Quantum field theory for the gifted amateur

Quantum field theory has given us such a radicallydifferent and revolutionary view of the physical world that we thinkthat more physicists should have the opportunity to engage with it.T

Quantum Field Theory for the Gifted Amateur

Quantum Field Theory for the Gifted Amateur

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BRICK: Well, they say nature hates a vacuum, Big Daddy.BIG DADDY: That’s what they say, but sometimes I thinkthat a vacuum is a hell of a lot better than some of the stuffthat nature replaces it with

Tennessee Williams (1911–1983) Cat on a Hot Tin Roof

Quantum field theory is arguably the most far-reaching and beautifulphysical theory ever constructed It describes not only the quantum vac-uum, but also the stuff that nature replaces it with Aspects of quantumfield theory are also more stringently tested, as well as verified to greaterprecision, than any other theory in physics The subject nevertheless has

a reputation for difficulty which is perhaps well-deserved; its ers not only manipulate formidable equations but also depict physicalprocesses using a strange diagrammatic language consisting of bubbles,wiggly lines, vertices, and other geometrical structures, each of whichhas a well defined quantitative significance Learning this mathematicaland geometrical language is an important initiation rite for any aspir-ing theoretical physicist, and a quantum field theory graduate course isfound in most universities, aided by a large number of weighty quantumfield theory textbooks These books are written by professional quan-tum field theorists and are designed for those who aspire to join them

practition-in that profession Consequently they are frequently thorough, seriousminded and demand a high level of mathematical sophistication.The motivation for our book is the idea that quantum field theory istoo important, too beautiful and too engaging to be restricted to theprofessionals Experimental physicists, or theoretical physicists in otherfields, would benefit greatly from knowing some quantum field theory,both to understand research papers that use these ideas and also tocomprehend and appreciate the important insights that quantum fieldtheory has to offer Quantum field theory has given us such a radicallydifferent and revolutionary view of the physical world that we thinkthat more physicists should have the opportunity to engage with it.The problem is that the existing texts require far too much in the way

of advanced mathematical facility and provide too little in the way ofphysical motivation to assist those who want to learn quantum fieldtheory but not to be professional quantum field theorists The gapbetween an undergraduate course on quantum mechanics and a graduatelevel quantum field theory textbook is a wide and deep chasm, and one

of the aims of this book is to provide a bridge to cross it That beingsaid, we are not assuming the readers of this are simple-minded folk who

vi Preface

can be fobbed off with a trite analogy as a substitute for mathematicalargument We aim to introduce all the maths but, by using numerousworked examples and carefully worded motivations, to smooth the pathfor understanding in a manner we have not found in the existing books

We have chosen this book’s title with great care.1Our imagined reader

1

After all, with the number of chapters

we ended up including, we could have

called it ‘Fifty shades of quantum field

theory’.

is an amateur, wanting to learn quantum field theory without (at leastinitially) joining the ranks of professional quantum field theorists; but(s)he is gifted, possessing a curious and adaptable mind and willing toembark on a significant intellectual challenge; (s)he has abundant cu-riosity about the physical world, a basic grounding in undergraduatephysics, and a desire to be told an entertaining and intellectually stim-ulating story, but will not feel patronized if a few mathematical nicetiesare spelled out in detail In fact, we suspect and hope that our book willfind wide readership amongst the graduate trainee quantum field theo-rists who will want to use it in conjunction with one of the traditionaltexts (for learning most hard subjects, one usually needs at least twobooks in order to get a more rounded picture)

One feature of our book is the large number of worked examples, whichare set in slightly smaller type They are integral to the story, and fleshout the details of calculations, but for the more casual reader the guts

of the argument of each chapter is played out in the main text Toreally get to grips with the subject, the many examples should providetransparent demonstrations of the key ideas and understanding can beconfirmed by tackling the exercises at the end of each chapter Thechapters are reasonably short, so that the development of ideas is kept

at a steady pace and each chapter ends with a summary of the key ideasintroduced

Though the vacuum plays a big part in the story of quantum field ory, we have not been writing in one In many ways the present volumerepresents a compilation of some of the best ideas from the literatureand, as a result, we are indebted to these other books for providingthe raw material for many of our arguments There is an extensive list

the-of further reading in Appendix A where we acknowledge our sources,but we note here, in particular, the books by Zee and by Peskin andSchroeder and their legendary antecedent: the lectures in quantum fieldtheory by Sidney Coleman The latter are currently available online asstreamed videos and come highly recommended Also deserving of spe-cial mention is the text by Weinberg which is ‘a book to which we areall indebted, and from which none of us can escape.’2

2T.S Eliot on Ulysses.

