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)-a1803;
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,
FdP(16)-a1410 [non-symmetric gravity];
Stoica IJMPD(16)-a1601 [degenerate metric, zero-length extra dimensions];
Collion & Vaugon a1709 [non-Ricci-flat geometries, including matter].

**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 AHEP(13)-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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