Equations of motion in Double Field Theory: from classical particles to quantum cosmology

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by  Nahomi Kan

N. Kan 1
Equations of motion in Double Field Theory: from classical
particles to quantum cosmology
Nahomi Kan1(a), Koichiro Kobayashi2(b), and Kiyoshi Shiraishi3(b)
(a)Yamaguchi Junior College, Hofu-shi, Yamaguchi 747–1232, Japan
(b)Yamaguchi University, Yamaguchi-shi, Yamaguchi 753–8512, Japan
Abstract
The equation of motion for a point particle in the background field of double field
2
theory is considered. We find that the motion is described by a geodesic flow in the
1
doubledgeometry. Inspiredbyanalysisontheparticlemotion,weproposeamodified
0
2 model of quantum string cosmology, which includes two scale factors. The report is
based on Phys. Rev. D84 (2011) 124049 [arXiv:1108.5795].
n
a
J 1 Introduction

9
2 Double Field Theory (DFT) [1] is the theory of the massless field with a higher symmetry of spacetime
including dual coordinates. Through this theory, Hull, Zwiebach, and Hohm clarified the T-duality
]
h symmetry of the massless field, and new symmetry related to the theory. More recently Jeon, Lee, and
t Park studied the structure of DFT using projection-compatible differential geometrical methods in [2].
-
p Thepresentreportconsistsoftwoparts. Inthefirstpart,themotionoftheparticleinthebackground
e fieldinDFTisinvestigated. Weshowthatthegeodesicinthe2D-dimensionaldoubled-spacetimecannot
h
be the geodesic in the D-dimensional spacetime. The geodesic equation in the D-dimensional spacetime
[
is found to be the geodesic flow equation. In the second part, we consider the string cosmology with

1 a bimetric model inspired by the constraint method discussed in the first part. Our method for the
v
restriction on the metrics functions well, at least in the present reduced model for cosmology.
3
2
0 2 Review of projection-compatible approach
6
.
1 CoordinatesarecombinedwithdualcoordinatestobeXA =(x˜ , xµ)T,wherethesuffixesA,B,...range
0 a
over 1,2,...,2D, while µ,ν,... as well as a,b,... range over 1,2,...,D. The constantmetric is assumed
2
1 to be expressed as the following 2D 2D matrix,
×
:
v 0 δa
Xi ηAB = δµb 0ν . (1)
(cid:18) (cid:19)
r
a Thesuffixesareentirelyraisedandloweredbythisconstantmetric. Ofcourse,ηACη =δA issatisfied.
CB B
The generalized metric is defined as follows:
gab gaσb
HAB = bµσgσb gµν− bµρgσρνσbσν . (2)
(cid:18) − (cid:19)
Here, g and b are the metric in D dimensions and the antisymmetric tensor, respectively. It should
µν µν
be noted that AB satisfes AC =δA .
CB B
H H H
The following projection matrices are defined on the basis of the existence of two kinds of metrics.
P 1(η+ ), P¯ 1(η ) whichsatisfy P2 =P, P¯2 =P¯, PP¯ =P¯P =0. Fromthese, one can derive
the≡id2entitiHes P(∂≡P2)P −=HP¯(∂ P¯)P¯ =0 or P B(∂ )PC =P¯ B(∂ )P¯C =0.
A A D A BC E D A BC E
H H
Now,theprojection-compatiblederivativeisdefined. Inotherwords,boththemetricsare“covariantly
constant,” i.e., η = = 0. So, the covariant derivative of the projection of an arbitrary
A BC A BC
∇ ∇ H
1Emailaddress: [email protected]
2Emailaddress: [email protected]
3Emailaddress: [email protected]
2 Double Field Theory
tensor coincides with the projection of the covariant derivative of the tensor. Jeon et al.[2] found that
the covariant derivatives including the following connection have the character,
Γ 2P DP¯ E∂ P +2(P¯ DP¯ E P DP E)∂ P . (3)
ABC [A B] C DE [A B] [A B] D EC
≡ −
They also obtained the action for the generalized metric, which was previously found by Hohm et al. [1]
1 1
S = dxdx˜e 2d AB∂ CD∂ AB∂ CD∂ 2∂ d∂ AB +4 AB∂ d∂ d ,
− A B CD B D AC A B A B
8H H H − 2H H H − H H
Z (cid:18) (cid:19)
from the consideration of the projection-compatible geometrical quantities. Here, e 2d =√ ge 2φ and
− −
−
φ is the dilaton field. If we set all the derivatives on the fields with respect to the dual coordinate zero
(∂˜a =0), the action for the effective theory for the zero-mode field in string theory is obtained as
1
S = dx√ ge 2φ R+4(∂φ)2 H2 ,
−
− − 12
Z (cid:20) (cid:21)
where the three-form field H =db is the field strength of the Kalb-Ramond 2-form b .
