Symmetry and conservation.... which is first?

In summary, Noether's theorem states that for every symmetry of the Lagrangian, there is a conserved quantity. This is equivalent to the statement that for every conserved quantity, there is a symmetry of the Lagrangian. The concepts of symmetry and conservation are equally fundamental and can be seen as two sides of the same coin. Noether's theorem has broad applications in physics and is a fundamental principle in understanding the relationships between symmetries and conservation laws.
  • #1
Eiren
1
0
I have a question.

According to Noether's theorem,
"For each symmetry of the Lagrangian, there is a conserved quantity."

But soon I thought that I can also prove
"For each conserved quantiry, there is a symmetry of the Lagrangian."

Actually I can prove the second statement if I start prove from back, although it seems unnatural...So... Symmetry and Conservation.
Which is first?
Which concept is more fundamental than another?
(the image is from http://www.people.fas.harvard.edu/~djmorin/chap6.pdf )
 

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  • #2
Neither is more fundamental, it depends on your point of view. They are equivalent statements. Such statements appear in many different contexts and if you have come to the level of Noether's theorem you should already be familiar with them.
 
  • #3
That's a matter of methodology, I guess. I think in the early days the conservation principles were considered to be more fundamental, but nowadays the symmetries are considered to be the fundamental thing. You start from some symmetry principles, which restricts the form of the Lagrangian you can write down, and from that the conservation laws are derived. It's not always a clear cut; e.g. in general relativity you have to use more principles than just this.

Hope this helps.
 
  • #4
Yes, it's a one-to-one-relationship: Each one-parameter Lie symmetry of a Hamiltonian system leads to the conservation of the generator of this one-parameter Lie group and vice versa any conserved quantity is the generator of a one-parameter Lie symmetry. Noether has been even more general, including also gauge symmetries for systems with constraints. For a nice review, see

http://arxiv.org/abs/hep-th/0009058
 
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Likes Ravi Mohan

1. What is symmetry?

Symmetry is a fundamental concept in mathematics and science that refers to the balance and proportion of an object or system. It is characterized by the presence of identical or similar elements on different sides of an imaginary line or point.

2. How does symmetry relate to conservation?

Symmetry is closely related to conservation, as it represents a balance or preservation of certain properties or quantities. In physics, symmetry principles are used to explain the conservation of energy, momentum, and other fundamental laws.

3. Why is symmetry important in science?

Symmetry is important in science because it allows us to simplify and understand complex systems and patterns. It also plays a key role in predicting and explaining physical laws and phenomena.

4. Can symmetry be broken?

Yes, symmetry can be broken in certain situations. For example, in particle physics, the symmetry between matter and antimatter is broken, leading to the dominance of matter in the universe. In crystal structures, symmetry can also be broken due to defects or impurities.

5. Which comes first, symmetry or conservation?

Symmetry and conservation are closely intertwined and it is difficult to determine which comes first. Some scientists argue that symmetry is a fundamental concept that leads to conservation laws, while others argue that conservation laws dictate the types of symmetry that are possible. Ultimately, both concepts are essential in understanding the natural world.

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