Galilean invariance and conserved quantities

[mtak0114]

Hi I have a simple question

what is the conserved quantity corresponding to the symmetry of galilean invariance?

and Lorentz invariance?

cheers

M

 

[Vanadium 50]

Energy, momentum and angular momentum in both cases.

 

[cortiver]

Energy, momentum and angular momentum in both cases.

Um, really? I'm pretty sure they correspond to time translation, spatial translation and rotation respectively...

As I understand it (but I may be wrong), not every symmetry in the laws of physics leads to a conserved quantity, only those that leave the Lagrangian unchanged. Hence there is no conserved quantity associated with Galilean or Lorentz invariance.

 

[Meir Achuz]

Momentum for Galileo; momentum and energy for Lorentz. Angular momentum follows from rotational invariance.

 

[Vanadium 50]

Gallilean and Lorentz invariance describe how translations through space, time and angle affect (or, in this case, don't) the Lagrangian.

 

[dx]

Momentum for Galileo; momentum and energy for Lorentz. Angular momentum follows from rotational invariance.

Nope. Ordinary momentum is associated with translations of space, not Galilean transformations of spacetime. Momentum and energy, considered together as four-momentum is associated with translations in spacetime, not Lorentz transformations.

[...]not every symmetry in the laws of physics leads to a conserved quantity, only those that leave the Lagrangian unchanged.Hence there is no conserved quantity associated with Galilean or Lorentz invariance.

Wrong. All laws of nature have Lorentz symmetry. If γ is a spacetime path of a system, then L(γ) where L is a lorentz transformation is also a possible spacetime path of the system (relativity). Therefore these two paths must have the same action S[γ] = S[L(γ)]. A simpe geometric argument shows that this leads to the following conserved quantity for a Lorentz transformation in the xt plane: -xE + tP_x*, and similarly for yt and zt planes. This can be thought of as a 'spacetime angular momentum'.

*Formally, this can be obtained from Noether's theorem. The killing vector of lorentz transformations in the xt plane is μ = x∂_t + t∂_x, and the consrved quantity is dS⋅μ = -xE + tP_x.

 

[Vanadium 50]

This may be a little advanced, but fundamentally what is happening is that Lorentz boosts, translations and rotations form what is called the "Poincaré Group". These transformations are, of course, intertwined - a rotation and a translation is equivalent to a (different) translation and (different) rotation, and if you boost something, after a period of time it's now translated.

You can do the same thing with Newtonian mechanics, and I am pretty sure what you get is the Euclidian Group. (I am less sure because this doesn't describe the world we live in, so I haven't thought about it very hard).

In both cases, the conserved quantities associated with these symmetries are energy, momentum and angular momentum. However, the algebraic expressions for these quantities (i.e. the way they are interrelated) are different, because the symmetries are different. Quantities that are invariant (which is different than being conserved) are also different.

One can pick off these conserved quantities one by one, but it's often more efficient to exploit the whole symmetry in one go.

If this is more confusing than enlightening, just ignore it.

 

[Meir Achuz]

Nope. Ordinary momentum is associated with translations of space, not Galilean transformations of spacetime. Momentum and energy, considered together as four-momentum is associated with translations in spacetime, not Lorentz transformations.

You're right. I was careless in my thinking-- too early in the AM.

 

[mtak0114]

*Formally, this can be obtained from Noether's theorem. The killing vector of lorentz transformations in the xt plane is μ = x∂t + t∂x, and the consrved quantity is dS⋅μ = -xE + tPx.

Is this essentially why we don't care of the corresponding conserved quantity to these symmetries because they are some combination of other more "fundamental" quantites i.e. energy and momentum?