ij
3 The geodesic equation is not the equation of motion for a
particle
Next, we consider the equation of motion for a particle. The geodesic equation is given by the following
expression Uµ Uν =0, where Uµ dxµ =x˙µ, s being a parameter.
∇µ ≡ ds
The corresponding equation in the projection compatible geometry of Jeon et al. is considered to be
UA UB =UA(∂ UB+Γ B UC)=0, (4)
A A A C
∇
whereUA =(U˜ ,Uµ)T = dXA. FromtheprojectspaceansatzP¯U =0,weareforcedtouseU˜ =g Uν.
a ds a aν
Moreover,we set ∂˜g =0 as in the interpretation of DFT. Now, we find that the above equation reads
1
Uµ∂ Uν + gνµ(∂ g +∂ g )UρUσ =0. (5)
µ ρ µσ σ µρ
2
It is obviously different from the usual geodesic equation in general relativity (or differential geometry).
In general, it is understood that the usage of the projection has a problem.
4 Projection and geodesic flow
The following Lagrangianis adopted, and the mechanics derived from it are considered:
1
L= X˙AX˙B+λAP¯ X˙B. (6)
AB AB
2H
Here, λA is an undecided multiplier. The Euler-Lagrangeequation leads to the constraint P¯X˙ =0.
We find that the Hamiltonian is defined as
1
H = AB(p λCP¯ )(p λDP¯ ). (7)
A CA B DB
2H − −
The multiplier can be determined from the Hamilton equation as λ = p +P MB, where MB is an
A A AB
arbitrary vector. When this is substituted into the above Hamiltonian, we obtain a new Hamiltonian
1
H = PABp p . (8)
⋆ A B
2
Using the new Hamiltonian, we obtain
∂H ∂H 1 1
X˙A = ⋆ =PABp , p˙ = ⋆ = ∂ PBCp p = ∂ BCp p . (9)
∂p B A −∂XA −2 A B C −4 AH B C
A
N. Kan 3
These equations describe the geodesic flow in the system. The combined equation is found to be
1
X¨A =X˙C(∂ PAB)p PAB∂ PCDp p . (10)
C B B C D
− 2
Now, let us take the condition ∂˜a = 0 for the correspondence with DFT. If we consider p˜a = 0, we
obtainx¨µ+ µ x˙ρx˙σ =0,the geodesicequationinausualD dimensionalspacetime. We haveobtained
ρσ
thegeodesicequationintheD-dimensionalspacetimefromthegeodesicflowinthe2D-dimensionalspace
(cid:8) (cid:9)
described by the generalized metric with natural assumptions.
5 A simple bi-metric model
We apply a similar method to a modified model for cosmology, which is related to the string cosmology
[3]. Inthe modelhere,we considertwometrics, g andg˜. Thoughourmodeldescribesa bi-metrictheory,
the degree of freedom is to be mildly restricted. For simplicity of the discussion, we consider b = 0.
µν
The cosmologicalaction we consider is
S =−λ2s dτ 18Tr(M′ηM′η)+Φ′2+e−2ΦV , with MAB ≡ G1˜ G1 , (11)
Z (cid:20) (cid:21) (cid:18) (cid:19)
where G and G˜ are the spatial parts of the metrics and Φ 2d. Here, we add the constant potential V
≡
totheLagrangianandλ istheconstantthatrepresentsthe scaleofstringtheory[3]. Theprimedenotes
s
differentiation with respect to τ.