It is a pleasure to acknowledge the help we have received from ious sources in writing this book Particular mention is due to S¨onkeAdlung at Oxford University Press who has helped steer this project tocompletion No authors could wish for a more supportive editor and

var-we thank him, Jessica White and the OUP team, particularly Mike gent, our eagle-eyed copy editor We are very grateful for the commentsand corrections we received from a number of friends and colleagues whokindly gave up their time to read drafts of various chapters: Peter Byrne,Claudio Castelnovo, John Chalker, Martin Galpin, Chris Maxwell, Tom

Nu-Preface vii

McLeish, Johannes M¨oller, Paul Tulip and Rob Williams They deserve

much credit for saving us from various embarrassing errors, but any that

remain are due to us; those that we find post-publication will be posted

on the book’s website:

http://www.dur.ac.uk/physics/qftgabook

For various bits of helpful information, we thank Hideo Aoki, Nikitas

Gi-dopoulos, Paul Goddard and John Singleton Our thanks are also due to

various graduate students at Durham and Oxford who have unwittingly

served as guinea pigs as we tried out various ways of presenting this

material in graduate lectures Finally we thank Cally and Katherine for

their love and support

TL & SJBDurham & OxfordJanuary 2, 2014

4.3 The kinetic energy and the tight-binding Hamiltonian 43

5.2 A charged particle in an electromagnetic field 52

6 A first stab at relativistic quantum mechanics 59

6.3 Feynman’s interpretation of the negative energy states 61

7 Examples of Lagrangians, or how to write down a theory 64

III The need for quantum fields 71

8.1 Schr¨odinger’s picture and the time-evolution operator 72

8.3 The death of single-particle quantum mechanics 758.4 Old quantum theory is dead; long live fields! 76

12.2 Noether’s current for complex scalar field theory 111

12.3 Complex scalar field theory in the non-relativistic limit 112

14.2 Electromagnetism is the simplest gauge theory 129

14.3 Canonical quantization of the electromagnetic field 131

IV Propagators and perturbations 143

16.3 Turning it around: quantum mechanics from the

propagator and a first look at perturbation theory 149

xii Contents

17.3 Finding the free propagator for scalar field theory 158

18.2 Some new machinery: the interaction representation 16718.3 The interaction picture applied to scattering 168

20.1 Another theory: Yukawa’s ψ†ψφ interactions 188

20.3 The transition matrix and the invariant amplitude 192

V Interlude: wisdom from statistical physics 195

22.2 Linking things up with the Gell-Mann–Low theorem 20322.3 How to calculate Green’s functions with diagrams 204

Contents xiii

23 Path integrals: I said to him, ‘You’re crazy’ 210

23.1 How to do quantum mechanics using path integrals 210

23.3 The propagator for the simple harmonic oscillator 217

24.3 The generating functional for scalar fields 223

26.3 Breaking a continuous symmetry: Goldstone modes 240

27.1 Coherent states of the harmonic oscillator 247

28 Grassmann numbers: coherent states

xiv Contents

30.1 Fractional statistics `a la Wilczek:

30.3 Fractional statistics from Chern–Simons theory 271

VIII Renormalization: taming the infinite 273

31 Renormalization, quasiparticles and the Fermi surface 27431.1 Recap: interacting and non-interacting theories 274

32 Renormalization: the problem and its solution 285

33 Renormalization in action:

33.1 How interactions change the propagator in perturbation

33.2 The role of counterterms: renormalization conditions 297

34.6 Application 3: the Kosterlitz–Thouless transition 309

35 Ferromagnetism: a renormalization group tutorial 31335.1 Background: critical phenomena and scaling 31335.2 The ferromagnetic transition and critical phenomena 315

Contents xv

36.2 Massless particles: left- and right-handed wave functions 323

36.5 The non-relativistic limit of the Dirac equation 332

37.4 Why are there four components in the Dirac equation? 339

38.1 Canonical quantization and Noether current 341

38.4 Local symmetry and a gauge theory for fermions 346

39.1 Quantum light and the photon propagator 348

40.2 Example 2: Spin sums and the Mott formula 356

41 The renormalization of QED and two great results 360

41.1 Renormalizing the photon propagator: dielectric vacuum 361

41.2 The renormalization group and the electric charge 364

41.3 Vertex corrections and the electron g-factor 365

xvi Contents

43.5 Finding the excitations with propagators 389

44.2 The ground state is made of Cooper pairs 402

XI Some applications from the world

46.4 Breaking symmetry with a non-abelian gauge theory 430

47.1 The symmetries of Nature before symmetry breaking 434

50 Instantons, tunnelling and the end of the world 457

50.1 Instantons in quantum particle mechanics 458

To begin at the beginning

Dylan Thomas (1914–1953)

Beginnings are always troublesome

George Eliot (1819–1880)

Every particle and every wave in the Universe is simply an excitation

of a quantum field that is defined over all space and time

That remarkable assertion is at the heart of quantum field theory It

means that any attempt to understand the fundamental physical laws

governing elementary particles has to first grapple with the fundamentals

of quantum field theory It also means that any description of

compli-cated interacting systems, such as are encountered in the many-body

problem and in condensed matter physics, will involve quantum field

theory to properly describe the interactions It may even mean, though

at the time of writing no-one knows if this is true, that a full theory

of quantum gravity will be some kind of quantum upgrade of general

relativity (which is a classical field theory) In any case, quantum field

theory is the best theory currently available to describe the world around

us and, in a particular incarnation known as quantum electrodynamics

(QED), is the most accurately tested physical theory For example, the

magnetic dipole moment of the electron has been tested to ten significant

figures

The ideas making up quantum field theory have profound

conse-quences They explain why all electrons are identical (the same

ar-gument works for all photons, all quarks, etc.) because each electron is

an excitation of the same electron quantum field and therefore it’s not

surprising that they all have the same properties Quantum field theory

also constrains the symmetry of the representations of the permutation

symmetry group of any class of identical particles so that some classes

obey Fermi–Dirac statistics and others Bose–Einstein statistics

Inter-actions in quantum field theory involve products of operators which are

found to create and annihilate particles and so interactions correspond

to processes in which particles are created or annihilated; hence there

a spinor or a tensor This concept of a field, an unseen entity whichpervades space and time, can be traced back to the study of gravity due

