1. Purpose and Scope

This note is intentionally narrow: 1. establish the “kernel as bi-half-density” semantics for spacetime propagators in QFT, 2. isolate what is canonical (half-density kernels, identity delta kernel) versus what is a convention (scalarization choices such as \(\sqrt{\lvert g\rvert}\)), 3. give one explicit computation that can later be promoted (densitized scalar field).

BV/BRST/field-space half-densities are only flagged as outlook here; a full treatment would require additional dedicated sources and is beyond scope.

2. Kernels on a Manifold: half-densities make the identity canonical

Let \(M\) be a \(D\)-dimensional manifold. A half-density is a section of \(|\Lambda^D T^\ast M|^{1/2}\).

The key operational point (as in the main paper’s kernel-composition spine) is: if an operator acts on half-densities, then its Schwartz kernel is naturally a bi-half-density [BatesWeinstein1997] [Hormander2003] \[ K_A(x,y)=a(x,y)\,|dx|^{1/2}|dy|^{1/2}, \] and composition is intrinsic: \[ (A\circ B)(x,z)=\int_M K_A(x,y)\,K_B(y,z), \] because the product in the intermediate variable \(y\) is a density.

Derivation HD-D1.1 (Identity kernel). The identity operator on half-densities has kernel \[ K_{\mathrm{Id}}(x,y)=\delta^{(D)}(x-y)\,|dx|^{1/2}|dy|^{1/2}, \] which is canonical: it does not require choosing a background density/volume form. Coordinate-free statement: \(K_{\mathrm{Id}}\) is the distributional section of the external tensor product \(|\Lambda^D T^M|^{1/2} \boxtimes |\Lambda^D T^M|^{1/2}\) over \(M \times M\), characterized by the pairing property \(\int_M K_{\mathrm{Id}}(x,y)\,f(y) = f(x)\) for all half-densities \(f\).

Derivation HD-D1.1a (Normalization witness: why \\(\varepsilon^{-D/2}\\) appears). In local coordinates on \(\mathbb{R}^D\), a standard approximate identity is the Gaussian family \[ \rho_\varepsilon(x)=\frac{1}{(2\pi\varepsilon)^{D/2}}\exp!\Bigl(-\frac{|x|^2}{2\varepsilon}\Bigr), \qquad \varepsilon>0. \] The exponent \(D/2\) is forced by normalization: by the change of variables \(x=\sqrt{\varepsilon}\,u\), \[ \int_{\mathbb{R}^D}\rho_\varepsilon(x)\,d^Dx =(2\pi\varepsilon)^{-D/2}\,\varepsilon^{D/2}\int_{\mathbb{R}^D}e^{-|u|^2/2}\,d^Du =1. \] Thus \(\rho_\varepsilon\rightharpoonup\delta^{(D)}\) as \(\varepsilon\to0^+\), and the diagonal delta kernel is the distributional limit of families whose normalization scales as \(\varepsilon^{-D/2}\).

3. Worked computation: densitized scalar field \(\psi=|g|^{1/4}\phi\)

Consider a real scalar field on a fixed Lorentzian/Euclidean background \((M,g)\) [Wald1984] with quadratic action. (The half-density conjugation \(\psi = |g|^{1/4}\phi\) is a purely density-bundle operation and is signature-independent; the specific operator \(P\) and its Green function properties depend on the metric signature.) \[ S[\phi]=\frac12\int_M d^Dx\,\sqrt{|g|}\;\phi\,P\,\phi, \qquad P=-\nabla^2 + m^2 + \xi R \ \ (\text{example}). \] The pairing for which \(P\) is symmetric is \[ (\phi_1,\phi_2)_g=\int d^Dx\,\sqrt{|g|}\;\phi_1\phi_2. \]

Define the densitized field (a half-density in coordinates) \[ \psi := |g|^{1/4}\phi, \qquad\text{so}\qquad \phi=|g|^{-1/4}\psi. \] Then the action becomes \[ S[\phi] =\frac12\int d^Dx\;\psi\;\widetilde P\;\psi, \qquad \widetilde P := |g|^{1/4}P|g|^{-1/4}, \] so the pairing is now just the coordinate density \(d^Dx\).

