(2+3) dimensional geometrical dual of the complex Klein-Gordon equation

0.15 MB·English

by  Benjamin Koch

(2+3) dimensional geometrical dual of the complex Klein-Gordon equation
Benjamin Koch
Institut fu¨r Theoretische Physik,
Johann Wolfgang Goethe - Universit¨at,
D–60438 Frankfurt am Main, Germany
(Dated: January 22, 2009)
InthispaperitisshownthatanequivalenttothecomplexKlein-Gordonequationcanbeobtained
fromthe(2+3)dimensionalEinsteinequationscoupledtoaconservedenergymomentumtensor. In
anexplicittoymodelwegivematchingconditionsforwhatcorrespondstothephase,theamplitude,
and themass of the complex wave function.
9
0 PACSnumbers: 04.50.Cd,04.62.+v,03.65.Ta
0
2
The purpose of this paper is to show that one can I. THE COMPLEX KLEIN-GORDON
n
find an analog to the complex Klein-Gordonequation in EQUATION IN CURVED SPACETIME
a
J curvedspace-timebyconsideringmatterwhichismoving
2 in an higher dimensional space-time. This is motivated The relativistic Klein-Gordon equation in curved
2 on the one hand by the fact that there are alternative spacetime [20] is
approaches to quantum mechanics. Early such attempts

] have been discussed within the so called Bohmian me- 1 m2
h ∂ gˆgˆµν∂ Φ(x,t) = Φ(x,t) , (1)
p chanics [1, 2]. Similarities between a higher dimensional √ gˆ µ − ν −~2
- wave equation and the non-relativistic quantum theory − (cid:16)p (cid:17)
nt have been pointed out in [3]. It was also suggested that where gˆ stands for the determinant of the metric gˆµν.
a quantum field theory might emerge from a chaotic clas- This equation for the complex wave function Φ can be
u sical theory with friction [4, 5]. For an overview see [6]. expressed in terms of the real function SQ(x,t) and the
q On the other hand a further motivation comes from the realfunctionρ(x,t)bydefiningΦ(x,t)=√ρexp(iSQ/~).
[ existingideasonanadditionaltimedimension. Thecon- This gives two coupled differential equations

2 sequencesofmorethanonetimedimensionswerestudied
√ρ
5v isnion[7,w8a,s9u,se1d0,t1o1i]n.teFruprrtehterg,eoadneasidcdsitinionfoaulrtidmimeednimsioenns- (cid:31)√ρ = ~2 (∂µSQ)(∂µSQ)−m2 , (2)
3 as null paths in the higher dimensions [12]. There is 0 = ∂ (cid:0) gˆρgˆµν(∂νS ) .(cid:1) (3)
µ Q
6 also an intersection of both motivations. Some papers −
4 (cid:16)p (cid:17)
. relate a quantum field theorie on the horizon of a black In the first equation we used the abbreviation (cid:31)√ρ
01 hmoelnestioonaalcosrpraecseptoimndein[1g3c,la1s4s]i.caSlitmheiloarryeiffnetchtse ahrigehfeorudnid- ∂µ √−gˆgˆµν∂ν√ρ /√−gˆ for the covariant d’Alember≡t
operator.
8
in the context of supergravity and string theory involv- (cid:0) (cid:1)
0 Theredefinitionofthecomplexwavefunctioninterms
ing a holographic principle and extra time dimensions
: oftherealfunctionsS andρdoesnotchangethemean-
v [15, 16, 17]. In a purely conceptual discussion an extra Q
ing or the interpretation of Eq. (1). The function S is
Xi time dimension is suggested as an alternative to quanti- usually understood as the phase of the quantum wQave
zation [18]. In a stochastic approach [19], the classical
r function.
a Langevin equation for an auxiliary time coordinate t˜is
used in order to obtain Euclidian quantum field theory
in the limit t˜ . All those ideas can be seen as moti-
II. GENERAL RELATIVITY WITH ONE
→∞
vationforthis paper,wherethe structureofthe complex ADDITIONAL TIME DIMENSION
Klein-Gordonequationisfoundfromclassicaldifferential
geometry with one additional time dimension.