to Kepler and ultimately Newton, though neither used the term andthe idea of action-at-a-distance between two gravitationally attractingbodies seemed successful but nevertheless utterly mysterious Euler’sfluid dynamics got closer to the matter by considering what we wouldnow think of as a velocity field which modelled the movement of fluid atevery point in space and hence its capacity to do work on a test parti-cle imagined at some particular location Faraday, despite (or perhapsbecause of) an absence of mathematical schooling, grasped intuitivelythe idea of an electric or magnetic field that permeates all space andtime, and although he first considered this a convenient mental picture

he began to become increasingly convinced that his lines of force had anindependent physical existence Maxwell codified Faraday’s idea and theelectromagnetic field, together with all the paraphernalia of field theory,was born

x µ

φ(x µ )

Fig 1 A field is some kind of

ma-chine that takes a position in

space-time, given by the coordinates x µ , and

outputs an object representing the

am-plitude of something at that point in

spacetime Here the output is the

scalar φ(x µ ) but it could be, for

ex-ample, a vector, a complex number, a

spinor or a tensor.

Thus in classical physics we understand that gravity is a field, tromagnetism is a field, and each can be described by a set of equationswhich governs their behaviour The field can oscillate in space and timeand thus wave-like excitations of the field can be found (electromagneticwaves are well-known, but gravity waves are still to be observed) Theadvent of quantum mechanics removed the distinction between what hadbeen thought of as wave-like objects and particle-like objects Thereforeeven matter itself is an excitation of a quantum field and quantum fieldsbecome the fundamental objects which describe reality

Quantum field theory is undoubtedly important, but it is also riously difficult Forbidding-looking integrals and a plethora of funnysquiggly Feynman diagrams are enough to strike fear in many a heartand stomach The situation is not helped by the fact that the many ex-cellent existing books are written by exceedingly clever practioners whostructure their explanations with the aspiring professional in mind Thisbook is designed to be different It is written by experimental physicistsand aimed at the interested amateur Quantum field theory is too in-teresting and too important to be reserved for professional theorists.However, though our imagined reader is not necessarily an aspiring pro-

noto-0.4 Special relativity 3

fessional (though we hope quite a few will be) we will assume that (s)he

is enthusiastic and has some familiarity with non-relativistic quantum

mechanics, special relativity and Fourier transforms at an

undergradu-ate physics level In the remainder of this chapter we will review a few

basic concepts that will serve to establish some conventions of notation

Quantum fields are defined over space and time and so we need a proper

description of spacetime, and so we will need to use Einstein’s special

theory of relativity which asserts that the speed c of light is the same

in every inertial frame This theory implies that the coordinates of an

event in a frame S and a frame ¯S (moving relative to frame S at speed

v along the x-axis) are related by the Lorentz transformation

where γ = (1 − β2)−1/2 and β = v/c Because the speed of light sets

the scale for all speeds, we will choose units such that c = 1 For similar

include the factors of ~ and c so that the reader can make better contact with what they already know, and will give notice when we are going to remove them.

A good physical theory is said to be covariant if it transforms

sen-sibly under coordinate transformations.2 In particular, we require that

2 A good counterexample is the component ‘shopping vector’ that con- tains the price of fish and the price of bread in each component If you ap- proach the supermarket checkout with the trolley at 45 ◦ to the vertical, you will soon discover that the prices of your shopping will not transform ap- propriately.

two-quantities should be Lorentz covariant if they are to transform

ap-propriately under the elements of the Lorentz group (which include the

Lorentz transformations of special relativity, such as eqn 1) This will

require us to write our theory in terms of certain well-defined

mathe-matical objects, such as scalars, vectors and tensors.3

3 We will postpone discussion of tensors until the end of this section.

• Scalars: A scalar is a number, and will take the same value in

every inertial frame It is thus said to be Lorentz invariant

Examples of scalars include the electric charge and rest mass of a

particle

• Vectors: A vector can be thought of as an arrow In a particular

basis it can be described by a set of components If the basis

is rotated, then the components will change, but the length of

the arrow will be unchanged (the length of a vector is a scalar)

In spacetime, vectors have four components and are called

four-vectors A four-vector is an object which has a single time-like

component and three space-like components Three-vectors will be

displayed in bold italics, such as x or p for position and momentum

respectively The components of three-vectors are listed with a

Roman index taken from the middle of the alphabet: e.g xi, with

i = 1, 2, 3 Four-vectors are made from a time-like part and a

space-like part and are displayed in italic script, so position in

These are written with c = 1.

• the energy-momentum four-vector p = (E, p),

• the current density four-vector j = (ρ, j),

• the vector potential four-vector A = (V, A).

