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Physical quantity that changes sign with improper rotation
In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under continuous rigid transformations such as
Pseudovector
In high energy physics, a pseudovector meson or axial vector meson is a meson with total spin 1 and even parity (+) (usually denoted JP = 1+). Compare
Pseudovector_meson
Scalar quantity, changing sign in mirrored coordinates
not. A pseudoscalar, when multiplied by an ordinary vector, becomes a pseudovector (or axial vector); a similar construction creates the pseudotensor. A
Pseudoscalar
Calculus of vector-valued functions
over a line. In more advanced treatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fields
Vector_calculus
Boson with spin 1
A pseudovector boson is a vector boson that has even parity, whereas "regular" vector bosons have odd parity. There are no fundamental pseudovector bosons
Vector_boson
Meson with total spin 1 and odd parity
mesons contrast with the pseudovector mesons, which also have a total spin 1 but instead have even parity. The vector and pseudovector mesons are also dissimilar
Vector_meson
Geometric object that has length and direction
vector-like mathematical objects that describe physical quantities, such as pseudovectors and tensors, transform in a similar way under changes of the coordinate
Euclidean_vector
Operator in quantum field theory
In physics, the Pauli–Lubański pseudovector is an operator defined from the momentum and angular momentum, used in the quantum-relativistic description
Pauli–Lubanski_pseudovector
Direction and rate of rotation
position vector r of a moving particle. Here, orbital angular velocity is a pseudovector whose magnitude is the rate at which r sweeps out angle (in radians per
Angular_velocity
Rate of change of angle
waves). Angular frequency (or angular speed) is the magnitude of the pseudovector quantity angular velocity. Angular frequency can be obtained by multiplying
Angular_frequency
Vector of length one
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase
Unit_vector
Physical quantity
decreases counterclockwise. In three dimensions, angular acceleration is a pseudovector. In two dimensions, the orbital angular acceleration is the rate at which
Angular_acceleration
Turning force around an axis
around which it is being determined. In three dimensions, the torque is a pseudovector; for point particles, it is given by the cross product of the displacement
Torque
Mathematical operation on vectors in 3D space
permutation of the basis vectors. Therefore, the cross product is a pseudovector. In connection with the cross product, the exterior product of vectors
Cross_product
Equations describing classical electromagnetism
the electric field, E, a vector field, and the magnetic field, B, a pseudovector field, each generally having a time and location dependence. The sources
Maxwell's_equations
Conserved physical quantity; rotational analogue of linear momentum
angular momentum for a point particle is classically represented as a pseudovector r × p, the cross product of the particle's position vector r (relative
Angular_momentum
Group of flat spacetime symmetries
W_{\mu }W^{\mu }} where W μ {\textstyle W_{\mu }} is the Pauli–Lubanski pseudovector; they serve as labels for the representations of the group. The Poincaré
Poincaré_group
Angular momentum in special and general relativity
momentum is the cross product of position x with momentum p to obtain a pseudovector x × p, or alternatively as the exterior product to obtain a second order
Relativistic_angular_momentum
Austrian physicist (1900–1958)
conjecture Schwinger-Lüders-Pauli Theorem Pauli group Pauli-Lubanski pseudovector Postulating the neutrino Spouses Käthe Deppner (m. 1929; div. 1930)
Wolfgang_Pauli
Pseudovector field describing the local rotation of a continuum near some point
In continuum mechanics, vorticity is a pseudovector (or axial vector) field that describes the local spinning motion of a continuum near some point (the
Vorticity
Line or vector perpendicular to a curve or a surface
product of tangent vectors (as described in the text above), it is a pseudovector. When applying a transform to a surface it is often useful to derive
Normal_(geometry)
Physical quantity, density of magnetic moment per volume
