The right mathematical question

01/08/2009

After my post on the Higgs field (see here) I would like to explain why there is no reason to be afraid. The point is the right mathematical question to be asked.  So, let me state why, from a strict mathematical standpoint, small perturbation theory is not the whole story. Let us consider the differential equation

\partial_t\psi=(H+\lambda V)\psi.

The exact solution of this equation is the function \psi(t;\lambda). So, one can ask : What happens when \lambda goes to zero? This is a proper mathematical question and the answer, when it exists, is a Taylor series. This is the most celebrated small perturbation theory and the terms of this Taylor series are computed directly from the given differential equation.

Of course, I can also ask what happens to the function \psi(t;\lambda) when \lambda goes to infinity. This is perfectly legal from a mathematical standpoint. This dual limit, when exists, produces an asymptotic series that has a development parameter 1/\lambda. This is a strong perturbation theory when each term is computed directly from the given differential equation.

Indeed, one can build a machinery for this case and prove the very existence of this technique that extends perturbation theory beyond the realm of a research of a small parameter. Rather, one can consider the case of differential equations with a large parameter and solve it producing an analysis of these equations in a range of the parameter space not reachable with small perturbation theory.

Physics is a lucky case for this mathematical question as it is all built on differential equations and having such a technique permits to analyze them in situations never reached before other than with computers.

I would like to emphasize that this is applied mathematics but the solutions one obtains can be interesting for physics. Of course, mathematics cannot be questioned except when is wrong.  But when it is right any discussion  is somewhat grotesque.


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