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Quantum Field Theory

Gregory Moreau∗

Laboratoire de Physique Theorique, Bat. 210, Universite Paris-sud 11

F-91405 Orsay Cedex, France

Abstract

The present notes provide exclusively a synthesis of the formalism presented in the lec-

ture series “Quantum Field Theory” (part of the Major Course “Statistical and Quantum

Mechanics”) at the 1st year Master General Physics of the Paris-Saclay University. In

these lectures, I provide an explicit and pedagogical approach to the so-called second

quantisation, for the simplest case of a scalar field (spinless). All the non-trivial calcu-

lations, including their intermediate steps, are performed explicitly. There are several

ways to treat the second quantisation: the canonical quantisation is presented in these

lectures. The final goal is to be able to calculate the transition amplitude for the simplest

1→ 2-body particle decay, at the level of one loop with respect to quantum corrections.

L.Euler E.Noether

V.A.FockG.-C.Wick

O.Klein

W.Gordon

W.R.Hamilton R.Feynman

E.SchrodingerJ.-L.Lagrange

∗[email protected]

Contents

Motivations 4

1 Analytical Mechanics 5

1.1 Rigid body coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Principle of least action . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.3 Quantum description . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Quantisation of the scalar field 10

2.1 The Lagrangien and its Hamiltonian . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2 Euler-Lagrange equation . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Hamiltonian density . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.4 Canonical commutation relation . . . . . . . . . . . . . . . . . . . 10

2.2 Decomposition of the field . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Canonical commutator . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Decomposition of the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 14

3 Fock space 16

3.1 Recalling the Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 1D Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.2 Generalisation to N H.O.’s . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Multi-particle states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.2 Normal ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.3 Physical/Quantum interpretation . . . . . . . . . . . . . . . . . . . 19

3.2.4 Normalisation of states . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Complex scalar field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.2 Physical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.3 Noether’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Propagator 24

4.1 Solving the generic K.-G. equation . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Introducing the time-ordering . . . . . . . . . . . . . . . . . . . . . . . . . 26

2

5 Evolution operator 28

5.1 Structure of interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2 Working out the physical states . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2.1 S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Reaction amplitudes 33

6.1 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.1.1 Wick contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.1.2 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.2 A simple decay example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.2.1 First order contribution . . . . . . . . . . . . . . . . . . . . . . . . 34

6.2.2 Next order terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.3 Amplitude calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.3.1 Quantum state normalisation . . . . . . . . . . . . . . . . . . . . . 37

6.3.2 Momentum/position integrations . . . . . . . . . . . . . . . . . . . 38

6.3.3 Generalisation to Feynman rules . . . . . . . . . . . . . . . . . . . 39

3

Motivations

Quantum Mechanics must be extended to (special) relativity framework :

→ consider relativistic fundamental equations “energy in

particular”→ write a covariant formalism

→ go beyond potential description [implying the transfer of information at an infinite

speed] “untenable”

The Quantum Field Theory (QFT) (introducing the creation/annihilation operators)

further :

• allows a systematic amplitude calculation for any Feynman diagram“rigorous

complete

formalism”

• includes the possibility of absorption/creation of elementary particles

• provides the correct interpretation for probability density

Overview of the fundamental variables

Classical physics :Newton’s law

x, p (momentum), t

⇓

⇒

Special relativity :Covariant form

xµ, pµ (4-vectors)

⇓

Quantum Mechanics :Schrodinger equation

X, P (operators in space H) ⇒

Quantum Field Theory :Klein-Gordon and Dirac equations

φ(xµ), ψ(xµ), Aµ(xµ) (wave functions in H)

↓Second quantisation :

φ(xµ), ψ(xµ), Aµ(xµ) (fields in Fock space)

4

1 Analytical Mechanics

1.1 Rigid body coordinates

A non-relativistic system is described by spatial coordinates qr(t), time derivatives (could

be velocities) qr(t). The ‘Lagrangian mechanics’ defines mathematically the action,

A =

∫ t2

t1

dt L

(qr(t), qr(t), t

).

1.1.1 Principle of least action

The principle: Among all trajectories (δqr(t) infinitesimal variations) that join qr(t1) to

qr(t2) (δqr(t1) = δqr(t2) = 0 at boundaries), the system follows the one for which A is

stationary (δA = 0; minimum).

The consequence: Euler-Lagrange equations i.e. Equations of Motion (EOM),

d

dt

(∂L

∂qr

)=∂L

∂qr

[A] = [E.T ] (as in QFT)

Exercise 1: Show that the Euler-Lagrange equations for L1D = 12mx

2−V (x) are nothing

else but the Newton’s second law(q ≡ x, r = 1).

m x︸︷︷︸a(x)

= −V ′(x)︸︷︷︸Fx

The momenta ‘conjugate’ to qr is defined by,

pr =∂L(qr, qr)

∂qr. (1)

(case of Exercise 1: p = mx is the canonical momentum)

1.1.2 The Hamiltonian

Hamiltonian definition (to be checked in Exercise 2):

H(qr, pr) = pr qr (qr, pr)− L(qr, qr(qr, pr))︸︷︷︸assumes Eq.(1) is invertible

. (2)

5

Notice that pr qr uses the Einstein’s convention of implicit summation over r,∑

r pr qr.

By differentiation,

dH = dpr qr + prdqr −(∂L

∂qr︸︷︷︸pr via Eq.(1)

dqr +∂L

∂qrdqr

)

⇔ ∂H

∂pr= qr,

∂H

∂qr= −pr .

EOM from the ‘Hamiltonian mechanics’: the alternative analytical branch.

Exercise 2: Apply the Hamilton’s EOM to the Lagrangian of Exercise 1.

x =∂H

∂p=

p

m, −mx =

∂H

∂x= V ′(x) (Newton’s law)

as H1D = (mx)x− 1

2mx2 + V (x) =

1

2mx2 + V (x) =

p2

2m+ V (x) .

Poisson bracket of two any dynamical variables :

[f1, f2]P =∂f1

∂qr

∂f2

∂pr− ∂f1

∂pr

∂f2

∂qr, ∀L

(i) ⇒ [qr, ps]P =∂qr∂qt

∂ps∂pt− ∂qr∂pu

∂ps∂qu

= δtrδts = δrs (3)

(ii) ⇒ [qr, H]P =∂qr∂qs

∂H

∂ps− ∂qr∂ps

∂H

∂qs= δsr qs = qr

(iii) ⇒ [pr, H]P =∂qr∂qs

∂H

∂ps− ∂pr∂ps

∂H

∂qs= −δsr(−ps) = pr

Possible definition :

f ≡ df

dt=∂f

∂t︸︷︷︸for explicit f(t) dependence

+ [f,H]P (4)

Analytical formalism generalisable to special relativity:

{4-force fµ = dpµ

dτ

4-velocity vµ

6

1.1.3 Quantum description

(1): extendable to pr

(2): valid for P ,X and L1D = P 2

2m − V (X)

as well as H1D = P 2

2m + V (X) and Hφ = i~∂tφ(~x, t) (Schrodinger’s equation)

(3): operator expression of Heisenberg principle of uncertainty, [Xr, Ps] = i~δrs

(4): Ehrenfest theorem,

d

dt< O >|φ(t)> =

1

it< [O, H >|φ(t)> + <

∂O∂t

>|φ(t)> with < ~x|φ(t) >= φ(~x, t) .

The Eq.(3) and Eq.(4) could be extended from classical to quantum analytical mechanics

with the replacements: [ , ]P → −i[ , ](−) and 1 → ~.

1.2 Field theory{Relavistic extension (xν , ...)

Quantum mechanics (φ, ...)⇒ introduce fields φ(xν)

Action defined now as,

A =

∫d4x︸︷︷︸

treat space/time on same footing

L(φA(xν), ∂µφ

A(xν)

).

Remarks:

- no xν explicit L dependence since all physics through fundamental fields

Example : no V 6= V (φ) (no other source/sink of energy,...)

- d4x and L : independently Lorentz invariant

- φA is a function of xν

- A,L are

{functional of φA

function of xν

Principle of least action with : δφA(xν)’s between boundaries xν1 , xν2 .