The four-dimensional derivative operator ∂ µ is also a combination of a time-like part and a space-like part, and is defined 5 by

5

Though strictly ∂ µ refers only to the

µth component of the four-vector

oper-ator ∂, rather than to the whole thing,

we will sometimes write a subscript (or

superscript) in expressions like this to

indicate whether coordinates are listed

with the indices lowered or with them

is significant, as we will now describe.

A general coordinate transformation from one inertial frame to anothermaps {xµ} → {¯xµ}, and the vector aµ transforms as

The jargon is that aµ transforms like a contravariant vector and

∂φ/∂xµ ≡ ∂µφ transforms like a covariant vector,7 though we will

7

These unfortunate terms are due

to the English mathematician J J.

Sylvester (1814–1897) Both types of

vectors transform covariantly, in the

sense of ‘properly’, and we wish to

re-tain this sense of the word ‘covariant’

rather than using it to simply label one

type of object that transforms properly.

Thus we will usually specify whether

the indices on a particular object are

‘upstairs’ (like a µ ) or ‘downstairs’ (like

∂ µ φ) and their transformation

proper-ties can then be deduced accordingly.

avoid these terms and just note that aµ has its indices ‘upstairs’ and

∂µφ has them ‘downstairs’ and they will then transform accordingly.The Lorentz transformation (eqn 1) can be rewritten in matrix form





¯t

¯x

The Lorentz transformation changes components but leaves the length

of the four-vector x unchanged This length is given by the square root9

In general, the four-vector inner product10 is

10 Note that other conventions are sible and some books write a · b =

pos-−a 0 b 0 + a · b and define their metric tensor differently This is an entirely le- gitimate alternative lifestyle choice, but it’s best to stick to one convention in a single book.

so that we can lower or raise an index by inserting the metric tensor

The form of the metric tensor in eqn 13 allows us to write

and hence

a · b = gµνaµbν = aµbµ= aµbµ (16)Note also that a · b = gµνaµbν and gµν= gµν

Exercise: You can check that

g µν gνρ= δρµand also that

Λ µν= g µκ Λ κ

ρ g ρν Example 0.2

(i) An example of an inner product is

p · p = p µ pµ= (E, p) · (E, p) = E2− p2= m2, (17) where m is the rest mass of the particle.

(ii) The combination ∂ µ x ν = ∂x∂xνµ = δ ν

µ and hence the inner product ∂ µ x µ = 4 (remember the Einstein summation convention).

(iii) The d’Alembertian operator ∂ 2 is given by a product of two derivative op- Named in honour of the French

math-ematician Jean le Rond d’Alembert (1717–1783) In some texts the d’Alembertian is written as ✷ and in some as ✷ 2 Because of this confusion,

we will avoid the ✷ symbol altogether.

erators (and is the four-dimensional generalization of the Laplacian operator).

j is a ‘mixed tensor of second rank’, 11 and one can check

11 It has two indices (hence second

rank) and one is upstairs, one is

down-stairs (hence mixed).

that it transforms correctly as follows:

¯ j

i = ∂ ¯xi

(ii) The antisymmetric symbol or Levi-Civita symbol 12 ε ijkℓ is defined in four

12

This is named after Italian

physi-cist Tullio Levi-Civita (1873–1941).

Useful relationships with the

Levi-Civita symbol include results for

dimensions by (i) all even permutations ijkℓ of 0123 (such as ijkℓ = 2301) have

ε ijkℓ = 1; (ii) all odd permutations ijkℓ of 0123 (such as ijkℓ = 0213) have ε ijkℓ =

−1; (iii) all other terms are zero (e.g ε 0012 = 0) The Levi-Civita symbol can be defined in other dimensions 13 We will not treat this symbol as a tensor, so the

13 The version in two dimensions is

Jean Baptiste Joseph Fourier (1768–

14 Getting these right is actually

impor-tant: if you have (2π) 4 on the top of an

equation and not the bottom, your

an-swer will be out by a factor of well over

two million.

0.6 Electromagnetism 7

Example 0.4

The Dirac delta function δ(x) is a function localized at the origin and which

has integral unity It is the perfect model of a localized particle The integral of a

d-dimensional Dirac delta function δ (d) (x) is given by

Z

ddx δ(d)(x) = 1 (26)

It is defined by Z

ddx f (x)δ(d)(x) = f (0) (27) Consequently, its Fourier transform is given by

˜ (d) (k) =

Z

d d x e ik·x δ (d) (x) = 1 (28) Hence, the inverse Fourier transform in four-dimensions is

Z d 4 k (2π) 4 e−ik·x= δ(4)(x) (29)

In this book we will choose15the Heaviside–Lorentz16system of units 15 Although SI units are preferable for

many applications in physics, the desire

to make our (admittedly often cated) equations as simple as possible motivates a different choice of units for the discussion of electromagnetism in quantum field theory Almost all books

compli-on quantum field theory use Heaviside– Lorentz units, though the famous text- books on electrodynamics by Landau and Lifshitz and by Jackson do not.

16 These units are named after the glish electrical engineer O Heaviside (1850–1925) and the Dutch physicist

En-H A Lorentz (1853–1928).

