Modified Kaluza-Klein Theory, Quantum Hidden Variables and 3-Dimensional Time

0.21 MB·English

by  Xiaodong Chen

Modified Kaluza-Klein Theory, Quantum Hidden Variables and 3-Dimensional Time
Xiaodong Chen
∗
(Dated: February 1, 2008)
In this paper, the basic quantum field equations of free particle with 0-spin, 1-spin (for cases of
massless and mass > 0) and 1 spin are derived from Einstein equations under modified Kaluza-
2
Klein metric, it shows that the equations of quantum fields can be interpreted as pure geometry
properties of curved higher-dimensional time-space . One will find that if we interpret the 5th and
6thdimensionas“extra”timedimension,theparticle’swave-functioncanbenaturallyinterpretedas
asingleparticlemovingalonggeodesicpathin6-dimensionalmodifiedKaluza-Kleintime-space. As
theresult,thefundamentalphysicaleffectofquantumtheorysuchasdouble-slitinterferenceofsingle
particle, statistical effect of wave-function, wave-packet collapse, spin, Bose-Einstein condensation,
Pauli exclusive principle can be interpreted as “classical” behavior in new time-space. In the last
5 partofthispaper,wewillcouplingfieldequationsof0-spin,1-spinand 12-spinparticleswithgravity
0 equations.
0
2 PACSnumbers: 03.65.-w,04.50.+h,04.62.+v

n
a
I. INTRODUCTION modified Kaluza-Klein equation. Thus, we shows that
J
the equations of quantum fields can be interpreted as
5
pure geometry properties of curved higher-dimensional
1 Kaluza-Klein’s theory[1] showed that five-dimensional
time-space. If we assume that the 5th and 6th “Kaluza
general relativity contains both Einstein’s 4-dimensional
2 like” dimensions are time dimensions, then the physical
theory of gravity and Maxwell’s theory of electromag-
v effects of these extra dimensions will show basic behav-
netism[2]. Kaluza gave physicist the hope of unify-
4 iorofquantumparticles. Wave-functionofsingleparticle
ing matter and geometry. Currently, people extended
3 becomesgeodesicpathin6-dimensionalmodifiedKaluza-
0 Kaluza-Klein’sidea to a possible “theoryofeverything,”
Klein time-space. In section (II), 0-spin single free par-
1 10-dimensionalsuperstrings. Oneofmaindifficultyofthe
ticle equation (Klein-Gordon equation) with mass > 0
0 original Kaluza-Klein’s theory was: trying to make the
is derived through modified Kaluza-Klein metric. Sec-
5 theorytofitonourmacrocosmobservationthatwhyhad
0 tion (III) will discuss the detail of using the two extra
no fifth dimension been observed in nature? This cause
/ time dimensions to interpret basic quantum effects such
h the later efforts of different versions of Kaluza-Klein’s
asdouble-slitinterferenceexperiment,statisticaleffectof
p theorysuchascompactified,projectiveandnoncompact-
wave-function, Bose-Einstein condensation, Pauli exclu-
- ified Kaluza-Klein theory.
t sive principle. In section (IV), Maxwell’s theory of elec-
n
In another totally different area, Bohm [3] and etc.
a tromagnetism will be re-derived from modified Kaluza-
triedtofindquantumhiddenvariablestointerpretquan-
u Kleinequations. The equationof1-spin’sfreesinglepar-
q tum physics under the language of “classical” physics. ticlewithmassm >0willalsobederived,themasspart
0
: There are many difficulties Quantum Hidden Variable
v ofU(1)gaugefieldwillbenaturallyincludedasderivative
(QHV) theory had to solve. One of main issue is: in
i of 6th dimension. In section (V) , we will obtain Dirac
X quantum physics, even a single particle can show non- field equation of single particle with 1 spin through 6-
r localeffect,butthereisnosuchthinginclassicalphysics dimensional Einstein equations. In sec2tion (VI), we will
a
orrelativitytheory. Forinstance,toexplainthe interfer- coupling field equations of 0-spin,1-spinand 1-spinpar-
ence pattern in double-slit interference experiment, one 2
ticles with gravity equations.
has to accept the fact that: a single photon (or elec-
tron)hasto passboth slits atthe sametime. Ingeneral,
in quantum physics, a single particle can spread out in
II. EQUATIONS OF 0-SPIN FREE PARTICLE
large area – occupy many different spatial locations at
the same time.
The original Kaluza metric can be written as follows
In 1999, X.Chen [4] proposed that using extra time
[2]:
dimension, we can explain why a single particle shows
at two different locations at the same time. The paper
g +κ2φ2A A κφ2A
didn’t derive any basic quantum equations. In this pa- (gˆ )= αβ α β α , (1)
per, single free-particle equation for 0-spin, 1-spin (for AB (cid:18) κφ2Aβ φ2 (cid:19)
massless and mass > 0) and 1 spin will be derived by
2 wheretheαβ-partofgˆ withg (thefour-dimensional
AB αβ
metrictensor),theα4-partwithA (theelectromagnetic
α
potential), and the 44-part with φ (a scalar field). The
∗2073B Vestavia Park Ct, Birmingham, AL 35216, U.S.A.; Elec- four-dimensional metric signature is taken to be (+
−
tronicaddress: [email protected] )
−−
2
Inthissection,we’llonlyfocusonobtainingfieldequa- Substitute (7) and (6) in (4):
tion of 0-spin free particle, so we ignore vector field A ,
α R = (φ⋆∂ φ)(φ⋆∂ φ)
(equation with 1-spin particle will be discussed in sec- αβ − α β
tion (IV)). Furthermore, We will add one more extra Rα5 =R5α = (φ⋆∂αφ)(φ⋆∂5φ)
−
dimension than original Kaluza metric, i.e. totally 6- R = (φ⋆∂ φ)(φ⋆∂ φ)
55 5 5
−
dimensional time-space with conditions:
R =R =0
α4 4α
R = φ(∂α∂ φ) φ(∂5∂ φ) (9)
∂ g =0, ∂ g = ia g , g = 1 (2) 44 α 5
4 44 5 44 5 44 55 − −
− −
and Ricci scalar becomes
where a is constant. For free particle,we ignoregravity
5
field, then gαβ = δαβ, 6-dimensional time-space metric R=gABRAB =0 (10)
becomes:
Here we used equation (7).
