INTRODUCTION TO QUANTUM FIELD THEORY

INTRODUCTION TO QUANTUM FIELD THEORY

Lectures at HDM 2017

University of Stellenbosch, 26^th November – 2^nd December 2017

C.A. Dominguez

Department of Physics & Centre for Theoretical & Mathematical Physics University of Cape Town

Lecture 3

PARTICLE PROPAGATION:

particles created/annihilated in Fock space particles propagated in Minkowski space

GREEN FUNCTION

• D(x , t) ψ(x , t) =f (x , t) ^e.g. D(x , t) ≡ ² – (1/v²) 𝛛 ² / 𝛛t²

• D(x , t) G(x , t | x″ ,t″) = δ⁽³⁾ (x - x″) δ(t – t″)

• ψ(x , t) =  d³x″ G(x , t | x″ , t″) f (x″ , t″)

PARTICLE PROPAGATION IN QFT

• <0| φ(y) φ^†(x) |0>

x y

i Δ(x – y) ≡ <0| φ(y) φ^†(x) |0>

D(x) Δ(x – y) ∝ δ⁽⁴⁾(x – y)

PROPAGATOR IN MOMENTUM SPACE

(SCALAR PARTICLE SPIN=0) ---

XTransform

X y





i m

p p

p p e

y d

x

^{i p x y}



 









 

^{ }

2 0 2

) (

. 4

4

) 1 (

) ) (

2 ) (

(

PROPAGATOR IN MOMENTUM SPACE

(FERMION PARTICLE SPIN=1/2) ---

XTransform

X y

Type equation here.

Type equation here.

Type equation here.

𝑆_𝐹 (x – y) = ^{𝑑4 𝑝}

(2 π)⁴ 𝑒−𝑖 𝑝∙(𝑥 −𝑦) 𝑆_𝐹 (p) 𝑆_𝐹 (p) = ¹

γ^μ 𝑝_{μ − 𝑚0} = ^γ

μ 𝑝_{μ + 𝑚0} 𝑝² − 𝑚₀²

Type equation here.

PROPAGATOR IN MOMENTUM SPACE

(BOSON PARTICLE SPIN=1 e.g. photon)

D^μν (q²) = ^{− 𝟏}_𝒒𝟐 [ 𝒈_𝝁𝝂 - ^𝒒𝝁_𝒒^𝒒_𝟐^𝝂 ]

 

 





 

 ^T

i q q

D

y A

x A

y x

D i

 











2

2

1

) (

0

| ) ( )

(

| 0 )

(

CURRENTS

• Local bilinear functions of fields

• Definite quantum numbers

• e.g. Electromagnetic current:

) ( )

(

| )

(x e x x

J _{ QCD}   _ 

CURRENTS

• QUANTUM ELECTRODYNAMICS

• Matter Current J_μ (x , t)

•

» J ^μ (y) J^ν(x) V(q) = ^|𝑒|_𝑞2

» Δ_μν (x – y)

V(r) = ¹

4 π 𝑑³𝑞 𝑒^{−𝑖𝑞∙𝑥} V(q) V(q) = ^|𝑒|

𝑞² V(r) = ^|𝑒|

𝑟 COULOMB !

Coulomb’s Law : One photon exchange

FEYNMAN DIAGRAMAS 𝑞² = (p_f – p_i)² = (q₀² - 𝑞²) ≠ 0 !!!

Insisting on energy-momentum conservation virtual particles Feynman’s (practical) invention

THE QUANTUM VACUUM

• First Law of Thermodynamics (conservation of energy-momentum)

• Heisenberg’s Uncertainty Principle:

Δp _x Δx ^≈ ℏ

^etc.

• ΔΔΔΔΔ

Particle-Antiparticle production out of nothing

VACUUM

An e- e+ pair produced and reabsorbed in ~ 10^-21 s A proton-antiproton pair in ~ 10^-24 s

VACUUM CORRECTIONS

ANTIMATTER PRODUCTION

QUANTUM CHROMODYNAMICS

Fritzsch & Gell-Mann 1972

) (

3 ,

2 , 1 )

( 8 ,...

2 , 1

) ( )

( )

(

4 ) 1

( )

( )

( )

( ₀

colours N

a gluons

i

x x

x G

g

F F

x x

m x

x i

c b

i ab a

i

i i

a a

a a



































 

 

 

L

k j

ijk i

i

i G G g f G G

F   _{ }  _{ }  _{ }

CORRELATION FUNCTION IN QFT

• <0| J_μ (y) J_ν†(x) |0>

x y

) (

| ) ( )

( )

(

| )

(x x x J x x

J _{ QCD}  _  _{ HAD}  _

Q C D

) ( )

(

| ) (

) ( )

(

| ) ( )

(

0

| )) 0 ( )

( (

| 0 )

(

5 4

2

x i

x x

V

x i

x x

A x

J

J x

J T

e x d

i q

j i

i j

j i

i j

iqx



































 

^

HADRONIC

DATA AL

EXPERIMENT fields

Hadronic x

J

J x J T

e x d i

q ^iqx

/ )

(

0

| )) 0 ( )

( (

| 0 )

( ^{2 4}













^

QCD HAD

x y x y

CAUCHY’S THEOREM IN THE

COMPLEX ENERGY PLANE