Uncertainty Relation in Flat Space from Holography

Following the idea that information is the negative entropy, we propose that the information and entropy of an isolated system can convert into each other while the sum of them is an invariant for any physical process. The holographic principle is then reformulated in the way that this invariant is bounded by the Bekenstein-Hawking entropy of the system. It is found that Heisenberg’s uncertainty relation in quantum mechanics can be derived from this bound. PACS number(s): 03.65.Ta,04.60.-m,03.67.-a The concepts of information and entropy [1, 2] play more and more important roles in physics both in technical and theoretical aspects [3]. It seems that information is more fundamental than space as well as matter since each physical law must be written in terms of a certain type of information such as coordinate, curvature, energy, mass, etc.. Hence, in order to construct the unified law of all interactions in nature, one should inquire what are the informational contexts of space and matter and how to measure these physical quantities in a unified framework of information. Ever since the holographic principle was proposed [4, 5] the information and entropy manifest themselves as key concepts as a guide for constructing successful theory of quantum gravity (for recent review, see [6]). The information, known as how much uncertainty can be eliminated in light of an observer, can be defined as the deviation of the actual entropy of a system from the maximal entropy that the system may contain [7]. The feature of information is that an observer’s acquiring of information about an isolated system is always associated with decreasing of uncertainty on the system. Since information describes the observer’s capability to predict the outcome of physical system some time later, it is an observer-dependent quantity. ∗E-mail: [email protected]