In this section we will consider the geometric struc-
ture of Einsteins equations with one additional coordi-
nate t¯. A simple ansatz for the metric in 2+3 dimen-
The motivation of our approach is the idea that the sions will be made. We then show that it is possible
movementof a classicalparticle with respect to an addi- to choose the energy-momentum tensor T in such a
AB
tional but unobservable time dimension t¯could produce way that the classical equations of motion in the higher
effects which are usually described by quantum mechan- dimensional theory (after the additional coordinate t¯is
ics. As first application of this idea we will derive the integrated out) correspond to the relativistic quantum
structure of the relativistic complex Klein-Gordonequa- equation for a spinless particle. We use the coordinate
tion from a classical theory with one additional dimen- notation x = (t¯,x ) = (t¯,t,~x), where capital Latin in-
A µ
sion. dices A run from 0 to 4 and Greek indices run from 1 to
2
4. As starting point we take the Einstein equations in Hamilton-Jakobidefinitionoftheenergymomentumten-
(2+3) dimensions sor of a free particle in curved spacetime is
R 1g R=G T 1g Λ , (4) TAB = (∂ASH)(∂BSH), with (10)
AB − 2 AB D AB− 2 AB TA = (∂0S )(∂ S )+(∂µS )(∂ S ) ,
A H 0 H H µ H
where RAB is the higher dimensional Ricci tensor, R is where SH is th(cid:0)e density of Hamilton’s principal(cid:1)function
it’s contraction, and GD is the higher dimensional grav- [24, 25]. A priori, the t¯dependence of SH is not known
itational coupling constant. The term Λ is directly con- Therefore,wemakeaseparationansatzwhereS canbe
H
nected to the cosmological constant Λ4. Our ansatz for decomposedintoaclassicalpartS˜whichonlydependson
the higher dimensional metric is the observable coordinates t,~x and a part B(t¯)√ρ which
depends on all coordinates t¯,t,~x:
g dxAdxB =α¯2(t¯)ρ(t,~x)dt¯2+g (t¯,x )dxµdxν. (5)
AB µν α
SH(t¯,t,~x)=ǫ1√ρB(t¯)+S˜(t,~x) , (11)
Later, the function α¯ will be separated into a constant
plus a small t¯dependent part with ∂0B(t¯) β(t¯). Instead of making the heroic but
≡
vain attempt to solve all those coupled equations simul-
α¯(t¯)=α +ǫ ω¯(t¯) . (6) taneously we restrain ourselves to a simpler scenario. In
0 0
this scenario we take the trace of the higher dimensional
Note that this metric is similar to the Kaluza-Klein Einstein equations and integrate over additional dimen-
metric for a vanishing electromagnetic field A = 0 sion t¯
µ
[3, 21, 22, 23]. However, in contrast to the Kaluza-Klein
1
approach the energy momentum Tensor T and Λ are dt¯R= − dt¯(2G TA 5Λ) . (12)
AB 3 D A−
not assumed to be identically zero and all the functions Z Z
have real values only. An other difference is that the Using the definitions (5, 10), and after a multiplication
functions in the metric (α¯2ρ and gµν) and the energy with √ρ/2 the scalar equation (12) reads
momentum tensor T are allowed to have a explicit t¯ −
AB
dependence. Withthemetric(5)thetwenty-fivecoupled G
(cid:31)√ρ dt¯=√ρ D dt¯(∂µS )(∂ S ) (13)
differential equations (4) read 3 H µ H
Z (cid:20) Z
(cid:18)−α¯2√ρ(cid:31)√ρ− (gλβg2˙λβ),0 + α¯˙gλ2βα¯g˙λβ − 65ΛZ dt¯− R dt¯2Rˆαα − GD3ǫ21 Z dt¯αβ¯22!#
gµβgλσg˙ g˙ 1 gαβg˙ dt¯ PαβP (Pα)2
λβ µσ dt¯ ∂ αβ + αβ α .
