Joos–Weinberg equation
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In relativistic quantum mechanics and
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 ...
, the Joos–Weinberg equation is a relativistic wave equations applicable to free particles of arbitrary
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 ...
, an integer for bosons () or half-integer for
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s (). The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields. 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 ...
is usually denoted by in quantum mechanics, however in this context is more typical in the literature (see
references Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. It is called a ''name'' ...
). It is named after H. Joos and Steven Weinberg, found in the early 1960s.


Statement

Introducing a matrix;; ; : \gamma^ symmetric in any two tensor indices, which generalizes the gamma matrices in the Dirac equation, NB: The convention for the four-gradient in this article is , same as the Wikipedia article. Jeffery's conventions are different: . Also Jeffery uses collects the and components of the momentum operator: . The components are not to be confused with
ladder operator In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raisin ...
s; the factors of occur from the gamma matrices.
the equation is : i\hbar)^\gamma^ \partial_\partial_\cdots\partial_ + (mc)^Psi = 0 or


Lorentz group structure

For the JW equations the representation of the Lorentz group is :D^\mathrm = D^\oplus D^. This representation has definite spin . It turns out that a spin particle in this representation satisfy field equations too. These equations are very much like the Dirac equations. It is suitable when the symmetries of charge conjugation,
time reversal symmetry T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal, : T: t \mapsto -t. Since the second law of thermodynamics states that entropy increases as time flows toward the future ...
, and
parity Parity may refer to: * Parity (computing) ** Parity bit in computing, sets the parity of data for the purpose of error detection ** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the r ...
are good. The representations and can each separately represent particles of spin . A state or quantum field in such a representation would satisfy no field equation except the Klein–Gordon equation.


Lorentz covariant tensor description of Weinberg–Joos states

The six-component spin-1 representation space, :D^\mathrm = D^\oplus D^ can be labeled by a pair of anti-symmetric Lorentz indexes, , meaning that it transforms as an antisymmetric Lorentz tensor of second rank B_, i.e. : B_\sim D^\oplus D^. The ''j''-fold
Kronecker product In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to ...
of decomposes into a finite series of Lorentz-irreducible representation spaces according to :\bigotimes_^j \left(D_i^\oplus D_i^\right) \to D^ \oplus D^ \oplus D^\oplus \cdots \oplus D^ \oplus D^ \oplus \cdots \oplus D^, and necessarily contains a D^\oplus D^ sector. This sector can instantly be identified by means of a momentum independent projector operator , designed on the basis of , one of the
Casimir element 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 ...
s (invariants) of the Lie algebra of the Lorentz group, which are defined as, where are constant matrices defining the elements of the Lorentz algebra within the D^ \oplus D^ representations. The Capital Latin letter labels indicate the finite dimensionality of the representation spaces under consideration which describe the internal angular momentum (
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 ...
) degrees of freedom. The representation spaces D^ \oplus D^ are eigenvectors to in () according to, : C^ \left D^\oplus D^\right= \left ( j_1(j_1+1) +j_2(j_2+1) \right ) \left D^ \oplus D^ \right Here we define: :\lambda^_ = j_1(j_1+1) +j_2(j_2+1), to be the eigenvalue of the D^ \oplus D^ sector. Using this notation we define the projector operator, in terms of : Such projectors can be employed to search through for D^\oplus D^, and exclude all the rest. Relativistic second order wave equations for any ''j'' are then straightforwardly obtained in first identifying the D^\oplus D^ sector in in () by means of the Lorentz projector in () and then imposing on the result the mass shell condition. This algorithm is free from auxiliary conditions. The scheme also extends to half-integer spins, s= j+\tfrac in which case the
Kronecker product In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors to ...
of with the Dirac spinor, :D^\oplus D^ has to be considered. The choice of the totally antisymmetric Lorentz tensor of second rank, , in the above equation () is only optional. It is possible to start with multiple Kronecker products of totally symmetric second rank Lorentz tensors, . The latter option should be of interest in theories where high-spin D^\oplus D^ Joos–Weinberg fields preferably couple to symmetric tensors, such as the metric tensor in gravity.


An Example

The :\left (\tfrac, 0 \right ) \oplus \left (0, \tfrac \right ) transforming in the Lorenz tensor spinor of second rank, : \psi_ = (1,0) \oplus (0,1)\otimes \left \left (\tfrac, 0 \right ) \oplus \left (0, \tfrac \right ) \right The Lorentz group generators within this representation space are denoted by \left ^_ \right , and given by: : \left ^_ \right = \left ^_ \right ^S+ _\,\, \left ^S_\right :\mathbf_=\tfrac \left (g_g_-g_g_ \right ), : M^_=\tfrac\sigma_=\frac gamma_\mu,\gamma_ where stands for the identity in this space, and are the respective unit operator and the Lorentz algebra elements within the Dirac space, while are the standard gamma matrices. The generators express in terms of the generators in the four-vector, : \left ^_ \right =i \left (g_g_-g_g_ \right ), as :\left _^\right =-2 \cdot ^ ^\rho_. Then, the explicit expression for the Casimir invariant in () takes the form, :\left ^ \right = -\frac \left (\sigma_\sigma_- \sigma_\sigma_-22 \cdot \mathbf_ \right), and the Lorentz projector on (3/2,0)⊕(0,3/2) is given by, : \left ^\right=\frac 1 8 \left (\sigma_\sigma_ + \sigma_ \sigma_ \right )-\frac 1 \sigma_ \sigma_. In effect, the (3/2,0)⊕(0,3/2) degrees of freedom, denoted by : \left ^_\pm \left(,\tfrac,\lambda\right)\right are found to solve the following second order equation, :\left ( _ p^2-m^2 \cdot _ \right )\left w^_\pm \left(,\tfrac 3 2, \lambda\right)\right=0. Expressions for the solutions can be found in.


See also

* Higher-dimensional gamma matrices *
Bargmann–Wigner equations :''This article uses the Einstein summation convention for tensor/spinor indices, and uses hats for quantum operators. In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles with non ...
, alternative equations which describe free particles of any spin * Higher spin theory


References

* {{DEFAULTSORT:Joos-Weinberg equation Quantum mechanics Quantum field theory Mathematical physics