Mathematical Physics — Foundations¶

Job: Package classical field theory (Lagrangian, Noether, Hamiltonian), quantization routes, QFT axiom pointers, and the mathematical objects of gauge theory.

Core¶

Classical field theory / Lagrangian¶

Spacetime $M$ (typically Lorentzian $(M,g)$ or Minkowski $\mathbb{R}^{1,d-1}$). Field $\varphi:M\to V$ (or section of a bundle $E\to M$).

Lagrangian density $\mathcal{L}(x,\varphi,\partial\varphi,\ldots)$; action $$ S[\varphi]=\int_M \mathcal{L}\,d\mathrm{vol}g. $$ Euler–Lagrange (first variation $\delta S=0$ for compactly supported variations): $$ \frac{\partial\mathcal{L}}{\partial\varphi}-\partial\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\varphi)}=0 $$ (schematic; jet-bundle form is coordinate-free).

Example	$\mathcal{L}$	EL
Scalar Klein–Gordon	$\tfrac12\partial_\mu\phi\partial^\mu\phi-\tfrac12 m^2\phi^2$	$(\square+m^2)\phi=0$
Maxwell	$-\tfrac14 F_{\mu\nu}F^{\mu\nu}$	$\partial_\mu F^{\mu\nu}=0$
Einstein–Hilbert	$R\sqrt{	g
Yang–Mills	$-\tfrac12\langle F\wedge *F\rangle$	$d_A^*F=0$

Jet bundles / variational bicomplex (pointer): coordinate-free EL and conservation laws via horizontal/vertical differentials.

Noether’s theorems¶

First theorem. Lie algebra of global variational symmetries $\Rightarrow$ on-shell conserved currents $j^\mu$ with $\partial_\mu j^\mu=0$ when EL hold. Charge $Q=\int_{\Sigma} j^0$ (under decay) is constant on Cauchy surfaces.

Second theorem. Local (gauge) symmetries $\Rightarrow$ differential identities among EL equations (dependencies); related to constraints in the Hamiltonian analysis.

Energy–momentum from spacetime translation symmetry: $T^{\mu\nu}$ with $\partial_\mu T^{\mu\nu}=0$ on-shell (Belinfante improvement for spin).

Hamiltonian field theory¶

Phase space: infinite-dimensional manifold of Cauchy data (e.g. $(\phi,\pi)$ on $\Sigma$), symplectic form $$ \omega=\int_\Sigma \delta\pi\wedge\delta\phi $$ (schematic). Hamiltonian $H$ from Legendre transform when nondegenerate; Poisson bracket $\{F,G\}$.

Constraints (Dirac): primary from degeneracy of Hessian $\partial^2\mathcal{L}/\partial\dot\phi^2$ (gauge theories); secondary from preservation in time. First class constraints generate gauge transformations; reduction $\mu^{-1}(0)/G$ (Marsden–Weinstein) yields physical phase space.

Covariant phase space: space of solutions of EL with symplectic current from boundary terms in $\delta L$ — elegant for GR/gauge.

Quantization paths¶

Route	Idea	Typical setting
Canonical	$\widehat q,\widehat p$ with $[\hat q,\hat p]=i\hbar$; Fock space for free fields	Free fields on static spacetimes; CCR/CAR
Geometric quantization	prequantum $U(1)$-bundle + polarization $\Rightarrow$ Hilbert space	Compact phase space; Chern class $[\omega/2\pi]$ integral
Deformation quantization	star product $\star_\hbar$ on $C^\infty(M)[[\hbar]]$ with first commutator $= i\hbar\{\,,\}$	Kontsevich for Poisson manifolds
Path integral	“$\int\mathcal{D}\phi\, e^{iS/\hbar}$”	Heuristic / constructive via Euclidean measures
Algebraic	net of $C^$/$W^$ algebras of locals	AQFT (Haag–Kastler)

Free field success: Gaussian measures / Fock representations well-defined; interacting $d=4$ still largely open constructively.

