首先，為了凸顯複數空間的延伸在討論軌跡與機率詮釋間的關係之重要性，吾人先探討複數力學作為結合古典力學與量子力學的力學體系，是如何利用複數空間的延伸來解釋與證明其他半古典量子理論中所提出的現象與假設。經由複數力學的分析，可以確定由量子漢彌爾頓理論所提出的轉折點的確提供了複數平面上的邊界條件。關於波姆力學中所假設之粒子具有不規則運動速度，經由複數力學證實其源自於複數形式波函數的虛部部分。另外，吾人提出非Hermitian量子力學是可由複數力學所涵蓋之。
量子機率透過複數力學可證明其與古典複數機率投影在實數空間相等義。此結果引導出量子力學是滿足古典統計力學的重要結論。本論文利用複數軌跡建立了量子流體，並推導出在複數空間中量子流體所滿足的流體連續方程式、動量方程式與能量方程式。透過量子流體詮釋，吾人發現從虛部部分的能量守恆可以推導出波恩對於量子機率的假設。因此，吾人透過軌跡的詮釋可以將古典統計理論帶入量子力學中。例如:粒子的速度決定了其在某特地地方存在的可能性，而此結果證明和量子力學所提出的可能性相符合。
另一方面，本研究提出當量子流體的軌跡流JT投影至實數軸的結果與量子機率流Jqm相符合。於是，粒子可能的運動情形可由軌跡流有具體視覺化的呈現，以取代抽象的量子機率流Jqm。藉由軌跡詮釋，伴隨著粒子運動軌跡的波函數可以藉由其量子作用函數S所重現，其中實部SR決定了波函數的相位，而虛部SI決定的其振幅。關於波函數，本篇論文以幾何的觀點分析其在複數平面上的所具有特色，發現波函數振幅的曲度是由位勢屏障的分佈所決定。最後吾人在複數平面上重現了漢彌爾頓的古典波，並分析其可能具有的動態情形。

The complex space dilation plays a crucial role for exploring the relationship between the trajectory interpretation and the probability interpretation. To verify the importance of extending the space structure to a complex domain, we discuss in this dissertation that how complex mechanics can explain and verify the phenomena and postulates provided by other classical approaches. We confirm that the turning point provided by the conventional quantum Hamilton-Jacobi formalism represents the boundary condition in the complex plane. The irregular fluctuating motion postulated in Bohmian mechanics has been verified that comes from the imaginary part of the complex wave function. Furthermore, we point out that the non-Hermitian Hamiltonian mechanics can be embodied in the formulation of complex mechanics.
The quantum probability is shown to be equivalent to the classical complex probability by projecting onto real subspace such that quantum mechanics satisfies classical statistical mechanics. We establish the quantum fluid in terms of the trajectory interpretation in complex space. The continuity equation, momentum equation, and energy equation of the quantum fluid in complex space are derived. According to the imaginary part of the energy conservation, we are able to verify the Born's postulate of quantum probability via the quantum fluid interpretation. Therefore, we confirm that the trajectory interpretation brings the classical statistical theory into quantum mechanics. For example, the particle's velocity determines the possibility of the particle being at a specific region which is compatible with the possibility presented by quantum mechanics.
On the other hand, the quantum probability density current Jqm is shown to be equivalent to the trajectory flow current JT if we project JT onto the real axis. We can concretely present instead of using the abstract Jqm to visualize the probability and the potential motions that a particle could have. By giving quantum action function S associated with trajectories, we are able to reconstruct the wave function, where SR determines the phase and SI gives the magnitude. We also discuss the characteristics of the wave function in the complex plane from a geometric standpoint. It shows that, the curvature of the wave function is determined by the distribution of the potential barrier. We generalize Hamilton's classical wave program in the complex plane, and present the dynamic evolution that the quantum field might have.

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