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.