QHE: rotational invariance, no terms linear in E or B

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SUMMARY

The discussion centers on the concept of rotational invariance in the context of three-dimensional space (d = 3 + 1 dimensions) and its implications for terms involving electric field (E) and magnetic field (B). It is established that due to rotational invariance, terms linear in E or B cannot be formulated, as vectors change direction under rotation. Instead, only terms involving the magnitudes of these vectors, such as E.E and B.B, maintain invariance, as their lengths remain unchanged regardless of rotation.

PREREQUISITES
  • Understanding of rotational invariance in physics
  • Familiarity with vector mathematics
  • Knowledge of electric and magnetic field concepts
  • Basic grasp of dimensional analysis in theoretical physics
NEXT STEPS
  • Study the implications of rotational invariance in quantum field theory
  • Explore vector calculus, focusing on the properties of vector magnitudes
  • Investigate the role of electric and magnetic fields in classical electromagnetism
  • Review David Tong's notes on Quantum Hall Effect (QHE) for deeper insights
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This discussion is beneficial for theoretical physicists, students of electromagnetism, and researchers focusing on quantum field theories, particularly those interested in the Quantum Hall Effect and its mathematical foundations.

binbagsss
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'Let’s first see what all of this means in the context of d = 3 + 1 dimensions. If we have rotational invariance then we can’t write down any terms linear in E or B. The first terms that we can write down are instead ...'

Why is this? I don't understnad . My thoughts would be pictruing the set up needing to be rotationally invariant, and since E and B are perpendicular a linear term alone wouldn't do this, but this wouldn't explaiin why E.E and B.B are invariant, I don't really know what I'm talking about, as you can tell

many thanks in advancehttp://www.damtp.cam.ac.uk/user/tong/qhe/five.pdf , page 145 David Tong notes QHE , chapter 5 (eq. 5.5)
 
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binbagsss said:
Why is this?

Because vectors are not invariant under rotations: the rotation makes them point in a different direction. But the magnitudes of vectors are invariant under rotations, since rotation doesn't change the length of a vector, only its direction.
 
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