View source for Joos–Weinberg equation

{{Short description|Equation for arbitrary spin particles}}
{{technical|date=December 2016}}

{{quantum field theory}}

In [[relativistic quantum mechanics]] and [[quantum field theory]], the '''Joos–Weinberg equation''' is a [[relativistic wave equation]] applicable to [[free particle]]s of arbitrary [[Spin (physics)|spin]] {{math|''j''}}, an integer for [[boson]]s ({{math|''j'' {{=}} 1, 2, 3 ...}}) or half-integer for [[fermion]]s ({{math|''j'' {{=}} {{frac|1|2}}, {{frac|3|2}}, {{frac|5|2}} ...}}). The solutions to the equations are [[wavefunction]]s, mathematically in the form of multi-component [[spinor field]]s. The [[spin quantum number]] is usually denoted by {{math|''s''}} in quantum mechanics, however in this context {{math|''j''}} is more typical in the literature (see [[#References|references]]).

It is named after [[Hans Joos|Hans H. Joos]] and [[Steven Weinberg]], found in the early 1960s.<ref>{{Cite journal|last=Joos|first=Hans|date=1962|title=Zur Darstellungstheorie der inhomogenen Lorentzgruppe als Grundlage quantenmechanischer Kinematik|url=https://onlinelibrary.wiley.com/doi/10.1002/prop.2180100302|journal=Fortschritte der Physik|language=German|volume=10|issue=3|pages=65–146|doi=10.1002/prop.2180100302|bibcode=1962ForPh..10...65J }}</ref><ref name="Weinberg"/><ref name="Jeffery" />

==Statement==

Introducing a {{math|2(2''j'' + 1) × 2(2''j'' + 1)}} matrix;<ref name="Weinberg">{{cite journal|author=Weinberg, S.|journal=Phys. Rev.|volume=133|pages=B1318–B1332|year=1964|doi=10.1103/PhysRev.133.B1318|title=Feynman Rules ''for Any'' spin|issue=5B|bibcode=1964PhRv..133.1318W|url=http://theory.fi.infn.it/becattini/files/weinberg3.pdf|access-date=2016-12-28|archive-date=2022-03-25|archive-url=https://web.archive.org/web/20220325020742/http://theory.fi.infn.it/becattini/files/weinberg3.pdf|url-status=dead}}; {{cite journal|author=Weinberg, S.|journal=Phys. Rev.|volume=134|pages=B882–B896|year=1964|doi=10.1103/PhysRev.134.B882|title=Feynman Rules ''for Any'' spin. II. Massless Particles|issue=4B|bibcode=1964PhRv..134..882W|url=http://theory.fi.infn.it/becattini/files/weinberg2.pdf|access-date=2016-12-28|archive-date=2022-03-09|archive-url=https://web.archive.org/web/20220309040610/http://theory.fi.infn.it/becattini/files/weinberg2.pdf|url-status=dead}}; {{cite journal|author=Weinberg, S.|journal=Phys. Rev.|volume=181|pages=1893–1899|year=1969|doi=10.1103/PhysRev.181.1893|title=Feynman Rules ''for Any'' spin. III|issue=5|bibcode=1969PhRv..181.1893W|url=http://theory.fi.infn.it/becattini/files/weinberg3.pdf|access-date=2016-12-28|archive-date=2022-03-25|archive-url=https://web.archive.org/web/20220325020742/http://theory.fi.infn.it/becattini/files/weinberg3.pdf|url-status=dead}}</ref>

