Introduction to Quantum Field Theory and Quantum...

1

Introduction to Quantum Field Theoryand Quantum Statistics

Michael Bonitz

Institut fur Theoretische Physik und AstrophysikKiel University

April 9, 2020

Preliminary lecture notesContains unpublished results and cannot be used or distributed without

explicit permission

Contents

1 Canonical Quantization 71.1 Minimal action principle . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Classical mechanics of a point particle . . . . . . . . . . 121.1.2 Canonical momentum and Hamilton density of classical

fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Conservation laws in classical field theory . . . . . . . . . . . . . 15

1.2.1 Translational invariance. Energy and momentum con-servation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3 Field quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4.1 Application of canonical field theory . . . . . . . . . . . 231.4.2 Expansion in terms of eigenfunctions . . . . . . . . . . . 241.4.3 Quantization of the displacement field . . . . . . . . . . 26

1.5 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.5.1 Maxwell’s equations. Electromagnetic potentials. Field

tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.5.2 Lagrange density of the free electromagnetic field . . . . 341.5.3 Normal mode expansion of the

electromagnetic field . . . . . . . . . . . . . . . . . . . . 381.5.4 Quantization of the electromagnetic field . . . . . . . . . 40

1.6 EMF Quantization in Matter . . . . . . . . . . . . . . . . . . . 441.6.1 Lagrangian of a classical relativistic particle . . . . . . . 441.6.2 Relativistic particle coupled to

the electromagnetic field . . . . . . . . . . . . . . . . . . 461.6.3 Lagrangian of charged particles in an EM field . . . . . . 471.6.4 Quantization of the electromagnetic field coupled to charges 501.6.5 Quantization of the EM field in a dielectric medium or

plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521.7 Quantization of the Schrodinger field . . . . . . . . . . . . . . . 541.8 Quantization of the Klein-Gordon field . . . . . . . . . . . . . . 54

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4 CONTENTS

1.9 Coupled equations for the Schrodinger and Maxwell fields . . . . 541.10 Problems to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . 54

2 Second Quantization 552.1 Second quantization in phase space . . . . . . . . . . . . . . . . 55

2.1.1 Classical dynamics in terms of point particles . . . . . . 552.1.2 Point particles coupled via classical fields . . . . . . . . . 572.1.3 Classical dynamics via particle and Maxwell fields . . . . 582.1.4 Discussion: ensemble averages, fluctuations, quantum

effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.2 Quantum mechanics and first quantization . . . . . . . . . . . . 62

2.2.1 Reminder: State vectors and operators in Hilbert space . 622.2.2 Probabilistic character of “First” quantization.

Comparison to experiments . . . . . . . . . . . . . . . . 632.3 The ladder operators . . . . . . . . . . . . . . . . . . . . . . . . 64

2.3.1 One-dimensional harmonic oscillator . . . . . . . . . . . 652.3.2 Generalization to several uncoupled oscillators . . . . . . 69

2.4 Interacting Particles . . . . . . . . . . . . . . . . . . . . . . . . 702.4.1 One-dimensional chain and its normal modes . . . . . . . 702.4.2 Quantization of the 1d chain . . . . . . . . . . . . . . . . 732.4.3 Generalization to arbitrary interaction . . . . . . . . . . 752.4.4 Quantization of the N -particle system . . . . . . . . . . 78

2.5 Continuous systems . . . . . . . . . . . . . . . . . . . . . . . . . 802.5.1 Continuum limit of 1d chain . . . . . . . . . . . . . . . . 802.5.2 Equation of motion of the 1d string . . . . . . . . . . . . 81

2.6 Solutions of Problems . . . . . . . . . . . . . . . . . . . . . . . . 84

Bibliography 101

Preface

The statistical description of many–particle systems in nonequilibrium beganwith Ludwig Boltzmann’s famous kinetic equation [Bol72]. Since then, numer-ous theoretical methods have been developed to describe equilibrium statesand nonequilibrium processes in various fields, including fluids, dense plasmas,solids and nuclear matter.

For quantum many-particle systems a very fruitful and general approachis provided by the method of second quantization and quantum field theory.These approaches gave rise to very general methods involving ensemble av-erages and probability densities. The equations of motion of them are thesubject of quantum kinetic theory and the theory of Nonequilibrium Greenfunctions.

An entirely different approach is based on fluctuating quantities. A pow-erful first-principle method to the thermodynamic properties of many-bodysystems is quantum Monte Carlo which has been developed in a number ofvariants.