Using(9),(7),andleta = m0 wherem isrestmass
gαβ of particle, ~ is Planck co5nsta−nt~, Einstein e0quations (5)
(gˆAB)= g44  (3)
become:
1
 − 
(φ⋆∂ φ)(φ⋆∂ φ)=κT (11)
α β αβ
The 6-dimensional Ricci tensor and Christoffel sym- −
bols are defined in terms of the metric exactly as in four
m
dimensions: i 0φ⋆∂ φ=κT =κT (12)
~ α 5α α5
Rˆ = ∂ ΓˆC ∂ ΓˆC +ΓˆC ΓˆD ΓˆC ΓˆD ,
AB C AB − B AC AB CD− AD BC
ΓˆCAB = 21gˆCD(∂AgˆDB+∂BgˆDA−∂DgˆAB) . (4) (m~0)2 =κT55 (13)
where A, B.. run over 0,1,2,3,4,5.
κT4β =κT =0 (14)
The 6-dimensional Einstein equations are α4
Gˆ =κT , (5)
AB AB 1 m
∂α∂ φ ( 0)2φ=0 (15)
where TAB is 6-dimensional energy momentum tensor, −~2 α − ~
Gˆ Rˆ Rˆgˆ /2 is the Einstein tensor, Rˆ and Here we let T = 0, i.e. no 5-dimensional energy mo-
AB AB AB AB 44
≡ −
Rˆ = gˆ RˆAB are the 6-dimensional Ricci tensor and mentumtensor. Equation(15)isKlein-GordonEquation
AB
scalar respectively, and gˆ is the 6-dimensional metric for free 0-spin particle. Actually, it is reasonable to let
AB
tensor, A, B.. run over 0,1,2,3,4,5 . Tαβ = pαpβ, pα is momentum vector of particle, α run
Using metric (3), we getfollowing non-zeroChristoffel over (0,1,2,3), the solution of equations (11)–(15) is:
symbol: φ=e−i√κ(pαxα) (16)
Γˆ44α = 12g44∂αg44 , Γˆ4α4 =Γˆ44α aα in (8) becomes p~α.
Ifweletκ= (1)2,equation(16)becomeplanewave-
Γˆα = 1gαα∂ g , Γˆ4 = 1g44∂ g function of a si−ngl~e particle. We can see that if we de-
44 −2 α 44 45 2 5 44 scribequantumfieldinpuregeometry, i playsthesimilar
1 ~
Γˆ454 =Γˆ445 , Γˆ544 =−2g55∂5g44 (6) role as √8πG ’s role in 4-dimensional Einstein equation,
where G is gravational constant. The metric tensor be-
Start from here, through out this paper, capital Latin comes
indices A,B,C .. run over 0,1,2,3,5 (A,B <> 4), Greek g
αβ
indicesα,β ... runover0,1,2,3,andsmallLatinindices (gˆAB)= e−2~i(pαxα−m0x5)  (17)
a, b, ... run over 1, 2, 3.
1
Let  − 
The interval
gg4444==((φφ(⋆x(0x,0x,1x,1x,2x,2x,3x)3e)−eii((aa55xx55))))22 (7) ds2 =dx20−dx21−dx22−dx23+e−2~i(pαxα−m0x5)dx24−(d1x825)
ds2 becomes complex function, and in this paper, we al-
where φ⋆ is complex conjugation of φ, and ways choose c 1. The length can be define as
≡
φ=e−i(a0x0−a1x1−a2x2−a3x3) (8) dl= ds2 (19)
q| |
where a is constant. where ds is mod of ds.
α
| |
3
III. A NEW INTERPRETATION – TIME AS atthesametimet–Bobcannottravelfasterthanlight,
5TH AND 6TH DIMENSIONS so we have way to ensure that time is the same (ignore
the gravity of the earth). How could we interpret this?
A. Time as quantum hidden variable Letusaskaquestion,isthesametimetmeansthe same
time? If time is more than 1 dimension, there is another
hiddendimensionoftimet ,sothatBobinNewYorkat
Before we discuss the meaning of metric (17), let’s go θ
time (t,t ), Bob in Las Vegas at time (t,t ). Bob uses
back to basic quantum physics. In double-slit interfer- θ2 θ1
t t travelingfromLas Vegasto New York andthen
ence experiment of photons, it is well known that to get θ2− θ1
travelbackandforth,butwedon’thaveanyapparatusto
interference pattern, one has to assume that both slits
measure t , also sine it is t , not t, we can not use speed
affect each single photon, even if we make light beam θ θ
to measure the travel from to spatial location by t . If
weak enoughto emit almostjust 1 photoneachtime, we θ
Bob accidentally dead in Las Vegas at 9:02 am, he can
will still get interference fringes, but if we tried to mea-
not travel to New York through second time dimension
sure which slit photon passed, the interference fringes
t , so he will disappear in New York at 9:02 am, which
will be destroyed. If one trys to use path to describe the θ
is called wave-packagecollapse in quantum physics.
movement of a single photon, he has to say that: a sin-
To understand the above discussion better, we need
gle photon passes two slits at the same time ! It sounds
understand what will happen if there are extra time di-
against our common physics law, that’s why in current
mensions. In our 1-dimensional time world, time tells
quantum physics, we have to say that there is no path
us the order of events happened. We are doing things
for quantum particle, even in Fenyman’s path integral,
always in the time order. We can not travel to two dif-
we have to interpret that the path is imaginary path, it
ferent places at the same time because of 1 dimensional
is not the realpath of particle’s movement. Similar non-
time; Time is the only parameter to describe the motion
localizedbehaviorcanbefoundinBose-Einsteinconden-
ofobject. Ifthereisanextratimedimension,thatmeans
sation and Superconductivity theory, one has to assume
the order of a serial events happened for an object will
thatthewavefunctionofeachsingleparticlesspreadout
be describedby two parametersTheneventA and event
thewholelattice—eachsingleparticlesiseverywherein
Bwhichhappedatthesametimetin1dimensionaltime
lattice,itissocalledidentityparticles. Shouldwesatisfy
world, could actually happened at two different times in
theanswerfromtraditionalquantumphysics: thereisno
2-dimensionaltimeworld–AandBatthesametimet1,
path in quantum world? Or there are some mysterious
but different time t2. i.e. Serial events in 2 dimensional
paths which we haven’t understood yet? Quantum hid-
time, could be seen as parallel events if you only use 1
den variable theory tried to find the hidden variables so
dimensionclock. Thenewtimet2ishiddenorderofeven
thatwecandescribethequantumeffectbyusingclassical
happening in our 1-dimensional time world.
