Non-relativistic limit of the Lagrangian

[befj0001]

Why does the following Lagrangian not have the correct non-relativistic limit? It is correct except for the derivative of proper time with respect time. But that factor goes to 1 so why is the expression wrong?

$L = -(\frac{1}{2}mu^{\mu}u_{\mu} + qu^{\mu}A_{\mu})\frac{d\tau}{dt}$

 

[stevendaryl]

Why does the following Lagrangian not have the correct non-relativistic limit? It is correct except for the derivative of proper time with respect time. But that factor goes to 1 so why is the expression wrong?

$L = -(\frac{1}{2}mu^{\mu}u_{\mu} + qu^{\mu}A_{\mu})\frac{d\tau}{dt}$

Could you write down what you think is the non-relativistic limit?

 

[haushofer]

Here,

http://arxiv.org/abs/1206.5176?context=hep-th

such a limit is taken (page 7). The coupling to the gauge field here is however to cancel a divergent term, associated to the rest energy of the particle. How does it differ from your calculation? And as Steven indicates, what do you mean by 'non-relativistic limit'?

 

[befj0001]

Could you write down what you think is the non-relativistic limit?

I think the relativistivictic limit should be the expression above, but only what is inside the parenteses, still with the minus sign there. That is the classical expression for the lagrangian.

 

[befj0001]

I think the relativistivictic limit should be the expression above, but only what is inside the parenteses, still with the minus sign there. That is the classical expression for the lagrangian.

i.e., why is the lagranian in my first post not covariant?

 

[haushofer]

It is not covariant because it contains an explicit time derivative on tau. You can check directly, by performing an explicit coord.transfo., that such a term does not transform covariantly. The terms within parentheses are covariant.