Delta Objects as Half-Density Kernels: Identity, Stationary-Set Concentration, and Point Interactions

This note is a companion to the cornerstone manuscript. Statements are kept finite-dimensional unless explicitly labeled heuristic.

1. Half-densities and kernels (coordinate free)

Let \(M\) be a \(d\)-dimensional manifold and \(|\Omega|^{1/2}\) the half-density bundle [BatesWeinstein1997]. An operator \(K:\Gamma_c(|\Omega|^{1/2})\to \Gamma(|\Omega|^{1/2})\) has a natural Schwartz kernel [Hormander2003] \(\mathsf K\in \mathcal D'(M\times M;\;|\Omega|^{1/2}\boxtimes|\Omega|^{1/2}),\) so that \((K\psi)(x)=\int_M \mathsf K(x,y)\,\psi(y),\) is coordinate invariant: \(\mathsf K(x,y)\psi(y)\) is a density in \(y\) valued in a half-density at \(x\).

Scalarizing kernels (writing \(\int dy\) with a scalar integrand) implicitly chooses a reference density/half-density; the half-density formalism keeps this choice explicit.

2. Delta as the identity kernel (and near-diagonal scaling)

The identity operator on half-densities has Schwartz kernel \(\mathsf K_{\mathrm{Id}}(x,y)=\delta^{(d)}(x-y)\,|dx|^{1/2}|dy|^{1/2}.\)

Worked scaling computation (the \(d/2\) exponent)

Introduce near-diagonal coordinates \(y=x+\varepsilon v\). Then \(\delta^{(d)}(x-y)=\delta^{(d)}(\varepsilon v)=\varepsilon^{-d}\delta^{(d)}(v)\) and \(|dy|^{1/2}=\varepsilon^{d/2}|dv|^{1/2}\), so \(\mathsf K_{\mathrm{Id}}(x,x+\varepsilon v) =\varepsilon^{-d/2}\,\delta^{(d)}(v)\,|dx|^{1/2}|dv|^{1/2}.\) Thus the universal \(\varepsilon^{-d/2}\) normalization exponent is already present in the identity delta kernel, once kernels are treated as half-densities.

Remark 2.1 (Why d/2 is universal). The exponent \(d/2\) is not a computational accident. Composing two half-density kernels requires integrating over an intermediate point: \(\int \mathsf{K}_1(x,z)\,\mathsf{K}_2(z,y)\,dz\). Each kernel carries \(|dz|^{1/2}\); their product gives \(|dz|\), which pairs with the integration measure to produce a scalar. The \(\varepsilon^{-d/2}\) prefactor is precisely what makes near-diagonal kernel scaling compatible with composition — the kernel analog of the \((N+1)/2\) normalization exponent in the time-sliced path integral (companion note, Section 4.5). For a comprehensive treatment showing this exponent is forced across temporal composition, Van Vleck determinant in curved space, heat-kernel diffusion, and renormalization thresholds, see [PathIntegralNormalization].

3. Delta on the stationary set: $\delta(\nabla f)$ and determinant weights

## 3.1 One-dimensional identity ($\delta(f’)$) Let \(f:\mathbb R\to\mathbb R\) have finitely many nondegenerate critical points \(x_i\) (so \(f'(x_i)=0\), \(f''(x_i)\neq 0\)). Then, as distributions, \(\delta(f'(x))=\sum_i \frac{\delta(x-x_i)}{|f''(x_i)|}.\) So \(\delta(f')\,dx\) is a density supported at stationary points with weights \(1/|f''|\).

3.1a $\delta(f’)$ versus $\delta’$: delta of a derivative vs derivative of delta

The notation \(\delta(f')\) above means: apply the Dirac delta distribution \(\delta(\cdot)\) to the function \(f'(x)\), thereby localizing to the stationary set \(f'(x)=0\). It should not be confused with \(\delta'\), the distributional derivative of \(\delta\), defined by duality: \(\langle \delta',\varphi\rangle := -\langle \delta,\varphi'\rangle = -\varphi'(0).\) So \(\delta'\) is the distribution that probes derivatives of test functions at a point (“value of the derivative at zero”, up to sign), whereas \(\delta(f')\) is a stationary-set localization distribution.

