AdS String: Classical Solutions and Moduli Dynamics

0.16 MB·English

by  Antal Jevicki

AdS String: Classical Solutions
and Moduli Dynamics
Antal Jevicki 1, Kewang Jin
Department of Physics
Brown University, Box 1843
0
Providence, RI 02912, USA
1
0
2

n
a
J 1 Introduction

9
2
In what follows we present a review of recently developed techniques for generating

]
classical solutions of strings in Anti-de-Sitter space and for studying their classical
h
t dynamics. The initial arguments for the relevance of semiclassical calculations in
-
p
AdS spacetime were given by Gubser, Klebanov and Polyakov [1] who described the
e
h rotating, folded strings and the associated Yang-Mills operators. A generalization
[
to the N-spike configuration was given by Kruczenski [2]. In a rotating frame, the

1 original configuration of [2] was static, as such it corresponds to a ground state of a
v
moregeneral dynamical solution. Its study andfull specification isof definite interest.
1
0 The spike configurations are associated with Minkowski worldsheets, they pro-
3
vide information about the semiclassical spectrum of the AdS string. The recently
5
. developed Wilson loop/Scattering amplitude duality requires classical solutions with
1
0 Euclidean worldsheets which enter the construction of minimal area surfaces with
0 polygonal boundary conditions [3]. The reduction technique that we have devel-
1
: oped for the AdS problem is seen to apply both to Minkowski and Euclidean cases
v
[4, 5, 6, 7]. It is based on a reduction originally introduced by Pohlmeyer [8] for non-
i
X
linear sigma models and applied by de Vega and Sanchez [9] to strings in de-Sitter
r
a spacetime. The reduction to the invariant fields provides a bridge to integrable equa-
tions of sinh-Gordon and more generally Toda type. These equations have soliton
type solutions and can be studied through the inverse scattering method [10].
One can see in [4, 11] that a soliton configuration with a singularity at its location
translates into a spike at the string level. This connection turns out to be most
fruitful for the construction of dynamical multi-spike solutions. We summarize the
construction given in [6] where a string solution associated with the most general
N-soliton solution was given. We describe here in some detail the particular cases
1Based on talks given at “Tenth workshop on non-perturbative Quantum Chromodynamics”,
l’Institut Astrophysique de Paris, France, June 8-12, 2009; “Summer school on AdS/CFT corre-
spondence and its applications”, Tihany, Hungary, August 24-28, 2009.
1
corresponding to one and two solitons and evaluate the energies. These provide
information on the fluctuation spectrum of the AdS string and more generally its
moduli dynamics.
It is useful to draw a parallel with an analogous construction found for the case
of strings moving on R Sn. In the n = 2 case one had the association (due to
×
Hofman and Maldacena [12]) between solitons of sine-Gordon and magnons of the
string. In [13] the moduli space description was presented, given by the N-body
problem of Calogero (or Ruijsenaars-Schneider type [14]). Our present work implies
an analogous description in the case of AdS .
3
Regarding Euclidean worldsheets, the Pohlmeyer reduction turned out to be very
useful for construction of minimal surfaces relevant to the Alday-Maldacena construc-
tion [15, 16]. We review here a simple configuration based on the sinh-Gordon soliton
which translates into a solution originally constructed by Berkovits and Maldacena
[17] representing a cusp near the boundary of AdS space, which can be simulated by a
Higgs field cutoff in Yang-Mills theory [18]. For the problem of constructing minimal
surfaces in AdS it turns out that one needs not soliton but instanton type solutions
of sinh-Gordon and more generally Toda equations. Contrary to solitons no exact
techniques exist for the instanton case. We describe an approximate series method
for construction of such solutions [7].
The content of this review is as follows. In section two, we review the Pohlmeyer
reduction of classical strings in AdS. In section three, we present the most general
N-spike solution in AdS associated with N solitons of sinh-Gordon. In the concrete
3
examples of one and two solitons, we exhibit explicitly the corresponding one and two
spikedynamicalsolutions. Fortheseweshowtherelationshipbetween solitonenergies
and the corresponding string energies and momenta. In section four, we discuss an
interesting correspondence between solitons and cusps with Euclidean worldsheets.
Finally, we give a series solution to the B Toda equations which are relevant to the
2
Pohlmeyer reduction in AdS .
