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Geometric structure
{\displaystyle {\mathrm {Spin} }(n).} Clifford bundle Clifford module bundle Orthonormal frame bundle Spin geometry Spinor Spinor representation Friedrich, Thomas (2000)
Spinor_bundle
Non-tensorial representation of the spin group
details, see Dirac spinor, Weyl spinor, Majorana spinor, and spinor bundle. One major mathematical application of the construction of spinors is to make possible
Spinor
Concept in differential geometry
a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in
Spin_structure
First-order differential linear operator on spinor bundle, whose square is the Laplacian
case of the Atiyah–Singer–Dirac operator acting on sections of a spinor bundle. For a spin manifold, M, the Atiyah–Singer–Dirac operator is locally defined
Dirac_operator
Subspace defined by a polynomial of degree 2 over a field
variety of Projective pure spinors, or simple spinor variety, of dimension m(m + 1)/2. (Another description of the pure spinor variety is as OGr + ( m
Quadric_(algebraic_geometry)
infinite rank vector bundle; this is the symplectic spinor construction due to Bertram Kostant. A section of the symplectic spinor bundle Q {\displaystyle
Symplectic_spinor_bundle
Connection on a spinor bundle
differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine
Spin_connection
Double cover Lie group of the special orthogonal group
corresponding to a given point group. Clifford algebra Clifford analysis Spinor Spinor bundle Spin structure Table of Lie groups Anyon Orientation entanglement Pin
Spin_group
Formula for spinors
curvature, and results on spinors and spin structures. Given a spin structure on a pseudo-Riemannian manifold M and a spinor bundle S, the Lichnerowicz formula
Lichnerowicz_formula
a spinor bundle. In fact, on a Spin manifold, every Clifford module is obtained by twisting the spinor bundle. The notion "Clifford module bundle" should
Clifford_module_bundle
Property of a differential manifold that includes complex structures
algebra on spinors. A spinor is said to be a pure spinor if it is annihilated by half of a set of generators of the Clifford algebra. Spinors are sections
Generalized_complex_structure
Algebra based on a vector space with a quadratic form
Hypercomplex number Octonion Paravector Quaternion Spin group Spin structure Spinor Spinor bundle Also known as a geometric algebra (especially over the
Clifford_algebra
Symmetry of physical laws under a charge-conjugation transformation
the spinor bundle, depending on the local choice of a coordinate frame. Put another way, a spinor field is a local section of the spinor bundle, and
C-symmetry
spin structure on orientable Riemannian manifolds. A metaplectic structure on a symplectic manifold allows one to define the symplectic spinor bundle
Metaplectic_structure
Special tangential structure
mathematics, these are used to describe spinor bundles and spinors, which in physics are used to describe spin, an intrinsic angular momentum of particles
Spinc_structure
Area of differential geometry and topology
geometry Symplectic topology Spinor Spinor bundle Spin manifold Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press
Spin_geometry
Concept in mathematics
bundle – Continuous surjection satisfying a local triviality condition Spinor bundle – Geometric structure Tensor field – Assignment of a tensor continuously
Tensor_bundle
Fiber bundle
distribution is integrable, Frobenius theorem applies, producing a foliation. Spinor bundle All of these constructions are due to Ehresmann (1941-3). Attributed
Associated_bundle
4-manifold invariants
Spin(4) on which U(1) acts by multiplication. We have K = c 1 ( W + ) = c 1 ( W − ) {\displaystyle K=c_{1}(W^{+})=c_{1}(W^{-})} . The spinor bundle W
Seiberg–Witten_invariants
is even if it is finite. Darboux theorem Symplectic frame bundle Symplectic spinor bundle Symplectic vector space Maurice de Gosson: Symplectic Geometry
Symplectic_basis
Study of vector bundles, principal bundles, and fibre bundles
A} and spinor field ψ {\displaystyle \psi } . In this case the four-manifold must admit a SpinC structure, which defines a principal SpinC bundle P {\displaystyle
Gauge_theory_(mathematics)
Matrices used for Lorentz group spinors
of the (co)tangent bundle and a Weyl spinor bundle, the construction carries over to a differentiable manifold with a spinor bundle. The σ {\displaystyle
Infeld–Van der Waerden symbols
Infeld–Van_der_Waerden_symbols
Type of derivative in differential geometry
a spinor is equivalent to the vanishing of the Lie derivative along the same Killing vector of all the spinor bi-linear quantities. While a spinor that
Lie_derivative
Relativistic wave description of fermions
relates a four-component spinor to its charge conjugate. As a 2×2 differential equation acting on a complex two-component spinor, resembling the Weyl equation
Majorana_equation
Continuous surjection satisfying a local triviality condition
