MANY BODY PHYSICS - HT 2007 Meeting 3 pt.1: Mattuck Chapter 0-2 1
1 The Many-Body Problem for Everybody 1.1 The many body problem Systems of many interacting bodies. Examples: Nucleons in a nucleus Electrons in an atom Atoms in molecule/solid Electrons in metal. 1.2 Solution methods Ignore interactions Canonical transformation - Use coordinates in which interactions are small Methods of quantum field theory - Propagators, Feynman diagrams,... 2
1.3 Quasi particles and collective excitations Quasi particle Systems composed of strongly interacting real particles can act as if composed by weakly interacting quasi particles. Typically Quasi particle = real particle + colud of other particles (1) Examples Positive ion moving together with a cloud of negative ions in classical liquid. Electron in a cloud of virtual photons in QED interactions. Electron with cloud of hole/lifted out electrons in electron gas. Quasi particle behaves like individual particle with the properties Effective mass m from free energy relation ɛ = p2 2m, not necessarily equal to real particle mass m. Self energy ɛ self = ɛ quasi particle ɛ real particle Lifetime τ Collective wavelike motion of all particles in system. Ex- Collective excitations amples Phonons Plasmons Magnons 3
2 Propagators Describes the average behavior of typically one or two particles from which the most important physical properties can be calculated. No particle propagator The probability (amplitude in Q.M.) P (t 2, t 1 ) that a system with no particle added to it at time t 1 will have no emerged particle at time t 2. Can be used for calculating the ground state energy and grand partition function of the system. One particle propagator The probability (amplitude) P (r 2, t 2, r 1, t 1 ) that a particle put to interact with a system at position r 1 (or possibly with momentum p 1 ) at time t 1 will be observed at position r 2 (or with momentum p 2 ) at time t 2. Can be used to calculate energy and lifetime of quasi particles. It gives the distribution of particle momentum, position and spin. Two particle propagator Same as one particle propagator, but with 2 particles both entering the system at time t 1 and being observed at time t 2. Can be used to calculate energy and lifetime of collective excitations. Can also be used to calculate magnetic susceptibility, electrical conductivity etc. 4
2.1 Methods to calculate propagators 1. Solve their differential equations. 2. Make an infinite series expansion of the propagator, which can be evaluated using different approximations. E.g. If all interactions are weak and of the same magnitude, make the summation over all terms containing a fixed number of interactions up to desired order. If one interaction type is stronger than the others, the sum over terms containing this interaction will dominate. Performing the sum over all terms with the strong interaction is called partial summation. 5
3 Classical Quasi Particles Free particle propagator under constant force F (from F = m r and ṙ(t 1 ) = 0) ( P 0 (r 2, t 2, r 1, t 1 ) = δ r 2 r 1 F ) 2m (t 2 t 1 ) 2. (2) The propagator if interactions are turned on can have a maximum P max at R = r 2 r 1 such that and R = i.e. we have a classical quasi particle. F 2m (t 2 t 1 ) 2 (3) P max e (t 2 t 1 )τ, (4) 6
4 Feynman diagrams Example: Classical Pinball. Classical perturbation series of one particle propagator for independent events with interaction probabilities P (A), P (B),... P (r 2, r 1 ) = P 0 (r 2, r 1 ) + P 0 (r A, r 1 )P (A)P 0 (r 2, r A ) + P 0 (r B, r 1 )P (B)P 0 (r 2, r B ) +... + P 0 (r A, r 1 )P (A)P 0 (r A, r A )P (A)P 0 (r 2, r A ) +... (5) can be translated into a diagram using a dictionary, where each word P (r j, r i ), P 0 (r j, r i ), P (A) and so can be translated into a unique symbol. From these symbols the picture (Feynman diagram) of a unique physical process can be assigned to each term in the series expansion. The processes in the Feynman diagrams are in general virtual processes rather than real physical processes, since they do not conserve energy and they may violate the Pauli exclusion principle. Features of the Feynman diagrams Shows physical meaning of the term they represent. A tool to keep track of which sets of diagrams have been summed over. The essential characteristic of many partial sums is the structure or topology of the diagrams. 7
5 Discussion topics What is the gain in using Feynman diagrams rather than mathematical expressions? What about nonanalytical behavior? 8