physics language (including “path” concept). No quan-
Imagine if we only have 1 dimensional knowledge of
tum hidden variable theory gives us a satisfied answer
the world but the actually world is 3-dimension, so we
yet.
use ruler to measure space and we do not have concept
Let’s re-phrase the statement we discussed: To use
ofdirection,thenifwepickuptwopointsonacirclewith
pathtodescribeasingleparticle’smotion,wehavetosay:
angleθ,(0<θ < π)andourthemeasurementstartfrom
1particlecanoccupied2(ormore)spatiallocationsatthe 2
centerofcircle, byusing 1-dimensionallanguage,we will
same time. It is against our common sense – how can a
think that the two points are the same point since the
particle shows up in two locations at the same time and
distance is the same.
it is stillthe same particle ? If yousee Bobat LasVegas
Extending the above discussions about Bob to quan-
at 9:00 am Jan. 1st, 2004 central time, your brother see
tum world: a single particle can occupy more than one
Bob at New York at 9:00 am Jan. 1st 2004central time,
spatial locations at the time t, if we change the state of
how can you tell that you and your brother are seeing
the particle in one location X(x ,x ,x ) at time t, its
the same Bob, not Bob’s twin brother? The way to find 1 2 3
state in another location Y(y ,y ,y ) will be changedat
out is: if you break Bob’s left hand one minute later(at 1 2 3
the same time t. If we localized the particles location at
9:01 am) in Las Vegas (as an extreme way to affect the
X(x ,x ,x ) at time t, the particle can not be shown at
physicalstateoftheobject),thenifyourbrotherseethat 1 2 3
location Y(y ,y ,y ) at time t anymore. These are the
Bob’slefthandsuddenlybrokenat9:01aminNewYork, 1 2 3
real phenomena observed in quantum world! It is only
then he is the same Bob (his physical state affected by
valid in quantum world, since wave-lengthλ= h, where
yourbehavior),butiftheBobwhoyourbrotherseeingis p
stillwithaperfectlefthand,thenhemustbe Bob’stwin pis momentum ofparticle,inmacrocosmworld,pis too
brother,notBobhimself. Supposethatyourbrotherdoes big and λ is too small. To interpret this by using lan-
see Bob’s left hand suddenly broken in New York, then guage of “classical” physics, we need two hypotheses:
there must be something wrong in our time-space. First 1) There is at least one extra time dimension in our
wearesureyouandyourbrotherareseeingthesameBob world, the new time dimension (or dimensions) acts like
(brokenleft hand), then youandyourbrother areseeing Kaluza’s 5th dimension, it is a loop wrapped aroundthe
Bob at different spatial locations (x ,x ,x )(Las Vegas rest 4-dimension time-space.
1 2 3
andNewYork),thenyouandyourbrotherareseeingBob 2)Aparticle’smotionisdeterminedbyitslocalcurved
4
geometry properties of multiple time dimensions + 3 di- where α runs over 0,1,2,3.
mensional spaces; A particle moves along geodesic path
im
in its local curved time-space. x5 = − 0τ2+constant (24)
Ifthefirsthypothesesistrue,then“aparticleshowsin 2~
more than one locations” at the same time t is because
and
of extra time dimensions. The particle moves from one
location to another location at the same time t by extra x4 =e~i(pαxα−m0x5)τ +constant (25)
timedimensiont . Later,wewillprovethat“thepathof
2
particle’s motion by extra time dimension” is plan-wave theconstantsin(23),(24),(25)aredeterminedbyinitial
and it is geodesic in 6-dimensional time-space. value of xA. Then:
Should we assume the extra time dimension is a
“stsamyaslli”nltohoep?samIteisloncoattinoenceastsatrimy.eItf tihneaplla2rtnidcleanadlw3aryds ddxτ4 =e~i(pαxα−m0x5)(1+ ~i(pαddxτα −m0ddxτ5)τ)
tsiammeedaismaennsoiobnjesctt2,int3c,ltahsesincatlhpehpyasrictisc,lesoweivllebneihfa2vnedthoer =e~i(pαxα−m0x5)(1+(~i)2(p20−p2a−m20)τ2)
3rdtimedimensionisbig,aslongasthewholeloopofex- (26)
tradimensionaltimet andt alwaysstayinonelocation
2 3
ateachfirstdimensionaltimet,wewillnotseeanyeffect where a runs over 1,2,3. After use the relation: p2 =
0
of extra dimensional time. The curvature of local time- p2+p2+p2+m2, equation (26) becomes:
1 2 3 0
space is in quantum level – “small”. The reason we can
naloltobsejeecetxs’trmaottiimoneidnimouernmsioancroincomsmacwroocroldsmariescboellceacutisvee: ddxτ4 =e~i(pαxα−m0~x5) (27)
motionofenormousparticles,theeffectoftheextratime
Using similar algebra, we can get:
dimensionofeachparticlescounterparttoeachother. In
addition, in Einstein theory, gravity potential is caused
d2x4
bycurvedtime-space,weextendEinstein’sidea: aparti- =0 (28)
dτ2
cle’senergymakesitslocalspacecurved. Asweallknow
that,eveninnon-relativityquantumphysics,aparticle’s
Itiseasytoexaminethat,(23)(24)(25)arethesolutions
energyisalways>0. Iftheabovetwohypothesesarecor-
of (20). i.e. τ is affine parameter of geodesic and the
rect, then a particle’s local time-space is always curved:
equationsabovearesolutionsofgeodesic. Rewrite(23):
nocurvedtime-space,noquantumeffectofparticle,then
particle can not exist. Based on two hypotheses, we can dxα ipα
= − τ (29)
start interpret quantum phenomena. dτ ~
Rewrite (25) and (26):
B. Geodesic of 6-dimensional time-space
τ =e−~i(pαxα−m0x5)x4 (30)
givNesowuswtheea6re-drimeaednysitoonaulsmeetthreicraensudltesquinatsieocntsioonf0I-Is.piInt ddxτ4 =e−~i(pαxα−m0x5) (31)
particle’slocaltime-space. Byusingtheresultsinsection
II, we are able to show that wave-function of particle is Then using (29), (31) :
geodesic in 6-dimensional time-space.
