實時間量子問題中的費曼路徑積分，會把不同歷史的相位加總在一起。當作用量只保留二次項時，問題雖然比完整理論簡單，積分仍會快速振盪，不能只沿實數方向直接當成一般高斯積分處理。本文的目標，是把這個二次問題改寫成有限、可直接檢查的矩陣計算。
本研究的想法起於重新閱讀傅立葉光學。傅立葉光學不會要求實際系統同時處理無限孔徑與無限頻寬，而會先選取有限孔徑、有限頻率範圍與適當窗函數，再研究輸入如何傳到輸出。本文將這個直覺帶入路徑積分：先保留有限個局部模態，再對有限個模態係數積分，最後才檢查增加模態後是否接近已知的連續結果。
本文以 Gabor frame（加伯框架）表示場。Gabor frame 中的每一個局部函數稱為 Gabor atom（加伯原子）；它由高斯窗與振盪因子組成，因此同時保留時間位置與頻率資訊。把有限展開代回二次作用量後，可得到有限作用量矩陣。因為相距很遠的高斯窗重疊很小，矩陣中的遠距離元素會快速變小，整體便接近帶狀稀疏矩陣。
對有限矩陣所產生的振盪高斯，本文使用 Lefschetz thimble（萊夫謝茲陡降路徑）處理。最簡單的一維例子顯示，若二次項為正，可把積分方向旋轉四十五度，使純振盪因子變成會下降的高斯；若二次項為負，則向相反方向旋轉。這與一次改變共同時間變數的 Wick rotation（威克旋轉）不同：後者在膨脹背景中不保證所有項同時下降，而前者可在有限矩陣分解後逐方向檢查。
本文以自由粒子、諧振子及 de Sitter 背景中的單一模態作為檢查。這些例子說明，固定有限模態時，方法給出明確的有限維高斯問題；增加模態後，可把結果與已知傳播核及單邊界高斯振幅比較。數值測試也顯示，局部模態所形成的近帶狀稀疏矩陣，比受測的稠密全域矩陣能處理更大的自由度。
本文不主張重新定義完整重力路徑積分，也不把選定例子的結果延伸成所有曲時空的一般定理。本文的貢獻，是把傅立葉光學所啟發的有限孔徑想法、局部 Gabor frame、有限作用量矩陣與 Lefschetz thimble 整理成一套簡單、可計算且可逐步檢查的二次洛倫茲方法。

Lorentzian path integrals provide a direct real-time formulation of quantum dynamics, but their oscillatory nature makes both formal and numerical treatments difficult, especially in time-dependent cosmological backgrounds. This thesis develops a finite-mode framework for the quadratic Lorentzian sector. The basic strategy is to replace the formal infinite-dimensional functional integral by a sequence of finite-dimensional representations and to examine how the resulting propagators behave as the representation is enlarged. Localized finite modes are used to construct a finite quadratic action, after which the corresponding oscillatory Gaussian integral can be treated within a controlled finite-dimensional setting. The formulation is examined through standard systems with known analytic behavior and is then applied to a representative cosmological background. The calculations show that the finite-mode construction reproduces the expected qualitative and quantitative behavior as the resolution is increased, while retaining a matrix structure that is suitable for numerical computation. The method is intended as a practical finite-resolution procedure rather than a new definition of the complete gravitational path integral or a general convergence theorem for arbitrary interacting theories.

[1] R. P. Feynman, “Space-Time Approach to Non-Relativistic Quantum Mechanics,” Reviews of Modern Physics 20, 367–387 (1948).
[2] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York (1965).
[3] J. Feldbrugge, J.-L. Lehners, and N. Turok, “No Smooth Beginning for Spacetime,” Physical Review Letters 119, 171301 (2017).
[4] J. Feldbrugge, J.-L. Lehners, and N. Turok, “Lorentzian Quantum Cosmology,” Physical Review D 95, 103508 (2017).
[5] E. Cordero, F. Nicola, and L. Rodino, “Gabor Representations of Evolution Operators,” Transactions of the American Mathematical Society 367, 7639–7663 (2015).
[6] E. Cordero, F. Nicola, and L. Rodino, “Gabor Wave Packets and Evolution Operators,” arXiv:1407.0997 [math.FA] (2014).
[7] G. E. Pfander, “Gabor Frames in Finite Dimensions,” in Finite Frames: Theory and Applications, Birkhäuser (2013).
[8] T. Kashiwa, S. Sakoda, and S. Zenkin, “Ordering, Symbols and Finite-Dimensional Approximations of Path Integrals,” Progress of Theoretical Physics 92, 669–685 (1994).
[9] S. Albeverio, R. Høegh-Krohn, and S. Mazzucchi, Mathematical Theory of Feynman Path Integrals, 2nd ed., Springer, Berlin (2008).
[10] S. I. Trapasso, “A Time-Frequency Analysis Perspective on Feynman Path Integrals,” Journal of Fourier Analysis and Applications 26, 68 (2020).
[11] A. A. Hari and S. Givli, “A New Method for the Calculation of Functional and Path Integrals,” Scientific Reports 13, 13852 (2023).
[12] Z.-G. Mou, P. M. Saffin, A. Tranberg, and S. Woodward, “Real-Time Quantum Dynamics, Path Integrals and the Method of Thimbles,” Journal of High Energy Physics 06, 094 (2019).