How does spin relate to the symmetry of a particle?

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Discussion Overview

The discussion centers on the relationship between spin and the symmetry properties of identical particles, specifically addressing why particles with half-integer spins exhibit anti-symmetry while those with integer spins are symmetric. The scope includes theoretical aspects and references to established theorems in quantum mechanics.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the relationship between spin and symmetry in identical particles, specifically regarding half-integer versus integer spins.
  • Another participant references the spin statistics theorem, explaining that symmetry and anti-symmetry pertain to many-particle systems and how the wave function behaves under particle exchange.
  • A different participant suggests that Weinberg's work provides a comprehensive proof applicable to any spin, contrasting it with other texts that focus on simpler cases.
  • One participant shares a resource by John Baez that may offer additional insights into the topic.

Areas of Agreement / Disagreement

Participants express differing views on the proofs and treatments of the spin statistics theorem, with no consensus on a singular approach or understanding of the topic.

Contextual Notes

Some limitations include the complexity of proofs, the varying levels of treatment in different texts, and the potential for differing interpretations of symmetry in quantum mechanics.

kregg87
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Why are particle with half integer spins anti-symmetric while integer spin particles are symmetric? Or in other words, how does spin relate to the symmetry of indentical particles?
 
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This is the spin statistics theorem. Google will find many references (it's hard to overstate its importance) but I am not aware of any proofs that work for an I-level thread.

One thing you do want to be clear about (you may already know this, but it's not certain from your post): The symmetry and anti-symmetry we're referring to is a property of many-particle systems. If the wave function of the entire system changes sign when any two particles are exchanged, we say that is is anti-symmetric, while if the wave function is unchanged we say it is symmetric.
 
I'd say the proof in Weinberg's, Quantum theory of fields is the best one in the sense that it treats the general case of any spin. That's A level. In most other textbooks you find the treatment of the most simple cases of spin 0, 1/2, and 1 (relevant for the elementary particles in the Standard Model), which is I level. In Germany the introduction to QFT is in the advanced undergraduate course (leading towards an MSc degree).
 

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