Mathematical Physics

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18 domain-specific abstractions whose origin domain is Mathematical Physics.

• Amplituhedron — A positive geometric object whose canonical differential form encodes scattering amplitudes in planar maximally supersymmetric Yang–Mills theory without making locality and unitarity manifest inputs.
• Antisymmetrizer — A linear projection that averages all particle-label permutations with their signs, extracting the totally antisymmetric component required for identical fermions.
• Current algebra — An infinite-dimensional Lie algebra of Lie-algebra-valued functions on a manifold, arising physically from equal-time commutators of conserved current densities.
• De Donder–Weyl theory — A covariant Hamiltonian formulation of classical field theory treating space and time coordinates symmetrically through polymomenta.
• Deformation quantization — A quantization method that replaces the commutative product of classical observables with a formal parameter-dependent noncommutative star product whose zeroth-order limit is classical multiplication and first-order commutator recovers the Poisson bracket.
• Gamma matrices — Matrices satisfying the Clifford anticommutation relations for a spacetime metric, used to represent spinors and linearize relativistic wave operators.
• Geometric quantization — A construction that seeks a quantum Hilbert space and observables from a classical symplectic phase space while preserving its geometric structures.
• Grassmann number — An element of an exterior algebra generated by anticommuting variables, with odd generators squaring to zero.
• Inverse problem for Lagrangian mechanics — The problem of determining whether a given system of differential equations is equivalent to Euler–Lagrange equations for some Lagrangian and, if so, constructing one.
• Inversion transformation — A conformal coordinate transformation that maps a nonzero point to a reciprocal radial position and, with translations and rotations, extends Poincaré symmetry toward the conformal group.
• Kramers–Kronig relations — Hilbert-transform relations connecting real and imaginary parts of a causal linear response function through analyticity in the upper complex-frequency half-plane.
• Mirror symmetry (string theory) — A duality pairing Calabi–Yau geometries whose associated string compactifications are physically equivalent while exchanging complex and symplectic geometric data.
• Noncommutative quantum field theory — A quantum-field-theory framework defined on a spacetime whose coordinate algebra does not commute.
• Scalar field — A function assigning one scalar quantity to every point of a space or spacetime region, invariant under coordinate changes appropriate to a scalar.
• Spin network — A labeled graph whose edges carry group representations and vertices carry invariant intertwiners, representing gauge-invariant quantum states or tensor contractions.
• Stochastic quantization — Represent a Euclidean quantum field measure as the stationary limit of an auxiliary-time stochastic process, allowing field correlation functions to be obtained as equilibrium stochastic averages.
• Topological quantum field theory — A quantum field theory whose observables depend only on topological structure, mathematically formalized as a symmetric monoidal functor from a cobordism category to vector spaces or related algebraic categories.
• Wigner–Weyl transform — An invertible correspondence between phase-space functions and quantum operators that underlies the quasiprobability formulation of quantum mechanics.