− 4 − α¯ 0 √ρα¯ 2√ρ α¯2 − 9α¯2
(cid:19) Z R (cid:18) (cid:19)
α¯2ρ α¯2ρ
= G T TA + Λ , (7) The sourcetermonthe righthandside containsthe ten-
(cid:20) D(cid:18) 00− 3 A(cid:19) 3 (cid:21) sor Pµν =(g˙µν gµνgαβg˙αβ)/2. Apartfrom the last two
−
termsthisequationhasalwaysthesameρdependenceas
Eq.(2), whichjustifies the ansatz(11). Thoselastterms
gλβ(g˙ ρ ρ g˙ )+ ∂λ(gλµg˙µδ) havetheformofasource,whichindicatesthatat¯depen-
λβ ,δ ,β δλ
4ρ − 2 dence of g can lead to particle production. This result
(cid:18) µν
∂ (gλµg˙ ) gλσgµβg˙ g g˙µβg is not too surprising because it is known that already a
δ λµ σδ µβ,λ µβ,δ
+ +
normal time dependence of the metric can lead to parti-
− 2 4 4
(cid:19) cleproduction[26]. However,weareprimarilyinterested
=G T , (8)
D 0δ in problems with a seemingly constant (with respect to
t¯) four dimensional spacetime. Therefore, we decompose
Rˆδβ − (√√ρ)ρ;δ;β + 2α¯12ρ α¯˙gα˙¯δβ −g¨δβ pthaertfogˆuµrνdainmdeanstiroancaellessusbt¯m-veatrriiacbglµeνpianrttoγaµνt¯(it¯n,xdαep)endent
(cid:20) (cid:18)
gµνg˙ g˙ g (t¯,xα)=gˆ +ǫ2γ (t¯,xα) . (14)
+gλµg˙ g˙ µν δβ µν µν 2 µν
δλ βµ
− 2
(cid:19)(cid:21) Nowweassumethatperturbationsduetot¯aresmalland
g g
= G T δβTA + δβΛ . (9) expand Eq. (13) to lowest order in ǫ 1 (i = 0,1,2).
D δβ − 3 A 3 i ≪ |
With Eq. (11) one finds up to order (ǫ )
h (cid:16) (cid:17) i O i
XH˙e,rteh,ewceodveanroiatnetddaedrievraivtiavteivienwthitehfroeusrpedcitmteont¯siaosna∂l0Xsub≡- (cid:31)√ρ= √ρ ~2GD(∂µS˜)(∂ S˜) (15)
snpoarcmeaalsfo∇urµXdim≡enXsi;oµn,aalnfdor(cid:31)m√oρf t=he(√Riρc)c;;iµµt.enRˆsδoβr wishtihche ~2 (cid:20) 3 ~2µ
5Λ 3Rˆα .
only contains derivatives of gµν with respect to xµ. The − 6 − α
(cid:16) (cid:17)(cid:21)
3
The second to last term of Eq. (13) vanished, since it is the definitions (11, 14)
proportional to the trace of γ , and the last term van-
µν
ished because it is of higher order in ǫi. A comparison 0 = ǫ1 ∂ ρ∂µ(S˜+ǫ ρβ) β (21)
of Eq. (15) with the complex Klein-Gordon equation (2) √ρ µ 1
(cid:20)
shows that both equations are the same if two identi- (cid:16) (cid:17)
ǫ2
fications are made. The first identification relates the 2gˆµαγ˙ (∂ S )(∂νS )
να µ H H
Hamilton principal function S˜ to the quantum phase S −2
Q ǫ2ǫ2 β2 β2
by + 2 1gˆαβγ˙ +ǫ2∂
2 αβα2 0 0 α2
0 (cid:18) 0(cid:19)
ǫ
G + 1gˆαβγ √ρ(∂νS )β+ (ǫ3) .
S ~ DS˜ . (16) 2 αβ,ν H O i
Q
≡ r 3 i
Onlythe firstandthe lasttermsurviveinafirstorderǫ
i
Thesecondidentificationrelatesthe massmtothefunc- expansion and the continuity equation simplifies to
tions Rˆµ and Λ by
µ
0 = ∂ gˆρ(gˆµν∂ S˜) + (ǫ ) . (22)
µ ν i
− O
m2 ~2 5Λ 3Rˆα . (17) (cid:16)p (cid:17)
≡ 6 − α With the identification (16) this is exactly the second
(cid:16) (cid:17) Klein-Gordon equation (3). Up to now we have used
theclassicalbutt¯dependenthigherdimensionalEinstein
One can already see that such an identification makes
equations (4) and the covariant continuity equation (20)
only senseif the scalarcurvaturein the four dimensional
subspaceRˆµ does notdepend onx (This is for instance and expanded them for small perturbations around the
µ µ t¯-independent solution. From this we have found two
the case in de-Sitter spacetimes. One also has to check
matching rules (16, 19) such that the complex Klein-
that a solutionof Eq.(15) allowsto solveall the twenty-
Gordon equations in a curved spacetime (2, 3) can be
five equations (7-9) in the ǫ expansion. The first check
i
exactly identified with the classical equations (15, 22).
