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研究生： 顏綜佑
Yen, Tseng-You
論文名稱： 四維時空COHERENT STATE描述和路徑積分
FOUR-DIMENSIONAL SPACE-TIME COHERENT STATE REPRESENTATIONS AND PATH INTEGRALS
指導教授： 陳泉宏
Chen, Chuan-Hung
共同指導教授: 江祖永
Kong, C.W. Otto
學位類別： 碩士
Master
系所名稱： 理學院 - 物理學系
Department of Physics
論文出版年： 2012
畢業學年度： 100
語文別： 英文
論文頁數： 33
中文關鍵詞： coherent states 、Poincaré-Snyder relativity 、fiducial unit vector 、Schrödinger wavefunction 、familiar coherent states 、Fock states 、quantum action principle 、propagator 、classical action
外文關鍵詞： coherent states, Poincaré-Snyder relativity, fiducial unit vector, Schrödinger wavefunction, familiar coherent states, Fock states, quantum action principle, propagator, classical action
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    • 我們推廣由J. Klauder在canonical Galilean dynamics下的量子力學建立的coherent state representation and path integral架構到建立在3 + 1 dimensional Minkowski space-time上的稱為Poincaré-Snyder relativity的canonical dynamics下的量子力學的架構。為了自然的物理圖像是在相空間的波函數是以各方向皆以相同形式圍繞對應古典相空間的點，coherent states是被定義成對應時空方向的四個沒有contravariant或covariant轉換向量型式的煙滅算符的特徵態。正如我們在論文裡的分析及陳述，這樣的架構保有了所有別的基本的特性。接著重要的事是Klauder由變分原理所建議的 quantum action principle以coherent states代入會得到和古典action有相同形式的運動方程。

    We extended the coherent state representation and path integral formulation by J. Klauder for the quantization of canonical Galilean dynamics to a formulation on the 3 + 1 dimensional Minkowski space-time the quantization of canonical dynamics under Poincaré-Snyder relativity. In order to have a natural physical picture with phase space wavefunction isotropic around the corresponding classical phase space point, the coherent states have to be de ned as eigenstates of four annihilation operators to the space-time directions that do not transform as components of a contravariant or a covariant vector. As we analyzed and elaborated in the dissertation, all other basic features of the formulation sustain. An important one among the latter is the quantum action principle suggested by Klauder | application of the variation principle restricted to the set of coherent states gives the correct equation of motion.

    1 Introduction 1 1.1 Poincaré-Snyder relativity and four-dimensional space-time with G(1; 3) symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Canonical mechanics with Poincare-Snyder relativity . . . . . . . . . 3 1.3 Quantum mechanics with Poincare-Snyder relativity . . . . . . . . . . 4 1.4 Outline of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Review of Klauder's Coherent State Representations and Path In- tegrals Formulation 6 2.1 Introducing the coherence states as representation of Heisenberg commutation relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Relation to the familiar coherent states and Fock states . . . . . . . . 10 2.3 The quantum action principle . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Conventional path integral . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Coherent states path integral . . . . . . . . . . . . . . . . . . . . . . 15 2.6 Coherent states path integral in terms of complex parameters . . . . 17 3 Coherent States on the Canonical Phase Space of Four-Dimensional Space-Time 20 3.1 Coherent states labeled by a space-time point . . . . . . . . . . . . . 20 3.2 Description of the coherent states in terms of complex parameters . . 23 3.3 Relationship to Fock states . . . . . . . . . . . . . . . . . . . . . . . . 25 4 Action Functional and Path Integral Formulation based on the Co- herent States 27 4.1 Action functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Path integral formulation based on the coherent states picture . . . . 29 5 Conclusion 31 6 Bibliography 33

    [1] J. Klauder, Continuous Representations and Path Integrals, Revisited .
    [2] O.C.W. Kong and H.-Y. Lee, NCU-HEP-k036, arXiv:0909.4676 [gr-qc].
    [3] O.C.W. Kong and H.-Y. Lee, NCU-HEP-k037, arXiv:1010.3515 [gr-qc].
    [4] A. Das and O.C.W. Kong, Phys. Rev. D73 (2006) 124029.
    [5] O.C.W. Kong, Phys. Lett. B665 (2008) 58 ; see also arXiv:0705.0845 [gr-qc] for
    an earlier version with some di erent background discussions.
    [6] O.C.W. Kong, NCU-HEP-k031, arXiv:0906.3581[gr-qc].
    [7] R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications, Dover
    (2005).
    [8] J. R. Klauder and E. C. G. Sudarshan, Fundamental Of Quantum Optics, W. A.
    Benjamin (1968).
    [9] A. O. Barut and L. Girardello, Commun. math. Phys. 21 (1971) 41.
    [10] J. J. Sakurai, Modern Quantum Mechanics, Addison-Wesley (1994).

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