Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lindblad-Deformed Spectral Geometry: Heat-Kernel Asymptotics and Effective Spectral Dimension

Soumadeep Maiti

Abstract

We introduce a Lindblad-deformed spectral geometric framework in which bounded dissipative data deform a standard spectral triple through the Dirac operator D_gamma = D - igammaSigma, where Sigma = (1/2) sum_k L_k^dagger L_k is constructed from Lindblad jump operators {L_k}. The associated positive operator Q_gamma = D_gamma^* D_gamma = D^2 + gamma^2 Sigma^2 - i*gamma [D, Sigma] is identified as the correct spectral-geometric observable. For smooth endomorphism-valued Lindblad data, Q_gamma is of Laplace type and admits a standard heat-kernel asymptotic expansion with dissipation-modified even Seeley-DeWitt coefficients. For the scalar deformation L = sqrt(gamma) f with f in C^infty(M) real-valued, we prove that the first-order Duhamel correction to the heat trace K_gamma(sigma) = Tr(exp(-sigma Q_gamma)) vanishes identically, so that the first nontrivial dissipative effect appears at order gamma^4. We identify the exact Duhamel-level decomposition of the O(gamma^4) correction into a direct W_2 insertion and a quadratic W_1 x W_1 term. In the round S^2 model we determine the explicit deformed operator and extract the leading local asymptotic contribution of the W_2 sector. We define the effective scale-dependent spectral dimension d_{s,eff}(sigma,gamma) = -2 d/d(log sigma) log K_gamma(sigma) and identify its leading perturbative deformation.

Lindblad-Deformed Spectral Geometry: Heat-Kernel Asymptotics and Effective Spectral Dimension

Abstract

We introduce a Lindblad-deformed spectral geometric framework in which bounded dissipative data deform a standard spectral triple through the Dirac operator D_gamma = D - igammaSigma, where Sigma = (1/2) sum_k L_k^dagger L_k is constructed from Lindblad jump operators {L_k}. The associated positive operator Q_gamma = D_gamma^* D_gamma = D^2 + gamma^2 Sigma^2 - i*gamma [D, Sigma] is identified as the correct spectral-geometric observable. For smooth endomorphism-valued Lindblad data, Q_gamma is of Laplace type and admits a standard heat-kernel asymptotic expansion with dissipation-modified even Seeley-DeWitt coefficients. For the scalar deformation L = sqrt(gamma) f with f in C^infty(M) real-valued, we prove that the first-order Duhamel correction to the heat trace K_gamma(sigma) = Tr(exp(-sigma Q_gamma)) vanishes identically, so that the first nontrivial dissipative effect appears at order gamma^4. We identify the exact Duhamel-level decomposition of the O(gamma^4) correction into a direct W_2 insertion and a quadratic W_1 x W_1 term. In the round S^2 model we determine the explicit deformed operator and extract the leading local asymptotic contribution of the W_2 sector. We define the effective scale-dependent spectral dimension d_{s,eff}(sigma,gamma) = -2 d/d(log sigma) log K_gamma(sigma) and identify its leading perturbative deformation.

Paper Structure

This paper contains 15 sections, 5 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Heat kernels and Seeley--DeWitt asymptotics
  4. Existing generalisations
  5. The Lindblad deformation and the operator $\mathcal{Q}_\gamma$
  6. The triviality of the naive Lindblad heat trace
  7. Heat-kernel asymptotics for $\mathcal{Q}_\gamma$
  8. Scalar power counting
  9. Scalar Lindblad deformation and the vanishing proposition
  10. Structure of the first nonzero correction
  11. The $S^2$ toy model
  12. Effective scale-dependent spectral dimension
  13. Discussion and outlook
  14. Derivation of $\mathcal{Q}_\gamma$: sign verification
  15. Duhamel expansion to order $\gamma^4$

Key Result

Proposition 3.2

Under (H1)--(H2), the operator $\mathcal{D}_\gamma := \mathcal{D} - i\gamma\Sigma$ is closed on $\operatorname{Dom}(\mathcal{D})$ with adjoint $\mathcal{D}_\gamma^* = \mathcal{D} + i\gamma\Sigma$, and the positive self-adjoint operator $\mathcal{Q}_\gamma := \mathcal{D}_\gamma^*\mathcal{D}_\gamma$ i $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (13)

  • Definition 2.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 5.1: Vanishing of the first Duhamel correction
  • proof
  • Remark 5.2
  • Corollary 5.3
  • Corollary 5.4: Coefficient-level rigidity in the scalar case
  • proof
  • ...and 3 more