We introduce the quadratic analog of the tensor Bogolyubov representation of the CCR. Our main result is the determination of the structure of these maps: each of them is uniquely determined by two arbitrary complex-valued Borel functions of modulus [Formula: see text] and two maps of [Formula: see text] into itself whose inverses induce transformations that map the Lebesgue measure [Formula: see text] into measures [Formula: see text] absolutely continuous with respect to it. Furthermore, the Radon–Nikodyn derivatives [Formula: see text], of these measures with respect to [Formula: see text], must satisfy the relation [Formula: see text] for [Formula: see text]-almost every [Formula: see text]. This makes a surprising bridge with the hyperbolic sine and cosine defining the structure of usual (i.e. first-order) Bogolyubov transformations. The reason of the surprise is that the linear and quadratic commutation relations are completely different.

中文翻译：

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重新归一化的白噪声平方 (RSWN) 代数的张量 Bogolyubov 表示

我们介绍了 CCR 的张量 Bogolyubov 表示的二次模拟。我们的主要结果是确定这些映射的结构：它们中的每一个都由模数 [公式：参见文本] 的两个任意复值 Borel 函数和 [公式：参见文本] 的两个映射唯一地确定到自身，其逆诱导将 Lebesgue 测度 [公式：参见文本] 映射为相对于它绝对连续的测度 [公式：参见文本] 的转换。此外，Radon-Nikodyn 导数 [Formula: see text]，关于 [Formula: see text] 的这些测度，必须满足 [Formula: see text] 的关系 [Formula: see text] - 几乎每个 [Formula: see text]见正文]。这与定义通常（即一阶）Bogolyubov 变换结构的双曲正弦和余弦形成了令人惊讶的桥梁。