Download - Dissertacao Elisa Ferreira
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FICHA CATALOGRFICAPreparada pelo Servio de Bibl ioteca e Informaodo Instituto de Fsica da Universidade de So Paulo
Ferreira, Elisa Gouva Mauricio
Anlogos cosmolgicos dos efeitos Schwinger estticoe oscilatrio. So Paulo, 2009.
Dissertao (Mestrado) Universidade de So Paulo. Instituto de Fsica, Departamento de Fsica Matemtica.
Orientador: Prof. Dr. Lus Raul Weber Abramo
rea de Concentrao: Fsica
Unitermos: 1. Cosmologia; 2.Universo primordial;
3. Teoria quntica de campo.
USP/IF/SBI-083/2009
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c= = kB = 1
kB
ou h.c.
ou
ou ,
Dou ;
gDD=DD
Re/Im
[A, B] =AB BA {A, B} =AB + BA
g=diag (1, 1, 1, 1)
+ = 2g
5 =i0123
= i2[, ]
pq= pq
(x y) = 0, se x < y1, se x > y
D (x y) = 1(2)D
dDqeiq(xy)
(x y; m2)
i(2)4 d
4qeiq(xy) (q0) (q2 m2)
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Treaq
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4%
0, 001%
23%
73%
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ds2 =dt2 a2 (t)
dr2
1 kr2 + r2
d2 + sen2d2
dl2
=gdx
dx
.
dl
ds2 = 0
cdt= dxfis= a (t) dl ,
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dxfis
a (t0) = 1
(r,,)
d=dt/a (t)
x =
(t,r,,) g
g=
1 0 0 00 a
2(t)1kr2 0 0
0 0 a2 (t) r2 0
0 0 0 a2 (t) r2sen2
.
E3
S3
S2,1
k= 0
E3
k= +1
S3
VS3 = 4a
2 (t)
k=
1
S
2,1
[a (t) 0]
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v= H0l ,
H0
v=a (t)
a (t)l
=H(t) l ,
H0 = 70, 5 1, 3 k m s1 Mpc1 .
tem
(rem, em, em)
ds2 = 0
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tobs=t0
tem
dt
a (t)=
robs=r0=0
rem
dr1 kr 2 .
tem+ tem
t0 +t0
tema (tem)
= tobsa (tobs)
.
= 1/t
=1
obsem
= a (tem)a (tobs)
= emobs
.
z=obs em
em=
a (tobs)
a (tem) 1 .
z >0
z
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tH H1 .
RH=ctH=cH1
.
tH= 14 109 anos RH 4, 27 106 Mpc
G R 12
gR = 8GT,
G
R
T
T= ( +p) uu pg ,
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u
u = (1, 0, 0, 0) T
= diag (,p,p,p)
T =,,
1
2,, V()
,
12
,,+ V () ,
p 12
,, V() , u ,/,,,
,,>0
(t)
a
a
2=H2 =
8G
3 k
a2,
a
a= H+ H2 = 4G
3 ( + 3p) ,
+ 3H( +p) = 0 ,
T; = 0
;;+ V () = 0 ,
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dE=T dSpdV
= E/V
d= T dS ( +p)dVV
.
V a3
dV/V = 3da/a
S = 0
H2
1 = total ka2H2
,
total =ii i i/crit
crit=3H2
8G,
crit = 1, 88 h
2 1029g cm3 = 8, 098 h2 1011eV4
H0
total = 1 + k/a
2H2
total >1 k= +1 , Universo fechado total = 1 k = 0 , Universo plano total
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total 1, 020, 02
total = m + rad + m
b dm rad
a (t)
(t)
p (t)
p= .
= 0
= 1/3
= 1
(a) =0a (t)3(1+) ,
0
p = 0
(a) = m0a
3
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p=
(a) =0
p= /3
(a) =
rad0 a
4
E = h
a1
= E/V T4rad Trad a1
a (t) =a0t2/3(1+) .
a (t) =
am0 t2/3 ,
arad0
t1/2 ,
a0 eHt , const. cosmol. ,
H=
8G0 /3
p=
= 8GV()
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ti = 0
p(t) =a (t)
t
0
dt
a (t)
,
p
p
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e(t) =a (t)
t
dt
a (t),
e<
e
H1
m= 0m(1 + z)
3 ,
,
rad = 0rad(1 + z)
4 ,
= 0(1 + z)
3(1+) , energia escura ,
1 1/3
zeq =critm
0rad 1 2, 26 104mh275
z=
m1/3
1 ,
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h275 H0/75 km s1M pc1
0, 33 z 1, 33 m 0, 3 0, 7
z > zeq 104
zeq > z > z
z < z
z
z O (1)
rad(T) =n
i=1
igi
2
30
T4 ,
gi
i= 1, 7/8
rad
a4
Trad(z) =T0rad(1 + z) .
1043s
T 1032K 1019GeV
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1032s
108 109K
H2
H3
He4
z 104
z= 1400
z = 1100
z= 2 103
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H0
72km s1Mpc1
He3
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T = 2, 725 0, 002K
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1044
= 1
( 1) a2H2 =k .
( 1) H2a2
( 1) a
( 1) a2
(0 1)
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| 1|t=tpl|0 1|
a2pla20
T20
T2pl.
Tpl 1019GeV T0
1013GeV
1064
1016
(0 1) O (1) ( 1)
1 =
kmpl/S2/3T2
S=4
3
s
H30 1090 ,
| 1|t=tpl =m2plT2pl
1
S2/3 =
1
S2/3 1060 ,
( 1)
a (t)t2/3
(tdec)
(ti= tdec,t0)=
H1
(ti= tdec,t0)
tdect0
1/3 102 .
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= s
(0 dec),
(t) =
tdt/a
360
N= 4D3/3
4 (D/2)3 =
8
3 106 .
106
106
ct01028cm
t= ti
aia0
1028cm
cti
t0ti
aia0
t0ti
T0Ti
1028 ,
t=tpl = 10
43s t0
tpl
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1016
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1028K
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ti = tpl 1043
s
H1
[ (ti= tdec,t0)]. >(H1).
(ti = tdec,t0) a
d
dt a
H1= a >0 .
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H = Ha
+ 3p
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| 1| 1060
1
tf
| 1|t=tf
aiaf
2| 1|t=ti =e2N | 1|t=ti ,
| 1|
0= 1
Sf=Z3Si ,
S = Sf
Z = 1030
S= 1090
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S (aT)3
afTf 1030aiTi
af
ai=eN
1030
Ti
Tf.
R= H1
| 1|1/2 =
a2
k
1/2.