The 6-dimensional geodesic equation is : ddxxα4 = ddxταddxτ4 = −i~pαe−~2i(pαxα−m0x5)x4 (32)
d2xA dxB dxC
dτ2 +ΓABC dτ dτ =0 (20) Similarly:
weqhuearteioτnibseacffiomneesp:arameter. Using equation (6 ), above ddxx45 = im~0e−2~i(pαxα−m0x5)x4 (33)
d2xα dx4dx4
+Γα =0 (21) Thus:
dτ2 44 dτ dτ
dd2τx24 +2Γ44αddxτ4ddxτα =0 (22) ds=r1− (gααpα~pα2 −m20)e−~4ipAxAx24e−~ipAxAdx4
where α run over 0,1,2,3,5. =e−~i(pαxα−m0x5)dx4
Tofindthesolutionfor(20),weneedfindtherelations (34)
between xA and τ. let:
ipα where A,B runs over 0,1,2,3,5, and α runs over 0,1,2,3
xα = −2~ τ2+constant (23) andp5 = m0, so6-dimensionalgeodesicbecomes plane
−
5
wave-function by dx . In addition, from (23), one can As we discussed before, P’s geodesic is not a straight
4
see: line, if it passes (X,t) more than once through 2nd and
dxα pα 3rd time dimensions, then particle A has better chances
= =v (35) tomeetP.ThechancesofmeetingPat(X,t)isdepended
dt p0
on how many (x ,x ) of particle P “pass” (X,t). i.e.,
4P 5P
where we let speed of light c = 1, v is “classical” speed the possibility of A meet P at (X,t) is proportional to
of particle. thedensityof(x4P,x5P)passthrough(X,t). Fromequa-
Re-write equation (32) as: tion (37), in a small area xα, the density (x4)2 is
△ |△ |
proportional to ψ2 , since the possibility of finding P in
△xα = i2p~αe−~2ipAxA(g44)2△(x4)2 = i2p~αe2~ipAxA△(x4)2 △litxyαofisfinpdroinpgorPti|oinna|lxto disenpsriotpyorotfioxn4a,ltthoenψ2th.eNpoowssiwbe-
(36) △ α | |
get the same conclusion as the statistical interpretation
It gives us
of wave-function! We don’t need considering the density
|△(x4)2|=|−p2αi~e−~2ipAxA||△xα| mofaxk5e bmeectaruicsee:quwaelsch(3o)s,euandspeerctiahliscmooertdriinc,attehesygsetoemdestioc
2i~ only depends on x4 as we saw in (34).
=|−pα ψ2||△xα| (37) In double slits experiment, a particles path is splited
into two paths, the particle will stay in path 1 in some
where ψ = e−~i(pαxα−m0x5) is wave-function. We need ofx4, and stay in path2 in the same portionof x4, after
passthedoubleslits,thetwopathswillinterferenceeach
this equation in next sub-section.
other, by equation (37), we have
Finally, from (25), one can see that: in vaccum, with-
out any particle, pA = 0, then dx4 = dτ. In relativity, 2~
proper time τp is affine parameter, time t has relation- |△(x4)2|=|pα△xα||ψpath1+ψpath2)|2 (38)
ship with τ : dτ = (1 v2)dt. Time t becomes τ in
p p p
−
special frame that vp=0. Now we see similarity between Atsomespatialpoints(onthescreen), ψ +ψ )
path1 path2
| |
x4 and x0 – under certain condition, they both become becomeszero,atthosepoints,equation(38)couldnotbe
affineparameter. Equation(34)tellsusthatx4alonecan trueforanynon-zero (x4)2 ,itmeansthatthedensity
|△ |
notkeepdsatfixedvalue,whenx4 unchange,westillcan of x4 is always zero at those points – the particle does
changeotherx0 andx5 tomakeparticle’sdschange(the not go to those points. That explains the minima in
other three spactial coordinate dx1, dx2, dx3 should de- interference fringes.
pend on time coordinates), so we have three parameters Equation (38) also tells that at the same first dimen-
to describe particle’s motion (or “order” of events) – 3 sion time t, X can be different value for different x4, so
dimensional time. the particle’s position X is uncertain in 4-dimensional
time-spacelanguage,butitisunique in3+3dimensional
time-space , i.e. at each 3-dimensional time (x ,x ,x ),
0 4 5
C. Quantum effects of 6-dimensional time-space particle P is always in 1 spatial location X(x ,x ,x ).
1 2 3
We can imagine that, if we modify metric g to the
44
Beforewegofurther,weneeddiscussthemeasurement combination of different ψ(pα), then particle can be at
i
of particle. different momentums at the same time t.