In mathematics, and particularly topology, a fiber bundle (Commonwealth English: fibre bundle) is a space that is locally a product space, but globally
Fiber_bundle
Canonical subbundle
Symplectic structure Symplectic geometry Symplectic group Symplectic spinor bundle Habermann, Katharina; Habermann, Lutz (2006), Introduction to Symplectic
Symplectic_frame_bundle
algebra Complex spin structure Conformal manifold Conformally flat manifold Dirac operator Poincaré metric Spin group Spin structure Spinor bundle Ahlfors, L
Clifford_analysis
Concept in differential geometry
pure spinor field. If M is a spin manifold, then Hol(ω) ⊂ SU(n) if and only if M admits at least two linearly independent parallel pure spinor fields
Holonomy
operators. Orthonormal frame bundle Spinor Spin manifold Spinor representation Spin geometry Spin structure Clifford module bundle Penrose, Roger (2004). The
Clifford_bundle
quasinormal mode no-cloning theorem quantum entanglement spinor, spinor group, spinor bundle Dirac sea Spin foam Poincaré group gamma matrices Dirac adjoint Wigner's
List of mathematical topics in quantum theory
List_of_mathematical_topics_in_quantum_theory
Relates 2 second-order elliptic operators on a manifold with the same principal symbol
1-forms. If M is an oriented spin manifold with Dirac operator ð, then one may form the spin Laplacian Δ = ð2 on the spin bundle. On the other hand, the Levi-Civita
Weitzenböck_identity
American Jewish mathematician
1959) Chern's conjecture (affine geometry) Supermanifold Symplectic spinor bundle "Bertram Kostant, professor emeritus of mathematics, dies at 88". MIT
Bertram_Kostant
Relativistic wave equation describing massless fermions
Majorana spinor is again a pair of Weyl spinors, but this time arranged so that the left-handed spinor is the charge conjugate of the right-handed spinor. The
Weyl_equation
Classical field theories on fiber bundles
spinor fields. Scalar electrodynamics/chromodynamics: coupling of scalar and gauge fields. Quantum electrodynamics/chromodynamics: coupling of spinor
Covariant classical field theory
Covariant_classical_field_theory
Special tangential structure
mathematics, these are used to describe spinor bundles and spinors, which in physics are used to describe spin, an intrinsic angular momentum of particles
Spinh_structure
Fiber bundle whose fibers are group torsors
In the mathematical area of topology, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product
Principal_bundle
American video streaming service
also announced a bundle including its other U.S. streaming services Hulu (ad-supported version) and ESPN+, marketed as The Disney Bundle, initially for
Disney+
Structure group sub-bundle on a tangent frame bundle
G} , is a principal G {\displaystyle G} -subbundle of the tangent frame bundle F M {\displaystyle {\text{F}}M} (or GL ( M ) {\displaystyle \operatorname
G-structure_on_a_manifold
Branch of algebraic topology
K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory
Topological_K-theory
Vector bundle associated with conformal manifolds
bundle, the tractor bundle, of M. The spin group of S O ( p + 1 , q + 1 ) {\displaystyle SO(p+1,q+1)} admits a fundamental representation, the spin representation
Local_twistor
Quantum mechanics taking into account particles near or at the speed of light
The 2-spinor ψ+ corresponds to a particle with 4-momentum (E, p) and charge q and two spin states (σ = ±1/2, as before). The other 2-spinor ψ− corresponds
Relativistic quantum mechanics
Relativistic_quantum_mechanics
space Γ {\displaystyle \Gamma } of square integrable sections of the spinor bundle over X {\displaystyle X} . Moreover, Connes observed that this distance
Spectral_triple
Assignment of a tensor continuously varying across a region of space
manifold Jet bundle – Construction in differential topology Ricci calculus – Tensor index notation for tensor-based calculations Spinor field – Geometric
Tensor_field
Relativistic quantum mechanical wave equation
the adjoint spinor ψ ¯ ( x ) {\displaystyle {\bar {\psi }}(x)} . Meanwhile, the adjoint Dirac equation is acquired by varying the spinor ψ ( x ) {\displaystyle
Dirac_equation
American mathematician
Commutation theorem for traces Metaplectic group Symplectic group Symplectic spinor bundle Shale, D. (1962). "Linear symmetries of free boson fields". Trans. Amer
Irving_Segal
the natural projections. Using spinor index notation, the Penrose transform gives a bijection between solutions to the spin ± n / 2 {\displaystyle \pm n/2}
Penrose_transform
Complex four-component spinor
covariants. Charge conjugation transforms the positive-energy spinor into the negative-energy spinor. Charge conjugation is a mapping (an involution) ψ ↦ ψ c
Plane-wave solutions to the Dirac equation
Plane-wave_solutions_to_the_Dirac_equation
Fiber bundle induced by a map of its base space
mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle π : E → B {\displaystyle
Pullback_bundle
Australian telecommunication company
