Understanding 4-Momentum in Special Relativity

[nikolafmf]

Hello,

I am studing elementary particle physics and want to ask something, just to check if I have understood properly. So, as I understand, this is true about four-momentum in special relativity:

1. The square of the sum of particles' four momenta is invariant under Lorentz transformations.
2. The four momentum is NOT invariant under Lorentz transformations.
3. The four momentum is conserved.

This seems to be extremely usefull in particle physics. I hope the statements are true. Are they? What about proving them? Where can I find proofs?

 

[PeterDonis]

1. The square of the sum of particles' four momenta is invariant under Lorentz transformations.

I assume that what you mean by this is: take the sum of the 4-momenta of the particles going into an interaction, and take the squared norm of the resulting 4-vector; take the sum of the 4-momenta of the particles coming out of the same interaction, and take the squared norm of that resulting 4-vector; the two results will be the same. Yes, this is true.

2. The four momentum is NOT invariant under Lorentz transformations.

The four-momentum of what?

3. The four momentum is conserved.

The four-momentum of what?

 

[nikolafmf]

"I assume that what you mean by this is: take the sum of the 4-momenta of the particles going into an interaction, and take the squared norm of the resulting 4-vector; take the sum of the 4-momenta of the particles coming out of the same interaction, and take the squared norm of that resulting 4-vector; the two results will be the same. Yes, this is true."

That would mean conserved, not invariant. A textbook made me cautious to make diference between the two. So, I mean this: take the sum of the 4-momenta of the particles going into an interaction, and take the squared norm of the resulting 4-vector; then do the same for the same particles in DIFFERENT coordinate system, into which we go by some Lorentz transformation. You must get the same number?

For the third and the second question, I meant the same as the first, the sum of the four momenta of isolated system of particles. Not the square of the sum, but just the sum. It should be not invariant, but should be conserved, right?

 

[PeterDonis]

That would mean conserved, not invariant.

Yes, fair point.

I mean this: take the sum of the 4-momenta of the particles going into an interaction, and take the squared norm of the resulting 4-vector; then do the same for the same particles in DIFFERENT coordinate system, into which we go by some Lorentz transformation. You must get the same number?

Yes. The squared norm of any 4-vector is a Lorentz scalar and is therefore invariant.

For the third and the second question, I meant the same as the first, the sum of the four momenta of isolated system of particles. Not the square of the sum, but just the sum.

The sum is a 4-vector, not a scalar. The usual terminology for 4-vectors, 4-tensors, etc. is "covariant", not "invariant", meaning that their components transform according to the appropriate laws ("4-vector", "4-tensor" etc. are really terms that denote transformation laws). So the sum of the 4-momenta of all the particles going into an interaction will be covariant, since it's a 4-vector.

The sum of 4-momenta of all the particles going in will be the same as the sum of 4-momenta of all the particles coming out--that is, you will get the same 4-vector in both cases. (This assumes an isolated interaction with no other forces, no external potentials, etc.) So the sum will be conserved.

 

[nikolafmf]

OK, thank you very much.