= Vacuum energy sign of a fermionic oscillator
{title2=$H=E(b^\dagger b-\tfrac12)$}
A fermionic mode with <canonical anticommutation relations> $\{b,b^\dagger\}=1$ has the symmetrically ordered <Hamiltonian operator> $H=\tfrac E2(b^\dagger b-bb^\dagger)=E(b^\dagger b-1/2)$. Its empty-state zero-point contribution is $-E/2$. A <quantum harmonic oscillator> with bosonic <canonical commutation relations> contributes $+E/2$ instead. For a <Dirac field>, sum over both particle and <antiparticle> modes and over <spin>: there are four independent oscillators per spatial <momentum> in four spacetime dimensions. Comparing bosonic and fermionic <vacuum energies> requires a common <regularization in quantum field theory>, matching degrees of freedom and a stated vacuum-energy subtraction convention.
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