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Path Integral Implementation of Relational Quantum Mechanics Path Integral Implementation of Relational Quantum Mechanics Jianhao M. Yang ( [email protected] ) Qualcomm (United States) Research Article Keywords: Relational Quantum mechanics, Path Integral, Entropy, Inuence Functional Posted Date: February 18th, 2021 DOI: https://doi.org/10.21203/rs.3.rs-206217/v1 License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License Version of Record: A version of this preprint was published at Scientic Reports on April 21st, 2021. See the published version at https://doi.org/10.1038/s41598-021-88045-6. Path Integral Implementation of Relational Quantum Mechanics Jianhao M. Yang∗ Qualcomm, San Diego, CA 92121, USA (Dated: February 4, 2021) Relational formulation of quantum mechanics is based on the idea that relational properties among quantum systems, instead of the independent properties of a quantum system, are the most fundamental elements to construct quantum mechanics. In the recent works (J. M. Yang, Sci. Rep. 8:13305, 2018), basic relational quantum mechanics framework is formulated to derive quantum probability, Born’s Rule, Schr¨odinger Equations, and measurement theory. This paper gives a concrete implementation of the relational probability amplitude by extending the path integral formulation. The implementation not only clarifies the physical meaning of the relational probability amplitude, but also gives several important applications. For instance, the double slit experiment can be elegantly explained. A path integral representation of the reduced density matrix of the observed system can be derived. Such representation is shown valuable to describe the interaction history of the measured system and a series of measuring systems. More interestingly, it allows us to develop a method to calculate entanglement entropy based on path integral and influence functional. Criteria of entanglement is proposed based on the properties of influence functional, which may be used to determine entanglement due to interaction between a quantum system and a classical field. Keywords: Relational Quantum mechanics, Path Integral, Entropy, Influence Functional I. INTRODUCTION to see if a quantum theory constructed based on rela- tional properties can address some of the unanswered fun- Quantum mechanics was originally developed as a damental questions mentioned earlier. Such reconstruc- physical theory to explain the experimental observations tion program was initiated [10] and had some successes, of a quantum system in a measurement. In the early days for example, in deriving the Schr¨odinger Equation. of quantum mechanics, Bohr had emphasized that the de- Recently, a similar reformulation of quantum mechan- scription of a quantum system depends on the measuring ics was proposed [12, 13]. The reformulation is based apparatus [1–3]. In more recent development of quan- on two basic ideas. 1.)Relational properties between the tum interpretations, the dependency of a quantum state two quantum systems are the most fundamental elements on a reference system was further recognized. The rela- to formulate quantum mechanics. 2.)A physical measure- tive state formulation of quantum mechanics [4–6] asserts ment of a quantum system is a probe-response interaction that a quantum state of a subsystem is only meaningful process. Thus, the framework to calculate the probabil- relative to a given state of the rest of the system. Sim- ity of an outcome when measuring a quantum system ilarly, in developing the theory of decoherence induced should model this bidirectional process. This implies the by environment [7–9], it is concluded that correlation in- probability can be derived from product of two quanti- formation between two quantum systems is more basic ties with each quantity associated with a unidirectional than the properties of the quantum systems themselves. process. Such quantity is defined as relational probability Relational Quantum Mechanics (RQM) further suggests amplitude. Specifically, the probability of a measurement that a quantum system should be described relative to outcome is proportional to the summation of probability another system, there is no absolute state for a quan- amplitude product from all alternative measurement con- tum system [10, 11]. Quantum theory does not describe figurations. The properties of quantum systems, such as the independent properties of a quantum system. In- superposition and entanglement, are manifested through stead, it describes the relation among quantum systems, the rules of counting the alternatives. As results, tradi- and how correlation is established through physical in- tional quantum mechanics formulations can be rediscov- teraction during measurement. The reality of a quantum ered but with new insights on the origin of quantum prob- system is only meaningful in the context of measurement ability. Schr¨odinger Equation is recovered when there is by another system. no entanglement in the relational probability amplitude The idea that relational properties are more basic than matrix [12]. On the other hand, when there is change in the independent properties of a quantum system is pro- the entanglement measure, the quantum measurement found. It should be considered a starting point for con- theory is obtained [13]. In essence, quantum mechan- structing the formulation of quantum mechanics. How- ics demands a new set of rules to calculate measurement ever, traditional quantum mechanics always starts with probability from an interaction process. an observer-independent quantum state. It is of interest Although the concept of relational probability ampli- tude is useful to derive the quantum probability, its phys- ical meaning is not obvious to understand. It is desirable to find an explicit calculation of the relational probabil- ∗ [email protected] ity amplitude. It turns out that the path integral method 2 can be used to achieve this goal and is briefly described in another apparatus A′. In an ideal measurement to mea- Ref. [12]. In this paper, the significance of the path inte- sure an observable of S, the apparatus is designed in such gral implementation of relational quantum mechanics is a way that at the end of the measurement, the pointer fully developed. Besides providing the physical meaning state of A has a distinguishable, one to one correlation of relation probability amplitude, the path integral for- with the eigenvalue of the observable of S. mulation also has interesting applications. For instance, The definition of Observer is associated with an appa- it can describe the history of a quantum system that has ratus. An observer, denoted as , is a person who can interacted with a series of measuring systems in sequence. operate and read the pointer variableO of the apparatus. As a result, the double slit experiment can be elegantly Whether or not this observer (a person) is a quantum explained from the formulation developed here. More sig- system is irrelevant in our formulation. An observer is nificantly, the coordinator representation of the reduced defined to be physically local to the apparatus he asso- density matrix derived from this implementation allows ciates with. This prevents the situation that can in- us to develop a method to calculate entanglement entropy stantaneously read the pointer variable of the apparatusO using path integral approach. We propose a criterion on that is space-like separated from . O whether there is entanglement between the system and In the traditional quantum measurement theory pro- external environment based on the influence functional. posed by von Neumann [14], both the quantum system This enables us to calculate entanglement entropy of a and the measuring apparatus follow the same quantum physical system that interacts with classical fields, such mechanics laws. After the interaction during the mea- as an electron in an electromagnetic field. Since entan- surement process, both systems encode information each glement entropy is an important concept in quantum in- other, allowing an observer to infer measurement results formation theory, the method described here may lead of S by reading pointer variable of A. Quantum mea- to new insight on the information aspect of a quantum surement is a question-and-answer bidirectional process. system that interacts with classical fields. The measuring system interacts (or, disturbs) the mea- The paper is organized as following. We first briefly re- sured system. The interaction in turn alters the state of view the relational formulation of quantum mechanics in the measuring system. As a result, a correlation is es- Section II. In Section IIIA the path integral implementa- tablished, allowing the measurement result for S to be tion of the relational probability amplitude is presented. inferred from the pointer variable of A. It is shown to be compatible with the traditional path A Quantum State of S describes the complete informa- integral quantum mechanics, particularly on the defini- tion an observer can know about S. From the exam- tion of the influence functional. Section IIIB generalizes ination on the measurementO process and the interaction the formulation to describe the history of quantum state history of a quantum system, we consider a quantum for the observed system that has interacted with a se- state encodes the information relative to the measuring ries of measuring systems in sequence. The formulation system or the environment that the system previously is applied to explain the double
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