How do we measure a particle in 6-dimensional time- Why we always only get 1 “dot” at interference
space? We use an apparatus to measure a particle P, fringes for each photon ? Why 1 photon can not pro-
first the apparatus need meet particle P, that means at duce 2 “dots” on the screen? To produce a “dot”
least one particle A of apparatus must meet the par- on the screen, the screen has to interact with photon,
ticle P at some points of 6-dimensional time-space; Or the interaction localized the photon and change pho-
in other words, particle A’s geodesic and particle P’s ton’s local curvature of time-space of photon. Let’s
geodesic must cross each other at some 6-dimensional say the interaction happened on 6-dimensional coordi-
points. At those points, the 6-dimension coordinates of nate A (x ,x ,x ,x ,x ,x ), which is spatial location
0 0 1 2 3 4 5
particlePequalsto6-dimensionalcoordinatesofparticle X (x ,x ,x ) at 3-dimensional time (x ,x ,x ). After
0 1 2 3 0 4 5
A. If at spatial location X(x ,x ,x ) at time t (through theinteraction,thecurvatureoflocaltime-spaceofparti-
1 2 3
this paper, we always use X as 3-space dimension and clePchanged,particlePcannotmovetoanotherscreen
t as first time dimension) , the 2nd and 3rd time coor- location X (x ,x ,x ) through x , the original geodesic
1 1 2 3 4
dinate of particle P is (x ,x ), and the 2nd and 3rd ofthe particleiscutby interaction. Thatcorrespondsto
4P 5P
time coordinate of particle A is (x ,x ), if x =x wave-packetcollapse in quantum physics.
4A 5A 4P 4A
6
or x = x , then even if P and A both show at loca- There are two most important properties of 3-
5P 5A
6
tion X at the same time t, they do not meet (they are dimensional time:
differentpointsin6-dimensionaltime-space),soAandP 1) a particle can occupy more than 1 locations at the
cannot“see”eachother. Therefore,thechanceoffindP same 1st dimensional time t. 2) Many particles can oc-
at(X,t)isnotdetermined,itisratherastatisticalresult. cupy the same location X at the same 1st dimensional
6
time t. andfromthe relationgˆ gˆαβ =δα,wehaveinversemet-
αβ β
The first property gives us the non-local results of ric:
quantum physics and statistical interpretation of wave-
function. The secondpropertywillgiveusBose-Einstein gαβ Aα
condensation. Considering two particles occupy location gˆAB = Aβ −1+A2  (41)
−
X at the same t, but their (x4,x5) are always different (cid:0) (cid:1)  −1
at (X,t), then those two particles can not “see” each
other, and they can not interact with each other ( un- the αβ , α4 , and 44-components of equations (5) be-
− −
less through 3rd particle). In a very small 3-D ball with come:
many particles inside, if we can find a distribution of
2nd and 3rd time dimension of those particles such that G = 1TEM , αF =0 , F Fαβ =0 (42)
(x ,x ) <> (x ,x ) ) always true at any time t and αβ 2 αβ ∇ αβ αβ
4 5 i 4 5 j
at any X (i, j are indices of particle), then those par-
ticles do not interact each other. If such distribution ThesecondofaboveequationsisMaxwellequation. The
of (x ,x ) exists, we get Bose-Einsteincondensationfor third of equation is true for plane electromagnetic wave-
4 5 i
those particles. Does such distribution exist? If at time function which is the case of single free-photon. If we
t, projection of the 6-dimensionalgeodesic to 3-D ball is ignore gravity (for free photon) in first item, Equation
always a loop (U(1) symmetry) at fixed time t, we can (42) is the equations for single 1-spin massless free par-
easilyputmanyloopstoSO(3)withoutcrosseachother, ticle.
so all particles do not “meet” each other, we get Bose-
Einsteincondensation. Butifthegeodesicoftheparticle
is not a loop, for example, if the geodesic of particle has B. 1-spin free particle with mass >0
SU(2) symmetry, as we know, we can find a map from
SU(2) to SO(3) which is double cover, i.e. one X only
For 1-spin particle with mass m >0, we let
contains two different points of SU(2), suppose the two 0
different points (x ,x ,x ) , (x ,x ,x ) from different
particle, then only02 p4art5iciles c0an4put5ijn a small space Aˆα =Aαe~im0x5 (43)
without meet with each other, this will explain why we
havePauliExclusivePrincipleforSU(2)field,whichonly where we choose ~=1. and
allows two electrons in the same “location”, one with 1
spin, the other one with 1 spin. We will talk abou2t Aˆ5 =0 (44)
−2
particle with 1 spin later.
2 where m is rest mass of particle, x is 6th dimension
0 5
coordinate. 6-dimensional metric for 1-spin free particle
IV. EQUATIONS OF 1-SPIN FREE PARTICLE become:
g +Aˆ Aˆ Aˆ
A. Massless 1-spin free particle αβ α β α
(gˆAB)= Aˆβ 1  (45)
1
The original 5-dimensional Kaluza field equations can  − 
be written as [2] :
Let Fˆ ∂ Aˆ ∂ Aˆ , and A, B runs over 0,1,2,3,5
AB A B B A
≡ −
κ2φ2 1 ( A, B <> 4). Energy momentum tensor
G = TEM [ (∂ φ) g (cid:3)φ] ,
αβ 2 αβ − φ ∇α β − αβ
αF = 3∂αφF , (cid:3)φ= κ2φ3 F Fαβ (39) TˆAB ≡gABFˆCDFˆCD/4−FˆACFˆBD (46)
αβ αβ αβ
∇ − φ 4
where A,B runover(0,1,2,3,5,A,B <>4). So the αβ ,
−
where G R Rg /2 is the Einstein tensor, α4 ,and44-componentsof6-dimensionalEinsteinequa-
TEM αgβ F≡ Fαγβδ/−4 FαγβF is the electromagnetic tio−ns (5) become:
αβ ≡ αβ γδ − α βγ
energy-momentum tensor, and F ∂ A ∂ A .
αβ α β β α
≡ − 1
Herethecylinderconditionisbeingapplied,whichmeans G = Tˆ , αFˆ m2Aˆ =0
dropping all derivatives with respect to the fifth coordi- αβ 2 αβ ∇ αβ − 0 β
nate. 1Fˆ Fˆαβ 1m2Aˆ Aˆα =0 (47)
For massless 1-spin free particle, if we write 6- 4 αβ − 2 0 α
dimensional metric as
wherewechoose~=1. Thisisequationsfor1-spinsingle
g +A A A particle with mass > 0. As we see above, the particle of
αβ α β α
(gˆAB)= Aβ 1  (40) 1-spin obtains its mass from derivative of 6th dimension
1 (third time dimension).