quantity of magnetic moment per unit volume. It is represented by a pseudovector M. Magnetization can be compared to electric polarization, which is the
Magnetization
Mnemonic for 3D vectors orientations and rotations
rotating body is commonly represented by the rotational velocity, a pseudovector along the axis of rotation. The length of the vector gives the speed
Right-hand_rule
Type of physical quantity
proper rotation followed by reflection. This is a generalization of a pseudovector. To evaluate a tensor or pseudotensor sign, it has to be contracted with
Pseudotensor
Polish theoretical physicist
was a Polish theoretical physicist. He developed the Pauli–Lubanski pseudovector in relativistic quantum mechanics. Lubanski obtained the degree of magister
Józef_Lubański
component being examined and their interconnection. While torque is a Pseudovector, volume only by definition exists in three Euclidean dimensions, must
Torque_density
Quantity in electromagnetism
Although the magnetic field, B {\displaystyle \mathbf {B} } , is a pseudovector (also called axial vector), the vector potential, A {\displaystyle \mathbf
Magnetic_vector_potential
Four-dimensional number system
the number of basis vectors, and each bivector can be identified as a pseudovector. There are several advantages for placing quaternions in this wider setting:
Quaternion
Classification of irreducible representations of the Poincaré group
{\displaystyle ~C_{2}=W^{\alpha }\,W_{\alpha }~,} where W is the Pauli–Lubanski pseudovector. The eigenvalues of these operators serve to label the representations
Wigner's_classification
Relativistic quantum mechanical wave equation
}={\bar {\psi }}\gamma ^{\mu }\gamma ^{5}\psi .} This transforms as a pseudovector, meaning that its spatial part is odd under parity transformations. Classically
Dirac_equation
Physical phenomenon
the interaction, and in this case pseudovectors and vectors are added. The Gamow–Teller transition is a pseudovector transition, that is, the selection
Beta_decay_transition
Representation of a tensor in Euclidean space
another. Note the cross product of two vectors is a pseudovector, while the cross product of a pseudovector with a vector is another vector. Other identities
Cartesian_tensor
Intrinsic quantum property of particles
Holstein–Primakoff transformation Kramers' theorem Pauli equation Pauli–Lubanski pseudovector Rarita–Schwinger equation Representation theory of SU(2) Spin angular
Spin_(physics)
Symmetry of spatially mirrored systems
rotationally invariant. vectors ( P = −1 ) and axial vectors (also called pseudovectors) ( P = +1 ) which both transform as vectors under rotation. One can
Parity_(physics)
acceleration ωa Change in angular velocity per unit time rad/s2 T−2 pseudovector angular momentum L Measure of the extent and direction an object rotates
List_of_physical_quantities
Circulation density in a vector field
(1-form); curl takes a vector field (1-form) to a pseudovector field (2-form); div takes a pseudovector field (2-form) to a pseudoscalar field (3-form)
Curl_(mathematics)
Branch of mathematics
product Kronecker delta Levi-Civita symbol Multilinear form Pseudoscalar Pseudovector Spinor Tensor Tensor algebra, Free algebra Tensor contraction Symmetric
Multilinear_algebra
Broad concept generalizing scalars in mathematics and physics
vector-like mathematical objects that describe physical quantities, such as pseudovectors and tensors, transform in a similar way under changes of the coordinate
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Property of space that quantifies the magnetic influence at a given location
magnetic fields on moving charges More precisely, magnetic field is a pseudovector field due to its properties under inversion. The letters B and H were
Magnetic_field
angular velocity can be represented by a pseudovector because second rank tensors are dual to pseudovectors in three dimensions. Since the angular velocity
Angular_velocity_tensor
Choice of reference for distinguishing an object and its mirror image
orientation of a surface normal. An oriented plane can be defined by a pseudovector. For any n-dimensional real vector space V we can form the kth-exterior
Orientation_(vector_space)
Subatomic particle; made of equal numbers of quarks and antiquarks
