论文标题
量子动力学的仪表图片
The Gauge Picture of Quantum Dynamics
论文作者
论文摘要
尽管当地的汉密尔顿人表现出当地时间动力学，但在施罗丁（Schrödinger）图片中，这种位置并不明确，因为波函数振幅不遵守局部运动方程。我们表明，通过将量子力学的全球单一不变性计入局部仪表不变性，可以在运动方程中明确实现几何局部性。也就是说，期望值$ \langleψ| a |ψ\ rangle $在全球统一转换下是不变的，该转换作用于波函数$ |ψ\ rangle \ to u | cin \ rangle $和操作员$ a \ u a \ u a a u a a u a u a u a u a u^\ dgagger $，我们可以向我们表明，可以衡量这个本地的范围。为此，我们用局部波函数的集合$ |ψ_j\ rangle $替换了波函数，每个空间$ j $一个。选择空间斑块的集合以覆盖空间。例如我们可以选择贴片是单量子位或晶格上最近的邻居站点。与相邻的空间补丁对相关的本地波形$ i $和$ j $通过动态统一转换$ u_ {ij} $相互关联。局部波形是局部的，因为它们的动态是局部的。也就是说，本地波函数的运动方程$ |ψ_j\ rangle $ and connections $ u_ {ij} $在太空中明确是本地的，仅取决于附近的汉密尔顿术语。 （局部波形是多体波函数，具有与通常的波函数相同的希尔伯特空间维度。）我们称这张量子动力学的图片为量规图片，因为它表现出局部仪表不变性。单个空间斑块的局部动力学与相互作用图片有关，在该图片中，汉密尔顿的相互作用仅由附近的汉密尔顿术语组成。我们还可以将显式区域概括为包括本地电荷和能量密度的地方。
Although local Hamiltonians exhibit local time dynamics, this locality is not explicit in the Schrödinger picture in the sense that the wavefunction amplitudes do not obey a local equation of motion. We show that geometric locality can be achieved explicitly in the equations of motion by "gauging" the global unitary invariance of quantum mechanics into a local gauge invariance. That is, expectation values $\langle ψ|A|ψ\rangle$ are invariant under a global unitary transformation acting on the wavefunction $|ψ\rangle \to U |ψ\rangle$ and operators $A \to U A U^\dagger$, and we show that it is possible to gauge this global invariance into a local gauge invariance. To do this, we replace the wavefunction with a collection of local wavefunctions $|ψ_J\rangle$, one for each patch of space $J$. The collection of spatial patches is chosen to cover the space; e.g. we could choose the patches to be single qubits or nearest-neighbor sites on a lattice. Local wavefunctions associated with neighboring pairs of spatial patches $I$ and $J$ are related to each other by dynamical unitary transformations $U_{IJ}$. The local wavefunctions are local in the sense that their dynamics are local. That is, the equations of motion for the local wavefunctions $|ψ_J\rangle$ and connections $U_{IJ}$ are explicitly local in space and only depend on nearby Hamiltonian terms. (The local wavefunctions are many-body wavefunctions and have the same Hilbert space dimension as the usual wavefunction.) We call this picture of quantum dynamics the gauge picture since it exhibits a local gauge invariance. The local dynamics of a single spatial patch is related to the interaction picture, where the interaction Hamiltonian consists of only nearby Hamiltonian terms. We can also generalize the explicit locality to include locality in local charge and energy densities.
