Creation and Annihilation Operators in Quantum Field Theories
In quantum field theories and many-body problems one works with sums of terms where the are complex numbers, whereas the are creation rsp. annihilation operators, which enhance (rsp. reduce) the eigenvalues of the number operator, by 1, in analogy to the harmonic oscillator. The indices reflect the behaviour of spacetime and are higher-order entities, e.g. quadrupels for particles without spin. The number operators assume all non-negative integer values, and the nontrivial commutation relations are where is the commutator bracket while is the well-known Kronecker delta.
This is true for bosons, whereas for fermions the commutator must be replaced by the anticommutator, As a consequence, in the fermionic case the number operator has only the eigenvalues 0 and 1.
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