Conjugate momenta :

Πa=δL(φ, ∂φ)

δφAwith functional derivative

δφB(t, ~y)

δφA(t, ~x)= δBAδ

3(~x− ~y) .

7

Hamilton density : (⇒ Hamilton’s equation)

H(φA,ΠA, ~∇φA) = ΠAφA − L(φA, ∂µφ

A(ΠA, φA, ~∇φA)) . ((2) bis)

Euler-Lagrange equations (demonstration through the Gauss theorem):

∂µ∂L

∂(∂µφA)=

∂L∂φA

∀A (5)

Poisson bracket generalisation (& application to φ,Π):

[φA(t, ~x),ΠB(t, ~y)]P =

∫d3z

(δφA(t, x)

δφC(t, z)

δΠB(t, y)

δΠC(t, z)− δφA(t, x)

δΠC(t, z)

δΠB(t, y)

δφC(t, z)

)=

∫d3z δACδ

3(x− z) δCB δ3(y − z)

= δAB δ3(x− y) (6)

(d3z to sum, as C summation, and not introduce new dependence from denominators)

Eq.(4) extendable via f → f(xν).

Exercise 3: For the Lagrangian L = −14FµνF

µν−jµAµ of an electromagnetic field, derive

the momenta then H.

∀µ Πµ =δL

δ(∂oAµ)= −1

4(

δ

δ(∂oAµ)[∂αAβ − ∂βAα]Fαβ + Fαβ

δ

δ(∂oAµ)Fαβ)

= −1

4([δoαδ

µβ − δ

oβδµα]Fαβ + Fαβ

δ

δ(∂oAµ)Fαβ)

= −F oµ

so Πo = 0 and Πi = −F oi = −(±Ei) = F (−∂iφ− Ai) , φ = Ao

⇒ H definable in gauge Ao = 0

{no Πo to introduce (no field conjugate)

H = ΣiA2i − L

Exercise 4: Find H for L = (∂µφ)†∂µφ−m2φ†φ− V (φ, φ†) for complexe scalar fields.

Πφ =δLδφ

=δ

δ(∂oφ)[δoφ

†∂oφ] = ∂oφ† and similarly Πφ† = ∂oφ

so H = (∂oφ†)∂oφ+ (∂oφ)∂oφ

† − [(∂oφ†)∂oφ− (∂iφ)†∂iφ−m2|φ|2 − V ]

= |φ|2 + |~∇φ|2︸︷︷︸(~∇φ)†.~∇φ

+m2|φ|2 + V

8

Quantum description (second quantisation)

φ→ φ, H → H, Π→ Π ex : −i[φA(t, x), ΠB(t, y)] = δAB δ3(x− y) (7)

Noether’s theorem (relativistic quantum framework) applicable to internal/external sym-

metries:

τ

{xµ → x′µ = xµ + δxµ

φA(xν)→ φ′A(x′ν) = φA(x) + δφA(x), δA = 0 under τ ⇒

∂µJ

µ = 0

Jµ = ∂L∂(∂µφA)

δφA − TµνδxνTµν= ∂L

∂(∂µφA)∂νφA − gµνL

9

2 Quantisation of the scalar field

2.1 The Lagrangien and its Hamiltonian

The lagrangian density for a wave function (spinless particle)

A =

∫d4x L , L =

1

2∂µφ∂µφ−

1

2m2φ2 ≡ 1

2[(∂oφ)2 − (~∇φ)2]− 1

2m2φ2 (8)

describes the kinimatics and mass of a “scalar field”.

2.1.1 Dimensions

In the natural unit system (~ = c = 1),

[A] = [~] = 1⇒ [L] = [E]4 ⇒ [φ] = [E] ([m] = [E]) .

[better not start from the probability density]

2.1.2 Euler-Lagrange equation

Eq.(5) :∂L∂φ

= ∂ρ[∂L

∂(∂ρφ)]−m2φ = [

1

2δρµ∂

µφ︸︷︷︸∂ρφ

+”]−m2φ

Quantum relativistic Equation of Motion:

(� +m2)φ = 0 (Klein-Gordon) (9)

2.1.3 Hamiltonian density

H = Π(φ)φ− L =1

2(∂oφ)2 +

1

2(~∇φ)2 +

1

2m2φ2 [see Section 1.2] (10)

since Π(φ) =∂L∂φ

=∂L

∂(∂oφ)= ∂oφ [see Section 1.2] (11)

2.1.4 Canonical commutation relation

Apply Eq.(6) ⇒ Eq.(7).

[φ(x, t), Π(y, t)] = iδ3(~x− ~y) ((7) bis)

Eq.((7) bis) → gradual development of the canonical second quantisation.

...covariance of Eq.((7) bis)?

10

• non zero for ~x = ~y ⇒ xµ = yµ ⇔ dxµ = 0⇒ dxµdxµ = ds2 = 0

•

(Π = ∂oφ is the time-component of a 4-vector (∂µφ)

φ scalar field : φ(xµ) = φ′(x′µ)

•∫d4xδ3(x− y) = dt and d4x scalar so δ3 time-component of a4-vector (like dxµ)

Eq.((7) bis) : V o1 = V o

2 , and same discussion in other frame : V′o

1 = V′o

2 .

2.2 Decomposition of the field

In terms of integrable functions (on R): any fonction can be Fourier decomposed. Hence

the field φ(xν) solution of Eq.(9) can be rewritten as,

φ(xν) =1

(2π)3/2

∫d4p δ(p2 −m2)A(pµ)e−ip

µxµ (12)

We use the notation p2 for the Lorentz’s square: p2 = pµpµ.

Fourier transformation with a sign convention different for

[xo

poand

[xi

pidue to

the metric (+−−−).

The introduction of δ(4) in the A definition permits φ to satisfy the Klein-Gordon equa-

tion,

(� +m2)φ =1

(2π)(3/2)

∫d4p δ(p2 −m2)A(pµ) (� +m2)e−ip

oxo+ipixi︸︷︷︸([−ipo]2 − [ipi]2 +m2)︸︷︷︸−[po2−Σipi

2 ]=−p2

e−ipµxµ

= 0

Real scalar φ = φ†(xν)∫d4p δ(p2 −m2)A(pp)e−ip

µxµ =

∫d4p δ(p2 −m2)A†(pρ)eip

µxµ (13)

=

∫ ∫ ∫ ∫ +∞

−∞d4(−p) δ((−p)2 −m2) A†(pρ)︸︷︷︸

if =A(−pρ) equality OK(*)so indeed A†(p)=A(−p)

e−i(−pµxµ)

(*) After the change of variable :

p′µ = −pµ and renaming p′µ → pµ as integration variable.

Let us now rewrite φ in a useful form, using the step function introduced as follows,

11

θ(x) =

1 for X > 012 for X = 0

0 for X < 0

⇔ θ(−x′) =

1 − x′ > 012 − x′ = 0

0 − x′ < 0

⇔ θ(−x) =

0 x > 012 x = 0

1 x < 0

, ∀x

so that θ(x) + θ(−x) = 1, ∀x. Introducing this sum in Eq.(12),

φ(xν) =1

(2π)32

∫d4p δ(p2 −m2)θ(po)A(p)e−ip

µxµ

+1

(2π)32

∫d4p δ(p2 −m2)θ(−po)A(p)e−ip

µxµ

+1

(2π)3/2

∫ ∫ ∫ ∫d4(−p)δ((−p)2 −m2)θ(−po)A(−[−p])ei(−pµ)xµ

φ(xν) =1

(2π)3/2

∫d4p δ(p2 −m2)θ(po)[A(p)e−ip

µxµ + A(−p)︸︷︷︸A†(p) from Eq.(13)

eipµxµ ] (14)

For a simpler form ⇒ eliminate the “po” dependence.

Let us recall the mathematical formula,

δ[f(x)] = Σnδ(x− xn)dfdx |x=xn

for |df(x)

dx|x=xn 6= 0

and apply it to

f(x) ≡ f(po) = po2 − ~p2 −m2 = pµpµ −m2

with the choice of the δ ‘function’ variable vanishing for po = ±√~p2 +m2=± Ep. |Ep|

is fixed so we take Ep > 0 as a convention throughout those lectures.