Derivation HD-D1.2 (Explicit form of the conjugated kinetic operator). For \(P_{\mathrm{kin}}=-\nabla^2\) one has in coordinates \[ \nabla^2\phi=|g|^{-1/2}\partial_\mu!\Bigl(\sqrt{|g|}\,g^{\mu\nu}\partial_\nu\phi\Bigr), \] hence \[ \widetilde P_{\mathrm{kin}} =-|g|^{1/4}\nabla^2|g|^{-1/4} =-|g|^{-1/4}\partial_\mu!\Bigl(\sqrt{|g|}\,g^{\mu\nu}\partial_\nu\bigl(|g|^{-1/4}\,\cdot\,\bigr)\Bigr). \] Assuming compact support (or boundary conditions killing boundary terms), \[ \int d^Dx\;\psi_1\,\widetilde P_{\mathrm{kin}}\psi_2 =\int d^Dx\;\sqrt{|g|}\,g^{\mu\nu}\, \partial_\mu\bigl(|g|^{-1/4}\psi_1\bigr)\, \partial_\nu\bigl(|g|^{-1/4}\psi_2\bigr), \] which is manifestly symmetric under \((1\leftrightarrow 2)\).

Derivation HD-D1.3 (Conformal metric expansion; D=4 simplification in the conformal class). As a worked expansion, take a conformally flat background \(g_{\mu\nu}=e^{2\sigma(x)}\delta_{\mu\nu}\) (Euclidean for simplicity). Then \(\sqrt{|g|}=e^{D\sigma}\), \(|g|^{1/4}=e^{D\sigma/2}\), \(g^{\mu\nu}=e^{-2\sigma}\delta^{\mu\nu}\), and one finds \[ \Delta_g f =|g|^{-1/2}\partial_\mu!\bigl(\sqrt{|g|}\,g^{\mu\nu}\partial_\nu f\bigr) =e^{-2\sigma}\bigl(\partial^2 f+(D-2)\,\partial\sigma\cdot\partial f\bigr). \] Setting \(\phi=|g|^{-1/4}\psi=e^{-D\sigma/2}\psi\) and expanding derivatives (the product rule produces a cross term \(-2(\partial\sigma)(\partial\psi)\) from differentiating the exponential prefactor) gives the conjugated operator \[ \widetilde\Delta\psi:=|g|^{1/4}\Delta_g|g|^{-1/4}\psi =e^{-2\sigma}\Big( \partial^2\psi -2\,\partial\sigma\cdot\partial\psi -\frac{D}{2}(\partial^2\sigma)\,\psi +\frac{D(4-D)}{4}(\partial\sigma)^2\,\psi \Big), \] so the kinetic operator \(\widetilde P_{\mathrm{kin}}=-\widetilde\Delta\) contains a term proportional to \(D(4-D)(\partial\sigma)^2\), which cancels at \(D=4\) in this conformal ansatz.

Scope disclaimer: this is recorded only as a checked simplification for \(\widetilde\Delta\) in this metric class; it is not, by itself, a dimension-selection claim or a conformal-invariance statement. A symbolic coefficient/sign check (SymPy) confirms the expansion.

Interpretation: - the metric half-density \(\lvert g\rvert^{1/4}\lvert dx\rvert^{1/2}\) is a scalarization gauge (a choice of reference half-density) on a fixed background, - writing the field as \(\psi\) makes the “half-density prioritary” viewpoint explicit: both the field and the kernels live naturally as half-density objects.

Remark HD-D1.3a (Half-density conjugation is distinct from conformal coupling). The conjugated Laplacian \(\widetilde\Delta=|g|^{1/4}\Delta_g|g|^{-1/4}\) acts on sections of the half-density bundle \(|\Lambda^D T^M|^{1/2}\), which carry conformal weight \(D/2\). The Yamabe operator \(Y=-\Delta_g+\frac{D-2}{4(D-1)}R\) acts on sections of the conformal density bundle of weight \((D-2)/2\). These are sections of *different line bundles, which is the primary conceptual reason the operators differ; the coefficient comparison below makes the mismatch explicit. Their first-order, Laplacian, and potential terms all differ.

