1. =δ(x− y),S(φ)=∫ dx( ∂µφ∂µφ− m φ 2),S(φ)=S(φ0)+∫ dx |φ0 (φ(x

1.
δj(x)
δj(y)
= δ(x − y),S(φ) =
R
dx
R
1
1 2 2
µ
∂ φ∂ φ − 2 m φ ,S(φ) = S(φ0) +
2 µ
δS
dx δφ(x) | φ0 (φ(x) −
φ0(x)) + ....
R
R
δS
δφ(x)
δφ(x)
2. δφ(y) = dx ∂ µφ∂ µ δφ(y) − m2 φ(x) δφ(y) = dx(∂ µφ∂ µδ(x − y) − m2 φ(x)δ(x − y))
=−∂ µ∂ µφ(y) − m2 φ(y)
3.
4.
5.
1R
2
d x dx ′ j(x)∆F (x−x ′)j(x ′) δZ0(j)
Z0(j)
=
e
, δj(y)
=
R
′
δj(x )
dxdx ′ j(x)∆F (x − x ′) δj(y) = Z0(j) dxj(x)∆F (x − y)
δ 2Z0(j)
δj(z)δj(y)
δZ0(j) R
δj(z)
R
e
1R
2
dx dx ′ j(x)∆F (x−x ′)j(x ′)R
dxj(x)∆F (x − y) + Z0(j) dxδ(x − z)∆F (x − y)
R
R
=Z0(j) dxj(x)∆F (x − z) dxj(x)∆F (x − y) + Z0(j)∆F (z − y)
=
δ 2Z0(j)
|
= ∆F (z
δj(z)δj(y) j =0
6. Li = e µφ
− y)
1. <q ′, t ′|q, t > =
2.
R
q(t ′)= q ′
q(t)=q
Dqe
Q n=N R
dqn
n=0
i
h
R
d τL(q,q̇)
q
m
e
2πhiε
i PN
ε n=1
h
q
1
m
2
n+1 − qn
ε
2
−V (qn)
,tn > tn−1... ⋗ t0
q(τ1)q(τ2)
3. <q ′, t ′|q̂ (τ1)q̂ (τ2)|q, t > =<q ′, t ′|q̂ (τ1)|q(τ1), τ1 > <q(τ1), τ1|q̂ (τ2)|q, t > =
i R
R q(t ′)= q ′
d τL(q,q̇)
h
Dqe
q(τ1)q(τ2) si τ1 > τ2
q(t)=q
4. <q ′, t ′|q̂ (τ2)q̂ (τ1)|q, t > =
q(t ′)=q ′
q(t)=q
R
Dqe
i
h
R
d τL(q,q̇)
q(τ1)q(τ2), si τ1 < τ2
5. T q̂ (τ1)q̂ (τ2) = θ(τ1 − τ2)q̂ (τ1)q̂ (τ2) + θ(τ2 − τ1)q̂ (τ2)q̂ (τ1)
6. <q ′, t ′|T q̂ (τ1)q̂ (τ2)|q, t > =
R
∂ψ
q(t ′)=q ′
q(t)= q
7. ψ(t) = U (t, t ′)ψ(t ′), i ∂t = HI (t)ψ,i
′
8. U (t, t ) =
P
n
n=0
9. U (1)(t, t ′) = −i
R
λ U
t
t′
(n)
(t, t
Dqe
i
h
∂U (t, t ′)
∂t
∂U (n+1)(t, t ′)
),i
∂t
′
R
dτL(q,q̇ )
q(τ1)q(τ2)
= λĤI (t)U (t, t ′)
= ĤI (t)U (n)(t, t ′)
dt1ĤI (t1)
R t2
R
(2)
′
2 t
10. U (t, t ) = (−i) t ′ dt2ĤI (t2) t ′ dt1ĤI (t1)
R
t
′
11. U (t, t ) = T exp −i t ′ dt1ĤI (t1)
1.
i3
j(x1)∆F (x1 −
3!
y1)j(x2)∆F (x2 − y2)j(x3)∆F (x3 − y3)κ(3)(y1, y2, y3) = W (3)
(3)
2. Gc (x1, x2, x3) = ∆F (x1 − y1)∆F (x2 − y2)∆F (x3 − y3)κ(3)(y1, y2, y3)
3. φ(3)
c (x) =
δW (3)[j]
i δj(x)
i2
= 2! ∆F (x − y1)j(x2)∆F (x2 − y2)j(x3)∆F (x3 − y3)κ(3)(y1,2, y3)
Parámetros de Feynman:
1
=
a
d
Z
∞
dte
−at
0
1
1
=
2 + m2 (k + p)2 + m2
k
d
(2π)
d k
11
=
ab
Z
0
d
∞
dt1dt2
d k
(2π)d
Z
∞
dt1dt2 e−(at1 +b t2)
0
e−(at1 +b t2) a = k 2 + m2, b = (k + p)2 + m2
Trazas
1. {γ µ , γ ν } = 2η µν
1
2. 2 (Tr(γ µγν ) + Tr(γνγ µ)) = δ µν Tr(1) = 4δ µν = Tr(γ µγν )
µ
ν
3. Tr(γ µγνγαγ β ) = Tr(Sγ µγνγαγ βS −1) = Tr(Sγ µS −1SγνS −1SγαS −1Sγ βS −1) = L−1 µ̄L−1ν̄
α
β
L−1ᾱL−1 β̄ Tr γ µ̄γ ν̄γ ᾱγ β̄
µ
4. S γ µ S −1 ≡L−1ν γ ν
5. Tr γ µ̄γ ν̄γ ᾱγ β̄ = Aη µνη αβ + Bη µαη νβ + Cη µβη αν = Aη βµη να + Bη βνη µα + Cη βαη νµ,
A=C
6. Tr(γ µ̄γ ν̄γ αγα) = Aη αµηαν + Bη ανηαµ + A4η νµ = Aη µν + Bη µν + 4Aη µν
1
1
7. γ αγα = ηαβγ αγ β = 2 ηαβ {γ α , γ β } = 2 ηαβ 2η αβ = 4
8. Tr(γ µ̄γ ν̄γ αγα) = 4 × 4η µν = (5A + B)η µν , 5A + B = 16
9. µ = ν = α = β = 0,4 = 2A + B,3A = 12,A = 4, B = −4
10. Tr(γ µγ νγ αγ β ) = 4η βµη να − 4η βνη µα + 4η βαη νµ
11. Tr(γ 5 γ µγ νγ αγ β ) = Aǫ µναβ ,γ 5 = iγ 0 γ 1 γ 2 γ 3, Tr(γ 5 γ 0 γ 1 γ 2 γ 3) = −4i = A