Quantum Field Theory Using Path Integralsshakes76/Files/qft_minibook.pdf · Why Quantum Field...

Quantum Field Theory Using Path Integrals

- A Mini Book -

Beta Version

Shekhar Suresh Chandra1

August 4, 2005

1Monash University. Email: [email protected]

mailto:[email protected]

Preface

In my Quantum Field Theory honours course at Monash University, it was required of me to complete

all relevant exercises and missing elements of Zee’s book [1]. As a result, I decided to write a mini-book

that did exactly this, together with any further understanding I gained through reading other various

books on the subject. The book contains a justification for studying the subject, an introduction, as well

as proofs and crucial elements worked through in detail. I believe it should provide a nice introduction

to undergraduate students endevouring to or currently studying quantum field theory.

iii

Acknowledgements

I wish to thank my lecturer and honours supervisor, Associate Professor Michael Morgan for various

learning materials and sharing his knowledge on the subject. Also to my family, for their support

throughout the year.

v

Contents

Preface iii

Acknowledgements v

1 Why Quantum Field Theory? 1

2 Classical Field Theory 3

2.1 The Action Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 The Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 The Euler-Lagrange Equation for Particle Mechanics . . . . . . . . . . . . . . . . . 4

2.2.2 Newton’s Law from the Euler-Lagrange Equation for Particles . . . . . . . . . . . 5

2.2.3 The Euler-Lagrange Equation for Fields . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Path Integral Formulation of Quantum Mechanics 9

3.1 Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Particle in a Time-Independent Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 From Particles to Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 The Vacuum and Free Field Theory 17

4.1 The Forms of the Scalar Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 The Free Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2.1 Free Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2.2 Solving for the Free Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 Origin of Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 The Phi-4 Scalar Field Theory 23

5.1 Wick Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

vii

viii CONTENTS

5.2 Schwinger Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Spin-1 Vector Fields 27

7 Dirac Equation 29

8 Fermion Fields 33

9 The Feynman Rules 35

10 Quantum Electro-Dynamics (QED) 37

A Formal Solution to Schrodinger’s Equation 39

B Proof of the Completeness Relation 41

C Proof of Normalisation Convention 43

D The Gaussian Integral 45

E The Generalised Gaussian Integral 47

Chapter 1

Why Quantum Field Theory?

Quantum Field Theory arises as a necessity of providing a truly relativistic quantum theory and while

bridging the philiphosical gap between the particle theory of electromagnetic radiation and the classical

field theory electromagnetic radiation. In quantum mechanics, even the Dirac Equation fails to treat time

and space on a totally equal footing, as one needs to in order to consider space-time. Finally, one needs a

theory which can also handle the creation and annihilation of particles and intermediate particles (ones

involved in the processes between the initial and final stages of a reaction), together with the relativistic

creation of particles from energy.

Quantum Field Theory provides a way of quantising fields in order to bridge this gap, as well as treating

space and time on an equal footing. It provides the most complete, relativistic quantum theory avaible to

us. The genaral approach taken will be on two fronts. On the first front we shall begin by introducing the

classical theory of fields. Then on the second we shall introduce the formulation of quantum mechanics

based on path integrals. Finally, we shall combine these two main ideas into the Quantum Field Theory.

1

2 CHAPTER 1. WHY QUANTUM FIELD THEORY?

Chapter 2

Classical Field Theory

Classical Field Theory (henceforth referred to as CFT) has existed long before its quantum cousin. It

has been successful in describing radiation and many other field based phenomena. CFT and standard

Newtonian mechanics is built on a simple idea of least action. We shall look at the particle theory first

and then make our leap to fields. First we need to define the term action.

2.1 The Action Principle

Action is defined as the total energy expended in order to evolve the system. In CFT and Newtonian

mechanics, this action is extremised. In order words, a classical system will evolve in such a way is to

minimise the total energy expended. For example, in a system involving a mass attached to a spring, the

extremized action can be used to determine the equations of motion of the mass (i.e., Hooke’s Law). In

this book we shall denote the action with the symbol S. This concept in classical mechanics is known

as the Principle of Least Action and we shall use it to distinguish a classical system from a quantum

mechanical one.

2.2 The Lagrangian Formalism

We now describe the Principle of Least Action in a more precise way. To begin with, consider a particle

with kinetic energy K and residing in potential V . Let q be the position (in an arbitary co-ordinate

system) and q(t) the path of the particle. Then we define the expended energy, which we shall call the

Lagrangian L, is defined as the residue of the kinetic and potential energies, i.e.

L = K − V

3

4 CHAPTER 2. CLASSICAL FIELD THEORY

Note that since Kinetic and potential energies will be functions of the path q(t) and velocity (i.e. q(t)),

then

L = L(q(t), q(t)).

The total residue over time is defined as the action S, i.e.

S =∫ t2

t1

L(q(t), q(t))dt.

Classically, the motion of the object will occur in accordance with the Principle of Least Action, given

by

δS = δ

∫ t2

t1

L(q(t), q(t))dt = 0. (2.1)

In other words, the motion of the object will occur so that the total residue over time (i.e. the action) is

extremized, which is usually minimized.

2.2.1 The Euler-Lagrange Equation for Particle Mechanics

This extremization can of the action can be writen in a more convenient and workable form known as

the Euler-Lagrange Equation. We show this by starting with the assumption that there is no variation

at the end points

δq(t1) = δq(t2) = 0. (2.2)

Then equation (2.1) becomes

δS =∫ t2

t1

δL(q(t), q(t))dt = 0. (2.3)

Now, using the chain rule

δL (q(t), q(t)) =(∂L

∂qδq +

∂L

∂qδq

).

Hence equation (2.3) becomes

δS =∫ t2

t1

(∂L

∂qδq +

∂L

∂qδq

)dt = 0. (2.4)

Now, using the chain rule again in order to acquire δq in second term and factor it out

δq(t) =dq

dtδt =

d

dt

dq

dtδt =

d

dtδq.

Substituting back into equation (2.4) we get

δS =∫ t2

t1

(∂L

∂qδq +

∂L

∂q

d

dtδq

)dt = 0. (2.5)

2.2. THE LAGRANGIAN FORMALISM 5

The δq cannot be factored out yet because of the ddt operating on it so the second term needs to be

further manipulated. From the product rule, the second term in equation 2.5 can be acquired from

d

dt

(∂L

∂qδq

)=

d

dt

(∂L

∂q

)δq +

∂L

∂q

d

dtδq.