H10 a0ai
H1I =a0af
afai
H1I =a0af
eNH1I ,
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a0= 1
N ln
afHIa0H0
= ln
HITfH0
,
a= T1
TT GU 1029K HI T2T GU/mpl 1056km/s/Mpc
H0 10km/s/Mpc N 60 efolds
H1
n (tT GU) 1082m3
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n (tf) e3N1082m3 = e3001082m3 5 1049m3 15pc3
n/s
ni/nf (Ti/Tf)3 (ai/af)3 e3N
< 1/3
= 1
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S[] =
d4xgL
=
d4xg
1
2 + V ()
,
L
g
g V()
V () = 1
2m2
V () = (
2
M2
)
2
V() =
4
T =,,
1
2,, V()
.
u ,/,,
12
,,+ V() =1
22 + V()
p 12
,, V () =12
2 V () ,
,
+ 3H + V () = 0 ,
V () =dV/d
g=a3 (t)
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H2 =8G
3
1
22 + V ()
,
M2pl= 1/G
p 0 p 2 V () .
H2 13m2pl
V()
3H V () ,
m2pl = 1/8G
3H ,
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2 V () m2pl
3
V
V
2 1
3H
1
3
V
H2
1 .
=m2pl
2
V
V
2= H
H2 =
1
2m2pl
2
H2
sr =1
3
V
H2 =m2pl
V
V
,
sr
= sr= /H
a
a=H2 (1 ) ,
a >0
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1
dt
a (t)=
da
aH = 1
H +
da
aH .
= 1H (1 + ) + O (2) ,
i f
Ntf
ti
H dt Hf
i
d
1m2pl
f
i
V
Vd ,
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N (i/mpl)2
i
mpl
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V
V () =1
2m2
2
V () =4 ,
a (t)tp
V() =V0exp
16
p
mpl
.
V ()
= 4 (f1 1)
a (t) exp
(t/t0)
f
V () =V0[1 + cos (/f)]
V () =1
2m22 ,
+
12
+ m221/2
+ m2= 0 .
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= d
d,
()
d
d=
12 ( + m22)
1/2 + m2
.
(, )
2 m22 ,
(, ) 0
(, )
= 0
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3H + m2= 0
H2 =1
6
m2
m2pl2 .
= =2m2pl
2 ,
|| 2mpl( mpl)
m mpl
(t) =i m mpl
2
3t ,
a (t) =aiexp1
6
m
mpl
it 1
6m mplt
2 .
N=1
4
2im2pl
12
,
N 70 e folds
i 17 mpl
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Treaq
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V () = 12m2
mpl
3H
mpl/2
2/2 m22/2
= reaq 1
3H
+ m2 0 ,
(t) (t)sen(mt) ,
=12
m22 ,p = 0 ,
a (t) = am0t
2/3
a3
H= 23
t
(t) = mpl
3mt mpl
20N,
N=mt/2
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Treaq
i =
L =12
()2 V () +1
2
i
(i)2 m2i(0) + g2i 2 m2i
2i
i
g2i 2i.
gi m2i
i V () =
12
m2 ( )2
V () = 1
2m22
= 0
g
2
g
m2 m2 g2 ,
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H m ,
+ 3H + total + m2= 0 ,
=g2
2
8m,
=
g2m
8 .
(t) = 0exp 1
2
dt (3H+ )=
0t
e12totalt ,
H= 2
3t
total = +
n=
m e(3H+total)t .
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d
dt
a3
= totala3 , ddt
na
3
= totalna3 .
m m m m
total < H
total H
total > H
1total < H
1
i
total H
Treaq 0, 2
100
g
1/4totalmpl ,
g
102 103
m106mpl
Treaq < 10
6GeV
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105
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g g(0) + g,
|g| g(0)
ds2 =
g(0)() + g(, x)
dxdx ,
=ds20+ ds2 ,
ds20=g(0)() dx
dx =a2 ()
d2 ijdxidxj
,
ds
2 = g(, x) dxdx
ds20
g
g00
g00(, x) = 2a2 () (, x) .
g0i
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g0i(, x) = 2a2 () (B,i+ Si) ,
Si S
,ii = 0 S
,ii = 0
Si
S,i
i = 0
gij
gij(, x) = 2a2 () (2ij+ 2E,ij+ Fi, j+ Fj, i+ hij) (, x) ,
(, x)
E(, x)
Fi(, x)
hij(, x)
hii= 0, hij, i = 0 .
gescalar =a2
2 B,iB,i 2 (ij Eij)
.
ds2 =a2 ()
(1 + 2) d2 + 2B,iddxi + [(1 2) ij 2E,ij] dxidxj
.
gvetorial =a2
0 SiSi Fi,j+ Fj,i
,
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ds2 =a2 ()
d2 + 2Siddxi (ij Fi,j Fj,i) dxidxj
.
g tensorial =a2
0 0
0 hij
,
ds2 =a2 () d2 (ij hij) dxidxj .
x x =x + ,
0
i
i =i+ ,i ,
i
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g(x) =
x
xx
xg(x
)
g(0)(x) + g g(0), g(0),
=g(0)
(x
) + g(x
) ,
g
g=g g(0), g(0), g(0), ,
g(0)
(x
) g(0)
(x
) g(0)
,
.
g(0)00 = a
2 () g
(0)ij =
a2 () ij
g00 = g00 2a
a0
,
g0i= g0i+ a2
i+
0,i ,
gij =gij+ a2
2a
aij
0 + 2,ij+
i, j+
j,i
.
= 1a
a0
, B=B + 0 ,
= +a
a0 , E=E+ .
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Si=Si+
i , Fi = Fi+ i .
hij
1a
a
B E , aa
B E .
vi=Si Fi .
B= 0 , E= 0 , Si= 0 ,
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ds2 =a2 ()
(1 + 2) d2 (1 2 Fi,j Fj,i hij) ijdxidxj
.
=0 = 0
i = 0
Tij ij
= , = , i= E
i,
g0 = 0
= 0 , B= 0 .
= +C1(x
j )
a , xi =xi + C1,i
xj d
a + C2,i
xj
,
C1 C2
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G0(0)0 =
3H2a2
, G0(0)i = 0 , G
i(0)j =
1
a2
2H + H2
ij,
T0(0)i = 0 , Ti(0)
j ij.
G= 8GT.
G
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G00 = 2 3H
+ H
= 4Ga2T00 ,
G0i =
+ H
,i= 4Ga2T0i ,
Gij =
+ H (2 + ) +
2H + H2
+
1
22 ( )
ij
1
2( ),i,j
= 4Ga2Tij.
