Pauli–Lubanski pseudovector
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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the Pauli–Lubanski pseudovector is an operator defined from the momentum and
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
, used in the quantum-relativistic description of angular momentum. It is named after
Wolfgang Pauli Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics fo ...
and
Józef Lubański Józef Kazimierz Lubański (1914 – 8 December 1946) was a Polish theoretical physicist. Life and works Lubanski obtained the degree of magister philosophies at Wilna in 1937. He then worked for two years as an assistant in theoretical phy ...
, It describes the spin states of moving particles. It is the generator of the
little group In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...
of the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
, that is the maximal subgroup (with four generators) leaving the eigenvalues of the
four-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
vector invariant.


Definition

It is usually denoted by (or less often by ) and defined by: where * \varepsilon_ is the
four-dimensional A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
totally antisymmetric
Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for some ...
; * J^ is the relativistic angular momentum tensor operator (M^); * P^ is the
four-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
operator. In the language of
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
, it can be written as the
Hodge dual In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the al ...
of a
trivector In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -vectors (a ...
, \mathbf = \star(\mathbf \wedge \mathbf). Note W_0 = \vec \cdot \vec, and \vec = E \vec- \vec \times \vec. evidently satisfies P^W_=0, as well as the following
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
relations, \begin \left ^\mu, W^\nu\right&= 0, \\ \left ^, W^\rho\right&= i \left( g^ W^\mu - g^ W^\nu\right), \end Consequently, \left _, W_\right= -i \epsilon_ W^ P^. The scalar is a Lorentz-invariant operator, and commutes with the four-momentum, and can thus serve as a label for irreducible unitary representations of the Poincaré group. That is, it can serve as the label for the
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
, a feature of the spacetime structure of the representation, over and above the relativistically invariant label for the mass of all states in a representation.


Little group

On an eigenspace S of the
4-momentum operator In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimensio ...
P with 4-momentum eigenvalue k of the Hilbert space of a quantum system (or for that matter the ''standard representation'' with interpreted as
momentum space In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all '' position vectors'' r in space, and ...
acted on by 5×5 matrices with the upper left 4×4 block an ordinary Lorentz transformation, the last column reserved for translations and the action effected on elements p (column vectors) of momentum space with appended as a ''fifth'' row, see standard texts) the following holds: * The components of W with P^\mu replaced by k^\mu form a Lie algebra. It is the Lie algebra of the Little group L_kof k, i.e. the subgroup of the homogeneous Lorentz group that leaves k invariant. * For every irreducible unitary representation of L_k there is an irreducible unitary representation of the full Poincaré group called an
induced representation In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" represent ...
. * A representation space of the induced representation can be obtained by successive application of elements of the full Poincaré group to a non-zero element of S and extending by linearity. The irreducible unitary representation of the Poincaré group are characterized by the eigenvalues of the two Casimir operators P^2 and W^2. The best way to see that an irreducible unitary representation actually is obtained is to exhibit its action on an element with arbitrary 4-momentum eigenvalue p in the representation space thus obtained. Irreducibility follows from the construction of the representation space.


Massive fields

In
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, in the case of a massive field, the
Casimir invariant In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator ...
describes the total
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
of the particle, with
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
W^2 = W_\mu W^\mu = -m^2 s(s + 1), where is the
spin quantum number In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe th ...
of the particle and is its
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, i ...
. It is straightforward to see this in the
rest frame In special relativity, the rest frame of a particle is the frame of reference (a coordinate system attached to physical markers) in which the particle is at rest. The rest frame of compound objects (such as a fluid, or a solid made of many vibratin ...
of the particle, the above commutator acting on the particle's state amounts to ; hence and , so that the little group amounts to the rotation group, W_\mu W^\mu = -m^2 \vec\cdot\vec. Since this is a
Lorentz invariant In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of v ...
quantity, it will be the same in all other
reference frame In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin (mathematics), origin, orientation (geometry), orientation, and scale (geometry), scale are specified by a set of reference point ...
s. It is also customary to take to describe the spin projection along the third direction in the rest frame. In moving frames, decomposing into components , with and orthogonal to , and parallel to , the Pauli–Lubanski vector may be expressed in terms of the spin vector = (similarly decomposed) as \begin W_0 &= P S_3, & W_1 &= m S_1, & W_2 &= m S_2, & W_3 &= \frac S_3, \end where E^2 = P^2 c^2 + m^2 c^4 is the
energy–momentum relation In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy (which is also called relativistic energy) to invariant mass (which is also called rest mass) and momentum. It i ...
. The transverse components , along with , satisfy the following commutator relations (which apply generally, not just to non-zero mass representations), \begin[] _1, W_2&= \frac \left(\left(\frac\right)^2 - \left(\frac\right)^2\right) S_3, & [W_2, S_3] &= \frac W_1, & [S_3, W_1] &= \frac W_2. \end For particles with non-zero mass, and the fields associated with such particles, _1, W_2= \frac m^2 S_3.


