论文标题
欧几里得$ ϕ^4_2 $理论是相互作用的bose气体的限制
The Euclidean $ϕ^4_2$ theory as a limit of an interacting Bose gas
论文作者
论文摘要
我们证明，当气体的密度变大并且相互作用的范围很小时，我们以二维为二维的复杂欧几里得场理论是在二维中以二维的限制。田间理论受到负面规律性的分布的支持，这需要通过不同的质量和能量反应来重新归一化。我们获得了重新归一化的还原密度矩阵的相对分区函数和均匀收敛的收敛性。 The proof is based on three main ingredients: (a) a quantitative analysis of the infinite-dimensional saddle point argument for the functional integral introduced in [32] using continuity properties of Brownian paths, (b) a Nelson-type estimate for a general nonlocal field theory in two dimensions, and (c) repeated Gaussian integration by parts in field space to obtain uniform control on the renormalized correlation functions.作为我们证明的副产品，在两个和三个维度上，我们还将平均场限制的结果从[32,56]扩展到满足波尔加因提出的最佳可集成性条件的无界相互作用势[13]。
We prove that the complex Euclidean field theory with local quartic self-interaction in two dimensions arises as a limit of an interacting Bose gas at positive temperature, when the density of the gas becomes large and the range of the interaction becomes small. The field theory is supported on distributions of negative regularity, which requires a renormalization by divergent mass and energy counterterms. We obtain convergence of the relative partition function and uniform convergence of the renormalized reduced density matrices. The proof is based on three main ingredients: (a) a quantitative analysis of the infinite-dimensional saddle point argument for the functional integral introduced in [32] using continuity properties of Brownian paths, (b) a Nelson-type estimate for a general nonlocal field theory in two dimensions, and (c) repeated Gaussian integration by parts in field space to obtain uniform control on the renormalized correlation functions. As a byproduct of our proof, in two and three dimensions we also extend the results on the mean-field limit from [32,56] to unbounded interaction potentials satisfying the optimal integrability conditions proposed by Bourgain [13].