:<math> \gamma^{\mu_1 \mu_2 \cdots \mu_{2j}}</math>

symmetric in any two tensor indices, which generalizes the gamma matrices in the Dirac equation,<ref name="Jeffery">{{cite journal | author =E.A. Jeffery
 | year =1978
 | title =Component Minimization of the Bargman–Wigner wavefunction
 | journal =Australian Journal of Physics
 | volume =31
 | issue =2
 | pages =137
 | location = Melbourne
 | publisher = CSIRO
 | bibcode =1978AuJPh..31..137J
 | doi=10.1071/ph780137
 | doi-access =free
 }} '''NB:''' The convention for the [[four-gradient]] in this article is {{math|∂<sub>μ</sub> {{=}} (∂/∂''t'', &nabla;
)}}, same as the Wikipedia article. Jeffery's conventions are different: {{math|∂<sub>μ</sub> {{=}} (−''i''∂/∂''t'', &nabla;
)}}. Also Jeffery uses collects the {{math|''x''}} and {{math|''y''}} components of the momentum operator: {{math|''p''<sub>±</sub> {{=}} ''p''<sub>1</sub> ± ''ip''<sub>2</sub> {{=}} ''p''<sub>''x''</sub> ± ''ip''<sub>''y''</sub>}}. The components {{math|''p''<sub>±</sub>}} are not to be confused with [[ladder operator]]s; the factors of {{math|±1, ±''i''}} occur from the [[gamma matrices]].</ref><ref>{{cite journal
 | author = Gábor Zsolt Tóth
 | year = 2012
 | title = Projection operator approach to the quantization of higher spin fields
 | arxiv = 1209.5673
 | doi=10.1140/epjc/s10052-012-2273-x
 | volume=73
 | pages = 2273
 | journal=The European Physical Journal C
| bibcode=2013EPJC...73.2273T
 | s2cid = 119140104
 }}</ref> the equation is<ref>{{cite journal
 | author = D. Shay
 | year = 1968
 | title = A Lagrangian formulation of the Joos–Weinberg wave equations for spin-''j'' particles
 |journal=Il Nuovo Cimento A
 |volume=57 |issue=2 |pages=210–218
 | doi=10.1007/BF02891000 
|bibcode = 1968NCimA..57..210S | s2cid = 117170355
 }}</ref>

:<math>[(i\hbar)^{2j}\gamma^{\mu_1 \mu_2 \cdots \mu_{2j}} \partial_{\mu_1}\partial_{\mu_2}\cdots\partial_{\mu_{2j}} + (mc)^{2j}]\Psi = 0 </math>

or

{{NumBlk|:|
{{Equation box 1
|title=
|indent =
|equation = <math>[\gamma^{\mu_1 \mu_2 \cdots \mu_{2j}} P_{\mu_1}P_{\mu_2}\cdots P_{\mu_{2j}} + (mc)^{2j}]\Psi = 0 </math>
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
|{{EquationRef|4}}}}

==Lorentz group structure==

{{main|Representation theory of the Lorentz group}}

For the JW equations the [[Representation theory of the Lorentz group|representation of the Lorentz group]] is<ref name="Jaroszewicz, Kurzepa">{{cite journal |author1=T. Jaroszewicz |author2=P.S Kurzepa | year = 1992 | location = California, USA | title = Geometry of spacetime propagation of spinning particles | journal = Annals of Physics | doi=10.1016/0003-4916(92)90176-M | volume=216 |issue=2 | pages=226–267 |bibcode = 1992AnPhy.216..226J }}</ref>

:<math>D^\mathrm{JW} = D^{(j,0)}\oplus D^{(0,j)}.</math>

This representation has definite spin {{math|''j''}}. It turns out that a spin {{math|''j''}} 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]], and [[Parity (physics)|parity]] are good.

The representations {{math|''D''<sup>(''j'', 0)</sup>}} and {{math|''D''<sup>(0, ''j'')</sup>}} can each separately represent particles of spin {{math|''j''}}. 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,

:<math>D^\mathrm{JW} = D^{(1,0)}\oplus D^{(0,1)}</math>

can be labeled by a pair of anti-symmetric Lorentz indexes, {{math|[''αβ'']}}, meaning that it transforms as an antisymmetric Lorentz tensor of second rank <math>B_{[\alpha\beta ]},</math> i.e.

