Spontaneous symmetry breaking in the standard model

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Discussion Overview

The discussion revolves around the concept of spontaneous symmetry breaking within the framework of the standard model of particle physics. Participants explore the implications of Lorentz invariance on the vacuum expectation values of scalar, spinor, and vector fields, and the mathematical formalism related to these concepts.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that only scalar fields are relevant in the context of spontaneous symmetry breaking due to their vacuum expectation values, while spinor and vector fields are excluded because they would violate Lorentz invariance.
  • Others propose that if a spinor or vector field had a nonzero expectation value, it would introduce a preferred direction, contradicting the requirement for a rotationally invariant vacuum state.
  • A participant seeks a more formal mathematical explanation regarding the vacuum expectation value of spinor fields, questioning how Lorentz invariance leads to the conclusion that \(\langle 0|\psi_\alpha|0\rangle=0\).
  • There is a query about the behavior of propagators, specifically why they do not vanish despite having Lorentz or spinor indices, suggesting a complexity in the relationship between vacuum expectation values and propagators.

Areas of Agreement / Disagreement

Participants express differing views on the implications of Lorentz invariance for various field types, and the discussion remains unresolved regarding the formal mathematical proofs and the behavior of propagators.

Contextual Notes

Limitations include the need for further clarification on the mathematical formalism and the assumptions underlying the claims about vacuum expectation values and propagators.

synoe
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In the standard model, the Lagrangian contains scalar and spinor and vector fields. But when we consider spontaneous symmetry breaking, we only account for the terms contain only scalar fields, " the scalar potential", in the Lagrangian. And if the scalar fields have vacuum expectation value, then we recognize the symmetry is broken spontaneously. Why don't we have to contain spinors and vectors? I think it's related to Lorentz invariance.
 
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synoe said:
Why don't we have to contain spinors and vectors? I think it's related to Lorentz invariance.
If a spinor or vector had a nonzero expectation value, it would single out a preferred direction. The vacuum state must be rotationally invariant (Lorentz invariant too) so this is not possible.
 
Thank you, Bill_K.
Could you explan more formally or mathematically by using language of quantum theory?
I would like to proof \langle 0|\psi_\alpha|0\rangle=0 from Lorentz invariance of the vacuum. Your explanation seems that the vacuum expectation vales is supposed to be Lorentz invariant but I think Lorentz invariance means U^\dagger(\Lambda)|0\rangle=|0\rangle.

And another question occurred. Your explanation seems that the propagator, the vacuum expectation value of two point function also vanishes. Why propagators don't vanish even if they have Lorentz indices or spinor indices?
 
synoe said:
Thank you, Bill_K.
Could you explan more formally or mathematically by using language of quantum theory?
I would like to proof \langle 0|\psi_\alpha|0\rangle=0 from Lorentz invariance of the vacuum. Your explanation seems that the vacuum expectation vales is supposed to be Lorentz invariant but I think Lorentz invariance means U^\dagger(\Lambda)|0\rangle=|0\rangle.

And another question occurred. Your explanation seems that the propagator, the vacuum expectation value of two point function also vanishes. Why propagators don't vanish even if they have Lorentz indices or spinor indices?

See post#35 in
www.physicsforums.com/showthread.php?t=172461

Sam
 

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