δ[p2 −m2] =δ(po − Ep)|2po|po=Ep

+δ(po + Ep)

|2po|po=−Ep(15)

Using the p.x = pµxµ notation for the invariant scalar product of Lorentz (with an

implicit sum on µ = 0, ..., 4) and inserting Eq.(15) into Eq.(14) gives rise to:

φ =1

(2π)3/2

∫d4p

1

2po[θ(po)δ(po − Ep) + θ(po)δ(po + Ep)]

(A(p)e−ix.p +A†(p)eix.p

)

φ(xν) =

∫d3p

1

(2π)3/22Ep

(A(pµ)e−ix.p + A†(pµ)eix.p

)

φ(xν) =

∫d3p√

(2π)32Ep

(a(pµ)e−ix.p + a†(pµ)eix.p

)with a(pµ) =

A(pµ)√2Ep

. (16)

12

Remark :

p2 = m2 (with E2p) or p2

o − ~p2 = m2 ⇔ p2o = E2

p︸︷︷︸as obtained here for the φ solution

It gives the 4-momentum relation to the mass/energy (mean/central/classical value

< P 2o >= E2

p), being relativistic, from which there are possible quantum fluctuations.

Eq.(11) : Π(φ)(xν) =

∫d3p i

√Ep

2(2π)3

(− a(pµ)e−ix.p + a†(pµ)eix.p

)(17)

2.2.1 Canonical commutator

Eq.(16) ⇒ dim a ≡ [E]−3/2.

[ap, a†p′ ] = δ3(~p− ~p′)

[ap, ap′ ] = 0

[a†p, a†p′ ] = 0

(18)

[⇒ (demonstration just below)

⇐ (demonstration: exercise)[φ(t, x), Π(t, y)] = iδ3(~x− ~y)

Exercise 5: Prove the implication ‘⇒’ above.

[φ(t, x), Π(φ)(t, y)] =

∫4-momentum

pµ with po = Ep︷︸︸︷d3p√

(2π)32Ep

∫d3p′ iEp′√

(2π)3√

2

([ape

−ix.p,−ap′e−ip′.y] + [ape

−ix.p, a†p′eip′.y]

+[a†peix.p,−ap′e−ip′.y] + [a†peix.p, a

†p′e

ip′.y]

)

=

∫ +∞

−∞

d3p

(2π)3√

2Ep

i√Ep√2

[ei~p.(~x−~y)e−iEpt+iEpt + e−i~p.(~x−~y)eiEpt−iEpt

]= i

1

2

1

(2π)3

[(2π)3δ3(~x− ~y) + (2π)3δ3(~y − ~x)

]= i δ3(~x− ~y)

Exercise 6: Prove that [φ(t, x), ∂iφ(t, y)] = 0.

∂iφ(t, y) =

∫d3p√

(2π)32Ep(ipi)(+ape

−ix.p − a†peix.p)

13

[φ, ∂iφ] = ... = −∫ +∞

−∞

d3p

(2π)3

ipi2Ep

ei~p.(~x−~y)−∫ ∫ ∫ ∫ −∞

+∞

d3(−p)(2π)3

(−i)(−pi)2 Ep︸︷︷︸

= E−~p ≡ (~p2 +m2)1/2

e−i~p.(~x−~y) = 0

2.3 Decomposition of the Hamiltonian

Eq.(10):

H =

∫d3x[

1

2(∂oφ)2 +

1

2~∇φ.~∇φ+

1

2m2φ2]

Exercise 7: Express H in terms of a, a†.

• Two times the first term of H reads as,∫d3x (∂oφ)2 =︸︷︷︸

Eq.(17)

∫d3x(−1)

∫d3p√2(2π)3

E1/2p

∫d3p′√2(2π)3

E1/2p′

(−ape−ix.p + a†peix.p)(−ap′e−ip

′.x + a†p′eip′.x)

using now,∫

d3x(2π)3 e

i~p.~x = δ3(~p) , “same mass

so same Ep:

Ep = Ep′”∫d3x(∂oφ)2 =−

∫d3p

√Ep2

∫d3p′

√Ep′

2(apap′ e

−i(Ep+Ep′ )t δ3(~p+ ~p′)− apa†p′ ei(Ep′−Ep)t δ3(~p− ~p′)

− a†pap′ ei(Ep−Ep′ )t δ3(−~p+ ~p′) + a†pa†p′ e

i(Ep+Ep′ )t δ3(−~p− ~p′))

= −∫d3p

Ep2

( (?)︷︸︸︷a(~p)a(−~p) e−i2Ept − a(~p)a†(~p)− a†(~p)a(~p) + a†(~p)a†(−~p)e2iEpt

).

(?)

~p′ = ±~p and a(pµ) = a(Ep, ~p) ≡︸︷︷︸

Ep = Ep(±~p)a(~p)

Ep′ = Ep

• Two times the second term of H reads as,∫d3x ~∇φ.~∇φ =

∫d3x

∫d3p ipi√(2π)32Ep

∫d3p′ ip′i√(2π)32Ep

(ape−ix.p−a†peix.p)(ap′e−ip

′.x−a†p′eip′.x)

Eq.(16) :

∫d3pi ipi√(2π)32Ep

(ape−ix.p − a†peix.p)

= −∫d3p

1

2

pipi

Ep(−a(~p)a(−~p)e−2iEpt − a(~p)a†(~p)− a†(~p)a(~p)− a†(~p)a†(−~p)e2iEpt)

14

Difference with starting point of previous calculation :√Ep → pi

E1/2p

.

• Two times the third term of H reads as,∫d3x m2φ2 =︸︷︷︸

Eq.(16)

∫d3p√

(2π)32Ep

∫d3p ′√

(2π)32Epm2

∫d3x(ape

−ix.p + a†peix.p)(ap′e

−ip′.x + a†p′eip′.x)

= m2

∫d3p

1

2Ep(a(~p)a(−~p)e−2iEpt + a(~p)a†(~p) + a†(~p)a(~p) + a†(~p)a†(−~p)e2iEpt)

⇒∫d3x(~∇2φ+m2φ2)

=

∫d3p

1

Ep[~p2 +m2]

(a(~p)a(−~p)e2iEpt + a(~p)a†(~p) + a†(~p)a(~p) + a†(~p)a†(−~p)e2iEpt

)⇒ H =

∫d3p

Ep2

(a(~p)a†(~p) + a†(~p)a(~p)

)(19)

Exercise 8: Calculate [H, a(~p)], [H, a†(~p′)] .

[H, a(~p)] =︸︷︷︸Eq.(19)

1

2

∫d3p Ep

[a(~p)a†(~p) + a†(~p)a(~p), a(~p)

]

= −1

2(Ep′a(~p′) + Ep′a(~p′)) = −Ep′ a(~p′)

(20)

[H, a†(~p′)] =1

2

∫d3p Ep

[[apa

†p, a†p′ ] + [a†pap, a

†p′ ]

]=

1

2(Ep′a

†(~p′) + a†(~p′)Ep′)

= Ep′ a†(~p′)

(21)

15

3 Fock space

3.1 Recalling the Harmonic Oscillator

In non-relativistic Quantum Mechanics.

3.1.1 1D Hamiltonian

H =1

2m(P 2 +m2ω2X2︸︷︷︸

V (X)

) ω : pulsation of H.O. with dimension T−1

Defining the dimensionless operators,

a =1√

2m~ω︸︷︷︸[M(MV 2)TT−1]1/2 = [MV ]

(P − imωX), a† =1√

2m~ω(P † + imωX)†

the Hamiltonian can be rewritten as,

H =1

2~ω(a†a+ aa†) OH.1

[X,P ] = i~⇒ [a, a†] = 1 OH.2

[H, a] =~ω2

[a†a+ aa†, a]

=~ω2

([a†, a]a+ a[a†, a]) = −~ωa

...and similarly, [H, a†] = ~ωa†. OH.3

Notice the dimension [~ω] = [E].

OH.1 and OH.2 : H = ~ω(a†a+1

2) OH.4

H|n >To Normalise= E|n >TN⇒

Ha|n >TN= (En − ~ω)a|n >TNHa†|n >TN= (En + ~ω)a†|n >TN (creates a quantum)

“still eigenstate with eigenvalues plus a quantum”

Remark : Energy conservation so a, a† appear together as e.g. in OH.4.

