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First-order differential linear operator on spinor bundle, whose square is the Laplacian
mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such
Dirac_operator
Relativistic quantum mechanical wave equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including
Dirac_equation
study of Dirac operators, and Dirac type operators in analysis and geometry, together with their applications. Examples of Dirac type operators include
Clifford_analysis
Generalized function whose value is zero everywhere except at zero
In mathematical analysis, the Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized
Dirac_delta_function
British physicist (1902–1984)
Paul Adrien Maurice Dirac (/dɪ.ˈræk/, dih-RAK; 8 August 1902 – 20 October 1984) was a British theoretical physicist who is considered to be one of the
Paul_Dirac
Generators of the Clifford algebra for relativistic quantum mechanics
\left\{\gamma ^{0},\gamma ^{1},\gamma ^{2},\gamma ^{3}\right\}\ ,} also called the Dirac matrices, are a set of conventional matrices with specific anticommutation
Gamma_matrices
Theoretical model of the vacuum
The Dirac sea is a theoretical model of the electron vacuum as an infinite sea of electrons with negative energy. It was first postulated by the British
Dirac_sea
Vector differential operator
spherical coordinates Dirac operator Maxwell's equations Nabla symbol Navier–Stokes equations Notation for differentiation Quabla operator Table of mathematical
Del
Operator in quantum mechanics
becomes +iħ preceding the 3-momentum operator. This operator occurs in relativistic quantum field theory, such as the Dirac equation and other relativistic
Momentum_operator
while the (absolute value of) Dirac operator retains the metric. On the other hand, the phase part of the Dirac operator, in conjunction with the algebra
Spectral_triple
Mathematical result in differential geometry
that this integrality could be explained if it were the index of the Dirac operator (which was rediscovered by Atiyah and Singer in 1961). The Atiyah–Singer
Atiyah–Singer_index_theorem
Generalization of the Dirac equation
In mathematical physics, the Dirac equation in curved spacetime is a generalization of the Dirac equation from flat spacetime (Minkowski space) to curved
Dirac equation in curved spacetime
Dirac_equation_in_curved_spacetime
Operator generalizing the Laplacian in differential geometry
first order operator d + δ {\displaystyle \mathrm {d} +\delta } is the Hodge–Dirac operator. When computing the Laplace–de Rham operator on a scalar function
Laplace–Beltrami_operator
Notation for quantum states
Bra–ket notation or Dirac notation is a mathematical notation for linear algebra and linear operators on complex vector spaces together with their dual
Bra–ket_notation
Geometric analogue of the Dirac equation
manifold using the Laplace–de Rham operator. In four-dimensional flat spacetime, it is equivalent to four copies of the Dirac equation that transform into each
Dirac–Kähler_equation
Typically linear operator defined in terms of differentiation of functions
Geometry of Dirac operators, p. 8, CiteSeerX 10.1.1.186.8445 Hörmander, L. (1983), The analysis of linear partial differential operators I, Grundl. Math
Differential_operator
Formula for spinors
Weitzenböck. The formula gives a relationship between the Dirac operator and the Laplace–Beltrami operator acting on spinors, in which the scalar curvature appears
Lichnerowicz_formula
Quantum field theory equations
TBDE requires a particular form of mathematical consistency: the two Dirac operators must commute with each other. This is plausible if one views the two
Two-body_Dirac_equations
Formulation of quantum mechanics on a Hilbert Space
mathematical physics, the Dirac–von Neumann axioms give a mathematical formulation of quantum mechanics in terms of operators on a Hilbert space. They
Dirac–von_Neumann_axioms
Algebra based on a vector space with a quadratic form
define the Dirac equation and introduce the Dirac operator. The entire Clifford algebra shows up in quantum field theory in the form of Dirac field bilinears
Clifford_algebra
Relativistic wave description of fermions
forms: As the Dirac equation written so that the Dirac operator is purely Hermitian, thus giving purely real solutions. As an operator that relates a