G0i = 16Ga2T0i ,
Gij = (vi,j+ vj,i)
+ 2
H(vi,j+ vj,i) =
16Ga2Tij(V) ,
Tij(V)
Gij =
h
ij+ 2Hh
ij 2hij
= 16Ga2Tij(T) ,
Tij(T)
Tij ij
= .
Tij
ij
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(, x) () + (, x) .
= +
0 ,
+
B E
.
V()
T00 =
a2
V, +
,
T0i = i
,
Tij =
2 + a2V,
ij.
2 3H
+ H
= 4G
+ a2V,
,
+ H = 4G ,
+ 3H +H + 2H2
= 4G
2 a2V,
.
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+ 2H
2 + 2H
H
= 0 ,
+ 2H + a2V,= 0 .
u a
, H/a ,
u 2u
m2eff
u= 0 .
/
u (x, ) =
d3k
(2)3/2uk() e
ik.x ,
x = l/a
k = akf is
u
k+
k2
uk= 0 .
m1eff
m1eff (aH)1
1 Caso
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k2 /
uk() =A+k e
ik + Ak eik ,
Ak
k(t,x) eikx
A+k sen
k
dt
a
+ Akcos
k
dt
a
.
/
2 Caso
k2 /
u
k
uk= 0 ,
u1,k= C1
W =u1,ku
2,k u
1,ku2,k=u21,k
d
d
u2,ku1,k
(=const.) .
u
k+ Q () uk= 0 ,
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uk C1+ C2
i
d
2 ,
= A
1a i
a2 () d
,
C1 C2 A
(t,x) A
1 Ha
a dt
,
a tp
p > 0
p= 1)
lp(t,x) =A(1)
1 p
p + 1
+ A(2)
1
t1+p.
p > 1
p < 1
R + 2H8G
1
2
+ H
.
= 0
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R = 2H8G
1
22 .
2
+23
H1 + 1 +
,
=p/
32
(1 + ) = + H 2 + 2 H H
,
2
(1 + ) = 0 ,
R
= 0 = 0 R
=const.
R
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2i = 0 ,
i+ 2
Hi = 0 .
i(, x) =Ci(x) a2 ,
h
ij+ 2Hh
ij 2hij = 0 ,
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hij
hij = 1
a ()
d3k
(2)3
k() ij(k) eikx + c.c.
k()
k+
k2 a
a
k= 0 .
hij
ij =ji ,ij =ijki = 0 .
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S=1
2
d4x (Lgrav+ Lm)
= 1
16G
d4x
g
R+ Lm
,
ds2 = N2 NiNi dt2 2Nidxidt gijdxidxj ,
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N
Ni gij
S= 116G
d4xg[NR(3) +NKijKij K2 8GNgijij 2V ()+ 8GN1 Niii ,
R(3)
gij Kij
Kij =1
2N1 (iNj+ jNi gij) ,
K=K
i
i =gijK
j
N
Ni
R(3) KijKij K2 2V() 8Gg ijij + 8GN2 Niii2 = 0 j
Kji Kji 8GN1 Niii i= 0 .
N2 = (1 + 2) , Ni= 0 , gij =a2 [ij(1 2) + Fi,j+ Fj,i+ 2hij] .
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v zR ,
z = 1/ = a
/H
R
ij
M2p4
ahij.
ij
hij
hij(, x) =h+(, x) +ij+ h (, x)
ij.
S(2) =1
2
dd3x (Lesc+ Ltens)
=1
2
dd3x
v2
v,iv,i+ z
zv2
+1
2
dd3x
=+,
2 ,i,i+ a
a2
.
V()
z
z
a
a
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v 2v z
zv= 0 ,
2 a
a= 0 ,
v
k+
k2 z
z
vk= 0 ,
k+
k2 a
a
k= 0 .
v(, x) =L(2)
v =v
,
(, x) =L(2)
=
,
H=
d3x
v
v+
L(2)
=1
2
d3x
2v+ v,iv,i
z
zv2
+1
2
d3x
2+ ,i,i
a
a2
.
iv
= [v, H] , i
v = [v, H] ,
i
= [, H] , i
= [, H].
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Ql(, x)
S[Ql] =
dd3x
Q
l
2 Ql,iQ,il +
Z
l
ZlQ2l
L(2)
,
Zl
LQl
L(Ql)
= 0 .
Q
l +
2 Z
l
Zl
Ql = 0 .
Ql Qll l ,
Ql(, x) ,l(, y)= ill(x y) , Ql(, x) , Ql(, y)
=
l(, x) ,l(, y)
= 0 .
Qk,l() ,k ,l()
= ll
k k
Ql(, x) =
d3k
(2)
3/2 Qk,l() ak,leikx + Qk,l() a
k,le
ikx
,
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ak,l a
k,l
ak,l, a
k,l
= ll
k k
,
ak,l, ak ,l
=
ak,l, ak,l
= 0 .
Qk,l()
Q
k,lQk,l Q
k,lQk,l = 2i ,
Qk,l()
Q
k,l+ k2 Z
l
Zl 2k,l()
Qk,l= 0 .
Hk=2k,l() a
kak ,
k,l()
|0
H|0 = E0 |0
ak,l |0 = 0 , bk,l |0 = 0 ,k .
ak
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ak,l
ak,l |0 = |1k, l , bk,l |0 = |1k, l
Nk= a
kak
0
|Nk()
|0
= 0 ,
n
|Nk()
|n
=nk ,
k .
Nk
|n
1
|0, 1 ak,l(1) |0, 1= 0
2
0, 2| Nk(1) |0, 2 = 0 .
2
1
ak,l(2)
ak,l(2)
=
k
k
k k
ak,l(1)
ak,l(1)
,
k k
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|k|2 |k|2 = 1 .
N
(1)k
2
= |k(1, 2)|2 .
i
|0, i
Qk,l() Q
k,l() i
Qk,l =rk,leik,l ,
rk,l k,l
Ek,l=1
2
Qk,l2 + 2k,l |Qk,l|2 =
1
2
r
k,l
2+
1
r2k,l+ 2k,lr
2k,l
,
r2k,l
k,l = 1
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Ek,l(i) =k,l
rk,l(i) =1/2k,l , r
k,l(i) = 0 .
Qk,l(i) = 1
k,leik,l(i) ,
Q
k,l(i) =i
k,leik,l(i) ,
k,l
0| Ql(, x) , Ql
,y
|0 =Gl(x, y) .
0| Ql(, x) , Ql
,y
|0 =
d3k
(2)3Qk,l() Q
k,l
Gk,l(,)eik(xy) .