We now define the “pseudo”-projection matrices P = η+M, P¯ = η−M and we wish to enforce
2 2
PM P =P¯M P¯ =0 using some constraints. Now, the LagrangianL with the constraint term is
′ ′ Λ
λ 1
LΛ = s M′ABM′AB +Λ¯ABP¯ACM′CDP¯DB+ΛABPACM′CDPDB Φ′2 e−2ΦV . (12)
2 −8 − −
(cid:20) (cid:21)
The Hamiltonian of the system becomes
2
4 λ 1 λ
H = ΠAB s P¯ACΛ¯ P¯DB+PACΛ PDB Π2 + se 2ΦV , (13)
Λ −λ − 2 CD CD − 2λ Φ 2 −
s (cid:20) (cid:21) s
(cid:0) (cid:1)
where the conjugate momentum for M and Φ are represented by ΠAB and Π , respectively. We
AB Φ
consider simplification by using the assumed relation, P2 P and P¯2 P¯. The symbol is used to
≃ ≃ ≃
indicate this assumed approximationadopted by us. Finally, we obtain the Hamiltonian
8 1 λ
H ΠABP ΠCDP¯ Π2 + se 2ΦV . (14)
∗ ≡−λs BC DA− 2λs Φ 2 −
6 “Minisuperspace” version of the bi-metric model
Next, we examine the previous procedure of modification in the minisuperspace model. We suppose
A˜(τ)δab 0
M = . (15)
AB 0 A(τ)δ
µν
(cid:18) (cid:19)
The Hamiltonian for the minisuperspace version of our modified model is found to be
2 1 λ
H = (ππ˜+π˜π AπAπ A˜π˜A˜π˜) Π2 + se 2ΦV . (16)
∗ −λsD − − − 2λs Φ 2 −
A special solution can be found for the Hamilton equations. The solution is
1 2 2
A˜(τ)= exp C(τ τ ) , A(τ)=A exp C(τ τ ) +δ, (17)
0 0 0
A0 (cid:20)−√D − (cid:21) (cid:20)√D − (cid:21)
whereA ,τ ,andδ areconstants. Thesimilaritytotheknownstringcosmologicalsolution[3]isobvious,
0 0
up to the possible constant deviation δ in A. For the solution, we find that AA˜ 1 when τ + .
→ → ∞
4 Double Field Theory
7 Quantum cosmology of the bi-metric model
Quantum cosmologicaltreatment of the string cosmology has been widely studied [3]. In our model, the
minisuperspace Wheeler-DeWitt equation is obtained as
2 ∂ ∂ ∂ ∂ ∂ ∂ 1 ∂2 λ
2 A A A˜ A˜ + + se 2ΦV Ψ=0, (18)
(cid:20)λsD (cid:18) ∂A∂A˜ − ∂A ∂A − ∂A˜ ∂A˜(cid:19) 2λs∂Φ2 2 − (cid:21)
where Ψ is the wave function of the universe. To simplify the description of the system, we use the
following variables: x= √D lnA/A˜, y = √D lnAA˜. Up to the ordering, we have
4 4
1+e (4/√D)y ∂2 ∂ 1 e (4/√D)y ∂ ∂2
"− −2 ∂x2 − ∂y − −2 ∂y + ∂Φ2 +λ2se−2ΦV#Ψ=0. (19)
If we assume a solution of the form, Ψ(x,y,Φ) = X (x)Y (y)Z (Φ), we find the non singular real
k kK K
solution for Y (y) at y =0 as follows:
kK
1+√1 k2+√k2 2K2 1 √1 k2+√k2 2K2
YkK(y)=e−√k2−2K2yF − − , − − − ,1;1 e−2y , (20)
2 2 − !
where F(α,β,γ;z) is the Gauss’ hypergeometric function.
IfK = k,Y (y)hasamaximumaty =0(seeFigure1). Whenweconstructawavepacketforthe
k k
± ±
cosmologicalwave function [3], the peak of this wave packetin terms of parameter y is naturally located
at y =0. Thus, the approximate scale factor duality AA˜ 1 is expected even at the “beginning” of the
≃
quantum universe. The detailed investigation on the behavior of the universe is left for future research.
Figure 1: 3D-plot of Y (y).
kk
References
[1] C. Hull and B. Zwiebach, JHEP 0909 (2009) 099; JHEP 0909 (2009) 090.
[2] I. Jeon, K. Lee and J.-H. Park, JHEP 1104 (2011) 014.
[3] For a review, M. Gasperini and G. Veneziano, Phys. Rep. 373 (2003) 1.