A geometrical dual to relativistic Bohmian mechanics - the multi particle case

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by  Benjamin Koch

A geometrical dual to relativistic Bohmian mechanics - the multi particle case
Benjamin Koch
Pontificia Universidad Cat´olica de Chile,
Av. Vicun˜a Mackenna 4860,
Santiago, Chile
(Dated: January 27, 2009)
In this article it is shown that the fundamental equations of relativistic Bohmian mechanics for
multiple bosonic particles have a dual description in terms of a classical theory of curved space-
time. Wefurthergeneralizetheresultstointeractionswithanexternalelectromagneticfield,which
corresponds to theminimally coupled Klein-Gordon equation.
PACSnumbers: 04.62.+v,03.65.Ta
9
0
0 I. INTRODUCTION a summary of the ingredients of the many particle dBB
2
theory will be given. In the third section we will define

n a theory of curved space-time and proof that it is dual
Understanding quantum mechanics as it is usually
a to the many particle dBB theory. In the fourth section
taught is a mayor challenge for many physics students.
J
it is shown, how both theories can be coupled to an ex-
For most of them the problem is not the introduction of
7 ternal electromagnetic field. The fifth section contains
new mathematical tools, but the understanding of the
2
the non-relativistic limit (the Schr¨odinger equation). It
newconceptslikeobservables. Instandardquantumme-
is further discussedwhatthis limit correspondsto in the
] chanics observables and the corresponding uncertainty
c dBB theory and in its geometrical dual. In the sixth
are promoted to a fundamental principle, which can not
q
section issues like non-locality, differences to general rel-
befurtherunderstood. ButitwasshownbyDavidBohm
-
r thatthisdoesnotnecessarilyhavetobethecase[1,2]. In ativity, and the universality of the given matching con-
g ditions are addressed. In the seventh section the results
thedeBroglie-Bohm(dBB)interpretationitisexplained
[
that the uncertainty “principle” and the description by are summarized.

1 means of operators can be understood in terms of un-
v
controllable initial conditions and non-local interactions
6 II. RELATIVISTIC BOHMIAN MECHANICS
between an additional field (the “pilot wave”) and the
0 FOR MANY PARTICLES
measuring apparatus. This theory was further general-
1
4 izedtorelativisticquantummechanicsandquantumfield
Inthissectionweshortlylisttheingredientsforthein-
. theory with bosonic and fermionic fields [3, 4, 5, 6]. It is
1
terpretationofthemanyparticlequantumKlein-Gordon
wellknownthat(due toits non-locality)the dBBtheory
0
equationintermsofBohmiantrajectories. Foradetailed
9 isnotincontradictiontotheBellinequalities[7]. Due to
description of subsequent topics in the dBB theory like
0 its contextuality, the dBB theory is also not affected by
: the Kochen-Specker theorem [8]. particle creation, the theory of quantum measurement,
v
manyparticlestates,andquantumfieldtheorythereader
One mayor drawback of the dBB theory is that the
i
X pilot wave and the corresponding “quantum potential” is referred to [6]. Let 0 be the vacuum and n be an
| i | i
arbitrary n-particle state. The corresponding n-particle
r have to be imposed by hand without further justifica-
a wave function is [4]
tion. In a previous work [9, 10] it was shown that the
relativistic dBB theory for a single particle is dual to a
scalar theory of curved spacetime. In this dual theory ψ(x ; ... ;x )= Ps 0Φˆ(x )...Φˆ(x )n , (1)
1 n 1 n
√n!h | | i
the ominous “pilot wave” can be readily interpreted as
a well known physical quantity, namely a space-time de- where the Φˆ(x ) are scalar Klein-Gordon field operators
pendentconformalfactorofthemetric. Thisworkonthe j
and the symbol denotes symmetrization over all po-
single particle has many features in common with other Ps
sitions x . For free fields, the wave function (1) satisfies
publications on the subject [11, 12, 13, 14, 15]. How- j
the equation
ever, having a duality for the single particle case is not
enough, because the dBB interpretationonly is a consis-
n M2
tent quantum theory when it also has the many particle ∂m∂ +n ψ(x ; ... ;x )=0 . (2)
case. The manyparticletheoryisforexamplecrucialfor  j jm ~2  1 n
j
understanding the quantum uncertainty and for evading X
 
the non existence theorems [7, 8]. Therefore, we will The mass of a single particle is given by M. The index
generalizethe previousresults andpresentadualforthe j indicates which one of the n particles is affected and
relativistic many particle dBB theory. the index m is the typical space-time index in four flat
Thepaperisorganizedafollows: Inthesecondsection dimensions. The use of the Latin indices in contrast to
2
the usual Greek indices will allow in the next section For convenience one might try to choose s as the eigen-
allow a distinction between coordinates in curved space- time of the particle which is finally subject to a mea-
time (Greek) and coordinates in flat space-time (Latin). surement. The non-local nature of the dBB theory be-
Thewavefunction(1)allowsfurtherfortheconstruction comes obvious in the above equations of motion. The
of a conserved current trajectoryoftheparticlej isdeterminedfromthepoten-
tial Q(x ,...,x ), which depends on the positions of all
n 1 n
∂m(iψ∗←→∂ ψ)=0 . (3) other particles of the system.
