nLab Velo-Zwanziger problem




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Higher spin geometry

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fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

scalar bosons

matter field fermions (spinors, Dirac fields)

flavors of fundamental fermions in the
standard model of particle physics:
generation of fermions1st generation2nd generation3d generation
quarks (qq)
up-typeup quark (uu)charm quark (cc)top quark (tt)
down-typedown quark (dd)strange quark (ss)bottom quark (bb)
neutralelectron neutrinomuon neutrinotau neutrino
bound states:
mesonslight mesons:
pion (udu d)
ρ-meson (udu d)
ω-meson (udu d)
ϕ-meson (ss¯s \bar s),
kaon, K*-meson (usu s, dsd s)
eta-meson (uu+dd+ssu u + d d + s s)

charmed heavy mesons:
D-meson (uc u c, dcd c, scs c)
J/ψ-meson (cc¯c \bar c)
bottom heavy mesons:
B-meson (qbq b)
ϒ-meson (bb¯b \bar b)
proton (uud)(u u d)
neutron (udd)(u d d)

(also: antiparticles)

effective particles

hadrons (bound states of the above quarks)


in grand unified theory

minimally extended supersymmetric standard model




dark matter candidates


auxiliary fields



The Velo-Zwanziger problem is is a phenomenon in theoretical physics: the “evident” action functionals for fields which

  1. have spin higher than 1

  2. are in an irreducible massive representation of the Lorentz group

  3. have minimal coupling to electromagnetism

tend to have equations of motion which are not hyperbolic and hence may violate the usual causality relation of general relativity.


The original references are

  • G. Velo, D. Zwanziger, Phys. Rev. 186 (1969) 1337,

  • G. Velo, D. Zwanziger, Phys. Rev. 188 (1969) 2218;

  • G. Velo, Nucl. Phys. B 43 (1972) 389

  • M. Hortacsu, Phys. Rev. D 9, 928 (1974).

Adjustments of the action functional for spin massive 3/23/2-particles (Rarita-Schwinger field) by non-minimal coupling that evades the Velo-Zwanziger problem is discussed in

  • Massimo Porrati, Rakibur Rahman, Causal Propagation of a Charged Spin 3/2 Field in an External Electromagnetic Background, Phys. Rev. D80:025009, 2009 (arXiv:0906.1432)

Discussion for the higher string excitations that appear in string theory is for instance in

Last revised on March 26, 2020 at 15:46:36. See the history of this page for a list of all contributions to it.