Derivation of the Ten Einstein Field Equations from the Semiclassical Approximation to Quantum Geometrodynamics
Derivation of the Ten Einstein Field Equations from the Semiclassical Approximation to Quantum Geometrodynamics
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从量子几何动力学的半经典近似推导十个爱因斯坦场方程
DOI：
10.1103/physrev.177.1929
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发表时间：
1969
期刊：
影响因子：
--
通讯作者：
U. Gerlach
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文献类型：
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作者：
U. Gerlach
他的工作涉及爱因斯坦S引力理论的哈密顿-雅可比方程。其目的是证明所有十个爱因斯坦6eld方程都是波阵面相长干涉原理的直接结果。波前本身的传播是由爱因斯坦-哈密尔顿-雅各比方程决定的，这个方程在一个公式(哈密尔顿和雅各比的公式)中标志着可能是迄今为止用量子语言表述广义相对论的最远的一步。EGorts试图将广义相对论置于量子理论的框架内，从而对许多问题(引力坍塌、空间几何涨落、基本粒子与几何动力学激发之间的关系等)做出了努力。几何动力学的固有性质一再受挫；然而，关于6eld方程的结构，人们已经了解了很多。‘几何动力学的目前状态让人想起玻尔试图理解电子为什么不会塌缩到原子核中的时候，以及普朗克在零温度下论证简谐振子系综中的零点量子理论的时候。随后，他们的解释背后的特殊假设被赋予了物理基础，与粒子de Broglie*联系在一起。目前地址：俄亥俄州哥伦布市巴特尔纪念研究所。‘虽然还没有详细的几何动力学量子理论，但我们今天已经广泛地认识到量子几何动力学的定性特征和它的一些问题。见J.A.Wheeler，在8attelle Eencontrres：1N7 Lectlres in数学和物理中，由J.A.Wheeler和C.de Witt编辑(W.A.Benjamin，Inc.，纽约，1968)；J.A.Wheeler，爱因斯坦S愿景(Julius Springer-Verlag，柏林，1968)；J.A.Wheeler，《相对论，群，和拓扑学》，C.de Witt和B.De Witt(Gordon and Back Science Publisher，Inc.，纽约，1964)，第507页。~P.G.Bergmann，Phys.Rev.75,680(1949)；P.A.M.狄拉克，可以。J·数学。2,129(1950)；罗伊SoC。(伦敦)A246,326(1958)；QNantlm Mechuics讲座(学术出版社，纽约，1966)；Proc.罗伊SoC。(伦敦)A246,333(1958)；R.Arnowitt，S.Deser和C.W.Misner，Phys.修订版116,1322(1959)；摘自L.Witten编辑的《广义相对论动力学》(John Wiley&Sons，Inc.，纽约，1962)；A.佩雷斯，公牛。议员会(以色列)SF，179(1959)；Nuovo Cimento 26，53(1962)。‘M·普朗克，热火理论IP。布莱克斯顿的儿子和公司，费城，1914年)，第142,164页。能够产生干扰的波。在半经典极限下，这些德布罗意波可以用波和波前来描述。通过写下他的波浪方程式。恩，薛定谔对运动的这些和附加的几何方面，如相位、波长和频率，赋予了数学上的严密性。波动力学的短波极限是经典力学。介于两种极端描述(波和射线)之间的近似是半经典近似。它的数学基础是相函数S，哈密顿-雅可比(HJ)方程的解，以及波应力原理。
HIS work deals with the Hamilton-Jacobi equation for Einstein s theory of gravity. The aim is to show that all ten Einstein 6eld equations are a direct consequence of the principle of constructive interference of wave fronts. The propagation of the wave fronts themselves is determined by the Einstein-HamiltonJacobi equation, an equation which marks in one formulation (that of Hamilton and Jacobi) perhaps the furthest step to date in formulating general relativity in quantum language. The eGorts exerted in trying to put general relativity within the framework of a quantum theory and thus obtain answers to a number of problems' (gravitational collapse, fluctuations in the geometry of space, the relation between elementary particles and geometrodynamical excitations, etc.) inherent in geometrodynamics have been frustrated repeatedly; nevertheless, a great deal has been learned about the structure of the 6eld equations. ' The present state of geometrodynamics reminds one of the times when Bohr was trying to understand why an electron does not collapse into the nucleus, and when Planck was arguing for the zeropoint Quctuations' in an ensemble of simple harmonic oscillators at zero temperature. Subsequently the ud hoc assumptions underlying their explanations were given a physical basis by associating with a particle de Broglie *Present address: Battelle Memorial Institute, Columbus, Ohio. ' Although there does not yet exist a detailed quantum theory of geometrodynamics, we already today perceive in broad outhne the qualitative character of quantum geometrodynamics and a number of its problems. See J. A. Wheeler, in 8attelle Eencontres: 1N7 Lectlres in Mathematics and Physics, edited by J. A. Wheeler and C. De Witt (W. A. Benjamin, Inc. , New York, 1968); J. A. Wheeler, Einstein s Vision (Julius Springer-Verlag, Berlin, 1968); J. A. Wheeler, in Relativity, Groups, and Topology, edited by C. De Witt and B.De Witt (Gordon and Breach Science Publishers, Inc. , New York, 1964), p. 507. ~ P. G. Bergmann, Phys. Rev. 75, 680 (1949); P. A. M. Dirac, Can. J. Math. 2, 129 (1950); Proc. Roy. Soc. (London) A246, 326 (1958); Lectures on QNantlm Mechunics (Academic Press Inc., New York, 1966); Proc. Roy. Soc. (London) A246, 333 (1958); R. Arnowitt, S. Deser, and C. W. Misner, Phys. Rev. 116, 1322 {1959);in The Dynamics of General Relativity, edited by L. Witten (John Wiley & Sons, Inc. , New York, 1962);A. Peres, Bull. Res. Counc. (Israel) SF, 179 {1959);Nuovo Cimento 26, 53 (1962). ' M. Planck, The Theory of Heat Radeattore iP. Blakiston's Son and Company, Philadelphia, 1914),pp. 142, 164. waves that are capable of interference. In the semiclassical limit, these de Broglie waves are describable in terms of waves and wave fronts. By writing down his wave equati. on, Schrodinger gave mathematical rigor to these and additional geometrical aspects of motion, such as phase, wavelength, and frequency. The shortwavelength limit of wave mechanics is classical mechanics. An approximation that stands in between the two extreme descriptions (wave and ray) is the semiclassical approximation. Its mathematical basis is the phase functions S, the solutions of the Hamilton-Jacobi (HJ) equation, together with the principle of cort strlctiee irsterferertce of waves,