nLab Velo-Zwanziger problem

Contents

Context

Physics

Higher spin geometry

spin geometry, string geometry, fivebrane geometry …

Ingredients

Spin geometry

spin geometry

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)

String geometry

string geometry

Fivebrane geometry

Ninebrane geometry

ninebrane 10-group

Fields and quanta

fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

photon - electromagnetic field (abelian Yang-Mills field)
W, Z, B-boson - electroweak field (Yang-Mills field)
gluon - strong nuclear force (Yang-Mills field)
graviton - field of gravity
infraparticle

scalar bosons

Higgs boson, inflaton (scalar field)

matter field fermions (spinors, Dirac fields)

flavors of fundamental fermions in the standard model of particle physics:
generation of fermions	1st generation	2nd generation	3d generation
quarks ( $q$ )
up-type	up quark ( $u$ )	charm quark ( $c$ )	top quark ( $t$ )
down-type	down quark ( $d$ )	strange quark ( $s$ )	bottom quark ( $b$ )
leptons
charged	electron	muon	tauon
neutral	electron neutrino	muon neutrino	tau neutrino

bound states:
mesons	light mesons: pion ( $u d$ ) ρ-meson ( $u d$ ) ω-meson ( $u d$ ) f1-meson a1-meson	strange-mesons: ϕ-meson ( $s \bar s$ ), kaon, K-meson ( $u s$ , $d s$ ) eta-meson ( $u u + d d + s s$ ) charmed heavy mesons*: D-meson ( $u c$ , $d c$ , $s c$ ) J/ψ-meson ( $c \bar c$ )	bottom heavy mesons: B-meson ( $q b$ ) ϒ-meson ( $b \bar b$ )
baryons	nucleons: proton $(u u d)$ neutron $(u d d)$

(also: antiparticles)

effective particles

Goldstone bosons

hadrons (bound states of the above quarks)

meson
- scalar meson
  - σ-meson
  - pion
  - kaon
  - D-meson
  - B-meson
- vector meson
  - ω-meson
  - ρ-meson
  - f1-meson
  - a1-meson
  - b1-meson
  - h1-meson
  - K*-meson
- tensor meson
- quarkonium
  - charmonium
  - ϒ-meson
exotic meson
- XYZ meson
- tetraquark
baryon
- nucleon
  - proton
  - neutron
  - chemical element
    - carbon
    - nitrogen
- Lambda baryon
- pentaquark

solitons

in grand unified theory

minimally extended supersymmetric standard model

superpartners

bosinos:

gravitino - field of supergravity (Rarita-Schwinger field)
gaugino - super Yang-Mills field
gluino

sfermions:

squark

dark matter candidates

WIMP
axion

Exotica

auxiliary fields

Idea
References

Idea

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

have spin higher than 1
are in an irreducible massive representation of the Lorentz group
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.

References

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 $3/2$ -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

Massimo Porrati, Rakibur Rahman, Augusto Sagnotti, String Theory and The Velo-Zwanziger Problem, Nucl. Phys. B846:250-282, 2011 (arXiv:1011.6411)

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