The present notes have been used in my lectures on quantum statisticsand quantum field theory of charged particle systems at Kiel University since2003. The book has benefited a lot from the students actively participatingin the lectures over the years and from the members of my group, in particu-lar Karsten Balzer, Sebastian Bauch, Tobias Dornheim, Simon Groth, AlexeiFilinov, Martin Heimsoth, Sebastian Hermanns, David Hochstuhl, Jan-PhilipJoost, Lasse Rosenthal, Niclas Schlunzen, Tim Schoof and Hauke Thomsen.They contributed parts of the various sections, solutions to problems and fig-ures. Moreover, through continuous discussions, they also helped to improvethe presentation and corrected many errors.

Kiel, April 2020 Michael Bonitz

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6 CONTENTS

Chapter 1

Introduction to canonicalquantization

Many physical systems are well described by continuum models, i.e. in terms offields. Examples are the displacement field q(r, t) of particles in a continuousmedium such as a gas, a liquid or a plasma. Another exmample is the elec-tromagnetic field described by the electric and magnetic field strength E(r, t),B(r, t) which are governed by Maxwell’s equations. Furthermore, also particlescan be described by fields. In fact, quantum mechanics has changed our pic-ture of the micro-world in such a way that point particles, i.e. concentration ofmatter in an infinitesimally small space point is meaningless. Instead, matteris delocalized in space which is described by probability densities. These againare field-like continuous functions which again obey equations of field theory.Depending on the particle type, this gives rise to the “Schrodinger field” whichis associated with non-relativistic particles or with the “Klein-Gordon field”and the “Dirac field”, in the case of relativistic particles.

1.1 Lagrange functional and minimal action

principle

A very elegant way to derive equations of motion of physical systems is theminimal action principle. We are now going to generalize this principle toarbitrary fields and derive general equations of motion – the Euler-Lagrangeequations for the Lagrange functional. A remarkable property of this approachis that it yields, in a straightforward and general way, the basic conservationlaws of any physical theory and their relation to the intrinsic symmetries ofthe system.

7

8 CHAPTER 1. CANONICAL QUANTIZATION

Let the state of a general continuous system be completely described by afinite number M of fields,

Φ(r, t) = Φ1(r, t), . . .ΦM(r, t), (1.1)

which can be considered as independent generalized “variables” of the system.The Φk can be scalar or vector fields, real or complex and are defined in the vol-ume V and time interval ti ≤ t ≤ tf which form a four-dimensional space timeelement ∆Ω. In fact, the theory should be Lorentz invariant, i.e. symmetricwith respect to space and time variables. We can make this symmetry moreexplicit by introducing the four vector notation for coordinates and derivatives(summation over identical subscript-superscript index pairs is implied)

xµ =(x0, ~x

)= (ct, r) , µ = 0, 1, 2, 3, (1.2)

xµ = gµνxν =

(x0,−~x

), (1.3)

where gµν is the metric tensor

gµν = gµν =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

, (1.4)

and a scalar product of two 4−vectors is given by aµbµ = aµbµ = a0b0−~a~b and

is a Lorentz scalar (Lorentz invariant). We will also need the metric tensorwith mixed indices,

gµν = gνµ =

1 0 0 00 1 0 00 0 1 00 0 0 1

= δµ,ν . (1.5)

Finally, we also note the way how to transform two-component tensors withvarious superscript and subscript properties,

F µν = gµαgνβFαβ, (1.6)

Fµν = gµαgνβFαβ, (1.7)

F νµ = gµαF

αν , (1.8)

which is straightforwardly generalized to more complicated tensor quantities.

1.1. MINIMAL ACTION PRINCIPLE 9

Using the 4-vectors we can construct two four-dimensional differential op-erators according to

∂µ =∂

∂xµ=

(∂

∂ct, ~∇), (1.9)

∂µ =∂

∂xµ=

(∂

∂ct,−~∇

), (1.10)

giving rise to a four-dimensional divergence or gradient which will be usedfrequently. For example, the 4-divergence of a 4-vector Bµ is a Lorentz skalar

∂µBµ =

∂

∂xµBµ =

∂

∂ctB0 + ~∇B = inv, (1.11)

as is the 4-dimensional generalization of the Laplace operator, the D’Alambertoperator

∂µ∂µ =

∂

∂xµ∂

∂xµ=

1

c2

∂2

∂t2+ ~∇(−~∇) ≡ = inv. (1.12)

Note the sign change in front of the spatial part in Eq. (1.12) which does notoccur in Eq. (1.11).

