Joos–Weinberg equation

This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (December 2016) (Learn how and when to remove this message)

Quantum field theory
Feynman diagram
History
Background Field theory Electromagnetism Weak force Strong force Quantum mechanics Special relativity General relativity Gauge theory Yang–Mills theory
Symmetries Symmetry in quantum mechanics C-symmetry P-symmetry T-symmetry Lorentz symmetry Poincaré symmetry Gauge symmetry Explicit symmetry breaking Spontaneous symmetry breaking Noether charge Topological charge
Tools Anomaly Background field method BRST quantization Correlation function Crossing Effective action Effective field theory Expectation value Feynman diagram Lattice field theory LSZ reduction formula Partition function Path Integral Formulation Propagator Quantization Regularization Renormalization Vacuum state Wick's theorem Wightman axioms
Equations Dirac equation Klein–Gordon equation Proca equations Wheeler–DeWitt equation Bargmann–Wigner equations Schwinger-Dyson equation Renormalization group equation
Standard Model Quantum electrodynamics Electroweak interaction Quantum chromodynamics Higgs mechanism
Incomplete theories String theory Supersymmetry Technicolor Theory of everything Quantum gravity
Scientists Adler Anderson Anselm Bargmann Becchi Belavin Bell Berezin Bethe Bjorken Bleuer Bogoliubov Brodsky Brout Buchholz Cachazo Callan Cardy Coleman Connes Dashen DeWitt Dirac Doplicher Dyson Englert Faddeev Fadin Fayet Fermi Feynman Fierz Fock Frampton Fritzsch Fröhlich Fredenhagen Furry Glashow Gell-Mann Glimm Goldstone Gribov Gross Gupta Guralnik Haag Hagen Han Heisenberg Hepp Higgs 't Hooft Iliopoulos Ivanenko Jackiw Jaffe Jona-Lasinio Jordan Jost Källén Kendall Kinoshita Kim Klebanov Kontsevich Kreimer Kuraev Landau Lee Lee Lehmann Leutwyler Lipatov Łopuszański Low Lüders Maiani Majorana Maldacena Matsubara Migdal Mills Møller Naimark Nambu Neveu Nishijima Oehme Oppenheimer Osborn Osterwalder Parisi Pauli Peccei Peskin Plefka Polchinski Polyakov Pomeranchuk Popov Proca Quinn Rouet Rubakov Ruelle Sakurai Salam Schrader Schwarz Schwinger Segal Seiberg Semenoff Shifman Shirkov Skyrme Sommerfield Stora Stueckelberg Sudarshan Symanzik Takahashi Thirring Tomonaga Tyutin Vainshtein Veltman Veneziano Virasoro Ward Weinberg Weisskopf Wentzel Wess Wetterich Weyl Wick Wightman Wigner Wilczek Wilson Witten Yang Yukawa Zamolodchikov Zamolodchikov Zee Zimmermann Zinn-Justin Zuber Zumino
v t e

In relativistic quantum mechanics and quantum field theory, the Joos–Weinberg equation is a relativistic wave equation applicable to free particles of arbitrary spin j, an integer for bosons (j = 1, 2, 3 ...) or half-integer for fermions (j = 1⁄2, 3⁄2, 5⁄2 ...). The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields. The spin quantum number is usually denoted by s in quantum mechanics, however in this context j is more typical in the literature (see references).

It is named after Hans H. Joos and Steven Weinberg, found in the early 1960s.^[1][2][3]

Introducing a 2(2j + 1) × 2(2j + 1) matrix;^[2]

${\displaystyle \gamma ^{\mu _{1}\mu _{2}\cdots \mu _{2j}}}$

symmetric in any two tensor indices, which generalizes the gamma matrices in the Dirac equation,^[3][4] the equation is^[5]

${\displaystyle [(i\hbar )^{2j}\gamma ^{\mu _{1}\mu _{2}\cdots \mu _{2j}}\partial _{\mu _{1}}\partial _{\mu _{2}}\cdots \partial _{\mu _{2j}}+(mc)^{2j}]\Psi =0}$

or

For the JW equations the representation of the Lorentz group is^[6]

${\displaystyle D^{\mathrm {JW} }=D^{(j,0)}\oplus D^{(0,j)}.}$

This representation has definite spin j. It turns out that a spin 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 are good.

The representations D^{(j, 0)} and D^{(0, j)} can each separately represent particles of spin j. A state or quantum field in such a representation would satisfy no field equation except the Klein–Gordon equation.

