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Apply the Grassmann Gaussian integral to a regulated finite collection of field components. Up to a field-independent measure normalization and phase,In the continuum this is a formal functional determinant, including spinor and spacetime indices. Its meaningful definition requires a regulator and boundary conditions. A complex Dirac field supplies a determinant, not the inverse square root obtained for a real commuting field.

The comparison is clean after Wick rotation. A free real scalar field with positive Euclidean operator $P_E=-{\partial }_E^2+m^2$ hasTaking a logarithm gives $-\frac{1}{2}\mathrm{Tr}\log P_E$ for the scalar and $+\mathrm{Tr}\log K_E$ for the Dirac field. The opposite statistics sign is the determinant counterpart of the minus sign for a closed fermion loop. The magnitude also differs because a Dirac field has several independent spin and antiparticle degrees of freedom.

The vacuum energy sign of a fermionic oscillator makes the comparison explicit. A real scalar mode contributes $+E/2$; each independent fermionic oscillator contributes $-E/2$. There is one oscillator per scalar momentum, but a massive Dirac field has two particle and two antiparticle oscillators. Thus the vacuum energy densities are formallyEquivalently, the large Euclidean-time vacuum functional behaves as $\log Z_E\sim -\beta V{\rho }_{\mathrm{vac}}$. Both displayed vacuum energies are ultraviolet divergent; a common regularization in quantum field theory and the appropriate renormalization are needed before comparing them. The zero-point vacuum energies of bosons and fermions have opposite signs, but do not cancel without matching masses and degrees of freedom. Normal ordering removes an additive vacuum constant in nongravitating flat-space theory. Coupling to the metric in general relativity makes that constant contribute to the cosmological constant, so it cannot simply be discarded without a renormalization condition.