Bell's Inequality
Introduction
Bell's theorem is a fundamental result in quantum mechanics that shows the incompatibility of local hidden variable theories with the predictions of quantum mechanics. The theorem is named after physicist John Bell, who derived an inequality (now known as Bell's inequality) that, if violated, would imply the impossibility of local hidden variables.
Background
The concept of local hidden variables was introduced to address the apparent randomness and indeterminacy in quantum mechanics. In this context, "local" refers to the principle of locality, which states that an object can only be influenced by its immediate surroundings. "Hidden variables", on the other hand, are hypothetical variables that are not directly observable but are assumed to determine the outcomes of quantum measurements.
Bell's theorem was a response to the Einstein-Podolsky-Rosen (EPR) paradox, a thought experiment that questioned the completeness of quantum mechanics. The EPR paradox suggested that if quantum mechanics was a complete theory, it would have to allow for "spooky action at a distance", a concept that was deeply unsettling to many physicists, including Einstein.
Bell's Inequality
Bell's inequality is a mathematical inequality involving the probabilities of different outcomes in a quantum mechanical system. It is derived under the assumption of local hidden variables. The violation of this inequality in experiments is taken as evidence against local hidden variable theories.
The original form of Bell's inequality, known as Bell's original inequality or simply Bell's inequality, involves the correlation function of two spin-1/2 particles in a singlet state. The inequality can be generalized to other quantum systems and other types of measurements, leading to a family of inequalities known as Bell inequalities.
Experimental Tests
Experimental tests of Bell's inequality have been performed using various systems, including photons, electrons, and atoms. These experiments typically involve measuring the properties of pairs of entangled particles in different directions.
The results of these experiments have consistently shown violations of Bell's inequality, providing strong evidence against local hidden variable theories. However, these experiments are subject to various loopholes that could potentially allow for alternative explanations.
Implications and Interpretations
The violation of Bell's inequality has profound implications for our understanding of the physical world. It suggests that either the principle of locality is violated, or there are hidden variables that are non-local.
Various interpretations of quantum mechanics, such as the Copenhagen interpretation, the many-worlds interpretation, and the Bohmian interpretation, offer different ways of understanding the implications of Bell's theorem.