j jm
Theequations(5-9)arethebuildingblocksofthemany
j
X
particledBBtheory. ThefunctionsP,S,andQthatap-
This is still standard quantum mechanics, in order to pear in those equations depend on the 4 n coordinates
arrive at the dBB interpretation one rewrites the wave xm. It is therefore possible to introduce×a single 4n di-
function ψ = PeiS/~ by introducing a real amplitude mjensional coordinate
P(x ; ... ;x ) and a real phase S(x ; ... ;x ). Doing
1 n 1 n
this, the complex equation (2) splits up into two real xL =(x0,x1,x2,x3; ... ;x0,x1,x2,x3) , (11)
1 1 1 1 n n n n
equations
which has a capital Latin index and contains the space-
n
2MQ (∂mS)(∂ S) nM2 , (4) time positions of all n particles. One further observes
≡ j jm − that in all equations every summation over a particle
j
X index j is accompanied by a summation over the space-
n
0 ∂ P2(∂mS) , (5) time index m of the corresponding particle. This allows
≡ jm j to replace ∂ ∂ and ∂m ∂L. Thus, one can
j jm → L j →
X (cid:0) (cid:1) rewrite the equations for the many particle case (4-9) as
where Q(x ; ... ;x ) is the quantum potential, and
1 n
P(x1; ... ;xn) is the pilot wave. The first equation can 2MQ ≡ (∂LS)(∂LS)−nM2 with (12)
be interpreted as a classical Hamilton-Jacobi equation ~2 ∂L∂ P
L
Q ,
with the additional potential Q and the second equation
≡ 2M P
takes the form of a conservedcurrent. The quantum po-
0 ∂ P2(∂LS) , (13)
tentialin equation(4) is givenfrom~, the particle mass, ≡ L
and the pilot wave by pL Md(cid:0)xL ∂(cid:1)LS , (14)
≡ ds ≡−
n ~2 ∂jm∂jmP d2xL (∂NS)(∂L∂ S)
Q= . (6) N
= with (15)
2M P ds2 M2
j
X
d dxL
This is the only way that the ~ enters into the dBB the- ∂L .
ds ≡ ds
ory. In the dBB interpretation one postulates the exis-
tenceofparticletrajectoriesxm(s)whosemomentumpm
j j
satisfies the relation III. A 4×n DIMENSIONAL THEORY OF
dxm CURVED SPACE-TIME
pm =M j ∂mS . (7)
j ds ≡− j
We will nowshowthat the equationsofthe manypar-
Now one can derive this expression with respect to ds
ticle dBB theory (12-15) have a dual description in a
and use the identity
scalar theory of curved space-time in 4 n dimensions.
