Equation 1 · Part 2 · No Particle Without a Cosigner

Symbol k

\tilde a_j = \sum_k \left( \alpha_{jk} a_k + \beta_{jk} a_k^\dagger \right).

k

What this part means

k appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

Its job in the formula

k appears in the bound of this sum. The bound states where the repeated operation starts, ends, or which values it includes.

Full expressionSymbol kArticle meaning

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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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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