 − 
7
V. EQUATIONS OF 1-SPIN FREE PARTICLE LetEˆ ∂ Kˆ ∂ Kˆ ,andA,Brunsover0,1,2,3,5
2 AB ≡ A B− B A
( A, B <> 4). Define energy momentum tensor for half
Equationsfor 1-spinfreeparticlesareDiracequations: spin particle:
2
Tˆ g Eˆ EˆCD/4 EˆCEˆ (55)
(iγν∂ m)ψ =0 (48) AB ≡ AB CD − A BD
ν
−
where A,B run over (0,1,2,3,5, A, B <> 4), so the AB-,
Hereψ isfour-componentcomplexwavefunction,ν runs A4-, and 44-components (A, B <> 4) of 6-dimensional
over 0,1,2,3. γi, i =0,1,2,3 are 4 4 complex constant Einstein equations (5) become:
×
matrices, satisfying the relation
1
γlγk+γkγl =2gklI, k,l=0,1,2,3. (49) GAB = TˆAB , ∂A(∂BKˆB) ∂B∂BKˆA =0
2 −
1
where I is the unit 4 4 matrix, and Eˆ EˆAB =0 (56)
gkl =diag(1, 1, 1, 1)is the me×tric tensor. 4 AB
− − −
Dirac equations are not equations of single particle, it
Toderivetheequationabove,weusedtherelationbelow:
has 4-solutions corresponding to different states of elec-
tron. The wave-function of Dirac particle is:
φ0 ∂BgBB(∂AKˆB−∂BKˆA)=∂A(∂BKˆB)−∂B∂BKˆA (57)
φ
ψ = 1  (50)
φ For free particle, it is reasonable to assume that each
2
 φ  components of Kˆ satisfied plane-wave condition:
 3 
where φα (α=0,1,2,3)is 4-components of the 1stsolu- ( ∂α∂ m2)Kˆ =0 (58)
α α
− −
tion Dirac equation.
Accordingtoourhypothesesthatdifferentsatesofpar- where α,β runs over 0,1,2,3, or equivalently:
ticle will have different local geometry metric, so the 4-
solutions should have different metrics. Let’s start from ∂B∂ Kˆ =0 (59)
B A
first solution with spin 1 and positive energy: For ψ, in
2
x3 representation, Dirac equation can be re-write as: where A, B run over 0,1,2,3,5. so the second equation of
equations (56) become
∂ φ +∂ φ i∂ φ +∂ φ +im φ =0 (51)
0 0 1 3 2 3 3 2 0 0
−
Note: Start from this section, we will always choose ~= ∂A(∂BKˆB)=0 forallA=0,1,2,3,5 (60)
1.
Now for particle with half-integer spin, Let i.e. ∂BKˆB does not depended on x0,x1,x2,x3,x5, so it
is reasonable to let ∂BKˆ equals zero. (Can not be a
B
Kˆ0 =Cg00φ0eim0x5 constant other than zero becaues Kˆ contains a plane-
Kˆ =Cg φ eim0x5 wavefunction partofcondition(59)). Togetherwith the
1 11 3
Kˆ = iCg φ eim0x5 third equation of equations (56), we have
2 22 3
−
Kˆ =Cg φ eim0x5
3 33 2 ∂BKˆ =0, Eˆ EˆAB =0 (61)
Kˆ =Cg φ eim0x5 (52) B AB
5 55 0
are the solutions of equation (56). Now we have 3 un-
where φ is the α components of equation (50 ), C is
α
known functions: φ ,φ ,φ of (52), we have above two
constant to be determined, g is element of usual 4- 0 2 3
αα
equations, plus we need choose constant C to make the
dimensional metric (1,-1,-1.-1), g = 1. m is rest
55 0
− first equation of (56) becomes reasonable, and for φ ,
mass of the particle. α
they should be normalized. Substitute φ into above
Forsinglefreeparticlewith 1-spinandpositiveenergy, α
2 Kˆ , the first equation of (61) becomes Dirac equation
we choose metric as below: A
(51), and the solutions of (61) are:
g +Kˆ Kˆ Kˆ Kˆ Kˆ
αβ α β α α 5
(gˆAB)= Kˆβ 1 Kˆ5  φ = m0+p0 , φ =0,
 Kˆ5Kˆβ Kˆ5 −1+Kˆ5Kˆ5  0 r 2m0 1
(53) m +p p
0 0 3
where α,β runs over 0,1,2,3, and φ2 =r 2m m +p ,
0 0 0
gαβ Kˆα m0+p0 p1+ip2
gˆAB = Kˆβ 1+Kˆ−AKˆA Kˆ5  (54) φ3 =r 2m0 m0+p0 (62)
− −
(cid:0) (cid:1) Kˆ5 1 (63)
 − − 
8
The above φ is the solution of equations (56) and it is reasonable because: in this paper, the basic idea is that
α
also first solution of Dirac equation (51). We choose the local metric of time-space determined the state of par-
constant C in equations (52) as ticle, for a free single particle with one state (not com-
bination of states), its local metric of time-space should
(m +p )2m only be one of solutions of Dirac equation. Thus, we can
0 0 0
C = (64)
p p saythatweobtainedthesingleparticleequationsforthe
3
particle with 1-spin in this section. Note: the metric
2
Substitute (64), (63) into (55),then: we got for 1-spin particle is similar to the metric for
2
1-spin particle except that, 1-spin has nonzero Kˆ com-
Tˆab =papbe−~2ipcxc (65) ponents. It causes the non2-diagonal elements be5tween
5th dimension and 6th dimension. It indicates that for
where a,b,c runs over 0,1,2,3,5 and p = m . The re-
5 0 integer spin particle, the 5th dimension loops around 4-
sult is just we expected: the Energy-Momentum tensor
dimensionaltime-space,butforhalfintegerspinparticle,
becomes the product of two 5-dimensional momentum
the 5th dimension loops around all other 5-dimensional
vector (p ,m ) times the square of plane wave function.
