Why the LSZ Formula? Exploring QFT Texts

[Lapidus]

In some introductory QFT texts there is no mentioning of the LSZ formula, in others it takes a central place. Some authors draw Feynman diagrams and calculate S-matrix without ever bringing up correlation functions and the LSZ formula (which gives the relationship between S-matrix and correlation function).

Why is that so? What is gained by introducing correlation functions and the LSZ formula?

thanks

 

[vanhees71]

The LSZ reduction formula connects time-ordered Green's functions, which can be calculated perturbatively with help of Feynman diagrams in a systematic way, including renormalization for higher-order calculations, with observable quantities as are transition probabilities or cross sections. The LSZ reduction formula is thus a very central and important part of quantum field theory. I do not know, how to explain how to calculate observable quantities from QFT without it.

 

[Lapidus]

Thanks for answering vanhees71.

But what I find curious is that in many texts, at least introductory texts (such as Lahiri, Zee, Mandl, Tong, Griffiths,etc.), S-matrices, transition probabilities and cross sections are computed with Feynman diagrams but without correlation functions and the LSZ formula.

Again my question, why correlation functions and the LSZ formula?
What is the benifit gained here?

 

[vanhees71]

It depends a bit on what you like to do. Some texts give handwaving arguments to come as quickly as possible to applications without developing much foundational arguments to arrive at Feynman-diagram rules to calculate matrix elements. This is legitimate to begin with and may help to develop some intuitive understanding of the quantities measured in high-energy physics.

The backside is that such texts do not give a clear understaning of the theoretical foundation of the whole picture gained in the last 50 years about quantum field theory, which can be seen as a rich toolbox to understand a whole lot of physics on the same footing, reaching from elementary-particle physics (the standard model of elementary particle physics as the paradigmatic application of relativistically covariant vacuum quantum field theory), nuclear and hadron physics (effective hadronic models, based on (approximate) chiral symmetry of QCD) to condensed-matter theory (non-relativistic many-body quantum field theory), critical phenomena and phase transitions (renormalization group techniques etc.).

The foundation of this theoretical framework is the space-time structure (Minkowski for the (special) relativistic and Galilei for the non-relativistic case, respectively) and the basic structure of quantum theory (Hilbert space of states and operator algebra for observables). From this point of view you derive from a minimal set of assumptions finally observable quantities like S-matrix elements (cross sections), transport coefficients etc. and for this purpose the Green's functions of various kinds (which can be seen as generalized correlation functions) are the most convenient quantities. The LSZ reduction is at the heart of scattering theory, defining the notion of "asymptotically free states" as good as one can. Despite the fact that there is no complete mathematical proof for the existence of any non-trivial realistic relativistic QFT in (1+3) dimensions, its successes using perturbative techniques (including the very efficient diagram formulation) speak for themselves, and to understand really what a cross section means, in my opinion, there's now other way than to go through the endeaver to study the Green's-function techniques and the LSZ reduction formalism.

A very good source for relativistic vacuum QFT are Weinberg's three volumes 'The Quantum Theory of Fields'. There he explains why relativistic QFT looks the way it looks. As I said before, maybe it's however better to look at a somewhat more intuitive approach first to get familiar with the subject. There I'd suggest Ryder as a solid source. I cannot recommend Zee, which on the first glance sounds funny and well-written but I don't think that one can really understand what's going on in QFT. Peskin/Schroeder isn't too bad, e.g., it gives a very nice explanation of scattering theory using the wave-packet approach, which is not so well described in most other textbooks that give usually only some hand-waving arguments and give scattering theory the touch of a "difficult subject". This book is, however, not too good at other places like renormalization theory, where sometimes the approach is too sloppy (e.g., there appear dimensionful quantities in logarithms; the self-energy in the non-linear sigma model is renormalized at the branch cut etc.). As I said, imho Weinberg's 3 vols. make the best textbook for relativistic QFT.

 

[A. Neumaier]

Thanks for answering vanhees71.

But what I find curious is that in many texts, at least introductory texts (such as Lahiri, Zee, Mandl, Tong, Griffiths,etc.), S-matrices, transition probabilities and cross sections are computed with Feynman diagrams but without correlation functions and the LSZ formula.

Again my question, why correlation functions and the LSZ formula?
What is the benifit gained here?

LSZ and correlation functions give the theoretical explanation why the diagram rules work (and precisely which version works) as they do, in the most general case.

This can be seen from Weinberg's book. he first derives the rules for a scalar field in a heuristic way (in Chapter 6), then again in Section 9.4 from the path integral, and finally in Chapter 10.2-3 in terms of LSZ. This last version is the most general situation and works even in cases (such as the nonlinear sigma model or nonabelian gauge theories)
where the former two derivations become ambiguous or outright false.

 

[Lapidus]

Vanhees71 and A.Neumaier, thanks for your posts.

Let's read Weinberg!

 

[DrDu]

As far as I understand it, work before LSZ was based on Low's adiabatic switching technique where one tries to correlate the ground state of the interacting theory to the one of the non-interacting theory. Formally this approach fails due to infinite renormalization whence the two ground states become orthogonal to each other so that the one cannot be derived from the other.

 

[sumeetkd]

Thanks to the pro's for regularly replying.
Much appreciated.