×
d n dxm Asgeneralizationoftheprevioussingleparticleapproach
= j ∂ . (8) [10]wewilldefine asetupwherethe momentumofevery
jm
ds ds
j particle is defined in the particles own four dimensional
X
space-time. Such a n-particle theory is therefore defined
This gives the equation of motion for all n relativistic
in a 4 n dimensional space-time. Following the nota-
particles in the dBB interpretation ×
tion in (11) the coordinates in curved space-time will be
d2xm n (∂lS)(∂m∂ S) denoted as
j i j il
= . (9)
ds2 M2 xˆΛ =(xˆ0,xˆ1,xˆ2,xˆ3; ... ;xˆ0,xˆ1,xˆ2,xˆ3) . (16)
i 1 1 1 1 n n n n
X
By using equation (4) this can be further simplified to The theory can be formulated by starting from the n-
particle action
d2xm
M j =∂mQ . (10)
ds2 j [gˆ ] = dαdxˆ4n gˆ
Λ∆ x
S | |
Theinfinitesimalparametersisnotnecessarilytime,be-
causeeverysingleparticlecarriesitsownreferenceframe. α Rˆ+κpˆM s(Rˆ+κˆM) . (17)
L −P L
n o
3
Here,gˆis the determinantofthe metric,α isa Lagrange A. The geometrical dual to the first dBB equation
multiplier, is a symmetrization operator between dif-
s
ferent partiPcles xλ and xλ, Rˆ is the Ricci scalar, ˆ is
i j LM The definition of the metric (20) allowsto reformulate
the matterLagrangian,andκisthe couplingconstantof
the action in terms of the separate functions φ and g
LD
thistheory. Avariationof(17)withrespecttoαensures
[16]. Keepinginmindthesymmetrizationcondition(18)
that only the part of Rˆ +κˆM which is symmetric un- the action (17) reads
der the exchange of two parLticle coordinates xλ and xλ
i j
survives
[φ,g ] = dx4n g (23)
LD
S | |
s Rˆ+κˆM =Rˆ+κˆM . (18) Z2(4n p1)
P L L − (∂Lφ)(∂ φ)+φ2(R+κ ) .
h i 1 2n L LM
This means thatthatthe fields thatarea solutionofthe (cid:20) − (cid:21)
problem have to be symmetric under exchange of two We areprimarilyinterestedinstudying a flatMinkowski
particle coordinates xλ and xλ. It also means that all backgroundspace-timeg =η . Thiscanbeachieved
i j LD LD
the different particles agree on their definition of what by adding an additional Lagrange condition term to the
is the xµ direction. Therefore, if one wants to perform action (17) that demands a vanishing Weyl curvature
j
a coordinate transformation in a single four dimensional
subspace, one has to perform the same transformation 1
C = R (g R g R )
in all other four dimensional subspaces. After having ΣΛΞ∆ ΣΛΞ∆− 2n 1 Σ[Ξ β]Λ− Λ[Ξ ∆]Σ
derivedthe symmetry relation(18) the remaining action 1 −
+ Rg g (24)
is (4n 1)(2n 1) Σ[Ξ ∆]Λ
− −
= 0 .
[gˆ ]= dxˆ4n gˆ Rˆ+κˆ . (19)
Λ∆ M
S | | L
Z p (cid:16) (cid:17) Such a condition also appears in the scalar theories
of curved space-time suggested by Gunnar Nordstro¨m
In order to describe the local conformalpart of this the-
[17, 18, 19]. Like in standard general relativity one fur-
oryseparatelyonesplitsthemetricgˆupintoaconformal
ther imposes that the metric has a vanishing covariant
function φ(x) and a non-conformal part g
derivative
2
gˆ =φ2n−1g . (20)
ΛΓ LG ( g) =0 . (25)
Σ ΛΞ
∇
The index notation for tensors used here is explained in
From the conditions (24, 25) it follows that the metric
table(I)anditallowstowritetheequationsinD =4 n
× gLG has only 1 on the diagonal, while all other entries
dimensionseitherwithoneindex(capitalletters),orwith ±
vanish
two indices (lower case letters). A further distinction
is made between indices that are shifted by the metric
g =η . (26)
LG LG
gˆ (Greek) and indices that are shifted by the metric g
(Latin). In this section we will only use the single index
Thus, g = 1, R =0, and therefore the action (23) sim-
| |
plifies to
Indexnotation Shifted by gˆ Shifted by g
2(4n 1)
Single index XˆΛ;...|{Λ:1...4n} XL;...|{L:1...4n} [φ]= dx4n − (∂Lφ)(∂Lφ)+κφ2 M .
