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Abstract:We establish a rigorous non-perturbative closure framework for the three-dimensional (3D) $\phi^4$ scalar field theory near criticality, bypassing the long-standing limitations of perturbative Feynman diagrammatic expansions. By slicing the 3D Euclidean space along a spatial axis, the exact dressed theory is mapped onto a matrix operator evolution governed by the infinite-dimensional symplectic Lie algebra $\mathfrak{sp}(\infty)$. We generalize the classical Stroh formalism and Barnett-Lothe integral invariants from anisotropic elasticity to the quantum field Hilbert space. Crucially, we prove that the dressed operator-valued Barnett-Lothe tensors satisfy the algebraic identity $\hat{\mathbf{S}}^2 + \hat{\mathbf{H}}\hat{\mathbf{L}} = -\hat{\mathbf{I}}$ exactly and non-perturbatively, regardless of the coupling strength. By coupling this symplectic invariance with the Källén-Lehmann spectral representation and the Schwinger-Dyson equations, a novel "Symplectic Bootstrap" master spectral integral equation is formulated. The framework exhibits exact dimensional reduction, naturally degenerating to the Onsager exact solution ($\eta = 1/4$) in 2D and the Gaussian triviality limit ($\eta = 0$) in 4D. Solving the transcendental bootstrap equation under the strong-coupling conformal fixed point yields the non-perturbative anomalous dimension $\eta \approx 0.0363$, matching the state-of-the-art conformal bootstrap numerical limits. Finally, a profound holographic duality between the statistical state equations and the post-buckling bifurcation of fluctuating soft matters is discussed.

Submission history

From: Yu-Xin Xie [view email]
[v1] Wed, 17 Jun 2026 03:13:53 UTC (8 KB)