Abstract From the previous chapters on linear response theory and Green’s functions, it is clear that complete calculations of thermal averages of time-dependent phenomena in quantum field theory are a rather formidable task. Even the basic imaginary time-evolution operator Û (τ ) itself is an infinite series to all orders in the interaction Û(ϊ) . One simply faces the problem of getting lost in the myriads of integrals, and not being able to maintain a good physical intuition of which terms are important. In 1948 Feynman solved this problem as part of his seminal work on quantum electrodynamics by inventing the ingenious diagrams that today bear his name. The Feynman diagrams are both an exact mathematical representation of perturbation theory to infinite order and a powerful pictorial method that elucidate the physical content of the complicated expressions. In this chapter, we introduce the Feynman diagrams for the case of non-interacting particles in an external potential. Our main example of their use will be the analysis of electron-impurity scattering in disordered metals.
Abstract From the previous chapters on linear response theory and Green’s functions, it is clear that complete calculations of thermal averages of time-dependent phenomena in quantum field theory are a rather formidable task. Even the basic imaginary time-evolution operator Û (τ ) itself is an infinite series to all orders in the interaction Û(ϊ) . One simply faces the problem of getting lost in the myriads of integrals, and not being able to maintain a good physical intuition of which terms are important. In 1948 Feynman solved this problem as part of his seminal work on quantum electrodynamics by inventing the ingenious diagrams that today bear his name. The Feynman diagrams are both an exact mathematical representation of perturbation theory to infinite order and a powerful pictorial method that elucidate the physical content of the complicated expressions. In this chapter, we introduce the Feynman diagrams for the case of non-interacting particles in an external potential. Our main example of their use will be the analysis of electron-impurity scattering in disordered metals.
At the end of the last chapter, we considered the electron-electron interaction. Here, we found two terms: the Hartree energy and the exchange energy. These work fine if we can solve for the energy exactly. However, as was remarked, this can only be done for a few electron systems. For more meaningful systems, we need to develop an approximation technique for the interacting Green’s functions. The standard approach to this is via the interaction representation, which gives us a prescription for developing the perturbation series. As one might infer from our previous discussion, this will lead to multiple, messy integrals. Fortunately, the Feynman diagrams give us an elegant and pictorial method of looking at the various terms in the perturbation series. As a result, they also give us a much more intuitive method of examining the various terms.
In Chapter 9 we applied the second quantization to the Hartree–Fock method. In this chapter we provide the basic aspects of the thermal Green function method, which is convenient for systematic perturbative expansion beyond the mean-field approximation. Several problems in quantum statistical mechanics can be elegantly and compactly described in terms of the so-called Matsubara Green functions (Matsubara (1955); Abrikosov et al. (1963)). Hence, if on one hand, the density matrix description, often used in the previous chapters, has a more direct physical interpretation, on the other hand the Green function method is far more powerful and takes advantage of a handy diagrammatic perturbation expansion. In the following chapters, we rely on the Green function for a more complete treatment of some many-body problems. To be self contained we introduce here this technical chapter with a short presentation of the by now standard Green function method. For a comprehensive treatment we refer the reader to the many books available and in particular to Abrikosov et al. (1963). The Matsubara Green function The main aim of quantum statistical mechanics is the evaluation of the partition function (see Eqs. (6.31) and (6.32)), e.g. in the grand canonical ensemble In the case of quantum gases one can evaluate Ƶ exactly as we did in Chapter 7. In general, the evaluation of the partition function is a difficult task and it is useful to use perturbation theory to obtain approximate solutions. To set up a perturbation theory, one begins by splitting the Hamiltonian as H = H 0 + H i, where H 0 represents the “easy” part, which can be solved exactly. In the second step, one could naturally think of expanding the exponential of the interacting part H i in order to write the partition function as a perturbative series. However, such a procedure requires extreme care due to the non-commutative nature of operators in quantum mechanics. The way of solving this problem is via the “ time ” ordering of operators in each term of the perturbative expansion. It can be shown, as we shall see in the present chapter, that the quantum statistical average of any time-ordered sequence of operators with respect to the unperturbed Hamiltonian H 0 can be expressed in terms of elementary quantum statistical averages of two field operators. Such elementary averages are called Green functions and in this section we introduce the basic definitions concerning them.