S 1 2n L
Double index Xˆiλ;...|{i:1... n } Xil;...|{i:1... n } Z (cid:20) − (cid:21)(27)
The Euler-Lagrangeequation for this action is
TABLEI: Indexconvention
2(4n 1)∂L∂ φ
L
notation, but all results can be immediately translated − =κ M . (28)
1 2n φ L
into the double index notation used in the previous sec- −
tion. The inverse of the metric (20) is AnExtensionofthe HamiltonJacobimatterLagrangian
ˆ can be defined by subtracting a mass term Mˆ2 for
M
gˆΛΓ =φ−2n2−1gLG . (21) Levery particle on finds
Ibnediidceenstwifiiethda∂ˆΛlowe∂rLG.rFereokmanthdisafloolwloewrsRfoomreaxnaminpdleextchaant LˆM == p(ˆ∂ΛˆΛpˆSΛ−)(n∂ˆMˆSG2 ) nMˆ2 (29)
theadjointderiv≡ativesarenotidentical,inbothnotations H Λ H − G
= φ2n−−21 (∂LS )(∂ S ) nM2
H L H − G
∂ˆΛ =gˆΛΣ∂ˆΣ =φ−2n2−1gLS∂S =φ−2n2−1∂L . (22) = φ2n−−21 (cid:0)M . (cid:1)
L
4
TheHamiltonprinciplefunctionS definesthelocalmo- the covariant derivatives in (35-37), one needs to know
H
mentum pˆΛ = Mˆ dxˆΛ/dsˆ= ∂ˆΛS . Plugging this La- the Levi Civita connection
G H
−
grangian into equation (28) gives 1
ΓΣ = gΣΞ(∂ g +∂ g ∂ g ) (38)
Λ∆ 2 Λ ∆Ξ ∆ ΞΛ− Ξ Λ∆
2(4n 1)∂L∂ φ
κ(1−−2n) φL =(∂LSH)(∂LSH)−nMG2 (30) = 21φ−2n2−1 (∂Lφ2n2−1)δDS +(∂Dφ2n2−1)δLS
Now one can see that this is exactly the first dBB equa- h(∂Sφ2n2−1)ηLD .
−
tion (12) if one identifies
i
With this, the condition (35) reads
φ(x) = P(x) , (31)
ˆ (∂ˆΛS )=φ−2n4n−1∂ φ2(∂LS ) =0 . (39)
SH(x) = S(x) , ∇Λ H L H
2(4n 1) With the matching condition(cid:2)s (31), the(cid:3)above equation
κ = − /~2 ,
is identical to the second Bohmian equation (13).
1 2n
−
M = M2 .
G
D. The geometrical dual to the dBB equation of
Note that the matching conditions demand like in the
motion
single particle case [10] a negative coupling κ.
The total derivative (8) is generalized to
B. The geometrical dual to the third dBB equation
d dxˆΛ dxL d
= ∂ˆ =φ2/(1−2n) ∂ =φ2/(1−2n) .
Λ L
dsˆ dsˆ ds ds
The stress-energy tensor corresponding to the matter (40)
Lagrangian(29) is Applying this to the momentum Mˆ (dxˆΛ/dsˆ)=(∂ˆΛS )
G H
gives the equation of motion
TˆΛ∆ = 2(∂ˆΛS )(∂ˆ∆S )+gˆΛ∆ (∂ˆS )2 nMˆ2 .