:<math> B_{[\alpha\beta ]}\sim D^{(1,0)}\oplus D^{(0,1)}. </math>

The ''j''-fold [[Kronecker product]] {{math|''T''<sub>[''α''<sub>1</sub>''β''<sub>1</sub>]...[''α<sub>j</sub>β<sub>j</sub>'']</sub>}} of {{math|''B''<sub>[''αβ'']</sub>}}

{{NumBlk|:|<math>T_{[\alpha_1\beta_1] \cdots [\alpha_j\beta_j]} = B_{[\alpha_1\beta_1]} \otimes \cdots \otimes B_{[\alpha_j\beta_j]} = \bigotimes_{i=1}^j B_{[\alpha_i\beta_i]},</math>|{{EquationRef|8A}}}}

decomposes into a finite series of Lorentz-irreducible representation spaces according to

:<math>\bigotimes_{i=1}^j \left(D_i^{(1,0)}\oplus D_i^{(0,1)}\right) \to D^{(j,0)} \oplus D^{(0,j)} \oplus D^{(j,j)}\oplus \cdots \oplus D^{(j_k, j_l)} \oplus D^{(j_l,j_k)} \oplus \cdots \oplus D^{(0,0)},</math>

and necessarily contains a <math>D^{(j,0)}\oplus D^{(0,j)}</math> sector. This sector can instantly be identified by means of a momentum independent projector operator {{math|''P''<sup>(''j'',0)</sup>}}, designed on the basis of {{math|''C''<sup>(1)</sup>}}, one of the [[Casimir element]]s (invariants)<ref>{{cite book|author1=Y. S. Kim |author2=Marilyn E. Noz |title=Theory and applications of the Poincaré group|date=1986|publisher=Reidel|location=Dordrecht, Holland|isbn=9789027721419}}</ref> of the Lie algebra of the [[Lorentz group]], which are defined as,

{{NumBlk|:|<math>\begin{cases}
\left[C^{(1)}\right]_{AB} = \frac 1 4 {\left[M^{\mu\nu}\right]_A}^{C}\left[M_{\mu\nu}\right]_{CB} \\[6pt]
\left[C^{(2)}\right]_{AB} =\frac 1 4 \varepsilon_{\mu\nu\lambda\eta} {\left [M^{\mu\nu} \right ]_A}^C \left[M^{\lambda\eta}\right]_{CB}
\end{cases} \qquad  A,B,C=1,\ldots,(2j_1+1)(2j_2+1) 
</math>|{{EquationRef|8B}}}}

where {{math|''M<sup>μν</sup>''}} are constant {{math|(2''j''<sub>1</sub>+1)(2''j''<sub>2</sub>+1) × (2''j''<sub>1</sub>+1)(2''j''<sub>2</sub>+1)}} matrices defining the elements of the Lorentz algebra within the <math> D^{(j_1, j_2)} \oplus D^{(j_2, j_1)}</math> representations. The Capital Latin letter labels indicate<ref name="Delgado-Banda-Kirchbach 2015">{{cite journal|author1=E. G. Delgado Acosta |author2=V. M. Banda Guzmán |author3=M. Kirchbach |title=Bosonic and fermionic Weinberg-Joos (j,0) ⊕ (0,j) states of arbitrary spins as Lorentz tensors or tensor-spinors and second-order theory|journal=The European Physical Journal A|date=2015|volume=51|issue=3|pages=35 |doi=10.1140/epja/i2015-15035-x|arxiv = 1503.07230 |bibcode = 2015EPJA...51...35D |s2cid=118590440 }}</ref> the finite dimensionality of the representation spaces under consideration which describe the internal angular momentum ([[Spin (physics)|spin]]) degrees of freedom.

The representation spaces <math> D^{(j_1, j_2)} \oplus D^{(j_2, j_1)}</math> are eigenvectors to {{math|''C''<sup>(1)</sup>}} in ({{EquationNote|8B}}) according to,

:<math> C^{(1)} \left[ D^{(j_1,j_2)}\oplus D^{(j_2,j_1)}\right] = \left ( j_1(j_1+1) +j_2(j_2+1) \right ) \left[ D^{(j_1,j_2)} \oplus D^{ (j_2,j_1)} \right], </math>

Here we define:

:<math>\lambda^{(1)}_{(j_1,j_2)} = j_1(j_1+1) +j_2(j_2+1),</math>

to be the {{math|''C''<sup>(1)</sup>}} eigenvalue of the <math> D^{(j_1, j_2)} \oplus D^{(j_2, j_1)}</math> sector. Using this notation we define the projector operator, {{math|''P''<sup>(''j'',0)</sup>}} in terms of {{math|''C''<sup>(1)</sup>}}:<ref name="Delgado-Banda-Kirchbach 2015" />