Ground state (of lowest energy) noted |o > (normalised) .

It must respect, a|o >= 0 OH.5

OH.4: H|o >= ~ω(a†a+ 12)|o >= ~ω

2 |o >

16

Then a†|o >︸︷︷︸noted |1 >TN as first excitation

has eigenvalue energy ~ω2 + ~ω = ~ω(1

2 + 1)

and a†|1 >︸︷︷︸|2 >TN since second excitation

has eigenvalue energy ~ω(1

2+2) to generalise : |n >TN with energy ~ω(

1

2+ n)︸︷︷︸

“En” OH.6

Remark : It turns out that there is no other eigenvalue of H.

Hence generalizing: |n >TN= (a†)n|o > and after normalization,

|n >=1√n!

(a†)n|o > OH.7

if < o|o >= 1 (always possible to set). It is easy to convince oneself by checking for first

states (< n|n >= 1) but the systematic (∀n) demonstration is longer.

Annihilation/creation operator actions on initial/final normalised states |n > can be

deduced:

• a†|n >=√

(n+ 1) 1√(n+1)!

a†(a†)n|o >=√n+ 1|n+ 1 > OH.8

• so < n|aa†|n >=< n|√n+ 1a|n+ 1 >⇔ 1 + ||a|n > ||2 = (n+ 1) < n+ 1|n+ 1 >

⇔ 1 + ||N |n− 1 > ||2 = (n+ 1)⇔ N2 = n⇒ N =√n.

Operator number : N = a†a since

N|n >= a†a|n >= a†√n|n− 1 >=

√n√

(n− 1) + 1|n >= n|n > .

3.1.2 Generalisation to N H.O.’s

H = ΣNi=1

1

2m(P 2

i +m2ω2X2i )

H = Σi~ωi2

(a†iai + aia†i )

Exercise 9: Show that [H, ai] = −~ωiai.(OH.3)

[H, aj ] = [Σi~ωi2

(a†iai + aia†i ), aj ] =

~ωj2

[(a†jaj + aja†j), aj ] = −~ωjaj

since aj commutes with all ai for i 6= j as belongs to different H’s.

Same individual energy spectrum (OH.6). Globally, H is additive (separate variables)

so its eigenvalues and eigenstates are,

E = ΣNi=1Ei = ΣN

i=1~ωi(1

2+ ni) = Σ

~ωi2

+ Σ~ωini

17

{|E >} = {|n1 > ⊗|n2 > ⊗...⊗ |nN >} (set of ni values )

Example:

H |2 >1 ⊗|o >2 ⊗...⊗ |o >N =

[(~ω(a†1a1 +

1

2)⊗ 1⊗ ...⊗ 1) +HO.H.

2 + ...

]|2 >1 |o >2 ...

=

(~ω(2 +

1

2) +

~ω2

+ ...

)|2 >1 |o >2 ...

= ~ω(2 +N

2) |2 >1 |o >2 ...

= E |2 >1 |o >2 ...

N = ΣNi = Σia†iai OH.10

3.2 Multi-particle states

3.2.1 Analogy

A free scalar field system is formally a continuous version of a multi-H.O.,

N =

∫a†papd

3p

⇔ OH.1 (generalised to N →∞)Eq.(18) ≡ OH.2Eq.(20), Eq.(21) ≡ OH.3OH.1 & OH.2⇒ OH.4

⇒ similar operators and same eigenstates |np >

ap|np >=√np|np − 1 > in a single Hp

Decomposition : Eq.(12) introduces a multi-Hilbert space description of the known multi-

H.O., which allows to find out the energy eigenstates and eigenvalues.

3.2.2 Normal ordering

This part treats a subtlety due to the infinite number of H.O.’s. Let us rewrite H again

using the a(†) commutation relation :

Eq.(19) : H =

∫d3p

Ep2

(apa†p + a†pap) =︸︷︷︸

Eq.(18) : compact notation for ...⊗ 1⊗ ap ⊗ 1⊗ ...

∫d3p

Ep

2(2a†pap + δ3(~op))

δ3(~op) is undefined mathematically and physically accounts for the infinite contribution

of finite ground state energies as in OH.1 : Σ∞i=1~ωi2 ∈ E.

Way out : only energy differences are physical (as classically: ~F = −~∇V ) so one

can introduce a prescription redefining the energy absolute scale. This is the so-called

18

“Normal ordering rearrangement”

Apply to H ≡

1.- Write all anihilation operators to the right of creation operators

in products, as if they were commuting with each other.

2.- Treat a and a† with usual commutation rules of Eq.(18), again.

Here it gives (for the step 1),

: H :=

∫d3p

Ep

2(a†pap + : apa

†p :︸︷︷︸

a†pap

) =

∫d3p Ep a

†pap (22)

which has indeed redefined the ground energy, : H : |o >=∫d3p Ep a

†pap|o >= 0 ,

in a minimal way from Eq.(19): only eliminating the δ(0) terms. (same a(†) formalism).

Remark :

< Ψ| : H : |Ψ >=

∫d3p < Ψ|a†pap|Ψ > Ep =

∫d3p Ep ||ap|Ψ > ||2 ≥ 0

Recall that Ep is positive by convention. This confirms (not proving) that Eo = 0 is well

the ground/minimal energy : any E ≥ Eo = 0.

Remark : Eq.(20) - Eq.(21) remain true for :H: (factor 2 compensates one term less).

3.2.3 Physical/Quantum interpretation

Exercise 10: (approximation)

: H : |E > =

∫d3p Ep(...⊗ 1⊗ a†pap ⊗ 1⊗ ...)(...|o > ⊗|2p1 > ⊗|3p1+δp > ⊗|o > ⊗...)

= (2Ep1 + 3Ep1+δp)︸︷︷︸whole energy

|E >

The term 2Ep1 corresponds to two times the energy of particle with momentum ~p1. |E >

is interpreted as a global state with 2 particles of Energy Ep1 and 3 particles of energy

Ep1+δp.

Hence |E > belongs to an Hilbert space of multi-particle states: Fock Space.

Therefore, e.g. in Hp, a†p|o >p=√

1|1p > shows that a†p has the effect of “creating“ a

particle, of momentum ~p, which is the now form of the (energy) ‘quantum’.

19

3.2.4 Normalisation of states

|np >=a†np√n!|op > generalisable in Hp ⇐ OH.3, OH.1

< 1p|1p′ > =< ap|apa†p′ |op′ >=< o|δ3(~p− ~p′) + a†p′ap|o >

= δ3(~p− ~p′)< o|o >~p︸︷︷︸=1

< 1p|2p′ > =< o|ap1√2!a†p′a

†p′ |o >=

1√2< o|(δ3 + a†p′ap)a

†p′ |o >= 0

So < np|n′p′ >= δnn′ δ3(~p− ~p′).

3.3 Complex scalar field

The Lagrangian of the complex scalar field (considered in Exercise 4) reads as

L = ∂µφ†∂µφ−m2φ†φ (23)

⇒︸︷︷︸EOM(φ,φ†)

K.-G. equation (for φ and φ†)

Remark : L† = L

Defining two real fields as,

φ(xν)=1√2

(φ1(xν) + iφ2(xν))

Eq.(23) can be re-expressed, omitting the “ˆ” symbols for simplicity.

Exercise 11:

L =1

2(∂µφ1 + i∂µφ2)†(∂µφ1 + i∂µφ2)−m2 1

2(φ1 + iφ2)†(φ1 + iφ2)

L =1

2[∂µφ

†1∂

µφ1+i∂µφ†1∂

µφ2−i∂µφ†2∂µφ1+∂µφ

†2∂

µφ2]−m2

2[φ†1φ1+iφ†1φ2−iφ†2φ1+φ†2φ2]

as [φ1, φ2] = 0 (as φ1,2) and φ†1,2 = φ1,2 taken Hermitian (can be seen as starting from

complex functions then quantised: φ1,2 real then Hermitian). Hence,

L = Σ2i=1[

1

2∂µφi∂

µφi −m2

2φ2i ] (24)

⇒ K.-G. Eq.(9) for φ1 and φ2. So the choice of physical degrees of freedom φ, φ† or

φ1, φ2 are equivalent at this level of free fields.