The physical properties of the system will then be determined by the fieldsΦ and their time and space derivatives, Φ, Φ, . . . and Φ′,Φ′′, . . . . There is noa-priori rule how the dependence of the Lagrange density on these functionsshould look like. This dependence could be non-local, i.e. L(r, t) could dependon the fields at different space time points, e.g. contain terms of the form∫dt∫d3r K(r, t; r, t)Φ(r, t) or second or higher powers in the fields, products

or other nonlinear combinations. Furthermore, L could, in principle, depend onspace and time explicitly, L = L(r, t; Φ, ...). An explicit time dependence wouldbe reasonable only if the system undergoes a time evolution which is controlledfrom the outside and not an intrinsic process described by the dynamics of thefields. Similarly, an explicit space dependence could be expected if the systemis subject to an external potential which leads to different physics in differentspace points. If this is not the case an explicit r and t dependence may be ruledout. Finally, the fundamental requirement to a theory of physical processesshould be its intrinsic simplicity. Therefore, we will start to construct theLagrange density using only local expressions and only linear dependencies onall fields and their space and time derivatives1.

1Interestingly, all experimentally verified theories which are known so far can be derivedfrom Lagrangians of this simple structure.


Thus we choose the following structure of the Lagrange density as anansatz:

L(r, t) = L[Φ(r, t), Φ(r, t),Φ′(r, t)] = L[Φ(r, t),Φµ(r, t)], (1.13)

where everywhere Φ is understood as a vector with M components (1.1) and Φµ

denotes the four-dimensional derivative vector (gradient), ∂µ = (∂tΦ,−∇Φ).Using L we find a central quantity of theoretical physics – the action – definedby

S =

∫dt

∫d3rL =

∫Ω

d4xL(r, t) (1.14)

The Lagrange function gives the possibility to derive the physical equationsof motion in a very general way based on the Minimal action principle: Thosefields Φ which obey the physical equations of motion and fulfill boundary con-ditions at two space-time points 1 and 2 minimize the action S. We will callthese particular realizations of the fields Φphys. This principle can be turnedaround: Those fields which minimize the action and fulfill the boundary con-ditions are the “true” physical fields Φphys. This latter formulation shows thepower and beauty of this principle: it provides a completely general approachto physical systems, independently of the particular area and specific kind offields involved.

Let us now find the extremum of the action 2. To this end we compute thevariation of S around the physical fields, i.e. in the “point” Φ = Φphys, andput it equal to zero,

0 = δS =

∫ 2

1

d4x

δLδΦ

δΦ +δLδΦµ

δΦµ

. (1.15)

Assuming that the fields Φ are continuous functions we can change the orderof differentiation and variation, i.e. δΦµ = ∂µδΦ. In order to express thevariation ∂µδΦ by δΦ we perform a partial integration of the second term inEq. (1.15) neglecting the terms at the boundaries of the integral by makinguse of the boundary condition δΦ(1) = δΦ(2) = 0. As a result we obtain

0 =

∫ 2

1

d4x

δLδΦ− ∂µ

δLδΦµ

δΦ. (1.16)

Since this equation should be fulfilled for arbitrary fluctuations δΦ we have to

2In practically all cases, finding the extremum is sufficient. See also problem 1.


require that the term in the parantheses vanishes

0 =δLδΦ− ∂µ

δLδΦµ

= (1.17)

=δLδΦ− ∂

∂t

δLδΦ− ~∇ δL

δ~∇Φ.

These are the famous Euler-Lagrange equations, the general equations ofmotion of the field(s) Φ. Note that the integral (1.16) has to be understood asa scalar product of two M−dimensional vectors. Since the fluctuations of theindividual fields δΦ1, . . . δΦM are independent (since the fields are assumed tobe independent variables), vanishing of δS requires that all M terms in theparantheses vanish simultaneously, so (1.18) is equivalent to

0 =δLδΦk

− ∂µδLδΦk,µ

= (1.18)

=δLδΦk

− ∂

∂t

δLδΦk

− ~∇ δLδ~∇Φk

, k = 1, . . .M.

Let us briefly discuss this result.

1. The variational principle is of course a postulate which cannot be proven,similar to Newton’s equations, Maxwell’s equations or the Schrodingerequation. Vice versa, assuming the validity of one of the latter equationsone can show the existence of a Lagrangian which obeys the minimalaction principle for this particular system. However, starting from theminimal action principle we have a universal principle method yieldingall these equations at once.

