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A Guide to Feynman Diagrams in the Many-Body Problem
By Richard D. Mattuck
Dover Publications, Inc.Copyright © 1976 McGraw-Hill, Inc.
All rights reserved.
Feynman Diagrams, or how to Solve the Many-Body Problem by means of Pictures
1.1 Propagators—the heroes of the many-body problem
We have seen that many-body systems consisting of strongly interacting real particles can often be described as if they were composed of weakly interacting fictitious particles: quasi particles and collective excitations. The question now is, how can we calculate the properties of these fictitious particles—for example, the effective mass and lifetime of quasi particles? There are various ways of doing this (see appendix A but the hero roles in the treatment of the many-body problem are played by quantum field theoretical quantities known as Green's functions or propagators. These are essentially a generalization of the ordinary, familiar undergraduate Green's function. They come in all sizes and shapes—one particle, two particle, no particle, advanced, retarded, causal, zero temperature, finite temperature—an assortment to suit every situation and taste.
There are three reasons for the immense popularity propagators are enjoying these days. First of all, they yield in a direct way the most important physical properties of the system. Secondly, they have a simple physical interpretation. Thirdly, they can be calculated in a way which is highly systematic and 'automatic' and which appeals to one's physical intuition.
The idea behind the propagator method is this: the detailed description of a many-body system requires in the classical case the position of each particle as a function of time, r1(t), r2 (t), ..., rN, (t), or in the quantum case, the time-dependent wave function of the whole system, Ψ (r1 · r2, ..., rN, t). A glance at Fig. 0.2B shows that this is an extremely complicated business. Fortunately, it turns out that in order to find the important physical properties of a system it is not necessary to know the detailed behaviour of each particle in the system, but rather just the average behaviour of one or two typical particles. The quantities which describe this average behaviour are the one-particle propagator and two-particle propagator respectively, and physical properties may be calculated directly from them.
Consider the one-particle propagator first. It is defined as follows: We put a particle into the interacting system at point r1 at time t1 and let it move through the system colliding with the other particles for a while (i.e., let it 'propagate' through the system). Then the one-particle propagator is the probability (or in quantum systems, the probability amplitude—see §3.1) that the particle will be observed at the point r2 at time t2. (Note that instead of putting the particle in at a definite point, it is sometimes more convenient to put it in with definite momentum, say p1, and observe it later with momentum p2.) The single-particle propagator yields directly the energies and lifetimes of quasi particles. It also gives the momentum distribution, spin and particle density and can be used to calculate the ground state energy.
Similarly, the two-particle propagator is the probability amplitude for observing one particle at r2, t2 and another at r4, t4 if one was put into the system at r1, t1 and another at r3, t3 (see Fig. 0.2B). This also has a wide variety of talents, giving directly the energies and lifetimes of collective excitations, as well as the magnetic susceptibility, electrical conductivity, and a host of other non-equilibrium properties.
There is also another useful quantity, the 'no-particle propagator' or so-called 'vacuum amplitude' defined thus: We put no particle into the system at time t1, let the particles in the system interact with each other from t1 to t2, then ask for the probability amplitude that no particles emerge from the system at time t2. This may be used to calculate the ground state energy and the grand partition function, from which all equilibrium properties of the system may be determined.
1.2 Calculating propagators by Feynman diagrams: the drunken man propagator
There are two different methods available for calculating propagators. One is to solve the chain of differential equations they satisfy—this method is discussed briefly in appendix M. The other is to expand the propagator in an infinite series and evaluate the series approximately. This can be carried out in a general, systematic, and picturesque way with the aid of Feynman diagrams.
Just to get an idea of what these diagrams are, consider the following simple example (see Fig. 1.1). A man who has had too much to drink, leaves a party at point 1 and on the way to his home at point 2, he can stop off at one or more bars—Alice's Bar (A), Bardot Bar (B), Club Six Bar (C), ..., etc. He can wind up either at his own home 2, or at any one of his friends' apartments, 3, 4, etc. We ask for the probability, P (2,1), that he gets home. This probability, which is just the propagator here (with time omitted for simplicity), is the sum of the probabilities for all the different ways he can propagate from 1 to 2 interacting with the various bars.