is to compare Eq. (15) with Eq. (7). It turns out that
both equations are only consistent if
III. CRITICAL POINTS
Rˆµ =Λ . (18)
µ
We will now try to point out some checks, criticism,
Consequently, the mass identification (17) simplifies to
and limitations of the idea and the derivation presented
here.
~2 Is this quantum mechanics?
m2 Λ . (19)
≡ 3 No, finding a dual to the complex Klein-Gordon equa-
tion does not mean that one has automatically derived
This shows that a mass term can only be defined if the quantum mechanics. To make such a statement one
four dimensionalbackgroundis curveddue to a nonzero would have to construct a self-consistent philosophical
cosmologicalconstantΛ. Asimilar resultwas previously and mathematical theory like the pilot wave theory of
obtainedformassivegravitonsinfourdimensionalspace- Bohm [1, 2]. Such a construction goes beyond the scope
time [27]. In the case of an exatly flat Minkowski back- of this paper.
ground the mass term of this toy model is zero. Apart Why is the second time not visible?
fromthevanishingmass,anexactlyflatMinkowskiback- The easiest way to explain this, is with a compactifica-
ground is a solution of the toy model. The remaining tion of the hidden time such that t¯ = t¯+T¯. As long
twenty-four checks will be postponed to the subtopic “Is the“radius”isjustshortenough,thiswouldonlyleadto
this theory consistent with all Einstein- and conserva- violations of Lorentz invariance on the small scale T¯. A
tion equations?” in the discussion. One also sees that similar construction for an extra but compact time vari-
for matching several different masses (m ) one has to able (imaginary or not) can be found in [19, 28]. Thus
j
impose severalconstants (Λ ). thereareseveralviabletheoreticalmodelsthatallowand
j
have extra time dimensions. Within those models there
Since Eq. (2) was found, the next step is to derive the
existhoweverexperimentalbounds onthe sizeofthe ad-
second Klein-Gordon equation (3). For this we consider
ditional time dimension T¯ [7, 11].
thecovariantconservationlawfortheenergy-momentum
What is the interpretation of the higherdimen-
tensor T
AB
sional metric?
The strongestassumption(5) for the higher dimensional
0= TB =∂ TB ΓD TB+ΓB TD . (20)
∇B A B A − AB D BD A metric gAB is that g00 ρ. This can be justified by a
∼
classical probability argument: The idea of this new ap-
(cid:0) (cid:1)
Nowwetakethe0componentofthisequationandapply proach becomes most clear for a system that does not
4
changeinthe observabletime directiont. Ina determin- andafterusingEqs.(18,26)theremainingnineindepen-
isticsystem,theposition~xandvelocity~v =d~x/dτ ofthe dent equations in (9) read
particle are known as soon as the initial position ~x , the
0
initialvelocity~v0,andthepropagationtimeτ areknown. dt¯(√ρ);δ;β = dt¯ G (∂ S˜)(∂ S˜) p p (27)
D δ β δ β
Not knowing the initial conditions one has to deal with √ρ − −
a probability density f(~x0,~v0,τ). Without loss of gen- Zgδβ(G (∂µS˜)(∂ S˜Z) Λh)+ g(cid:16)δβ (G pµp Λ)] (cid:17).
erality we choose some fixed starting point which leaves − 3 D µ − 4 D µ−
a probability density g(~v ,τ). In a deterministic system
0 It is hard to prove in general that those equations and
the initial velocity can be calculated as a function of the
the Klein-Gordon equations can always be consistently
actual velocity and the time ~v = ~v (v (τ),τ). There-
0 0 ~x solved. However, we can show that this is the case for
fore, the whole system can also be described by a statis-
the most simple solution of the Klein-Gordon equation.
tical probability density h(~v (τ)). The probability ρ(~x)
x ThemostsimplesolutionofthefirstKlein-Gordonequa-
of finding the particle at a point~x will than be obtained tion (15) is: ρ=const, S˜ p , and pµp =Λ/G . For
,µ µ µ D
by an integration over the proper time variable ≡
thissolutionthenine independentequations(27)areob-
viously fulfilled since both sides of the equation vanish.
