超静态时空中含时质量标量场Fock量子化中含时标度的判定准则
Criteria for the determination of time dependent scalings in the Fock quantization of scalar fields with a time dependent mass in ultrastatic spacetimes
摘要：
众所周知，对于克莱因-戈登场，存在无限多个正则对易关系的非等价Fock表示，因此也存在无限多个非等价的量子理论。出现这种模糊性并阻止导出鲁棒结果的上下文是，例如，宇宙学扰动的量子分析。在这些情况下，通常情况下，一个适当的缩放场的时间相关的功能导致一个辅助静态背景下的描述，虽然非平稳性仍然显示在一个时间相关的质量。对于这样一个场的描述，并假设空间截面的紧致性，我们最近证明了在三个或更少的空间维度上，空间对称性的自然实现和酉时间演化的标准能够选择唯一一类酉等价真空，从而选择唯一一类Fock表示。在这项工作中，我们成功地扩展我们的唯一性结果的考虑所有可能的字段描述，可以达到一个依赖于时间的正则变换，特别是，涉及一个缩放的字段的时间的函数。这种正则变换修改了系统的动力学，并在其量子描述中引入了进一步的模糊性，超过了Fock表示的选择。值得注意的是，对于任何紧凑的空间流形在小于4维，我们表明，我们的标准消除任何可能的非平凡缩放的字段以外的其他导致在辅助静态背景的描述。此外，我们表明，要么没有依赖于时间的场动量的重新定义是允许的，或者，如果可能发生这种情况，重新定义不引入任何Fock表示，不能得到一个酉变换。
英文摘要：
For Klein-Gordon fields, it is well known that there exist an infinite number of nonequivalent Fock representations of the canonical commutation relations and, therefore, of inequivalent quantum theories. A context in which this kind of ambiguities arises and prevents the derivation of robust results is, e.g., in the quantum analysis of cosmological perturbations. In these situations, typically, a suitable scaling of the field by a time dependent function leads to a description in an auxiliary static background, though the nonstationarity still shows up in a time dependent mass. For such a field description, and assuming the compactness of the spatial sections, we recently proved in three or less spatial dimensions that the criteria of a natural implementation of the spatial symmetries and of a unitary time evolution are able to select a unique class of unitarily equivalent vacua, and hence of Fock representations. In this work, we succeed to extend our uniqueness result to the consideration of all possible field descriptions that can be reached by a time dependent canonical transformation which, in particular, involves a scaling of the field by a function of time. This kind of canonical transformations modify the dynamics of the system and introduce a further ambiguity in its quantum description, exceeding the choice of a Fock representation. Remarkably, for any compact spatial manifold in less than four dimensions, we show that our criteria eliminate any possible nontrivial scaling of the field other than that leading to the description in an auxiliary static background. Besides, we show that either no time dependent redefinition of the field momentum is allowed or, if this may happen, the redefinition does not introduce any Fock representation that cannot be obtained by a unitary transformation.
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