merged with SpinTel, transferring all of its users. In 2014, SpinTel launched an NBN service bundle. In 2015, it was discovered that SpinTel has accidentally
SpinTel
Infinite-dimensional group in topology
-connected cover of a spin group. A string manifold is a manifold with a lifting of its frame bundle to a string group bundle. This means that in addition
String_group
X_{K}\,} on the orthonormal frame bundle of its natural lift X ^ {\displaystyle {\hat {X}}\,} defined on the bundle of linear frames. Generalisations
Kosmann_lift
Die-cast metal top toys
guys". Spin Fighters were sold two to a package, one gold and one black. Power Launchers and Battle Arenas were available separately or bundled. The tops
Spin_Fighters
General relativity in M-theory
a real Majorana spinor representation, whose dimension is half that of the Dirac representation. When k is even there is a Weyl spinor representation,
Higher-dimensional supergravity
Higher-dimensional_supergravity
Second-rank tensor in quantum chromodynamics
the spacetime with values in the adjoint bundle of the chromodynamical SU(3) gauge group (see vector bundle for necessary definitions). Throughout this
Gluon_field_strength_tensor
Generalization of the Dirac equation
{\displaystyle \Psi } is a spinor field on spacetime. Mathematically, this is a section of a vector bundle associated to the spin-frame bundle by the representation
Dirac equation in curved spacetime
Dirac_equation_in_curved_spacetime
Concept in differential geometry
differentiable principal bundle or vector bundle with a connection. Let G be a Lie group and P → M be a principal G-bundle on a smooth manifold M. Suppose
Exterior_covariant_derivative
Math/physics concept
formulated subsequent to Cartan's initial work. In particular, on a principal bundle, a principal connection is a natural reinterpretation of the connection
Connection_form
Isomorphism between the tangent and cotangent bundles of a manifold
isomorphism) is an isomorphism between the tangent bundle T M {\displaystyle \mathrm {T} M} and the cotangent bundle T ∗ M {\displaystyle \mathrm {T} ^{*}M} of
Musical_isomorphism
Approach to general relativity
general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis
Tetrad_formalism
Specification of a derivative along a tangent vector of a manifold
contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically
Covariant_derivative
System of moving vectors in differential geometry
affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along
Parallel_transport
Candidate unified theory of physics
on any globally hyperbolic Lorentzian spin manifold ( M ^ , g ) {\displaystyle ({\hat {M}},g)} with spinor bundle S M ^ {\displaystyle S{\hat {M}}} , one
Causal_fermion_systems
Modern theory of gravitation that combines supersymmetry and general relativity
Majorana spinor. This Majorana spinor can be reexpressed as a complex left-handed Weyl spinor and its complex conjugate right-handed Weyl spinor (they're
Supergravity
Construct allowing differentiation of tangent vector fields of manifolds
simplest methods of defining differentiation of the sections of vector bundles. The notion of an affine connection has its roots in 19th-century geometry
Affine_connection
Clifford algebra in 4 dimensions
to the Majorana spinor and the ELKO spinor, which cannot (i.e. they are electrically neutral), as they explicitly constrain the spinor so as to not interact
Dirac_algebra
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
In differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space)
Hopf_fibration
Mathematical operation on vector spaces
Dimension Exterior form Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable
Tensor_product
Expression that may be integrated over a region
fiber at p {\displaystyle p} of the dual bundle of the k {\displaystyle k} th exterior power of the tangent bundle of M {\displaystyle M} . That is, β {\displaystyle
Differential_form
Machinery used to spin cotton
Cotton-spinning machinery is machines which process (or spin) prepared cotton roving into workable yarn or thread. Such machinery can be dated back centuries
Cotton-spinning_machinery
Moduli space of the Seiberg–Witten equations
bundles W ± ↠ M {\displaystyle W^{\pm }\twoheadrightarrow M} , called associated spinor bundles (whose sections are called (anti) self-dual spinors)
Seiberg–Witten_moduli_space
Affine connection on the tangent bundle of a manifold
the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the (pseudo-)Riemannian metric and is torsion-free
Levi-Civita_connection
Array of numbers describing a metric connection
frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is
Christoffel_symbols
Tensor in differential geometry
with the methods of constructing more exotic geometric objects, such as spinor fields. The complicated formula defining R i j {\displaystyle R_{ij}} in
Ricci_curvature
Exterior algebraic map taking tensors from p forms to n-p forms
perspectives, making contacts to the use of the two-spinor language in modern physics such as spinor-helicity formalism or twistor theory. The Hodge star