Majorana_equation
Type of Dirac operator eigenspinor
indicates those twistor spinors which are also eigenspinors of the Dirac operator. The term is named after Wilhelm Killing. Another equivalent definition
Killing_spinor
Measure of curvature in differential geometry
found that on a spin manifold, the difference between the square of the Dirac operator and the tensor Laplacian (as defined on spinor fields) is given exactly
Scalar_curvature
British-Lebanese mathematician (1929–2019)
that this integrality could be explained if it were the index of the Dirac operator (which was rediscovered by Atiyah and Singer in 1961). The first announcement
Michael_Atiyah
Physical constant in quantum mechanics
and Dirac again introduced special symbols for it: K {\textstyle K} in the case of Schrödinger, and h {\textstyle h} in the case of Dirac. Dirac continued
Planck_constant
Operator in quantum mechanics
position operator should necessarily be Dirac delta distributions, suppose that ψ {\displaystyle \psi } is an eigenstate of the position operator with eigenvalue
Position_operator
Type of differential operator
a weakly elliptic first-order operator, such as the Dirac operator can square to become a strongly elliptic operator, such as the Laplacian. The composition
Elliptic_operator
Solution method for linear differential equations
previous work by Ecalle and Voros. An application to the non-self-adjoint Dirac operator followed and this has made possible the rigorous justification of the
WKB_approximation
Concept in differential geometry
realizing the  genus as the index of a Dirac operator – a Dirac operator is a square root of a second order operator, and exists due to the spin structure
Spin_structure
Area of differential geometry and topology
differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fundamental
Spin_geometry
Fermion discretization with four doublers
version of the Dirac–Kähler fermion. The naively discretized Dirac action in Euclidean spacetime with lattice spacing a {\displaystyle a} and Dirac fields ψ
Staggered_fermion
Symbol used to indicate the del operator
the operator of the train carries out the Stepping Back procedure. Del, treating the mathematics of the vector differential operator Dirac operator Del
Nabla_symbol
Operators useful in quantum mechanics
as second quantization. They were introduced by Paul Dirac. Creation and annihilation operators can act on states of various types of particles. For example
Creation and annihilation operators
Creation_and_annihilation_operators
Method of calculating chiral anomalies
values in the Lie algebra g . {\displaystyle {\mathfrak {g}}\,.} The Dirac operator (in Feynman slash notation) is D / = d e f ∂ / + i A / {\displaystyle
Fujikawa_method
Property of particles related to spin
all other fundamental interactions. Chirality for a Dirac fermion ψ is defined through the operator γ5, which has eigenvalues ±1; the eigenvalue's sign
Chirality_(physics)
Spectrum of eigenvalues
In mathematics, a Dirac spectrum, named after Paul Dirac, is the spectrum of eigenvalues of a Dirac operator on a Riemannian manifold with a spin structure
Dirac_spectrum
Mathematical description of fermions
In physics, and specifically in quantum field theory, a Dirac spinor is a mathematical construction that is used to describe some of the fundamental particles
Dirac_spinor
Periodic distribution ("function") of "point-mass" Dirac delta sampling
In mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic generalized function with the formula Ш T
Dirac_comb
Quantum operator for the sum of energies of a system
However, in the more general formalism of Dirac, the Hamiltonian is typically implemented as an operator on a Hilbert space in the following way: The
Hamiltonian (quantum mechanics)
Hamiltonian_(quantum_mechanics)
of a Dirac operator. For example, the vacuum expectation value of the baryon number is given by the spectral asymmetry of the Hamiltonian operator. The
Spectral_asymmetry
Lattice fermion discretisation
Euclidean spacetime lattice with spacing a {\displaystyle a} by the overlap Dirac operator D ov = 1 a ( ( 1 + a m ) 1 + ( 1 − a m ) γ 5 s i g n [ γ 5 A ] ) {\displaystyle
Overlap_fermion
Description of a quantum-mechanical system
unviable. This was fixed by Dirac by taking the so-called square root of the Klein–Gordon operator and in turn introducing Dirac matrices. In a modern context
