Dissertacao Elisa Ferreira

7/25/2019 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)


6/206


7/206


8/206


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10/206


11/206


12/206

Treaq


13/206


14/206


15/206


16/206

4%

0, 001%

23%

73%


17/206


18/206

ds2 =dt2 a2 (t)

dr2

1 kr2 + r2

d2 + sen2d2

dl2

=gdx

dx

.

dl

ds2 = 0

cdt= dxfis= a (t) dl ,


19/206

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]


20/206

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


21/206

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


22/206

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 ,


23/206

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 ,


24/206

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


25/206

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


26/206

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()


27/206

ti = 0

p(t) =a (t)

t

0

dt

a (t)

,

p

p


28/206

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


30/206

1032s

108 109K

H2

H3

He4

z 104

z= 1400

z = 1100

z= 2 103


31/206


32/206

H0

72km s1Mpc1

He3


33/206

T = 2, 725 0, 002K


34/206

1044

= 1

( 1) a2H2 =k .

( 1) H2a2

( 1) a

( 1) a2

(0 1)


35/206

| 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 .


36/206

= 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


37/206

1016


38/206

1028K


39/206

ti = tpl 1043

s

H1

[ (ti= tdec,t0)]. >(H1).

(ti = tdec,t0) a

d

dt a

H1= a >0 .


40/206

H = Ha

+ 3p


41/206

| 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


42/206

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 ,


43/206

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


45/206

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 ,


47/206

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


48/206

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 ,


49/206

N (i/mpl)2

i

mpl


50/206

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 .


51/206

= d

d,

()

d

d=

12 ( + m22)

1/2 + m2

.

(, )

2 m22 ,

(, ) 0

(, )

= 0


52/206

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


53/206

Treaq


54/206

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


55/206

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 ,


56/206

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 .


57/206

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


58/206


59/206

105


60/206


61/206

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


62/206

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

,


63/206

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


64/206

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+ .


65/206

Si=Si+

i , Fi = Fi+ i .

hij

1a

a

B E , aa

B E .

vi=Si Fi .

B= 0 , E= 0 , Si= 0 ,


66/206

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


67/206

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


68/206

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


69/206

(, 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,

.


70/206

+ 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


71/206

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 ,


72/206

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


73/206

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


74/206

2i = 0 ,

i+ 2

Hi = 0 .

i(, x) =Ci(x) a2 ,

h

ij+ 2Hh

ij 2hij = 0 ,


75/206

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 .


76/206

S=1

2

d4x (Lgrav+ Lm)

= 1

16G

d4x

g

R+ Lm

,

ds2 = N2 NiNi dt2 2Nidxidt gijdxidxj ,


77/206

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] .


78/206

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


79/206

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].


80/206

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

,


81/206

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


82/206

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


83/206

|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


84/206

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) .


85/206

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)


87/206

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


88/206

E2

B2 >0

E B


89/206

Ecrit 1.3 1018V /m


90/206


91/206

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 .


92/206

(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


93/206

= 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


94/206

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


95/206

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 .


96/206

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

.


97/206

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

.


98/206


99/206

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


100/206

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)


101/206

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


102/206

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 (+) + () ,


105/206

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 .


106/206

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


107/206

B

F

Ecrit

1.3

1018V /m

|eE| m2


108/206

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 .


109/206

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


110/206

|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)


111/206

N(t) = 1(2)3

d3k Nk(t) .

=, p = i

,

p i

p i

| [, , t]

| =D2 (, ) (, ) ,

D2= DD

(, ) = , | .


112/206

[, , 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)


113/206

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

.


114/206

= 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 ,


115/206

|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

()


116/206

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

()


117/206

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


118/206

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.


119/206

Nk(t)

= cosh2 rk.

k = 0, 1

kz = 0, 01 m2 = 0, 2


120/206

Ec=

m2c3

e 1, 4 1016V/cm

E(t) =E0sen(0t)

0


121/206

E(t) = E0/ cosh

2 (k0t) A (t) =E0k0

tanh(k0t)


122/206


123/206


124/206

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


125/206

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 |


126/206

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


127/206

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


128/206

= (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


129/206

= 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


130/206

= 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


131/206

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 .


132/206

=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


133/206

eE m2g () ,

eE m0c

sinh(0/4mc2),

46, 6GeV/c


134/206


135/206

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


136/206

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


137/206

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.


138/206

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


139/206

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 ,


140/206

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() ,


141/206

(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


142/206

tj

tj+1


143/206


144/206


145/206

(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)]


146/206

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)


147/206

(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


148/206

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


149/206

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)


150/206

(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

,


151/206

(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)

,


152/206

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


153/206

Nn(t) = sinh2 (nt) , com n=

n

inn

l0

2.

= 3 103kg/m3

/2= 2GHz

x = 1nm

V = 3cm 3cm 0, 1mm


154/206

3 108W


155/206

9 1 8 cm3

103

104


156/206


157/206

N >103


158/206


159/206

H/mpl 105


160/206

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 .


161/206

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

.


162/206

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


163/206

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 ,


164/206

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 .


165/206

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.


166/206

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


167/206

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


168/206

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


169/206

Ninfl =

|k

|2

V () = 4/4


170/206

(, 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 .


171/206

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 ,

Dissertacao Elisa Ferreira

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Transcript of Dissertacao Elisa Ferreira