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Gk,l(, )
= hc k=
Z
l/Zl
PQl(k) =
1
22
k3 limhc
Gk(, ) = 1
22
k3
|Qk,l()
|2
|=hc .
PQl kns1 ,
ns
ns 1 d ln
PQ
d ln k .
Qk,l
vk()
v= zR ,
k aH
R
PR= k3
22 vk()z 2
.
R
k aH
= hc khc =
z/z H
hc(k) thc(k)
vk z
k z/z1/2 ( H)
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PR= k3
22z2 (t)
z(t)
z(thc(k))
2|vk(thc(k))|2 .
H
z(thc(k))a (thc(k)) vk(thc(k))
PR k3
22a2 (thc(k)) |vk(ti)|2 .
|vk(ti)| = |k|1 k2
PR
k3
H2
k2
1
k
=H2 ,
a1 (thc(k)) = H/k
H2
H2
ns
ns= 1
2sr
6= 1
2
4 ,
, |sr| , 1
ns 1
Pog = k3
224
M2p
=+,
()a 2
.
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r 12
PogPR = 8 .
V()
>0
ns= 1 + 2
N ,r=
4
N ,
= 2
ns 0, 94 r 0, 11
ns= 0, 94+0,0140,013 ,68% CL ,
r
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E2
B2 >0
E B
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Ecrit 1.3 1018V /m
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90/206
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1 2
=1+ i2
2,
S=
R
d4xL = d
4x(L (1) + L (2))
= 12
d4x
D (D
) m2 ,
=
D= + ieA e m
Lint= ie
2 [() A
A (
)] + e2
A
2
.
S = 0
DD + m2= 0 ,
(t,x) = 1
(2)3
d3kk(t) e
ik.x .
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(x) = 1
(2)3
d3k
2k
ake
ikx + bkeikx
,
ak bk
S=
dt
R3
d3kkk+ 2k(t) kk
,
2k(t) =k2 + m2 + 2iekA
+ e2A2 .
k+ 2k(t) k= 0 .
pk= Lk
k
= k
pk= Lk
k
= k,
H=R3
d3kHk=R3pkk+pkk Lk
=
R3
pkp
k+
2k(t) k
k
,
k
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= k kp p= L
= = pkpk= k,
[ (x, t) , p (y, t)] =i(x y) , [ (x, t) , (y, t)] = [p (x, t) , p (y, t)] = 0[k(t) , pk(t)] =(k k) , [k(t) , k(t)] = [pk(t) , pk(t)] = 0 .
Hk= pkpk+
2k(t)
kk .
ak
bk ak bk
ak, ak
= (k k) , [ak, ak] =
ak, a
k
= 0
bk, bk
= (k k) ,
bk, bk
=
bk, bk
= 0
.
Hk=k(t)
akak+ bkbk
.
|0
H|0 = E0 |0
ak |0 =bk |0 = 0 ,
ak
bk
ak b
k
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ak |0 = |1k , bk |0 =1k .
Nk= a
kak
Nk=bkbk
0| Nk |0 = 0 , n| Nk |n =nk , k .
(x)
(iD m) (x) = 0 ,
e m
L = (x) (iD m) (x)=i eA
Lint
,
0
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i m= eAi
+ m=
eA
,
(x) = 1
(2)3
d3k
k(t) /m
r=1,2
dr(k) u(r) (k) eik.x (+)
+fr(k) v(r) (k) eik.x
()
,
dr fr (+)
()
k0
0
r
(k m) u (k) = 0 , (k+ m) v (k) = 0 .
44
22
2 2
= L(0)
=i0 =i , = L
0 =i0 .
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H=
d3x
i
2
0D0 D00 L
=
d3
x
1
2 D0 D0 ,
Hk=
r
k(t)
dr(k) dr(k) + fr(k) fr(k)
,
k(t) =
(p eA)2 + m2
f(k) , f (k)
=
,
Hk=
r
k(t)
dr(k) dr(k) + fr(k) fr(k)
.
(x, t) ,(y, t)= (x y)(x, t) , (y, t)= (x, t) ,(y, t)= 0
H=1
2
D0
D0
.
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d(r)k ,d(s)k
= rs(k k) ,
d(r)k ,
d(s)k
=
d(r)k , d(s)k
= 0
f(r)k ,f(s)k
= rs(k k) ,
f(r)k ,
f(s)k
=
f(r)k , f(s)k
= 0
.
Hk =
r
k(t)
d(r)k d
(r)k +
f(r)k f
(r)k
.
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aink |0 = 0, bink |0 = 0aoutk |0 = 0, boutk |0 = 0 .
S
|0in=S|0out .
in = SoutS1 .
S= limt+
U(t) =Texp
i
d4xLint(x)
,
U(t)
Hint(t) =
d3xHint=
d3xLint ,
P0 = |0out|0in|2 = |0in| S| 0in|2 = |S0|2 ,
S0
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Sb= Texp
i
d4xLint(x)
=Texp
ie
2 (A
A) + ieA2,
S0,b(A) =
n=0(ie)n
2nn!
n
i=1 dxi0| T(A A) + ieA2 (x1) |0 . . . 0| T(() A A ()) ieA2 (xn) |0 .
S0
S0,b(A) =Det1 (I b) = exp [T r ln (I b)] ,
b
x| b |x = 0in| T(() A A ()) |0in (i)
+ ieA2 0in| T |0in (ii)
.
(ii)
(i)
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b=e2A2 + 2ePA
P2 m2 + i I b=(P eA)2 m2 + i
P2 m2 + i = G
G0,
G0 G
S0,b(A) = Det1
(P eA)2 m2 + iP2 m2 + i
= exp
T r (PeA)2m2+i
P2m2+i
.
Sf=Texp ie d4x (x) (x) A(x) ,
S0,f(A) =
n=0
(ie)nn!
ni=1
dxi
0in| (x1) (x1) A(x1) . . . (xn) (xn) A(xn) |0in ,
S0,f(A) =
Det [I f] f=eA (x) 1Pm+i
0| k(xk) l(xl) |0 = iSF(x y)kl
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S0,f(A) =Det
I eA (x) 1
P m + i
=exp
T r ln
[P
m] 1P
eA (x)
m + i
.