4
2 Pohlmeyer reduction
In general, sting equations in AdS spacetime (in conformal gauge) are described by
d
the non-compact nonlinear sigma model on SO(d 1,2). Defining the AdS space as
d
−
Y2 = Y2 Y2 +Y2 + +Y2 = 1, the action reads
− −1 − 0 1 ··· d−1 −
√λ
S = dτdσ ∂Y ∂Y +Λ(σ,τ)(Y Y +1) , (1)
4π · ·
Z (cid:16) (cid:17)
where τ,σ are the Minkowski worldsheet coordinates, the equations of motion are
∂∂Y (∂Y ∂Y)Y = 0, (2)
− ·
2
with z = (σ τ)/2, z = (σ + τ)/2 and ∂ = ∂ ∂ , ∂ = ∂ + ∂ . In addition to
σ τ σ τ
− −
guarantee the conformal gauge we have to impose the Virasoro conditions
∂Y ∂Y = ∂Y ∂Y = 0. (3)
· ·
It was demonstrated a number of years ago (by Pohlmeyer [8]) that nonlinear
sigma models subject to Virasoro type constraints can be reduced to integrable field
equations of sine-Gordon (or Toda) type. This reduction is accomplished by concen-
trating on SO(d 1,2) invariant sub-dynamics of the sigma model. The steps of the
−
reduction were well described in [9, 4] and consist in the following. One starts by
identifying first an appropriate set of basis vectors for the string coordinates
e = (Y,∂Y,∂Y,B , ,B ), i = 1,2, ,d+1, (4)
i 4 d+1
··· ···
where B form an orthonormal set B B = δ ,B Y = B ∂Y = B ∂Y = 0.
i i j ij i i i
· · · ·
Defining the scalar field α and two sets of auxiliary fields
α(z,z) ln[∂Y ∂Y], (5)
≡ ·
2
u B ∂ Y, (6)
i i
≡ ·
v B ∂2Y, (7)
i i
≡ ·
one can derive the equations of motion
d+1
∂∂α eα e α u v = 0, (8)
− i i
− −
i=4
X
∂u = (B ∂B )u , (9)
i j i j
·
j=i
X6
∂v = (B ∂B )v . (10)
i j i j
·
j=i
X6
In the case of AdS , there is only one vector B so that ∂u = 0, ∂v = 0. Therefore,
3 4
the equation of motion for the scalar field α is simplified to be
∂∂α eα e αu(z)v(z) = 0. (11)
−
− −
This is the generalized sinh-Gordon equation with two (anti)-holomorphic functions
u(z) and v(z). In order to get the standard sinh-Gordon equation, we first shift the
field
α(z,z) = αˆ(z,z)+ln u(z)v(z), (12)
−
and then do a (conformal) change of variablesp
dz = 2u(z)dz, dz = 2v(z)dz, (13)
′ ′
−
such that the equation of moption for αˆ satisfies p
∂′∂′αˆ(z′,z′) sinhαˆ(z′,z′) = 0. (14)
−
3
3 N-spike solution
We had the fact that the classical strings in AdS can be reduced to a generalized
3
sinh-Gordon model coupled to two arbitrary functions u(z) and v(z) which together
represent a free scalar field. These functions are central to the string theory interpre-
tation of the sinh-Gordon equation, they represent the freedom of performing general
conformal transformations which are the symmetry of the conformal gauge string.
The general solution to the sinh-Gordon equation (14) with N solitons is well
known, it can be obtained by the inverse scattering method [6, 10] and written in the
form
4ζ ∂(ϕ +ϕ )
1 2
αˆ(z ,z ) = ln , (15)
′ ′
i ϕ ϕ
1 2
h − i
where ζ is the spectral parameter and the components of wavefunction ϕ are
N
λ
ϕ1(ζ,z′,z′) = − ζ +jζ (1−A)−jl1λl eiζz′−iz′/4ζ, (16)
j
(cid:16)jX,l=1 (cid:17)
N
λ λ λ
ϕ2(ζ,z′,z′) = 1+ ζ +jζ ζ +j lζ (1−A)−lk1λk eiζz′−iz′/4ζ, (17)
j j l
(cid:16) j,Xl,k=1 (cid:17)
with the definitions
λ λ
Aij = ailalj, ail = i l , λk = ck(0)eiζkz′−iz′/4ζk. (18)
ζ +ζ
i l
l
X p
Herec (0)andζ aretwosetsofconstantsrelatedtotheinitialpositionsandmomenta
k k
of N solitons.