− H H H − G d2xˆΛ 1
(cid:16) (cid:17)(32) Mˆ dsˆ2 = Mˆ (∂ˆ∆SH)∂ˆ∆(∂ˆΛSH) , (41)
According to the Hamilton-Jacobi formalism the deriva-
tives of the Hamilton principle function (S ) define the or equivalently the equation of motion in the Minkowski
H
momenta coordinates xL
pˆΛ (∂ˆΛSH) . (33) d2xL = (∂NSH)(∂L∂NSH) . (42)
≡− ds2 M2
Therefore, with the prescription (22) and the match- Using (31) one sees that the equation of motion (42) is
ing condition (31) one sees immediately that the third dual to the equation of motion of relativistic Bohmian
Bohmian equation (14) is fulfilled. mechanics (15). This is however almost a triviality be-
cause the equations (15), (41), and (42) are all derived
from the same mathematical prescription (8). But in
C. The geometrical dual to the second dBB curved space-time there is an other equation of motion
equation
in addition to the equation of motion (41), the geodesic
equation
In orderto find the dual to the secondBohmianequa-
tion one has to exploit that the stress-energytensor (32) d2xˆΛ +ΓˆΛ dxˆ∆dxˆΣ =pˆΛ f(xˆ) . (43)
is covariantly conserved dsˆ2 ∆Σ dsˆ dsˆ ·
Here, f(xˆ) is some arbitrary scalar function. For this
ˆ TˆΛ∆ =0 . (34)
∇Λ equation of motion it is on the contrary not trivial to
show that it is consistent with (15). To proof this, one
This is true if the following relations are fulfilled
can proceed a follows: First one uses on the left hand
ˆ (∂ˆΛS ) = 0 , (35) side of (43) the equations (41) and (38). By using the
∇Λ H relations (35) and (37) one can show that the result is
(∂ˆΛSH)ˆ∆(∂ˆΛSH) = 0 , (36) somescalarfunction times (∂ˆΛS ). This proofsthatthe
∇ H
(∂ˆΛS )ˆ (∂ˆ∆S ) = 0 . (37) geodesic equation (43) is consistent with equation (41).
H Λ H
∇
Since it was already shown that (41) is dual to (15), one
In addition to the covariant conservation of momentum can conclude that also (43) is dual to (15).
(35)andtheconservationofsquaredmomentum(36)the Thus, all the equations of the dBB theory (12-15) have
tensor nature of (32) also demands (37), which will be a dual description (30, 33, 39, and 43) in the presented
important in the next subsection. In order to calculate theory of higher dimensional curved space-time.
5
IV. INTERACTION WITH AN EXTERNAL dinates (11) and the interacting equations (45-48) read
FIELD
2MQ (∂LS+eAL)(∂ S+eA ) nM2 (49)
L L
≡ −
0 ∂ P2(∂LS+eAL) , (50)
Now, the results of the previous sections will be gen- ≡ L
eralizedto interactions with an external electromagnetic pL Md(cid:0)xL (∂LS+e(cid:1)AL) , (51)
field. ≡ ds ≡−
d2xL
M = ∂LQ+eπ FLK . (52)
ds2 K
A. An external field in the dBB theory
B. An external field in the 4×n dimensional
theory of curved space-time
Couplingnbosonicparticles(2)withchargeetoanex-
ternal electromagnetic field A is achieved by replacing
m
In the interacting case,the classical4 n dimensional
the partial derivative with a gauge covariant derivative
×
∂ ∂ + ieA /~ in the Klein-Gordon Lagrangian. theory of curved space-time is analogous to the discus-
m m m
→ sion in section III. The only difference appears in the
The resulting equation of motion is
definition of the canonical momentum
n " ∂jm∂jm+ M~2 − e2~A22j!ψ+ i∂jm(ψ~2Ajm)#=0, πˆΛ =MˆGddxˆsˆΛ =−(∂ˆΛSH +eAˆΛ) , (53)
j
X (44) insteadofthefreemomentumpˆΛ. Withthisreplacement
whereAm is the electromagneticpotentialAm evaluated the equations (30,39,and 33) transform to
j
at the position of particle j. By again rewriting the n-
2(4n 1)∂L∂ φ
particle wave function ψ = Pexp(iS) the two equations − L (∂LS +eAL)(∂ S +eA )
H L H L
(4, 5) generalize to κ(1 2n) φ ≡
−
nM2 , (54)
− G
2MQ ≡ n (∂jmS+eAmj )(∂jmS+eAjm)−nM2,(45) 0 ≡ φ1−4n2n∂L P2(∂LSH +eAL) (5.5)
Xj One immediately sees that wit(cid:0)h the identification(cid:1)s (31)
n the above equations (53-55) are dual to the equations
0 ∂ P2(∂mS+eAm) , (46)
≡ jm j j (49-51) In order to check the equation ofmotion (52) we
Xj (cid:0) (cid:1) use (40) and find
where the quantum potential Q in the first equation is d2xˆΛ =φ1−42n d2xL +πˆL (...) . (56)
given by (6). In the presence of an external force, the dsˆ2 ds2 ·
dBB definition for the particlesmomentum (7) now con-
Using this and the identity (37) in its canonical version
tains the canonicalmomentumπm insteadofthe normal
momentum pm j πˆΛˆΛπˆ∆ =0 one can verify that the geodesic equation
j ∇
d2xˆΛ πˆ∆ πˆΣ
πm =Mdxmj (∂mS+eAm) . (47) dsˆ2 +ΓˆΛ∆ΣMGMG =πˆΛ·(...) , (57)
j dτ ≡− j j
is consistent with (52).