QFT axioms (pointer)¶

Wightman axioms (Minkowski): Hilbert space $\mathcal{H}$ with unitary Poincaré rep $U$; vacuum $\Omega$; operator-valued distributions $\phi(f)$ on Schwartz space; domain/invariance; spectrum condition $p\in \overline{V}^+$; locality $[\phi(f),\phi(g)]=0$ for spacelike separated supports; cyclicity of $\Omega$. Wightman reconstruction: reconstruction of fields from vacuum correlation distributions satisfying positivity, Poincaré covariance, spectrum, locality, cluster.

Osterwalder–Schrader (Euclidean): reflection positivity, Euclidean invariance, ergodicity, growth — reconstruct Wightman theory via analytic continuation (OS reconstruction theorem). Constructive QFT often builds Euclidean measures first.

Algebraic QFT: isotone net $\mathcal{O}\mapsto \mathfrak{A}(\mathcal{O})$ of $C^*$-algebras; locality (Einstein causality); covariance; vacuum state / GNS. Locality without pointlike fields.

Gauge theory — mathematical objects (overview)¶

Object	Role
Principal $G$-bundle $P\to M$	global gauge configuration space
Connection $A\in\Omega^1(P,\mathfrak{g})$	gauge potential
Curvature $F_A=dA+\tfrac12[A,A]$	field strength
Associated bundle $P\times_G V$	matter fields as sections
Gauge group $\mathcal{G}=\mathrm{Aut}(P)$	local symmetries
Yang–Mills action	$\int\langle F\wedge *F\rangle$
Chern–Weil classes	characteristic classes from invariant polynomials of $F$

Details: gauge-theory.md.

See it¶

Klein–Gordon quantization sketch. Mode expansion $\phi(t,x)=\int\bigl(a(k)u_k+a^\dagger(k)\overline{u_k}\bigr)$ on Minkowski; CCR $[a(k),a^\dagger(k')]=\delta$; Hamiltonian $\int\omega_k a^\dagger a$; vacuum $a(k)|0\rangle=0$. Two-point Wightman function $\langle 0|\phi(x)\phi(y)|0\rangle$ is the positive-frequency Pauli–Jordan / Wightman distribution — seeds the axioms.

Symplectic reduction (charged particle / Kaluza–Klein pointer). First-class constraint $\Phi=0$ and gauge orbits: physical observables are gauge-invariant functions on the constraint surface. Electromagnetism: Gauss law $\nabla\cdot E-\rho=0$ as constraint; reduced data are gauge classes of $A$ + $E$.

Check¶

State Noether’s first theorem carefully (what must be invariant?).

A continuous family of transformations of independent/dependent variables leaving the action invariant up to a total divergence (variational symmetry) yields a current conserved when the EL equations hold. Invariance of $\mathcal{L}$ alone is sufficient but not necessary (quasi-invariance / boundary terms allowed).

Why does geometric quantization need a polarization?

Prequantum Hilbert space $L^2(M,L)$ is “too large” (depends on all phase-space directions). Polarization selects a Lagrangian distribution of directions to depend on (e.g. real: $q$-representation; Kähler: holomorphic sections) so states match quantum half-dimensionality.

Pitfalls¶

On-shell vs off-shell: Noether currents conserve only on-shell; off-shell identities differ (especially with gauge).
Legendre transform may fail for gauge/gravitational systems — Dirac–Bergmann algorithm required; not every field has unconstrained Hamiltonian density.
Wightman fields are distributions: products at coinciding points undefined without renormalization — the interaction problem.
Signature: Euclidean OS $\neq$ Lorentzian real-time path integrals without analytic continuation control.

Refs¶

Abraham–Marsden, Foundations of Mechanics; Deligne–Freed; Gelfand–Fomin (calculus of variations); Streater–Wightman; Haag; Woodhouse, Geometric Quantization; Kontsevich deformation quantization.

Next¶

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

Example	\(\mathcal{L}\)	EL
Scalar Klein–Gordon	\(\tfrac12\partial_\mu\phi\partial^\mu\phi-\tfrac12 m^2\phi^2\)	\((\square+m^2)\phi=0\)
Maxwell	\(-\tfrac14 F_{\mu\nu}F^{\mu\nu}\)	\(\partial_\mu F^{\mu\nu}=0\)
Einstein–Hilbert	$R\sqrt{	g
Yang–Mills	\(-\tfrac12\langle F\wedge *F\rangle\)	\(d_A^*F=0\)