Section
Mathematical physics
Navigation for Lorentzian geometry, relativistic field equations, and their symmetry and spin structures.
Core idea
This section collects geometric and analytic structures used in classical relativistic field theory. The present convention is
and
Thus on Minkowski spacetime . Every operator formula linked below uses this convention unless it explicitly says otherwise.
Spacetime geometry and symmetry
- Pseudo-Riemannian manifold
- Lorentzian manifold
- Time orientation
- Causal curve
- Chronological and causal future
- Strong causality
- Causal diamond
- Cauchy hypersurface
- Globally hyperbolic spacetime
- Global hyperbolicity and Cauchy hypersurfaces
- Smooth splitting of globally hyperbolic spacetimes
- Minkowski vector space
- Minkowski spacetime
- Lorentz group
- Poincaré group
- Poincaré algebra
Supersymmetry
Scalar operators and equations
- Laplace–Beltrami operator
- d’Alembert operator
- Normally hyperbolic operator
- Connection form of a normally hyperbolic operator
- Cauchy problem for normally hyperbolic operators
- Advanced and retarded Green operators
- Existence of advanced and retarded Green operators
- Wave equation
- Klein–Gordon operator
- Klein–Gordon equation
- Conformal coupling of a scalar field
Quantum chaos and fractal uncertainty
- Compact hyperbolic surface
- Geodesic flow
- Anosov flow
- Horocycle flow
- Unique ergodicity
- Hyperbolic Poisson kernel
- Hyperbolic plane wave
- Laplace–Beltrami eigenfunction
- Semiclassical measure
- Fractal uncertainty principle
- Higher-dimensional line-porous fractal uncertainty principle
- Uniform mass lower bound for hyperbolic-surface eigenfunctions
- Complete fractal uncertainty and quantum chaos batch index
Spinors and first-order fields
- Spinor module
- Classification of complex Clifford modules
- Dirac spinor
- Weyl spinor
- Majorana spinor
- Majorana–Weyl spinor
- Lorentzian spin structure
- Lorentzian spinor bundle
- Chirality operator
- Gamma matrices
- Clifford slash notation
- Lorentzian Dirac operator
- Cauchy problem for the Lorentzian Dirac operator
- Minkowski Dirac operator
- Dirac equation
- Dirac–Klein–Gordon factorization
- Riemannian spin Dirac operator
Internal gauge symmetry and exceptional algebra
- Standard Model gauge group
- Standard Model Lie algebra
- Weak hypercharge
- Georgi–Glashow embedding
- Standard Model exterior-algebra representation
- Standard Model fermion generation
- Right-handed neutrino gauge singlet
- Standard Model gauge group as an stabilizer intersection
- Complete E7 and exceptional Jordan-algebra paper index
Analytic boundary
Riemannian scalar and Dirac operators are elliptic; Lorentzian wave operators are normally hyperbolic, and Lorentzian Dirac operators have the null cone as their characteristic set. Compact-spectrum statements from Riemannian geometry therefore do not transfer to relativistic evolution problems. Conversely, Cauchy evolution and finite propagation speed belong to the Lorentzian setting and require causal hypotheses such as global hyperbolicity.
References
- Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record.
- H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989. Publisher record.