20

Then repeating the φ1,2 decomposition of Eq.(12) and formalism of Section 3.3, one

finds identical relations,

[aip , a†ip′

] = δ3(~p′ − ~p), [aip, aip′ ] = [a†ip, a†ip′ ] = 0 with i=1,2 and [a

(†)1p , a

(†)2p ] = 0.

((18) bis)

from L = L1(φ1) + L2(φ2)⇒ H = H1 +H2.

Let us now define the operators,{ap = 1√

2(a1p + ia2p)

ap = 1√2(a1p − ia2p)

⇔

{a†p = 1√

2(a†1p − ia

†2p)

a†p = 1√2(a†1p + ia†2p)

and write φ(†) as Fourier decompositions,

φ =

∫d3p√

(2π)32Ep(a1pe

−ix.p + a†1peix.p + i[a

−ix.p2p + a†2pe

ix.p])1√2

φ =

∫d3p√

(2π)32Ep(ape

−ix.p + a†peix.p) ⇒ φ† =

∫d3p

(2π)3Ep(ape

−ix.p + a†peix.p)

with,

[ap, a†p′ ] =

1

2[a1p + ia2p, a

†1p − ia

†2p′ ] = δ3(~p− ~p′)

[ap, ap′ ] = [a†p, a†p′ ] = 0

[ap, a†p′ ] =

1

2[ap − ia2p, a

†1p′ + ia†2p′ ] = δ3(~p− ~p′)

[ap, ap′ ] = [a†p, a†p′ ] = 0

and [a(†)

p , a(†)p′ ] = 0 (25)

Since,

[ap, ap′ ] =1

2[a1p + ia2p, a1p′ − ia2p′ ] = 0 = [a†p, a

†p′ ]

[ap, a†p′ ] =

1

2[a1p + ia2p, a

†1p′ + ia†2p′ ] = 0 = [a†p, ap′ ] .

3.3.1 Hamiltonian

Since Eq.(3) as Eq.(8): L, as Eq.(19):

H =1

2

∫d3p Ep[(a1pa

†1p + a†1pa1p) + (a2pa

†2p + a†2p + a†2pa2p)] .

To be compared with,

aa† + a†a =1

2(a1a

†1 − ia1a

†2 + ia2a

†1 + a2a

†2 + a†1a1 + ia†1a2 − ia†2a1 + a†2a2)

21

aa† + a†a =1

2(a1a

†1 + ia1a

†2 − ia2a

†1 + a2a

†2 + a†1a1 − ia†1a2 + ia†2a1 + a†2a2)

so that,

H =1

2

∫d3p Ep (apa

†p + a†pap + apa

†p + a†pap) ((26) bis)

: H : =

∫d3p Ep (a†pap + a†pap) (26)

which is comparable to Eq.(22), : H : =∫d3p Ep(Np + Np).

Eq.((26) bis )⇒ [: H :, ap′ ] = −Ep′ap′ since [a(†)p , ap′ ] = 0 and [: H :, ap′ ] = −Ep′ ap′ .

Similarly :

[: H :, a†p′ ] = Ep′a†p′ [: H :, a†p′ ] = Ep′ a

†p′ . (27)

3.3.2 Physical interpretation

ΣH.O. in H ⇒ 2 Hilbert spaces for 2 species of particles (in Fock space), a and a creating

different particles ⇒ same construction of quantum states,

|np >=a†np√n!|op >

|np >=a†np√n!|op >

Since HH.O.ap +HH.O.

apis like HH.O.

p +HH.O.p+δp.

Remark : As the “p” index, another subscript could label some species of particles.

One could have chosen φ1,2 as the two species of particles (or other orthonormal com-

binations, with relations as in Eq.(18)...) which would have been equivalent in the free

case but with potentially distinct physical implications when interactions are introduced

– as we are going to describe in the next section.

3.3.3 Noether’s theorem

L in Eq.(23) is invariant under a global ‘gauge’ transformation.

Remark : The following discussion remains true for the real gauge transformations

of nature (θ = θ(x)) arising when including the interactions in L.{φ(xν)→ e−iqθφ(xν)

φ†(xν)→ eiqθφ†(xν)

22

τ

xµ → xµ′ = xµ

φ(xν)→ φ′(xν′) ' φ(xν)−iqθφ(xν)︸︷︷︸δφ

and δφ† = iqθφ†

Jµ =∂L

∂(∂µφ)δφ+

∂L∂(∂µφ†)

δφ† − Tµνδxν

= ∂µφ†(−iqθφ) + ∂µφ(iqθφ†)

= iqθ[−∂µφ†φ+ ∂µφφ†]

Jµ is thus defined up to a factor and a constant so the q-charge as well (only relative

charges, among several fields, do matter and only interactions allow their determina-

tions). Even classically: the electric charge is ‘absolute’ by convention in the Maxwell’s

equations since a q-redefinition can be done with(~B~E

unit redefinition e.g. in ~∇. ~E =qρ

εo.

Now the global charge conservation reads as,

dQ

dt= 0 with Q =

∫d3x Jo

= iq

∫d3x [(∂oφ)φ− (∂oφ†)φ]

= q

∫d3p [a†(~p)a(~p)︸︷︷︸

Np

− a†(~p)a(~p)︸︷︷︸Np

]

= qN + (−q) ˆN

Conclusion: {a annihilates an anti-particle

a† creates an anti-particle

⇒ φ(xν) annihilates a particle / creates an anti-particle i.e. decreases the global Q-

charge by an amount ‘q’.

Similarly φ†(xν) increases Q by ‘q’ (+q instead of −q).

Remark : In Quantum ElectroDynamics (QED), φ, φ†(x) field operators imposed as

φ1, φ2(x) would not allow a charge description since one would get,

Q = iq

∫d3p [a†1(~p)a2(~p)−a†2(~p)a1(~p)] for same Q (origin: Jo(L) and still φ = φ1+iφ2) .

23

4 Propagator

4.1 Solving the generic K.-G. equation

The K.-G. equation with a source (interaction) term is, (� + m2)φ(xν) = S(xµ) (for φ

real or complex scalar field). How to solve this equation ?

Let us introduce for that purpose the Green’s function, defined through,

(�x +m2)G(x− x′) = − δ4(x− x′) . (28)

Remark : Green’s function also permits to solve Maxwell’s (relativistic and quantum)

equations: K.-G. + S(x), neglecting the spin degrees of freedom.

Indeed if Eq.(28) solvable, i.e. if solution G exists, then one knows the solution of

the generic K.-G. equation for all integrable S(x) :

φ(x) = φo(x)−∫d4x′ G(x− x′) S(x′) (29)

where φo is the solution of the free/original K.-G. equation. Check :

(�(x) +m2)φ(xν) = (�x +m2)φo(x)︸︷︷︸=0

−∫d4x′ (�x +m2)G(x− x′)︸︷︷︸

−δ4(x−x′)

S(x′) = S(xµ) .

So let us find out G(x− x′) from Eq.(28), through its (inverse) Fourier transform:

G(xµ − x′µ) =

∫d4p

(2π)4e−ip.(x−x

′) G(pµ) ((29) bis)

with independent po and Ep (or p.p and m2) as not free EOM [only math. variable].

Eq.(28) gives the information,

−δ4(x−x′) =

∫d4p

(2π)4G(pµ)( �x︸︷︷︸

(∗)

+m2)e−ip.(x−x′) =

∫d4p

(2π)4e−ip.(x−x

′)(−p2+m2)G(pµ)

⇒ G(pµ) =1

p2 −m2≡ 1

po2 − ~p2 −m2

(*) ∂µ∂µ(−ipµxµ)eip.(x−x′) = (−ipµ)∂µ(−ipµxµ)e−ip(x−x

′) = −p2 e−ip.(x−x′).

The propagator will appear in the Feynman rules. Physically po2

can have quantum

fluctuations around E2p .