ρ(~x)= dτh(~v~x(τ),τ) . (23) A comparison with Eq. (18) further shows that n1 = 0.
Z Thus, we have shown that, at least in the discussed spe-
The key observation, which relates a probability to the cial case, all twenty-five Einstein equations can be ful-
extra time variable, is: The faster a particle moves at a filledconsistentlyinourtheory. Aslastconsistencycheck
certain point ~x the smaller is the probability of finding wehavetostudythefourremainingcontinuityequations
theparticleatthispointh(~v (τ),τ) 1/~v (τ)andhence from Eq. (20)
~x ~x
∼
1 1 TB =∂ gδνS˜ S˜ Γˆδ gγνS˜ S˜ (28)
ρ(x)∼ dτv (τ) = dτ dx . (24) ∇B µ δ ,µ ,ν − µν ,γ ,δ
Z x Z dτ +(cid:0)Γˆν (cid:1) gγδS˜(cid:16)S˜ + ρ,(cid:17)ν gγνS˜(cid:16)S˜ + ((cid:17)ǫ ) .
This probability is directly related to the 00 component δν ,µ ,γ 2ρ ,γ ,µ O i
(cid:16) (cid:17) (cid:16) (cid:17)
ofthe metricgAB. We alreadyassumedthatthe particle FortheabovesolutionoftheKlein-Gordonequationthis
moves mostly in parallel to the additional time coordi- isthe conservationlawforthefourmomentump S˜ .
µ ,µ
nate (dt¯/dτ dx/dτ > dt/dτ). In this limit the differ- Therefore, we have checked the consistency of a∼n ex-
≫
ential of the proper time τ can be approximated by the emplary solution of our theory with all its thirty equa-
differential of the additional time variable dτ dt¯√g00. tions (7-9,20). Because of the condition (26) the Klein-
≈
Under the assumption that some part of From the inte- Gordon equation on a flat Minkowski background can
gral in Eq. (24) one finds only result from our theory in an asymptotic region of
spacetime. Since energy always curves spacetime, this
1
ρ(x) dt¯ g . (25) what one would expect also from the standard interpre-
∼ dx 00
Z dt¯ tation of quantum mechanics and general relativity.
What is the classical limit?
This shows that a statistical interpretation of an unob-
The construction with its higher dimensional energy-
servable additional time leads naturally to the metric
momentum tensor T and the higher dimensional met-
ansatz (5). AB
ric g should (in the limit of large distancescales and
Is this theory consistent with all Einstein- and AB
large timescales) be consistent with the usual four di-
conservation equations?
mensional classical general relativity. In this limit the
SinceweonlyworkedwithasubsetoftheEinsteinequa-
ρ dependence and the terms proportional to ǫ can be
tions we haveto check whether this canlead to inconsis- i
omitted. However, a different normalization in the met-
tencies with the full set. From comparing Eq. (15) with
ricexpansion(14)isusefulsuchthatgˆ 3g0 /2. Now
Eq. (7) we already found that Rˆµ = Λˆ. The same con- µν → µν
µ Eq. (9) simplifies to
dition is found when comparing Eq. (15) with the trace
of Eq. (9). Because of this condition one can always de- g0 g0 Λ
compose the four dimensional Ricci tensor as Rˆµν =GD S˜,µS˜,ν µνS˜,αS˜,α + µν . (29)
− 2 ! 2
Rˆ =n gˆ +G p p , (26)
µν 1 µν D µ ν Foraninfinitelysmallnormalizingvolumeattheposition
where the four vector p and the number n do not de- oftheparticletrajectorywedenotethedensityoftheen-
pendonxµ. Pleasenoteµthatwedonotknow1thesolution ergy momentum tensor as S˜,µS˜,ν → p˜µp˜µδ3(~x−~xn)/E.