The six-component spin-1 representation space,

${\displaystyle D^{\mathrm {JW} }=D^{(1,0)}\oplus D^{(0,1)}}$

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

${\displaystyle B_{[\alpha \beta ]}\sim D^{(1,0)}\oplus D^{(0,1)}.}$

The j-fold Kronecker product T_{[α1β1]...[αjβj]} of B_[αβ]

${\displaystyle T_{[\alpha _{1}\beta _{1}]\cdots [\alpha _{j}\beta _{j}]}=B_{[\alpha _{1}\beta _{1}]}\otimes \cdots \otimes B_{[\alpha _{j}\beta _{j}]}=\bigotimes _{i=1}^{j}B_{[\alpha _{i}\beta _{i}]},}$

8A

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

${\displaystyle \bigotimes _{i=1}^{j}\left(D_{i}^{(1,0)}\oplus D_{i}^{(0,1)}\right)\to D^{(j,0)}\oplus D^{(0,j)}\oplus D^{(j,j)}\oplus \cdots \oplus D^{(j_{k},j_{l})}\oplus D^{(j_{l},j_{k})}\oplus \cdots \oplus D^{(0,0)},}$

and necessarily contains a ${\displaystyle D^{(j,0)}\oplus D^{(0,j)}}$ sector. This sector can instantly be identified by means of a momentum independent projector operator P^(j,0), designed on the basis of C⁽¹⁾, one of the Casimir elements (invariants)^[7] of the Lie algebra of the Lorentz group, which are defined as,

${\displaystyle {\begin{cases}\left[C^{(1)}\right]_{AB}={\frac {1}{4}}{\left[M^{\mu \nu }\right]_{A}}^{C}\left[M_{\mu \nu }\right]_{CB}\\[6pt]\left[C^{(2)}\right]_{AB}={\frac {1}{4}}\varepsilon _{\mu \nu \lambda \eta }{\left[M^{\mu \nu }\right]_{A}}^{C}\left[M^{\lambda \eta }\right]_{CB}\end{cases}}\qquad A,B,C=1,\ldots ,(2j_{1}+1)(2j_{2}+1)}$

8B

where M^μν are constant (2j₁+1)(2j₂+1) × (2j₁+1)(2j₂+1) matrices defining the elements of the Lorentz algebra within the ${\displaystyle D^{(j_{1},j_{2})}\oplus D^{(j_{2},j_{1})}}$ representations. The Capital Latin letter labels indicate^[8] the finite dimensionality of the representation spaces under consideration which describe the internal angular momentum (spin) degrees of freedom.

The representation spaces ${\displaystyle D^{(j_{1},j_{2})}\oplus D^{(j_{2},j_{1})}}$ are eigenvectors to C⁽¹⁾ in (8B) according to,

${\displaystyle C^{(1)}\left[D^{(j_{1},j_{2})}\oplus D^{(j_{2},j_{1})}\right]=\left(j_{1}(j_{1}+1)+j_{2}(j_{2}+1)\right)\left[D^{(j_{1},j_{2})}\oplus D^{(j_{2},j_{1})}\right],}$

Here we define:

${\displaystyle \lambda _{(j_{1},j_{2})}^{(1)}=j_{1}(j_{1}+1)+j_{2}(j_{2}+1),}$

to be the C⁽¹⁾ eigenvalue of the ${\displaystyle D^{(j_{1},j_{2})}\oplus D^{(j_{2},j_{1})}}$ sector. Using this notation we define the projector operator, P^(j,0) in terms of C⁽¹⁾:^[8]

${\displaystyle {\left[P^{(j,0)}\right]_{\left[\alpha _{1}\beta _{1}\right]\cdots \left[\alpha _{j}\beta _{j}\right]}}^{\left[\rho _{1}\sigma _{1}\right]\cdots \left[\rho _{j}\sigma _{j}\right]}={\left[\prod _{k,l}\left({\frac {C^{(1)}-\lambda _{(j_{k},j_{l})}^{(1)}}{\lambda _{(j,\,0)}^{(1)}-\lambda _{(j_{k},j_{l})}^{(1)}}}\right)\right]_{\left[\alpha _{1}\beta _{1}\right]\cdots \left[\alpha _{j}\beta _{j}\right]}}^{\left[\rho _{1}\sigma _{1}\right]\cdots \left[\rho _{j}\sigma _{j}\right]}.}$

8C

Such projectors can be employed to search through T_{[α1β1]...[αjβj]} for ${\displaystyle D^{(j,0)}\oplus D^{(0,j)},}$ and exclude all the rest. Relativistic second order wave equations for any j are then straightforwardly obtained in first identifying the ${\displaystyle D^{(j,0)}\oplus D^{(0,j)}}$ sector in T_{[α1β1]...[αjβj]} in (8A) by means of the Lorentz projector in (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, ${\displaystyle s=j+{\tfrac {1}{2}}}$ in which case the Kronecker product of T_{[α1β1]...[αjβj]} with the Dirac spinor,

${\displaystyle D^{\left({\frac {1}{2}},0\right)}\oplus D^{\left(0,{\frac {1}{2}}\right)}}$

has to be considered. The choice of the totally antisymmetric Lorentz tensor of second rank, B_[αiβi], in the above equation (8A) is only optional. It is possible to start with multiple Kronecker products of totally symmetric second rank Lorentz tensors, A_αiβi. The latter option should be of interest in theories where high-spin ${\displaystyle D^{(j,0)}\oplus D^{(0,j)}}$ Joos–Weinberg fields preferably couple to symmetric tensors, such as the metric tensor in gravity.