2. When accepting the validity of the minimal action principle, the maintasks of physical theories consist in deriving (or guessing) explicit ex-pressions for the Lagrange density.

3. A remarkable property of the variaional principle is that it yields local(space and time-dependent) solutions Φphys(r, t) from minimizing a singleglobal scalar function – the action which is an integral over a space-timevolume Ω.

4. The Euler-Lagrange equations (ELE) are Lorentz invariant as they con-tain only Lorentz scalars which is seen from the first equation (1.18).

5. The ELE do not depend on the choice of the boundary of the volume Ω.In fact, we expect the physical equations of motion to be independent of


time and space. On the other hand, if space or time are inhomogeneous,i.e. the Lagrangian explicitly depends on r or t we cannot exclude alsoa dependence on the chosen boundary points 1 and 2.

6. When deriving the ELE we implicitly have assumed that the Lagrangedensity is a sufficiently smooth functional of the fields and their time andspace derivatives and also that the time derivative and the gradient ofall fields are continuous.

7. When finding the extremum of the action given by the ELE we cannotrule out that this result corresponds to a maximum of S. Strictly speak-ing, we have to verify that the second variation of the action is negative(see problem 1).

1.1.1 Classical mechanics of a point particle

Before proceeding we consider the simplest case of a point particle of mass min a 1D external potential U(x). Then, the field is replaced by the coordinate,Φ(r, t) → q(t) and L → L where the Lagrangian is given by kinetic (T ) andpotential energy (V ) according to

L(q, q) = T − V =m

2q2 − U(q). (1.19)

Then the Euler-Lagrange equations (1.18) become

d

dt

∂L

∂q− ∂L

∂q= 0, (1.20)

yielding the equation of motion for the “physical” variable q(t)

mq = −∂V∂x

, (1.21)

i.e. we recover Newton’s equation. Knowing the Lagrange function, mechanicstells us how to obtain from it the momentum p and the hamilton functionH(q, p),

p =∂L

∂q= mq, (1.22)

H(q, p) = pq(p)− L[q, q(p)] =p2

2m+ U(q). (1.23)


Using the hamitonian we obtain an alternative form of the equations of motion:two first-order equations (Hamilton’s equations) for q and p,

p = −∂H∂q

= −∂U∂q

, (1.24)

q =∂H

∂p=

p

m. (1.25)

Finally, we recall another result of point mechanics, now extending the analysisto N particles: any function F depending only on the canonical variablesq1 . . . qN , p1 . . . pN has a simple time evolution given by the Poisson bracketwith the Hamiltonian,

F = F,H , (1.26)

F,H =N∑i=1

(∂F

∂qi

∂H

∂pi− ∂F

∂pi

∂H

∂qi

). (1.27)

1.1.2 Canonical momentum and Hamilton density ofclassical fields

We now return to the general problem described by the fields Φ1 . . .ΦM . Theexample of the point particle suggests to define a “momentum” πl(r, t), i.e.the field which is “canonically” adjoint to Φl, by defining, in analogy to (1.22),

πl(r, t) =δL

δΦl(r, t)= π0

l (r, t), l = 1, . . .M (1.28)

Thus the canonical momentum field follows from the functional derivative ofthe Lagrange density with respect to the time derivative of the field. Asindicated by the last equality, this can also be understood as the 0 componentof a 4-vector πµ defined as

πµl (r, t) =δL

δµΦl(r, t), (1.29)

but only the zero component of πµ has the physical meaning of a momentum.Yet the whole 4-vector can be conveniently used to make the expressions morecompact. By combining the 4-vectors πµl into a momentum vector of all fieldsby defining πµ = πµ1 , . . . π

µM we can rewrite Eq. (1.18) compactly as

δLδΦ− ∂µπµ = 0 (1.30)


Using again the analogy with the point mechanics, we can now intro-duce a Hamilton density H which is related to the hamiltonian by H(t) =∫d3rH(r, t). Generalizing Eq. (1.23) to fields we can write

H(r, t) = H[Φ, π] = πΦ[π]− L, (1.31)

where the time-derivative of Φ has to be eliminated to recover the dependenceof H on the two independent variables Φ and π. Interestingly, also in thecase of continuous fields there exist generalized Hamilton’s equations whichare analogous to those of point mechanics, Eqs. (1.24, 1.25). To verify this,we consider the variation of H, performing the variation of H and L using thedefinition of π and the Euler Lagrange equations (1.18),

