Mass dimension of coupling constant -- always an integer?

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Discussion Overview

The discussion centers around the dimensionality of coupling constants in quantum field theory, specifically whether these dimensions can be rational numbers or must always be integers. The context involves constructing a Lagrangian with interaction terms that include spin-1 particles and fermions, exploring implications for Lorentz invariance and causality.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether the dimension of a coupling constant can be a rational number, citing a specific example where the dimensions sum to 7/2, leaving 1/2 for the coupling.
  • Another participant asserts that any Lagrangian with an odd number of fermions cannot be a Lorentz scalar, implying that this leads to integer mass dimensions.
  • A different participant raises the question of the physical consequences of having an interaction term with an odd number of fermions.
  • One participant argues that even when considering theories beyond the Standard Model (SM), the Lagrangian would still not be a Lorentz scalar if it contains an odd number of fermions, thus maintaining the requirement for integer dimensions.
  • Another participant suggests that a term with an odd number of fermions could be treated as a pseudoscalar, but acknowledges that it would not transform correctly under gauge transformations, leading to mathematical inconsistencies.
  • One participant elaborates on the implications of having an odd number of fermion fields in the Hamiltonian, noting that it could violate microcausality and affect the Lorentz invariance of the S matrix.

Areas of Agreement / Disagreement

Participants generally agree that having an odd number of fermions in a Lagrangian leads to issues with Lorentz invariance and scalar properties. However, there is disagreement regarding the implications of these issues and whether rational dimensions for coupling constants can exist.

Contextual Notes

Participants express limitations in their arguments based on the constraints of quantum field theory and the properties of Lorentz invariance, but do not resolve the question of whether coupling constants can have rational dimensions.

guest1234
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Just a simple question -- can the dimension of coupling constant be a rational number or should it always be an integer?

The question arose when I was trying to construct a Lagrangian with an interaction term involving two spin-1 particles and a fermion. The dimensions add up to 7/2, which leaves 1/2 for the coupling.
 
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Any Lagrangian with an odd number of fermions will not be a Lorentz scalar. Therefore you will always end up with an even number of fermions leading to an integer mass dimension.
 
Thinking beyond SM, what are the physical consequences if the Lagrangian contains an interaction term with an odd number (i.e. only one) of fermions?
 
Going beyond the SM does not help (in fact, this was my assumption in my first post, in the SM there definitely are no such interactions), your Lagrangian will still not be a Lorentz scalar. You simply cannot do this within the confinements of QFT without breaking Lorentz invariance.
 
Well I was thinking whether it'd be pseudoscalar (that shouldn't be any big problem -- if we really want a parity violating theory). I just thought about gauge transformations, and yup, you were right -- the term wouldn't transform as a (pseudo)scalar but as a spinor. Adding scalar and spinor terms together doesn't make any sense, even mathematically.
Thanks anyways
 
Last edited:
If the Lagrangian has a term with an odd number of fermion fields, so will have the Hamiltonian, representing energy density. Then due to the canonical equal-time anticommutators for fermions, in general the commutator of the Hamiltonian at equal times in general won't commute, which violates the principle of microcausality, which is pretty bad, because the the time ordering appearing in the S matrix won't be Lorentz invariant anymore, and you'd loose causality (and perhaps unitarity) of the S matrix.

To keep the story short: It simply doesn't make much sense :-).
 

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