Basic Notions: Eugene Trubowitz (ETH) "Mathematical Methods of Many Body Quantum Field Theory"

Thu, 30/05/201916:00-17:15
Ross 70
Let (V,<, >) be a finite dimensional inner product space and K a self adjoint element of End(V ). It is an axiom of physics that the expected value of A in End(V ) in equilibrium at temperature T with respect to K is
the ration Tr(A exp (-K/T))/Tr(exp (-K/T)).
It is another axiom of physics that a ‘many body (Grassmann) quantum field’ appears by writing the expected value as a ‘Grassmann integral’. Then, ‘Feynmann diagrams’ naturally arise, forcing us to ‘renormalize’ (the technique of perturbative renormalization was invented by Poincare in the 1880’s) using a ‘(momentum space) renormalization group’ and to unavoidably encounter ‘superconductivity’. We will formulate and indicate the proofs of Theorems about the behavior of many body electron systems (quantum fields) such as the existence of an ‘interacting Fermi surface’ and the possibility of a ‘forward scattering instability’. All the jargon in parentheses will be logically and rigorously introduced from scratch - this is not a physics talk.
In the second lecture, to expose the ‘broken symmetry’ symptomatic of superconductivity, we take V = S(W). To write the expected value as an integral and obtain a ‘many body (bosonic) quantum field’ is now a subtle problem requiring its own ‘(position space) renormalization group’
analysis of highly oscillatory integrals. Two more (position space) renormalization groups are needed to uncover the ‘Goldstone boson’ accompanying the symmetry breaking. Here, thinking in terms of graphs is not very helpful. Again, all jargon in parentheses will be logically and rigorously introduced from scratch.
The only prerequisites for both talks is linear algebra and undergraduate analysis.