Massless fields

In general, in the case of non-massive representations, two cases may be distinguished. For massless particles, W^2 = W_\mu W^\mu = -E^\left((K_2 - J_1)^2 + (K_1 + J_2)^2\right) \mathrel\stackrel -E^2\left(A^2 + B^2\right) , where is the dynamic mass moment vector. So, mathematically, 2 = 0 does not imply 2 = 0.


Continuous spin representations

In the more general case, the components of transverse to may be non-zero, thus yielding the family of representations referred to as the ''cylindrical'' luxons ("luxon" is another term for "massless particle"), their identifying property being that the components of form a Lie subalgebra isomorphic to the 2-dimensional Euclidean group , with the longitudinal component of playing the role of the rotation generator, and the transverse components the role of translation generators. This amounts to a
group contraction In theoretical physics, Eugene Wigner and Erdal İnönü have discussed the possibility to obtain from a given Lie group a different (non-isomorphic) Lie group by a group contraction with respect to a continuous subgroup of it. That amounts to a ...
of , and leads to what are known as the ''continuous spin'' representations. However, there are no known physical cases of fundamental particles or fields in this family. It can be argued that continuous spin states possess an internal degree of freedom not seen in observed massless particles.


Helicity representations

In a special case, \vec is parallel to \vec ; or equivalently \vec \times \vec = \vec . For non-zero \vec this constraint can only be consistently imposed for luxons (
massless particle In particle physics, a massless particle is an elementary particle whose invariant mass is zero. There are two known gauge boson massless particles: the photon (carrier of electromagnetism) and the gluon (carrier of the strong force). However, glu ...
s), since the commutator of the two transverse components of \vec is proportional to m^2 \vec \cdot \vec \, . For this family, W^2 = 0 and W^\mu = \lambda \, P^\mu the invariant is, instead given by \left(W^0\right)^2 = \left(W^3\right)^2 , where W^0 = -\vec \cdot \vec , so the invariant is represented by the helicity operator W^0 / P . All particles that interact with the
weak nuclear force In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, ...
, for instance, fall into this family, since the definition of weak nuclear charge (weak
isospin In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions ...
) involves helicity, which, by above, must be an invariant. The appearance of non-zero mass in such cases must then be explained by other means, such as the
Higgs mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bein ...
. Even after accounting for such mass-generating mechanisms, however, the
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always ...
(and therefore the electromagnetic field) continues to fall into this class, although the other mass eigenstates of the carriers of the
electroweak force In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very differe ...
(the
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
and anti-
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
and
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
) acquire non-zero mass.
Neutrino A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
s were formerly considered to fall into this class as well. However, because neutrinos have been observed to oscillate in flavour, it is now known that at least two of the three mass eigenstates of the left-helicity neutrinos and right-helicity anti-neutrinos each must have non-zero mass.


See also

*Center of mass (relativistic) *
Wigner's classification In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ~ (~E \ge 0~)~ energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (Since thi ...
*
Angular momentum operator In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum prob ...
*
Casimir operator In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the Center (ring theory), center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared ang ...
*
Chirality Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from ...
*
Pseudovector In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...
*
Pseudotensor In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g. a proper rotation) but additionally changes sign under an orientation-reversing coordinat ...
*
Induced representation In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" represent ...


Notes


References

* * * * * * * * * * * * {{DEFAULTSORT:Pauli-Lubanski pseudovector Quantum field theory Representation theory of Lie algebras