δH =

∫d3r

πδΦ + Φδπ − δL

=

∫d3r


δΦδΦ− δL

δΦµ

δΦµ

=

∫d3r


δΦδΦ− πδΦ− δΦ~∇ δL

δ~∇Φ

=

∫d3r

Φδπ − πδΦ

≡∫d3r δH[Φ, π]. (1.32)

In the last term of the third line we have performed an integration by parts.The two terms containing derivatives of L are then replaced by the time deriva-tive of π using Eq. (1.18). Since H[Φ, π] is a functional of the two independentfields Φ and π, it is evident from the last equation that

π = −δHδΦ

, (1.33)

Φ =δHδπ

, (1.34)

i.e. we have obtained another variant of the equations of motion of the canon-ical field – the continuum generalization of Hamilton’s equations. This con-firms the consistency of the definitions of the canonical momentum and of theHamilton density.

Finally, we generalize another result of point mechanics – the time evolutionof any function F depending only on the canonical variables qi, pi of N whichis given by the Poisson brackets, cf. Eq.(1.27). To this end we introducethe spatial density f of F and compute the time derivative, using Hamilton’s

1.2. CONSERVATION LAWS IN CLASSICAL FIELD THEORY 15

equations (1.33), (1.34)

F (t) =

∫d3r

δf

δΦΦ +

δf

δππ

=

∫d3r

δf

δΦ

δHδπ− δf

δπ

δHδΦ

= F,H , (1.35)

where in the last line we introduced the continuum generalization of the Pois-son bracket. Compared to the discrete case, Eq. (1.27), here the sum over theparticles is replaced by a space integration.

1.2 Conservation laws in classical field theory

We have seen in the last section that minimizing the action yieds the physicalequations of motion.The key quantity in this approach is the Lagrange density.It is, therefore, important to know whether the definition of L is unique orwhether there exists any freedom in choosing it. In fact, the answer is givenby a very simple statement:

Theorem: The equations of motion (1.18) remain invariant under anytransformation of L of the form

L −→ L+ δL (1.36)

with δL(x) = ∂µWµ(x), (1.37)

where W is a continuous function of the space and time arguments and van-ishes at the boundary δΩ of the four-dimensional volume Ω.

Proof:Consider the variation of the action and its change under the above transfor-mation

δS =

∫ 2

1

d4x δL −→∫ 2

1

d4x δL+ ∂µWµ(x) . (1.38)

The additional term can be converted into a surface integral by using thefour-dimensional Gauss theorem∫ 2

1

d4x ∂µWµ(x) =

∮δΩ

dSµWµ = 0,


which is zero since W is assumed to vanish at the boundary and the theoremis proven.

So what does this transformation mean? First, the transformation δL(x)is understood as an infinitesimal transformation so its variation is the functionitself. Second, the particular form (1.37) means that the addition to L is a full4-divergence, i.e. W has to have the form of a 4-vector W µ = (W 0, ~w) andδL(x) is a Lorentz scalar of the form

δL(x) =∂W 0

∂t− div ~w. (1.39)

Thus there is a certain flexibility in defining L without changing the equationsof motion derived from minimzing the action3. Now the question is how toexploit this freedom. One way to produce such a transformation of L is tomake a transformation of the fields Φ. We will call a transformation of thefields which generates the change (1.37) of L a Symmetry transformation. Thismeans

Φ(x) → Φ(x) + δΦ(x),

L[Φ(x)] → L[Φ(x) + δΦ(x)] ≡ L[Φ(x)] + ∂µWµ[δΦ(x)]. (1.40)

Furthermore we require that the transformation is continuous and exists forall fields Φ, not just the physical fields Φphys which obey the Euler Lagrangeequations. Under these conditions there exists a generalized current densityjµ which obeys

Noether’s Theorem: 4 For any continuous symmetry transformation δΦ ofthe form (1.40) there exists a 4-current density of the form (1.41). For allphysical fields Φphys this current density has a vanishing 4-divergence, i.e.

jµ(x) = πµ(x)δΦ(x)−W µ(x), (1.41)

∂µjµ(x) = 0, (1.42)

where πµ is defined in (1.29).