3.1b $\delta’$ from point splitting (difference quotient of shifted deltas)

The distribution \(\delta'\) can be realized as a regulated point-splitting limit. Let \(\varepsilon\to 0\) and consider the shifted delta \(\delta(x+\varepsilon)\). For any test function \(\varphi\), \(\left\langle \frac{\delta(\,\cdot+\varepsilon)-\delta}{\varepsilon},\varphi\right\rangle =\frac{\varphi(-\varepsilon)-\varphi(0)}{\varepsilon} \xrightarrow[\varepsilon\to 0]{} -\varphi'(0) =\langle \delta',\varphi\rangle.\) Hence, in the sense of distributions, \(\frac{\delta(x+\varepsilon)-\delta(x)}{\varepsilon}\xrightarrow[\varepsilon\to 0]{}\delta'(x).\)

This gives a clean dictionary item for “probing the derivative at a point”: \(f'(0)=\langle -\delta', f\rangle.\) For the parallel smooth-function toy model (“difference quotient as divergence + subtraction”) and further remarks, see the companion notes.

3.2 Multi-dimensional identity ($\delta(\nabla f)$)

Let \(f:\mathbb R^n\to\mathbb R\) have finitely many nondegenerate critical points \(x_i\) (so \(\nabla f(x_i)=0\) and \(\det(\mathrm{Hess}\,f)(x_i)\neq 0\)). Then \(\delta^{(n)}(\nabla f(x)) =\sum_i \frac{\delta^{(n)}(x-x_i)}{|\det(\mathrm{Hess}\,f)(x_i)|}.\)

3.3 Stationary phase and square-root weights (amplitudes vs densities)

For the oscillatory integral \(I(\hbar)=\int_{\mathbb R^n} e^{\frac{i}{\hbar}f(x)}\,a(x)\,dx,\qquad \hbar\to 0^+,\) stationary phase gives amplitude contributions weighted by \(\frac{1}{\sqrt{|\det(\mathrm{Hess}\,f)(x_i)|}},\) up to a universal \(\hbar\)-dependent factor and a signature phase. Squaring amplitude weights produces the density weights in \(\delta^{(n)}(\nabla f)\). This is the finite-dimensional prototype of the slogan: amplitudes are half-densities; probabilities are densities.

3.4 Extremals in weak form: where $\delta$ and $\delta’$ appear in Euler–Lagrange

For an action \(S[q]=\int L(q,\dot q,t)\,dt\), the extremal condition is naturally distributional: for test variations \(\eta(t)\) of compact support, \(\delta S[q;\eta]=\int \Bigl(\frac{\partial L}{\partial q}-\frac{d}{dt}\frac{\partial L}{\partial \dot q}\Bigr)\eta(t)\,dt.\) If \(\delta S[q;\eta]=0\) for all \(\eta\), then the Euler–Lagrange expression vanishes as a distribution. Approximating \(\eta\) by bump functions converging to \(\delta(t-t_\ast)\) localizes the equation at \(t_\ast\) under regularity.

When \(\partial L/\partial \dot q\) has jumps (corners/impulses), the distributional derivative produces delta terms automatically; more generally, point-supported singularities are encoded by delta kernels and their derivatives (\(\delta,\delta',\ldots\)), depending on distributional order.

Example 3.4a (Free particle: smooth vs kinked trajectories). For \(L=\tfrac{m}{2}\dot q^2\), the Euler–Lagrange expression is \(E[q]=-m\ddot q\). If \(q(t)\) is smooth and \(\eta=\rho_\varepsilon(t-t_\ast)\) is a mollifier sequence, then \(\delta S[q;\eta]=-\int m\ddot q(t)\,\rho_\varepsilon(t-t_\ast)\,dt\to -m\ddot q(t_\ast)\) as \(\varepsilon\to0\), localizing the equation pointwise (the companion note’s Theorem 2.1). If instead \(q\) has a velocity jump \(\Delta v\) at \(t_\ast\), the distributional derivative \(\ddot q\) acquires a delta term: \(\ddot q=(\Delta v)\,\delta(t-t_\ast)+\{\text{smooth}\}\), and the forced equation \(m\ddot q=J\,\delta(t-t_\ast)\) gives \(m\,\Delta v=J\) — the impulse-matching condition of the companion note’s Theorem 3.2. The transition from smooth extremals to impulsive ones is thus captured entirely by the distributional order of the Euler–Lagrange expression.