(also known as the ‘rationalized Gaussian CGS’ system) which can be

obtained from SI by setting ǫ0= µ0= 1 Thus the electrostatic potential

c from these equations In addition, the fine structure constant α =

Note that we will give electromagnetic charge q in units of the electron

charge e by writing q = Q|e| The charge on the electron corresponds

to Q = −1

• In Chapter 1 we provide a formulation of classical mechanics which

is suitable for a quantum upgrade This allows us to talk aboutfunctionals and Lagrangians

• The simple harmonic oscillator, presented in Chapter 2, is known from basic quantum physics as an elementary model of anoscillating system We solve this simple model using creation andannihilation operators and show that the solutions have the charac-teristics of particles Linking masses by springs into a chain allows

well-us to generalize this problem and the solutions are phonons

• The next step is to change our viewpoint and get rid of wave tions entirely and develop the occupation number representationwhich we do in Chapter 3 We show that bosons are described bycommuting operators and fermions are described by anticommut-ing operators

func-• Already we have many of the building blocks of a useful theory

In Chapter 4 we consider how to build single-particle operatorsout of creation and annihilation operators, and this already gives

us enough information to discuss the tight-binding model of solidstate physics and the Hubbard model

Fig 1.1 Refraction of a light ray

through a slab of glass The ray finds

the path of least travel time from A to

B.

We begin with an example from the study of optics Consider the passage

of a light ray through a slab of glass as shown in Fig 1.1 The bending

of the light ray near the air/glass interface can be calculated using theknown refractive indices of air and glass using the famous Snell’s law ofrefraction, named after Willebrord Snellius, who described it in 1621,but discovered first by the Arabian mathematician Ibn Sahl in 984 In

1662, Pierre de Fermat produced an ingenious method of deriving Snell’slaw on the basis of his principle of least time This states that thepath taken between two points A and B by a ray of light is the paththat can be traversed by the light in the least time Because light travelsmore slowly in glass than in air, the ray of light crosses the glass at asteeper angle so it doesn’t have so much path length in the glass If thelight ray were to take a straight line path from A to B this would takelonger This was all very elegant, but it didn’t really explain why lightwould choose to do this; why should light take the path which took theleast time? Why is light in such a hurry?

Fermat’s principle of least time is cute, and seems like it is telling ussomething, but at first sight it looks unhelpful It attempts to replace

a simple formula (Snell’s law), into which you can plug numbers andcalculate trajectories, with a principle characterizing the solution of theproblem but for which you need the whole apparatus of the calculus ofvariations to solve any problems Fermat’s principle is however the key

to understanding quantum fields, as we shall see

t

Fig 1.2 A particle moves from A to B

in time τ and its path is described by

Newton’s laws of motion.

A somewhat similar problem is found in the study of dynamics Consider

a particle of mass m subject to a force F which moves in one spatialdimension x from point A to B, as shown in the spacetime diagram inFig 1.2 Here time is on the horizontal axis and space is on the verticalaxis The exact path x(t) that the particle takes is given by Newton’s

1.3 Functionals 11

laws of motion, i.e

This equation can be integrated to find x(t) However, when you stop

and think about it, this is a very quantum-unfriendly approach The

solution gives you the position of the particle at every time t from t = 0

to t = τ Quantum mechanics tells us that you might measure the

particle’s position at t = 0 and find it at A, and you might measure it

again at t = τ and find it at B, but you’d not be able to know precisely

what it did in between Having a method which lets you calculate x(t)

from a differential equation is not a good starting point

Dynamics need to be formulated in a completely different way if the

subject is to be generalized to quantum mechanics This is exactly what

Joseph-Louis Lagrange and William Rowan Hamilton did, although they J.-L Lagrange (1736–1813) was an

Italian-born French mathematician and physicist.

W R Hamilton (1805–1865) was an Irish mathematician and physicist.

had no idea that what they were doing would make dynamics more

quantum-friendly We will take a slightly different approach to theirs

and arrive at the final answer by asking ourselves how kinetic energy T

and potential energy V vary during the trajectory of the particle We

know that they must sum to the total energy E = T +V which must be a

constant of the motion But during the trajectory, the balance between

kinetic and potential energy might change

It is simple enough to write down the average kinetic energy ¯T during

the trajectory, which is given by

¯

T = 1τ

Z τ 0

Z τ 0

These two quantities must sum to give the total energy E = ¯E = ¯T + ¯V

However, what we want to do is to consider how ¯T and ¯V vary as you

alter the trajectory To do this, we need a little bit of mathematics,

which we cover in the next section

The expressions in eqns 1.2 and 1.3 are functionals of the trajectory x(t)

What does this mean?

Let us recall that a function is a machine (see Fig 1.3) which turns a

number into another number For example, the function f (x) = 3x2will

turn the number 1 into the number 3, or the number 3 into the number

27 Give a function a number and it will return another number

A functional is a machine which turns a function into a number You

feed the machine with an entire function, like f (x) = x2or f (x) = sin x

and it returns a number

12 Lagrangians

Example 1.1 Here are some examples of functionals.

• The functional F [f] operates on the function f as follows:

F [f ] =

Z 1 0

Hence, given the function f (x) = x 2 , the functional returns the number

F [f ] =

Z 1 0

G[f ] =

Z a

−a 5x4dx = 2a5 (1.7)

• A function can be thought of as a trivial functional For example, the tional F x [f ] given by

func-F x [f ] =

Z ∞

−∞ f (y)δ(y − x)dy = f(x), (1.8) returns the value of the function evaluated at x.

