Is minimal coupling procedure compatible with minimal action principle?
نویسنده
چکیده
When space-time is assumed to be non-Riemannian the minimal coupling procedure (MCP) is not compatible, in general, with minimal action principle (MAP). This means that the equations gotten by applying MCP to the Euler-Lagrange equations of a Lagrangian L do not coincide with the Euler-Lagrange equations of the Lagrangian obtained by applying MCP to L. Such compatibility can be restored if the space-time admits a connection-compatible volume element. We show how these concepts can alter qualitatively the predictions of the Einstein-Cartan theory of gravity. This Essay was awarded a Honorable Mention for 1994 by the Gravity Research Foundation. Supported by FAPESP. Minimal coupling procedure (MCP) provides us with an useful rule to get the equations for any physical field on non-Minkowskian manifolds starting from their versions of Special Relativity (SR). When studying classical fields on a non-Minkowskian manifold X we usually require that the equations of motion for such fields have an appropriate SR limit. There are, of course, infinitely many covariant equations on X with the same SR limit, and MCP solves this arbitrariness by saying that the relevant equations should be the “simplest” ones. MCP can be heuristically formulated as follows. Considering the equations of motion for a classical field in the SR, one can get their version for a non-Minkowskian space-time X by changing the partial derivatives by the X covariant ones and the Minkowski metric tensor by the X one. MCP is also used for the quantum analysis of gauge fields, where the gauge field is to be interpreted as a connection, and it is in spectacular agreement with experience for QED. Suppose now that the SR equations of motion for a classical field follow from an action functional via minimal action principle (MAP). It is natural to expect that the equations obtained by using MCP to the SR equations coincide with the Euler-Lagrange equations of the action obtained via MCP of the SR one. This can be better visualized with the help of the following diagram
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