Whose integral is ∫ t2

t1

d

dt

(∂L

∂qδq

)dt =

∫ t2

t1

(d

dt

(∂L

∂q

)δq

)dt+

∫ t2

t1

(∂L

∂q

d

dtδq

)dt,

and so [∂L

∂qδq

]t2

t1

=∫ t2

t1

(d

dt

(∂L

∂q

)δq

)dt+

∫ t2

t1

(∂L

∂q

d

dtδq

)dt.

Using the end point variation conditions of expression (2.2)[∂L

∂qδq

]t2

t1

= 0.

and hence the second term can be written as

−∫ t2

t1

(d

dt

(∂L

∂q

)δq

)dt =

∫ t2

t1

(∂L

∂q

d

dtδq

)dt.

Substituting this back into expression (2.5)

δS =∫ t2

t1

(∂L

∂q− d

dt

(∂L

∂q

))δqdt = 0.

Now this must be true in general, therefore

∂L

∂q− d

dt

(∂L

∂q

)= 0. (2.6)

Equation (2.6) is known as the Euler-Lagrange Equation of particle mechanics. We shall now extend this

into the realm of fields. We now look at a quick example of the Euler-Lagrange Equation in practice.

2.2.2 Newton’s Law from the Euler-Lagrange Equation for Particles

In classical or Newtonian mechanics, the kinetic energy in one dimension, K(x) is given as

K(x) =12mx2 =

12mv2, (2.7)

where v and x represent velocity. The potential can have different forms depending on the physical

system involved and so we shall denote it by V (x) to be general. Then our Lagrangian will be

L = K − V

=12mx2 − V (x). (2.8)


Then the terms of the Euler-Lagrange equations are

∂L

∂x= −dV (x)

dx(2.9)

d

dt

(∂L

∂q

)= mx (2.10)

Therefore the Euler-Lagrange equation yields

mx = −dV (x)dx

, (2.11)

which is Newton’s Law. In the case of a classical harmonic oscillator, the potential is V (x) = 12kx

2, and

so equation (2.11) becomes

mx = −kx, (2.12)

which is Hooke’s Law.

2.2.3 The Euler-Lagrange Equation for Fields

A field is essentially a set of numbers at each point in space-time [2]. So now instead of our Lagrangian

consisting solely of paths of the form q(t), our Lagrangian must consist of field configurations. These field

configurations are functions of both space and time, and not solely of time as before. As a consequence,

to acquire the action one has to now integrate over both space and time, making our Lagrangian for

the field a Lagrangian Density (for which we reserve the symbol L ). With this in mind, we can replace

our paths q(t) with the field configurations Φ(xµ) in the previous section (where xµ in this case is a

four-vector). Likewise q(t) is replaced with ∂µΦ(xµ), where µ index represents all of the four dimensional

space-time. It is possible to include more than the first derivative of the field ∂µΦ(xµ), but we shall do

this later when we consider more complex interactions. Hence our action for fields now becomes

S =∫

Ω

d4xL (Φ(xµ), ∂µΦ(xµ)), (2.13)

where Ω is the region in space-time we are interested in and the action is writen in a convention whereby

the integrand appears last. We now derive the Euler-Lagrange Equation for Fields.

We start by noting that the field will trace out a hypersurface1 of possible field configurations over

space-time as it evolves. In order words, the configurations will trace out a four dimensional surface in

a five dimensional space which contains space-time. This is the region Ω and its boundary will be ∂Ω.1A surface that extends beyond three dimensions. Essentially a subspace that has one less dimension than that of the

space it occupies.

2.2. THE LAGRANGIAN FORMALISM 7

Consider very small deviations in the field and its derivative, in such a way that these variation vanish

on the boundary ∂Ω (an important point we shall use later on) [3]. This gives our action as

δS = δ

∫Ω

d4xL (Φ(xµ), ∂µΦ(xµ)) (2.14)

=∫

Ω

d4xδL (Φ(xµ), ∂µΦ(xµ)). (2.15)

The Lagrangian Density, henceforth referred to as the Lagrangian (for convenience), can be expanded as

δL =∂L

∂ΦδΦ +

∂L

∂(∂µΦ)δ(∂µΦ). (2.16)

As with our particle case, we attempt to rewrite our second term in the above to factor out the δΦ. We

firstly rewrite δ(∂µΦ) using the fact that differentials and variations commute, hence

δ(∂µΦ) = ∂µ(δΦ). (2.17)

We then note that the second term is part of the product rule as in the following

∂µ

[∂L

∂(∂µΦ)δΦ]

= ∂µ

[∂L

∂(∂µΦ)

]δΦ +

∂L

∂(∂µΦ)∂µ(δΦ) (2.18)

∂L

∂(∂µΦ)∂µ(δΦ) = ∂µ

[∂L

∂(∂µΦ)

]δΦ− ∂µ

[∂L

∂(∂µΦ)δΦ]. (2.19)

We substitute the above back into equations (2.16) and (2.14) to get the following integrals

δS =∫

Ω

d4x∂L

∂ΦδΦ +

∫Ω

d4x∂µ

[∂L

∂(∂µΦ)

]δΦ−

∫Ω

d4x∂µ

[∂L

∂(∂µΦ)δΦ]. (2.20)

The last term in the above equation is in the form of a divergence, and from Gauss’s Theorem2, one can

write ∫Ω

d4x∂µ

[∂L

∂(∂µΦ)δΦ]

=∫

Σ=∂Ω

d4x

[∂L

∂(∂µΦ)δΦ]dΣµ, (2.21)

where the last term is now a surface integral. We can simplify this by remembering that we defined our

variation on the boundary to vanish. Hence, the above integral vanishes also, which leaves our action as

δS =∫

Ω

d4x

[∂L

∂Φ+ ∂µ

(∂L

∂(∂µΦ)

)]δΦ. (2.22)

Finally, recall that for classical mechanics we extremize the action, so the above becomes

δS =∫

Ω

d4x

[∂L

∂Φ+ ∂µ

(∂L

∂(∂µΦ)

)]δΦ = 0, (2.23)

2Also known as the Divergence Theorem.


and so the expression in the square brackets must be zero as the integral applies to arbitary instances of

space-time. Hence

∂L

∂Φ+ ∂µ

(∂L

∂(∂µΦ)

)= 0, (2.24)

which is the Euler-Lagrange Equation for Relativistic Fields. This completes our coverage of classical

field theory. We now turn our attention to quantum mechanics and its path integral formulation. We

shall return to the ideas of the this section when we combine both the classical field theory with the path

integral formulation of quantum mechanics into Quantum Field Theory.

Chapter 3

Path Integral Formulation of

Quantum Mechanics

Thirty-one years ago, Dick Feynman told me about his “sum over histories” version of quan-

tum mechanics. “The electron does anything it likes,” he said. “It just goes in any direction

at any speed, . . . however it likes, and then you add up the amplitudes and it gives you the

wavefunction.” I said to him, “You’re crazy.” But he wasn’t.