3.5 Van Vleck determinant: the propagator instance of the square-root Hessian

The square-root Hessian weight of Section 3.3 has a distinguished physical instance: the Van Vleck determinant [Hamilton1834] [VanVleck1928Correspondence] [Morette1951] in the semiclassical propagator.

For the short-time quantum propagator between positions \(q_i\) and \(q_f\) with time interval \(\Delta t\), stationary-phase evaluation of the path integral gives \(K(q_f,q_i;\Delta t) \;\propto\; \sqrt{D(q_f,q_i;\Delta t)}\;e^{(i/\hbar)\,S_{\mathrm{cl}}(q_f,q_i;\Delta t)},\) where \(S_{\mathrm{cl}}\) is the classical action on the extremal path and \(D(q_f,q_i;\Delta t) \;:=\; \left|\det\!\left(-\frac{\partial^2 S_{\mathrm{cl}}}{\partial q_f^a\,\partial q_i^b}\right)\right|\) is the Van Vleck determinant — a mixed Hessian (derivatives at the two endpoints of the classical path), as opposed to the full Hessian of \(f\) that appears in \(\delta(\nabla f)\). Despite this difference, it arises by the same stationary-phase mechanism: square-root Hessian weights at the amplitude level, confirming the “amplitudes are half-densities” pattern.

Example 3.5a (Free particle). For the free particle in \(d\) dimensions, \(S_{\mathrm{cl}}=m|q_f-q_i|^2/(2\Delta t)\), so \(D = (m/\Delta t)^d,\qquad \sqrt{D}=(m/\Delta t)^{d/2},\) reproducing the \((\Delta t)^{-d/2}\) normalization of Section 2.

Example 3.5b (Harmonic oscillator). For the harmonic oscillator (\(V=\tfrac12 m\omega^2 q^2\)) in \(d=1\), the classical action between \(q_i\) and \(q_f\) in time \(\Delta t\) is \(S_{\mathrm{cl}}=\frac{m\omega}{2\sin\omega\Delta t}\bigl[(q_f^2+q_i^2)\cos\omega\Delta t - 2q_f q_i\bigr]\), giving \(D = \left|\frac{m\omega}{\sin\omega\Delta t}\right|,\qquad \sqrt{D}=\sqrt{\frac{m\omega}{|\sin\omega\Delta t|}}.\) As \(\omega\Delta t\to 0\), \(\sin\omega\Delta t\approx\omega\Delta t\), recovering the free-particle result \(\sqrt{D}\to\sqrt{m/\Delta t}\). At \(\omega\Delta t=\pi\) (half-period), \(\sin\omega\Delta t\to 0\) and \(\sqrt{D}\to\infty\): this is the familiar caustic (focal point) where the semiclassical approximation breaks down because the classical flow focuses all initial momenta onto a single final point.

Remark 3.5c (Caustics and the Maslov index). At caustics (conjugate points where \(D\to\infty\)), the amplitude \(|D|^{1/2}\) diverges and the branch choice for \(\sqrt{D}\) becomes ambiguous. In the half-density framework the singularity is a projection artifact: the semiclassical state is a smooth half-density on the Lagrangian submanifold \(L\subset T^\ast M\), and the caustic occurs because the projection \(\pi\colon L\to M\) develops a fold [BatesWeinstein1997]. The branch ambiguity of \(\sqrt{D}\) is resolved by the metaplectic structure: half-densities transform under the double cover \(\mathrm{Mp}(2n)\) of \(\mathrm{Sp}(2n)\), and the Maslov index \(\mu\) — counting conjugate points along the classical path — records the accumulated phase correction \(\exp(-i\pi\mu/2)\). This makes the “amplitudes are half-densities” pattern of Section 3.3 precise: the Van Vleck prefactor is not a scalar but a section of the half-density bundle on \(L\), globally well-defined even through caustics.

