Commutation and Curvature

Posted by on Jan 30, 2013 in Science | 0 comments

The commutation of a pair of operators, or its failure, is a fairly broad topic in physics. By commutation, we mean whether or not or, put another way, If the equality holds, then the operators are said to commute. If it fails, then they don’t commute. Simple. In quantum mechanics, operators for position and the conjugate momentum fail to commute; their difference is proportional to Planck’s constant, and this is the famous Heisenberg Uncertainty Principle. In the following, I want to show how the failure of two operators for parallel transport to commute with one another is a...

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Parallel transport and curvature

Posted by on Jan 13, 2013 in Science | 0 comments

Parallel transport and curvature

In the most basic physics, we learn that the most basic equations are generally vector equations. This is true of Newton’s equations involving force, , and acceleration, , and equally true of the Maxwell equations that involve vectors for the electric and magnetic fields, , and . Moreover, we learn things like acceleration is proportional to force, force is proportional to charge, rotational acceleration is proportional to torque, and in consideration of the fact that all of these quantities are vectors, that the resultant vector is in the same direction as the vector it is...

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