Kaluza-Klein
Theories |

**In General** > s.a. higher-dimensional
gravity.

* __Idea__: A (4+*k*)-dimensional
general relativity (with matter in extended versions), and the usual Einstein-Hilbert
action; The extra dimensions have to be all spacelike,
otherwise one gets arbitrarily large causality violations and negative
4D energy density; The non-abelian theories often have torsion, if
they include fermions.

* __Dimensional reduction__:
An effective 4D theory can be obtained in a natural way if there are Killing
vector fields along the extra dimensions; One gets gravity coupled
to fields coming from the extra metric components (or, if there were
already matter fields, they end up coupled to gauge fields); If there are
no Killing vector fields, one does a harmonic mode expansion – this of
course is coordinate-dependent; There are ambiguities in the choice of fields
in 4D; Jordan proposed to solve
them by experiment; usually it is done getting some desired
form for the action; > s.a. spacetime models.

* __Motivation__: (1) One gets
from a purely metric theory all the known interactions in 4D; The field strengths
are components of the higher-dimensional Christoffel symbols; To
get strong and electroweak interactions, one needs at least 11D; (2)
Internal charges coupled to gauge fields are treated on the same footing as
energy-momentum; (3) It arises naturally in supergravity theories;
(4) Possible spacetime dependence of the gravitational and fine-structure constants
(this really depends on the 4D interpretation adopted); (5) Lowering of GUT scale from
10^{16} to 10^{3} GeV,
if the internal dimension is large enough (> see branes).

* __Problems__: Chiral fermions
(ok if there are matter fields); Renormalizability
(as usual for gravity); Failure to reproduce quantum numbers of particles; Appearance
of huge mass terms.

**Early Work**

* __History__: Kaluza suggested the use of an extra dimension; Klein that
it be compactified.

@ __"Constant internal radius"__: Nordström PZ(14)phy/07,
OFVSF(14)phy/07,
OFVSF(15)phy/07; Kaluza SPAW(21); Einstein & Grommer SUBH(23); Klein ZP(26);
Einstein SPAW(27), SPAW(27); Klein ZP(28); Einstein & Rosen PR(35);
Einstein & Bergmann
AM(38); Einstein, Bargmann & Bergmann in(41); Einstein & Pauli AM(43).

@ __Variable radius__: Jordan AN(48); Thiry CR(48), JMPA(51); in Lichnerowicz
55; Klein HPAS(56); Jordan ZP(59); in Wheeler 62; in DeWitt 65; Rayski APP(65);
Kerner AIHP(68).

@ __Non-Abelian__: in DeWitt 65; Kerner AIHP(68); Trautman RPMP(70).

@ __History__: in Pais 82; Krym TMP(99)gq/98;
O'Raifeartaigh & Straumann RMP(00)hp/98;
Straumann gq/00-MG9
[Pauli and non-abelian Kaluza-Klein]; van Dongen SHPMP(02)gq/00 [Einstein
1938–1943]; Ravndal a1309 [Oskar Klein's contributions]; > s.a. higher-dimensional gravity.

**And Other Higher-Dimensional Theories** > s.a. branes;
higher-dimensional gravity; teleparallel
theories.

@ __Kaluza-Klein modes in brane world__: Gu ap/03 [Kaluza-Klein
bursts and cosmic rays].

@ __With extra timelike dimensions__: Yndurain PLB(91); Iglesias & Kakushadze PLB(01)ht/00;
Ellis gq/01 [5D
Kaluza-Weyl space].

@ __With no extra dimensions__: Pavšič PLB(05)ht/04,
IJMPA(06)gq/05 [Clifford
space].

@ __Non-commutative__: Madore in(89)-a1510 [supplementary factor in the algebra], PRD(90);
Madore & Mourad ht/96; > s.a. non-commutative
geometry.

@ __Higher-order gravity theories__: Maeda & Dadhich PRD(07)ht/06,
Molina & Dadhich IJMPD(09)-a0804 [Einstein-Gauss-Bonnet gravity]; Dehghani & Assyyaee PLB(09)-a0811 [third-order
Lovelock gravity, and dark energy]; Başkal & Kuyrukcu GRG(13)-a1004 [dimensional reduction].

@ __Other variations__:
Cianfrani & Montani a0904-proc [based on an averaging procedure]; Kalinowski EPJC(14)-a1401, a1410 [non-symmetric gravity];
Stoica a1601 [degenerate metric, zero-length extra dimensions].

**Special Aspects** > see ADM formulation; black
holes; CPT; energy
positivity; models and phenomenology;
spacetime topology and topology change.

@ __Reduction__: Grumiller & Jackiw IJMPD(06)mp [conformally
flat spaces]; Lacquaniti & Montani MPLA(09)-a0906 [5D];
Gauntlett et al JHEP(09)
[*D* = 11 supergravity]; Chopovsky et al a1311 [Ricci-flat internal spaces and absence of KK particles].

**General References** > s.a. particle
physics;
spacetime.

@ __I__: Kaku 94.

@ __Textbooks and reviews__: Hermann 78; Ferrara & Ellis 82; de Sabbata & Schmutzer
83; Chodos CNPP(84); Mecklenburg FdP(84);
Lee 84; Freund PhyD(85);
Piran & Weinberg ed-86; Appelquist et al ed-87; Bailin & Love
RPP(87);
Blau et al in(87); Coquereaux & Jadczyk 88;
Duff ht/94;
Overduin & Wesson
PRP(97)gq/98;
Wesson 99 [5D]; Cianfrani & Montani a0904-proc.

@ __Mathematical aspects__: Wesson MPLA(95)
[Killing vector fields and covariance]; Betounes JGP(04)
[*R*_{ab}].

@ __Related topics__: Keyl CQG(97)
[quantum field theory on Kaluza-Klein background]; Kane PT(98)may
[fiction]; de Assis et al G&C(00)
[loop variables];
Ponce de León G&C(02)
[equation of motion]; Starks et al IJTP(06)
[5D, geometric
formulation]; Reifler & Morris a0707 [equivalence
with Yang-Mills theory]; Lukman et al NJP(11)-a1001 [effective two-dimensionality].

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