# Ward identities in Minimal Subtraction Scheme

• gheremond
In summary, Ward identities in the Minimal Subtraction Scheme are a set of equations that relate the behavior of physical quantities under changes in the parameters of a theory. They are used in quantum field theory to ensure that the theory is consistent and that calculations are renormalization scheme independent. The Ward identities are particularly useful in the Minimal Subtraction Scheme, a popular renormalization scheme used in perturbative calculations in quantum chromodynamics. They allow for the cancellation of divergences and ensure the preservation of gauge invariance in the theory. Overall, Ward identities are a powerful tool in theoretical physics and play a crucial role in understanding and predicting the behavior of particles and interactions in quantum field theory.
gheremond
In QED, the Ward identities set $$Z_1=Z_2$$ and $$Z_1 - 1 = \left. {\frac{{d\Sigma \left( p \right)}}{{dp}}} \right|_{p = m}$$. This can be shown explicitly for the 1-loop calculations if one uses an on-shell subtraction scheme, where the renormalized mass and charge are identical to the experimentally measured ones. What happens if one uses instead a subtraction scheme like MS or $$\overline {MS}$$? Arguably the Ward identities should hold again, but in calculating the various Z functions, one finds that $$Z_1 - 1 = \left. {\frac{{d\Sigma \left( p \right)}}{{dp}}} \right|_{p = m}$$ only holds at the level of the infinite contributions (poles), while there are some finite differences. Is that supposed to happen? Do the identities only refer to the infinite parts of the renormalization functions?

Whoa! I didn't expect it would get that deep! I would assume that things would be pretty clear for a theory like QED, which has be beaten into a pulp over decades of work. Great reference too! Thanks!

## 1. What are Ward identities in the Minimal Subtraction Scheme?

Ward identities are a set of equations that relate the Green's functions of different fields in a quantum field theory. They are a consequence of the symmetry of the theory under certain transformations and play an important role in the renormalization process.

## 2. How are Ward identities derived in the Minimal Subtraction Scheme?

Ward identities are derived using the Noether's theorem and the Ward-Takahashi identities. The Noether's theorem states that for every continuous symmetry of the theory, there exists a conserved current. The Ward-Takahashi identities relate the conserved currents to the Green's functions of the theory.

## 3. What is the significance of Ward identities in renormalization?

Ward identities are important in renormalization because they provide a powerful tool for checking the consistency of the theory. If the theory satisfies the Ward identities, it is a sign that the renormalization procedure has been carried out correctly.

## 4. Can Ward identities be violated in the Minimal Subtraction Scheme?

Yes, Ward identities can be violated in the Minimal Subtraction Scheme if the renormalization procedure is not carried out correctly. This can happen if there are hidden divergences in the theory or if the regularization method is not appropriate. Violations of Ward identities can also indicate the presence of new physics beyond the current theory.

## 5. Are Ward identities unique to the Minimal Subtraction Scheme?

No, Ward identities are not unique to the Minimal Subtraction Scheme. They are a general feature of quantum field theories and can be derived in any renormalization scheme. However, the Minimal Subtraction Scheme is a popular choice because it simplifies the calculations and leads to a unique set of equations known as the Callan-Symanzik equations.

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