[论文解读] Generalized Galilean Transformations and the Measurement Problem in the Entropic Dynamics Approach to Quantum Theory

David T. Johnson|arXiv (Cornell University)|May 6, 2011

Quantum Mechanics and Applications参考文献 45被引用 8

一句话总结

本文從熵动态框架出發，推導出量子力學，其中機率透過最大熵方法進行更新。它引入了廣義的伽利略變換，從資訊等價性產生引力勢，並在不假設的前提下推導出玻恩規則與薛定諤方程，透過基於資訊的動力學解決測量問題。

ABSTRACT

Quantum mechanics is an extremely successful and accurate physical theory, yet since its inception, it has been afflicted with numerous conceptual difficulties. The primary subject of this thesis is the theory of entropic quantum dynamics (EQD), which seeks to avoid these conceptual problems by interpreting quantum theory from an informational perspective. We begin by reviewing probability theory as a means of rationally quantifying uncertainties. We then discuss how probabilities can be updated with the method of maximum entropy (ME). We then review some motivating difficulties in quantum mechanics before discussing Caticha's work in deriving quantum theory from the approach of entropic dynamics. After entropic dynamics is introduced, we develop the concepts of symmetries and transformations from an informational perspective. The primary result is the formulation of a symmetry condition that any transformation must satisfy in order to qualify as a symmetry in EQD. We then proceed to apply this condition to the extended Galilean transformation. This transformation is of interest as it exhibits features of both special and general relativity. The transformation yields a gravitational potential that arises from an equivalence of information. We conclude the thesis with a discussion of the measurement problem in quantum mechanics. We discuss the difficulties that arise in the standard quantum mechanical approach to measurement before developing our theory of entropic measurement. In entropic dynamics, position is the only observable. We show how a theory built on this one observable can account for the multitude of measurements present in quantum theory. Furthermore, we show that the Born rule need not be postulated, but can be derived in EQD. Finally, we show how the wave function can be updated by the ME method as the phase is constructed purely in terms of probabilities.

研究动机与目标

• 透過基於資訊與機率的推論理論重構量子力學，以解決其基礎性問題。
• 展示量子理論可從最大熵原則應用於不完全資訊中推導而出。
• 透過將位置視為唯一可觀測量並從資訊更新推導玻恩規則，以處理測量問題。
• 在熵動力學中推廣伽利略變換，從資訊等價性產生非平凡的引力勢。
• 將貝葉斯定理與最大熵方法統一為單一更新形式的特例。

提出的方法

• 應用考克斯公理推導機率理論作為可信度的邏輯，奠定理性推論的基礎。
• 使用擴展的最大熵方法（ME）在約束下更新機率，廣義化貝葉斯定理。
• 透過在統計流形上定義資訊動力學，從熵動力學推導薛定諤方程。
• 在熵動力學中引入對稱性條件：變換必須保持資訊流形的結構。
• 將此對稱性條件應用於廣義伽利略變換，從資訊等價性推導出非平凡的引力勢。
• 將測量建模為透過ME方法進行資訊更新的過程，其中波函數相位由機率構建而成。

实验结果

研究问题

• RQ1 量子力學能否在不假設玻恩規則的前提下，從資訊理論基礎推導而出？
• RQ2 如何從資訊原則而非幾何不變性推導出量子理論中的對稱性？
• RQ3 廣義伽利略變換在從資訊產生類引力勢方面扮演何種角色？
• RQ4 在僅位置為可觀測量的熵動力學框架中，測量過程如何產生？
• RQ5 薛定諤方程與單位演化能否從機率的最大小熵更新中推導而出？

主要发现

• 玻恩規則被推導為熵動力學的結果，而非假設，方法為顯示機率密度 |ψ|² 是由最大小熵更新產生。
• 薛定諤方程從熵動力學框架推導而出，方法為將時間定義為資訊變化的度量，並施加對稱性條件。
• 在熵動力學中，廣義伽利略變換產生了從不同參考系間資訊等價性所衍生的非平凡引力勢。
• 波函數的相位完全由機率構建，無需額外假設，顯示量子干涉及發於資訊幾何。
• 測量被視為透過ME方法進行資訊更新的過程，其中波函數坍縮被解釋為對系統狀態的貝葉斯更新。
• 擴展的最大熵方法統一了貝葉斯定理與標準ME，顯示貝葉斯更新是通用ME形式的特例。