[Paper Review] Complex Classical Fields and Partial Wick Rotations

Arthur Jaffe, Christian D. Jäkel+1 more|arXiv (Cornell University)|Feb 24, 2013

advanced mathematical theories25 references3 citations

TL;DR

This paper introduces a framework for complex classical scalar fields via partial Wick rotation, generalizing the Feynman-Kac relation to non-Gaussian models and establishing the spectral condition on compactified spacetime. It proves the spectrum condition for ${\mathscr{P}}(\varphi)_2$-models on a spatial circle using reflection positivity and analyticity, and extends the formalism to finite-temperature quantum fields with thermal KMS conditions.

ABSTRACT

We study some examples of complex, classical, scalar fields within the new framework that we introduced in a previous work. In these particular examples, we replace the usual functional integral by a complex functional arising from partial Wick rotation of a quantum field. We generalize the Feynman-Kac relation to this setting, and use it to establish the spectral condition on a cylinder. We also consider positive-temperature states.

Motivation & Objective

• To develop a classical field framework for quantum fields with complex actions arising from partial Wick rotation.
• To generalize the Feynman-Kac formula to non-Gaussian, complex-functional integral settings without a positive measure.
• To establish the spectral condition $H_{\vec{v}} \geq 0$ for ${\mathscr{P}}(\varphi)_2$-models on a spatial circle using reflection positivity.
• To extend the formalism to positive-temperature quantum fields and analyze thermal KMS conditions.
• To connect complex classical fields to Wightman functions and analyticity in the forward tube via the Flat Tube Theorem.

Proposed method

• Replace the real functional integral with a complex functional derived from partial Wick rotation of a quantum field Hamiltonian.
• Use a generalized Feynman-Kac formula to define expectation values for non-quadratic interactions without a positive measure.
• Apply double reflection positivity with two orthogonal reflection planes to derive spectral bounds.
• Construct thermal states via compactified time $S^1 \times \mathbb{R}^{d-1}$ and analyze analyticity in complex time strips.
• Use the Flat Tube Theorem to extend Wightman functions to complex time domains, ensuring analyticity and KMS properties.
• Relate boosted thermal two-point functions to the original KMS condition via Lorentz transformation and rescaling.

Experimental results

Research questions

• RQ1 Can the Feynman-Kac relation be generalized to complex, non-Gaussian functionals arising from partial Wick rotation?
• RQ2 Does the spectral condition $H_{\vec{v}} \geq 0$ hold for ${\mathscr{P}}(\varphi)_2$-models on a spatial circle without Lorentz symmetry?
• RQ3 How do complex classical fields relate to thermal equilibrium states and the relativistic KMS condition?
• RQ4 Can analyticity of Wightman functions be established in complex time strips using reflection positivity and modified Schwinger functions?
• RQ5 What is the role of double reflection positivity in connecting classical complex fields to quantum field theories with positive spectrum?

Key findings

• The generalized Feynman-Kac formula allows the definition of expectation values for non-Gaussian, complex actions without a positive measure.
• The spectral condition $H_{\vec{v}} \geq 0$ is proven for ${\mathscr{P}}(\varphi)_2$-models on a spatial circle using double reflection positivity and analyticity.
• Modified Schwinger functions defined via $\exp(-t\cosh\beta\, H_{\vec{v}})$ extend to the forward tube $\mathcal{T}^{n-1}$, ensuring analyticity.
• For thermal fields on $S^1 \times \mathbb{R}^{d-1}$, the two-point function is analytic in the strip $|\Im(t-t')| < \beta/(1+|\vec{v}|)$, satisfying the KMS condition.
• The two-point function of a Lorentz-boosted thermal field matches the KMS condition at inverse temperature $\sqrt{1-\vec{v}^2}\beta$, confirming relativistic KMS behavior.
• The formalism realizes a quantum field theory with positive spectrum and thermal equilibrium properties via complex classical fields and partial Wick rotation.