–Freeman Dyson, 1980 [4]

We have seen in the classical theory of mechanics, the motion of an object occurs in such a fashion

so is to undergo the least action, i.e. it abides the Hamilton’s Principle of Least Action

δS = δ

∫ t2

t1

L(q(t), q(t))dt = 0 (3.1)

where the action is defined by

S =∫ t2

t1

L(q(t), q(t))dt (3.2)

Quantum Mechanically however, the quantum mechanical object explores every possible path (not just

the path of least action) in its motion. Richard Feynman developed this idea to formulate Quantum

Mechanics in a different way (by using path integrals in quantum mechanics developed by P.A.M Dirac)

and the definition (3.2) played a critical role. This formulation is useful in the theory of Quantum Fields,

as we shall see.

9

10 CHAPTER 3. PATH INTEGRAL FORMULATION OF QUANTUM MECHANICS

3.1 Free Particle

We now show that the expected value for a free particle evolving from q to q′ quantum mechanically

is described by the full classical Lagrangian in the following way

<q′|e−i bH(t′−t)/~|q> ≈ exp(i

~S(q, q)

)(3.3)

but where the particle explores all possible paths. We begin by noting that the non-relativistic quantum

mechanics is governed by the Schrodinger Equation

i~∂

∂t|Ψ(t)> = H(p, q)|Ψ(t)>

where |Ψ(t)> is the state vector in Hilbert space. The formal solution1 to this is

|Ψ(t)> = e−i bHt/~|Ψ(0)>

In the generalised co-ordinate system q, the above becomes

|q, t> = e−i bHt/~|q> (3.4)

Therefore the transitional amplitude for the particle at q propagating to q′, i.e. the probability density

of particle transiting from q to q′ is

<q′, t′|q, t> = <q′|e−i bH(t′−t)/~|q> (3.5)

This is known as the Feynman Kernal or Propagator. Using the path integral formalism for the propa-

gator, one can construct the Quantum Mechanical Transitional Amplitude from the classical Lagrangian

of the system, without the need for Hilbert Space state vectors and non-commuting Operators. In order

to do this, we must divide the time interval t′ − t into small sub-intervals, i.e.

tn = t+ n∆t

and so t′ − t = N∆t where n = 0, 1, . . . , N − 1. We substitute this into the propagator to get

<q′|e−i bHN∆t/~|q> (3.6)

Now at each sub-interval, a set of basis states is introduced using the completeness relation2∫ ∞

−∞|qn><qn|dqn = I (3.7)

1proof can be found in appendix A2proof can be found in appendix B


Hence relation (3.12) becomes

<qn+1|e−i bH∆t/~I|qn> =∫ ∞

−∞

dpn

2π~e−ip2∆t/2m~eipnqn+1/~e−ipnqn/~ (3.13)

=∫ ∞

−∞

dpn

2π~e−ip2∆t/2m~eipn(qn+1−qn)/~ (3.14)

The above relation is the form of a Gaussian. Making the change of variables

α = −∆tm~

(3.15)

J =1~(qn+1 − qn) (3.16)

Then the Gaussian Integral4 is

I =∫ ∞

−∞e

12 iαx2+iJxdx =

√2πiαe−

iJ22α

Now the transfer matrix can be written

<qn+1|e−i bH∆t/~|qn> =

√−im

2π~∆tei∆t m

2~

qn+1−qn

∆t

2

Substituting back into the kernel

<q′|e−i bHN∆t/~|q> =(−im

2π~∆t

)N/2 N−1∏n=0

∫ ∞

−∞dqn

N−1∏n=0

ei∆t m

2~

qn+1−qn

∆t

2

(3.17)

=(−im

2π~∆t

)N/2 N−1∏n=0

∫ ∞

−∞dqne

i∆t m2~PN−1

n=0

qn+1−qn

∆t

2

(3.18)

Now we proceed to take the limit as ∆t→ 0 and hence N →∞, the result being

lim∆t→0

<q′|e−i bH(t′−t)/~|q> =∫Dq(t)exp

(i

~

∫ t′

t

12mq2(t)dt

)Which is off the form of

Z ≡<q′|e−i bH(t′−t)/~|q> =∫Dq(t)exp

(i

~

∫ t′

t

L(q, q)dt

)(3.19)

=∫Dq(t)exp

(i

~S(q, q)

)(3.20)

3.2 Particle in a Time-Independent Potential

We apply the same proceedure as in the previous section, however this time we note that the Hamil-

tonian of the system is in fact

H = K + V (q) (3.21)4worked through in appendix D

3.2. PARTICLE IN A TIME-INDEPENDENT POTENTIAL 13

Letting λ = i(t′ − t)/~ and using the Hamiltonian, equation 3.5 can be writen as

<q′, t′|q, t> = <q′|e−λ( bK+bV )|q> (3.22)

Using the Trotter Formula

<q′|e−λ( bK+bV )|q> = limN→∞

<q′|(e−λ bK/Ne−λbV /N

)N

|q> (3.23)

the above can be expressed as


<q′|(e−λ bK/Ne−λbV /N

)(e−λ bK/Ne−λbV /N

). . .(e−λ bK/Ne−λbV /N

)|q> (3.24)

We need to divide the time interval into sub-divisions so that

tn = t+ n∆t

and so t′−t = N∆t where n = 0, 1, . . . , N−1. This means the now λ = iN∆t/~. This makes equation 3.24

into


<q′|(e−γ bKe−γ bV )N

|q>

= limN→∞

<q′|(e−γ bKe−γ bV )(e−γ bKe−γ bV ) . . .(e−γ bKe−γ bV ) |q>

= limN→∞

<q′|(e−γ bKe−γ bV ) I

(e−γ bKe−γ bV ) I . . . I

(e−γ bKe−γ bV ) |q> (3.25)

where γ = i∆t/~. We then proceed to take all possible paths. This is done by introducing the closure

relation 3.7. This makes the right hand side of equation 3.25 into

limN→∞

∫ ∞

−∞. . .