Types of mesons Type S L P J JP Pseudoscalar meson 0 0 − 0 0− Pseudovector meson 0, 1 1 + 1 1+ Vector meson 1 0, 2 − 1 1− Scalar meson 1 1 + 0 0+ Tensor
Meson
Concept in physics
three-dimensional parameter space the Berry curvature can be written in the pseudovector form Ω n ( R ) = ∇ R × A n ( R ) . {\displaystyle \mathbf {\Omega } _{n}(\mathbf
Berry connection and curvature
Berry_connection_and_curvature
shows that they are different kinds of quantities. In physics, the names pseudovector and axial vector are used to describe vectors that transform in the same
Antivector
Surname list
American bowler Józef Lubański (1915–1947), Polish physicist Pauli–Lubanski pseudovector Włodzimierz Lubański (born 1947), Polish football striker This page lists
Lubanski_(surname)
antisymmetric tensor ( J i j {\displaystyle J_{ij}} ) or, equivalently, as a pseudovector. As a tensor, the equation for the conservation of angular momentum for
Rotational_viscosity
Fluid flow revolving around an axis of rotation
bodies within a fluid flow due to vortices in the fluid Vorticity – Pseudovector field describing the local rotation of a continuum near some point Whirly
Vortex
Rotation composed with a reflection
symmetry plane), it is important to distinguish between vectors and pseudovectors (as well as scalars and pseudoscalars, and in general between tensors
Improper_rotation
Mathematical description of spacetime used in relativity
the space. The imaginary part, on the other hand, may consist of four pseudovectors, such as angular velocity and magnetic moment, which change their direction
Minkowski_spacetime
Second order tensor in vector algebra
two vectors and returns a scalar, while the cross product returns a pseudovector. Both of these have various significant geometric interpretations and
Dyadics
Family of linear transformations
depends on relative motion. In the rest frame of the particle, the spin pseudovector can be fixed to be its ordinary non-relativistic spin with a zero timelike
Lorentz_transformation
Algebraic structure designed for geometry
geometric algebra. Bivectors provide a more natural representation of the pseudovector quantities of 3D vector calculus that are derived as a cross product
Geometric_algebra
Ways to represent 3D rotations
+1 (initially), through (scalar + pseudovector) values to scalar −1 (at one full turn), through (scalar + pseudovector) values back to scalar +1 (at two
Rotation formulations in three dimensions
Rotation_formulations_in_three_dimensions
4D relativistic energy and momentum
{\displaystyle n} . Physics portal Four-force Four-gradient Pauli–Lubanski pseudovector Taylor, Edwin; Wheeler, John (1992). Spacetime physics introduction to
Four-momentum
Parameterization of a rotation into a unit vector and angle
|v| is the Euclidean norm of the 3-vector v. Homogeneous coordinates Pseudovector Rotations without a matrix Screw theory, a representation of rigid-body
Axis–angle_representation
Operator in quantum mechanics
operator (quantum mechanics) Relativistic wave equations Pauli–Lubanski pseudovector Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd
Momentum_operator
Topics referred to by the same term
True vector may refer to: A polar vector, one that is not a pseudovector (or axial vector). More formally, a true vector is a contravariant vector, see:
True_vector
Motion of a certain space that preserves at least one point
vector, called the rotation vector (although, strictly speaking, it is a pseudovector). Matrices, versors (quaternions), and other algebraic things: see the
Rotation_(mathematics)
Sum of directed areas in exterior algebra
They are related to complex numbers in two dimensions and to both pseudovectors and vector quaternions in three dimensions. They can be used to generate
Bivector
Vectors representing a particle spin state
the case of a single spin 1/2 particle. for example as described at Pseudovector under "Details" Griffiths, David J. (2005) Introduction to Quantum Mechanics(2nd
Eigenspinor
=\mathbf {u} \left(\mathbf {r} ,t\right)\,\!} m s−1 [L][T]−1 Velocity pseudovector field ω ω = ∇ × v {\displaystyle {\boldsymbol {\omega }}=\nabla \times
List of equations in fluid mechanics
List_of_equations_in_fluid_mechanics
Spinning motion in theoretical physics
angular momentum Mathisson–Papapetrou–Dixon equations Pauli–Lubanski pseudovector A. K. Raychaudhuri; S. Banerji; A. Banerjee (2003). General Relativity