Schrödinger_equation
American mathematician
area. In a series of three papersMikhael Gromov and Lawson used the Dirac operator and other techniques to prove global results about manifolds with positive
H._Blaine_Lawson
Quantum variations of random walks
)\otimes |0\rangle } Consider what happens when we discretize a massive Dirac operator over one spatial dimension. In the absence of a mass term, we have left-movers
Quantum_walk
Linear operator equal to its own adjoint
and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables
Self-adjoint_operator
Characteristic property of holomorphic functions
_{2}+\sigma _{2}\sigma _{1}=0} , so J 2 = − 1 {\displaystyle J^{2}=-1} ). The Dirac operator in this Clifford algebra is defined as ∇ ≡ σ 1 ∂ x + σ 2 ∂ y {\displaystyle
Cauchy–Riemann_equations
Intrinsic quantum property of particles
of the spin operators and introduced a two-component spinor wave-function. Pauli's theory of spin was non-relativistic. In 1928, Paul Dirac published his
Spin_(physics)
_{i=1}^{n}\alpha ^{i}{\frac {\partial }{\partial x^{i}}}} be the massless Dirac operator. Let g ( s ) {\displaystyle g(s)} be continuous and real-valued, with
Pokhozhaev's_identity
Lattice fermion discretisation
completely decouple from the system. Kaplan's (and equivalently Shamir's) DW Dirac operator is defined by two addends D DW ( x , s ; y , r ) = D ( x ; y ) δ s r
Domain_wall_fermion
Method of analysis applied to problems wave propagation
the d'Alembertian gives rise to the KdV hierarchy; analogously, the Dirac operator gives rise to the AKNS hierarchy. Wikimedia Commons has media related
Huygens–Fresnel_principle
Breakdown of parity at the quantum level
of the determinant of a Dirac operator changes sign as one circumnavigates the circle. The eigenvalues of the Dirac operator come in pairs, and the sign
Parity_anomaly
Branch of mathematics
) {\displaystyle L^{2}(M,S)} of square-integrable spinors, and the Dirac operator D {\displaystyle D} encodes the metric. This motivates the notion of
Noncommutative_geometry
Lattice fermion discretisation
continuum formulation in the continuum limit. The class of fermions whose Dirac operators satisfy this equation are known as Ginsparg–Wilson fermions, with notable
Ginsparg–Wilson_equation
Geometric structure
(2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1 page 53 Friedrich, Thomas (2000), Dirac Operators in
Spinor_bundle
Lattice fermion discretisation
a {\displaystyle a} . The twisted mass Dirac operator is constructed from the (massive) Wilson Dirac operator D W {\displaystyle D_{W}} and reads D tw
Twisted_mass_fermion
Algebraic invariant of topological spaces
of the Dirac operator vanishes. In 1983, Witten conjectured that in this situation the equivariant index of a certain twisted Dirac operator is at least
Elliptic_cohomology
Particle effect
Erwin Schrödinger in 1930 in his analysis of wave packet solutions of the Dirac equation for relativistic electrons in free space. These exhibit interference
Zitterbewegung
Pictorial representation of the behavior of subatomic particles
earlier. The rules for spin-1/2 Dirac particles are as follows: The propagator is the inverse of the Dirac operator, the lines have arrows just as for
Feynman_diagram
Second-order differential operator
d'Alembert operator (denoted by a box: ◻ {\displaystyle \Box } ), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (cf
D'Alembert_operator
Spin of an electron
are the gamma matrices (known as Dirac matrices) and i is the imaginary unit. A second application of the Dirac operator will now reproduce the Pauli term
Electron_magnetic_moment
Mathematical problem in spectral theory
as well as for other elliptic differential operators such as the Cauchy–Riemann operator or Dirac operator. Other boundary conditions besides the Dirichlet
Hearing_the_shape_of_a_drum
Symmetry of physical laws under a charge-conjugation transformation
(helicity eigenstates) correspond to eigenstates of the chiral operator. This allows the massless Dirac field to be cleanly split into a pair of Weyl spinors ψ
C-symmetry
Canadian mathematician
Meinrenken, Eckhard (1998-03-25). "Symplectic Surgery and the Spinc–Dirac Operator". Advances in Mathematics. 134 (2): 240–277. doi:10.1006/aima.1997.1701
Eckhard_Meinrenken