∫ ∞

−∞dqN−1 <q

′|(e−γ bKe−γ bV ) |qN−1><qN−1|

(e−γ bKe−γ bV ) |qN−2><qN−2|

. . . |q1><q1|(e−γ bKe−γ bV ) |q> (3.26)

Which can be simlified so that equation 3.25 can be writen


N−1∏n=1

∫ ∞

−∞dqn <q

′|(e−γ bKe−γ bV ) |qN−1><qN−1|

. . . |q1><q1|(e−γ bKe−γ bV ) |q> (3.27)

We write each of the Transfer Amplitude in above as

T (qn+1, qn) = <qn+1|(e−γ bKe−γ bV ) |qn>

We evaluate the transfer amplitude by introducing the momentum version of the completeness relation,

i.e. equation 3.11 (retaining the normalisation constants). Therefore the transfer matrix can be writen


as

T (qn+1, qn) = <qn+1|(e−γ bKe−γ bV ) I|qn>

=1

2π~

∫ ∞

−∞dpn <qn+1|

(e−γ bKe−γ bV ) |pn><pn|qn>

=1

2π~

∫ ∞

−∞dpn <qn+1|e−γ bKe−γV |pn><pn|qn>

=1

2π~

∫ ∞

−∞dpn <qn+1|e−γKe−γV |pn><pn|qn>

=1

2π~

∫ ∞

−∞dpne

−γKe−γV <qn+1|pn><pn|qn> (3.28)

Here we have used F (A)|A > = F (A)|A > above to change the operators to eigenvalues. Using the

relations between co-ordinate and momentum spaces, each of the bra-kets in the above can be determined

from the normalisation convention

<qn|pn> = eipnqn/~ , <pn|qn> = e−iqnpn/~

Thus equation 3.28 becomes the following from the above conventions

T (qn+1, qn) =1

2π~

∫ ∞

−∞dpne

−γKe−γV eipnqn+1/~e−iqnpn/~ (3.29)

To evaluate the integral, we write it in a more recognizable form. We see that if

K =p2

2m

Then equation 3.29 can be put into a complex Gaussian integral of the form of 3.1. By completing the

square

T (qn+1, qn) =1

2π~

∫ ∞

−∞dpnexp

(−1

2i∆tm~

p2n

)exp

(ipn

(qn+1 − qn)~

)exp (−γV )

=1

2π~

∫ ∞

−∞dpnexp

(−1

2i∆tm~

p2n + i

1~(qn+1 − qn)pn

)exp (−γV ) (3.30)

and conducting a change of variables

α = −∆tm~

(3.31)

J =1~(qn+1 − qn) (3.32)

Equation 3.29 can be put into the Gaussain Integral form of equation 3.1. Now if the potential is a spatial

function (i.e. V (q)), then exp (−γV ) is constant with respect to pn and the integral has the solution of

T (qn+1, qn) =1

2π~

√2πiαexp

(− iJ

2

2α

)exp (−γV )

3.3. FROM PARTICLES TO FIELDS 15

=1

2π~

√−2πim~

∆texp

(im(qn+1 − qn)2

2∆t~2

)exp (−γV )

=

√− im

2π~∆texp

(im∆t

2~

((qn+1 − qn)

∆t

)2)exp (−γV )

=

√− im

2π~∆texp

(im∆t

2~

[((qn+1 − qn)

∆t

)2

− V (q)

])(3.33)

Substitutioning this back into equation 3.8 (which is known as the Feynman Kernel) gives

Z = lim∆t→0

<q′|e−λ( bK+bV )|q> (3.34)

= limN→∞

(−im

2π~∆t

)N/2 N−1∏n=1

∫ ∞

−∞dqn

N−1∏n=0

exp

(i∆t~

[m

2

((qn+1 − qn)

∆t

)2

− V (q)

])(3.35)

= limN→∞

(−im

2π~∆t

)N/2 N−1∏n=1

∫ ∞

−∞dqnexp

(i∆t~

N−1∑n=0

[m

2

((qn+1 − qn)

∆t

)2

− V (q)

])(3.36)

=∫Dq(t)exp

(i

~

∫ t′

t

[12mq2(t)− V (q)

]dt

)(3.37)

3.3 From Particles to Fields

Since we are using classical fields in this path integral formulation of quantum mechanics, we simply

replace our full Lagrangian of the particle case with that of a scalar field. We have already seen how this

is done and its justification in section 2.2.3. Therefore, our action is of the form of equation (2.13) and

so our expectation value is

Z ≡∫Dq(t) exp

(i

~

∫d4xL (Φ)

). (3.38)

Now the form of the Lagrangian will be that of the previous section, i.e. K(∂Φ) + V (Φ). Writing this

standard Hamiltonian with our substitution, the Lagrangian becomes

L =12(∂Φ)2 − V (Φ). (3.39)

The form of the potential will be discussed in various theories described in later chapters (such as the Φ4

theory).

Chapter 4

The Vacuum and Free Field Theory

In our state vectors for the particle cases, we choose |q> and <q′| to represent the particle’s initial and

final states and <q′|e−i bH(t′−t)/~|q> progation of a particle from position q to q′. The same is applicable

to fields, except now the positions we used are replaced with configurations. We now look at a special

case where the initial and final states are the same and the ground state (i.e., |0> and < 0|). So the

statement

Z ≡<0|eiHT |0>=∫Dq(t) exp

(i

~

∫ T

0

dt

[12mq2 − V (q)

]), (4.1)

gives the transitional amplitude from the vacuum state to the vacuum state. This will allow us to measure

the energy of the vacuum state, consider also quantum flucuations within the vacuum, but not much else

that interests us. Let us consider a situation where we disturb this vacuum while dealing with fields.

We introduce to the vacuum state, a source, J(xµ) that provides the mechanism for the disturbance.

So the above equation now has the form

Z ≡<0|eiHT |0>=∫DΦ exp

(i

~

∫Ω

d4x

[12

(∂µΦ)2 − V (Φ) + JΦ])

. (4.2)

Before we move on to see how to evaluate this, we need to decide on the form of our potential V (Φ).

4.1 The Forms of the Scalar Potential

Conducting a power series expansion of the potential, one gets the expression

V (Φ) = 1 + Φ +Φ2

2+

Φ3

3!+

Φ4

4!+ . . . (4.3)

17

18 CHAPTER 4. THE VACUUM AND FREE FIELD THEORY

Now we demand that the potential is symmetric so that no negative infinities are encountered. This

means that the potential is reduced to

V (Φ) = 1 +Φ2

2!+

Φ4

4!+ . . . (4.4)

The quadratic term in the potential is the harmonic potential and will allow us to evaluate the integral

as a Gaussian. The quartic term on the other hand has no analytic solution, so if it is included, it will

need to be done perturbatively. This is discussed in the chapter on Phi-4 Theory.

