Pauli–Lubanski Pseudovector
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In
physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...
, the Pauli–Lubanski pseudovector is an operator defined from the momentum and
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
, 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 a pioneer of quantum mechanics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics "for the ...
and
Józef Lubański Józef Kazimierz Lubański (1914 – 8 December 1946) was a Polish theoretical physicist. He developed the Pauli–Lubanski pseudovector in relativistic quantum mechanics. Life and works Lubanski obtained the degree of magister philosophi ...
. It describes the spin states of moving particles. It is the generator of the little group of the
Poincaré group The Poincaré group, named after Henri Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our unde ...
, 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 i ...
vector invariant.


Definition

It is usually denoted by (or less often by ) and defined by: where * \varepsilon_ is the
four-dimensional Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only 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 i ...
operator. In the language of
exterior algebra In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
, 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 ...
of a
trivector In multilinear algebra, a multivector, sometimes called Clifford number or multor, 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 -ve ...
, \mathbf = \star(\mathbf \wedge \mathbf). Note W_0 = \vec \cdot \vec, and \vec = E \vec- \vec \times \vec where \vec is the generator of rotations and \vec is the generator of boosts. 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, ...
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, 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 P with 4-momentum eigenvalue k of the Hilbert space of a
quantum system Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
(or for that matter the ''standard representation'' with interpreted as momentum space 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 group representation, 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 "m ...
. * 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 Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, 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 operato ...
describes the total spin of the particle, with
eigenvalues In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
W^2 = W_\mu W^\mu = -m^2 s(s + 1), where is the
spin quantum number In physics and chemistry, the spin quantum number is a quantum number (designated ) that describes the intrinsic angular momentum (or spin angular momentum, or simply ''spin'') of an electron or other particle. It has the same value for all ...
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, ...
. 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 vibrati ...
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 a scalar expression whose value is invariant under any Lorentz transformation. A Lorentz scalar may be generated from, e.g., the scalar product of vectors, or by contracting tensors. While ...
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, orientation, and scale have been specified in physical space. It is based on a set of reference points, defined as geometric ...
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 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. At present the only confirmed massless particle is the photon. Other particles and quasiparticles Standard Model gauge bosons The photon (carrier of ...
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, 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. Isospin is also known as isobaric spin or isotopic spin. Isospin symmetry is a subset of the flavour symmetr ...
) 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 Mass generation, generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles ...
. 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 particles that can ...
(and therefore the electromagnetic field) continues to fall into this class, although the other mass eigenstates of the carriers of the electroweak force (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 half odd-intege ...
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 half odd-intege ...
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 half odd-intege ...
) acquire non-zero mass.
Neutrino A neutrino ( ; denoted by the Greek letter ) is an elementary particle that interacts via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass is so small ('' -ino'') that i ...
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

* * * * * * * *


Notes


References

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