[Paper Review] Convoluted generalized white noise, Schwinger functions and their continuation to Wightman functions

Sergio Albeverio, Hanno Gottschalk+1 more|ArXiv.org|Sep 21, 2004

Spectral Theory in Mathematical Physics17 references4 citations

TL;DR

This paper constructs Euclidean random fields via convolution of generalized white noise with kernels derived from pseudo-differential operators $(-\Delta + m_0^2)^{-\alpha}$ for $\alpha \in (0,1)$, establishes their Schwinger functions and truncated Schwinger functions, and performs analytic continuation to Wightman functions on Minkowski space. The key contribution is the rigorous derivation of relativistic quantum field theory distributions satisfying spectral, locality, and cluster properties, with partial results on reflection positivity and scattering theory.

ABSTRACT

We construct Euclidean random fields $X$ over $\R^d$, by convoluting generalized white noise $F$ with some integral kernels $G$, as $X=G* F$. We study properties of Schwinger (or moment) functions of $X$. In particular, we give a general equivalent formulation of the cluster property in terms of truncated Schwinger functions which we then apply to the above fields. We present a partial negative result on the reflection positivity of convoluted generalized white noise. Furthermore, by representing the kernels $G_\a$ of the pseudo--differential operators $(-\D + m^2_0)^{-α}$ for $α\in (0,1)$ and $m_0>0$ as Laplace transforms we perform the analytic continuation of the (truncated) Schwinger functions of $X=G_α* F$, obtaining the corresponding (truncated) Wightman distributions on Minkowski space which satisfy the relativistic postulates on invariance, spectral property, locality and cluster property. Finally we give some remarks on scattering theory for these models.

Motivation & Objective

• To construct Euclidean random fields $X = G * F$ by convolving generalized white noise $F$ with integral kernels $G$ derived from $(-\Delta + m_0^2)^{-\alpha}$.
• To study the Schwinger functions of these fields and characterize their cluster property using truncated Schwinger functions.
• To investigate reflection positivity for convoluted generalized white noise fields, yielding a partial negative result.
• To perform analytic continuation of (truncated) Schwinger functions to Wightman distributions on Minkowski space, ensuring relativistic invariance, spectral conditions, and locality.
• To explore scattering theory for the resulting models, particularly in the context of Haag–Ruelle theory and asymptotic behavior of states.

Proposed method

• Represent the kernel $G_\alpha = (-\Delta + m_0^2)^{-\alpha}$ for $\alpha \in (0,1)$ as a Laplace transform to enable analytic continuation.
• Use the Laplace representation to express truncated Schwinger functions as Laplace transforms, facilitating continuation to Minkowski space.
• Apply the analytic continuation procedure to transform Euclidean Schwinger functions into relativistic Wightman distributions satisfying the Wightman axioms.
• Characterize the cluster property in terms of truncated Schwinger functions to analyze long-distance behavior of the fields.
• Analyze the asymptotic behavior of states $\Psi(t)$ under time evolution, showing convergence to free states in the non-overlapping velocity-space case.
• Use Krein space structure to handle indefinite metric in the state space, ensuring consistency with modified Wightman axioms for fields in indefinite metric spaces.

Experimental results

Research questions

• RQ1 Can the Schwinger functions of convoluted generalized white noise fields be analytically continued to Wightman functions on Minkowski space while preserving relativistic invariance and spectral properties?
• RQ2 How does the cluster property manifest in the truncated Schwinger functions of these fields, and can it be characterized in a general form?
• RQ3 Is the reflection positivity property satisfied for convoluted generalized white noise fields, particularly for $\alpha \in (0,1)$?
• RQ4 What is the behavior of scattering states in the asymptotic region, and does the model support a Haag–Ruelle-type scattering theory?
• RQ5 Under what conditions do the Wightman distributions of the model satisfy the full set of Wightman axioms, especially positivity of the norm?

Key findings

• The analytic continuation of truncated Schwinger functions to Wightman distributions is achieved via Laplace representation of the kernel $G_\alpha = (-\Delta + m_0^2)^{-\alpha}$, ensuring relativistic invariance and spectral properties.
• The truncated Schwinger functions with 'sharp masses' are expressed as Laplace transforms, enabling the continuation to Minkowski space and the derivation of explicit Fourier-transformed Wightman distributions.
• For $\alpha = \frac{1}{2}$, the model admits stable one-particle states, allowing the application of Haag–Ruelle theory and establishing positivity of $\|\Psi(t)\|^2$ in the asymptotic region.
• The norm $\|\Psi(t)\|^2$ approaches $\|\Psi(0)\|^2_g$ as $t \to \pm\infty$, with convergence rate $\sim (1+|t|)^{-3/2}$ in general and faster for non-overlapping velocity-space supports.
• The model satisfies all Wightman axioms except for positivity of the norm in the state space, which fails or remains uncertain except when $F$ is Gaussian.
• The Wightman distributions fulfill the Hilbert-structure condition, allowing the construction of a Krein space with a continuous self-adjoint operator $\eta$ satisfying $\eta^2 = 1$ and $<\cdot,\cdot> = (\cdot, \eta \cdot)$ on the dense subspace $\underline{{\cal S}}$.