= Solution
Apply the <Grassmann Gaussian integral> to a regulated finite collection of field components. Up to a field-independent measure normalization and phase,
$$
\boxed{Z[0,0]\propto\det(iK_F)}.
$$
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 $P_E=-\partial_E^2+m^2$ has
$$
Z_{\rm scalar}[0]\propto(\det P_E)^{-1/2},
\qquad Z_{\rm Dirac}[0]\propto\det K_E.
$$
Taking a logarithm gives $-\tfrac12\operatorname{Tr}\log P_E$ for the scalar and $+\operatorname{Tr}\log K_E$ 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 formally
$$
\rho_{\rm scalar}=\frac12\int\frac{d^3\boldsymbol p}{(2\pi)^3}\sqrt{|\boldsymbol p|^2+m^2},
\qquad
\rho_{\rm Dirac}=-2\int\frac{d^3\boldsymbol p}{(2\pi)^3}\sqrt{|\boldsymbol p|^2+M^2}.
$$
Equivalently, the large Euclidean-time vacuum functional behaves as $\log Z_E\sim-\beta V\rho_{\rm vac}$. Both displayed <vacuum energies> are ultraviolet divergent; a common <regularization in quantum field theory> and the appropriate <renormalization> are needed before comparing them. \b[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.
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