Mathematical Physics

Job: bridge rigorous geometry and analysis with classical and quantum field theories — Lagrangians, gauge structure, quantization paths, and QFT axiom systems.

Core

Mathematical physics isolates the structures that make physical theories well-defined: symplectic/Poisson geometry for mechanics, principal bundles for gauge fields, operator algebras for quantum systems, and axiomatic constraints for QFT.

Layer Content
Classical fields Lagrangian densities; Euler–Lagrange; Noether
Hamiltonian phase space; constraints (Dirac); symplectic reduction
Gauge theory bundles, connections, YM, instantons
Quantization geometric, deformation, canonical, path integral (heuristic)
QFT axioms Wightman; Osterwalder–Schrader; AQFT pointer
Quantum mechanics Hilbert space axioms; spectral theory; \(C^*\)/KMS

Reading map

Page Focus
Foundations classical field theory, Noether, Hamiltonian, quantization paths, QFT axioms pointer, gauge objects overview
Gauge theory principal bundles, connections, curvature, Chern–Weil, Yang–Mills, instantons
Quantum Hilbert QM, spectral theorem, \(C^*\)-algebras/KMS, scattering
Differential geometry Manifolds, metrics, curvature, connections
Functional analysis Hilbert space, unbounded operators, spectra
PDE Field equations; hyperbolic evolution; energy estimates
Representation theory Symmetry groups; induced reps; little groups
SPDE Stochastic quantization / Euclidean constructive routes

Dependency order

  1. Differential geometry (manifolds, metrics, curvature, principal bundles).
  2. Functional analysis (Hilbert space, operators, spectra).
  3. PDE (field equations as PDE; hyperbolic evolution).
  4. Representation theory (symmetry; induced representations).
  5. This section; stochastic processes / SPDE for constructive/Euclidean routes.

Interfaces

Area Contact
GR Lorentzian geometry + Einstein–Hilbert action
Standard Model \(G\)-connections on principal bundles + matter in associated bundles
Condensed matter topological phases via bundle/index theory
Constructive QFT Euclidean axioms, lattice approximations, SPDE link
Symplectic geometry Hamiltonian mechanics; moment maps; reduction

Notation (paper-standard)

  • \(L(q,\dot q,t)\), \(S[\phi]=\int\mathcal{L}\); EL equations \(\delta S=0\)
  • \((M,\omega)\) symplectic; \(X_H\); \(\{f,g\}\); \(\mu:M\to\mathfrak{g}^*\) moment map
  • \(P\to M\) principal \(G\)-bundle; \(A\) connection; \(F_A=dA+A\wedge A\)
  • \(\mathcal{H}\), self-adjoint \(H\); \(U(t)=e^{-itH/\hbar}\); \(\omega\)-Wightman / OS axioms

See it

Noether \(\leftrightarrow\) conserved currents. Continuous symmetry of the action \(\Rightarrow\) on-shell conserved current \(j^\mu\); charge \(\int j^0\) generates the symmetry in the Hamiltonian picture (when well-defined). Gauge symmetries yield constraints / identities (Noether 2) rather than free charges in the same way — the dividing line between global and local symmetry.

Yang–Mills curvature. \(F_A\in\Omega^2(M,\operatorname{ad}P)\); Bianchi \(d_AF=0\); YM equation \(d_A^*F=0\) (or with matter current). Instantons on \(S^4\) realize nontrivial \(\pi_3(G)\) and enter Donaldson theory — gauge theory as differential topology.

Check

Why is the electromagnetic potential \(A\) a connection, not a globally defined 1-form on spacetime alone?

On nontrivial \(U(1)\)-bundles (Dirac monopole, Aharonov–Bohm), \(A\) is only local; transition functions \(e^{i\chi}\) glue \(A\mapsto A+d\chi\). Observables from holonomy / curvature \(F=dA\) are global.

Name one obstruction to making the path integral rigorous for interacting QFT in \(d=4\).

No nonperturbative continuum construction of \(\phi^4_4\) or YM\(_4\) with full Wightman/OS axioms and nontrivial scattering is known; renormalization group and triviality issues intervene. Lower-dimensional and lattice results exist; \(d=4\) YM mass gap is a Millennium problem.

What does symplectic reduction produce, schematically?

From \((M,\omega)\) with \(G\)-action and moment map \(\mu\), the reduced space \(\mu^{-1}(0)/G\) (when \(0\) regular and action free/proper) inherits a symplectic form — classical counterpart of “impose constraints and quotient by gauge.”

Pitfalls

  • Formal path integrals are not measures on naive field space without cutoffs; Osterwalder–Schrader reconstruction needs verified Euclidean axioms.
  • Gauge fixing introduces ghosts/BRST; physical Hilbert space is cohomology, not the full BRST complex.
  • Classical \(\neq\) quantum symmetry: anomalies cancel classical Noether conservation at the quantum level.
  • Unbounded operators: domains matter; “formally self-adjoint” \(\neq\) essentially self-adjoint.

Refs

  • Abraham–Marsden, Foundations of Mechanics
  • Deligne–Freed (QFT and strings IAS volumes)
  • Baez–Muniain, Gauge Fields, Knots and Gravity
  • Haag, Local Quantum Physics
  • Weinberg Vol. 1–2 (physics side with care)
  • Hamilton, Mathematical Gauge Theory

Next

Foundations · Gauge theory · Quantum · Differential geometry · Functional analysis · PDE · Representation theory