DRSN XII: Heat Kernel, Heat Trace, and Spectral Invariants (De Rerum Spectrale Natura, Report XII, Version 1.0)

DSRN XII: HEAT KERNEL, HEAT TRACE, AND SPECTRAL INVARIANTS De Rerum Spectrale Natura series REPORT XII (Version 1.0) J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 •Establishes the heat-kernel layer of the Drifted Spectral Renormalisation Network (DSRN). • Proves that heat traces of nonnegative self-adjoint operators with trace-class semigroup are spectral invariants and therefore rigid under bounded similarity (drift). •Identifies Seeley–DeWitt coefficients as spectrally rigid objects whenever heat-trace asymptotics exist. • Fixes precisely where effective, scale-dependent quantities arise only through truncation, projection, or other information-loss operations. •Contains no physical interpretation beyond the analytic heat-kernel framework fixed here. DSRN XII: Heat Kernel Theory, Trace–Class Semigroups, and Spectral Invariants under Drift J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal We develop the heat-kernel layer of the Drifted Spectral Renormalisation Network (DSRN). For nonnegative self-adjoint operators with trace-class heat semigroup we prove that the heat trace is a spectral invariant and therefore rigid under bounded similarity drift. We then recall the small-time heat-trace asymptotics for Laplace-type operators on compact manifolds and identify the resulting Seeley–DeWitt coefficients as spectrally rigid objects. Finally, we formalise where and how effective (scale-dependent) quantities arise only through truncation, projection, and other information-loss operations, in accordance with the DSRN Separation Principle fixed in DSRN XI. Keywords: heat semigroup; heat trace; trace class; spectral invariants; Seeley–DeWitt coefficients; zeta regularisation; Laplace-type operators; drifted similarity; effective truncation. CONTENTS I. Scope and Non-Claims 3 II. Analytic Setting 3 III. Heat Trace as a Spectral Invariant 3 IV. Drift Invariance of the Heat Trace 4 V. Heat Trace Asymptotics and Seeley–DeWitt Coefficients 5 VI. Zeta Regularisation from Heat Traces 6 VII. Where Effective Dynamics Enters 6 A. Truncation as information loss 6 B. Scale dependence 7 C. Projection and coarse-graining 7 ∗jp[email protected] 3 D. Summary 7 A. Sufficient Conditions for Heat-Trace Class 7 B. Notes on Laplace-Type Operators 8 Acknowledgement of Sources 8 References 8 I. SCOPE AND NON-CLAIMS This report fixes the heat-kernel layer of DSRN: (i) analytic hypotheses under which heat traces are well-defined, (ii) invariance of heat traces under drifted similarity families, and (iii) the status of Seeley–DeWitt coefficients as spectrally rigid data. Non-claims. No claim is made that heat-kernel truncations define fundamental dynamics. Any physical or dynamical use of truncated asymptotics is, by construction, an effective procedure and must be treated as such (cf. the DSRN Separation Principle fixed in DSRN XI). II. ANALYTIC SETTING Let H be a separable Hilbert space. Throughout, H denotes a nonnegative self-adjoint operator on H: H=H∗, H ≥0. We write {e−tH}t≥0for the heat semigroup generated by −H. Definition 1 (Heat trace class hypothesis).We say that H is heat-trace class if e−tH is trace class for every t > 0. Remark 2.In geometric settings, e.g. Laplace-type operators on compact manifolds, e−tH is trace class for t > 0and the heat trace admits a small-time asymptotic expansion; see [1,2]. III. HEAT TRACE AS A SPECTRAL INVARIANT We first recall the basic spectral-theoretic fact: whenever the heat trace exists, it depends only on the spectrum (with multiplicities) of H. 4 Theorem 3 (Heat trace depends only on spectrum).Assume H≥ 0is self-adjoint and heat-trace class. Then the scalar function ΘH(t) := Tr(e−tH) (t>0) is determined by the spectral data of H (equivalently, by the spectral measure of H ). Hence Θ H is a spectral invariant. Proof. By the spectral theorem, there exists a projection-valued measure E(λ)such that H=Z[0,∞) λ dE(λ), e−tH =Z[0,∞) e−tλ dE(λ). When e−tH is trace class, Tr ( e−tH )is well-defined and depends only on the spectral measure of H , hence only on spectral data. See [2] for standard details. Remark 4.If H has compact resolvent, then its spectrum is discrete with finite multiplicities accumulating only at +∞, and Tr(e−tH) = X n≥1 e−tλn, where {λn} are eigenvalues listed with multiplicity. This makes the dependence on spectral data explicit. IV. DRIFT INVARIANCE OF THE HEAT TRACE We now connect the heat trace to the drifted similarity framework fixed in DSRN XI. Definition 5 (Drifted similarity family (bounded)).Let {Us}s∈I be a family of bounded invertible operators on Hwith bounded inverses. Define Hs:= U−1 sHUs,Dom(Hs):=U−1 sDom(H). Theorem 6 (Heat-trace rigidity under bounded drift).Let H≥ 0be self-adjoint and heat-trace class. Assume Hs = U−1 sHUs with Us bounded and invertible and bounded inverse. If each Hs is also self-adjoint and heat-trace class,1then Tr(e−tHs) = Tr(e−tH )for all t > 0and all s. 1This holds automatically in many standard settings, but is stated explicitly to avoid hidden assumptions. 