SU（N）格点规范理论的正则变换和环表述
Canonical Transformations and Loop Formulation of SU(N) Lattice Gauge Theories
摘要：
我们构造正则变换，用一组基本环和弦通量算符及其正则共轭环和弦电场来重新表述SU（N）Kogut-Susskind格点规范理论。我们表明，作为SU（N）高斯定律的结果，所有SU（N）弦的自由度成为循环的，并从物理希尔伯特空间${\cal H}^p$解耦。以自洽的方式给出了整个格上初始SU（N）链算符与最终SU（N）圈弦算符之间的正则关系。用基本物理回路算符改写的Kogut-Susskind哈密顿量具有整体SU（N）不变性。没有规范字段。我们进一步表明，$(1/g^2)$磁场条款的格元创建和湮灭的基本格元环通量，而$(g^2)$电场条款描述所有他们的相互作用。在弱耦合（$g^2 \rightarrow 0$）连续极限下，SU（N）圈动力学可用具有最近邻相互作用的SU（N）自旋哈密顿量来描述.在最简单的SU（2）情形下，正则变换将SU（2）圈Hilbert空间映射到氢原子的Hilbert空间，分析了氢原子动力学对称群$SO(4,2)$在圈动力学和谱中的特殊作用.本文讨论了SU（2）规范不变氢原子圈基中的一个简单张量网络模型。
英文摘要：
We construct canonical transformations to reformulate SU(N) Kogut-Susskind lattice gauge theory in terms of a set of fundamental loop & string flux operators along with their canonically conjugate loop & string electric fields. We show that as a consequence of SU(N) Gauss laws all SU(N) string degrees of freedom become cyclic and decouple from the physical Hilbert space ${\cal H}^p$. The canonical relations between the initial SU(N) link operators and the final SU(N) loop & string operators over the entire lattice are worked out in a self consistent manner. The Kogut-Susskind Hamiltonian rewritten in terms of the fundamental physical loop operators has global SU(N) invariance. There are no gauge fields. We further show that the $(1/g^2)$ magnetic field terms on plaquettes create and annihilate the fundamental plaquette loop fluxes while the $(g^2)$ electric field terms describe all their interactions. In the weak coupling ($g^2 \rightarrow 0$) continuum limit the SU(N) loop dynamics is described by SU(N) spin Hamiltonian with nearest neighbour interactions. In the simplest SU(2) case, where the canonical transformations map the SU(2) loop Hilbert space into the Hilbert spaces of hydrogen atoms, we analyze the special role of the hydrogen atom dynamical symmetry group $SO(4,2)$ in the loop dynamics and the spectrum. A simple tensor network ansatz in the SU(2) gauge invariant hydrogen atom loop basis is discussed.
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