Spin_tensor
Representation of the symmetry group of spacetime in special relativity
section below, the already mentioned vector irrep, (1/2, 1/2), a pseudovector irrep, (1/2, 1/2) with parity inversion eigenvalue +1 (not −1), and
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
Projection of spin along the direction of momentum
exhibiting an analogous phenomenon Wigner's classification Pauli–Lubanski pseudovector Griffiths, David (2008). Introduction to Elementary Particles. Wiley-VCH
Helicity_(particle_physics)
Theoretical physics phenomenon
correctly, therefore the axis is also parallel to u×v. The magnitude of this pseudovector is neither interesting nor important, only the direction is, so it can
Wigner_rotation
Local rate of change in potential with respect to displacement
means the velocity field is conservative, or equivalently the vorticity pseudovector field ω is zero: ω = ∇ × v = 0 . {\displaystyle {\boldsymbol {\omega
Potential_gradient
Consequence of the conservation of angular momentum
g'={\frac {2m}{e}}{\frac {M}{J}}} (here we use the projections of the pseudovectors M {\displaystyle \mathbf {M} } and J {\displaystyle \mathbf {J} } onto
Einstein–de_Haas_effect
Relativistic correction
particle's instantaneous velocity. Like any angular velocity, ωT is a pseudovector; its magnitude is the angular speed the particle's frame precesses (in
Thomas_precession
Term from classical mechanics
areal velocity (also called sector velocity or sectorial velocity) is a pseudovector whose length equals the rate of change at which area is swept out by
Areal_velocity
Exterior product of vectors
In a vector space of dimension n, a blade of grade n − 1 is called a pseudovector or an antivector. The highest grade element in a space is called a pseudoscalar
Blade_(geometry)
time rate of change of angular velocity. In three dimensions, it is a pseudovector. In SI units, it is measured in radians per second squared (rad/s2),
Glossary_of_physics
List of definitions of terms and concepts related to civil engineering
The rate of change of angular velocity. In three dimensions, it is a pseudovector. In SI units, it is measured in radians per second squared (rad/s2),
Glossary_of_civil_engineering
Antisymmetric permutation object acting on tensors
symbol is a pseudotensor, the result of taking a cross product is a pseudovector, not a vector. Under a general coordinate change, the components of the
Levi-Civita_symbol
Quantum field theory equations
single particle exchange diagrams. For scalar, pseudoscalar, vector, pseudovector, and tensor exchanges those matrix structures are respectively 1 1 1
Two-body_Dirac_equations
Setting of relativistic physics in geometric algebra
_{1}\gamma _{2},\gamma _{2}\gamma _{3},\gamma _{3}\gamma _{1}\}} , four pseudovectors (trivectors) { I γ 0 , I γ 1 , I γ 2 , I γ 3 } {\displaystyle \{I\gamma
Spacetime_algebra
Properties underlying modern physics
describe spin in relativistic quantum mechanics, the Pauli–Lubanski pseudovector W μ = 1 2 ε μ ν ρ σ J ν ρ P σ , {\displaystyle W_{\mu }={\frac {1}{2}}\varepsilon
Symmetry_in_quantum_mechanics
Distinguished element of a Lie algebra's center
Casimir by direct computation. Harish-Chandra isomorphism Pauli–Lubanski pseudovector Clebsch–Gordan coefficients Oliver, David (2004). The shaggy steed of
Casimir_element
Vector in relativity
) {\displaystyle \mathbf {S} =(0,\mathbf {s} )} where s is the spin pseudovector. In quantum mechanics, not all three components of this vector are simultaneously
Four-vector
the rate of change of angular velocity. In three dimensions, it is a pseudovector. In SI units, it is measured in radians per second squared (rad/s2),
Glossary_of_engineering:_A–L
Representation theory of the symmetries of non-relativistic quantum space
{L}}+{\vec {P}}\times {\vec {C}}~,} somewhat analogous to the Pauli–Lubanski pseudovector of relativistic mechanics. More generally, in n + 1 dimensions, invariants
Representation theory of the Galilean group
Representation_theory_of_the_Galilean_group
VA, entities such as pseudovectors and pseudoscalars need to be bolted on, whereas in GA the equivalent bivector and pseudovector respectively exist naturally
Comparison of vector algebra and geometric algebra
Comparison_of_vector_algebra_and_geometric_algebra
