# A question about the Bosonization of the Thirring model

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Is there a way or sense in which one can Bosonize this kind of a Lagrangian,

$L = \bar{\psi}\gamma^\mu \partial _\mu \psi + f(x) \bar{\psi}\psi$

for $f(x)$ being some function on space-time.

• What is the most pedagogic reference which explains the Thirring Model/Sine-Gordon bosonization from scratch?
This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user user6818
asked Feb 20, 2014

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Try the lecture Classical Lumps and their Quantum Descendants (Lecture 6) from Sidney Coleman's Aspects of Symmetry - Selected Erice Lectures (Cambridge, 1988), pp. 185-264. I can think of no better place, since not only is Coleman a great expositor, but he also was the one who discovered this stuff.

This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user Pedro Lauridsen Ribeiro
answered Feb 20, 2014 by (580 points)
Does it deal with the example that I was talking of? (..I have tried reading one of Coleman's original papers and it was very hard to read - he used a lot of previous results from some Klaiber - and it became a mess...)

This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user user6818
Sorry, I should have said that my answer refers to your second question (i.e. the Thirring / sine-Gordon bosonization). As for your first question, one must have in mind that bosonization of fermion fields only works in 1+1 dimensions (this is missing from your question). That being said, it doesn't look much different from bosonization of free fermion fields, although your model may get messy depending on the form of $f$. Do you have a specific $f$ in mind?

This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user Pedro Lauridsen Ribeiro
Can you explain as to how do you think the 1+1 bosonization will work with that $f$? It seems you think it will! (...also it seems that Bosonization of these theories makes sense for higher-dimensions also - arxiv.org/pdf/hep-th/9509173v2.pdf )

This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user user6818
Your model is still free in the sense that the field equations are linear, but you no longer have a translation invariant vacuum state if $f$ is not constant. Nevertheless, you still have quasi-free reference states (i.e. such that all truncated $n$-point functions vanish for $n>2$) whose 2-point function have a short distance behavior similar to that of a vacuum state (the so-called Hadamard states). If one works with one of these states, the bosonization procedure shouldn't be that different from the standard one applied to Dirac fields in 1+1 dimensions.

This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user Pedro Lauridsen Ribeiro
Bosonization of Dirac (free) fields in 1+1 dimensions, on its turn, can be found in books on string theory such as Green-Schwarz-Witten or Polchinski.

This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user Pedro Lauridsen Ribeiro
I didn't understand your comment about Hadamard stats etc - can you give a reference? I checked Witten's lectures on Bosonization in the book "QFT and strings for physicists and mathematicians" (vol 2) - what I understand of his argument there is that the bosonization map between the fermionic operators and the bosonic operators is independent of the form of the interaction - that map is the same always as for free theories on both the sides.

This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user user6818
Then I think that the equivalent bosonized Lagrangian for my case with a non-trivial $f$ should be whatever is gotten by doing the usual replacements of the free fermionic operators by their free boson duals - right?

This post imported from StackExchange Physics at 2014-04-25 01:58 (UCT), posted by SE-user user6818

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