森林火灾蔓延的时空几何图与量子导航
A Spacetime Geometry picture of Forest Fire Spreading and of Quantum Navigation
摘要：
在定常洛伦兹时空中寻找零测地线的问题被认为等价于寻找一个兰德斯-芬塞尔结构的测地线。这后一个问题相当于找到在背景磁场中的黎曼流形上运动的带电粒子的运动，或者等价地，通过费马原理的推广，到策梅罗的问题，即在风的存在下使飞机的旅行时间极值化。在本文中，这三个等价扩展到包括最近的森林火灾的蔓延模型，使用惠更斯原理的形式。该构造还可以用于解决量子控制理论中的问题，其中人们寻求在最短时间内将具有其自身哈密顿量的量子力学系统的初始状态带到期望的最终状态的控制哈密顿量。相关的静止时空可以被认为是定义在一个扩展的量子相空间（苏里奥演化空间）上，量子凝视的空间是一个复射影空间，配备了它的Fubini-Study Kähler度规。这种时空观点可能会为我们理解量子引力提供相关的见解。
英文摘要：
The problem of finding null geodesics in a stationary Lorentzian spacetime is known to to be equivalent to finding the geodsics of a Randers-Finlser structure. This latter problem is equivalent to finding the motion of charged particles moving on a Riemannian manifold in a background magnetic field or equivalently, by a generalization of Fermat's principle, to Zermelo's problem of extremizing travel time of an aeroplane in the presence of a wind. In this paper this triad of equivalences is extended to include recent model of the spread of a forest fire which uses form of Huyghen's principle. The construction may also be used to solve a problem in quantum control theory in which one seeks a control Hamiltonia taking an initial state of a quantum mechanical system with its own Hamiltonian to a desired final state in least time. The associated stationary spacetime may be thought of as defined on an extended quantum phase space (Souriau's evolution space), the space of quantum stares being complex projective space equipped with its Fubini-Study Kähler metric. It is possible that this spacetime view point may provide insights relevant for our understanding of quantum gravity.
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