In Coleman's paper, "Quantum sine-Gordon equation as the massive Thirring Model" (link to Phys Rev D article), he pointed out that the massless Thirring Model is exactly scale invariant. More over, the massless part scales as $$H_0(x) \rightarrow \lambda^2 H_0(\lambda x)$$ What does it mean here by scale invariance, and why does it imply that the Hamiltonian scales as this?

Note the Thirring Model has Lagrangian $$\cal{L} = i\bar{\psi}\partial_{\mu}\gamma^{\mu}\psi - \frac{g}{2} j_{\mu} j^{\mu}$$


1 Answer 1


Scale invariance refers to invariance under scaling the coordinates i.e., $x^\mu \rightarrow \lambda\ x^\mu$ ($\mu=0,1$ in this case). One needs to associate a (naive) scaling dimension to the fields -- this is done as follows. Suppose that $$ \psi(\lambda\ x) = \lambda^\Delta\ \psi(x)\ . $$ Plug this into the action and use the kinetic term to figure out a value for $\Delta$. For your Lagrangian, one obtains $\Delta=-1/2$. Since $j_\mu$ is a fermion bilinear, it transforms as $j_\mu(\lambda\ x)= \lambda^{-1} j_\mu(x)$. Thus, the term $j_\mu j^\mu$ goes as $\lambda^{-2}$ -- this exactly cancels the $\lambda^2$ coming from the integration measure $d^2x$ in the action. This implies that the action is scale-invariant classically. Equivalently, one says that the coupling constant $g$ has zero scaling dimension. Quantum corrections can change this conclusion and Coleman says that this is true quantum mechanically as well -- that is what is meant by the adjective "exact". Suppose you add a mass term -- that will break scale-invariance as you can/must check.

  • $\begingroup$ Thank you! That is of great help. One more question: is there any relevant paper discussing scale invariance quantum mechanically? Also as I understand it, when adding a mass term one cannot obtain anomalous dimension from classical argument right? Then what is the rule to follow when one compute the anomalous dimension? (like requiring the Green's function to be invariant?) Correct me if there is something unclear. $\endgroup$
    – Kevin Ye
    Commented Feb 12, 2014 at 6:19
  • $\begingroup$ Anomalous dimension is by definition the quantum correction to the classical scaling dimension. One usually defines it for operators by computing the their two-point function. Search for conformal invariance on the web. Also try to read Coleman's Erice lectures available as the book "Aspects of Symmetry". $\endgroup$
    – suresh
    Commented Feb 12, 2014 at 13:15

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