{{NumBlk|:|<math>{\left[  P^{(j,0)}  \right]_{\left[\alpha_1\beta_1\right] \cdots \left[ \alpha_j\beta_j\right]}}^{\left[\rho_1\sigma_1\right]\cdots \left[ \rho_j\sigma_j\right]} = {\left[\prod_{k, l} \left( \frac{C^{(1)} -\lambda^{(1)}_{(j_k,j_l)}} {\lambda^{(1)} _{(j,\, 0)}-\lambda^{(1)}_{(j_k,j_l)}}\right)\right]_{\left[ \alpha_1\beta_1\right]\cdots \left[ \alpha_j\beta_j\right]}}^{\left[\rho_1\sigma_1\right]\cdots \left[ \rho_j \sigma_j\right]}.</math>|{{EquationRef|8C}}}}

Such projectors can be employed to search through {{math|''T''<sub>[''α''<sub>1</sub>''β''<sub>1</sub>]...[''α<sub>j</sub>β<sub>j</sub>'']</sub>}} for <math>D^{(j,0)}\oplus D^{(0,j)},</math> and exclude all the rest. Relativistic second order wave equations for any ''j'' are then straightforwardly obtained in first identifying the <math>D^{(j,0)}\oplus D^{(0,j)}</math> sector in {{math|''T''<sub>[''α''<sub>1</sub>''β''<sub>1</sub>]...[''α<sub>j</sub>β<sub>j</sub>'']</sub>}} in ({{EquationNote|8A}}) by means of the Lorentz projector in ({{EquationNote|8C}}) 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, <math>s= j+\tfrac{1}{2}</math> in which case the [[Kronecker product]] of {{math|''T''<sub>[''α''<sub>1</sub>''β''<sub>1</sub>]...[''α<sub>j</sub>β<sub>j</sub>'']</sub>}} with the Dirac spinor,

:<math>D^{\left (\frac{1}{2},0 \right )}\oplus D^{ \left (0,\frac{1}{2} \right)}</math>

has to be considered. The choice of the totally antisymmetric Lorentz tensor of second rank, {{math|''B''<sub>[''α<sub>i</sub>β<sub>i</sub>'']</sub>}}, in the above equation ({{EquationNote|8A}}) is only optional. It is possible to start with multiple Kronecker products of totally symmetric second rank Lorentz tensors, {{math|''A<sub>α<sub>i</sub>β<sub>i</sub></sub>''}}. The latter option should be of interest in theories where high-spin <math>D^{(j,0)}\oplus D^{(0,j)}</math> Joos–Weinberg fields preferably couple to symmetric tensors, such as the metric tensor in gravity.

====An Example ====
Source:<ref name="Delgado-Banda-Kirchbach 2015" />

The

:<math>\left (\tfrac{3}{2}, 0 \right ) \oplus \left (0, \tfrac{3}{2} \right )</math>

transforming in the Lorenz tensor spinor of second rank,

:<math> \psi_{[\mu \nu]} = [ (1,0) \oplus (0,1)] \otimes \left [ \left (\tfrac{1}{2}, 0 \right ) \oplus \left (0, \tfrac{1}{2} \right ) \right ].</math>

The Lorentz group generators within this representation space are denoted by <math> \left [M^{ATS}_{\mu\nu} \right ]_{[ \alpha\beta][\gamma\delta]},</math> and given by:

:<math> \left [M^{ATS}_{\mu\nu} \right ]_{[ \alpha\beta][\gamma\delta]}= \left [M^{AT}_{\mu\nu} \right ]_{[\alpha\beta][\gamma\delta]}{\mathbf 1}^S+ {\mathbf 1}_{[\alpha\beta][\gamma\delta]}\,\,
\left[M^S_{\mu\nu}\right], </math>
:<math>\mathbf{1}_{[\alpha\beta][\gamma\delta]}=\tfrac{1}{2} \left (g_{\alpha\gamma}g_{\beta\delta}-g_{\alpha\delta}g_{\beta\gamma} \right ), </math>
:<math> M^{S}_{\mu\nu}=\tfrac{1}{2}\sigma_{\mu\nu}=\frac{i}{4}[\gamma_\mu,\gamma_{\nu}], </math>

where {{math|'''1'''<sub>[''αβ''][''γδ'']</sub>}} stands for the identity in this space, {{math| '''1'''<sup>''S''</sup>}} and {{math|''M<sup>S</sup><sub>μν</sub>''}} are the respective unit operator and the Lorentz algebra elements within the Dirac space, while {{math|''γ<sub>μ</sub>''}} are the standard [[gamma matrices]]. The {{math|[''M<sup>AT</sup><sub>μν</sub>'']<sub>[''αβ''][''γδ'']</sub>}} generators express in terms of the generators in the four-vector,

:<math> \left [M^{V}_{\mu\nu} \right ]_{\alpha\beta}=i \left (g_{\alpha\mu}g_{\beta\nu}-g_{\alpha \nu}g_{\beta\mu} \right ), </math>

as

:<math>\left[M_{\mu\nu}^{AT}\right]_{[\alpha\beta][\gamma\delta]} =-2 \cdot {\mathbf{1}_{[\alpha\beta]}}^{[\kappa\sigma]} {\left[M^V_{\mu\nu}\right]_\sigma}^\rho{\mathbf 1}_{[\rho\kappa][\gamma\delta]}. </math>

Then, the explicit expression for the Casimir invariant {{math|''C''<sup>(1)</sup>}} in ({{EquationNote|8B}}) takes the form,

:<math>\left [C^{(1)} \right ]_{[\alpha\beta][\gamma\delta]}= -\frac{1}{8} \left (\sigma_{\alpha\beta}\sigma_{\gamma\delta}- \sigma_{\gamma\delta}\sigma_{\alpha\beta}-22 \cdot \mathbf{1}_{[\alpha\beta][\gamma\delta]} \right),</math>

and the Lorentz projector on (3/2,0)⊕(0,3/2) is given by,

:<math> \left[P^{\left(\frac 3 2, 0\right)}\right]_{[\alpha\beta][\gamma\delta]}=\frac 1 8 \left (\sigma_{\alpha\beta}\sigma_{\gamma\delta} + \sigma_{\gamma\delta} \sigma_{\alpha\beta} \right )-\frac 1 {12} \sigma_{\alpha\beta} \sigma_{\gamma\delta}. </math>

In effect, the (3/2,0)⊕(0,3/2) degrees of freedom, denoted by

:<math> \left[w^{\left(\frac{3}{2},0\right)}_\pm \left({\mathbf p},\tfrac{3}{2},\lambda\right)\right]^{[\gamma\delta]} </math>

are found to solve the following second order equation,

:<math>\left ( {\left[P^{\left(\frac 3 2, 0\right)} \right]^{[\alpha\beta]}}_{[\gamma\delta]} p^2-m^2 \cdot {{\mathbf 1}^{[\alpha\beta]}}_{[\gamma\delta]} \right )\left[ w^{\left(\frac 3 2, 0 \right)}_\pm \left({\mathbf p},\tfrac 3 2, \lambda\right)\right]^{[\gamma\delta]}=0.</math>

Expressions for the solutions can be found in.<ref name="Delgado-Banda-Kirchbach 2015" />

==See also==

*[[Higher-dimensional gamma matrices]]
*[[Bargmann–Wigner equations]], alternative equations which describe free particles of any spin
*[[Higher-spin theory|Higher spin theory]]

==References==

{{reflist}}

*{{Cite journal|author=V. V. Dvoeglazov
 |year=1993
 |title=Lagrangian Formulation of the Joos–Weinberg's 2(2''j''+1)–theory and Its Connection with the Skew-Symmetric Tensor Description
 |journal=International Journal of Geometric Methods in Modern Physics
 |volume=13
 |issue=4
 |pages=1650036
 |arxiv=hep-th/9305141
|doi=10.1142/S0219887816500365
 |bibcode=2016IJGMM..1350036D
 |s2cid=55918215
 }}

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