In particular, the zeroth-order (“potential”) terms generated by the half-density conjugation in the conformal class cannot be written as \(\xi R\) for any single value of \(\xi\). Matching the \(\partial^2\sigma\) coefficient gives \(\xi=D/(4(D-1))\), but the \((\partial\sigma)^2\) coefficient then has a residual mismatch of \(D/2\). Thus the \(D=4\) cancellation of the \((\partial\sigma)^2\) term in \(\widetilde\Delta\) is a property of the half-density weight, not of conformal invariance or conformal coupling (which corresponds to \(\xi_{\mathrm{conf}}=(D-2)/(4(D-1))\)).

The detailed cancellation pattern (which terms vanish, which acquire residual mismatches) is conformal-class-specific: it relies on the conformal structure (contracted Christoffel symbols proportional to \(\partial\sigma\)). However, the leading-order coefficient of the conjugation potential is universal across all metrics, as the following remark shows.

Remark HD-D1.3b (Normal-coordinate computation: the universal coefficient \\(1/6\\) matches conformal coupling only at \\(D=4\\)). For a general Riemannian \((M,g)\), the conjugation potential is \(V=|\nabla\omega|^2g - \Delta_g\omega\) with \(\omega=\tfrac{1}{4}\ln|g|\). In Riemannian normal coordinates centered at a point \(p\), the metric determinant expands as \(|g(x)|=1-\tfrac{1}{3}R{\mu\nu}x^\mu x^\nu + O(x^3)\), so \(\omega=-\tfrac{1}{12}R_{\mu\nu}x^\mu x^\nu+O(x^3)\). (In the conformally flat setting of Derivation HD-D1.3, \(\omega=(D/4)\sigma\); the present computation holds for arbitrary Riemannian metrics.) At \(p\): the gradient \(\nabla\omega\) vanishes (quadratic in \(x\)), the Hessian is \(-\tfrac{1}{6}R_{\alpha\beta}\), and the Laplacian is \(\Delta_g\omega=-\tfrac{1}{6}R\). Hence \[ V(p)=\tfrac{1}{6}R. \] This coefficient \(1/6\) is dimension-independent: it holds for all \(D\) and for all Riemannian metrics, at leading curvature order. Conformal coupling contributes \(\xi_{\mathrm{conf}}R\) with \(\xi_{\mathrm{conf}}=(D-2)/(4(D-1))\), and \(\xi_{\mathrm{conf}}=1/6\) if and only if \(D=4\). Thus, in \(D=4\) and only \(D=4\), the total curvature potential \((\xi_{\mathrm{conf}}-1/6)R\) of the conformally coupled half-density field vanishes at leading order — the half-density conjugation potential \(R/6\) matches the conformal coupling \(\xi_{\mathrm{conf}}R = R/6\) at leading (Ricci scalar) order; subleading curvature terms (\(R^2\), \(R_{\mu\nu}R^{\mu\nu}\), …) are not affected by this cancellation and retain \(D\)-dependent coefficients in general. (This is the same \(1/6\) that appears in the first Seeley-DeWitt coefficient \(a_1(x,x)=(\tfrac{1}{6}-\xi)R\) [Vassilevich2003].)

4. Propagators/Green functions as bi-half-density kernels

Let \(\widetilde G\) be the inverse kernel of \(\widetilde P\) with respect to the coordinate pairing: \[ (\widetilde P^{-1}f)(x)=\int \widetilde G(x,y)\,f(y)\,d^Dy, \qquad \widetilde P_x\,\widetilde G(x,y)=\delta^{(D)}(x-y). \] Then the corresponding canonical bi-half-density kernel is \[ K_{\widetilde P^{-1}}(x,y)=\widetilde G(x,y)\,|dx|^{1/2}|dy|^{1/2}. \]