We now want to see how a functional changes as you adjust the functionwhich is fed into it The key concept here is functional differentiation.Recall that a derivative of a function is defined as follows:

Example 1.2 Here are some examples of calculations of functional derivatives You can work through these if you want to acquire the skill of functional differentiation, or skip to the next bit of text if you are happy to take the results on trust.

»Z [f (y) + ǫδ(y − x)]pφ(y) dy −

Z [f (y)]pφ(y) dy

–

• The functional H[f] =Rabg[f (x)] dx, where g is a function whose derivative is

g ′ = dg/dx, has a functional derivative given by

δH[f ]

δf (x 0 ) = ǫ→0lim

1 ǫ

»Z g[f (x) + ǫδ(x − x 0 )] dx −

Z g[f (x)] dx

–

= lim

ǫ→0

1 ǫ

»Z (g[f (x)] + ǫδ(x − x 0 )g′[f (x)]) dx −

Z g[f (x)] dx

–

=

Z δ(x − x 0 ) g′[f (x)] dx

df ′

–

−

Z δ(y−x) ddy

„ dg(f ′ )

df ′

« (1.17) The term in square brackets vanishes assuming x is inside the limits of the

integral, and we have simply

Equation 1.19 can be easily generalized

to three dimensions and leads to a very useful result which is worth memoriz- ing, namely that if

I =

Z (∇φ)2d3x, then

14 Lagrangians

With these mathematical results under our belt, we are now ready toreturn to the main narrative How does the average kinetic energy andthe average potential energy vary as we adjust the particle trajectory?

We now have the main results from eqns 1.14 and 1.20 which are:

1 They might both decrease by the same

amount if the classical trajectory

max-imizes (rather than minmax-imizes) the

ac-tion, see below, but this case is not the

one usually encountered.

as

δδx(t)( ¯T [x] − ¯V [x]) = 0, (1.23)i.e that the difference between the average kinetic energy and the av-erage potential energy is stationary about the classical trajectory Thisshows that there is something rather special about the difference be-tween kinetic energy and potential energy, and motivates us to define aquantity known as the Lagrangian L as

The integral of the Lagrangian over time is known as the action S

S =

Z τ 0

and so the action has dimensions of energy×time, and hence is measured

in Joule-seconds This is the same units as Planck’s constant h, and wewill see later in this chapter why it is often appropriate to think ofmeasuring S in units of Planck’s constant Our variational principle(eqn 1.23) connecting variations of average kinetic energy and averagepotential energy can now be rewritten in a rather appealing way sincefor this problem S =Rτ

0(T − V ) dt = τ( ¯T [x] − ¯V [x]), so that

t

Fig 1.4 Small adjustments to the path

of a particle from its classical trajectory

lead to an increase in the action S.

δS

and this is known as Hamilton’s principle of least action.2It states

2 To be pedantic, the principle only

shows the action is stationary It could

be a maximum, or a saddle point, just

as easily as a minimum ‘Stationary

action’ would be better than ‘least

ac-tion’ But we are stuck with the name.

that the classical trajectory taken by a particle is such that the action isstationary; small adjustments to the path taken by the particle (Fig 1.4)only increase the action (in the same way that small adjustments to thepath taken by a ray of light from the one determined by Snell’s lawlengthen the time taken by the light ray)

1.4 Lagrangians and least action 15

Example 1.3

The Lagrangian L can be written as a function of both position and velocity Quite

generally, one can think of it as depending on a generalized position coordinate x(t)

and its derivative ˙x(t), called the velocity Then the variation of S with x(t) is

δS/δx(t) and can be written as

δS

δx(t) =

Z du

» δL δx(u)

δx(u) δx(t)+

δL

δ ˙x(u)

δ ˙x(u) δx(t) –

=

Z du

» δL δx(u) δ(u − t) + δL

» δ(u − t) δL

δ ˙x(u) – t f

δL

and hence the principle of least action (eqn 1.26) yields

δL δx(t) − ddt

δL

which is known as the Euler–Lagrange equation.

Leonhard Euler (1707–1783) In the words of Pierre-Simon Laplace (1749– 1827): Read Euler, read Euler, he is the master of us all.

The Lagrangian L is related to the Lagrangian density L by

The following example introduces the idea of a Lagrangian density, a

concept we will come back to frequently, but also provides a nice way to

derive the classical wave equation

ψ(x, t)x

dx

ρ dx

Fig 1.5 Waves on a string The placement from equilibrium is ψ(x, t) and the equation of motion can be de- rived by considering an element of the string of length dx and mass ρ dx The figure shows a short section in the mid- dle of the string which is assumed to be tethered at either end so that ψ(0, t) = ψ(ℓ, t).

dis-Example 1.4

Consider waves on a string of mass m and length ℓ Let us define the mass density

ρ = m/ℓ, tension T and displacement from the equilibrium ψ(x, t) (see Fig 1.5) The

kinetic energy T can then be written as T = 1Rℓ

0 dx ρ(∂ψ/∂t) 2 and the potential energy V = 12R0ℓdx T (∂ψ/∂x) 2 The action is then

„ ∂ψ

∂t

« 2

−T2

16 Lagrangians

As a final trick, let us put the Euler–Lagrange equation on a fully tivistic footing, bracing ourselves to use some four-vector notation andthe Einstein summation convention (see Section 0.4) If the Lagrangiandensity L depends on a function φ(x) (where x is a point in spacetime)and its derivative3 ∂µφ, then the action S is given by

rela-3

Recall from the argument in

Sec-tion 0.4 that the index µ in ∂ µ φ is

nat-urally lowered; see eqn 0.2.