Hodge_star_operator
Representation theory of an important group in physics
representation by associating a 4-component Dirac spinor ψ {\displaystyle \psi } with each particle. These spinors transform under Lorentz transformations generated
Representation theory of the Poincaré group
Representation_theory_of_the_Poincaré_group
Differential form of degree one or section of a cotangent bundle
cotangent bundle. Equivalently, a one-form on a manifold M {\displaystyle M} is a smooth mapping of the total space of the tangent bundle of M {\displaystyle
One-form
Construct in differenital geometry
mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any
Metric_connection
Universal construction in multilinear algebra
Dimension Exterior form Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable
Tensor_algebra
Mathematical function, in linear algebra
Dimension Exterior form Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable
Linear_map
University Press, p. 25 Roger Penrose; Wolfgang Rindler, Spinors and Space–Time: Volume 1, Two-Spinor Calculus and Relativistic Fields, Cambridge University
Holonomic_basis
Matrix operation which flips a matrix over its diagonal
Dimension Exterior form Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable
Transpose
Straight path on a curved surface or a Riemannian manifold
double tangent bundle TTM into horizontal and vertical bundles: T T M = H ⊕ V . {\displaystyle TTM=H\oplus V.} The double tangent bundle can be visualized
Geodesic
Algebraic operation on coordinate vectors
Dimension Exterior form Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable
Dot_product
Shorthand notation for tensor operations
Dimension Exterior form Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable
Einstein_notation
Representation of the supersymmetry algebra
a gauge boson, the highest component of a chiral or hypermultiplet is a spinor, the highest component of a gravity multiplet is a graviton. The names are
Supermultiplet
valve Spin wave Spinhenge@Home Spinodal decomposition Spinon Spinor Spinor bundle Spinoza Prize Spinplasmonics Spinthariscope Spintronics Spin–charge
Index_of_physics_articles_(S)
Long, narrow bundle of fiber
A roving is a long and narrow bundle of fiber. Rovings are produced during the process of making spun yarn from wool fleece, raw cotton, or other fibres
Roving
Study of curves from a differential point of view
Dimension Exterior form Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable
Differentiable_curve
Swedish video game developer
It also released smaller games as part of game jams organised by Humble Bundle and published the externally developed Cobalt and Cobalt WASD. Mojang Studios
Mojang_Studios
Method for specifying point positions
Dimension Exterior form Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable
Coordinate_system
Dimension Exterior form Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable
Tensors in curvilinear coordinates
Tensors_in_curvilinear_coordinates
Game series published by Capcom
Reconnaissance) is sent aboard the probe ship Seyfert to investigate. Dino Crisis Bundle is a compilation of Dino Crisis and Dino Crisis 2 that was released on Windows
Dino_Crisis
used in some parts of geometry. See: Spin group Spin-c group Spinor Pin group Pinors Spinor field Killing spinor Spin manifold Ricci, Gregorio; Levi-Civita
Glossary_of_tensor_theory
Night Neil Faulkner 2015 HN5683 Always Friends Neil Faulkner 2015 HN5684 Bundle Of Joy Neil Faulkner 2015 HN5685 Happy Anniversary Neil Faulkner 2015 HN5686
List of Royal Doulton figurines
List_of_Royal_Doulton_figurines
Special functions
the spin-weighted harmonics are U(1) gauge fields rather than scalar fields: mathematically, they take values in a complex line bundle. The spin-weighted
Spin-weighted spherical harmonics
Spin-weighted_spherical_harmonics
Operation in mathematics
Dimension Exterior form Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable
Tensor_contraction
Smooth manifold
J^{2}=-1} when regarded as a vector bundle isomorphism J : T M → T M {\displaystyle J\colon TM\to TM} on the tangent bundle. A manifold equipped with an almost
Almost_complex_manifold
Derivative used in gauge theories
associated bundle for the principal fiber bundle of the gauge theory; and, for the case of spinors, the associated bundle would be a spin bundle of the spin structure
Gauge_covariant_derivative
Operation that pairs a left and a right R-module into an abelian group
bundles T M , T ∗ M {\displaystyle TM,T^{*}M} are viewed as locally free sheaves on M. The exterior bundle on M is the subbundle of the tensor bundle
Tensor_product_of_modules
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
\kappa _{n}}.} The 4D version of the last relation appears in Penrose's spinor approach to general relativity that he later generalized, while he was developing
Kronecker_delta
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