1. Chalmers University of Technology Many Body Solid State Physics 2007 Mattuck chapter 5 - 7

2. Chalmers University of Technology Contents Quantum Vacuum: how to solve the equations of nothing Birds eye of Diagrams: start of the elementary part of the book (page 118->) Learning how to count: occupation number formalism Any questions ?!?

3. Chalmers University of Technology Quantum vacuum Meaning of the vacuum of amplitude

4. Chalmers University of Technology Vacuum amplitude R(t) =probability (amplitude) that if the system at t=0 is in the Fermi vacuum, then at t = t the system is in the Fermi vacuum =“no particle propagator” Fermi vacuum

5. Chalmers University of Technology Vacuum amplitude U(t) = time development operator

6. Chalmers University of Technology Pinball vacuum amplitude

7. Chalmers University of Technology Pinball vacuum amplitude OO O L L O G G G P= ++++… O O +

8. Chalmers University of Technology Quantum one-particle vacuum amplitude ZerothFirst Second Third “Vacuum polarisation” or “vacuum fluctuation” t

9. Chalmers University of Technology Quantum one-particle vacuum amplitude = - “Nevertheless it is important to retain such diagrams which violates conservation of particle number to prove the linked cluster theorem.”

10. Chalmers University of Technology Quantum one-particle vacuum amplitude Topological equivalence t1t1 t2t2 t3t3 t2t2 t2t2 t1t1 t1t1 t3t3 t3t3 t t1t1 t2t2 t3t3

11. Chalmers University of Technology Quantum one-particle vacuum amplitude + +++ +… + +

12. Chalmers University of Technology Quantum one-particle vacuum amplitude Linked cluster theorem All linked diagrams Which can be shown via entities like ++= x These gives us the possibility to get the ground state energy even when the perturbation in strong.

13. Chalmers University of Technology The many body case R = 1 + + + … = all diagrams starting and beginning in the ground state Again E 0 is only sum over linked diagrams We can get E 0 in some approximation, eg. Hartree-Fock: E 0 = W 0 + +

14. Chalmers University of Technology Bird’s eye view of MBP Field theoretic ingredientSignificance in MB theory Occupation number formalismExpress arbitrary state of MB system Creation and destruction operatorsPrimitive operators from which all MB operators can be built Single particle propagatorQuasi particle energies, momentum distribution and more Vacuum amplitudeGround state energy Two-particle propagatorCollective excitations, non equilibrium properties Finite temperature vacuum amplitudeEquilibrium thermodynamic properties Finite temperature propagatorTemperature dependent properties

15. Chalmers University of Technology Second quantization (again) A way to write the wave function in a compact way (no Slater determinant crap) A way to treat the particle type automatically (fermions and bosons) Can refer to any basis (momentum, real…) A way to vary particle number

16. Chalmers University of Technology Second quantization (again) Extended Hilbert space = No particleOne particleTwo particles…

17. Chalmers University of Technology