Thus, using (8, 45, 47, and the relation ∂ Am =
kn j
δ ∂ Am) one finds the equationof motion for all n par-
kj n j V. THE NON-RELATIVISTIC LIMIT
ticles in an external field Am
The dBB theory was originally developed for non rel-
d2xm
M j =∂mQ+eπ Fmn . (48) ativistic quantum mechanics. It will now be shown how
ds2 j jn thislimitcanbeobtainedfromtheinteractingrelativistic
version and its dual. Since in this limit spatial and tem-
HδjekrFe,mtnhewfiiethldFstmrenng=th∂tmenAsonris∂FnjmkAnm=. ∂OjmnAtnkh−e∂RknAHmjS o=f ptoorraeltudrenritvoattihveestwaroe-intrdeeaxtendotdaitffieorne.ntNlyevietrtisheilnesstsrutrcatnivse-
−
equation (48) appear two terms. The first term is the lation to the one-index notation is always possible with
quantum potential also present in equation (10) and the the help the index convention (I). Starting from the rel-
second term is the Lorentz force which is familiar from ativistic and interacting many particle equations (45-47)
classical electrodynamics. the non-relativistic limit can be obtained from a num-
Now on can apply again the formal rewriting of coor- ber of low energy approximations. At first, one wants to
6
achievethatthequantumphasedependsonlyononesin- a linear part containing the mass and a remaining part
gle time coordinate (whichwill be denoted t ) insteadof S˜ (58). Second, all the time coordinatest in the func-
1 H j
thentimecoordinatestj. Therefore,thequantumphase tions S˜H, φ, and A are synchronized (58-60) and the
S (or equivalently the Hamilton principal function SH) Coulomb gauge is chosen. Third, the time components
is redefined by oftheremainingfourvectorsaretakentothelimitwhere
they are much smaller than the mass M (63).
S(t ,~x ) Mt +S˜(t ,~x ) . (58) G
j j j 1 j In this limit one finds in the geometricaltheory that the
≡−
function φ plays a double role. On the one hand it is a
For the vector potential Am one chooses the Coulomb
j conformalfieldofthemetricwhichinteractswithmatter.
gauge andimposes a similar condition for the time coor-
On the other hand its square φ2(t,~x ) gives the proba-
dinates and indices j
bility density of finding the particles of the system in
A0(t ,~x )=A0(t ,~x ) andAi =0 . (59) the positions ~xj at the time t. When one intends to find
j k k 1 k j
normalizablesolutions(forinstanceofacentralpotential
For the pilot wave,the conditionfor the time coordinate A0 1/r) usually one imposes the boundary condition
∼
is analogously lim~x→∞φ2 = 0. This limit means that in the dual the-
ory the metric gˆ as to vanish asymptotically for far
Λ∆
P(tj,~xj) P(t1,~xj) . (60) regions.
≡
Now the equations (45, 46) read
VI. DISCUSSIONS
Q(P(t ,~x )) = S˜˙ + S˜˙2 eA + e2A20 (61)
1 j 0
− 2M − 2M In this section some interpretational issues are ad-
n (∂~ S˜)2 dressed. This aims to develop a physical understanding
j
,
− 2M of the presented duality.
j=1
X
eA S˜˙
0 = (∂0P2) 1 0 (62) A. Locality
− M − M!
eA˙ +S¨˜ n ∂~ S˜ QuantummechanicsinthedBBinterpretationisathe-
P2 0 + ∂~ P2 j ,
j ory which allows to talk about particle position at the
− M M !