24

It remains to Fourier transform G(p) to get G(x − x′) ⇒ problem to know how dpo-

integrate over the 2 poles at po = ±Ep[= f(~p)] of the integrant. (in general, integrability

of a pole: question of the convergence rapidity)

One has to introduce, F (p) ≡ 1

po2−(Ep−iε)2, and then take the limit ε → 0. This is

not necessary in quantum field theory where ε = Γ, Γτ = ~ (dimension [E.T ]). Feynman

prescription : introduce a complexity to reject the two poles outside the real axis via

defining, G(p) = limε→0

F (p).

F (p) =1

2[Ep(−iε)]

[po + (Ep − iε)po + (Ep − iε)

1

po − (Ep − iε)− po − (Ep − iε)po − (Ep − iε)

1

po + (Ep − iε)

]

Hence,

G(x− x′) =

∫d3p

(2π)3

ei~p.(~x−~x′)

2Ep

∫dpo

(2π)e−ip

o(t−t′) limε→0

[1

po − Ep + iε− 1

po + Ep − iε

].

Remark : ε < 0 possible and equivalent but changes the analytical calculation.

Useful mathematical result from a residue theorem application:

0 +

∫ +∞

−∞dX

e−iXt

X + iε= −2iπ θ(t)e−i(−iε)t

⇒ limε→0

∫ +∞

−∞dX

e−iXt

X + iε= −i2π θ(t) .

Let us use this formal result for our purpose: the first term (second integration) of the

G(x− x′) expression reads as,

limε→0

∫ +∞

−∞

dpo2π

e−ipo(t−t′)

po − Ep + iε= lim

ε→0

∫ +∞

−∞

dp′o

(2π)

e−ip′o(t−t′)

p′o + iεe−iEp(t−t′)

=e−iEp(t−t′)

2π(−i2π θ(t− t′)) .

The second term (second integration) of the G(x− x′) expression reads as,

limε→0

∫dpo2π

−e−ipo(t−t′)

po + Ep − iε= lim

ε→0(−1)

∫ +∞

−∞

(−dp′o)2π

−e−ip′o(t′−t)

−(p′o + iε)e−iEp(t′−t)

=e−iEp(t′−t)

2π(−i2π θ(t′ − t)) .

25

So that,

G(x− x′) = −i[ ∫ +∞

−∞

d3p

(2π)3e−iEp(t−t′)θ(t− t′)ei~p.(~x−~x′)

+

∫ +∞

−∞

−d3(−p)(2π)3

eiEp(t′−t)θ(t′ − t)ei(−~p).(~x+~x′)

]1

2Ep

G(x− x′) = −i∫ +∞

−∞

d3p

(2π)3

1

2Ep

[θ(t− t′) e−ip.(x−x′) + θ(t′ − t) e−ip.(x′−x)

](30)

with now po = Ep, from the change of variable.

4.2 Introducing the time-ordering

Let us now connect G(x− x′) to the quantised scalar field φ. For the real scalar field,

φ(x)|o > =

∫d3p√

(2π)32Epa†(~p) eix.p|o >

=

∫d3p√

(2π)32Epeix.p|1~p > .

Taking the Hermitian conjugate,

< o|φ†(x′) =

∫d3p′√

(2π)32Ep′e−ip

′x′ < 1~p| with φ†(x) ≡ φ(x)

so < o|φ(†)(x′)φ(x)|o > =

∫d3p

(2π)32Ep

∫d3p′√

(2π)32Ep′ei(p.x−p

′.x′)< 1~p′ |1~p >︸︷︷︸δ3(~p− ~p′)

=

∫d3p

(2π)32Ep.

One can thus link the quantised field with the Green function,

iG(x− x′) = θ(t− t′) < o|φ(x)φ(x′)|o > +θ(t′ − t) < o|φ(x′)φ(x)|o > .

Using x↔ x′.

iG(x− x′) =< o|τ [φ(x)φ(x′)]|o > = < o|τ [φ(x′)φ(x)]|o > (31)

with τ [φ1(xν)φ2(x′ν)] =

{φ1(xν)φ2(x′ν) if t > t′

φ2(x′ν)φ1(xν) if t′ > t.

26

Indeed, e.g. for t > t′ : iG(x− x′) = 1× < o|φ(x)φ(x′)|o > +0 =< o|τ [φ(x)φ(x′)]|o >.

Physical interpretation:

Still for the example case t > t′:

• φ(x)→∫d3p′a(~p′), at time t of φ(x), so the operator annihilates a particle to get

< o|o >6= 0

• φ(x′)|o >→∫d3p |1p > (up to coefficients) as seen, so it creates particle at time t′

of φ(x′)

So the matrix element < o|τ [φ(x)φ(x′)]|o >︸︷︷︸iG

can describe the amplitude of a particle be-

ing created at t′, propagating in space-time and being annihilated at posterior moment t.

It will turn out to enter the calculation of amplitudes of reactions involving interactions

(vertex). This can be expected intuitively, as after all, in any reaction, particles only

propagate freely or interact with other particles.

Remark : This Green function meaning differs from the interpretation in relativis-

tic quantum theory : Eq.(29).

Exercise 12: For the complex scalar field, using Eq.(30) and φ, φ†(a, ˆa) expressions, show

that,

iG(x− x′) =< o|τ [φ(xν)φ†(x′ν)]|o > = < o|τ [φ(x′ν)φ†(xν)]|o > . (32)

There is thus a unique Green funcion / propagator for a complex scalar field as well.

27

5 Evolution operator

5.1 Structure of interactions

So far, only free particles studied (in a boring world) with quadratic terms in L [mass or

kinetic terms] which basically create and annihilate a particle from vacuum. For events

to happen, some interactions need to be introduced.

Notice the possibility, not studied here, of a term like mφ†2φ1 ‘mixing’ 2 particles so that

a state rotation is then needed to obtain 2 independent states.

What do the interaction terms look like in general ?

{→ Lorentz invariant

→ L Hermitian (up to total derivatives)

For a real field φ, the Lagrangian of interactions reads generically as,

LI = −gφ3 − λφ4 −N∑k=5

λkφk

(k!)

{→ φ(x) Lorentz invariant by itself

→ φ = φ†

recalling that,

L = πAφA −H = πAφA − (Hkin + V ) .

The coupling constants determine the strength of the interactions : [g] = E, [λ] =

1, [λk] = E−1, E−2, ...

For a complex field φ, the Lagrangian of interactions contains generically,

L∗I 3 same terms or e.g. φ†φ2 (mixed)

→ φ† also invariant (another field)

φ(x) = φ′(x′) so φ†(x) = φ′†(x′)

→ +H.c.

.

The nature and structure of the interaction terms

define the specific theory (possibly hypothetical).

5.2 Working out the physical states

Quantum states (of a generic system) known

⇒

{eigenvalues of observables known, O|α >= α|α >transition amplitudes known, P = | < β|α > |2

Now the eigenstates of the Hamiltonian H (main observable and related to L) are gener-

ically hard to find out (in QFT too). Therefore, one is usually constrained to consider

28

an Hamiltonian, H = Ho+HI 3 interactions typically small, in order to develop pertur-

bation approachs. Here, Ho represents the free Hamiltonian. As usually in perturbation

theory, Ho and HI belong to the same H-space whose possible Ortho-Normal Basis is

noted

{...|n >p |n′p+δp > ...} = {|β >} = {|E >} .

The Schrodinger equation in Fock space, providing the quantum state evolution, is

i(~)d

dt|Ψ(t) >= (Ho +HI)|Ψ(t) > (33)

and for a free state,

i(~)d

dt|Ψo(t) >= Ho|Ψo(t) > (34)

As in Quantum Mechanics where the typical evolution goes like |Ψ(t) > ⊗|s(t) >, i.e.

there are simultaneous evolutions with time in Hx,p and in Hspin.

Let us introduce the evolution operator Uo(t) relating ‘−∞’ to ‘t’ for the free state:

|Ψo(t) >= Uo(t)|Ψo(−∞) > (35)

Let us then introduce the evolution operator U(t) relating ‘free’ states with ‘coupled’

states.

|Ψ(t) >= Uo(t)U(t)U †o (t)|Ψo(t) > (∀t) (36)

Eq.(35) and Eq.(36) synthesize into the relation,

|Ψ(t) >= Uo(t)U(t)U †o (t)Uo(t)|Ψo(−∞) > (|i > =|Ψo(−∞) >) (37)

Therefore, one needs to work out Uo(t) and U(t) to obtain the quantum states. Note the

unitarity of both operators, U †(o)(t) = U−1(o) (t), (demonstrated below) allowing to conserve

the quantum state normalisations:

< Ψo(t)|Ψo(t) >=< i|U †oUo|i > and < Ψ(t)|Ψ(t) >= (< Ψo(t)|UoU †U †o )(UoUU†o |Ψo(t) >).