Tb(A) =V + V
1
P
2
m2
+ iTb(A) ,
Tb(A) =V + V 1P2 m2 iTb(A) ,
V = 1
G0 1
G Tb(A) = Tb (A) Tf=0Tf0
Tf(A) =eA (x)
1 + Tf(A)
P
m + iTf(A) =eA (x) 1 + Tf(A)
P m i
,
T (A) T (A) =i (+) + () ,
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()b 2
P0 P2 m2()f 2 [P + m] (P0) (P2 m2) ,
= 0
Gr =
1P2m2+i
Det(P eA)2 m2 + iP2 m2 + i = 1 .
Sr = P+m
(P+i)2m2 = 1
Pmi
Det
I eA (x) 1
P m i
= 1 .
|S0(A)|2
|S0(A)|2 = expTr ln I T (A)(+)T (A) () ,
P0= e
d4x =
eV T
B(x) =x|
ln I T (A)(+)T (A) () |x , F(x) = tr x| ln
I T (A)(+)T (A) () |x ,
b(x) F(x)
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|S0(A)|2 = exp
d4x tr x| ln
(P2 m2 + i)2(P eA)2 m2 + i
|x
B(x)
.
ln D= 0
ds
s exp
i
D i
s
.
B
B(x) = 2Re0
ds
seis(m
2) x| exp
is
[P eA (x)]2 P2
|x .
F
F(x) =Re
0
ds
seis(m
2)
trx| exp
is
[P eA (x)]2 + e2
F P2
|x
,
[P eA (x)]2
+
e
2F
(i
eA + m)
2m
E2 B2 >0
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E= Ek ,
A3
(x) = Et ,
trx| eis[[PeA(x)]2+ e2F] |x=4cosh(eEs) d4p
(2)4d4k
(2)4ei(pk)x
ei P0P3
eE
p| eis
P20
P2Te2E2X20
|k e
i P
0P3
eE
=2eE
(2)2cosh (eEs)
d | eis(P20e2E2X20) |
(I)
,
[X0, P0] = i
[P eA (x)]2 + e2
F =e(i
P0P3eE )
P20 P2 e2E2t2
e(i
P0P3eE ) +
e
2F
,
i
P0 P X0 Q 2ieE 0
1/2 m0
P20
e2E2X20
P2 +20
4Q2 =
P2
2m20+
m20
2 Q2 .
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d | eis(P20e2E2X20) | =Tr exp
is
P2
2m20+
m020
2 Q2
=
n=0
exp[eEs (2n + 1)]
=1
2
1
senh(eEs).
B(x) = 1
2 (2)2
+
ds
s2
eE
1
senh(eEs)1
s
Re
ieism2
,
F(x) = 12 (2)2
+
ds
s2
eEcoth (eEs) 1
s
Re
ieism2
,
1/s e = 0
+
0
B =E2
22
n=1
(1)n+1n2
exp
nm
2
|eE|
,
F =E2
2
n=1
1
n2exp
nm
2
|eE|
.
expnm2|eE|
2m
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B
F
Ecrit
1.3
1018V /m
|eE| m2
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in out
t
t +
in = bout+ b
out
in = b out+ b out,
out =b in bout
out = b out+ bout.
x= y
|b|2 |b|2 = 1 ,
in
out in=fout+ foutin=fout+ fout ,
out = fin foutout = fout+ fout .
|n|2 + |n|2 = 1 .
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in(x) = 1
(2)3
d3k
2k
Ake
ikx + Bkeikx
,
in(x) = 1
(2)3
d3k
k/m
r=1,2
Dr(k) u(r) (k) eik.x (+)
+Fr (k) v(r) (k) eik.x
()
,
Ak Ak Bk B
k
Dr(k) D
r(k) Fr(k) F
r (k)
t
Ak |0in =Dr(k) |0in = 0Bk |0in =Fr(k) |0in = 0
.
ak a
k bk
bk fr(k) d
r(k) fr(k) b
r(k)
ak |0in = 0 , dr(k) |0in = 0 , bk |0in = 0 , fr(k) |0in = 0 .
Ak =x1kak+ x2kbkBk =x2kak+ x1kbk , ak=x1kAk+ x2kBkbk= x2kAk+ x1kBk , Dk =y1kdk+ y2kf
k
Fk =y2kd + y
1kf
k
,
dk =y1kDk y2kFk
fk = y2kDk+ y1kFk.
k= k
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|x1k|2 |x2k|2 = 1 .
|y1k|2 + |y2k|2 = 1 .
x1k= bx2k= b ey1k= fy2k= f
ix1k =kx1k+ i1
2
kk
x2k
ix2k = kx2k+ i 12
kk
x1k.
Nk(t) = 0| ak(t) ak(t) |0 =|b|
2 = Nk(t)0 , seNk(t1) = 0Nk(t1) + |b|2 (1 + 2 Nk(t1)) , seNk(t1) = 0
,
Nk(t1)
Nk(t) = 0| ak(t) ak(t) |0 =
2 |f|2 = Nk(t)0 , seNk(t1) = 0Nk(t1) + |f|2
1 Nk(t1)
Nk(t1)
,
seNk(t1) = 0 ,
Nk(t1)
Nk(t1)
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N(t) = 1(2)3
d3k Nk(t) .
=, p = i
,
p i
p i
| [, , t]
| =D2 (, ) (, ) ,
D2= DD
(, ) = , | .
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[, , t] =N(t)exp
d3x d3x
(x)
x, x
; t
x
,
x, x ; t
out0|0in
id
dt=H
=
d3x
+ (i
)
i
+ iebAi
i
+
m2 e2bA2
[, , t]
N(t)
iN
N =T r ,
= i2 + i2 ,
D2 d3xd3y+ d3x (a+b)== det()
1exp
d3xd3ya1b
.
det() = expTrln Tr =d3x (x)
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T r =
d3x
x, x
, t
,
2 =
d3y (x,y,t)
y, x
, t
,
= 2 + m2 + 2iebAi (x, t) i .
=k
k(kk) ,
Hk = 12
2
2k+
1
22k(t)
2 =i
t ,
k[, , t] =Nk(t) ek(t)
2k .
Nk(t) k(t)
iNkNk
= k, k= i2k+ i2k .
(t)
k(t)
k = i fkfk
,
fk
fk+ 2kfk= 0 .
N(t)
Nk(t) | = 1
Nk(t) =
2 (Re)
1/2
.
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= k 2(Rek)
1/2
ek(t)2k ,
k k
1|2 =
kdkdk
12
=
k
dkdk
2(Re1,k)
1/2 2(Re2,k)
1/2e
k(1,k(t)+2,k(t))2k ,
|1|2|2 = det
4(Re1)(Re2)
(1+ 2) (1+ 2)
.
|1|2|2 = det
f1,kf1,k f1,kf1,k
f2,kf
2,k f2,kf2,k
f1,kf2,k f2,kf1,k
f2,kf2,k f1,kf2,k .