Ties Euler characteristic of a closed even-dimensional Riemannian manifold to curvature
class. The Chern–Gauss–Bonnet theorem is derived by considering the Dirac operator D = d + d ∗ {\displaystyle D=d+d^{*}} The Chern formula is only defined
Chern–Gauss–Bonnet_theorem
Relates 2 second-order elliptic operators on a manifold with the same principal symbol
derivation inverse to θ on 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
Weitzenböck_identity
French mathematician (born 1948)
established a local version of the Atiyah-Singer families index theorem for Dirac operators, by introducing the Bismut superconnection which plays a central role
Jean-Michel_Bismut
Non-conservation of chiral current in physics
index theorem for Dirac operators. Roughly speaking, the symmetries of Minkowski spacetime, Lorentz invariance, Laplacians, Dirac operators and the U(1)xSU(2)xSU(3)
Chiral_anomaly
Connes approach a more fundamental object is a replacement for the Dirac operator in the form of a spectral triple, and an exterior algebra can be constructed
Quantum_differential_calculus
integral Dirac delta function Dirac comb Dirac measure Dirac operator Dirac algebra 5997 Dirac, an asteroid The various Dirac Medals Dirac (software) DiRAC supercomputing
List of things named after Paul Dirac
List_of_things_named_after_Paul_Dirac
Hypothetical particle with one magnetic pole
magnetic charge started with a paper by the physicist Paul Dirac in 1931. In this paper, Dirac showed that if any magnetic monopoles exist in the universe
Magnetic_monopole
Concept in theoretical mathematical physics
reasonable choice of this algebra, its representation and extended Dirac operator, the Standard Model of elementary particles can be recovered. In this
Quantum_spacetime
Unified field theory
dimension of the total space must be 2 mod 8, and the G-index of the Dirac operator of the compact space must be nonzero. The above development generalizes
Kaluza–Klein_theory
Textbook by Paul Dirac
influential monograph written by Paul Dirac and first published by Oxford University Press in 1930. In this book, Dirac presents quantum mechanics in a formal
The Principles of Quantum Mechanics
The_Principles_of_Quantum_Mechanics
Non-tensorial representation of the spin group
Hermitian metric on the complex representations of the real spin groups. A Dirac operator on each spin representation. If n = 2k is even, then the tensor product
Spinor
Symmetry in statistical physics
Jurkiewicz, J.; Krzywicki, A.; Petersen, C.; Petersson, B. (2001). "Dirac operator and Ising model on a compact 2D random lattice". Acta Physica Polonica
Kramers–Wannier_duality
Function acting on the space of physical states in physics
(x-y)} denotes the Dirac Delta. Let ψ be the wavefunction for a quantum system, and A ^ {\displaystyle {\hat {A}}} be any linear operator for some observable
Operator_(physics)
Raising and lowering operators in quantum mechanics
group SU(3) Many sources credit Paul Dirac with the invention of ladder operators. Dirac's use of the ladder operators shows that the total angular momentum
Ladder_operator
Fields giving rise to fermionic particles
relation as we do for the bosonic field) in order to make the operators compatible with Fermi–Dirac statistics. By putting in the expansions for ψ ( x ) {\displaystyle
Fermionic_field
Topics referred to by the same term
represented by the symbol Ð In mathematics and quantum physics, the Dirac operator is sometimes represented by a D with a slash through it This disambiguation
D with stroke (disambiguation)
D_with_stroke_(disambiguation)
Invariant of a quadratic form over a field of characteristic 2
correspond to a non-trivial value of the mod 2 Atiyah-Singer index of the Dirac operator. de Rham invariant, a mod 2 invariant of ( 4 k + 1 ) {\displaystyle
Arf_invariant
Formulation of the quantum many-body problem
quantization. The key ideas of this method were introduced in 1927 by Paul Dirac, and were later developed, most notably, by Pascual Jordan and Vladimir
Second_quantization
{\displaystyle \mathrm {GL} (4,\mathbb {R} )} . For instance, the Dirac operator in metric-affine gravitation theory in the presence of a general linear
Metric-affine gravitation theory
Metric-affine_gravitation_theory
Partial differential equation describing the evolution of temperature in a region
