In Non-Relativistic Lagrangian mechanics usually we didn't impose any constraint on the Action/Lagrangian, rather than to be respectively a functional/function (or 3 form on Manifolds).

In QFT and Relativistic field theory we usually impose that the Action should be a Lorentz scalar, and the Lagrangian density a Lorentz scalar (or scalar density more generally).

I understood the idea behind it is to have Lorentz covariance in the e.o.m. so that one cannot distinguish two inertial frames.

In non-Relativistic mechanics we never imposed it, it just happened to be so by construction that two frame related by a Galileian Boost sees the same physics and the saem e.o.m. In Relativistic mechanics we are concerned with this topic.

My first question is: Shouldn't we only impose invariance by Lorentz Boosts, instead of invariance by general Lorentz transformations, which include rotations?

Because if we impose the Action to be a Lorentz Scalar, we are imposing also rotational $ SO(3) $symmetries (not quasi-symmetries but symmetries!), just to have the same physics in two frames related by two lorentz boost. Isn't that too restrictive?

It is understandable to impose full Lorentz invariance for the actions of Free motions $L_{free}$, such as the Action for the free Klein-Gordon Field and the free Dirac field, because invariance by boost is imposed as a postulate, and rotational invariance is understandable as it reflect the isotropy of space.

But if we add a potential, in principle, I should be allowed to break at least rotational symmetries for a point particle system or a field system, because I could build a laboratory system which has a cilindrical symmetry or which doesn't have any rotational symmetry.

I couldn't find any exemples so I ask: Could you show some examples of a Relativistic field and a relativistic point particle, with a potential (interaction) which doesn't have rotational symmetry? (and or translational symmetry) In this case the Action shouldn't be a Lorentz scalar, should it?

My last question concerns Maxwell equations. The Action without sources is a Lorentz scalar, but I can construct a physical system which isn't rotationally symmetric, so in this case the rotational symmetry is broken by a source term in the Action? ($\rho$ non rotationally symmetric) Or by boundary conditions?

Imposing full Lorentz invariance of the Action (and Lorentz Covariance of the e.o.m.) is too strict to me as it would include rotational symmetry which seems unjustified, and also conserved angular momentum for every system. Should also Covariance of the e.o.m. be unjustified because it does include rotational symmetry?


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