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Equation 1 · Part 5 · No Particle Without a Cosigner

Symbol beta_jk

a~j=∑k(αjkak+βjkak†).\tilde a_j = \sum_k \left( \alpha_{jk} a_k + \beta_{jk} a_k^\dagger \right).
βjk\beta_{jk}

What this part means

the when any of the coefficients.

Its job in the formula

betaja_jk is one of the signed contributions combined to compute the quantity on the left.

Where the article explains it

When any of the coefficients βjk\beta_{jk} is nonzero, the vacuum of the a -decomposition is a many-particle state of the a~\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 itself.

The passage around this formula

…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: a~j=∑k(αjkak+βjkak†)\tilde a_j = \sum_k \left( \alpha_{jk} a_k + \beta_{jk} a_k^\dagger \right). When any of the coefficients βjk\beta_{jk} is nonzero, the vacuum of the a -decomposition is a many-particle state of the a~\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 itself. Fulling…

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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

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Sources cited in the surrounding passage

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