Berline, Nicole; Getzler, Ezra; Vergne, Michèle. Heat kernels and Dirac operators. Grundlehren der Mathematischen Wissenschaften, 298. Springer-Verlag
Heat_equation
View of quantum mechanics
interaction picture (also known as the interaction representation or Dirac picture after Paul Dirac, who introduced it) is an intermediate representation between
Interaction_picture
calculus Clifford algebra Clifford analysis the study of Dirac operators and Dirac type operators from geometry and analysis using clifford algebras. Clifford
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Dutch mathematician (1942–2010)
(2011), The heat kernel Lefschetz fixed point formula for the Spinc dirac operator, Boston: Birkhäuser, ISBN 978-0-8176-8247-7; Duistermaat, J. J. (1996)
Hans_Duistermaat
Fundamental solution to the heat equation, given boundary values
Berline, Nicole; Getzler, E.; Vergne, Michèle (2004), Heat Kernels and Dirac Operators, Berlin, New York: Springer-Verlag Chavel, Isaac (1984), Eigenvalues
Heat_kernel
Type of graph in mathematics and physics
is a scalar potential. Another example is the Dirac operator on a graph which is a matrix valued operator acting on vector valued functions that describe
Quantum_graph
René Descartes Dirac equation Dirac delta function Dirac comb Dirac spinor Dirac operator See also: List of things named after Paul Dirac Mathematics,
List of scientific laws named after people
List_of_scientific_laws_named_after_people
Connection on a spinor bundle
Ashtekar variables Dirac operator Cartan connection Levi-Civita connection Ricci calculus Supergravity Torsion tensor Contorsion tensor Dirac equation in curved
Spin_connection
Chinese-American mathematician (born 1949)
Mikhael; Lawson, H. Blaine Jr. (1983). "Positive scalar curvature and the Dirac operator on complete Riemannian manifolds". Publications Mathématiques de l'Institut
Shing-Tung_Yau
American mathematician (1924–2021)
of the Dirac operator, the general geometric construction of which was a notable new discovery. It is sometimes called the Atiyah–Singer operator in their
Isadore_Singer
Cocycle in an entire cyclic cohomology group
{\mathcal {A}}} . (c) A self-adjoint (unbounded) operator D {\displaystyle D} , called the Dirac operator such that (i) D {\displaystyle D} is odd under
JLO_cocycle
Condensed matter system
term Dirac matter refers to a class of condensed matter systems which can be effectively described by the Dirac equation. Even though the Dirac equation
Dirac_matter
Result about when a matrix can be diagonalized
functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix
Spectral_theorem
4-manifold invariants
field X {\displaystyle X} . The Clifford connection then defines a Dirac operator D A = γ ⊗ 1 ∘ ∇ A = γ ( d x μ ) ∇ μ A {\displaystyle D^{A}=\gamma \otimes
Seiberg–Witten_invariants
Study of vector bundles, principal bundles, and fibre bundles
spinor bundle and D / A {\displaystyle {D\!\!\!\!/}_{A}} is the induced Dirac operator of the induced covariant derivative ∇ A {\displaystyle \nabla _{A}}
Gauge_theory_(mathematics)
Theoretical framework in physics
scalar fields, Dirac fields, vector fields (e.g. the electromagnetic field), and even strings. However, creation and annihilation operators are only well
Quantum_field_theory
Process in quantum mechanical theories
quite Werner Heisenberg's route to obtaining quantum mechanics, but Paul Dirac introduced it in his 1926 doctoral thesis, the "method of classical analogy"
Canonical_quantization
American mathematician
articles, on topics including complex geometry, spin manifolds and the Dirac operator, and the theory of algebraic cycles. Half of her work has been in collaboration
Marie-Louise_Michelsohn
DIRAC OPERATOR
DIRAC OPERATOR
Boy/Male
Indian
Scholar
Boy/Male
Muslim
Scholar
Girl/Female
Indian, Sanskrit
Name of Lord Shiva; The Operator; One who Maintains Balance Between Life and Death
Girl/Female
Indian
Beautiful, Splendor, Derived from Indira - Goddess laxmis name
Surname or Lastname
English
English : from the Old Norse female personal name Gunvǫr, composed of the elements gunn ‘battle’ + vǫr, the feminine form of varr ‘defender’, or possibly from the Old Norse male personal name Gunnarr.English : occupational name for an operator of heavy artillery (see Gunn).Americanized spelling of German Gönner, a habitational name for someone from any of numerous places named Gönne.