S =

Z

By analogy with eqn 1.27, the action principle gives4

4 Remember that we are using the

Ein-stein summation convention, by which

twice repeated indices are assumed

summed Equation 1.35 is simply

the four-dimensional generalisation of

In this chapter, we have considered two variational principles: Fermat’sprinciple of least time and Hamilton’s principle of least action Onedescribes the path taken by a ray of light, the other describes the pathtaken by a classical particle They are very elegant, but why not stick

to using Snell’s law and Newton’s law? And why do they both work?The answer to both of these questions is quantum mechanics We willtalk about this in more detail later in the book, but the motion of aparticle (photon or billiard ball) going from A to B involves all possiblepaths, the sensible classical ones and completely insane ones You have

to sum them all up, but they are each associated with a phase factor, andfor most sets of paths the different phases mean that the contributionscancel out It is only when the phase is stationary that nearby paths allgive a non-cancelling contribution The wave function for a particle has

a phase factor5given by

5 Fermat’s principle of least time tells us

something about optical path length of

a ray, that is the difference between the

phase at the beginning and end of a ray.

By analogy with Hamilton’s principle

of least action of a classical mechanical

system, one can posit that the action S

is given by a constant multiplied by the

phase of a wave function This defines

the constant which is given the symbol

~ Thus we take the phase to be S/~.

Exercises 17

where S = R

L dt is the action, so a stationary phase equates to a

sta-tionary action (eqn 1.26) (Running the argument in reverse, the phase

factor for a photon of energy E is e−iEt/~, and so stationary phase

equates to stationary time, which is Fermat’s principle.) We will see

how this approach leads naturally to Feynman’s path-integral approach

later in the book (Chapter 23) But for now, notice simply that if the

action is stationary then the classical path which minimizes the action

is the one that is observed and everything else cancels

Snell’s law and Newton’s law are enough to solve classical systems

But neither allow the generalization to quantum systems to be performed

with ease Thus, to formulate quantum field theory (the grand task of

this book), we have to start with a Lagrangian picture Our next step is

to gain some insight into what happens in non-classical systems, and so

in the next chapter we will turn our attention to an archetypal quantum

system: the simple harmonic oscillator

Chapter summary

• Fermat’s principle of least time states that light takes a path which

takes the least time

• The Lagrangian for a classical particle is given by L = T − V

• Classical mechanics can be formulated using Hamilton’s principle

• Both Fermat’s and Hamilton’s principles show how the classical

paths taken by a photon or massive particle are ones in which the

phase of the corresponding wave function is stationary

δH[f ]

δf (x) =

∂g

∂f − ddx

∂g

∂f′, (1.42)

18 Lagrangians

where f′ = ∂f /∂y For the functional J[f ] =

R g(y, f, f′, f′′) dy show that

(1.4) Show that

δφ(x)δφ(y)= δ(x − y), (1.44)and

δ ˙φ(t)δφ(t0) =

d

dtδ(t − t0) (1.45)(1.5) For a three-dimensional elastic medium, the poten-

tial energy is

V = T2

Z

d3x (∇ψ)2, (1.46)

and the kinetic energy is

T =ρ2

∇2ψ = 1

v2

∂2ψ

∂t2, (1.48)where v is the velocity of the wave

(1.6) Show that if Z0[J] is given by

Z0[J] = exp

„

−12

Z

d4x d4y J(x)∆(x − y)J(y)

«,(1.49)where ∆(x) = ∆(−x) then

Simple harmonic oscillators

2.1 Introduction 19 2.2 Mass on a spring 19 2.3 A trivial generalization 23

The advent of quantum mechanics convinced people that things that had

previously been thought of as particles were in fact waves For example,

it was found that electrons and neutrons were not simply little rigid

bits of matter but they obey a wave equation known as the Schr¨odinger

equation This idea is known as first quantization To summarize:

First quantization: Particles behave like waves (2.1)

However, this is not the end of the story It was also realized that

things that had been previously thought of as waves were in fact

par-ticles For example, electromagnetic waves and lattice waves were not

simply periodic undulations of some medium but they could actually

behave like particles These phenomena were given particle-like names:

photons and phonons This idea is known as second quantization To

summarize:

Second quantization: Waves behave like particles (2.2)

Quite how these ideas link up is one of main themes in quantum field

theory which sees the underlying reality behind both waves and particles

as a quantum field However, before we get to this point, it is worth

spending some time reviewing second quantization in a bit more detail

as it is the less familiar idea In this chapter, we focus on the most

famous example of a wave phenomenon in physics: the oscillations of a

mass on a spring

mK

Fig 2.1 A mass m suspended on a spring, of spring constant K.