Quantum mechanical operator related to rotational symmetry
Jordan map (Schwinger's bosonic model of angular momentum) Pauli–Lubanski pseudovector Angular momentum diagrams (quantum mechanics) Spherical basis Tensor
Angular_momentum_operator
Optical phenomenon
y , g z ) {\displaystyle \mathbf {g} =(g_{x},g_{y},g_{z})} is a real pseudovector called the gyration vector, whose magnitude is generally small compared
Magneto-optic_effect
rotational force and is determined by a cross product. This makes it a pseudovector. τ = r × F {\displaystyle \tau =r\times F} where τ {\displaystyle \tau
Angular_mechanics
=\mathbf {I} \cdot {\boldsymbol {\omega }}} and the cross-product is a pseudovector i.e. if r and p are reversed in direction (negative), L is not. In general
List of equations in classical mechanics
List_of_equations_in_classical_mechanics
Formulation of quantum mechanics
defines the orbit orientation. The components of the angular momentum pseudovector are L i = ε i j k X j P k {\displaystyle L_{i}=\varepsilon _{ijk}X^{j}P^{k}}
Matrix_mechanics
Element of an exterior algebra
are pseudovectors; n-vectors are pseudoscalars. In the presence of a volume form (such as given an inner product and an orientation), pseudovectors and
Multivector
Study of the effects of forces on undeformable bodies
}}+{\boldsymbol {\omega }}\times {I{\boldsymbol {\omega }}}} where the pseudovectors τ and L are, respectively, the torques on the body and its angular momentum
Rigid_body_dynamics
Topological insulators of magnetic materials
{\displaystyle {\boldsymbol {\sigma }}_{\text{AHC}}^{\text{surf}}} is a pseudovector on the surface of the crystal, it must respect the surface symmetries
Magnetic topological insulator
Magnetic_topological_insulator
Tensor operator generalizes the notion of operators which are scalars and vectors
and when one wishes to provide this information, it is said to be a pseudovector.) Scalar, vector and tensor operators can also be formed by products
Tensor_operator
Application of Clifford algebra
different kinds of vector because of this, including Gibbs vectors, pseudovectors and contravariant vectors. The latter of these two, in plane-based GA
Plane-based_geometric_algebra
Meson with total spin 0 and odd parity
model-predicted mass around 250–300 MeV/c2. List of mesons Vector meson Pseudovector meson Pseudoscalar boson Compare the definition of the pseudoscalar meson
Pseudoscalar_meson
Mathematical description of fermions
null mass and pseudoscalar fields; the flagpole additionally has a null pseudovector field, whereas the Weyl spinors have a null antisymmetric tensor (a null
Dirac_spinor
Concept in chemistry and physics
sources of the violation of this rule: First, if dynamics depend on a pseudovector like the magnetic field or the rotation angular speed in the rotating
Microscopic_reversibility
General relativity equation
the mathematics of general relativity Geodesic equation Pauli–Lubanski pseudovector Test particle Relativistic angular momentum Center of mass (relativistic)
Mathisson–Papapetrou–Dixon equations
Mathisson–Papapetrou–Dixon_equations
Quantum mechanics taking into account particles near or at the speed of light
non-relativistic QM, the angular momentum operator is formed from the classical pseudovector definition L = r × p. In RQM, the position and momentum operators are
Relativistic quantum mechanics
Relativistic_quantum_mechanics
Formulations of electromagnetism
part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation
Mathematical descriptions of the electromagnetic field
Mathematical_descriptions_of_the_electromagnetic_field
Extension of the scalar spherical harmonics for use with vector fields
spherical harmonics are commonly used in physics to describe vector and pseudovector interactions, such as electromagnetic transitions, in atomic and nuclear
Vector_spherical_harmonics
Meson with total spin 0 and even parity
f0(1710) a0(1450) X(1110) f0(1200-1600) f01790 X(1810) List of mesons Pseudovector meson Scalar boson Ishida, M.Y. (1998). "Existence of σ(600)-particle
Scalar_meson
the magnetic field transforms under mirror reflection—as a field of pseudovectors. In electromagnetics, the term "magnetic field" is used for two distinct
Glossary_of_engineering:_M–Z
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