Equivalently, if \(G_g(x,y)\) denotes the usual scalar Green function for \(P\) defined with respect to the metric pairing \(\int \sqrt{|g|}\,d^Dy\) (so \((P^{-1}J)(x)=\int G_g(x,y)\,J(y)\,\sqrt{|g(y)|}\,d^Dy\)), then the kernels are related by \[ \widetilde G(x,y)=|g(x)|^{1/4}\,G_g(x,y)\,|g(y)|^{1/4}. \]

This is exactly the same “kernel as bi-half-density” structure used for QM propagators [BatesWeinstein1997] [deGosson2018ShortTimePropagators], now applied to spacetime Green functions in QFT. In the QFT-on-curved-spacetime context, this half-density form of the propagator kernel is discussed in [ParkerToms2009, §6.3].

Example HD-D4.3 (Flat-space massless propagator). On flat \(D\)-dimensional Euclidean space, the massless scalar propagator is \[ G(x,y)=\frac{\Gamma(D/2-1)}{(4\pi)^{D/2}\,|x-y|^{D-2}}. \] The exponent \(D/2\) in \((4\pi)^{D/2}\) is the same normalization exponent as in Derivation HD-D1.1a: it encodes the volume-scaling of the approximate identity. The singularity \(|x-y|^{2-D}\) has the same dimensional behavior as \(\varepsilon^{-D/2}\) when the regularization scale \(\varepsilon\) scales as \(|x-y|^2\) (the natural diffusion time scale). In flat space \(|g|=1\), so the bi-half-density kernel \(\widetilde G=G\); the propagator’s normalization is canonically tied to the identity kernel’s \(D/2\) scaling.

Remark HD-D4.1 (Doubling: densities live on \\(M\times M\\)). Half-density kernels also make the amplitude-vs-density doubling completely explicit. Let \(U_t\) be a (unitary) evolution operator on half-densities with kernel \(K_t(x,y)\). Then a density operator \(\rho_t=U_t\rho_0U_t^{-1}\) has a kernel satisfying \[ K_{\rho_t}(x,y) =\int_{M\times M} K_t(x,x’)\,K_{\rho_0}(x’,y’)\,\overline{K_t(y,y’)}. \] Here \(K_{\rho_0}(x’,y’)\) is the kernel of the density operator \(\rho_0\), which is itself a bi-density — equivalently, a product of two bi-half-density kernels — defined by \(K_{\rho}(x,y)=\rho_{\mathrm{scalar}}(x,y)\,|dx|^{1/2}|dy|^{1/2}\). So densities naturally propagate by a kernel on the doubled space \(M\times M\), built from the forward kernel and its conjugate. The composition in both slots is intrinsic (no background measure needed) precisely because \(K_t\) is a bi-half-density — this is the kernel-level origin of bra/ket (forward/backward) doubling in real expectation values.

Remark HD-D4.2 (Heat kernel as bi-half-density; trace without \\(\sqrt{|g|}\\)). The heat kernel \(K(t;x,y)\) of \(P\) on a Riemannian \((M,g)\) has the DeWitt-Schwinger short-time expansion [DeWitt1965] [Vassilevich2003] \[ K(t;x,y)\sim(4\pi t)^{-D/2}\,\Delta^{1/2}(x,y)\,e^{-s(x,y)/(2t)}\sum_{n\ge0}a_n(x,y)\,t^n, \] where \(s(x,y)\) is Synge’s world function (half the squared geodesic distance; not to be confused with the conformal factor \(\sigma\) of Derivation HD-D1.3), \(\Delta(x,y)\) the Van Vleck determinant (the Jacobian of the exponential map, normalized so \(\Delta(x,x)=1\)), and \(a_n\) the Seeley-DeWitt coefficients (\(a_0(x,x)=1\)). Since \(\Delta^{1/2}(x,y)\) is a bi-scalar and \(|g(x)|^{1/4}\) provides a half-density at \(x\), the combination \(|g(x)|^{1/4}\,\Delta^{1/2}(x,y)\,|g(y)|^{1/4}\) transforms as a bi-half-density. Define \[ \widetilde K(t;x,y):=|g(x)|^{1/4}\,K(t;x,y)\,|g(y)|^{1/4}. \] At coincidence (\(s=0\), \(\Delta=1\)), the leading behavior \(\widetilde K(t;x,x)\sim(4\pi t)^{-D/2}\) matches exactly the normalization of Derivation HD-D1.1a: the heat kernel is the time-parameterized approximate identity, \(\widetilde K(t)\to K_{\mathrm{Id}}\) distributionally as \(t\to0^+\). The trace is then \[ \mathrm{Tr}\,K(t)=\int_M\widetilde K(t;x,x)\,d^Dx, \] where the \(\sqrt{|g|}\) factor of the standard trace formula has been absorbed into the definition of \(\widetilde K\). For the massless operator \(P=-\nabla^2+\xi R\), the first Seeley-DeWitt coefficients at coincidence are \(a_1(x,x)=(1/6-\xi)R\) and \(a_2(x,x)\) is a linear combination of \(R^2\), \(R_{\mu\nu}R^{\mu\nu}\), \(\Box R\); these encode the curvature counterterms that appear in Section 5’s contact-term analysis.

