Noether current for exchange symmetry

• Ravi Mohan
In summary, the conversation discusses the exchange symmetry in Quantum Mechanics and the conservation of certain properties when this symmetry is demanded. Noether's theorem is not applicable in this case and there is no infinitesimal generator that is conserved. The exchange symmetry can be formulated in terms of a continuous and differentiable unitary group, but it must be done in a way that recovers the discrete symmetry. The American Journal of Physics has an article that discusses the explicit construction of the exchange operator in terms of position and momentum, but the link is not available.

Ravi Mohan

In Quantum Mechanics, when we exchange identical particles the physics doesn't change. I wonder what stuff is conserved when this symmetry is demanded.

That's not a discrete symmetry so Noether's theorem is not applicable and there is no infinitesimal generator which is conserved.

DrDu said:
That's not a discrete symmetry so Noether's theorem is not applicable
I think you mean "That's not a differentiable symmetry ..."

Ok got it. Thanks.
http://ajp.aapt.org/resource/1/ajpias/v64/i7/p840_s3?isAuthorized=no [Broken]

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It should be possible to formulate the exchange symmetry in terms of a continuous and differentiable unitary group, by means of mixing the particles instead of exchanging them entirely. Of course you'd have to do it in a way that recovers the discrete symmetry for a certain mixing angle.

Cheers,

Jazz

DaleSpam said:
I think you mean "That's not a differentiable symmetry ..."

Of course, thanks!

Jazzdude said:
It should be possible to formulate the exchange symmetry in terms of a continuous and differentiable unitary group, by means of mixing the particles instead of exchanging them entirely. Of course you'd have to do it in a way that recovers the discrete symmetry for a certain mixing angle.

Cheers,

Jazz

I remember having seen in American Journal of Physics an explicit construction of the exchange operator in terms of x and p.

DrDu said:
I remember having seen in American Journal of Physics an explicit construction of the exchange operator in terms of x and p.

I would like to read that paper. Can you give the link?

Ravi Mohan said:
I would like to read that paper. Can you give the link?

I fear not as I no longer have access to Am J Phys.

1. What is the Noether current for exchange symmetry?

The Noether current for exchange symmetry is a mathematical concept in physics that describes the conservation of a physical quantity related to the symmetry of exchanging two identical particles in a system. It is based on the Noether's theorem, which states that for every continuous symmetry of a system, there exists a corresponding conserved quantity.

2. How does the Noether current for exchange symmetry relate to particle exchange symmetry?

The Noether current for exchange symmetry is directly related to the exchange symmetry of particles in a system. This means that if the physical laws governing the system remain unchanged when two identical particles are exchanged, then there exists a conserved quantity associated with this symmetry, known as the Noether current.

3. What is the significance of the Noether current for exchange symmetry?

The Noether current for exchange symmetry is significant because it provides a way to understand and analyze the conservation laws in physical systems where there is an exchange symmetry present. It helps to explain the conservation of certain quantities, such as energy and momentum, in these systems.

4. Can the Noether current for exchange symmetry be applied to all physical systems?

Yes, the Noether current for exchange symmetry can be applied to all physical systems that exhibit exchange symmetry. This includes systems in classical mechanics, quantum mechanics, and field theory.

5. How is the Noether current for exchange symmetry calculated?

The Noether current for exchange symmetry is calculated using the Noether's theorem, which involves finding the continuous symmetry transformation of the Lagrangian of a system. This transformation will lead to the conserved quantity associated with the exchange symmetry, which is the Noether current.