The first way he can propagate is 'freely' from 1 to 2, i.e., without stopping at a bar. Call the probability for this free propagation P0 (2,1).
The second way he can propagate is to go freely from 1 to bar A (the probability for this is P0(A, 1)), then stop off at bar A for a drink (call the probability for this P (A)), then go freely from A to 2 (probability=P0(2, A)). Assume for simplicity that the three processes here are independent. Then the total probability for this second way is the product of the probabilities for each process taken separately, i.e., P0(A, 1) × P (A) × P0(2, A). (This is like the case in coin-tossing: since each toss is independent, the probability of first tossing a head, then a tail, equals the probability of tossing a head times the probability of tossing a tail.)
The third way he can propagate is from 1 to B to 2, with probability P0(B, 1)P (B)P0 (2, B). Or he could go from 1 to C to 2, etc., or from 1 to A to B to 2, or from 1 to A, come out of A, go back into A, then go to 2, and so on. The total probability, P (2,1) is then given by the sum of the probabilities for each way, i.e., the infinite series:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)
This is an example of a 'perturbation series', since each interaction with a bar 'perturbs' the free propagation of the drunken man.
Now, such a series is a complicated thing to look at. To make it easier to read, we follow the journal 'Classic Comics' where difficult literary classics are translated into picture form. Let us make a 'picture dictionary' to associate diagrams with the various probabilities as in Table 1.1. Using this dictionary, series (1.1) can be drawn thus:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)
Since, by dictionary Table 1.1, each diagram element stands for a factor, series (1.2) is completely equivalent to (1.1). However it has the great advantage that it also reveals the physical meaning of the series, giving us a 'map' which helps us to keep track of all the sequences of interactions with bars which the drunken man can have in going from 1 to 2.
The series may be evaluated approximately by selecting the most important types of terms in it and summing them to infinity. This is called partial summation. For example, suppose the man is in love with Alice, so that P (A) is large, and all the other P (X)'s are small. Then Alice's bar diagrams will dominate, and the series (1.2) may be approximated by a sum over just repeated interactions with Alice's Bar:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)
Using the above dictionary, this can be translated into functions:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)
Assume for simplicity that all P0(s, r) are equal to the same number, c, i.e., P0(2, 1)=P0(2, A) = P0(A, 1)=P0(A, A) = c. Then series (1.4) becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)
The series in brackets is geometric and can be summed exactly to yield 1/(1 - cP(A)), so that
P(2, 1) = c × (1/1 - CP(A)) = 1/c-1 - P(A) (1.6)
which is the solution for the propagator in this case.
Note that since each diagram element stands for a factor, we could have done calculation (1.5), (1.6) completely diagrammatically:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)
The partial summation method is extremely useful in dealing with the strong interactions between particles in the many-body problem, and it is the basic method which will be used throughout this book.
1.3 Propagator for single electron moving through a metal
The example here is just like the previous one, except that instead of a propagating drunken man interacting with various bars, we have a propagating electron interacting with various ions in a metal. A metal consists of a set of positively charged ions arranged so they form a regular lattice, as in Fig. 0.13A or a lattice with some irregularities, as in Fig. 0.13B. An electron interacts with these ions by means of the Coulomb force. The single particle propagator here is the sum of the quantum mechanical probability amplitudes (see §3.1) for all the possible ways the electron can propagate from point r1 in the crystal at time t1, to point r2 at time t2, interacting with the various ions on the way. These are: (1) freely, without interaction; (2) freely from r1, t1 (= '1' for short) to the ion at rA at time tA, interaction with this ion, then free propagation from the ion to point 2; (3) from 1 to ion B, interaction at B, then from B to 2, etc. Or we could have the routes 1-A-A-2, 1-A-B-2, etc. We can now use the dictionary in Table 1.1 to translate this into diagrams, provided the following changes are made: change 'probability' to 'probability amplitude', and change the meaning of the circle with an X to 'probability amplitude for an interaction with the ion at X'. When this is done, the series for the propagator can be translated immediately into exactly the same diagrams as in the drunken man case! That is, (1.2) is also the propagator for an electron in a metal, provided that we just use a quantum dictionary to translate the lines and circles into functions. The series can be partially summed, and from the resulting propagator we obtain immediately the energy of the electron moving in the field of the ions.