The inverse scattering method also gives us a procedure to generate string solu-
tions from the soliton solutions of the sinh-Gordon equation. Essentially the sinh-
Gordon configuration serves as a (time-dependent) potential in the Lax scattering
equations with scattering wavefunctions. The string coordinates are then directly ob-
tained from these wavefunctions. In this manner the Lax system provides an explicit
mapping between the N-soliton sinh-Gordon solutions and the dynamical N-spike
AdS string configurations. The fundamental one-to-one correspondence between the
solitons/singularities of the sinh-Gordon field and spikes of the string solution was
established in detail in our previous work [4].
We now shortly summarize the construction of the classical string solutions. From
the scattering problem associated with the Lax system one finds the scattering wave-
functions
φ1 = (1+i)e−2i(z′−z′) 8 −ve41αˆ(ϕ˜2 ϕ˜1)1 + 8 u e−41αˆ(ϕ˜2 +ϕ˜1)1 , (19)
− 2√2 nr u − − r−v −o
φ2 = +(1−i)e−2i(z′−z′) 8 −ve41αˆ(ϕ˜2 ϕ˜1)1 8 u e−41αˆ(ϕ˜2 +ϕ˜1)1 , (20)
2√2 nr u − − −r−v −o
4
where
λ λ λ λ
(ϕ˜ ϕ˜ )1 = 1 j (1 A) 1λ + j j l (1 A) 1λ . (21)
2 ± 1 ± 1 +ζ − −jl l 1 +ζ ζ +ζ − −lk k
± j,l ±2 j j,l,k ±2 j j l
X X
The subscript corresponds to the signs before 1.
± ± 2
The second set of linearly independent wavefunction is found to be
1 v
ψ1 = −2√2 r8 −u e14αˆ e−21(z′+z′)(ϕ˜2 −ϕ˜1)2+ +ie12(z′+z′)(ϕ˜2 −ϕ˜1)2−
un h i
+i 8 ve−14αˆ e−21(z′+z′)(ϕ˜2 +ϕ˜1)2+ −ie21(z′+z′)(ϕ˜2 +ϕ˜1)2 , (22)
−
r− h io
i v
ψ2 = −2√2 r8 −u e14αˆ e−21(z′+z′)(ϕ˜2 −ϕ˜1)2+ +ie12(z′+z′)(ϕ˜2 −ϕ˜1)2−
un h i
−i 8 ve−14αˆ e−21(z′+z′)(ϕ˜2 +ϕ˜1)2+ −ie12(z′+z′)(ϕ˜2 +ϕ˜1)2 , (23)
−
r− h io
where
λ λ λ λ
(ϕ˜ ϕ˜ )2 = 1 j (1 A) 1λ + j j l (1 A) 1λ . (24)
2 ± 1 ± ± j,l ±2i +ζj − −jl l j,l,k ±2i +ζj ζj +ζl − −lk k
X X
Similar to (21), the subscript corresponds to the signs before i.
± ± 2
The N-spike string solution is given by
Z1 = 1−4 ie2i(z′−z′) i(ϕ˜2 −ϕ˜1)1+ e−21(z′+z′)(ϕ˜2 +ϕ˜1)2+ −ie21(z′+z′)(ϕ˜2 +ϕ˜1)2
−
+(ϕ˜2 +ϕ˜1)1+ne−21(z′+z′)(ϕ˜2h−ϕ˜1)2+ +ie21(z′+z′)(ϕ˜2 −ϕ˜1)2 , i(25)
−
Z2 = 1+4 ie2i(z′−z′)h i(ϕ˜2 −ϕ˜1)1+ e−21(z′+z′)(ϕ˜2 +ϕ˜1)2+ +ie12(z′+izo′)(ϕ˜2 +ϕ˜1)2
−
+(ϕ˜2 +ϕ˜1)1+ne−21(z′+z′)(ϕ˜2h−ϕ˜1)2+ −ie12(z′+z′)(ϕ˜2 −ϕ˜1)2 . i(26)
−
h io
Let us summarize the properties of the general solution constructed this way. This
general string configuration is characterized by two arbitrary functions u(z), v(z) and
a discrete set of moduli representing the soliton singularities (coordinates). After
fixing the conformal frame only the soliton moduli remain giving a specification of
the dynamical string moduli. These represent general motions of the spikes and their
locations.