4. Delta at a point: point interactions as rank-one kernels

A point interaction [AlbeverioGesztesyHoeghKrohnHolden2005] is naturally the rank-one operator \(V=g\,|0\rangle\langle0|.\) In the half-density kernel calculus this is written as the bi-half-density distribution supported at \((0,0)\): \(\mathsf K_V(x,y)=g\;\delta^{(d)}(x)\,\delta^{(d)}(y)\,|dx|^{1/2}|dy|^{1/2}.\) This is the “projector-like delta” object underlying contact interactions.

Example 4.1 (1D delta potential: resolvent as a rank-one perturbation). In one dimension the rank-one structure is particularly explicit. For \(H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+g\,\delta(x)\) with \(g<0\) (attractive), the free resolvent at energy \(E=-\hbar^2\kappa^2/(2m)\) is \(G_0(x,x';E)=-\frac{m}{\hbar^2\kappa}\,e^{-\kappa|x-x'|}\). The rank-one perturbation formula gives \(G(x,x';E)=G_0(x,x';E)+\frac{g\,G_0(x,0;E)\,G_0(0,x';E)}{1-g\,G_0(0,0;E)}.\) The correction term factors as \(f(x)\cdot f(x')\) with \(f(x)=G_0(x,0;E)\) — this is the rank-one kernel in action: the point interaction contributes a term proportional to \(|0\rangle\langle0|\) in the resolvent. The denominator vanishes at \(\kappa=|g|m/\hbar^2\), yielding the unique bound state \(E=-mg^2/(2\hbar^2)\), and the residue at this pole factors as \(\psi_b(x)\,\psi_b(x')\) with \(\psi_b(x)=\sqrt{\kappa}\,e^{-\kappa|x|}\) — a rank-one projector \(|\psi_b\rangle\langle\psi_b|\) [AlbeverioGesztesyHoeghKrohnHolden2005]. In the half-density kernel language, the factored piece reads \((\sqrt{\kappa}\,e^{-\kappa|x|}|dx|^{1/2})\otimes(\sqrt{\kappa}\,e^{-\kappa|x'|}|dx'|^{1/2})\), manifestly a product of half-densities.

Example 4.1b (2D delta potential: logarithmic divergence and transmutation scale). In \(d=2\), the free Green function at the origin is \(G_0(0,0;E)=\frac{m}{2\pi\hbar^2}\ln(\Lambda/\kappa)\), logarithmically divergent in the UV cutoff \(\Lambda\). The rank-one perturbation formula of Example 4.1 still applies: the denominator \(1-g\,G_0(0,0;E)=0\) determines the bound state. Inserting the running coupling \(g(\Lambda)=2\pi\hbar^2/(m\ln(\Lambda/\alpha))\) from Remark 4.2, the \(\Lambda\)-dependent terms cancel: \(1-\frac{2\pi\hbar^2}{m\ln(\Lambda/\alpha)}\cdot\frac{m}{2\pi\hbar^2}\ln(\Lambda/\kappa) =1-\frac{\ln(\Lambda/\kappa)}{\ln(\Lambda/\alpha)} =0 \quad\text{iff}\quad \kappa=\alpha.\) The transmutation scale \(\alpha\) (dimension length\(^{-1}\)) is the sole physical parameter: it replaces the bare coupling \(g\) that has no well-defined continuum limit [Jackiw1991DeltaPotentials]. The bound state energy is \(E=-\hbar^2\alpha^2/(2m)\), fully determined by the RG-invariant scale. Compare Example 5.1a for the scalarization perspective on this same result.

Remark 4.2 (Self-adjoint extensions: the classification of point interactions). The one-dimensional delta potential of Example 4.1 is the simplest instance of a general classification [AlbeverioGesztesyHoeghKrohnHolden2005] [Derezinski2024]. The free Hamiltonian \(H_0=-\frac{\hbar^2}{2m}\nabla^2\) on \(\mathbb R^d\), initially defined on \(C_c^\infty(\mathbb R^d\setminus\{0\})\), is symmetric but not self-adjoint when \(d\leq3\). Its self-adjoint extensions — the mathematically well-defined point interactions — include a distinguished one-parameter family of spherically symmetric extensions in each admissible dimension:

• \(d=1\): parametrized by the coupling \(g\in\mathbb R\cup\{\infty\}\). The boundary condition is \(\psi'(0^+)-\psi'(0^-)=(2mg/\hbar^2)\psi(0)\); this is the \(\delta\)-subfamily with \(\psi\) continuous at the origin.
• \(d=2\): parametrized by a transmutation scale \(\alpha>0\) (dimensional transmutation). The bare coupling \(g(\Lambda)\sim 2\pi\hbar^2/(m\ln(\Lambda/\alpha))\) is logarithmically divergent and requires renormalization.
• \(d=3\): parametrized by the scattering length \(a\in\mathbb R\cup\{\infty\}\). The bare coupling requires power-law renormalization: \(1/g(\Lambda)=m(1/a-2\Lambda/\pi)/(2\pi\hbar^2)\).

For \(d\geq4\), no non-trivial self-adjoint extensions exist: \(H^2(\mathbb R^d)\) functions need not be continuous (Morrey embedding requires \(2\cdot 2>d\), i.e.\ \(d\leq3\)), so the point is invisible to the Laplacian and removing it does not change the operator closure [Derezinski2024]. The diagonal singularity of the free Green function controls which case applies: \(G_0(0,0;E)\) is finite (\(d=1\)), log-divergent \(\sim\ln(\Lambda/\kappa)\) (\(d=2\)), linearly divergent \(\sim\Lambda\) (\(d=3\)), or power-divergent \(\sim\Lambda^{d-2}\) with no remedy (\(d\geq4\)). The resolvent exponent \(d-2\) arises from the heat kernel exponent \(d/2\) of Section 2 via a Laplace transform in the spectral parameter. In each admissible dimension, the self-adjoint extension parameter is the RG-invariant contact datum that survives all regularization schemes — a concrete instance of the scalarization perspective of Section 5.

Remark 4.3 (Heat kernel exponent determines the extension classification). The connection between the half-density exponent \(d/2\) (Section 2) and the self-adjoint extension hierarchy above is made explicit by the Laplace representation of the Green function: \(G_0(0,0;E)\propto\int_0^\infty t^{-d/2}e^{-\kappa^2 t}\,dt\) where \(E=-\hbar^2\kappa^2/(2m)\). The integrand’s short-time singularity \(t^{-d/2}\) is precisely the identity-kernel scaling of Section 2. The integral equals \(\Gamma(1-d/2)\,\kappa^{d-2}\) (up to constants) and converges iff \(d<2\); this is why \(d=2\) is the critical dimension where the contact coupling is classically marginal and dimensional transmutation first appears. The same exponent that ensures coordinate-free composition of half-density kernels (the \(t^{-d/2}\) normalization) thus controls whether a point interaction can be defined without renormalization.

5. Where scales enter upon scalarization (and why RG invariants are natural candidates)

Half-density kernels are canonical; scalar representatives are not. Choosing a reference half-density \(\sigma_\ast\) identifies any half-density \(\psi\) with a scalar \(f\) via \(\psi=f\,\sigma_\ast\). If one insists that scalar representatives be dimensionless, then \(\sigma_\ast\) must carry a \(\text{length}^{d/2}\) constant.

In marginal cases (notably the 2D point interaction), renormalization generates an RG-invariant scale \(\kappa_\ast\) (dimensional transmutation). This suggests a conditional identification: if one adds a universality hypothesis that scalarization scales must be built from physical invariants, then RG-invariant scales are natural candidates to supply the missing \(\text{length}^{d/2}\) factors required by scalarization.

This note treats that identification as an organizing perspective, not as a theorem.

Remark 5.1 (The exponent $$d/2$$ as the unifying thread). The half-density weight \(d/2\) that fixes the identity-kernel scaling \(\varepsilon^{-d/2}\) (Section 2) is the same exponent that controls the heat-kernel convergence condition \(d<2\) (Remark 4.3) and the Sobolev embedding threshold \(H^k\hookrightarrow C^0\) iff \(k>d/2\) (the Morrey condition underlying Remark 4.2’s \(d\leq3\) restriction for \(k=2\)). These are not three independent coincidences but one geometric fact: the half-density bundle is the borderline object whose sections scale at the exact dimensional rate that separates convergent from divergent short-distance behavior. The scalarization scale \(\text{length}^{d/2}\) of this section is therefore determined by the same exponent that governs which point interactions are definable without renormalization.