1.4 Single-particle propagator for system of many interacting particles
We will now indicate in a qualitative way how the single-particle propagator may be calculated in a system of many interacting particles. The argument is general, but we may think in terms of the electron gas as illustration. The propagator will be the sum of the probability amplitudes for all the different ways the particle can travel through the system from r1, t1 to r2, t2. First we have free propagation without interaction. Another thing which can happen is shown in the 'movie', Fig. 1.2, which depicts a 'second-order' propagation process (i.e., a process with two interactions). (It should be mentioned here that unlike the drunken man case, the processes involved in Fig. 1.2 are not real physical processes, but rather 'virtual' or 'quasi physical', since they do not conserve energy, and they may violate the Pauli exclusion principle. The reason for this is that, as we shall see later on, the sequence in Fig. 1.2 (or the corresponding diagram (1.9)) is simply a convenient and picturesque way of describing a certain second-order term which appears in the perturbation expansion of the propagator. Hence Fig. 1.2 and diagram (1.9) are in reality mathematical expressions so we have to be careful not to push their physical interpretation too far (see §4.6).)
To represent this sequence of events diagrammatically, let us imagine that time increases in the upward-going direction and we use the following diagram elements:
[ILLUSTRATION OMITTED] (1.8)
(Note that the hole is drawn as a particle moving backward in time. The reason for this is in §4.2.) Then the probability amplitude for the above sequence of events can be represented by the diagram
[ILLUSTRATION OMITTED] (1.9)
The piece of diagram:
[ILLUSTRATION OMITTED] (1.10)
is called a 'self-energy part' because it shows the particle interacting with itself via the particle-hole pair it created in the many-body medium. Diagram (1.9) may be evaluated by writing a free propagator factor for each directed line, and an amplitude factor for each wiggly line (see Chapter 4, Table 4.3), analogous to the drunken man case.
Another sequence of events which can occur involves only one interaction (i.e., a 'first-order' process). It is a quick-change act in which the incoming electron at point r interacts with another electron at point r' and changes place with it. This is analogous to billiard ball 1 striking billiard ball 2 and transferring all its momentum to 2. The first-order process and its analogy are shown in Fig. 1.3. The sequence may be drawn diagrammatically
[ILLUSTRATION OMITTED] (1.11)
The diagrams in (1.8)–(1.11) are called Feynman diagrams after their inventor, Richard P. Feynman who employed them in his Nobel prize-winning work on quantum electrodynamics. They are used extensively in elementary particle physics.
The total single particle propagator is the sum of the amplitudes for all possible ways the particle can propagate through the system. This will include the above processes, repetitions of them, plus an infinite number of others. Thus we find
[ILLUSTRATION OMITTED] (1.12)
(Note: the interpretation of the 'bubble' diagram, just after the open oyster, will be discussed in chapter 4.)
We can see the direct connection between the one-particle propagator and the quasi particle by looking at all the diagrams at a particular time to (dashed line):
[ILLUSTRATION OMITTED] (1.13)
At to, we see that various situations may exist: there may be just the bare particle (a), or there may exist two particles plus one hole created by the second-order sequence (c), or three particles plus two holes in (d), etc. That is, the diagrams show all the configurations of particles and holes which may be kicked up by the bare particle as it churns through the many-body system. If we now compare with the picture of the quasi particle in Fig. 0.10, we see that the diagrams reveal the content of the ever-changing cloud of particles and holes surrounding the bare particle and converting it into a quasi particle.
Excerpted from A Guide to Feynman Diagrams in the Many-Body Problem by Richard D. Mattuck. Copyright © 1976 McGraw-Hill, Inc.. Excerpted by permission of Dover Publications, Inc..
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