α 0 time-space. The metric of 1-spin particle has symme-
Note: the above solutions is dervied under x3 represen- try of 4-dimensional time-space, but the metric of 1 has
tation of Pauli matrix, it means choose a special coordi- 2
symmetry of all 5-dimensions.
nate for our 6-dimensional metric. If we choose different
The similarity between the metric of single electron
representation, the relationship between Kˆ and φ in
A A (53) and the metric of a single photon (40) makes us
(52) will be different, but equations (56), (61) does not
easierto combine eletronand photoninto the same met-
dependent on the choice of coordinate, so we’ll still get
ric. If we use Klein’s idea [5]: the derivatives of its fifth
solutions of Dirac equation under new representation.
coordinate not equal 0. We can interpret the non-zero
If we let
derivatives of fifth coordinate as: The coupling between
electron and photon changed local time-space metric.
Kˆ =Cg φ eim0x5
0 00 1 Let
Kˆ =Cg φ eim0x5
1 11 2
Kˆ2 =iCg22φ2eim0x5 KˆA →KˆAe−iγx4 (69)
Kˆ = Cg φ eim0x5
3 − 33 3 where γ is a very small constant, and there is no Kˆ .
Kˆ =Cg φ eim0x5 (66) 4
5 55 1 Then
We can get localtime-space metric of single free particle
∂ K = iγK (70)
withspin 1,andthecorrespondingequationwillderive 4 A − A
−2
the second solution of Dirac equation. Similarly, we can
The metric gˆ of the coupling of electron-photon is:
AB
get the local metric of single free particle with negative
energy and spin 21: gαβ +(AαAβ +KˆαKˆβ) Aα+Kˆα KˆαKˆ5
KKˆˆ01 ==−CCgg0101φφ31eeiimm00xx55  AKβˆ5+KˆKβˆβ Kˆ5 1 −1+Kˆ5Kˆ5Kˆ5(71)
−
Kˆ2 =iCg22φ1eim0x5 where α,β runs over 0,1,2,3. Aα is components of vec-
Kˆ = Cg φ eim0x5 tor field of photon. Kˆα satisfied one of equations (52),
3 33 0
− (66),(67), (68). From above metric, one can obtain the
Kˆ =Cg φ eim0x5 (67)
5 55 3 couplingitem: ieA Kˆ and∂ ∂ ieA . Themet-
ν ν α α α
− → −
ric of interaction between electron and photon will be
and the third solution of Dirac equation. And
discussed in future.
Kˆ = Cg φ eim0x5
0 00 4
−
Kˆ = Cg φ eim0x5
1 − 11 0 VI. COUPLING OF QUANTUM FIELD
Kˆ2 = iCg22φ0eim0x5 EQUATIONS AND GRAVITY
−
Kˆ =Cg φ eim0x5
3 33 1
Kˆ =Cg φ eim0x5 (68) Kaluza’s original purpose is to unify gravity and
5 55 4
Maxwell equation. Since we already have above quan-
forsinglefreeparticlewithnegativeenergyandspin 1, tum field equations and they are all derived under 6-
−2
and the 4th solution of Dirac equation. dimensional Einstein equations, if we assume that the
As we see above,we didn’t getall four Dirac solutions gravitydoes not change the curvature of 5th and 6th di-
in one metric, instead, we get each solutions of Dirac mension, it is straight forward to combine gravity and
equation under each different metric, and each different quantum field equations for particle with 0-spin, 1-spin
metric corresponding to different state of particle. It is and 1-spin.
2
9
For 0-spin particle, the metric is the same Einstein tensor which is contributions from 5th and 6th
dimension:
g
αβ
(gˆAB)= φ2 1  , (72) GQAB ≡RAQB−RQgˆAB/2 (77)
 − 
and
where g is 4-dimensional metric of original Einstein
αβ
equation. The 5-dimension and 6-dimension are diago- RQ = (φ⋆∂ φ)(φ⋆∂ φ) (78)
nal, so we can separate the contribution of gravity part AB − A B
and 5th and 6th dimension part. Field equations be-
As we see that Klein-Golden equation becomes equation
comes:
(76).
For 1-spinparticle with mass > 0, the metric becomes
GE +GQ =8πGTE +TQ (73)
αβ αβ αβ αβ
GQ =TQ = (φ⋆∂ φ)(φ⋆∂ φ) (74)
5α 5α −(cid:3)φ Am2φ=B0 (75) gαβ +κ2AαAβ κAα
− 0 (gˆAB)= κAβ 1  (79)
GQ55 =T5Q5 =m20 (76) 1
 − 
where G is gravational constant. TE is original Ein-
αβ where κ = 4√πG. Field equations (47) should stay the
stein energy momentum tensor, TQ is quantum energy
AB same. If m0 = 0, the metric above becomes original
momentum tensor defined in section II: TAQB = pApB. Kaluza metric with scalar field φ=1.
GE is original Einstein tensor, GQ is quantum part of For 1-spin particle, the metric becomes:
αβ AB 2
g +κ2Kˆ Kˆ κKˆ κ2Kˆ Kˆ
αβ α β α α 5
(gˆAB)= κKˆβ 1 κKˆ5  (80)
κ2Kˆ Kˆ κKˆ 1+κ2Kˆ Kˆ
 5 β 5 − 5 5 
where κ = 4√πG. With gravity, we do not have simple the field equations of those particles become pure geom-
relationasequation(57),sincethederivativeofgααcould etry, that makes us easier to unify gravity and quantum
be non-zero,we can not get free particle Dirac equation. fields.
Instead, the equations below should still valid: Allofthosecanstaythesamewithoutinterpretingthe
5th and 6th dimensions are “Time” dimension. Why we
1
G = Tˆ , AEˆ =0 need call them extra time dimensions ? To answer this
AB AB AB
2 ∇
question, we face with a much bigger question, what is
1
Eˆ EˆAB =0 (81) time? It is too big question to answer here. Maybe we
AB
4 never get the answer. But we can show some critical
properties of time. First, Time makes us know the order
whereA,Brunover0,1,2,3,5,thedefinitionofTˆ ,Kˆ ,
AB A of events happening. We know what happened before
Eˆ is defined the same as in section V.