Girl/Female
Tamil
Beautiful, Splendor, Derived from Indira - Goddess laxmis name
Boy/Male
Indian
Old Arabic name
Boy/Male
Muslim
Old Arabic name
DIRAC OPERATOR
DIRAC OPERATOR
Boy/Male
Hindu
Liquid
Boy/Male
Arabic, Muslim
Iskaf is a Shoe-maker
Girl/Female
Hebrew American Biblical English
Wished-for child; rebellion; bitter. Famous Bearers: the Virgin Mary; Mary Magdalene; Mary, Queen...
Male
Italian
 Italian, Portuguese and Spanish form of Latin Demetrius, DEMETRIO means "loves the earth" or "follower of Demeter."
Boy/Male
American, British, English
Broad-spreading Oak
Girl/Female
Arabic, Muslim
She was a Narrator of Hadith Known as Qarsafah Al-zahliyah
Girl/Female
Arabic, Muslim
Traveller
Surname or Lastname
English
English : habitational name from a place in Northamptonshire named Flore, from Old English flÅr(e) ‘floor’, probably with reference to a lost tessellated pavement.Danish : from a short form of the personal name Florentz or the Frisian Flores (see Florence).
Boy/Male
Tamil
Sangram | ஸஂகà¯à®°à®¾à®®
War
Boy/Male
Arabic, Muslim, Pashtun
To Unite; To Get Together
DIRAC OPERATOR
DIRAC OPERATOR
DIRAC OPERATOR
DIRAC OPERATOR
DIRAC OPERATOR
n.
A laboratory.
n.
A small branching shrub (Dirca palustris), with a white, soft wood, and a tough, leathery bark, common in damp woods in the Northern United States; -- called also moosewood, and wicopy.
n.
The symbol that expresses the operation to be performed; -- called also facient.
n.
A steel cutting instrument, with a long bent shank set in a handle which rests against the shoulder of the operator. It is operated by a thrust movement, and used in paring the hoofs of horses.
n.
One who, or that which, operates or produces an effect.
n.
An instrument for writing by means of type, a typewheel, or the like, in which the operator makes use of a sort of keyboard, in order to obtain printed impressions of the characters upon paper.
n.
One who sends telegraphic messages; a telegraphic operator; a telegraphist.
n.
A dealer in stocks or any commodity for speculative purposes; a speculator.
n.
A quantity of explosives anchored in a channel, beneath the water, or set adrift in a current, and so arranged that they will be exploded when touched by a vessel, or when an electric circuit is closed by an operator on shore.
v. t.
Any contrivance, especially one having a directing edge, surface, or channel, for giving direction to the motion of anything, as water, an instrument, or part of a machine, or for directing the hand or eye, as of an operator
n.
One who performs some act upon the human body by means of the hand, or with instruments.