Extracting quantum field theory dynamics from an approximate ground state

TL;DR

The paper develops a convex, linear-programming framework to extract dynamical information from static equal-time correlators in quantum field theory. By recasting the Källén–Lehmann inversion into a discretized LP with positivity and normalization constraints, and using a variational ground state provided by Relativistic Continuous Matrix Product States, robust bounds on smeared spectral densities and real-time propagators are obtained, along with a posteriori error estimates. Applied to 1+1D φ^4 theory, the method yields a mass gap in excellent agreement with renormalized Hamiltonian truncation and Borel-resummed perturbation theory across couplings, validating the approach and its potential to recover dynamical data from a single ground-state slice. The work also introduces a mass-gap bootstrap procedure and discusses extensions to multi-operator, matrix-valued spectra and higher dimensions.

Abstract

We develop a linear-programming method to extract dynamical information from static ground-state correlators in quantum field theory. We recast the Källén-Lehmann inversion as a convex optimization problem, in a spirit similar to the recent approach of Lawrence [arXiv:2408.11766]. This produces robust estimates of the smeared spectral density, the real-time propagator, and the mass gap directly from an approximate equal-time two-point function, and simultaneously yields an \emph{a posteriori} lower bound on the correlation-function error. We test the method on the $1+1$-dimensional $φ^4$ model, using a variational approximation to the vacuum -- relativistic continuous matrix product states -- that provides accurate correlators in the continuum and thermodynamic limits. The resulting mass gaps agree with renormalized Hamiltonian truncation and Borel-resummed perturbation theory across a wide range of couplings, demonstrating that accurate dynamical data can be recovered from a single equal-time slice.

Figures (7)

• Figure 1: Left: Two spectral densities with identical one particle state but very different multiparticle continua. Right: Corresponding Euclidean two-point functions obtained via the Källén-Lehmann relation \ref{['eq:KL']}. The absolute difference between both correlators is of the order $10^{-4}$.
• Figure 2: (a)-(c) Top: illustrative projection of convex space of physically allowed spectral densities for the various relaxations/approximations described in the main text. Bottom: Relaxations on $C(x)$ or approximations on $\rho(s)$ of the problem \ref{['eq:LPreduced']}. (a) Approximation by discretizing in $s$, which can be controlled by taking a large number of variables $N_v$ for discretization in $s$. (b) Relaxation of original problem \ref{['eq:linprobinf']} by restricting to finite number of points on the correlator. (c) Relaxation by allowing a certain slack $\delta C$ on correlator.
• Figure 3: Smeared spectral densities ($\sigma^2=0.01$) obtained from the linear program \ref{['eq:LPreduced']} for $\phi^4$-theory at $g=2$. We use $N_c \sim 100$ constraints in the range $x\in[10^{-5},3]$ calculated on an RCMPS ground-state approximation with bond dimension $D=44$. For comparison, we show the mass-gap estimate from renormalized Hamiltonian truncation Elias-Miro:2017xxf, and the expected start of the multiparticle threshold at $9M^2$.
• Figure 4: Retarded propagator $G_R(t,0)$, see Eq. \ref{['eq:cost_realtime']}, obtained by numerically solving the linear program \ref{['eq:LPreduced']} for $\phi^4$-theory at coupling $g=2$ and by bootstrapping the RCMPS slack $\delta C$, until the upper and lower bound coincide. We took $N_c\sim 100$ logaritmically spaced points for the correlation function in the range $x\in [10^{-3},3]$.
• Figure 5: Mass gap estimated with the bootstrap approach for $g=2$ using the RCMPS value of the correlator in the domains $x\in [10^{-5},5]$, $x\in [10^{-5},3]$ and $x\in [10^{-5},1]$, as a function of the bond dimension. On the top right, we plot the corresponding output of the systematic error on RCMPS correlation functions. In gray, we also compare to the carefully extrapolated Renormalized Hamiltonian Truncation results of Elias-Miro:2017xxf. The gray dashed lines correspond to a $2\sigma$ distance from the mean, obtained from the estimated systematics reported in Elias-Miro:2017xxf.