4.2 The Free Field

Using the harmonic form of the potential, one gets the generating functional Z as

Z ≡<0|eiHT |0>=∫DΦ exp

(i

~

∫Ω

d4x

[12

(∂µΦ)2 − 12m2Φ2 + JΦ

]). (4.5)

Now the above is almost in the form of a generalised Gaussian integral, when we use the harmonic form

of the potential 12m

2Φ2 (refer to appendix E to see how is it done). This means that the Lagrangian has

the form

L =12

(∂µΦ)2 −m2Φ2 + JΦ (4.6)

=12

(∂µΦ∂µΦ)− Φm2Φ + JΦ. (4.7)

We have to turn the ∂µΦ∂µΦ into the form Φ∂2Φ, so we can complete the square and then do the

Gaussian integral as in the form described in appendix E, which can be viewed as the discretization of

spacetime and then evaluating it (albeit after alot of assumptions, details of which we will largely ignore).

Re-expressing is done by using integration by parts, which is the following∫∂µΦ∂µΦd4x =

∫∂µ (Φ∂µΦ) d4x−

∫Φ∂2Φd4x. (4.8)

The first term on the right is a divergence (and so can be converted to a surface integral) and require

that the field falls off sufficiently rapidly at the boundaries at infinity, and so vanishes, leaving∫∂µΦ∂µΦd4x = −

∫Φ∂2Φd4x. (4.9)

This means that the Lagrangian is

L = −Φ(∂2 −m2

)Φ + JΦ. (4.10)

So the result of the integration is

Z(J) = Z(0) exp(− i

2JA−1J

), (4.11)

where Z(0) is the constant when no source is present.

4.2. THE FREE FIELD 19

4.2.1 Free Propagator

Now equation (4.11) can be writen as

Z(J) = Z(0) exp

(− i

2

∑ik

JiA−1ik Jk

), (4.12)

which for the continous form is

Z(J) = Z(0) exp(− i

2

∫ ∫dxdyJ(x)A−1(x− y)J(y)

). (4.13)

The inverse of A however, requires a further look. The inverse and its matrix will satisfy

AA−1 = I, (4.14)

or also

AijA−1jk = δik. (4.15)

In the continuum limit, this is

(∂2 −m2

)∆F (x− y) = δ(4)(x− y), (4.16)

where we have substituted in for A and replaced the symbol A−1 with ∆F . We shall show that ∆F is

the Free Propagator.

4.2.2 Solving for the Free Propagator

It is best to show this in reciprocal space. Taking the Fourier transform of expression (4.16) we get

F4

[(∂2 −m2

)∆F (x− y)

]= F4

[δ(4)(x− y)

](4.17)

F4 [∂µ∂µ∆F (x− y)]−F4

[m2∆F (x− y)

]= 1 (4.18)

The first term in the above expression can be writen using the product rule as

∂µ(∂µ∆F e

−ikx)

= ∂2∆F e−ikx + ∂µ∆F

(∂µe

−ikx), (4.19)

and so using the integration by parts once, this term becomes∫Ω

dx4∂µ∂µ∆F e−ikx =

∫Ω

dx4∂µ(∂µ∆F e

−ikx)−∫

Ω

dx4∂µ∆F

(∂µe

−ikx)

(4.20)

=∫

Σ=dΩ

dΣµ∂µ∆F e−ikx −

∫Ω

dx4∂µ∆F

(−ikµe

−ikx)

(4.21)

= −∫

Ω

dx4∂µ∆F

(−ikµe

−ikx), (4.22)


where we have used the divergence theorem in the same way as for equation (2.20) in section 2.2.3.

Likewise a second integration by parts can be done to the remaining partial derivative as

−∫

Ω

dx4∂µ∆F

(−ikµe

−ikx)

=∫

Ω

dx4∂µ(∆F ikµe

−ikx)−∫

Ω

dx4∆F∂µ(−ikµe

−ikx)

(4.23)

=∫

Σ=dΩ

dΣµ(∆F ikµe

−ikx)−∫

Ω

dx4∆F

(ikµik

µe−ikx)

(4.24)

= −∫

Ω

dx4∆F

(−kµk

µe−ikx). (4.25)

After these integrations, one can see the fact that the Fourier transform has transformed differential

operator to an algebraic expression of the form

∂µ → −kµ, ∂µ∂µ → −kµk

µ = k2. (4.26)

Therefore our propagator becomes the following(k2 −m2

)∆F (k) = 1 (4.27)

∆F (k) =1

k2 −m2. (4.28)

It is however evident that the propagator has simple poles at k2 = m2. One can displace these poles by

using the iε prescription, which is simply just replacing m→ m− iε. This ensures the convergence of the

functional, and so the propagator now has the form

∆F (k) =1

k2 −m2 − iε, (4.29)

or in co-ordinate space

∆F (x− y) =∫

d4k

(2π)4e−ik(x−y)

k2 −m2 − iε, (4.30)

where the factor of (2π)4 as a convention in our Fourier transform.

Combining the results of the previous sections, the functional has the final form of

Z(J) = Z(0) exp(− i

2


)(4.31)

≡ Z(0) expiW (J), (4.32)

where we will discuss the exact nature of W (J) in the next section.

4.3 Origin of Force

We shall see that W (J) is associated with the energy of the interaction. If it is negative, then it is

energetically favourable for the fields to couple and attract, since the introduction of the disturbances has

4.3. ORIGIN OF FORCE 21

lowered the energy. We shall show that introducing the disturbance of the form J(x) = J1(x1) + J2(x2),

where each of the disturbances is a delta function source then the energy is lowered.

Chapter 5

The Phi-4 Scalar Field Theory

We have seen in the free field theory that the potential had the form of an harmonic potential and all

remaining terms were removed. Now we reintroduce one of the terms so our Lagrangian has the form

L =12

(∂µΦ)2 −m2Φ2 − λ

4!Φ4 + JΦ. (5.1)

This represents a generating functional which can no longer be done analytically. So we must resort to

perturbation theory to evaluate the integral. We shall cover two methods which both give the exact same

predictions. The first method is Wick Contraction and the other is the Schwinger Formalism. Before we

proceed, let us re-express the generating functional in a first order expansion in the powers of Φ4.

Z(J) =∫DΦ exp

(i

~

∫Ω

d4x

[12

(∂µΦ)2 − 12m2Φ2 + JΦ

])exp

(− i

~

∫Ω

d4xiλ

4!~Φ4

)(5.2)

=∫DΦ exp

(i

~

∫Ω

d4x

[12

(∂µΦ)2 − 12m2Φ2 + JΦ

])×

(1− i

~

∫Ω

d4xλ

4!~Φ4 +

12i

~

∫Ω

d4x

(λ

4!~

)2

Φ8 + . . .

). (5.3)

One can see that the above equation has the form

Z[J ] =1n!

(−iλ4!

)n ∫ ∫. . .