to Eq. (26) but we assume that it exists. In the ǫ ex- Thus,we foundthe fourdimensionalEinsteinfieldequa-
i
pansion the eight equations (8) do not create any new tions for a free particle [32]
conditions, since the LHS is of order ǫ2 and the RHS is
composed of a vanishing t¯boundary tierm plus a term Rˆ = GD p˜ p˜ + gµ0νp˜2 δ3(~x ~x )+gµ0νΛ . (30)
µν µ ν n
which is also of higher order in ǫi. In the same scheme E 2 ! − 2
5
Please note that in contrast to [33, 34] the gravitating ject to future studies.
matterisfreetopropagateintoallspaceandtimedimen-
sions and therefore the higher dimensional gravitational
coupling G is equal to the Newton constant G . This
D N
IV. SUMMARY
shows that classical general relativity can be obtained
fromtheansatz(4)afterintegratingoutt¯andtakingthe
limit of large time and distance scales. In this paper we study the higher dimensional Ein-
Isn’t the cosmological constant very small? steinequations(4)andthehigherdimensionalcontinuity
ThehigherdimensionalcosmologicaltermΛisontheone equation (20) with one additional time dimension t¯. For
hand related to the mass in the quantum equation and the higher dimensional metric we make the ansatz (5),
on the other hand to the four dimensional cosmological and for the higher dimensional principal function we
constantΛ . TheobservedcosmologicalconstantΛ is a make the ansatz (11). Then we make an expansion in
4 4
very small number [35] andtherefore inconsistentwith a the parameter ǫ to ensure a small t¯dependence. After
i
typically much bigger particle mass in Eq. (19). A pos- defining the quantum phase (16) and the quantum mass
sible way outofthe mass-cosmologicalconstantproblem (19) we find the structure of the Klein-Gordon equation
mightbe foundbyconsideringhigherordersintheǫ ex- in curved spacetime
i
pansion or by taking a possible t¯dependence of Λ into
account. Evenifthe issueremainsinourtoy model,this √ρ
(15): (cid:31)√ρ = (∂µS )(∂ S ) m2 + (ǫ ) ,
might be a good sign because also most of the conven- ~2 Q µ Q − O i
tional quantum theories have a problem in getting the
(22): 0 = ∂ (cid:2) gˆgˆµνρ(∂ S ) +(cid:3) (ǫ ) .
cosmologicalconstant right [36]. µ − ν Q O i
Which parameters had to be engineered? (cid:16)p (cid:17)
The presented mechanism only worked after demanding Thus, by constructing this toy model we explicitly
Eqs. (16, 19) to be true. The first equation (16) relates showed that it is possible to find a gravitational dual
the classical principal functional S˜ to the phase of the to the complex Klein-Gordon equation in curved space-
quantum wave function S . The second equation (19) time. This achievement of the model is accompanied
Q
relates the constant Λ to the mass of the quantum par- by a number of possible issues which have to be further
ticle. understood. First, the weak t¯dependence of the func-
What about spinors? tions S and α¯ is just an ansatz and not a necessity.
H
Oneofthegreatsuccessesofquantummechanicswasthe Second, except of the case with m=0 we could not find
predictionandexplanationofaparticlewithaLand´efac- a solution of all twentyfive Einstein equations. Third,
tor (g = 2) due to the Dirac equation. It will be a chal- havingagravitationaldualtothecomplexKlein-Gordon
lenging task to find the analog of the Dirac equation in equation does not mean that one also has a dual for rel-
ourpicture. However,ithas beenshownby[37]thatthe ativistic quantum mechanics. However, the toy model is
g =2factorisalsopresentinaKerr-Newmanspacetime. oneofseveral(verydifferent)theories[15, 16,17, 18,19]
Later, [28] found out how to relate the electro magnetic that make a formal connection between the quantum
field of a Dirac electron to the surrounding field of the structure of nature and an introduction of an additional
Kerr-Newman solution. Those progresses indicate that time dimension. Our conclusion is that this connection
therecouldbe awayto explainspin1/2withthe helpof could be more than a pure mathematical peculiarity, it
classical general relativity. might actually reflect the structure of spacetime.
