Physics – Condensed Matter – Statistical Mechanics
Scientific paper
Jun 2005
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2005cqgra..22.2307z&link_type=abstract
Classical and Quantum Gravity, Volume 22, Issue 11, pp. 2307 (2005).
Physics
Condensed Matter
Statistical Mechanics
Scientific paper
Jean Zinn-Justin's textbook Path Integrals in Quantum Mechanics aims to familiarize the reader with the path integral as a calculational tool in quantum mechanics and field theory. The emphasis is on quantum statistical mechanics, starting with the partition function Tr exp(-β H) and proceeding through the diffusion equation to barrier penetration problems and their semiclassical limit. The 'real time' path integral is defined via analytic continuation and used for the path-integral representation of the nonrelativistic S-matrix and its perturbative expansion. Holomorphic and Grassmannian path integrals are introduced and applied to nonrelativistic quantum field theory. There is also a brief discussion of path integrals in phase space. The introduction includes a brief historical review of path integrals, supported by a bibliography with some 40 entries.
As emphasized in the introduction, mathematical rigour is not a central issue in the book. This allows the text to present the calculational techniques in a very readable manner: much of the text consists of worked-out examples, such as the quartic anharmonic oscillator in the barrier penetration chapter. At the end of each chapter there are exercises, some of which are of elementary coursework type, but the majority are more in the style of extended examples. Most of the exercises indeed include the solution or a sketch thereof. The book assumes minimal previous knowledge of quantum mechanics, and some basic quantum mechanical notation is collected in an appendix.
The material has a large overlap with selected chapters in the author's thousand-page textbook Quantum Field Theory and Critical Phenomena (2002 Oxford: Clarendon). The stand-alone scope of the present work has, however, allowed a more focussed organization of this material, especially in the chapters on, respectively, holomorphic and Grassmannian path integrals.
In my view the book accomplishes its aim admirably and is eminently usable as a textbook for physics students. My only significant regret is that the author has given the inquisitive student - or indeed the instructor to whom the student's queries might be addressed - little guidance on where to seek more precise statements on the validity of the mathematical techniques. This issue arises even in a few places where the mathematics is not particularly advanced, such as complex contour deformations in finite-dimensional Gaussian integrals. There are also a few places where the text alludes without preparation to concepts that may not be familiar to a typical reader, such as Trotter's formula or the Fredholm determinant. I should be delighted if a future edition were able to address these concerns without compromising the present book's readability, perhaps by including judicious references also in the main text rather than just in the historical introduction.
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