Based on conservation laws for surface layer integrals for critical points of causal variational principles, it is shown how jet spaces can be endowed with an almost-complex structure. We analyze under which conditions the almost-complex structure can be integrated to a canonical complex structure. Combined with the scalar product expressed by a surface layer integral, we obtain a complex Hilbert space $(\mathfrak{h}, \langle . \vert . \rangle)$. The Euler–Lagrange equations of the causal variational principle describe a nonlinear time evolution on $\mathfrak{h}$. Rewriting multilinear operators on $\mathfrak{h}$ as linear operators on corresponding tensor products and using a conservation law for a nonlinear surface layer integral, we obtain a linear norm-preserving time evolution on bosonic Fock spaces. The so-called holomorphic approximation is introduced, in which the dynamics is described by a unitary time evolution on the bosonic Fock space. The error of this approximation is quantified. Our constructions explain why and under which assumptions critical points of causal variational principles give rise to a second-quantized, unitary dynamics on Fock spaces.

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因果变分原理的喷射空间复杂结构和Bosonic Fock空间动力学

基于因果变分原理的临界点的表面层积分的守恒律，表明了如何赋予射流空间一种几乎复杂的结构。我们分析了在什么条件下几乎复杂的结构可以集成到规范的复杂结构中。结合表面层积分表示的标量积，我们得到了一个复杂的希尔伯特空间$（\ mathfrak {h}，\ langle。\ vert。\ rangle）$。因果变分原理的Euler-Lagrange方程描述了$ \ mathfrak {h} $上的非线性时间演化。将$ \ mathfrak {h} $上的多线性算子重写为相应张量积的线性算子，并使用非线性表层积分的守恒律，我们得到了Bosonic Fock空间上线性守恒的时间演化。介绍了所谓的全纯近似，其中动力学是通过在玻色子Fock空间上的a时间演化来描述的。该近似误差被量化。我们的构造解释了为什么以及在哪些假设下因果变分原理的临界点在Fock空间上产生了第二量化的unit动力。