[∫DΦ exp

(i

~

∫Ω

d4x

[12

(∂µΦ)2 − 12m2Φ2 + JΦ

])(Φ4)n

]d4x . . . d4x.(5.4)

We now introduce the functional derivative, which acts as a normal derivative in almost every way, except

derivatives are done with respect to functionals instead of variables. The functional derivative will be

represented by the δ symbol and used in a similar fashion to partial derivatives. We use the functional

derivative to simplify the above expression in the following way

Z[J ] =1n!

(−iλ4!

)n ∫d4x

δ4

iδJ4. . .

∫d4x

δ4

iδJ4

[∫DΦ exp

(i

~

∫Ω

d4x

[12

(∂µΦ)2 − 12m2Φ2 + JΦ

])].(5.5)

23

24 CHAPTER 5. THE PHI-4 SCALAR FIELD THEORY

The functional derivatives can now be re-expressed, as it is an expansion of an exponential. This leads to

Z[J ] = exp(−iλ4!

∫d4x

δ4

iδJ4

)[∫DΦ exp

(i

~

∫Ω

d4x

[12

(∂µΦ)2 − 12m2Φ2 + JΦ

])]. (5.6)

We have already seen how to evaluate the integral inside the square brackets, so the above expression

simplifies to

Z[J ] = Z(0) exp(− iλ

4!

∫d4x

δ4

iδJ4

)[exp

(− i

2


)]. (5.7)

5.1 Wick Contraction

Wick contraction allows us to define expectation values in terms of 2 point Green Functions. From our

Gaussian Integral

<q1, q2, . . . , qn>=1

Z(0)

N∏k

∫ ∞

−∞dqk exp

(i

2qTAq

)q1q2 . . . qn = in

∑Perm

(A−1

ab A−1bc . . . A−1

pq

), (5.8)

where for our field case we can make the changes

A−1ij → ∆F (x− y), lim

k→∞

N∏k

∫ ∞

−∞dqk →

∫DΦ. (5.9)

We define the n-point Green function by

G(n)(x1, . . . , xn) =1

Z(0)

∫DΦ exp

(i

~

∫Ω

d4x

[12

(∂µΦ)2 − 12m2Φ2 + JΦ− iλ

4!~Φ4(x)

])Φ(x1) . . .Φ(xn)(5.10)

In the first order perturbation theory O(−iλ), the above becomes

1Z(0)

iλ

4!~

∫d4x

∫DΦ exp

(i

~

∫Ω

d4x

[12

(∂µΦ)2 − 12m2Φ2 + JΦ

])Φ(x1) . . .Φ(xn)Φ4(x) (5.11)

We shall now show an example of how Wick contraction arises for the case of the 4-point Greens function

in first order perturbation theory. We start by writing the definition of the 4-point Green function, noting

that ∫DΦΦ(x1) . . .Φ(xn) exp

(i

~

∫Ω

d4x

[12

(∂µΦ)2 − 12m2Φ2 + JΦ

])=(

δ

iδJ(x1)

). . .

(δ

iδJ(xn)

)∫DΦ exp

(i

~

∫Ω

d4x

[12

(∂µΦ)2 − 12m2Φ2 + JΦ

]). (5.12)

The definition is

G(4) =1

Z(0)iλ

4!~

∫d4x

∫DΦ exp

(i

~

∫Ω

d4x

[12

(∂µΦ)2 − 12m2Φ2 + JΦ

])×Φ(x1)Φ(x2)Φ(x3)Φ(x4)Φ4(x) (5.13)

5.2. SCHWINGER FORMALISM 25

=1

Z(0)iλ

4!~

(δ

iδJ(x1)

). . .

(δ

iδJ(x4)

)×∫d4x

∫DΦ exp

(i

~

∫Ω

d4x

[12

(∂µΦ)2 − 12m2Φ2 + JΦ

])Φ4(x) (5.14)

=iλ

4!~

(δ

iδJ(x1)

). . .

(δ

iδJ(x4)

)exp

(− i

2


)Φ4(x) (5.15)

5.2 Schwinger Formalism

26 CHAPTER 5. THE PHI-4 SCALAR FIELD THEORY

Chapter 6

Spin-1 Vector Fields

We consider the Proca Lagrangian for a massive spin-1 vector field, which has the form

L = −14FµνF

µν +12m2AµAµ +AµJ

µ, (6.1)

where the electromagnetic field tensor is defined as

Fµν = ∂µAν − ∂νAµ. (6.2)

If one substitutes the above Lagrangian into the Euler-Lagrange equation for fields one will get the

inhomogenous Maxwell equations. In order to determine the propagator for the spin-1 massive vector

field we need to re-express the first term of the Lagrangian into a quadratic form. We begin by expanding

the term and then simplifying using the 4-D divergence theorem as in section 2.2.3.

27

28 CHAPTER 6. SPIN-1 VECTOR FIELDS

Chapter 7

Dirac Equation

In order for a relativitic treatment, the appropriate relativitic equation for energy E must be used for

forming the Quantum Mechanical equation, namely

E2 = m2 + |~p|2 (7.1)

here we have used Natural Units (i.e. c = ~ = 1). Equation 7.1 can be rewriten as the difference of

squares

E2 − |~p|2 = m2 (7.2)

(E − |p|) (E + |p|) = m2 (7.3)

However, the momentum squared term in equation 7.1 will be rewritten in order to take into account of

spin and therefore equation 7.3 will be represented in a different way. The spin in Quantum Mechanics

is represented by the Pauli Spin Matrices

~σ ≡ (σx, σy, σz) (7.4)

where

σx =

0 1

1 0

, σy =

0 −i

i 0

, σz =

1 0

0 −1

.

and that the spin matrices follow the identity(~σ • ~A

)(~σ • ~B

)= ~A • ~B + i~σ •

(~A× ~B

)(7.5)

Setting ~A = ~B = ~p, we get

(~σ • ~p) (~σ • ~p) = ~p • ~p+ i~σ • (~p× ~p)

29

30 CHAPTER 7. DIRAC EQUATION

but ~p × ~p = 0 since our definition of momentum (i.e. p = −i∇), and ∇ × ∇ = 0. Hence the identity

becomes

(~σ • ~p) (~σ • ~p) = ~p • ~p

and the mass-energy equation (equation 7.2) can be rewritten as

E2 − (~σ • ~p) (~σ • ~p) = m2

which, as a difference of squares, becomes

(E − (~σ • ~p)) (E + (~σ • ~p)) = m2 (7.6)

Applying the Schrodinger Correspondence Rules and operating on a wavefunction to the above, gives

(i∂t + i (~σ • ∇)) (i∂t − i (~σ • ∇))Ψ = m2Ψ (7.7)

Now, in order to represent the above as first order in time and linear in Gradient operator, we separate

equation 7.7 as

(i∂t + i (~σ • ∇))[ 1m

(i∂t − i (~σ • ∇))Ψ]

