# Invariance of a Lagrangian

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I've just read that for a Lagrangian to be Lorentz Invariant the Lagrangian density cannot have second or higher derivatives.
Last day in class, a professor told us that, for a Lagrangian to be Lorentz Invariant, the Lagrangian density cannot have second or higher derivatives. Is this true?
Because one can write the KG lagrangian as $$\mathscr{L}=\phi(\square + m^2)\phi,$$ which have second derivatives.

And, where can I find a proof of this fact?

Thank you

## Answers and Replies

Science Advisor
"I've just read..." where?

It's simply not true. You can perfectly write down Lorentz-covariant terms with higher order derivatives (e.g. look at the equations of motion, as you mention!)

bhobba and vanhees71
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Summary:: I've just read that for a Lagrangian to be Lorentz Invariant the Lagrangian density cannot have second or higher derivatives.

Last day in class, a professor told us that, for a Lagrangian to be Lorentz Invariant, the Lagrangian density cannot have second or higher derivatives. Is this true?
Because one can write the KG lagrangian as $$\mathscr{L}=\phi(\square + m^2)\phi,$$ which have second derivatives.

And, where can I find a proof of this fact?

Thank you

The Lagrangian density for the Klein-Gordon equation is
$$\mathscr{L} = \frac 1 2 (\partial_{\mu} \phi)^2 - \frac 1 2 m^2\phi^2$$
Which involves only first-order derivatives.

Staff Emeritus
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The Lagrangian density for the Klein-Gordon equation is
$$\mathscr{L} = \frac 1 2 (\partial_{\mu} \phi)^2 - \frac 1 2 m^2\phi^2$$
Which involves only first-order derivatives.
Which is equivalent to the Lagrangian given by the OP by partial integration (and up to multiplication by a constant).

PeroK
Science Advisor
What could help is to play around with the action of a nonrelativistic point particle, what kind of actions are possible, what the order of derivatives is and what kind of boundary conditions you need.

Just write down (via an inner priduct) some invariant combinations of the position, velocity, acceleration etc. and see what you get upon varying.

vanhees71