In general the sinh-Gordon singularities behave as particles and follow interacting
particle trajectories. Through our explicit transformations this dynamics translates
into the spike dynamics of the AdS string. Concretely, given the trajectories of N
3
5
solitons x (t),i = 1,2, ,N, we can in principle by direct substitution (25, 26) with
i
···
σ (τ) construct the trajectories of N spikes by
i
Zi(τ) = Z (τ,σ (τ)), Zi(τ) = Z (τ,σ (τ)), (27)
1 1 i 2 2 i
where τ acts like the proper time. We therefore have a mapping where on the left
hand side the index i labels the string spikes while on the right side it denotes the
solitons/singularities. This construction is straightforward in principle, with the map
provided by the known wavefunctions of the scattering problem.
3.1 One-spike dynamical solution
Now we focus on two simple examples where the energies can be explicitly evaluated.
First, for one soliton, we choose the parameters to be
u(z) = 2, v(z) = 2, c (0) = 2ζ = iv˜ , (28)
1 1 1
− − −
where v˜ (1 v )/(1+v ) and v is the velocity of soliton on the worldsheet.2
1 1 1 1
≡ −
This will correspond to the one-soliton solution of the sinh-Gordon equation
p
σ v τ
α = ln 2tanh2 − 1 . (29)
1 v2
h − 1 i
(cid:2) (cid:3)
In terms of the worldsheet coordinates τ andpσ, the string solution is given by
eiτ
Z = e(σ τ)/v˜1 coshσ
1 −
e(σ τ)/v˜1 +e (σ+τ)v˜1
− −
(1 iv˜ )2n((1+v˜2)coshσ +2v˜ sinhσ)
+e (σ+τ)v˜1 − 1 1 1 , (30)
−
1 v˜4
− 1 o
ieiτ
Z = − e(σ τ)/v˜1 sinhσ
2 e(σ τ)/v˜1 +e (σ+τ)v˜1 −
− −
(1 iv˜ )2n((1+v˜2)sinhσ +2v˜ coshσ)
+e (σ+τ)v˜1 − 1 1 1 . (31)
−
1 v˜4
− 1 o
The momenta densities are calculated as
√λ √λ
Ptτ = 2π Im(Z˙1∗Z1), Ptσ = 2π Im(Z1′∗Z1), (32)
√λ √λ
Pθτ = 2π Im(Z˙2∗Z2), Pθσ = 2π Im(Z2′∗Z2), (33)
where λ is the coupling constant. One can explicitly check σ and σ vanish asymp-
Pt Pθ
totically at σ = as long as the string solution is regular. That is, in the singular
±∞
2Notice, in general, the worldsheet time τ is different from the global time t.
6
case when v = 0, we found nonvanishing momentum flow with τ dependence at the
1
boundary of the string [4].
By the current conservation ∂ τ ∂ σ = 0, we can calculate the energy and
τPt,θ − σPt,θ
angular momentum at any convenient τ. For instance, the energy and momentum
densities at τ = 0 are simplified to be
√λ coshσ +ǫ 1sinhσ 2
τ = e2σ/v˜1 cosh2σ +e 2σv˜1 −1 , (34)
Pt 2π(eσ/v˜1 +e σv˜1)2 − v
− 1
n (cid:16) (cid:17) o
√λ sinhσ +ǫ 1coshσ 2
τ = e2σ/v˜1 sinh2σ +e 2σv˜1 −1 , (35)
Pθ 2π(eσ/v˜1 +e σv˜1)2 − v
− 1
n (cid:16) (cid:17) o
where ǫ (1 v2) 1/2 is the energy of the soliton.
