Equation 1 · Part 10 · No Particle Without a Cosigner

Starting index or lower bound: k

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

k

What this part means

This label says where the repeated addition, multiplication, or accumulation starts. Read its value or condition together with the article’s description of the index.

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 expressionStarting index or lower bound: 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

Σ adds a collection of terms. Π multiplies them. The lower and upper labels tell you which terms belong to the collection.

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Sources cited in the surrounding passage

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