addition

The passage around this formula

The precise, non-negotiable content of the ambiguity is the Bogoliubov transformation. Given two ways of splitting a free field into positive- and negative-frequency modes — two choices of “particle,” in the ordinary sense of excitations above a chosen vacuum — the annihilation operators of one decomposition mix the creation and annihilation operators of the other: $\tilde a_j = \sum_k \left( \alpha_{jk} a_k + \beta_{jk} a_k^\dagger \right)$. When any of the coefficients $\beta_{jk}$ is nonzero, the vacuum of the a -decomposition is a many-particle state of the $\tilde a$ -decomposition, and vice versa: “how many particles are present” is a question whose answer depends on which set of modes was declared fundamental, not on the field configuration by…

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