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辐射主导与物质主导阶段中带规范固定项的麦克斯韦场：精确解与应力张量

Maxwell field with gauge fixing term in the radiation- and matter-dominant stages: exact solution and stress tensor

Xuan Ye, Yang Zhang

备注：
19 pages, add references, published
发表信息：
IntJMPD , 2450050 (2024) (21 pages)
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AI总结：

本文在辐射主导和物质主导阶段研究带规范固定项的麦克斯韦场，推导精确解并证明规则化真空应力张量为零，仅保留横向光子部分，且结果与规范固定项无关。

摘要：

作为先前在德西特空间中工作的延续，我们研究了膨胀宇宙辐射主导（RD）和物质主导（MD）阶段中带有一般规范固定（GF）项的麦克斯韦场。我们推导了精确解，进行了协变正则量子化，并得到了Gupta-Bleuler（GB）物理态中的应力张量，该应力张量与GF常数无关，并且在量子残余规范变换下保持不变。横向应力张量在所有平坦Robertson-Walker时空中是相似的，其真空部分正比于$k^4$，并在0阶绝热正则化后变为零。在RD和MD两个阶段中，纵向-时间应力张量由于GB态中纵向部分和时间部分之间的抵消而为零，GF应力张量的粒子部分也为零。在RD阶段，真空GF应力张量包含$k^4,k^2$发散，并通过2阶正则化变为零；然而，在MD阶段，它包含$k^4, k^2, k^0$发散，并通过4阶正则化变为零。因此，适当正则化的阶数不仅取决于场的类型，还取决于背景时空。总之，在RD和MD两个阶段中，如同在德西特空间中一样，总的规则化真空应力张量为零，仅保留横向光子部分，并且这一结论与GF项无关。

英文摘要：

We study the Maxwell field with a general gauge fixing (GF) term in the radiation-dominant (RD) and matter-dominant (MD) stages of expanding Universe, as a continuation to the previous work in de Sitter space. We derive the exact solutions, perform the covariant canonical quantization and obtain the stress tensor in the Gupta-Bleuler (GB) physical states, which is independent of the GF constant and is also invariant under the quantum residual gauge transformation. The transverse stress tensor is similar in all flat Robertson-Walker spacetimes, and its vacuum part is $\propto k^4$ and becomes zero after the 0th-order adiabatic regularization. The longitudinal-temporal stress tensor, in both RD and MD stages, is zero due to a cancelation between the longitudinal and temporal parts in the GB states, and so is the particle part of the GF stress tensor. The vacuum GF stress tensor, in the RD stage, contains $k^4,k^2$ divergences and becomes zero by the 2nd-order regularization, however, in the MD stage, contains $k^4, k^2, k^0$ divergences and becomes zero by the 4th-order regularization. So, the order of adequate regularization depends not only upon the type of fields, but also upon the background spacetimes. In summary, in both the RD and MD stages, like in de Sitter space, the total regularized vacuum stress tensor is zero, only the transverse photon part remains, and this holds independent of the GF term.

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