ФИАН / Nefedyev_Ktp

n=0 exp

−i Ztnn

V (τ)dτ

6= exp

n=0

−i Ztn n

V (τ)dτ ! .

N

t +1

N

t

+1

Y

T

X

S(t, t ) = T exp

i

t V (τ)dτ

∞

(−i)n

t T

{

V (t )V (t ) . . . V (t ) dt dt

. . . dt ,

0

−

Zt0

≡ n=0 n!

Zt0

1 2

n } 1 2

n

X

S(2)(t, t )

=

(−i)2

t

dt

t

dt T

V (t )V (t )

Zt0

Zt0

{

}

0

2!

1

2

1

2

1

t

t1

1

t

t

=

−

Zt0

dt1 Zt0

dt2V (t1)V (t2) −

Zt0

dt1 Zt1

dt2V (t2)V (t1)

2

2

0.02

0.925

0.01

0.920

0.915

5

10

15

20

0.910

-0.01

0.905

5

10

15

20

n

n

n

fn = (−1)nαnn! n≈1(−1)nαn√2πn

(−1)nen ln(αn).

e

n n . 1/α

n 1/α

n

n

n

Sn = Pm=0 fm

n

n n

Sn

α = 0.1

f(0.1)

n 1

|ii = |(k1σ1)(p1λ1)i

k1,2 σ1,2

|fi = |(k2σ2)(p2λ2)i,

p1,2 λ1,2

S

Sif = hf|S|ii = hf|S(0) + S(1) + S(2) + . . . |ii,

S

e

S(0) = 1

hf|S(0)|ii

hf|S(1)|ii = −iehe′γ′| Z

d4xψγ¯ µψAµ|eγi hγ′|Aµ|γi hγ′|0i + hγ′|γγ′′i = 0,

Aµ

S

iT (2)

hf|S(2)|ii ≡

√

if

(2π)4δ(4)(p1

+ k1 − p2

− k2),

2ω12ω22E12E2

T

(2)

if

S(x − y) ≡ −ih0|T (ψ(x)ψ¯(y))|0i, S(z) = Z

d4p

S(p)e−ipz.

(2π)4

Dµν (x − y) = ih0|T (Aµ(x)Aν (y))|0i, Dµν (z) = Z

d4k

Dµν (k)e−ikz.

(2π)4

O

O

±O(t2)O(t1),

t1

< t2

T (

(t1)

(t2)) =

O(t1)O(t2),

t1

> t2

Aµ(x) Aν (y)

N T

Aµ(x)Aν (y) =

[aµ(x)b† (y)],

x0 > y0

= −gµν Z

d3k 1

|

−

|

−

,

[aν (y)bµ†

(x)],

x0 < y0

(2π)3 2ω e−

x0

y0

ν

iω

+ik(x y)

¯

¯

ψ = a + b, ψ = a¯ + b,

−

i{bβ(y)¯bα(x)},

x0

< y0

Sαβ(x

y) =

−i{aα(x)¯aβ (y)}, x0

> y0 =

= −i(iγµ∂µ + m)αβ Z

d3p

1

e−iEp|x0−y0|+ip(x−y),

(2π)3

2Ep

p → −p

1

e−iEp|x0

−y0| = i Z

dp0

1

e−ip0(x0−y0) = i Z

dp0

1

e−ip0

(x0

−y0),

2Ep

2π

p02 − Ep2 + i0

2π

p2 − m2 + i0

Sαβ(x − y) = Z

d4p

(ˆp + m)αβ

e−ip(x−y),

(2π)4

p2 − m2 + i0

T (AB) = N(AB) + AB,

n

n + 1

t1 > t2 > . . . > tn > tn+1

T (A1 . . . AnAn+1) = T (A1 . . . An)An+1 = N(A1 . . . An)An+1

+

AiAjN(A1 . . . An)ijAn+1 + . . . ,

i6=j

X

n

i j N(A1 . . . An)ij

Ai Aj

N(A1 . . . Am)An+1

m

Xi

N(A1 . . . Am)An+1 = N(A1 . . . AmAn+1) +

AiAn+1N(A1

. . . Am)i.

=1

iTif

T