Remark HD-D4.2a (Trace anomaly and the index theorem from \\(a_{D/2}\\)). In even dimension \(D=2n\), the coefficient \(a_n(x,x)\) is a local curvature polynomial of mass dimension \(D\), and its integral \(\int_M a_n\,d^Dx\) is scheme-independent. Two fundamental results follow: the conformal trace anomaly \(\langle T^\mu{}\mu\rangle\propto a{D/2}(x,x)\) for quantum fields on curved backgrounds [Vassilevich2003], and the Atiyah-Singer index theorem, where the index of a Dirac-type operator equals \(\int_M a_{D/2}\,d^Dx\) (up to normalization) via the McKean-Singer supertrace formula (in the Dirac case, \(a_{D/2}\) is the \(\hat{A}\)-genus integrand; the general characteristic-class connection is beyond this note’s scope). In the half-density framework, both calculations are manifestly coordinate-free: the intrinsic trace of Remark HD-D4.2 absorbs the metric factor, and the topological content resides entirely in the Seeley-DeWitt coefficients — geometric invariants of \(P\), not of the coordinate choice.

Remark HD-D4.4 (Van Vleck determinant as the geometric half-density factor). The Van Vleck determinant \(\Delta(x,y)\) appearing in Remark HD-D4.2 satisfies the transport equation [DeWitt1965] \[ \nabla_\mu!\bigl(\Delta^{1/2}(x,y)\,\nabla^{\mu}s(x,y)\bigr)=D\,\Delta^{1/2}(x,y), \] where the derivative acts in the \(x\) variable. This means \(\Delta^{1/2}\) is the Jacobian converting coordinate volume to geodesic normal-coordinate volume: it is the half-density factor intrinsic to the geometry. Its appearance in the heat kernel expansion is therefore not accidental — it ensures the leading kernel \((4\pi t)^{-D/2}\Delta^{1/2}(x,y)\,e^{-s/(2t)}\) transforms as a proper bi-half-density under coordinate changes in both slots — without \(\Delta^{1/2}\), the coordinate-free composition of Section 2 would fail for the heat kernel.

Remark HD-D4.5 (Proper-time representation unifies the D/2 exponent). The propagator is the proper-time (Schwinger) integral of the heat kernel: \(G(x,y)=\int_0^\infty K(t;x,y)\,dt\). The exponent \(D/2\) recurs throughout: identity-kernel normalization \((2\pi\varepsilon)^{-D/2}\) (Derivation HD-D1.1a), heat-kernel short-time behavior \((4\pi t)^{-D/2}\) (Remark HD-D4.2), flat propagator prefactor \(\Gamma(D/2-1)/(4\pi)^{D/2}\) (Example HD-D4.3), and mass-counterterm UV degree \(\Lambda^{D-2}=\Lambda^{2(D/2-1)}\) (Remark HD-D5.3). The proper-time integral converts the heat kernel’s \(t^{-D/2}\) into the propagator’s \(\Gamma(D/2-1)\) via Schwinger parametrization, making explicit that all four manifestations trace back to the same half-density volume-scaling origin. (For a comprehensive treatment showing the \(d/2\) exponent is uniquely forced by semigroup composition under dimensional homogeneity — across temporal composition, Van Vleck determinant, heat kernel, and renormalization thresholds — see the companion note “Path Integral Normalization and the d/2 Exponent” [PathIntegralNormalization, in preparation].)