Let us first work out Uo(t). Eq.(34) and Eq.(35) lead to,

id

dtUo(t)|i >= HoUo(t)|i >; ∀|i > (Eq.(38) bis)

whose solution reads as,

Uo(t) = e−iHot (38)

29

Exercise 13 : Check out this solution.

id

dt[1− iHot+

(−1)2

2!H2o t

2 + ...]?︷︸︸︷= Ho[1− iHot+

(−1)2

2!H2o t

2 + ...]

Ho 6= Ho(t) as time-dependence in |Ψ(t) > (see Eq.(37)) being the usual prescription in

Quantum Mechanics (the other one would be equivalent).

⇒ i[0− iHo −1

22H2

o t+ ...] = Ho − iH2o t+ ...

which is a true relation (at first orders).

Exercise 14 : check Uo(t) unitarity at first order.

Uo U†o︸︷︷︸

U−1o

= (1− iHot+O(t2))(1 + iH(†)o t+O(t2))

= 1 + iHot− iHot+O(t2)

We now focus on U(t). The Eq.(33) and Eq.(37) allow us to write,

id

dt(Uo(t)U(t)|i >) = (Ho +HI)Uo(t)U(t)|i > ∀|i >

i(U ′o(t)U(t) + Uo(t)U′(t))|i > =

[HoUo(t)U(t) + iUoU′(t)]|i > =

idU(t)

dt= U †o (t)HIUo(t)U(t) (39)

Interpration of HI(t) = U †o (t)HIUo(t): < β|HI(t)|β′>=< β|U †o (t)HIUo(t)|β

′> .

HI 6= HI(t) used here [to have |Ψ(t) > in Eq.(37)] so HI turned on slowly from HI = 0

at t = −∞.

Solving Eq.(39):∫ ttodt1 : U(t)− U(to) = −i

∫ ttodt1HI(t1)U(t1)

t→ −∞ :dU(t)

dt= 0⇒ U(t) = 1 as U(t) implements the interaction (Eq.(37))

Hence, for to → −∞: U(t) = 1− i∫ t−∞ dt1HI(t1)U(t1).

One can then use an iterative approach,

30

U(t) = 1− i∫ t

−∞dt1HI(t1)[1− i

∫ t1

−∞dt2HI(t2)U(t2)]

= 1− i∫ t

−∞dt1HI + (−i)2

∫ t

−∞dt1

∫ t1

−∞dt2HI(t1)HI(t2)U(t2)

= 1− i∫ t

−∞dt1HI + (−i)2

∫ t

−∞dt1

∫ t1

−∞dt2HI(t1)HI(t2)

+ (−i)3

∫ t

dt1

∫ t1

−∞dt2

∫ t2

−∞dt3HI(t1)HI(t2)HI(t3)U(t3)

= ...

= 1 +∞∑n=1

(−i)n∫ t

−∞dt1...

∫ tn−1

−∞dtnHI(t1)...HI(tn)

Let us rewrite the third term.

(−i)2

2!

∫ t

−∞dt1

∫ t

−∞dt2 τ [HI(t1)HI(t2)] =

(−i)2

2!

[ ∫ t

−∞dt1

∫ t1

−∞dt2 θ(t1 − t2)HI(t1)HI(t2)

+

∫ t

−∞dt2

∫ t2

−∞dt1 θ(t2 − t1)HI(t2)HI(t1)

]where the θ step function informations are encoded into the integration boundaries.

After renaming t1 ↔ t2 in the 2nd term above, we obtain,

(−i)2

2!

∫ t

−∞dt1

∫ t

−∞dt2 τ [HI(t1)HI(t2)] = (−i)2

∫ t

−∞dt1

∫ t1

−∞dt2 HI(t1)HI(t2)

so that U(t) can be rewritten in a more compact form (convenient for the following), if

generalizing to higher terms, as

U(t) = 1 +

∞∑n=1

(−i)n

n!

∫ t

−∞...

∫ t

−∞dtn τ

[HI(t1)HI(t2)...HI(tn)

](40)

U(t) is thus solved and exhibits a possible perturbative treatment: HI(t) having typi-

cally small eigenvalues, the first terms of this series are expected to be dominant and

can thus be exclusively considered – in a first approximation.

General warning: in Quantum Fiel Theory, such series are in general not convergent

(even for ‘small’ HI ’s) which questions the validity of the QFT formalism in general and

its quantitative predictions as well (same comment for non-canonical quantisations).

This implies a non-trivial open question but the experimental tests on particle reaction

amplitudes have been passed brillantly so far so that the theory is still acceptable in its

present form.

31

5.2.1 S-matrix

S= limt→∞

U(t) (41)

This definition implies the unitarity of S as U(t) : S†S = 1. The evolution operator

U(t), and in turn S, can be rewritten as,

S = 1 +

∞∑n=1

(−i)n

n!

∫ +∞

−∞d4 x1...

∫ +∞

−∞d4xn τ

[HI(xµ1 )...HI(xµn)

](42)

or even in a neater form (rendering also the normal order explicit):

S = τ

[exp(−i

∫d4x : HI(x) :)

]

What is the motivation for considering the S-matrix element?

The interesting quantity is the amplitude of transition : < f |...|i >. When interactions

have ceased, the final state |f > can be taken among |β >’s i.e. free states (identical

when HI 6= 0 anyway). The interaction is not yet turned on for the initial state |i > so

that it could be as well chosen among the |β >’s.

< f |S|i >= limt→∞

< f |U(t)|i >= limt→∞

< f |U †o (t)Uo(t)U(t)|i >≡ Amplitude of reaction.

The quantum approach is as announced : U(t)⇒ |Ψ(t) > ⇒ Amplitude’s.

32

6 Reaction amplitudes

6.1 Wick’s theorem

6.1.1 Wick contraction

* Real field:

φ(x)φ(x′)=< o|τ [φ(xµ)φ(x′µ)]|o >︸︷︷︸iG(xµ−x′µ)

1 = φ(x′)φ(x)

φ(x)φ′(x′) =< o|θ(t− t′)φ(x)φ′(x′) + θ(t′ − t)φ′(x′)φ(x)|o >= 0

since < 1p|o′p >,< op|1′p > or < 1p|1′p > vanish.

* Complex field:

φ(x)φ†(x′)= < o|τ [φ(x)φ†(x′)]|o > 1 = φ†(x′)φ(x)

=< o|τ [φ(x)φ†(x)]|o > 1 = φ(x′)φ†(x) = φ†(x)φ(x′)

As for real field : φ(x)φ′†(x′) = 0

φ(x)φ(x′)= < o|τ [φ(x)φ(x′)]|o > 1 = 0 = φ(x′)φ(x)

since 0 =< 1p|op >=< op|1p > and at best < 1p|1p >= 0.

Similarly, φ†(x)φ†(x′)= < o|τ [φ†(x)φ†(x′)]|o > 1 = 0 = φ†(x′)φ†(x).

One can relate the time-order and normal-order of a field product through the Wick

contraction, as it will turn to be useful,

τ [φ1(x)φ2(x′)] = : φ1(x)φ2(x′) : +φ1(x)φ2(x′) (43)

Case φ1 6= φ2 trivial: (τ)[φ1(x)φ2(x′)] =: φ1(x) : : φ2(x′) : +0

Case φ1 = φ2 ⇒ [φ1, φ2] = 0⇒ no τ [ ] as well

Case φ2 = φ†1 ⇒ not trivial.

A regularisation method leads to the needed formula:

τ [: A(x1)B(x1).... : : A(x2)B(x2)... : · · · : A(xn)B(xn)... :]

= τ [AB...(x1) AB...(x2) . . . AB...(xn)]e.6t.c.(44)

where e. 6 t.c. stands for ‘no equal-time (Wick) contraction’.