Quantum field theory?
Questionsconcerningtheformulationofthesecondquan- Many thanks to Jorge Noronha, Silke Weinfurtner,
tizationin quantum field theory,interacting field theory, Goirgio Torrieri, Sabine Hossenfelder, Martin Kober,
or Yang Mills theories in this picture are hopefully sub- Nan Su, and Basil Sad for their comments and remarks.
[1] D.Bohm, Phys. Rev. 85, 166 (1951). [8] I. Bars, Class. Quant. Grav. 18 (2001) 3113
[2] D.Bohm, Phys. Rev. 85, 180 (1951). [arXiv:hep-th/0008164].
[3] O.Klein, Z. f. Physik 37, 895 (1926). [9] I. Bars and S. J. Rey, Phys. Rev. D 64 (2001) 046005
[4] G. ’t Hooft, Class. Quant. Grav. 16, 3263 (1999), [arXiv:hep-th/0104135].
gr-qc/9903084. [10] V. M. Villanueva, J. A. Nieto, L. Ruiz and J. Silvas, J.
[5] T.S.Biro,S.G.Matinyan,andB.Muller, Found.Phys. Phys. A 38 (2005) 7183 [arXiv:hep-th/0503093].
Lett.14, 471 (2001), hep-th/0105279. [11] I.Bars, Phys.Rev.D74,085019(2006),hep-th/0606045.
[6] H. Nikolic, Found. Phys. 37, 1563 (2007), [12] S.S.SeahraandP.S.Wesson, Gen.Rel.Grav.33,1731
quant-ph/0609163. (2001), gr-qc/0105041.
[7] G. R. Dvali, G. Gabadadze, and G. Senjanovic, (1999), [13] G. ’t Hooft, (1993), gr-qc/9310026.
hep-ph/9910207. [14] L. Susskind, J. Math. Phys. 36, 6377 (1995),
6
hep-th/9409089. [26] C. W. Bernard and A. Duncan, Ann. Phys. 107, 201
[15] J.M.Maldacena,Adv.Theor.Math.Phys.2,231(1998), (1977).
hep-th/9711200. [27] M. Novello and R. P. Neves, Class. Quant. Grav. 20
[16] E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998), (2003) L67.
hep-th/9802150. [28] A. Burinskii, (2007), arXiv:0712.0577 [hep-th].
[17] C. M. Hull, (1999), hep-th/9911080. [29] J. S.Bell, Physics 1, 195 (1964).
[18] S.Weinstein, Physicsworld Sept., 18 (2007). [30] G. Baym, Lectures on quantum mechanics (Perseus
[19] P. H. Damgaard and H. Huffel, Phys. Rept. 152, 227 Books Publishing, New York,1990).
(1987). [31] P. S. Wesson, (2002), gr-qc/0205117.
[20] N.D.BirrellandP.C.W.Davies, CambridgeUniversity [32] S. Weinberg, Gravitation and Cosmology (ISBN 0-471-
Press; ISBN:0 521 27858 9 , 43+ (1982). 92567-5, 1972).
[21] T.Kaluza, Sitzungsber.Preuss.Akad.Wiss.Phys.math. [33] N.Arkani-Hamed,S.Dimopoulos,andG.R.Dvali,Phys.
Klasse,996 (1921). Lett. B429, 263 (1998), hep-ph/9803315.
[22] O.Klein, Nature 118, 516 (1926). [34] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, and
[23] S.Wesson,Paul, Space-Time-Matter(WorldScientific G. R. Dvali, Phys. Lett. B436, 257 (1998),
PublishingCo.Pte.Ldt.;ISBN981-02-3588-7,NewYork hep-ph/9804398.
07661, 2000). [35] W.-M. e. a. Yao, Journal of Physics G 33, 1+ (2006).
[24] W.Hamilton, DublinUniversityReviewpp.,795(1833). [36] S. Weinberg, Rev.Mod. Phys. 61, 1 (1989).
[25] L. Landau and L. Lifshitz, Elsevier, Amsterdam; ISBN [37] B. Carter, Commun. Math. Phys. 10, 280 (1968).
0750628960 (1975).