In this chapter, we introduce and explain one of the fundamental tools in the study of dynamic critical phenomena, namely dynamic perturbation theory. Inevitably, large parts of Chapter 4 need to be rather technical. Complementary to the straightforward iterative method for the solution of non-linear Langevin equations presented in Section 3.2.3, we now describe the more elegant and efficient field-theoretic techniques. Yet both for the derivation of general properties of the perturbation series and for the sake of practical calculations, the response functional and the following elaborations on Feynman diagrams, cumulants, and vertex functions prove indispensable. As is the case with every efficient formalism, once one has become acquainted with the Janssen–De Dominicis response functional for the construction of dynamic field theory, and therefrom the perturbation expansion in terms of vertex functions, it serves to save a considerable amount of rather tedious work. For the purely relaxational O ( n )-symmetric models A and B, we derive the fluctuation-dissipation theorem within this formalism, and then systematically construct the perturbation series and its diagrammatic representation in terms of Feynman graphs. In order to reduce the required efforts to a minimum, the generating functionals first for the cumulants, and then, motivated by Dyson's equation for the propagator, for the one-particle irreducible vertex functions are examined. As an example, the relevant vertex functions for models A and B are explicitly evaluated to two-loop order. Furthermore, alternative formulations of the perturbation expansion are discussed, and the Feynman rules are given in both the frequency and the time domain. The results of this chapter provide the foundation for the renormalization group treatment in the subsequent Chapter 5.
Develops the many-body one-particle Green function, and explains its physical interpretation in terms of the spectral function, self-energy, and quasiparticle. lifetime. Its application in angle-resolved photoemission spectroscopy is presented in detail. The time-evolution operator in the interaction picture is derived, and time-ordering and adiabatic switching-on are introduced as precursor tools to construct the Feynman–Dyson many-body perturbation theory. A detailed account of Wick’s theorem, normal ordering, and contractions is outlined. Feynman diagrams are constructed, and the emergence of the infinite Dyson series from irreducible diagrams is outlined. Two-particle Green function and the particle-hole excitation spectra are developed. Diagramatic application of RPA for interacting systems is described. The finite-temperature Matsubara Green function is introduced and developed, together with its Fourier series representation and the evaluation of Matsubara sums.
1. First and second quantization 2. The electron gas 3. Phonons: coupling to electrons 4. Mean field theory 5. Time evolution pictures 6. Linear response theory 7. Transport in mesoscopic systems 8. Green's functions 9. Equation of motion theory 10. Transport in interacting mesoscopic systems 11. Imaginary time Green's functions 12. Feynman diagrams and external potentials 13. Feynman diagrams and pair interactions 14. The interacting electron gas 15. Fermi liquid theory 16. Impurity scattering and conductivity 17. Green's functions and phonons 18. Superconductivity 19. 1D electron gases and Luttinger liquids A. Fourier transformations B. Exercises C. Index
Feynman diagrams have fascinated physicists and philosophers since they were introduced to the world about 70 years ago. Clearly, they help in calculation; they have allowed nearly impossible problems to be solved with relative ease. This is agreed by all, but that is probably where the consensus ends. Are they pictures of physical processes? Are they just devices for keeping track of mathematical formulae, that do the real work? Are they some sort of mix of both?They are almost as famous as representations of the Bohr atom, even though most people could not give even a sketchy account of what they do. In spite of this they have taken on a role as a cultural icon. Feynman used them to decorate his own van (Fig. 1). The van was even used in an episode of the popular TV series The Big Bang Theory. They have been used as art work in everything from posters, to coffee mugs, to wallpaper. But none of this is particularly helpful in understanding how they work.If we ask working physicists who use Feynman diagrams what it is they are doing, we get a variety of responses. One is puzzlement; a shrug of indifference. Even those very skilled at using Feynman diagrams might be hard pressed to explain (to a philosopher’s satisfaction) what’s happening. After all, most of us can ride a bike but would be similarly hard pressed to explain in any detail what we’re doing. A more common response might go like this: The diagram, or series of diagrams, is a schematic picture of some physical process, say, two electrons interacting, which we then link to mathematical terms. After writing out the mathematical terms in the form of a series, we calculate and get the probability of an interaction happening. This sounds good initially, but runs into trouble when we ask what a “picture of a physical process” could mean? This is, after all, the quantum world where things do not move continuously through space and time. It seems doubtful the diagrams could be diagrams of things in the physical realm. They are connected to physical reality and they are connected to wonderfully accurate predictions. And