Talk:Lattice quantum field theory

Reviewer B

Comments on "Lattice quantum field theory" by Gernot Munster

• p.6: The Euclidean path integral sort of drops out of the blue sky. A reference to the book "Quantum Physics" by Glimm and Jaffe might be helpful.

• Somewhere in the Section "Euclidean Field Theory" a reference to the theorem of Osterwalder and Schrader, providing the theoretical basis for the Euclidean method, would be in order.

• p.9: The introduction of the lattice constant a is somewhat misleading. It is certainly not a parameter to be inserted in a computer program for an evaluation (e.g. via Monte Carlo) of the path integral. It should be made clear that it is a derived quantity determined by the dynamics. This is actually explained later, in the Section "Continuum Limit", but it would be appropriate to mention it right away.

• p. 16: I think it is worth mentioning that the strong coupling expansion has a finite radius of convergence, whereas perturbation theory is expected to be divergent and at best asymptotic.

• pp. 18-19: It would be important to bring out the difference between `scaling' and `asymptotic scaling' (Eq. (5)). It should further be made clear that the former, essential for the existence of a continuum limit, is numerically quite well established, wheras the latter has (to the best of my knowledge) never been confirmed.

Moreover: Greens --> Green's, analytical --> analytic, Brillouin zone --> first Brillouin zone.

Further comment (Dec. 3rd, 2010): The reference to the Osterwalder-Schrader theorem should be

K.~Osterwalder and R.~Schrader, Commun.\ Math.\ Phys.\ {\bf 31} (1973) 83; Commun.\ Math.\ Phys.\ {\bf 42} (1975) 281.

It is important to cite both papers because the first one contains an error which is corrected in the second one.

Reviewer B (User:):

Further comment (Dec. 3rd, 2010): The reference to the Osterwalder-Schrader theorem should be

K.~Osterwalder and R.~Schrader, Commun.\ Math.\ Phys.\ {\bf 31} (1973) 83; Commun.\ Math.\ Phys.\ {\bf 42} (1975) 281.

It is important to cite both papers because the first one contains an error which is corrected in the second one.

SLAC derivative

A classical alternative to the finite differences approach is the so-called SLAC derivative introduced by J. M. Rabin in his 1981 Ph.D. Thesis at SLAC. As far as I know, it doesn't lead to fermion doubling, which is a significant advantage over Wilson's approach, at least from a conceptual point of view. It looks like it has sunk into oblivion today, but I don't know if it is for the good reasons... I am using it in my own works, with some success, it seems. Wouldn't it be interesting to mention it here, too?

Rabin, J. M. (1981). Long Range Interactions In Lattice Field Theory. Ph.D. Thesis SLAC-0240, Stanford (California, United States), 1981. http://www.slac.stanford.edu/pubs/slacreports/slac-r-240.html

As an example, here is a usage I made of it lately:

Fauvel, S. (2010). Quantum Ethics: A Spinozist Interpretation of Quantum Field Theory. CreateSpace Independent Publishing Platform, Scotts Valley (California, United States), 2013. http://quantum-ethics.org/Quantum%20Ethics.php