5. Contact terms and counterterms as diagonal delta kernels (kernel-level remark)

In QFT, divergences are removed by adding local counterterms to the action, e.g. \(\delta m^2\,\phi^2\), \(\delta Z\,(\partial\phi)^2\), curvature couplings, etc. At the operator/kernel level this means: the kinetic operator \(P\) is modified by local (differential) operators, and therefore its kernel acquires distributions supported on the diagonal \(x=y\).

In the half-density kernel language the diagonal object is canonical: \[ K_{\mathrm{Id}}(x,y)=\delta^{(D)}(x-y)\,|dx|^{1/2}|dy|^{1/2}. \] Multiplication counterterms correspond to \(c(x)\,K_{\mathrm{Id}}(x,y)\), and derivative counterterms correspond to derivatives acting on the delta kernel (e.g. \(\partial_x\delta^{(D)}(x-y)\) and higher).

Remark HD-D5.1 (Derivative of the diagonal delta kernel; coordinate-free identity). The slogan “\(\partial_x\delta(x-y)=-\partial_y\delta(x-y)\)” can be seen cleanly, without a connection, in the half-density kernel calculus. The identity kernel \(K_{\mathrm{Id}}\) is invariant under diagonal diffeomorphisms \((\Phi\times\Phi)\), so for any vector field \(V\) on \(M\) one has \[ (\mathcal L_{V_x}+\mathcal L_{V_y})\,K_{\mathrm{Id}}=0, \qquad\text{hence}\qquad \mathcal L_{V_x}K_{\mathrm{Id}}=-\mathcal L_{V_y}K_{\mathrm{Id}}, \] where \(\mathcal L\) is the Lie derivative acting on half-densities. In local coordinates, taking \(V=\partial/\partial x^\mu\) recovers \(\partial_{x^\mu}\delta^{(D)}(x-y)=-\partial_{y^\mu}\delta^{(D)}(x-y)\). This is the kernel-level mechanism behind “moving derivatives between slots” in integration by parts and in contact-term identities.

Remark HD-D5.2 (Point splitting produces δ' contact terms). Point splitting makes the simplest derivative contact term explicit: in one dimension, \[ \frac{\delta(x+\varepsilon)-\delta(x)}{\varepsilon}\xrightarrow[\varepsilon\to 0]{}\delta’(x), \qquad \langle\delta’,\varphi\rangle=-\varphi’(0). \] This limit illustrates “locality = diagonal support” at the kernel level: derivative counterterms are distributional derivatives of \(K_{\mathrm{Id}}\), and their form is fixed by the half-density structure independently of scheme-dependent finite-subtraction conventions.

Remark HD-D5.3 (Mass counterterm scaling from the D/2 exponent). The mass counterterm \(\delta m^2\) is a scalar multiple of \(K_{\mathrm{Id}}\). Its UV degree of divergence follows from the heat kernel at coincidence: the propagator at \(x=y\) is formally \(\int_0^\infty K(t;x,x)\,dt\), and Remark HD-D4.2 gives \(K(t;x,x)\sim(4\pi t)^{-D/2}\) as \(t\to 0^+\). The UV divergence arises from small proper times (large momenta \(k^2\sim 1/t\)); introducing a proper-time cutoff at \(t_{\min}=\Lambda^{-2}\) removes the divergent piece \(\int_0^{\Lambda^{-2}} t^{-D/2}\,dt\sim\Lambda^{D-2}\) for \(D>2\). Thus the same \(D/2\) exponent that normalizes the identity kernel (Derivation HD-D1.1a) controls the UV degree of divergence of the mass correction: \(\Lambda^{D-2}=\Lambda^{2(D/2-1)}\).

