[Paper Review] On Wick Power Series Convergent to Nonlocal Fields

A. G. Smirnov, M. A. Soloviev|ArXiv.org|Apr 4, 2001

Algebraic and Geometric Analysis5 references3 citations

TL;DR

This paper establishes the asymptotic commutativity of nonlocal quantum fields defined by infinite Wick power series of a generalized free field, using analytic properties of vacuum expectation values in x-space and the Cauchy–Poincaré theorem. The convergence is proven for analytic test functions, extending Borchers' equivalence classes and generalizing relative locality to nonlocal fields.

ABSTRACT

The infinite series in Wick powers of a generalized free field are considered that are convergent under smearing with analytic test functions and realize a nonlocal extension of the Borchers equivalence classes. The nonlocal fields to which they converge are proved to be asymptotically commuting, which serves as a natural generalization of the relative locality of the Wick polynomials. The proposed proof is based on exploiting the analytic properties of the vacuum expectation values in x-space and applying the Cauchy--Poincare theorem.

Motivation & Objective

• To extend Borchers' equivalence classes of local fields to nonlocal fields via infinite Wick power series of a generalized free field.
• To establish the convergence of such series under smearing with analytic test functions, particularly in the context of indefinite metric and zero-mass fields.
• To prove that the limiting nonlocal fields are asymptotically commuting, generalizing the relative locality of Wick polynomials.
• To develop a method based on analyticity of vacuum expectation values in x-space, applicable even when the analyticity domain is empty.
• To provide a framework for analyzing nonlocal quantum fields in gauge theories and models with singular UV behavior, including string/M-theory and AdS/CFT contexts.

Proposed method

• Uses the analytic continuation of the two-point function $\mathbf{w}(z)$ to complex space-time, with $u(\tau)$ as an indicator function for growth in imaginary directions.
• Applies the Bargmann–Hall–Wightman theorem to define the extended analyticity domain $\mathbb{T}^{\text{ext}}$ under complex Lorentz transformations.
• Employs the Cauchy–Poincaré theorem to ensure consistency of functional extensions across different regularization parameters $\tau$.
• Estimates integrals over boundary surfaces using decay properties of test functions in generalized Gelfand–Shilov spaces $\mathcal{E}^\beta$ and $S^{1,B}$.
• Uses convexity and growth bounds of the indicator function $\beta$ to control exponential terms and ensure absolute convergence.
• Relies on the monotonicity of $u(\tau)$ and integral estimates over spheres to bound the difference functionals $\mathbf{b}_{V_{n-}}\mathbf{W}^K - \mathbf{b}_{\tau_j V_{n-}}\mathbf{W}^K$.

Experimental results

Research questions

• RQ1 Can infinite Wick power series of a generalized free field converge to nonlocal fields under analytic test functions, even when the analyticity domain of the vacuum expectation values is empty?
• RQ2 Does the limiting nonlocal field satisfy a generalized form of locality, such as asymptotic commutativity, in the absence of strict locality?
• RQ3 How can the convergence of such series be rigorously established using analytic properties of vacuum matrix elements in x-space, rather than momentum-space estimates?
• RQ4 To what extent does this method extend to zero-mass fields and theories with indefinite metric, where standard positivity-based approaches fail?
• RQ5 Can this framework be applied to nonlocal models in string theory, M-theory, and AdS/CFT, particularly in the context of UV finiteness and CPT invariance?

Key findings

• The infinite series $\sum_{k=0}^\infty d_k :\!\phi^k\!:\!(x)$ converge to well-defined nonlocal fields when smeared with analytic test functions in $\mathcal{E}^\beta$ and $S^1$ spaces.
• The limiting nonlocal fields are asymptotically commuting, which generalizes the relative locality of Wick polynomials and ensures compatibility with CPT invariance and spin-statistics theorems.
• The convergence is guaranteed if the coefficient series $\sum_K L^{|K|} |D_K| < \infty$ for all $L > 0$, which holds under the condition $\sum_K L^{|K|} |D_K| < \infty$ derived from Lemma 4 and Theorem 1.
• The indicator function $u(\tau)$, defined as the supremum of $|\mathbf{w}(z)|$ over $\text{Re}\,z = 0$, $\text{Im}\,z = (-\tau, 0, \dots, 0)$, serves as the optimal bound for the growth of $\mathbf{w}(z)$ and is infinitely differentiable and strictly increasing.
• The method applies to generalized free fields in arbitrary dimension $d$, including zero-mass cases, and extends to quasianalytic test function classes not satisfying the strict localizability condition $\int_1^\infty \frac{\beta(s)}{s^2} ds < \infty$.
• The functional extensions to $\mathcal{E}^\beta$ and $S^1$ are consistent across different regularization parameters $\tau$, ensuring the existence of a unique continuous extension of the series.