AB and what happened after. In the world 1-dimensional
time, we know one person can only do one thing at a
time (depends on how do you define one thing). But in
VII. DISCUSSIONS AND CONCLUSIONS quantum world, as we discussed in section III, quantum
non-localeffectmakesaparticlecanshowindifferentlo-
In the most part of this paper, we derived quantum cations at the same time, the distance of those locations
field equations of 0-spin particle, massless 1-spin parti- canbequitelargeasitisshowninBell’sinequalityexper-
cle,1-spinparticlewithmass>0and 1-spinparticlesby iment. That makes us to question that if there is more
2
using 6-dimensional metrics. The equations are derived than 1 dimensional time in the world? Second, when
naturally as pure geometry properties of 6-dimensional we are talking about time in common life, we usually
time-space. Mass is included as derivative of 6th dimen- talk about “proper time” τ, actually time plays “affine
sion. Metric of 1-spin particle has the same format as parameter” in our time-space. In relativity, time is no
2
metric of 1-spin except that the 6th component of vec- longer as “affine parameter”, time is “affine parameter”
tor field is not zero. Particle’s wave-function becomes ofgeodesiconlywhenwechooseaspecialreferenceframe.
geodesic of6-dimensionaltime-space. As the conclusion, As we see in section III, if we choose 5th dimension as
10
parameter, the particle’s geodesic path is wave-function, special coordinates. Can we choose different coordinate
it also naturally shows that the possibility to meet the tomakemetriconlycontainingrealparts? Weknowthat
particle is proportion to square of wave-function. If no a complex function can be described as a real function
particleexists–invacuum,5thdimensionbecomesaffine with two components. Is it possible that the reason we
parameter. That makes us believe that 5th dimension is have complex metric is just because of trying to derive
time. Third, the extra dimensional time makes us re- current quantum equations; it is possible that in future,
define the meaning of “meet” of two particles. In the wecanmakethe metric whichonly containsrealpartby
new definition that two particle can meet each other if including second and third time dimensions in geodesic?
andonlyifall6coordinateshavethesamevalueforboth
Now let’s try to understand spin. As we see in 6-
particle, i.e. their 6-dimensional geodesics have at least
dimensional time-space metric of 0-spin particle, 5th di-
one point crossing each other. That makes us to obtain
mensionis diagonal. There is no components between 4-
the results of Bose-Einstein condensation. Actually the
dimensionandotherdimensions. Intime-spacemetricof
two properties of extra time dimension :
1-spin particle, 5th dimension wrapped around 4 dimen-
1) a particle can occupy more than 1 locations at the
sionaltime-space,nocomponentsbetween5thdimension
same 1st dimensional time t. 2) Many particles can oc-
and 6th dimension. In time-space metric of 1-spin par-
cupy the same location X at the same 1st dimensional 2
ticle, the time-space geometry become more complicate,
time t.
5th dimension wrapped around all other 5 dimensional
They makes us to understand most of basic quantum
time-space. In our knowledge of 4-dimensional world, if
phenomena. In addition, if we interpret 5th dimension
an objects moves around some space dimensions, it is a
as space, we have to face the same problem as Kaluza:
rotationmovement. Itisreasonabletointerpretthatthe
to make 5thdimension small. As we discussedin section
spinistheparticlerotatingtheotherdimensionsthrough
III,wedonotneedmakethisassumptionaslongasthat
5th dimension.
in macrocosm world, the metric of 6-dimensional time
“localized”all3 time dimensions —the collective effects As a conclusion, this paper shows us that we can
of enormous particles inside each objects in macrocosm describe quantum particle fields by using pure geome-
world make all 3 time dimensions “moves” in the same try methods. All we needs are proper metrics and 6-
behavior. dimensional Einstein field equations. The two hypothe-
But why the 6th dimension is also time dimension. ses we used plus modified Kaluza metric are very good
First, it is the considerationof symmetry. Since we have candidatestointerpretquantumphenomena. Themeth-
3-dimensional space, and we also need at least 2 dimen- ods are simple in both logical and mathematical. We
sionaltime,andweneedtotally6-dimensionaltime-space alsodemostratethe potentialstounify gravityandother
to derive all those equations, why not the 6th dimension quantum fields by using just 6-dimensional time-space,
is time, which makes world 3D-Time + 3D-Space ? Sec- sinceallthosefieldscanbederivedbyEinsteinequation.
ond, as one notices that the metric is complex in this Finally, using pure geometry —Einstein equation to de-
paper. That makes the definition of interval ds becomes scribequantumnparticlesisdifferentfromcurrentgauge
complex. The geodesic is complex function too and only fieldtheory,insectionV,wedonotuseallDiracsolutions
relatedto 5th dimension. Actually the metric in original (the 4-solutions), instead each time we only need one of
Kaluza theory could be complex too since A is plane DiracsolutionsInotherwords,themethodsweareusing
α
electromagnetic wave. But how do we understand com- is focusing on interactions between single particles, it is
plex interval? How do we understand complex geodesic particle theory rather than fields theory. We will discuss
? To deriveallthe results inthis paper basedonselecta interactions between different particles in future.
[1] T. Kaluza, Zum Unit¨atsproblem der Physik, Sitz. Preuss. [4] X. Chen , A new interpretation of quan-
Akad.Wiss. Phys.Math. K1(1921) 966. tum theory - Time as hidden variable,
[2] J. M. Overduin, P.S.Wilson, Kaluza-Klein Gravity, http://xxx.lanl.gov/abs/quant-ph/9902037
http://arxiv.org/abs/gr-qc/9805018 [5] O. Klein, Quantentheorie und fu¨nfdimensionale Rela-
[3] D. Bohm, Phys.Rev., 85, 166, (1983), Phys.Rev., 85, tivit¨atstheorie , Zeits. Phys.37 (1926) 895.
180, (1983), Quantum Implications , ed. B. J. Hiley and
F. D. Peat, London:Routledge and Kegan Paul (1987).