Noether current for exchange symmetry

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Discussion Overview

The discussion revolves around the conservation laws associated with exchange symmetry in Quantum Mechanics, particularly in the context of Noether's theorem and the nature of symmetries. Participants explore whether exchange symmetry can be treated as a continuous symmetry and its implications for conservation laws.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant questions what is conserved when demanding exchange symmetry for identical particles, noting a lack of clarity from their professor and online resources.
  • Another participant asserts that exchange symmetry is not a discrete symmetry, suggesting that Noether's theorem does not apply and implying there is no conserved infinitesimal generator.
  • A similar viewpoint is expressed, emphasizing that the symmetry in question is not differentiable, which may affect the applicability of Noether's theorem.
  • One participant proposes that exchange symmetry could potentially be formulated as a continuous and differentiable unitary group by mixing particles rather than exchanging them entirely, contingent on recovering the discrete symmetry for a specific mixing angle.
  • A participant recalls having seen an explicit construction of the exchange operator in terms of position and momentum variables, indicating interest in further exploration of this topic.
  • There is a request for a link to the mentioned paper, but another participant indicates they no longer have access to it.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of Noether's theorem to exchange symmetry, with some asserting it is not applicable while others explore the potential for a continuous formulation. The discussion remains unresolved regarding the implications of these differing perspectives.

Contextual Notes

There are limitations regarding the definitions of symmetry being discussed, particularly the distinction between discrete and continuous symmetries, and the implications for conservation laws are not fully explored or agreed upon.

Ravi Mohan
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In Quantum Mechanics, when we exchange identical particles the physics doesn't change. I wonder what stuff is conserved when this symmetry is demanded.
I asked my professor but he didnt/couldnt answer. Google is no help either.
 
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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.
I find this helpful too
http://ajp.aapt.org/resource/1/ajpias/v64/i7/p840_s3?isAuthorized=no
 
Last edited by a moderator:
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.
 

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