非交换量子场论的SO(1,1)×SO(2)公理化：新定理与应用

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本文主要探讨了非交换量子场论（Noncommutative Quantum Field Theory, NCQFT）的公理化表达，特别是在相对论框架下的进一步发展。首先，作者在《核物理学B》(Nuclear Physics B) 的950卷2020期114846号文章中，回顾了在SO(1,1)×SO(2)不变性的量子场论背景下，如何成功地证明了场算符的不可约性，这是量子场论基石之一，意味着理论中的场不能进一步分解为更简单的组成部分。接着，文章深入研究了广义Haag定理的推导和证明。Haag定理是量子场论中的重要概念，它关系到粒子间的相互作用以及不同区域之间的独立性。在非交换量子场论中，这个定理的推广揭示了新的物理现象和边界条件，这些对于理解量子系统的全局性质至关重要。 Reeh-Schlieder定理，另一个核心的量子场论原理，也得到了验证和独特解读。该定理确保了真空态在整个希尔伯特空间中具有稠密的存在性，这意味着任何可观测量都可以通过对真空态进行操作来得到。在非交换量子场论中，这一特性可能需要重新审视，因为它可能会受到量子纠缠和其他非经典效应的影响。在SO(1,3)不变性理论中，作者发现对广义Haag定理的运用产生了更具深度的结果。这表明，与传统的交换量子场论相比，非交换性可能导致不同的空间时间结构和物理规律，从而对四点Wightman函数（用于描述基本粒子相互作用的数学工具）的等式性提出了新的挑战或限制。总结来说，这篇论文不仅深化了我们对非交换量子场论的理解，还提供了关于量子场论基础定理在非经典情况下的新见解。它为构建一个更加全面和严谨的非交换量子场论框架做出了贡献，这对于理论物理学，特别是量子引力和量子宇宙学等领域具有重要的理论意义。读者可以在线访问ScienceDirect或Elsevier的网站获取全文，以便进一步探索这些新颖的数学和物理成果。

4 M. Chaichian et al. / Nuclear Physics B 950 (2020) 114846

is NC QFT. We prove that classical results, such as the irreducibility of the set of ﬁeld operators,

the theorems of Reeh and Schlieder [1–4] remain valid in the noncommutative case. It should

be emphasized that the results obtained in this paper do not depend on the SO(2) invariance of

the theory in the v

ariables x

and x

and therefore can be extended to more general cases. The

irreducibility of the set of ﬁeld operators remain valid in any theory, which is translation invariant

in commutative variables, if only eq. (27)is fulﬁlled. The ﬁrst theorem of Reeh and Schlieder is

valid, if the Wightman functions are analytical in the v

ariables x

and x

in the primitive domains

of analyticity (“tubes”).

In the SO

(1, 3) invariant theory new consequences of the generalized Haag’s theorem are

found, without analogues in NC QFT. At the same time it is proven that the basic physical

conclusion of Haag’s theorem is valid also in the SO(1, 1) invariant theory, and it is sufﬁcient

that spectrality

, local commutativity condition and translational invariance be fulﬁlled only for

the transformations concerning the commutating coordinates. The analysis of Haag’s theorem

reveals essential distinctions between commutative and noncommutative cases, more precisely

between the SO(1, 3) and SO(1, 1) invariant theories. In the commutative case, the conditions

(59) and (60), whose consequence is generalized Haag’

s theorem, lead to the equality of Wight-

man functions in two theories up to four-point ones. In the present paper it is shown that in the

SO(1, 1) invariant theory, unlike the commutative case, only two-point Wightman functions are

equal and it is sho

wn that from the equality of two-point Wightman functions in two theories it

follows that if in one of them the current is equal to zero, it is equal to zero in the other as well

and under weaker conditions than the standard ones. It is also shown that for the deri

vation of

eq. (60)it is sufﬁcient to assume that the vacuum vector is a unique normalized vector, invariant

under translations along the axis x

. It is proven that from the equality of four-point Wightman

functions in two theories, the equality of their elastic scattering amplitudes follows and, owing

to the optical theorem, the equality of total cross sections as well. In derivation of this result the

local commutativity (LCC) is not used.

The study of W

ightman functions leads still to new nontrivial consequences also in the com-

mutative case.

The paper is arranged as follows. In section 2 the basic properties of Wightman functions in

space-space NC QFT are formulated; in section 3 the irreducibility of the set of ﬁeld operators

is proven; in section 4 generalizations of the theorems of Reeh and Schlieder to NC QFT are

obtained; section 5 is devoted to generalized Haag’

s theorem; in section 6 it is shown that in the

commutative case, the conditions of weak local commutativity (WLCC) and of local commuta-

tivity (LCC), which are valid in the noncommutative case ((24) and (22)), appear to be equivalent

to the usual WLCC and LCC, respectively.

2. Basic properties of Wightman functions in space-space NC QFT

As in the commutative case, we assume that every vector from the space of the complete set

of all physical states, J , can be approximated with arbitrary accuracy by the vectors of the

··· ϕ



. (8)

In other words the vacuum vector 

is cyclic, i.e. the axiom of cyclicity of vacuum is fulﬁlled.

A partial result on the subject had been previously communicated in [40].

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