= mΨ (7.8)

we label the square bracket term as

Ψ(R) =[ 1m

(i∂t − i (~σ • ∇))Ψ]

(7.9)

Using the above, equation 7.8 becomes equation 7.10 and equation 7.9 to get equation 7.11, as noted

below

mΨ = (i∂t + i (~σ • ∇))Ψ(R) (7.10)

mΨ(R) = (i∂t − i (~σ • ∇))Ψ (7.11)

Note that both of the above equations are first order in time and linear in the gradient, but we now have

two wavefunctions to consider. We need to recombine these two equations to form just one. We shall do

this by adding and subtracting the equations to give

(i∂t + i (~σ • ∇))Ψ(R) + (i∂t − i (~σ • ∇))Ψ = m(Ψ + Ψ(R)

)(7.12)

(i∂t + i (~σ • ∇))Ψ(R) − (i∂t − i (~σ • ∇))Ψ = m(Ψ−Ψ(R)

)(7.13)

We endevour to write the equation in terms of just one wavefunction, so we need to combine the wave-

functions Ψ & Ψ(R). For this, we begin by rewriting the above two equations by expanding and collecting

like terms, giving

i∂t

(Ψ(R) + Ψ

)+ i (~σ • ∇)

(Ψ(R) −Ψ

)= m

(Ψ + Ψ(R)

)(7.14)

i∂t

(Ψ(R) −Ψ

)+ i (~σ • ∇)

(Ψ(R) + Ψ

)= m

(Ψ−Ψ(R)

)(7.15)

31

Let

ΨA =(Ψ(R) + Ψ

), ΨB =

(Ψ(R) −Ψ

)Therefore equations 7.14 & 7.15 can be rewriten as

i∂tΨA + i (~σ • ∇) ΨB = mΨA (7.16)

i∂tΨB + i (~σ • ∇) ΨA = −mΨB (7.17)

Which is in a form that can be expressed as a matrix equation i∂t i (~σ • ∇)

i (~σ • ∇) i∂t

ΨA

ΨB

= −m

−ΨA

ΨB

(7.18)

Bringing everything to the left hand side and factorising in terms of the wavefunction matrix we get i∂t i (~σ • ∇)

i (~σ • ∇) i∂t

ΨA

ΨB

+m

−1 0

0 1

ΨA

ΨB

= 0 (7.19)

i∂t i (~σ • ∇)

i (~σ • ∇) i∂t

ΨA

ΨB

−m

1 0

0 −1

ΨA

ΨB

= 0 (7.20)

i∂t i (~σ • ∇)

i (~σ • ∇) i∂t

−m

1 0

0 −1

ΨA

ΨB

= 0 (7.21)

The matrix involving the gradient and the partial derivative can be expressed as i∂t i (~σ • ∇)

i (~σ • ∇) i∂t

= i∂t

1 0

0 1

+ i

0 ~σ

~σ 0

• ∇ (7.22)

= i∂t + i

0 ~σ

~σ 0

• ∇ (7.23)

Hence we get i∂t + i

0 ~σ

~σ 0

• ∇ −m

1 0

0 −1

ΨA

ΨB

= 0 (7.24)

Letting

α =

0 ~σ

~σ 0

, β =

1 0

0 −1

, Ψ =

ΨA

ΨB

We the Dirac Relativistic Equation for Spin-Half Particles in Free Space

(i∂t + iα • ∇ −mβ)Ψ = 0. (7.25)

32 CHAPTER 7. DIRAC EQUATION

The above can be writen in a slightly different way using a single matrix γµ, giving

(iγµ∂µ −m) Ψ = 0, (7.26)

where γµ has the form

γ0 = β, γi = βαi (i = 1, 2, 3). (7.27)

The Important Features of the Equation are

1. It is first order in time.

2. Wavefunction has four components due to the nature of the construction, not necessarily due to

the 4-dimensional nature of space-time.

Chapter 8

Fermion Fields

The main feature of the Fermions different to that of other fields and particles is the exclusion principle.

This principle means that the Fermions cannot be in the same quantum state and that the wavefunctions

are anti-symmetric under particle interchange (i.e., they are anti-commutative). Our path integral has to

be modified to account for this exclusion principle. Luckily a set of anti-commuting variables are available

to us, which are called Grassmann Variables. These variables obey the following set of properties (where

we shall use the η symbol to denote Grassmann variables)

1. They anti-commute:

[ηi, ηj ]+ = ηiηj − ηjηi = 0 ∀i, j. (8.1)

2. Taylor Expansion is non-zero for only upto the first order:

f(η) = a+ bη. (8.2)

3. The derivatives are:

d

dη1 = 0,

dηj

dηi= δij . (8.3)

4. Integration properties are: ∫dη1 = 0,

∫dηη = 1. (8.4)

An example of using the above properties is the following, consider∫dηf(η) =

∫dη(a+ bη) = a

∫dη1 + b

∫dηη = b. (8.5)

33

34 CHAPTER 8. FERMION FIELDS

Lets now convert our Gaussian integral to reflect these new variables also. The Gaussian shall have the

form∫ ∫dηdηeηaη =

∫ ∫dηdη (1 + ηaη) =

∫dη

∫dη1 +

∫dη

∫dηηaη =

∫dηaη = a = eln a, (8.6)

where η and η are independent complex Grassmann variables. Note the positive sign in the exponent.

Applying this to our generalised Gaussian we obtain

N∏n=1

∫dηn

∫dηn exp

(ηTAη

)= det[A] = exp (+Tr lnA) . (8.7)

Now we can write our quantum mechanical action using these variables to denote Fermionic or Dirac

fields with Grassmann spinor sources η and η

Z[η, η] =∫Dη

∫Dη exp

(i

~

∫d4x

[ψ (iγµ∂µ −m)ψ + ηψ + ηψ

])(8.8)

= C det (iγµ∂µ −m) exp(−iη (iγµ∂µ −m)−1

η), (8.9)

where the Dirac field satisfies

(iγµ∂µ −m)S(x− y) = δ(4)(x− y). (8.10)

Using the property of Fourier transform found in section 4.2.2, namely ∂µ → −ipµ (where we shall

use p instead of k in the case of fermions), and using the iε prescription, our Fermionic Propagator in

co-ordinate space is

S(x) =∫ ∞

−∞

d4p

(2π)4eipx

γµpµ −m+ iε. (8.11)

Chapter 9

The Feynman Rules

We have seen how Wick Contraction of section 5.1 can be used to determine the expectation value of

the various sources involved in scattering.

The rules can be found in QFT Notes.