33

6.1.2 Wick’s theorem

(Demonstration for a unique field in [Peskin & Schroeder])

τ [φ1(x1)φ2(x2)...φn(xn)] =: φ1(x1)...φn(xn) :

+ : φ1φ2φ3...φn : +permutations

+ : φ1φ2φ3φ4...φn : +permutations

+ ...

+

{: φ1φ2...φn−1φn : +permutations (if n even)

: φ1φ2...φn−2φn−1 φn : +permutations (if n odd)

(45)

Case of all φn’s distinct (from each other): τ needless and only the first term on the

right hand side is non-zero, and without normal order anymore. It is thus a simple case.

6.2 A simple decay example

6.2.1 First order contribution{Interaction : LI = −gφ†2(x)φ2(x)φ1(x)

Free part : Lo(φ1) + Lo(φ2)

⇒ : H : = : Ho(a(†)1 ) + Ho(a(†)

2 ) + HI : with HI = g : φ†2φ2φ1(xµ) : since only the

“−L” part of the Hamiltonian is non-zero as there is no new π = ∂L∂(∂oφ) contribution.

The perturbation method can be used with the developed decomposition of the free

fields. The reaction considered is the particle P1 decays into the particle P2 and its

anti-particle P2, with respective 4-momenta defined as,

P1(kµ)→ P2(pµ) + P2(p′µ)

which is a kinematically open channel under the condition that M1 > 2m2, where M1

(m2) is the mass of the particle P1 (P2). This condition arises from the 4-momentum

conservation [see later on].

A(P1 decay) =< f |S|i >'< 1P2p 1P2

p′ |1 + (−i)∫ +∞

−∞d4x τ [g : φ†2(xµ)φ2(xµ)φ1(xµ) :]|1P1

k > + . . .

↓

τ [gφ†2(x)φ2(x)φ1(x)]e. 6t.c.

↓

g : φ†2(x)φ2(x)φ1(x) :

34

Example of 2 equal order-permutations:

∫ ∫ ∫d4x1d

4x2d4x3 : φ†2φ2φ1|x1 φ

†2φ2φ1|x2 φ

†2φ2φ1|x3 :

=

∫ ∫ ∫d4x2d

4x1d4x3 : φ†2φ2φ1|x2 φ

†2φ2φ1|x1 φ

†2φ2φ1|x3 :

=

∫ ∫ ∫d4x1d

4x2d4x3 : φ†2φ2φ1|x2 φ

†2φ2φ1|x1 φ

†2φ2φ1|x3 :

Therefore including the order-permutations and position-permutations is equivalent to

muliply by a factor 3! which compensates exacty the 1/3!.

A(3) = (−ig)3 < 1(2)p 1

(2)p′ |∫d4x1d

4x2d4x3

{: φ†2φ2φ1|x1 φ

†2φ2φ1|x2 φ

†2φ2φ1|x3 :

} ∑config.permut.

|1(1)k >

(47)

An easy systematic listing of all the possible configuration-permutations uses the Feyn-

man diagrams. One of the terms corresponds to the following triangular loop.

k

P2(p− q)

P2P1(q)

p′

p

P1

P2

P2

x3

x2

This term reads as,

A(3) 3 (−ig)3 < 1(2)p 1

(2)p′ |∫d4x1d

4x2d4x3 : φ†2+(x2)φ2+(x3)φ1−(x1) : |1(1)

k >

× φ†2(x1)φ2(x2)φ2(x1)φ†2(x3)φ1(x2)φ1(x3)

(48)

36

From now on, we will concentrate on this term to illustrate the amplitude calculation.

However, we mention that other A(3) terms exist (1-Particle Reducible) :

P2

P2

P1

P2

P2

P1

P2

P2

P1

P2

P2

6.3 Amplitude calculation

6.3.1 Quantum state normalisation

Problem 1 : the different dimension of different states, a†p|o >= |1p >Problem 2 : < 1p|1p >= δ3(~o) is undefined

Solution : define any state within a finite volume V , and then take the limit V → ∞at the end of calculation.

|1p >=

(√(2π)3

Va†p

)|o > (Eq.(49) bis)

⇒ < 1p|1p′ >=< o|apa†p′ |o >(2π)3

V=

(2π)3

Vδ3(~p− ~p′) . (49)

At this level there is still the δ3(~o) problem, as e.g. in Quantum Mechanics (linear

combinations to address it) for free states/wave functions. Nevertheless, Eq.(49) can be

written as,

< 1p|1p >=(2π)3

V× limV→∞

[1

(2π)3

∫Vd3x e−i~o.~x

]= 1

37

6.3.2 Momentum/position integrations

With the above normalisation prescriptions, one has,

φ1−(xµ)|1(1)k > =

∫d3q√

(2π)32Eqa1qe

−iq.x√

(2π)3

Va†1k|o >

=

∫d3q√2Eq

e−iq.x√V

(δ3(~q − ~k) + a†1ka1q

)|o >=

1√2EkV

e−ik.x|o >

φ2−(xµ)|1(2)p >= 1√

2EpVe−ip.x|o >

φ†2−(xµ)|1(2)p′ >= 1√

2Ep′Ve−ip

′.x|o >⇒ < 1(2)p′ |(φ

†2−)† = 1√

2EpVeip′.x < o|

(50)

which allows to write the amplitude at first order,

Eq.(46) : A(1) = (−ig)

∫d4x < o(2)| < o(2)|o(1) >

e−ik.x√2EkV

eip.x√2EpV

eip′.x√

2Ep′V(51)

and the term considered for the higher order, using Eq.(51) versus Eq.(46),

Eq.(48) : A(3) 3 (−ig)3

∫d4x1d

4x2d4x3√

V 32Ek2Ep2Ep′e−ik.x1+ip.x2+ip′.x3

×iG2(−x1 + x2) iG2(x1 − x3) iG1(x2 − x3) .

Using Eq.((29) bis ), one can write,

iG2(−x1 + x2) iG2(x1 − x3) iG1(x2 − x3) =∫d4q1 d4q2 d4q3

(2π)4 (2π)4 (2π)4 e−iq1.(x2−x1)−iq2.(x1−x3)−iq3.(x2−x3) iG2(q1) iG2(q2) iG1(q3) .

Once more, as in Eq.((29) bis ), the qo’s can be different from Eq. We thus get,

A(3) 3 (−ig)3

∫d4q1d

4q2d4q3√

V 32Ek2Ep2Ep′δ4(k − q1 + q2)δ4(−p+ q1 + q3) δ4(−p′ − q2 − q3)︸︷︷︸

δ4(k−p−p′)

× iG2(qµ1 ) iG2(qµ2 ) iG1(qµ3 )

A(3) 3 (−ig)3

∫d4q1d

4q3√V 32Ek2Ep2Ep′

δ4(−p+q1+q3)δ4(k−p−p′) iG2(qµ1 ) iG2(qµ1−kµ) iG1(qµ3 )

A(3) 3 (−ig)3

∫d4q

(2π)4δ4(k− p− p′) iG2(p− q) iG2([p− q]− k) iG1(q)

(2π)4√V 32Ek2Ep2Ep′

(52)

The 3 propagators induce the 4-momentum conservation as well as for the intermediate

particle, exchanged in the loop, which is off-shell (QFT result).

38

6.3.3 Generalisation to Feynman rules

The obtained results generalise as follows. The amplitude for a given term (fixed dia-

gram contribution) at order (n), for the transition |i >→ |f >, generically reads as,

A(n)if = (2π)4 δ4

(∑i

ki −∑f

pf

)Πi

1√2EiV

Πf1√

2EfViM(n)

if

where the last amplitude part can be directly obtained via the Feynman rules given

in the following table.

Name (interpretation) Diagram pattern example (iM) content

Vertex “− i g” if LI = −g φ†2φ2φ1

(coupling) “− i4! λ” if LI = −λ φ4

Internal leg

q

iG(qµ) = i/(q2 −m2)

(propagator)

Loop

q

∫d4q / (2π)4

(exchange)

The Feynman rules are thus powerful in the sense that those avoid to apply the ‘heavy’

Wick’s theorem for calculating transition amplitudes.

39

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