量子引力的路径积分形式

量子引力的路径积分理论Path Integral Theory of Quantum Gravity本文讨论量子力学与广义相对论的结合问题，旨在通过以经典广义相对论的作用量为基础得出量子引力的路径积分形式.This paper discusses the combination between quantum mechanics and general relativity, aiming to obtain the path integral formulation of quantum gravity based on the action of classical general relativity.关键词：量子引力，路径积分，作用量Key words: quantum gravity, path integral, actionPACC: 0460，0455，03651.引言Introduction自量子力学和广义相对论诞生以来，把两者结合成为既具有量子特征，又能描述弯曲时空的量子引力理论是人们追求的理想.然而，引力的特殊性造成了这一问题的复杂.其主要原因是来自于引力的自作用引起的波函数的非线性，这一方面破坏了量子力学的基础——态的叠加原理；另一方面，也导致了量子引力论缺乏相应的背景时空.引力量子化的这些早期尝试所遭遇的困难反映了一个很基本的事实，那就是许多不同的量子理论可以具有同样的经典极限，在一个本质上是量子化的物理世界中，理想的做法应该是从量子理论出发，在量子效应可以忽略的情形下对理论作“经典化”，而不是相反. 类似于量子力学，量子引力理论也应该有正则量子化、协变量子化和路径积分三种形式.霍金早在上世纪70年代就曾指出，路径积分或许是通向量子引力理论的捷径.但作为路径积分的核心，时空坐标和作用量的选择至关重要.下面的分析指出，选择合适的作用量和“时空间”坐标，可以避免以上困难，得出令人满意的结果.It has been a pursuing ideal since the birth of quantum mechanics and general relativity to combine them to form quantum gravity theory with the feature of quantum characteristics as well as the description of curved space-time. However, the particularity of gravitation makes this problem complex. The main reason is the non-linearity of wave function caused by gravitationalself-interaction, which on the one hand breaks the basis of quantum mechanics, that is the superposition principle; on the other hand leads to the lacking of corresponding space-time background. All the difficulties encountered in the early experience of gravity quantization reflect a basic fact that numbers of different quantum theories may have the same classical limit. The ideal method is to make the theories classic under the condition of ignoring quantum effects starting from quantum theories, but not the reverse. Being similar to quantum mechanics, quantum gravity theory should have three forms as well, that are, canonical quantization, covatiant quantization, and path integral. Hocking pointed that path integral may be the shortcut to quantum gravity theory as early as in the 1970s. But the selection of space-time coordinates and actions are most important, which are the cores of path integral. It will be indicated in the following analysis that selecting an appropriate action and space-time coordinate can avoid above difficulties and reach satisfying results.2. 作用量和时空坐标的选择Selections of Actions and Space-time Coordinates在广义相对论中存在两种作用量：There are two actions in general relativity,一种是标量作用量：（1）one is scalar quantity action,其中（2）in which是引力场的作用量.式中R是黎曼曲率标量，d∑是四维度量空间的体元；（3）is gravitational field action. In the formula, R represents the scalar quantity of Riemannian Curvature; d∑represents volume element in the four-dimension metric space.是物质场的作用量. is material field action式中的m L 是物质场的拉格朗日函数。

由这种形式的作用量再加上适当的表面项即能推导出爱因斯坦方程.但是，按照这种形式定义的拉格朗日量中含有g μν的二阶导数，所以不适合用它将场方程改写为拉格朗日运动方程形式，同时也就不适合作为量子引力路径积分的作用量.但是，在广义相对论中还有另一种形式的作用量，这来自于引力场方程的拉格朗日形式：In the formula, m L represents Lagrange function of material field. The Einstein's equation can be derived from this action with some appropriate surface terms. However, the Lagrangian defined through this form includes second derivative of g μν. Therefore, it can not be used to revise the field equation to Lagrange's equation of motion and can not be used as the action of path integral in quantum gravity as well. But, there is another form of action in the general relativity, which is originated from Lagrangian form of gravitational field equation.（4） 式中m g L L L +=，其中 In the formula m g L L L +=,（5）是引力场的拉格朗日函数，m L 是物质场的拉格朗日函数。

注意到，g L 虽然不是标量，但它不再含有μνg 高于一阶的导数。

这样，通过引力场方程的拉格朗日形式可以看出，只要我们把L 作为经典的作用量，而把μνg 或 μνg 看做广义“空间坐标”，再把x α看做广义“时间坐标”，那么μνα,g 或,g μνα即是广义“速度”，其中μνα,g 是逆变速度， ,g μνα是协变速度.这样，我们就可将此作用量与广义“时间”和广义“坐标”作为量子引力的路径积分形式的出发点，在这种量子引力理论中，时间是αx （这里面含有普通时间t 和其他三维普通空间i x ,3,2,1=i ）。

广义坐标则是μνg 或 μνg . is Lagrange function of gravitational field, m L is Lagrange function of material field. It should be noted that although g L is not a scalar quantity , it does notinclude a higher derivative μνg any more. It can be perceived through Lagrangian form of gravitational field equation that once taking L as the classical action, μνg or μνg as the generalized space coordinate, and x α asgeneralized time coordinate, μνα,g or ,g μνα is the generalized velocity, in which μνα,g is contravariant velocity and ,g μνα is covariant velocity. Thus, this action and generalized space coordinate and time coordinated can be taken as the starting points of path integral in quantum gravity. In this quantum gravity theory, αx is time(which includes north time t and other ordinary three-dimensional space i x , 3,2,1=i ), μνg or μνg is generalized coordinate.3. 量子引力路径积分的基本假设Fundamental Assumptions for path integral of Quantum Gravity若已知α'x 时，坐标为'g μν处的量子引力几率幅为'ψ，则量子引力规律应能预言α'x 之后任一“时刻” αx 在“位置” μνg 的量子引力几率幅ψ.那么，几率幅从()''',g x αμνψ出发演化到()x αμνψg ,的几率幅可用一个函数（传播函数）()'',;,K g x g x ααμνμν来描写。