35

36 CHAPTER 9. THE FEYNMAN RULES

Chapter 10

Quantum Electro-Dynamics (QED)

37

38 CHAPTER 10. QUANTUM ELECTRO-DYNAMICS (QED)

Appendix A

Formal Solution to Schrodinger’s

Equation

Theorem 1 Show that

|q, t> = e−i bHt/~|q>

We start with the Schrodinger Equation

i~∂

∂t|Ψ> = H|Ψ>

which can be writen as1|Ψ>

∂

∂t|Ψ> =

H

i~Notice that ... ∫ |Ψ>

|Ψ0>

d|Ψ>|Ψ>

= − iH~

∫ t

t0

dt (A.1)

[loge(|Ψ>)

]|Ψ>

|Ψ0>= − iH

~

[t]t

t0(A.2)

loge(|Ψ>)− loge(|Ψ0>) = − iH~

(t− t0) (A.3)

loge

(|Ψ>|Ψ0>

)= − iH

~(t− t0) (A.4)

|Ψ>|Ψ0>

= e−icH~ (t−t0) (A.5)

|Ψ> = e−icH~ (t−t0)|Ψ0> (A.6)

J

39

40 APPENDIX A. FORMAL SOLUTION TO SCHRODINGER’S EQUATION

42 APPENDIX B. PROOF OF THE COMPLETENESS RELATION

Appendix C

Proof of Normalisation Convention

Theorem 3 Show that

<qn|pn> = <pn|qn>∗ = eipnqn/~

We begin by writing the state vector as

|Ψ> = I |Ψ>

Then we substitute the identity with the co-ordinate completeness relation

|Ψ> =∫ ∞

−∞dqn|qn><qn|Ψ>

Now we project the state vector onto the momentum representation p and we get

<pn|Ψ> =∫ ∞

−∞dqn <pn|qn><qn|Ψ> (C.1)

Since < qn|Ψ> is the co-ordinate representation of the state vector, we denote it by |Ψq >. Similarly

<pn|Ψ> is the momentum representation of the state vector, we denote it by |Ψp>. Hence equation C.1

becomes

|Ψp> =∫ ∞

−∞dqn <pn|qn> |Ψq> (C.2)

The above is of the form of a Fourier Transform, as the co-ordinate space is being transformed into

momentum or reciprocal space. Fourier Transform is of the form

F [f(q)] =∫ ∞

−∞f(q)e−ipq/~dq

Therefore, in order for equation C.2 to be true

<pn|qn> = e−ipnqn/~ (C.3)

43

44 APPENDIX C. PROOF OF NORMALISATION CONVENTION

For the case of <qn|pn>, it will be the coomplex conjugate of equation C.3. This can be shown by taking

the momentum completeness relation instead of the co-ordinate completeness relation, giving

|Ψ> =∫ ∞

−∞dpn|pn><pn|Ψ>

Projecting the state vector onto co-ordinate representation to acquire the momentum equivalent relation

to equation C.1, namely

<qn|Ψ> =∫ ∞

−∞dpn <qn|pn><pn|Ψ>

Re-writing the above in a similar way for the <pn|qn> case, giving

|Ψq> =∫ ∞

−∞dqn <qn|pn> |Ψp> (C.4)

Now this is the inverse Fourier Transform, which has the form

F−1[f(p)] =∫ ∞

−∞f(p)eipq/~dp

Hence

<qn|pn> = <pn|qn>∗ = eipnqn/~ (C.5)

J

Appendix D

The Gaussian Integral

Theorem 4 Show that

I =∫ ∞

−∞e

12 iαx2+iJxdx =

√2πiαe−

iJ22α

J

45

46 APPENDIX D. THE GAUSSIAN INTEGRAL

Appendix E

The Generalised Gaussian Integral

Starting with the Gaussian Integral of the form

I =N∏n

∫ ∞

−∞dxn exp

(−1

2xTAx + JTx

)(E.1)

We take analogous form the above in one-dimension (indicated by tildes) to complete the square in above

I =∫ ∞

−∞dx exp

(−1

2xax+ Jx

)(E.2)

=∫ ∞

−∞dx exp

[−a

2

(xx− 2Jx

a

)](E.3)

=∫ ∞

−∞dx exp

[−a

2

(x− J

a

)2

− JJ

2a

](E.4)

=∫ ∞

−∞dx exp

[−1

2

(x− J

a

)a

(x− J

a

)+JJ

2a

](E.5)

where x′ =(x− J

a

). Hence, in the matrix form, it is

I =N∏n

∫ ∞

−∞dxn exp

(−1

2x′TAx′ +

12JTA−1J

)(E.6)

where x′ =(x− JA−1

)and assuming A is diagonalisable of the form

UTAU = I (E.7)

where U is some diagonalising matrix. Then equation (E.6) becomes

I =N∏n

∫ ∞

−∞dxn exp

(−1

2x′Tx′

)exp

(−1

2Tr lnA

)exp

(12JTA−1J

)(E.8)

47

48 APPENDIX E. THE GENERALISED GAUSSIAN INTEGRAL

=N∏n

∫ ∞

−∞dxn exp

(−1

2x′Tx′

)exp

(Tr(lnA)−1/2

)exp

(12JTA−1J

)(E.9)

=N∏n

∫ ∞

−∞dxn exp

(−1

2x′Tx′

)exp

((ln(det(A))−1/2

)exp

(12JTA−1J

)(E.10)

=N∏n

∫ ∞

−∞dxn exp

(−1

2x′Tx′

)√1

det(A)exp

(12JTA−1J

)(E.11)

where we have used the result

Tr lnA =N∑

i=1

ln(λi) = ln

(N∏

i=1

λi

)= ln(det(A)) (E.12)

We already know that

N∏n

∫ ∞

−∞dxn exp

(−1

2x′Tx′

)=√

(2π)N (E.13)

since it is a series of one-dimensional Gaussian integrals. So the result for the Generalised Gaussian

Integral is

I =N∏n

∫ ∞

−∞dxn exp

(−1

2x′Tx′

)exp

(−1

2Tr lnA

)exp

(12JTA−1J

)(E.14)

=

√(2π)N

det(A)exp

(12JTA−1J

)(E.15)

J

Bibliography

[1] Zee A.,

Quantum Field Theory in a Nutshell,

Princeton University Press, 2003.

[2] Lahiri A. and Pal P.B.,

A First Book of Quantum Field Theory,

Narosa Publishing House and CRC Press, 2001.

[3] Ryder L.H.,

Quantum Field Theory, 2nd Ed..

Cambridge University Press, 1996.

[4] Dyson F.,

Some Strangeness in the Proportion, edited by H. Woolf,

AddisonWesley, Reading, MA, 1980, p. 376.

49

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