Xj=1 cost of non-local interactions. The non-locality becomes
obvious by looking at the equation of motion for a free
where derivatives with respect to t are denoted with a
1
particle in the dBB theory (10). The movement of this
dot. The non-relativistic limit consists now in assuming
“free”particle is not free but it is governedby the quan-
that the mass M is a much larger quantity than all the
tum potential Q, which simultaneously depends on the
time components of the four vectors
positions of all the other particles of the system. It is
M S˜˙, M eA , M ∂0P . (63) often argued that such a non-locality can not have any
≫ ≫ 0 ≫ P dual in a (by construction) local theory such as a geo-
metricaltheory of curved space-time. Therefore the first
In this limit the above equations give question is:
“Howcananon-localtheoryhaveaduallocaltheory?”
0 = S˜˙ +eA + n (∂~jS˜)2 ~2 ∂~j2P , (64) The reason is that in the presented theory of curved
0
2M − 2PM! space-time, every single particle is living in its own four
j=1
X dimensionalspace-time. Therefore,thepositionsofndif-
0 = ∂0(P2)+ n ∂~j P2∂~MjS˜ . (65) fdeirmenentspiaorntaicllsepsaccoer-rteismpeo.ndIntosoomneessienngsleepthoeinitnitnrotdhuec4t×ionn
!
j=1
X of additional dimensions seems to be a common feature
By replacing ψ = PeiS˜/~ one sees that this is the many ofaclassicalunderstanding ofquantumtheories(see the
AdS/CFT dualities or the many worlds interpretation).
particleSchr¨odingerequationwiththepotentialeA . Fi-
0 This higher dimensional construction helps around the
nally, with the same relations (59,63) the equation (47)
non-locality argument but it brings a new problem:
gives the original momentum definition in the Bohmian
interpretation
p~ =∂~ S˜ . (66) B. Four dimensional space-time
j j
Due to the matching conditions (31), in the geomet- “How does the 4 n dimensionaltoy model cope with
×
rical dual the non-relativistic limit means the following: thefactthateveryacceptedphysicaltheoryisformulated
First,theHamiltonprinciplefunctionS issplitupinto in four dimensions?”
H
7
We don’t know the answer to this question yet. There- themassM . Fixingthoserelationsforcesarelationbe-
G
fore, it is up to now only prooven that the presented tween the particle number n, the Planck constant ~ and
duality exists mathematically, but a better physical un- the coupling of the geometrical theory κ
derstanding is certainly desirable. Part of the answer
might lie in the symmetrization condition in the ac-
tion (17), which corresponds to the symmPestrization in κ= 2(4n−1)/~2 .
1 2n
bosonic many particle quantum mechanics. Since all so- −
lutions have to be invariant under the exchange of two
sub-coordinates xµ and xµ it is clear that all those sub- This has the consequence that if one gives ~ a fixed nu-
i j
coordinates xµ have to have the same interpretation. mericalvalue,thenthecouplingofthegeometricaltheory
i κ runs from 6/~2 to 4/~2, depending on the number
This symmetry forbids for instance to rotate the x and
− −
of particles (n = 1,..., ) which are part of the sys-
y coordinates for a single particle without rotating the
∞
tem. Reversely demanding a fixed geometrical coupling
coordinates for the other particles. In this sense the in-
would result in a running Planck constant ~. This be-
terpretationofcommonsensecoordinatesin4dimensions
havior seems to be the only unexpected (and undesired)
is unique.
peculiarityofthechosenmatchingconditions(31). More
optimistically, one might also see the scale dependence
C. Gravity of the coupling κ as a feature of the toy model which
it shares with effective quantum field theories and even
some approaches to gravity [20].
“How is this geometrical theory related to THE geo-
VII. SUMMARY
metricaltheory - generalrelativity?” It is very tempting
to think about such a relation because the action (17) is
verysimilartotheEinsteinHilbertactionandthemetric In this paper we showed that the equations of the
condition (25) is the same as in general relativity. How- free relativistic dBB theory for many particles (12-15)
everthepresentedtheoryinthishigherdimensionalform have a dual description in a 4 n dimensional theory
canNOTdescribefourdimensionalgravity. Thepossible of curved space-time. For th×e translation between
connectionbetweenbothgeometricaltheoriesstillhasto the two theories a single set of matching conditions
be explored. was defined (31). The result was then generalized to
interactions with an external electromagnetic field.