1 ≡ − 1 −
Introduce a cutoff Λ for the σ integration, up to the subleading term, the energy
and angular momentum are
Λ √λ 1
E = τdσ e2Λ +Λ , (36)
Pt ∼ 2π 4(1+ǫ 1)
Z Λ −1
− h i
Λ √λ 1
S = τdσ e2Λ Λ . (37)
Pθ ∼ 2π 4(1+ǫ 1) −
Z−Λ h −1 i
Therefore, the difference between E and S can be calculated as
√λ Λ cosh[2ǫ σ] √λ 8πS
E S = 1 dσ ln +ln(1+ǫ 1) ǫ 1 . (38)
− 2π 1+cosh[2ǫ σ] ∼ 2π √λ −1 − −1
Z−Λ 1 h i
Measuring the energy from the infinite GKP string, the dispersion relation can be
written as [21]
√λ 1 1+ǫ 1
E S = E + ln −1 ǫ 1 (39)
− 0 2π 2 1 ǫ 1 − −1
h − −1 i
with the excitation energy
√λ 1 1+ǫ 1
E1 (ǫ ) E S E = ln −1 ǫ 1 . (40)
spike 1 ≡ − − 0 2π 2 1 ǫ 1 − −1
h − −1 i
The inverse power in the soliton energy shows a similarity with the case of giant
magnons on R S2 [12] where the excitation energy of the string (and the giant
×
magnon) is equal to the inverse of the sine-Gordon soliton energy.3 Here in the AdS
case we find extra logarithmic terms pointing to a more complex dynamical system
governing the spike dynamics in AdS as compared with R S2.
×
3There one uses a time-like gauge t=τ.
7
3.2 Two-spike dynamics
Our next explicit example is for two solitons, where the parameters are chosen to be
ζ +ζ ζ +ζ i i
c (0) = 2ζ 1 2, c (0) = 2ζ 1 2, ζ = v˜ , ζ = v˜ 1, (41)
1 1ζ ζ 2 2ζ ζ 1 −2 1 2 2 2−
1 2 1 2
− −
where v˜ (1 v )/(1+v ) and v are the magnitudes of the velocities of the
1,2 1,2 1,2 1,2
≡ −
two solitons moving towards each other. The special case v = v = v corresponds to
1 2
p
the two-soliton solution used in [4]
vcoshX coshT 2
α = ln 2 − , (42)
vcoshX +coshT
h (cid:16) (cid:17) i
where X 2γσ,T 2vγτ.
≡ ≡
The exact expressions for the momenta densities are lengthy to write them down,
but the leading terms at τ = 0 are
√λ 1 1 v˜ 2 1 1 v˜ 2 1
τ − 1 e2σ + − 2 e 2σ + , (43)
Pt ∼ 2π 4 1+v˜ 4 1+v˜ − 2
1 2
h (cid:16) (cid:17) (cid:16) (cid:17) i
√λ 1 1 v˜ 2 1 1 v˜ 2 1
τ − 1 e2σ + − 2 e 2σ . (44)
Pθ ∼ 2π 4 1+v˜ 4 1+v˜ − − 2
1 2
h (cid:16) (cid:17) (cid:16) (cid:17) i
In the special case where v = v = v, the dispersion relation is
1 2
√λ 8πS 1+ǫ 1
E S = ln +ln − 2ǫ 1 + . (45)
−
− 2π √λ 1 ǫ 1 − ···
−
h − i
For different velocities, using the symmetric ρ regularization, we obtain the result
√λ 8πS 1 1+ǫ 1 1 1+ǫ 1
E S = ln + ln −1 + ln −2 ǫ 1 ǫ 1 . (46)
− 2π h √λ 2 1−ǫ−11 2 1−ǫ−21 − −1 − −2 i
Following the definition of spike energy (40), the excitation energy of the two-spike
solution is given by the sum of two individual spike energies
E2 (ǫ ,ǫ ) = E1 (ǫ )+E1 (ǫ ). (47)
spike 1 2 spike 1 spike 2
In the center of mass frame v = v = v, the energy is E2 (ǫ) = 2E1 (ǫ). For this
1 2 spike spike
special case, the result is also obtained in [21].
4 Solitons and Cusps: Euclidean worldsheet
We have described in Introduction the relevance of Euclidean worldsheet solution.
The simplest example of such a solution is the cusp near the boundary of AdS. We
8
describe how this solution can be simply found from the one-soliton solution of sinh-
Gordon using an Euclidean version of the inverse scattering method.
