Here I consider quantum many-body theory, as it spans from quantum field theory to nuclear physics to ultra cold atomic gases to solid state physics to quantum chemistry.
Here is my tentative list of the ten most important key concepts.
n.b. I am not concerned with key techniques (e.g. path integrals, renormalisation group, Green's functions, Feynman diagrams, imaginary time (KMS), numerical methods....). That is a separate topic. Unfortunately, in most many-body theory texts the concepts are lost in the midst of all the technical details. Furthermore, concepts are what experimentalists need to know and understand.
This is the overarching concept that underscores almost all the others. Reality is stratified and at each length, energy, and time scale distinctly new phenomena, interactions and entities can emerge. More is different.
2. Effective Hamiltonians
At each strata there is some Hamiltonian that describes the "particles" and interactions between them. The parameters in this Hamiltonian can sometimes be determined from experiment.
There are two cases. In the simplest case, due to adiabatic continuity, the quasi-particles have the same quantum numbers and statistics as the constituent particles (e.g., Fermi liquid theory). But, their mass and spin-g factors can be significantly different from the free particles. The second and more exciting case is where the quasi-particles have different quantum numbers and/or statistics to
the constituent particles (e.g. Fractional Quantum Hall effect, spinons, and Luttinger liquid).
This explains how the effective interactions at one strata are related to those at a lower strata (or equivalently how high energy virtual processes lead to low energy real interactions). Examples range from weak interactions in a Fermi liquid to asymptotic freedom in QCD to the van der Waals attraction between two neutral atoms.
5. Incoherent excitations
Sum rules provide a means to understand the redistribution of spectral weight. Quasi-particles don't exhaust the total spectral weight.
This also connects to Hubbard bands, bad metals, quantum decoherence, and the condensate fraction for superfluids and superconductors.
6. Spontaneously Broken Symmetry
This leads to new states of matter and new collective excitations, including Goldstone bosons (e.g. phonons and magnons) and massive particles such as the Higgs boson. This is also connected to rigidity, but that does have a classical analogue too.
7. Emergent energy scales
These can be orders of magnitude different (always smaller?) from the energy scales in the underlying Hamiltonian. Often they reflect non-perturbative effects, as in the BCS energy gap or the Kondo temperature.
8. Emergent length scales
9. The fluctuation-dissipation theorem
This means we can calculate linear transport (i.e. non-equilibrium) properties from fluctuations in thermal equilibrium. Inelastic scattering experiments can then be related to dynamical correlations.
10. Topology sometimes matters
Examples include topological excitations (e.g. vortices), topological terms in the action (e.g. for quantum spin chains and the Haldane conjecture), topological insulators, and topological order in the Quantum Hall effect. Topology is also somehow involved in anomaly cancellation.
Piers Coleman's article (focussing on condensed matter) provides a nice discussion of some of these concepts.
What do you think? What should I add or subtract from this list?
Some might suggest entanglement and/or quantum criticality or there own current favourite hot topic. However, I still remain to be convinced that these are key organising principles that are essential for understanding broad ranges of systems, rather than currently fashionable exotica.
I might include the Mott insulator, but I am not clear how that is relevant beyond solid state physics.
Is the list helpful? I am considering spending more time on this: perhaps a blog post on each concept, then a colloquium style talk, and a short tutorial review article.
I think the only mathematical entity needed to understand and illustrate all of this is a spectral function.
Any feedback would be appreciated.