Mathematical characterization of the physical vacuum for a linear Bose-Einstein field
Mathematical characterization of the physical vacuum for a linear Bose-Einstein field
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线性玻色-爱因斯坦场物理真空的数学表征
DOI：
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发表时间：
1962
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通讯作者：
I. Segal
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作者：
I. Segal
物理场的数学处理是量子场论中最具挑战性的基本问题之一。在“相互作用”场的CSE中，这个问题实质上是基本理论的表述之一，但在“自由”场的CSE中，由于它们的结构与“相互作用”表示中的相互作用场的结构在数学上的同一性而在物理上得到了很好的启发，在“自由”场的CSE中，这个结构得到了经验证明，根据通常的假设，在+/-时代，必要的基本公式现在就在手边。从目前最保守和最笼统的立场来看，这样的场在数学上是一个某种程度上结构化的C*-代数，是“可观测的”(可表示，但不是以任何唯一的方式表示为一致封闭的自我：希尔伯特空间上有界线性算子的伴随代数)，而物理状态是该代数上的某些线性形式，具有与给定状态相对应的期望值形式的传统解释。这一假设完全独立于关于时空性质或理论的群不变性的任何假设。本文讨论如何用严格的数学和物理意义来刻画这类一般类型的场的物理真空的问题。理论物理中将真空定义为“最低能量状态”的传统表述可以立即用数学方法来表述，但这种表述方式并不独特，除非用于某些形式上的目的，否则是无效的。众所周知的量子场论的分歧本质上意味着传统的相互作用场理论中的能量在数学上是高度模糊的。我们不太熟悉，但同样麻烦的是，假定能量在其上作为自伴算子的希尔伯特空间在传统理论中没有明确的表述。这两个困难都与量子场[1]的正则变量存在许多不等价的表示有关，尽管在自由场的相对透明的情况下没有非平凡的发散。然而，对自由场的传统描述要么涉及数学上的模棱两可，要么涉及缺乏物理解释的技术要求。上述更保守的方法避免了由于
The mthemticl treatment of the physical vcuum is one of the most challenging nd bsic problems of quantum field theory. In the cse of "interacting" fields this problem is in substantial prt one of the formulation of the underlying theory, but in the cse of "free" fieldswhich re theoretically enlightening s well s physically relevant by virtue of the mthemticl identity of their structure with that of an interacting field in the "interaction" representation t a prticulr time, this structure being empirically mnifested, according to the usual postulates, at the times +/-the necessary basic formulations are now at hand. To take the presently most conservative and general position, such a field is mathematically a somewhat structured C*-algebra of "observables" (representable, but not at all in any unique way, as a uniformly closed self:adjoint algebra of bounded linear operators on a Hilbert space), and the physical states are certain linear forms on this algebra, having the conventional interpretation of the expectation value form corresponding to the given state. This assumption is entirely independent of any assumptions as to the nature of space-time, or groupinvariance of the theory. The present paper is concerned with the problem of characterizing, in terms which are both mathematically rigorous and physically meaningful, the physical vacuum for such general types of field. The conventional formulation of the vacuum in theoretical physics as "the state of lowest energy" can be immediately transcribed mathematically, but in such a nonunique manner as to be ineffective except for certain formal purposes. The well-known divergences of quantum field theory signify essentially that the energy in conventional theories of interacting fields is mathematically highly ambiguous. It is less familiar, but equally troublesome, that the Hilbert space on which the energy is supposed to uct as self-adjoint operator, has no explicit formulation in the conventionul theories. Both of these difficulties are connected with the existence of many inequivalent representations for the canonical variables o a quantum field [1], although in the relatively transparent case of a free field there are no nontrivial divergences. Nevertheless, the conventional description of the free field involves either mathematical ambiguity or technical requirements lacking in physical interpretation. The more conservative approach described above avoids the problem posed by the ambiguity of the