Start with AdS metric in Poincar´e coordinates
3
dy2 +dy2 +dr2
ds2 = − 0 1 , (48)
r2
the single cusp solution sitting at the boundary r = 0 can be found using the ansatz
y = escoshθ, y = essinhθ; r = esw(s). (49)
0 1
One can check explicitly that w(s) = √2 solves the equation of motion [3]. This
single cusp solution can be regularized to some finite boundary at r = r [17] with
c
the same ansatz (49) while w(s) is now implicitly given by the equation
es w+√2 1/√2 1
= . (50)
r w √2 1+w
c
(cid:16) − (cid:17)
The cusp is approached at s . In many circumstances, the conformal gauge
→ −∞
solutions are desirable [18, 19]. Choosing the ansatz
y = f(τ)coshσ, y = f(τ)sinhσ; r = g(τ), (51)
0 1
one finds that the functions f(τ) and g(τ) are solved to be
tanh[τ/√2]
f(τ) = √2eτr , (52)
c
2+√2tanh[τ/√2]
1
g(τ) = 2eτr , (53)
c
2+√2tanh[τ/√2]
where the cusp is approached at τ = 0.
To develop the general method for constructing AdS string solutions in conformal
gauge, we employ the technique developed in [4, 16] based on Pohlmeyer reduction
and inverse scattering. Now for Euclidean worldsheets, all one needs to change is
τ iτ. In the case of AdS , the reduction map is given in terms of the embedding
3
→ −
coordinates by
1
e2α(z,z) = ∂Y ∂Y, (54)
2 ·
e 2α
B = − ǫ Yb∂Yc∂Yd, (55)
a abcd
2
1 1
p = B ∂2Y, p = B ∂2Y. (56)
2 · −2 ·
9
The equation of motion for the scalar field α(z,z) turns out to be
∂∂α(z,z) e2α +p(z)p(z)e 2α = 0, (57)
−
−
with a holomorphic function p(z).
In order to construct the string solution, one needs to solve two SL(2) scattering
problems. Eachoftheseproblemshastwolinearlyindependent solutionsψL ,a = 1,2
α,a
and ψR , a˙ = 1,2 which are normalized as
α˙,a˙
ψL ψL ǫβαψL ψL = ǫ , (58)
a ∧ b ≡ α,a β,b ab
ψR ψR ǫβ˙α˙ψR ψR = ǫ . (59)
a˙ ∧ b˙ ≡ α˙,a˙ β˙,b˙ a˙b˙
The string solution can be written in terms of a combination of the wavefunctions
Yaa˙ = YY−1++YY2 YY1 −YY0 = ψαL,aM1αβ˙ψβR˙,a˙, M1αβ˙ = 10 01 . (60)
(cid:18) 1 0 −1 − 2(cid:19)aa˙ (cid:18) (cid:19)
Inaseries ofpapers[4, 5,6], it isshown that solitonsinthefield theorycorrespond
to cusps of the string. For the construction of the single cusp string solution (51),
one uses the one-soliton solution of the sinh-Gordon equation
1 1 τ
α(z,z) = ln tanh2 , (61)
2 2 √2
h i
together with the holomorphic function p(z) = i/2.
The Dirac wavefuntions are solved to be
ψL ψL ψL ψL
ΩL = 1,1 1,2 = 1 2 , (62)
(cid:18)ψ2L,1 ψ2L,2(cid:19) ψ2L ψ1L!
ψR ψR ψR ψR
ΩR = 1,1 1,2 = 1 2 , (63)
(cid:18)ψ2R,1 ψ2R,2(cid:19) ψ2R ψ1R!
where
ψL = ei58π c1eσ−2τ (√2−1)e−√τ2 −ie√τ2 −c2e−σ−2τ (√2+1)e−√τ2 +ie√τ2 , (64)
1
(cid:0) (cid:0) 2 sin(cid:1)h[√2τ] (cid:0) (cid:1)(cid:1)
π σ τ τ qτ σ τ τ τ
ψL = e−i8 c1e −2 (√2−1)e−√2 +ie√2 +c2e− −2 (√2+1)e−√2 −ie√2 , (65)
2
(cid:0) (cid:0) 2 sin(cid:1)h[√2τ] (cid:0) (cid:1)(cid:1)
ψR = ei58π c3eσ+2τ (√2+1)e−√τ2 +qie√τ2 −c4e−σ+2τ (√2−1)e−√τ2 −ie√τ2 , (66)
1
(cid:0) (cid:0) 2 sin(cid:1)h[√2τ] (cid:0) (cid:1)(cid:1)
ψR = e−iπ8 c3eσ+2τ (√2+1)e−√τ2 −qie√τ2 +c4e−σ+2τ (√2−1)e−√τ2 +ie√τ2 . (67)
2
(cid:0) (cid:0) 2 sin(cid:1)h[√2τ] (cid:0) (cid:1)(cid:1)
q
10