That's probably a dumb question but what are the criteria for picking the lagrangian that leads to a certain wave equation? It is not enough to simply impose that Hamilton's principle leads to the correct wave equation so I am looking for the general principles.(adsbygoogle = window.adsbygoogle || []).push({});

To be specific, consider Schroedinger's equation.

The lagrangian that is given in textbooks is

[tex] {\cal L} = {i \over 2} ( \phi^* \partial_t \phi - \phi \partial_t \phi^*) - {1 \over 2} \partial_x \phi^* \partial_x \phi + V(x) \phi \phi^* [/tex]

and the variation wrt phi^* gives the equation for phi whereas the variation wrt phi gives the equation for phi^*.

Fine, but one could have picked

[tex] {\cal L} = i \phi \partial_t \phi - {1 \over 4} (\partial_x \phi)^2 + {1 \over 2} V(x) \phi^2 + {\rm terms \, in \,} \phi^*[/tex]

instead and it seems to me that this would also lead to the correct equations (it's easy to write the correct eqs for phi^*).

So what is wrong with this lagrangian?

There *is* a problem because the conjugate momentum is phi (instead of being i phi^* with the usual lagrangian) and of course, the canonical quantization rule does not make sense if the conjugate momentum is equal to the field itself.

One problem is that my lagrangian does not have the phase invariance [itex] \phi \rightarrow e^{i \alpha} \phi [/itex] so that there is no symmetry current associated to this symmetry.

So what is the fundamental principle in determining a lagrangian? Obtaining a lagrangian which yields the correct eoms is necessary but not sufficient.

Thanks

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# What is wrong with this lagrangian

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