1 2
f2,k= kf1,k+ kf1,k ,
f1,k=kf2,k kf2,k ,
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|n|2 |n|2 = 1 .
f1,kf
1,k f1,kf1,k
f2,kf
2,k f2,kf2,k
P0,b= det
1
|k|2
= det
1
1 + |k|2
= det
1
1 + Nk(t)0
.
fk
fk
A= (0, 0, 0, A (t)) = (0, 0, 0, Et)
d2kd2
+
+ 2
2()
k= 0 ,
ebEtkzebE
(k2+m2)
ebE
Dp(z)
k() =kD(1+i)/2[(1 + i) ] + kD(1+i)/2[ (1 + i) ] .
k k
()
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Q/2
1 ,
Q 34
1
2
d
d
2 1
2
d2
d2 ,
k+
2 () Qk= 0 .
wkb,k= 12
exp
d
Q2= 12 12 1 + 2
2 5
2
2
1 + 2
1 ,
k() kwkb,k() + kwkb,k()
= k2
expi ini
d + k2
exp+i ini
d ,
ini< 0
/
|Q/2|
|0in /
|0out /
+
|0in /
k = 1 k = 0
lim
/k = wkb,k k k
= 0
ini
d
()
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ini
d
=
2
1 +
2
ini
1 +
2ini
+ ln +1 + 2
ln ini +1 + 2ini
,
1
ini
d 1
2
|| + |ini|2
+
2
||
ln || + ln |ini|
.
wkb,k= 1
2
ei(||+|ini|2)/2 ||1/2i||/(2) |ini|i/2 .
k= k
2
12
+ i2
ei(1i)/4 (1 i)(1+i)/2 2ei|ini|2/2 |ini|i/2 k
wkb,k+
+
ke
i(1+i)/2 + k(1 + i)(1+i)/2
2ei|ini|
2/2 |ini|i/2 kwkb,k ,
k = 1 k = 0 k k
k=12
+ i2
2ei(1i)/4 (1 i)(1+i)/2 1
2ei|ini|
2/2 |ini|i/2 ,
k= kei(1+i)/2 .
+
|k|2 = 1 + e, |k|2 = e ,
N(t)
P0b
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P0b= det
1
1 + e
= exp
Tr ln1 + e .
Tr d3x
P0b= exp
V TbE
2
22
n=1
(1)n+1n2
exp
nm2bebE
= exp [V T b] .
Nk(t)0= e = exp
k+ m2
eE
.
x1k =ei(k2k) sinh rk,
x2k =eik cosh rk,
rk k k rk(0) = k(0) = 0
2rk =kk
cos2k,
2k = 2k kk
sen2kcoth 2rk,
= k 12
kk
sen2ktanh 2rk.
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Nk(t)
= cosh2 rk.
k = 0, 1
kz = 0, 01 m2 = 0, 2
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Ec=
m2c3
e 1, 4 1016V/cm
E(t) =E0sen(0t)
0
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E(t) = E0/ cosh
2 (k0t) A (t) =E0k0
tanh(k0t)
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0 m eE m2
0 A (x) = (0, 0, 0, A (t))
ddt+ 2k(t)k(t) = 0 ,
2k(t) = k2+ m
2 + (k3 eA (t))2
B(x) = x| ln
I T (A)(+)T (A) ()
|x ,
T (A) T (A)
V = 2k3A (t) + e2A2 (t)
d2
dt2+ 2
(t) =V(t) (t) .
|t|
P0=eW =e
d
4
x B
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dW
dV =
dt
d4k d4k k| T (A)(+)T (A) () |k ei(pp)x ,
dW
dV =
1
2
d3p
(2)2|| T ||2 ,
=
p2 + m2
|
k0
|| T ||
(t) eit + beit, t aeit, t + .
(t) =0(t)
dt G (t t) V (t) (t) .
(t) =eit + 1
i
dtei(tt
)V (t) (t)
=eit +f
e+it ,
f =| T |
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b=
i| T | , (a 1) =
i| T | .
dW
dV =
1
(2)3
d3k |b|2 .
t
k2k
= e A (t) (k3 eA (t))
m2 + k2+ (k3 eA (t))23/2,
|t|
E0
k2k < eE0m2 + k2
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a= (+)expi+ dt [ (t) ]
b= ()exp2i 0 dt [ (t) ]
.
dW
dV =
1
(2)3
d3k |()|2 .
(t)
(t)
(t)
(t)
t
ei + ei = 0 .
ei ei =
(t)
(t)
ei ei .
= (t)2(t)
( e2i)
= (t)2(t)
( e2i) .
|t| (t) 0
, 0
, constante
e2i
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= (t) (t)(t)
(0) (t) = (t)1/2
, (0) (t) = 0.
(a 1) i
= | T |
| T | = i
expi
+
dt
2 + V (t)1/2
1 .
| T |
(0) (t)
= +
t
dt (t)
2 (t)e2i(t
) .
|| T || =
|()|
dW
dV = d3p
(2)3 +
dt (t)2 (t)e
2i(t)2
.
T /2
T /2
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= dW
dV dT
= limT
1
T
d3p
(2)3
+T /2
T /2
dt (t)
2 (t)e2i(t
) .
E(t) =E0sen(0t)
A (t) = E0
0cos(0t) .
(t) = m2 +p2+ p3
eE00
cos 0t2
1/2
.
(t)
(t) =
t
0
dt (t)
=t
2
0
d (0t)
2
m2 +p2+
p3 eE0
0cos 0t
21/2
+
t
2
d (0t)
2
m2 +p2+
p3 eE0
0cos 0t
21/2
(t)
=t + (t) ,
(t)
(t)2(t)
e2i(t) =
n= cnein0t
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= limT
1
T
+n=
d3p
(2)3
+T /2
T /2
dtcnein0te2i(t)
=0 d3p
(2)2
+
n=n 2
0 |cn|2
=0
d3p
(2)2|c|20
=0eE
2
d3p
(2)2|c|20
,
/0 m/0 1
= 2
0
c = eE2
c
f(0t) = (t)2 (t)
e2i(t) =
n=cne
in0t
n 2/0
c = 1
2
dx f(x) ei 2
0x
, x= 0t .
f(x) =senxp3 eE00 cos x
m2 +p2+
p3 eE00 cos 0t2e2i(t) ,
c=
dx
2
senx
p3 eE00 cos x
m2 +p2+ p3 eE00 cos 0t2
e2i[t+(t)]
exp2i0
x
0
dx
m2 +p2+
p3 eE00
cos 0t
21/2 ,
|c|2
2
2/0
c= 2iImc1 c1
x00 = A + iB c1
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1
dy
2yexp
y3/2
=
2
3
du
2ueu =
2
3i ,
= 19
0
d3p(2)2
e2A cos2 B .
p3 cos
2 B
x0=1
2+ isinh1
0eE m2 +p21/2 y
.
x0
A=2 (m2 +p2)
eE
1
0
dy
1 y21 + y2 20
m2 +p2
/e2E2
z
,
g (z) = 4
1
0
dy 1 y21 + z2y2
.
g (0) = 1
(m2 +p2) = (eE/0)
2 u2
= (eE)3
1820
du uexpeE
20u2g (u)
,
=m0/eE
(0/m) (eE/m
2)
u2g (u)
eE
20 u2
g (u) eE
20 2
g () =
m2
eEg () 1 .
