Abstract
Non-relativistic quantum mechanics is reformulated here 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. This idea, combining with the emphasis that measurement of a quantum system is a bidirectional interaction process, leads to a new framework to calculate the probability of an outcome when measuring a quantum system. In this framework, the most basic variable is the relational probability amplitude. Probability is calculated as summation of weights from the alternative measurement configurations. The properties of quantum systems, such as superposition and entanglement, are manifested through the rules of counting the alternatives. Wave function and reduced density matrix are derived from the relational probability amplitude matrix. They are found to be secondary mathematical tools that equivalently describe a quantum system without explicitly calling out the reference system. Schrödinger Equation is obtained when there is no entanglement in the relational probability amplitude matrix. Feynman Path Integral is used to calculate the relational probability amplitude, and is further generalized to formulate the reduced density matrix. In essence, quantum mechanics is reformulated as a theory that describes physical systems in terms of relational properties.
Highlights
Quantum mechanics is one of the most successful physical theories and has been experimentally confirmed extensively, there are many fundamental questions still left unanswered
The entanglement measure quantifies the difference between time evolution and quantum measurement. We show that such framework to calculate probability amplitude can be explicitly implemented using Feynman Path Integral[18]. The impacts of this framework are fundamental as it is the basis for deriving the formulations that are mathematically equivalent to the laws in traditional quantum mechanics
We assume observers share the same information of the relational matrix, and focus on formulating quantum mechanics based on the relational probability amplitude
Summary
Quantum mechanics is one of the most successful physical theories and has been experimentally confirmed extensively, there are many fundamental questions still left unanswered. What is novel in our approach is a new framework for calculating the probability of an outcome when measuring a quantum system Such a framework is fundamental in deriving basic laws of quantum mechanics, so we briefly describe it here. We show that such framework to calculate probability amplitude can be explicitly implemented using Feynman Path Integral[18] The impacts of this framework are fundamental as it is the basis for deriving the formulations that are mathematically equivalent to the laws in traditional quantum mechanics. The formulation for calculating the probability of finding the system in an eigenstate is equivalent to Born’s rule, but with a new insight: the fact it is an absolute square of a complex number is a consequence that a quantum measurement is a bidirectional process. On the other hand, when there is change in the entanglement measure, quantum measurement theory is obtained
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