As a psychologist, I am trying to find a way to grasp the meaning of Noether's theorem and to conceptualize relationships among what is conserved, the laws of symmetry, and the role of fields in this process. I'm not at all certain I'm even asking a meaningful question. Any thoughts would be welcome.

Noether's theorem says (loosely) "for every symmetry of a physical system there is an associated conserved charge". I have left this statement rather non-technical, but this is the basic idea.

 

• Time translation symmetry gives conservation of energy.
  
• Space translation symmetry gives conservation of momentum.
  
• Rotation symmetry gives conservation of angular momentum.
  

 

and so on...

 

If you have some specific questions I suggest you ask them. I don't understand your opening question.

Sorry. Does Noether's theorem assume a closed universe? Thanks.

Noether's theorem says (loosely) "for every symmetry of a physical system there is an associated conserved charge". I have left this statement rather non-technical, but this is the basic idea.

 

• Time translation symmetry gives conservation of energy.
• Space translation symmetry gives conservation of momentum.
• Rotation symmetry gives conservation of angular momentum.

 

and so on...

 

If you have some specific questions I suggest you ask them. I don't understand your opening question.

Sorry. Does Noether's theorem assume a closed universe? Thanks.

 

You have local and global statements about symmetries and conservation laws. Noether's theorem tells us that symmetry leads to a locally conserved current which implies a global conservation law.

 

If the space-time is non-compact, then you need that the currents fall off sufficiently fast at infinity in order to integrate. If this is what you are asking?

Yes. Thank you.

You have local and global statements about symmetries and conservation laws. Noether's theorem tells us that symmetry leads to a locally conserved current which implies a global conservation law.

 

If the space-time is non-compact, then you need that the currents fall off sufficiently fast at infinity in order to integrate. If this is what you are asking?

Sorry. Does Noether's theorem assume a closed universe? Thanks.

 

I have some specific questions - as you remark,

"Time translation symmetry gives conservation of energy.

Space translation symmetry gives conservation of momentum.

Rotation symmetry gives conservation of angular momentum."

 

What I do not understand is why each of these is the case. Does it have to do with the nature of the equations for kinetic energy and/or E=M(c*c), where E and M are essentially equivalent, and the other variable, velocity, involves time? And velocity is the first differential of the function for displacement over time. But, were that the case, why would not displacement symmetry - change in position - likewise give conservation of energy? I'm just not following why these specific conservations arise paired with their symmetries.

 

Thanks - jwatersphd

 

 

 

 

I'm just not following why these specific conservations arise paired with their symmetries.

 

You have to examine Lagrangians and their symmetries to get at this. You can do it for simple mechanical models in 1d to get at energy and linear momentum.