5 Proof. Under bounded similarity, the spectra coincide: Spec ( Hs ) = Spec ( H ). By Theorem 3, the heat trace depends only on spectral data. Hence Tr(e−tHs) = Tr(e−tH). Corollary 7 (Rigidity of derived spectral quantities).Any quantity defined uniquely from Θ H ( t ) = Tr(e−tH)(when defined) is invariant under bounded drift. In particular: •Mellin transforms of ΘHdefining zeta functions (where convergent/continued), •coefficients extracted from heat-trace asymptotics (when such asymptotics hold). V. HEAT TRACE ASYMPTOTICS AND SEELEY–DEWITT COEFFICIENTS We now specialise to the standard geometric case where heat-trace asymptotics exist. Let ( M, g ) be a smooth compact Riemannian manifold of dimension d without boundary (or with boundary, with suitable boundary conditions; omitted here for brevity). Let E→M be a smooth vector bundle and let P be a Laplace-type operator on sections of E . Set H := P (or H := P + m2 to ensure strict positivity). Theorem 8 (Small-time heat trace expansion).Let P be a Laplace-type operator on a compact d-manifold. Then, as t↓0, Tr(e−tP )∼(4πt)−d/2 ∞ X n=0 an(P)tn/2, where the coefficients an ( P )are the Seeley–DeWitt coefficients, determined locally by the symbol of Pand the geometry. Remark 9.The coefficients an ( P )can be expressed as integrals over M of universal polynomials in curvature and bundle data. A systematic derivation and explicit formulae are standard; see [1]. Corollary 10 (Spectral rigidity of Seeley–DeWitt coefficients).Assume Ps = U−1 sP Us is a bounded drifted similarity family satisfying the hypotheses of Theorem 6. Then every coefficient an ( Ps )equals an(P). Proof. By Theorem 6, Tr ( e−tPs ) = Tr ( e−tP )for all t > 0. If both admit asymptotic expansions of the form in Theorem 8, equality of the functions implies equality of all coefficients in the asymptotic expansion, hence an(Ps)=an(P). 6 VI. ZETA REGULARISATION FROM HEAT TRACES Heat traces can be used to define spectral zeta functions via Mellin transform. Definition 11 (Spectral zeta function via heat trace).Assume H > 0is self-adjoint and heat-trace class. Define for ℜ(s)large ζH(s) := Tr(H−s) := 1 Γ(s)Z∞ 0 ts−1Tr(e−tH)dt, whenever the integral converges, and extend by analytic continuation when possible. Proposition 12 (Drift invariance of zeta function where defined).Under the hypotheses of Theorem 6, and assuming ζHs is defined by the heat-kernel Mellin transform (with the same continuation procedure), one has ζHs(s)=ζH(s). Proof. The defining Mellin integral depends only on Tr ( e−tH ). By Theorem 6this trace is invariant under drift, hence so is ζH, wherever the definition/continuation is valid. Remark 13.Consequently, zeta-regularised determinants (when defined) are drift-invariant: detζ(H) := exp(−ζ′ H(0)). Standard references include [1,2]. VII. WHERE EFFECTIVE DYNAMICS ENTERS This section implements, in the heat-kernel context, the DSRN Separation Principle fixed in DSRN XI: spectral invariants are rigid; dynamics is effective. A. Truncation as information loss Given the expansion Tr(e−tP )∼(4πt)−d/2X n≥0 an(P)tn/2, any finite truncation Tr(e−tP )≈(4πt)−d/2 N X n=0 an(P)tn/2 discards the remainder O ( t(N+1)/2−d/2 )as t↓ 0. This truncation is an explicit loss of information and defines an effective object. 7 B. Scale dependence Truncations are typically combined with a scale parameter. For example, in spectral-action-type constructions one introduces a cutoff Λand keeps only a finite number of terms in an asymptotic expansion in Λ. Such procedures are, by construction, not spectral invariants unless taken to infinite order. C. Projection and coarse-graining If one replaces H by a subspace Heff (finite-dimensional truncation, low-mode projection, bandlimiting, etc.), then the resulting operator Heff generally does not retain full spectral data of H . Therefore, s-dependence may appear at the effective level even when Spec(Hs)is rigid. D. Summary Any nontrivial s -dependence in quantities derived from heat-kernel technology must be traced to one (or more) of: •truncation of asymptotic expansions, •explicit scale/cutoff dependence, •projection to reduced degrees of freedom, •regularisation choices that discard spectral information. This is the precise sense in which DSRN permits emergent dynamics while preserving spectral rigidity under drift. Appendix A: Sufficient Conditions for Heat-Trace Class We record standard sufficient conditions under which e−tH is trace class for t>0. Proposition 14 (Compact resolvent implies trace-class heat semigroup).Let H≥ 0be self-adjoint with compact resolvent. Then e−tH is trace class for every t > 0. Proof. Compact resolvent implies purely discrete spectrum with finite multiplicities accumulating only at + ∞ . Hence e−tH has eigenvalues e−tλn with λn→ ∞ , and Pne−tλn<∞ for t > 0, proving trace class. 8 Appendix B: Notes on Laplace-Type Operators For Laplace-type operators on compact manifolds, compactness of the resolvent is standard and so is the heat-kernel expansion. See [1,2]. ACKNOWLEDGEMENT OF SOURCES The heat-kernel and spectral invariant background is standard in [ 1 , 2 ]. Semigroup foundations are standard in [3]. [1] P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, 2nd ed., CRC Press, 1995. [2] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989. [3] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983.