Remark HD-D5.3a (Derivative counterterm scaling from the Seeley-DeWitt hierarchy). The wave function renormalization counterterm \(\delta Z\,(\partial\phi)^2\) involves a derivative of the diagonal delta kernel (Remark HD-D5.1). Its UV degree follows from the next Seeley-DeWitt coefficient: the \(a_1(x,x)\,t\) contribution shifts the proper-time integrand from \(t^{-D/2}\) (mass counterterm, Remark HD-D5.3) to \(t^{1-D/2}\), giving \(\int_0^{\Lambda^{-2}}t^{1-D/2}\,dt\sim\Lambda^{D-4}\). Each subsequent coefficient \(a_n\) reduces the divergence by \(\Lambda^2\), so that in \(D=4\) the mass counterterm is quadratic (\(\Lambda^2\)), the wave function counterterm is logarithmic (\(\ln\Lambda\)), and all higher counterterms are finite — the standard renormalizability count for a scalar field, organized entirely by the \(D/2\) exponent.

Remark HD-D5.3b (Dimensional regularization and the D/2 pole structure). In dimensional regularization, the proper-time integral for the \(a_n\) contribution becomes \(\int_0^\infty t^{n-D/2}e^{-m^2 t}\,dt=\Gamma(n+1-D/2)/m^{2(n+1-D/2)}\). The \(\Gamma\)-function pole at \(D=2(n+1)\) is the dim-reg counterpart of the cutoff divergence \(\Lambda^{D-2(n+1)}\) of Remarks HD-D5.3–HD-D5.3a: the mass counterterm (\(n=0\)) has its pole at \(D=2\), the wave function counterterm (\(n=1\)) at \(D=4\), and each subsequent counterterm at \(D=2(n+1)\). The hierarchy is scheme-independent: whether one uses a proper-time cutoff or dimensional continuation, the same \(D/2\) exponent determines which counterterms diverge in a given dimension [Vassilevich2003].

Remark HD-D5.4 (Functional determinant from the heat-kernel trace). The spectral \(\zeta\)-function \(\zeta_P(s)=\mathrm{Tr}\,P^{-s}=\Gamma(s)^{-1}\int_0^\infty t^{s-1}\,\mathrm{Tr}\,K(t)\,dt\) inherits its pole structure from the small-\(t\) asymptotics of the heat-kernel trace: the Seeley-DeWitt expansion gives simple poles at \(s=D/2-n\) (\(n=0,1,2,\ldots\)), again controlled by the \(D/2\) exponent. The functional determinant \(\det P:=\exp(-\zeta_P’(0))\) is finite, coordinate-independent, and — in the half-density framework — requires no extraneous volume-form choice: the trace \(\mathrm{Tr}\,K(t)=\int_M\widetilde K(t;x,x)\,d^Dx\) is intrinsic since \(\widetilde K\) absorbs the \(\sqrt{|g|}\) factor (Remark HD-D4.2) [Vassilevich2003].

6. Link to the half-density scale program (where Planck-area enters conditionally)

On a fixed background \((M,g)\), the metric provides a natural reference half-density \(|g|^{1/4}|dx|^{1/2}\). The Planck-area program begins only when we ask for a background-free scalarization convention that turns half-density amplitudes into universal dimensionless numbers. A separate note develops that stronger hypothesis ladder; it is not needed for the present kernel/propagator semantics.

This paper’s role is only to show that half-densities are not a QM quirk: the same kernel semantics is already present in standard QFT propagator definitions, once the hidden measure conventions are made explicit.

7. Outlook: BV half-densities

Gauge theories suggest a second, deeper appearance of half-densities: the BV formalism treats the integrand as a (half-)density on an (odd) symplectic space of fields/antifields, and the quantum master equation expresses independence of gauge-fixing choices [Costello2011]. This note does not develop BV beyond this remark; doing so responsibly would require additional authoritative sources and a separate dedicated treatment.

References

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