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Second quantization or canonical quantization in quantum field theory and many-body systems is the collective organizing and accounting of an infinity of quantum excitations and their interactions through quantum field operators.

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What happens if we expand out fields in terms of different functions?

This is done so that the $S$-matrix for the fields is manifestly invariant under translations. I will illustrate the case explicitly for a real scalar field $\phi$, with creation and annihilation oper …
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