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=E2
2
1
g () + 12
g ()exp
m
2
eE g ()
.
1
1
g ()
g ()
g (z) 4
sinh1
14
E2
2 exp
m2
eE
, 1
E2
8
eE2m0
4m/0, 1 .
2m/0
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eE m2g () ,
eE m0c
sinh(0/4mc2),
46, 6GeV/c
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x3
A= (0, 0, 0, A (t)) , com A (t) = E00
sen(0t) .
k+ 2k(t) k= 0 ,
2k(t) =k2 + m2 + 2ek3
E00
sen(0t) (ie)2
E00
sen(0t)
2.
=k2+ m2 +
k3+
eE00
sen(0t)
2
z= 0t/2
d2kdz2
+
0+ 2 cos2z+
1
2 cos4z
k= 0 ,
200=
k23+
e2E202
= 4eE0k3
20
= 2eE0
k3+
eE00
sen(0t)
k3 k3 = 0
A
=k3A3= 0
k3A3
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k+
k2 + m2 +
e2E2020
sen2 (0t)
k= 0 ,
2k(t) =k2 + m2 +
e2E2020
sen2 (0t) .
0t= z
d 2kdz2
+ (Ak 2qcos2z) k= 0 ,
Ak = (m2+k2)
20+ 2q
q =
eE0220
2
Ak=
(m2+k2)20
2q q=
eE0220
2
k= u (z, ) eNz ,
N k
2+m2
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Aq
A= 2q
eE00
0 q 1
eE020
4 m2
201
1 ,
0/m 1
eE020
> mc
0senh (0/4mc2) mc
00/4mc2 4m
2
20.
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k
A (tj ) = 0 , j = 1, 2, 3...
A (t) = E0
0sen(0t)
0tj =j tj
k2 + m2
A (t)
/2 1
|Q/2| 1
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tj
nk
nk
tj
tj
k+
k2 + m2 + e2E20(t tj)2
k= 0 ,
(t tj)
=
eE0(t tj ) =(k2+m2)eE0
d2kd2
+
+ 2j
2()
k= 0 ,
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tj
t1
wkb,k= 1
2exp
i
ini
d
,
(1)k () (0)k wkb,k() + (0)k wkb,k() ,
t2 (0)k
(0)k
(0)k 2 (0)k 2 = 1
k = 1 k = 0
lim
/
(0)k =wkb,k
N(1)k =
(0)k 2 ,
(0)k 2 = e ,
tj
tj
k
tj
(j)k () (j)k wkb,k() + (j)k wkb,k() ,
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(j)k
(j)k tj1 < t < tj
tj < t < tj+1 k
(j+1)k () (j+1)k wkb,k() + (j+1)k wkb,k() ,
(j+1)
k
(j+1)
k
tj < t < tj+1
(0)k (0)k
(j+1)k () (0)k (j)k () + (0)k jk () ,
(j+1)k(j+1)k
= (0)k (0)k(0)k
(0)k
(j)k(j)k
.
nj+1k =(j+1)k 2
=njk+(0)k
2
1 + 2njk
+ (0)k
(j)k
(0)k
(j)k .
=njk+ e 1 + 2njk+ (0)k (j)k (0)k (j)k termode fase .
nj
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tj
tj+1
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(t, x)
L =12
t
2
x
2 m22
,
S=1
2 dt
I(t)
dx
t 2
x2
m22 ,
I(t) = [0, l (t)]
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Dirichlet : (t, 0) = (t, l (t)) = 0,
Neumann :
x (t, 0) =
x (t, l (t)) = 0,
Mista(DirichletNeumann) : (t, 0) = x (t, l (t)) = 0 , ou
x (t, 0) = (t, l (t)) = 0.
t2
x2
m2
(t, x) = 0, com (t, 0) = (t, l (t)) = 0 ,
2x {n(t, x)}
2xn(t, x) =2n(t) n(t, x) , 2n(t)
[0, l0]
2l0
sen
nl0
x
(n/l0)
2
n(t, x) =
2
l (t)sen[n(t) x] ,
n(t) =
nl(t)
qn(t)
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(t, x) =
n
qn(t) n(t, x) .
m
qn
qn+ 2n(t) qn 2
m
Mnm(t) qm+
m
Mmn(t) Nnm(t)
qm= 0 ,
2n(t) =
2n(t) + m
2
Mnm(t) Nnm(t)
Mnm(t) =
I(t)
dxn(t, x) m(t, x) =
l(t)l(t)
(1)n+m 2nmm2n2se n =m
0 se n= m
Nnm(t) =
I(t)
dxn(t, x) m(t, x) =
k
MnkMmk.
Mnm(t)
n(t) Mnm(t)
Mnm(t)
Mnm Mmn Mnm= 2Mnm
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E (t,x) = 0 , B (t,x) = 0
E (t,x) = tB (t,x) , B (t,x) =tE (t,x) .
x
E= 0 x E= 0 B= 0 x B= 0 ,
E(T E) (t,x) x = 0
B(T M) (t,x) x = 0
A(T E)
A(T M)
.A(TE/TM) = 0 , A(TE/TM) (t,x) = 0 ,
A(T E) (t,x) x= 0 A(T M) (t,x) x= 0
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S(t0)
xespelho=l (t) t0
xespelho=l (t0) (t t0) + l (t0) l (t0)
t0 S
(t0)
S(t0)
S
(t0)
tA(T E)
t
= 0, x
= 0, x
= 0 ,
xA(T M)
t
= 0, x
= 0, x
= 0 ,
t =(vx+ t) =
d/dt x =(x+ vt) = 1/
1 v2
A(T E)
t, x= l (t) ,x
= 0 ,
(x+ vt)A(T M)
t, x= l (t) ,x
= 0 ,
(lx=l (t) , ly, lz)
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(t,x) = 0, com
|paredes= 0, modo TE(x+ vt) |x=l(t)=x|paredes estticas, modo TM .