Before discussing interpretationalissues, the limit of the
D. The matching conditions non-relativistic Schr¨odinger equation is derived. The
important question whether such dualities can also be
Thematchingconditions(31)arenotunique,butthey foundfor,self-interactingtheories,fermions,orquantum
were chosen in order to have the most direct connection field theory will be subject of future studies.
between three functions. Those are the quantum phase
S connected to the Hamilton principle function S , the The author wants to thank M. A. Diaz, J.M. Isidro,
H
quantum amplitude P connected to the conformal met- H. Nikolic, D. Rodriguez, and C. Valenzuela, for very
ric function φ, and the quantum mass M connected to helpful comments and discussions.
[1] D.Bohm, Phys.Rev. 85, 166 (1951). [8] S. Kochen and E. P. Specker, Journal of Mathematics
[2] D.Bohm, Phys.Rev. 85, 180 (1951). and Mechanics 17, 59 (1976).
[3] P. R. Holland and J. P. Vigier, Nuovo Cim. B 88, 20 [9] B. Koch, arXiv:0801.4635 [quant-ph].
(1985); P. R. Holland, Phys. Lett. A 128, 9 (1988); [10] B. Koch, arXiv:0810.2786 [hep-th].
P.R. Holland, Phys. Rept. 224, 95 (1993). [11] E. Santamato, J. Math. Phys. 25, 2477 (1984); E. San-
[4] H. Nikolic, Found. Phys. Lett. 17, 363 (2004) tamato, Phys. Rev.D 32, 2615 (1985).
[arXiv:quant-ph/0208185]; H. Nikolic, Found. Phys. [12] F. Shojai and A. Shojai, Int. J. Mod. Phys. A 15,
Lett. 18, 123 (2005) [arXiv:quant-ph/0302152]; 1859 (2000) [arXiv:gr-qc/0010012]; A. Shojai, Int. J.
H. Nikolic, Found. Phys. Lett. 18, 549 (2005) Mod. Phys. A 15, 1757 (2000) [arXiv:gr-qc/0010013];
[arXiv:quant-ph/0406173]. F. Shojai and A. Shojai, Pramana 58, 13 (2002)
[5] W.Struyveand H. Westman,AIPConf. Proc. 844, 321 [arXiv:gr-qc/0109052]; F. Shojai and A. Shojai,
(2006) [arXiv:quant-ph/0602229]. arXiv:gr-qc/0404102.
[6] H. Nikolic, Found. Phys. 37, 1563 (2007) [13] R. Carroll, arXiv:gr-qc/0406004; R. Car-
[arXiv:quant-ph/0609163]. roll, arXiv:quant-ph/0406203; R. Carroll,
[7] J. S. Bell, Speakable and Unspeakable in Quantum Me- arXiv:math-ph/0701007; R. Carroll, arXiv:0705.3921
chanics (CambridgeUniversityPress;ISBN0521368693, [gr-qc].
1988). [14] R. Carroll, arXiv:math-ph/0703065; M. Novello,
8
J. M. Salim and F. T. Falciano, arXiv:0901.3741 [gr-qc]. Phys. Lett. B 479, 411 (2000) [arXiv:gr-qc/9907081].
[15] J. M. Isidro, J. L. G. Santander and P. F. de Cordoba, [17] G. Nordstrom, Ann.d.Phys. 40, 872 (1913).
arXiv:0808.2351[hep-th];S.Abraham,P.F.deCordoba, [18] G. Nordstrom, Ann.d.Phys. 42, 533 (1913).
J. M. Isidro and J. L. G. Santander, arXiv:0810.2236 [19] F. Ravndal,arXiv:gr-qc/0405030.
[hep-th]. S. Abraham, P. F. de Cordoba, J. M. Isidro [20] M. Reuter and F. Saueressig, Phys. Rev. D 65, 065016
and J. L. G. Santander, arXiv:0810.2356 [hep-th]. (2002) [arXiv:hep-th/0110054].
[16] J. A. Helayel-Neto, A. Penna-Firme and I. L. Shapiro,