Consider a mass on a spring (as shown in Fig 2.1), one of the simplest

physical problems there is Assume we have mass m, spring constant

K, the displacement of the mass from equilibrium is given by x and the

momentum of the mass is given by p = m ˙x The total energy E is the

sum of the kinetic energy p2/2m and the potential energy 1

2Kx2

In quantum mechanics, we replace p by the operator −i~∂/∂x, and

we then have the Schr¨odinger equation for a harmonic oscillator

20 Simple harmonic oscillators

The solutions to this equation can be obtained by a somewhat involvedseries-solution method and are given by

ψn(ξ) = √1

2nn!

 mωπ~

1/4

Hn(ξ)e−ξ2/2, (2.4)

where Hn(ξ) is a Hermite polynomial and ξ = p

mω/~x As shown

in Fig 2.2, these eigenfunctions look very wave-like However, they

do have a ‘particle-like’ quality which is apparent from the eigenvalues,which turn out to be

En=



n +12

if you are adding a lump, or particle, of energy Can we make this vaguefeeling any more concrete? Yes we can Moreover, we can do it quiteelegantly and without dirtying our hands with Hermite polynomials!

Fig 2.2 Eigenfunctions of the simple

Fig 2.3 The ladder of energy levels for

the simple harmonic oscillator.

To accomplish this, we start with the Hamiltonian for the simple monic oscillator written out as follows:

x − i

mωpˆ

 ˆ

1

2mω

2

ˆ

x − mωi pˆ

 ˆ

2 , which is nearly ˆH but hasthe correction −~ ω

2 due to the zero-point energy being subtracted Thefactorization can therefore be made to work and we realize that theoperators ˆx − i

ˆ

a =

rmω2~

ˆ

ˆ

x −mωi pˆ



„ ~

mω+

~ mω

a†ˆa +12



The active ingredient in this Hamiltonian is the combination ˆa†ˆa If

ˆ

a†a has an eigenstate |ni with eigenvalue n, then ˆˆ H will also have an

eigenstate |ni with eigenvalue ~ω(n +12), so that we have recovered the

eigenvalues of a simple harmonic oscillator in eqn 2.5 However, we need

to prove that n takes the values 0, 1, 2, The first step is to show that

n ≥ 0 We can do that by noting that

n = hn|ˆa†a|ni = |ˆa|ni|ˆ 2≥ 0 (2.15)Next, we have to show that n takes only integer values, and we will

do that below but beforehand let us introduce a bit of notation to save

some writing We define the number operator ˆn by

Number of what? The quantity n labels the energy level on the ladder

(see Fig 2.3) that the system has reached, or equivalently the number

of quanta (each of energy ~ω) that must have been added to the system

when it was in its ground state We can rewrite the Hamiltonian as

ˆ

H = ~ω

ˆ

n +12



and therefore

ˆH|ni =



n +12



so that |ni is also an eigenstate of the Hamiltonian Thus |ni is a

convenient shorthand for the more complicated form of ψn(ξ) shown in

eqn 2.4 The next examples show that the eigenvalue n indeed takes

integer values

22 Simple harmonic oscillators

Example 2.2 This example looks at the property of the state defined by ˆ a † |ni One of the things

we can do is to operate on this with the number operator:

ˆ

nˆ a†|ni = ˆa†aˆ ˆ a†|ni (2.20) Using the commutator in eqn 2.11 gives ˆ aˆ a † = 1 + ˆ a † ˆ a and hence

ˆ

nˆ a † |ni = (n + 1)ˆa † |ni (2.21)

The above example shows that the state ˆa†|ni is an eigenstate of ˆH butwith an eigenvalue one higher than the state |ni In other words, theoperator ˆa† has the effect of adding one quantum of energy For thisreason ˆa† is called a raising operator

Example 2.3 This example looks at the property of the state defined by ˆ a|ni We can operate on this with the number operator:

Example 2.4 Question: Normalize the operators ˆ a and ˆ a † Solution: We have shown that ˆ a|ni = k|n − 1i, where k is a constant Hence, taking the norm of this state (i.e premultiplying it by its adjoint) gives

|ˆa|ni|2= hn|ˆa†ˆ a|ni = |k|2hn − 1|n − 1i = |k|2, (2.24) where the last equality is because the simple harmonic oscillator states are normalized (so that hn − 1|n − 1i = 1) However, we notice that ˆa † ˆ a = ˆ n is the number operator and hence

hn|ˆa†ˆ a|ni = hn|ˆ n|ni = n (2.25) Equations 2.24 and 2.25 give k = √n (This assumes k to be real, but any statecontains an arbitrary phase factor, so we are free to choose k to be real.)

In the same way, we have shown that ˆ a † |ni = c|n + 1i, where c is a constant Hence,

|ˆa†|ni|2= hn|ˆaˆa†|ni = |c|2hn + 1|n + 1i = |c|2, (2.26) and using ˆ aˆ a † = 1 + ˆ a † ˆ a = 1 + ˆ n, we have that

hn|ˆaˆa † |ni = hn|1 + ˆ n|ni = n + 1, (2.27) and hence c = √

n + 1 (choosing c to be real) In summary, our results are:

ˆ a|ni = √

ˆ

a†|ni = √

n + 1|n + 1i (2.29)