t >0
n(t, x) = 1
2(3)n (t)
2
lxcos
nx
lxx
2
lycos
ny
lyy
2
lzcos
nz
lzz
,
t 0
n(t, x) = 1
2(3)k
2
l (t)cos
kx
l (t)x
2
lycos
ky
lyy
2
lzcos
kz
lzz
,
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(kx, ky, kz) > 0
(3)k
(3)
(3)n (t) =nxl (t)
2+ny
ly
2+ny
ly
2
k2
.
nx, ny, nz = 1, 2,...
n = (nx, ny, nz) m(t,x)
qn+ (3)2n (t) qn= 2 (t)
m
Mnm(t) qm (t)m
Mnmqm
2 (t) l2 (t)m
Nnm(t) qm m
Nnm (t) l
2 (t) (t) Qnm
qm
(t) l2 (t)m
Nnmtqm ,
Mnm=l (t)
l(t)
0
dx
ly
0
dy
lz
0
dzn(t,x)
l (t) m(t,x)
=
(1)nx+mx 2nxmxm2xn2x nymynzmz, se nx=mx ,
nymynzmz, se nx=mx , (t) =l (t) /l (t) ,
Nnm=
l(t)
0
dx
ly
0
dy
lz
0
d z v n(t,x) m(t,x) ,
Qnm=l2 (t)
l(t)
0
dx
ly
0
dy
lz
0
dz
v (3)2m v
n(t,x) m(t,x) + 2v
n(t,x) m(t,x)
,
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nx = 0
lx ly, lz
tin tout l (t)
t < tin t > tout tin tout +
in: inn = n(t < tin) =const = 0, Mnm(t < tin) = 0 ,out: outn = n(t > tout) =const = 0, Mnm(t > tout) = 0 ,
l (t) =l0[1 + sen(cavt)] , 1 .
cav = 2
inn
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Nn(t) = sinh2 (nt) , com n=
n
inn
l0
2.
= 3 103kg/m3
/2= 2GHz
x = 1nm
V = 3cm 3cm 0, 1mm
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3 108W
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9 1 8 cm3
103
104
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N >103
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H/mpl 105
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S[] = 12
d4x
gg.
ds2 = a2 () (d2 ijdxidxj)
S[] =
1
2 d
4x a2 () 2
ijij .
a1 ()
(, x) = 1
a ()
d3k
(2)3/2k() e
ikx ,
k = k
k= 12 Rk + iIkS=
1
2
d
R3+
d3k
R
k
2+
I
k
2 2a
a
Rk
R k +
Ik
I
k
+
a2
a2 k2
Rk2
+
Ik2
.
R,I
k
R,Ik
pR,Ik = LR,I
k
=R,I
k a
aR,Ik .
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H=
R3+
d3k
1
2
pRk2
+ 2a
aRkp
Rk + k
2
Rk2
+1
2
pIk2
+ 2a
aIkp
Ik+ k
2
Ik2
,
1
2
d
d
a
a
Rk2
+a
a
Ik2
,
S=1
2
d
R3+
d3k
R
k
2+
I
k
2
a
a k2
Rk2
+
Ik2
,
2k() =k
2 a/a
H=
R3+
d3k
1
2
pRk2
+
a
a k2
Rk2
+1
2
pIk2
+
a
a k2
Ik2
.
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V() = 12
m22
S=1
2
d
R3+
d3k
R
k
2+
I
k
2 k2 + a2m2 aa
Rk2
+
Ik2
,
2k() = k
2 +
a2m2 a/a
a () = 1HI
.
2k() =k2 1
2
2 m
2
H2I
.
m mpl
t2
1/2
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k+ 2k()
2k= 0 ,
(, x) ,p
, x
= ix x
,
(, x) =
d3k
(2)3/2
k() ake
ikx + k() ake
ikx
,
ak a
k
ak, a
k
=
k k
,
kk
k k=i ,
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k()
Hk=2k() a
kak .
ak |0 = 0 ,k .
ak |0 = |1k ,
1 |0, 1
2
0, 2| Nk(1) |0, 2 = 0 .
ak(2)
ak(2)
=
k
k
k k
ak(1)
ak(1)
,
k k
|k|2 |k|2 = 1 .
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N(1)k
2
=
|k(1, 2)|2 .Nk(t1) + |b|2 (1 + 2 Nk(t1)) , seNk(t1) = 0
,
k() = C1H(1) (k) + C2H(2) (k) ,
9
4m
2
H2.
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z k 1
k() expi
z2
4
,
Q/2 1 ,
k+
2k() Q
k= 0 .
Q2= 142
1 6 (k/)21 (k/)23 ,
= (m2/H2I) 2
m2 1025
2
k
k
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k() kwkb,k() + kwkb,k()
= k
2exp
i
ini
d
+ k
2exp
+i
ini
d
,
wkb,k= 1
2kexp
kd
.
k
C1 C2
2k() =k
2
2/2
k= 1
k = 0
wkb,k() = 1
2k
1 i
k
exp(ik) ,
k 0
|Q/2| 1/42
|Q/2| 1/8
() =2
ka () ,
lH() =a2
a ,
k
k 0
lH
lH
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lH
k 0
lH
V () =m22/2
a (t) =am0 t
2/3
a ()1/
< < 0
a () 2
0 <
Q2= 14
1 6 (k)21 (k)23 0 , k ,
r 0
k 0
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Ninfl =
|k
|2
V () = 4/4
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(, m)
L =12
1
2m22 +
1
2
12
m22 1
2g222 +
1
2R2 ,
g R
V () 1
2m2 ( )2
1
2g222
12
g222 12
2 12
g222
mpl
V () =
1
2m
2
2
(x, t) =
d3k
(2)3/2k(t) e
ikx ,
k+ 3a
ak+
k2 + m2 R+ g22
k= 0 .
= 0
mm
m m
k/a m
m H
a= 1
L =12
1
2m22 +
1
2
12
g222 .
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p=
L0 =0= ,
p
p
kk pk= kpk (t,x) , (t,y)
= i(x y) ,
[k, pk ] =
k k
(, x) = 1
(2)3/2
d3k
akk(t) e
ikx + akk(t) e
ikx
,
ak, a
k
=
k k
,
k(t)
kk kk= 2i ,
k+
k2 + m2+ g
22
k= 0 ,