_n$ is real, however, one already needs four coefficients to be non-vanishing to satisfy equations (2.A.17)&(2.A.18):

If $A_m^*A_n$ is real, then choose all $C$'s to vanish except for $C_k, C_l, C_m$ and $C_n$, and choose these four coefficients all to have different phases.

Weinberg indeed omits the remaining details on how to avoid non-generic cases (like your counterexample) where Equations (2.A.17) and/or (2.A.18) just happen to not be satisfied. Because of the freedom in choosing the distinct phases and the absolute value of the four non-vanishing coefficients, there are infinitely many ways to avoid the non-generic cases, which is probably why Weinberg refrained from giving a definite way.

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