yet, mystery remains.The following articles on Feynman diagrams address several timely issues, both historical and contemporary. These are metaphysical and epistemological questions related to models and how we represent the world. The papers are united by these themes, but otherwise are independent of one another. They are arranged alphabetically and can be read in any order.James Robert Brown, “How Do Feynman Diagrams Work?” takes the view that they are not representations or models of the world at all. Like others, he stresses that they cannot be pictures of reality, since, thanks to the Heisenberg relations, there can be no such thing as a trajectory. Rather than seek some other more abstract type of representation, he introduces a distinction between descriptive and prescriptive flow charts. The latter could be a representation of physical reality, but not the former. A prescriptive flow chart is more like a recipe to bake a cake. He takes Feynman diagrams to be this sort of thing.Mauro Dorato and Emanuele Rossanese, “The Nature of Representation in Feynman Diagrams,” also reject the idea that Feynman diagrams can be considered to be pictures or depictions of actual physical processes. But unlike Brown, they do not abandon the thought that they can be models in some sense. They appeal to a famous account by RIG Hughes, known as the Denotation, Deduction, and Interpretation theory of models, where “models” are to be interpreted as merely inferential, non-representational devices constructed in given social contexts by the community of physicists.Letitia Meynell, “Picturing Feynman Diagrams and the Epistemology of Understanding,” assumes that Feynman diagrams are pictorial representations but agrees with Brown and Dorato and Rossanese that they do not represent actual physical processes. She then takes up the following puzzle: If Feynman diagrams represent states of affairs, but do not do so truthfully, what can their epistemic value be? She argues that Feynman diagrams have been epistemically powerful (at least in part) because, as pictorial representations, they facilitate an understanding of QED, and particle physics more generally. Appealing to Feynman’s own remarks, she draws out what it might mean to have an understanding of something that is not factive. Although her approach allows for a thin sense of substantively non-factive epistemic success, it is continuous with a factive sense of understanding that is more familiar in the sciences.Michael Stöltzner, “Feynman Diagrams: Modeling between Physics and Mathematics,” analyzes Feynman diagrams within the context of recent debates about models in science and against the backdrop of other diagrammatic methods in mathematical physics when dealing with infinite or asymptotic series. This allows one to avoid the dichotomy of whether Feynman diagrams represent mathematical or physical objects, or a mere tool mediating between them. Along those lines one does not, however, obtain a universal answer to the question as to what Feynman diagrams represent. A single Feynman diagram, in actual scientific practice, can stand for a single mathematical expression or for a physical phenomenon depending on whether the diagram stands for a single term in an infinite series or for a subseries that is given a physical interpretation. Stöltzner’s reading of the representation problem also derives support from the historical fact that Feynman was initially motivated by the Breit-Schrödinger model of a quivering electron and from interpreting this model within the context of stochastic processes.Adrian Wüthrich, “The Exigencies of War and the Stink of a Theoretical Problem,” approaches these topics from an historian’s point of view, shedding light on the issues by focussing on debates in this historical context. As he put it in his subtitle, Wüthrich’s aim is to understand the genesis of Feynman’s electrodynamics as mechanistic modelling at different levels. Feynman’s idiosyncratic method of dealing with quantum electrodynamic phenomena has been cast as an instance of a non-rigorous, modular and rule-based style of theorizing. Given such a characterization, it seems natural that significant changes in Feynman’s style were triggered by work he was assigned during World War II, where quick fixes may have counted more than derivations from first principles. Indeed, Feynman’s theoretical proposals of an improved electrodynamics do bear traces of Feynman’s war-related work, which immediately preceded it. However, Wüthrich explains, the attempt to apply the vocabulary of the “new mechanistic philosophy” to lesser known pieces of archival evidence brings to the fore a new kind of shift in Feynman’s style. Feynman’s disposition for such a shift may also have been fostered during the war, but it was forced upon him by a specific theoretical difficulty, which was largely unrelated to his war-time research.These articles contain some technicalities, but they should all be sufficiently accessible to general readers who would like to know more about Feynman diagrams and how they relate to theories, models, mathematics, and the world. The best philosophy of science is intimately linked to and draws upon current science and its history. It pays close attention to details: the details of theorizing, the details of models and how they function, the details of experimentation and measurement, and so on. We hope this collection of five articles on Feynman diagrams, by focussing closely on detail, will contribute to a better understanding of these strange but highly productive ciphers and symbols.