arXiv 1902.002622019-02gr-qc

混合圈量子宇宙学中费米子哈密顿量的渐近对角化

Asymptotic diagonalization of the fermionic Hamiltonian in hybrid loop quantum cosmology

Beatriz Elizaga Navascués, Guillermo A. Mena Marugán, Santiago Prado

备注：: 12 pages, matches published version
发表信息：: Phys. Rev. D 99, 063535 (2019)
DOI：: 10.1103/PhysRevD.99.063535

摘要：

我们利用混合圈量子宇宙学的自由度，分裂几何场和物质场的自由度，从而建立一个具有良好量子性质的物质场的量子场论。我们研究这个问题，在一个膨胀，平坦的宇宙学与非均匀扰动，并集中讨论狄拉克场，最低限度地耦合到宇宙学背景，并作为一个扰动。在截断扰动中最低非平凡阶的作用之后，必须为物质内容定义正则变量，为此通常采用混合均匀背景和扰动的正则变换。这些可能的定义中的每一个都与对完整系统的哈密顿量的不同物质贡献相关联，一般来说，完整系统可能包含类创生变量和类湮灭变量的二次项，以及随后的费米子粒子和反粒子对的产生和毁灭。我们确定的费米子正则变量的选择，其中的相互作用部分的哈密顿量可以忽略不计，如所需的大粒子/反粒子波数的渐近制度。最后，我们研究了这种选择的量子动力学，对量子态施加总的哈密顿约束，并假设它们的引力部分不受费米子的存在的显着影响。这样，我们得到了一个薛定谔方程的费米子自由度的量子期望值的几何，导致渐近对角海森堡关系和Bogoliubov演化变换，在相关的正常有序哈密顿没有分歧。

英文摘要：

We use the freedom available in hybrid loop quantum cosmology to split the degrees of freedom between the geometry and the matter fields so as to build a quantum field theory for the matter content with good quantum properties. We investigate this issue in an inflationary, flat cosmology with inhomogeneous perturbations, and focus the discussion on a Dirac field, minimally coupled to the cosmological background and treated as a perturbation. After truncating the action at the lowest nontrivial order in perturbations, one must define canonical variables for the matter content, for which one generally employs canonical transformations that mix the homogeneous background and the perturbations. Each of these possible definitions comes associated with a different matter contribution to the Hamiltonian of the complete system, that may, in general, contain terms that are quadratic in creationlike variables, and in annihilationlike variables, with the subsequent production and destruction of pairs of fermionic particles and antiparticles. We determine a choice of the fermionic canonical variables for which the interaction part of the Hamiltonian can be made as negligible as desired in the asymptotic regime of large particle/antiparticle wave numbers. Finally, we study the quantum dynamics for this choice, imposing the total Hamiltonian constraint on the quantum states and assuming that their gravitational part is not affected significantly by the presence of fermions. In this way, we obtain a Schrödinger equation for the fermionic degrees of freedom in terms of quantum expectation values of the geometry that leads to asymptotically diagonal Heisenberg relations and Bogoliubov evolution transformations, with no divergences in the associated normal-ordered Hamiltonian.

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