Formalism Rosetta
Concepts first, formalism second — but the formalism still matters when you open a book. The same object wears different clothes in the operator, path-integral, and diagrammatic languages. This table is a translator: find the row, read across.
A living reference — rows are added as topics need them.
| Object | Operator (second quantization) | Path integral (functional) | Diagrammatic |
|---|---|---|---|
| Building block | creation / annihilation operators c†, c | Grassmann (or complex) fields ψ̄, ψ | external & internal lines |
| Free dynamics | H₀ = Σₖ ε(k) c†ₖ cₖ | quadratic action ψ̄(∂τ + ε)ψ | the bare line G₀ |
| Interaction | U n↑ n↓ | quartic term in the action S | the vertex |
| Partition function Z | Tr e^(−βH) | ∫ 𝒟[ψ̄,ψ] e^(−S) | exp(Σ connected vacuum diagrams) |
| Propagator G | −⟨T c(τ) c†(0)⟩ | ⟨ψ(τ) ψ̄(0)⟩ functional average | dressed line G = G₀ + G₀ Σ G |
| Self-energy Σ | Dyson: G⁻¹ = G₀⁻¹ − Σ | 1PI part of the two-point function | sum of one-particle-irreducible insertions |
| Susceptibility χ | ⟨T A(τ) A(0)⟩ (connected) | δ²lnZ / δh² | bubbles + ladders (Bethe–Salpeter) |
| Expectation ⟨Ô⟩ | Tr(ρ Ô) | ∫ 𝒟φ O e^(−S) / Z | diagrams with one O insertion |
See these objects at work in the Green's functions and diagrammatics method pages, or read the ethos on why we stay formalism-agnostic.