Source:^[8]

The

${\displaystyle \left({\tfrac {3}{2}},0\right)\oplus \left(0,{\tfrac {3}{2}}\right)}$

transforming in the Lorenz tensor spinor of second rank,

${\displaystyle \psi _{[\mu \nu ]}=[(1,0)\oplus (0,1)]\otimes \left[\left({\tfrac {1}{2}},0\right)\oplus \left(0,{\tfrac {1}{2}}\right)\right].}$

The Lorentz group generators within this representation space are denoted by ${\displaystyle \left[M_{\mu \nu }^{ATS}\right]_{[\alpha \beta ][\gamma \delta ]},}$ and given by:

${\displaystyle \left[M_{\mu \nu }^{ATS}\right]_{[\alpha \beta ][\gamma \delta ]}=\left[M_{\mu \nu }^{AT}\right]_{[\alpha \beta ][\gamma \delta ]}{\mathbf {1} }^{S}+{\mathbf {1} }_{[\alpha \beta ][\gamma \delta ]}\,\,\left[M_{\mu \nu }^{S}\right],}$

${\displaystyle \mathbf {1} _{[\alpha \beta ][\gamma \delta ]}={\tfrac {1}{2}}\left(g_{\alpha \gamma }g_{\beta \delta }-g_{\alpha \delta }g_{\beta \gamma }\right),}$

${\displaystyle M_{\mu \nu }^{S}={\tfrac {1}{2}}\sigma _{\mu \nu }={\frac {i}{4}}[\gamma _{\mu },\gamma _{\nu }],}$

where 1_[αβ][γδ] stands for the identity in this space, 1^S and M^S_μν are the respective unit operator and the Lorentz algebra elements within the Dirac space, while γ_μ are the standard gamma matrices. The [M^AT_μν]_[αβ][γδ] generators express in terms of the generators in the four-vector,

${\displaystyle \left[M_{\mu \nu }^{V}\right]_{\alpha \beta }=i\left(g_{\alpha \mu }g_{\beta \nu }-g_{\alpha \nu }g_{\beta \mu }\right),}$

as

${\displaystyle \left[M_{\mu \nu }^{AT}\right]_{[\alpha \beta ][\gamma \delta ]}=-2\cdot {\mathbf {1} _{[\alpha \beta ]}}^{[\kappa \sigma ]}{\left[M_{\mu \nu }^{V}\right]_{\sigma }}^{\rho }{\mathbf {1} }_{[\rho \kappa ][\gamma \delta ]}.}$

Then, the explicit expression for the Casimir invariant C⁽¹⁾ in (8B) takes the form,

${\displaystyle \left[C^{(1)}\right]_{[\alpha \beta ][\gamma \delta ]}=-{\frac {1}{8}}\left(\sigma _{\alpha \beta }\sigma _{\gamma \delta }-\sigma _{\gamma \delta }\sigma _{\alpha \beta }-22\cdot \mathbf {1} _{[\alpha \beta ][\gamma \delta ]}\right),}$

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

${\displaystyle \left[P^{\left({\frac {3}{2}},0\right)}\right]_{[\alpha \beta ][\gamma \delta ]}={\frac {1}{8}}\left(\sigma _{\alpha \beta }\sigma _{\gamma \delta }+\sigma _{\gamma \delta }\sigma _{\alpha \beta }\right)-{\frac {1}{12}}\sigma _{\alpha \beta }\sigma _{\gamma \delta }.}$

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

${\displaystyle \left[w_{\pm }^{\left({\frac {3}{2}},0\right)}\left({\mathbf {p} },{\tfrac {3}{2}},\lambda \right)\right]^{[\gamma \delta ]}}$

are found to solve the following second order equation,

${\displaystyle \left({\left[P^{\left({\frac {3}{2}},0\right)}\right]^{[\alpha \beta ]}}_{[\gamma \delta ]}p^{2}-m^{2}\cdot {{\mathbf {1} }^{[\alpha \beta ]}}_{[\gamma \delta ]}\right)\left[w_{\pm }^{\left({\frac {3}{2}},0\right)}\left({\mathbf {p} },{\tfrac {3}{2}},\lambda \right)\right]^{[\gamma \delta ]}=0.}$

Expressions for the solutions can be found in.^[8]

• Higher-dimensional gamma matrices
• Bargmann–Wigner equations, alternative equations which describe free particles of any spin
• Higher spin theory

• V. V. Dvoeglazov (1993). "Lagrangian Formulation of the Joos–Weinberg's 2(2j+1)–theory and Its Connection with the Skew-Symmetric Tensor Description". International Journal of Geometric Methods in Modern Physics. 13 (4): 1650036. arXiv:hep-th/9305141. Bibcode:2016IJGMM..1350036D. doi:10.1142/S0219887816500365. S2CID 55918215.