By D.E. Newland
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Great for engineers and pros, too simple for mathematicians.
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Extra resources for An Introduction to Random Vibrations, Spectral & Wavelet Analysis: Third Edition
A, >+» = 0. 5) t--- N,(x) ,Y~(x) ,Y2(x) = L e-•••• and p2 (p) = I: e'•••, 0 whence they evidently commute. 5) only through the almost "accidental" cancellation of two operators, each of which may diverge in the field-theory limit when N = co . We now show that in that limit the operators in fact no longer cancel, by evaluating the commutator using form (b) for the density operators. It is a matter of only some minor manipulation to obtain the important new result: [Ho, p,(±p) ] = ±pp,(±p), o,...
1. Introduction The crucial observation that the low energy excitations of one-dimensional metals are well described in terms of 1+1 massless Dirac fermions dates back to Tomonaga1 and it was one of the earliest examples of emerging Quantum Field Theory (QFT) description in condensed matter physics; an idea which, later on, found several applications including the recent one to graphene (which is described by Dirac fermions in d = 2 + 1). Subsequently Luttinger,2 unaware of the Tomonaga results, proposed a model, nowadays called Luttinger model, based on the same ideas of Tomonaga: the quasi-particle excitations are described in terms of two kinds of spinless fermions with linear dispersion relation, and a Dirac sea is introduced ﬁlling the inﬁnite states with negative energy.
S. Wightman, for pointing out to us that P. ' 'P. Jordan, Z. Physik 93, 464 ( 1935); 98, 759 (1936); 99, 109 (1936); 102, 243 ( 1936); lOS, ll4 (1937); 105, 229 ( 1937). M. Born and N. Nagendra-Nath, Proc. Ind. Acad. Sci. 3, 318 ( 1936). A. Sokolow, Phys. Z. der Sowj. 12, 148 ( 1937). July 8, 2013 14:17 BC: 8875 — Luttinger Model: The First 50 Years and Some New Directions This page intentionally left blank divider1 July 8, 2013 14:18 BC: 8875 — Luttinger Model: The First 50 Years and Some New Directions Chapter II Lattice, Dynamical and Nonlinear Effects divider2 July 8, 2013 14:17 BC: 8875 — Luttinger Model: The First 50 Years and Some New Directions This page intentionally left blank divider1 August 12, 2013 10:46 BC: 8875 — Luttinger Model: The First 50 Years and Some New Directions 03 25 LUTTINGER MODEL AND LUTTINGER LIQUIDS∗ VIERI MASTROPIETRO University of Rome Tor Vergata, Mathematics Department, Viale della Ricerca Scientifica 00133, Rome, Italy The Luttinger model owes its solvability to a number of peculiar features, like its linear relativistic dispersion relation, which are absent in more realistic fermionic systems.