Example 5.1a (d=2 scalarization: transmutation scale as the natural unit). In \(d=2\), the scalarization exponent is \(d/2=1\): expressing kernels as scalars requires choosing one physical length. Remark 4.2 establishes that the 2D point interaction generates a unique RG-invariant scale \(\alpha\) (dimension \(\text{length}^{-1}\)) via dimensional transmutation. The bound-state condition (Remark 4.3, Laplace representation at \(d=2\)) reduces to \(\kappa=\alpha\), where \(\kappa^2=-2mE/\hbar^2\). Expressed in units of \(\alpha\), all scalarized quantities are dimensionless: the bound-state condition becomes \(\kappa/\alpha=1\), manifestly coupling-independent and RG-invariant. This is the concrete content of “RG invariants supply scalarization scales”: the single RG-invariant output of 2D renormalization provides exactly the single \(\text{length}^{d/2}=\text{length}^1\) reference that scalarization demands.

6. Outlook

1. ~~Relate determinant weights to Van Vleck type.~~ Addressed: Section 3.5 makes the connection explicit.
2. Clarify which parts of the “functional delta \(\delta(\delta S)\)” story survive rigorous regularization and which remain heuristic.
3. Extend the half-density treatment to spacetime (Lorentzian) propagators and distributional kernels in field theory.

References

1. [Hamilton1834] W. R. Hamilton, “On a General Method in Dynamics,” Philosophical Transactions of the Royal Society, Part II, 247–308 (1834). OA: https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/GenMeth.pdf. (Hamilton’s principal function \(S(q,Q,t)\); the mixed Hessian \(\partial^2 S/\partial q\,\partial Q\) appears here a century before Van Vleck takes its determinant for the semiclassical propagator prefactor.)
2. [VanVleck1928Correspondence] J. H. Van Vleck, “The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics,” Proceedings of the National Academy of Sciences of the United States of America 14(2) (1928), 178–188. DOI 10.1073/pnas.14.2.178.
3. [BatesWeinstein1997] Sean Bates and Alan Weinstein, “Lectures on the Geometry of Quantization,” Berkeley Mathematics Lecture Notes, vol. 8, AMS, 1997. ISBN 978-0-8218-0798-9. OA: https://math.berkeley.edu/~alanw/GofQ.pdf. (Canonical reference for half-density formalism in geometric quantization; half-density kernels and composition.)
4. [Hormander2003] Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, 2003. DOI 10.1007/978-3-642-61497-2. (Schwartz kernel theorem; distributional calculus for PDE Green functions.)
5. [AlbeverioGesztesyHoeghKrohnHolden2005] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, 2nd ed., AMS Chelsea Publishing, 2005. ISBN 978-0-8218-3624-4. (Canonical reference for point interactions in quantum mechanics; self-adjoint extensions, delta potentials.)
6. [Morette1951] C. Morette, “On the Definition and Approximation of Feynman’s Path Integrals,” Phys. Rev. 81, 848–852 (1951). DOI 10.1103/PhysRev.81.848. (Van Vleck determinant in path integral; semiclassical expansion weights.)
7. [Derezinski2024] J. Dereziński, “Point potentials on Euclidean space, hyperbolic space and sphere in any dimension,” arXiv:2403.17583 (2024). (Self-adjoint extensions of the Laplacian restricted to \(\mathbb R^d\setminus\{0\}\); essential self-adjointness for \(d\geq4\); complete treatment for all \(d\).)
8. [Jackiw1991DeltaPotentials] R. Jackiw, “Delta-function potentials in two- and three-dimensional quantum mechanics,” MIT-CTP-1937 (Jan 1991). Reprinted in M.A.B. Bég Memorial Volume (World Scientific, 1991), pp. 25–42. OA mirror: https://www.physics.smu.edu/scalise/P6335fa21/notes/Jackiw.pdf. (Renormalization of contact interactions in d=2,3; dimensional transmutation.)