Proof:The 4-divergence of the change of the Lagrange density associated with the

3This situation is similar to gauge invariance in electrodynamics where different formsof the electromagnetic potentials may be chosen without changing the physical observables(the electromagnetic field and the quantum-mechanical probability density)

4The theorem is due to Emmy Noether who formulated it in 1918.


transformation δΦ is the total variation of L

∂µWµ = δL =

δLδΦ

δΦ + πµδΦµ

=

δLδΦ− ∂µπµ

δΦ + ∂µ (πµδΦ) . (1.43)

Here we used δΦµ = ∂µδΦ, and the derivative of πµ in the last term on ther.h.s. is compensated by the second term in the parantheses. So far these wereidentical transformations valid for arbitrary fields Φ. Now we specialize tothe physical fields which obey the Euler Lagrange equations (1.18) which justappear in the parantheses, i.e. the first term on the r.h.s. of (1.43) vanishesfor Φ = Φphys, and we may rewrite (1.43)

0 = ∂µ (πµδΦ−W µ) = ∂µjµ.

Thus we have confirmed the vanishing of the 4-divergence of the 4-currentdensity and obtained the explicit form of this density which exactly agreeswith the statement of the theorem.

1.2.1 Translational invariance. Energy and momentumconservation

We now consider the simplest symmetry transformation – an infinitesimalspace time transformation

xµ −→ xµ + aµ

Φ (xµ) −→ Φ (xµ + aµ) . (1.44)

The corresponding symmetry transformation δΦ is then obtained by Taylorexpanding the fields in the shifted arguments around the original value

δΦ = Φ (xµ + aµ)− Φ (xµ) = aµ∂µΦ(x) + . . . , (1.45)

where second order and higher terms are neglected. This is a superpositionof four orthogonal translations – one in time and three in space. Owing toindependence of these translations we can consider (any) one of them choosingµ = α, where α = 0, . . . 3. Further, we may rescale the coordinate system suchthat the shift aα = 1. Then the symmetry transform and its derivative aresimply

δΦ ≈ ∂αΦ(x), (1.46)

δΦµ ≈ ∂αΦµ(x). (1.47)


Let us now compute the Noether current associated with translations, i.e.with the symmetry transfrom (1.46). The variation of the Lagrange density Lwill now be equivalent to space-time variation of its arguments Φ and Φµ, i.e.

δL(x) =δLδΦ

δΦ +δLδΦµ

δΦµ =

=δLδΦ

∂αΦ +δLδΦµ

∂αΦµ ≡ ∂αL(x). (1.48)

Since for a symmetry transformation, the variation of L can be written as a4−divergence, cf. Eq. (1.40), we can rewrite [transforming to a lower derivativeusing Eq. (1.3)]

∂µWµ ≡ δL(x) = ∂αL(x) = ∂µ [gµαL(x)] . (1.49)

Comparing the left and right sides of this equation we can identify the four4−vectors W µ0 . . .W µ3, corresponding to the translation in α−direction. Wecombine them into a 4×4 tensor, W µα(x) ≡ gµαL(x). According to Eq. (1.41)this yields four Noether currents jµ0 . . . jµ3 which we again combine into atensor which is called canonical energy-momentum tensor

T µαc (x) = πµ(x)∂αΦ(x)− gµαL(x) (1.50)

with the associated four conservation laws

∂µTµαc (x) = 0 (1.51)

Separating the time and space components (k = 1, 2, 3) this system can berewritten as

∂

∂ctT 0αc (x) +

∂

∂xkT kαc (x) = 0, (1.52)

or, splitting the α values into a time (α = 0) and space (α = j = 1, 2, 3) part,

∂

∂ctT 00c (x) +

∂

∂xkT k0c (x) = 0, (1.53)

∂

∂ctT 0jc (x) +

∂

∂xkT kjc (x) = 0, j = 1, 2, 3, (1.54)

where Eq. (1.54) is a system of three equations.Equations (1.53) and (1.54) are four coupled local conservation laws con-

necting temporal changes of the tensor components T 00c and T 0j

c with the di-vergence of spatial flux terms. As for any local conservation law we can find


the associated global balance equation by integrating over space. Then weobtain

d

dtPα = 0, α = 0, 1, 2, 3 (1.55)

with Pα(t) =

∫d3r T 0α

c (r, t), (1.56)

where we transformed the flux term using the Gauss theorem∫V

d3r∂

∂xkT kαc (x) =

∮∂V

dSkTkαc (x) = 0, (1.57)

and assumed that the tensor components T kαc vanish at the system boundary∂V . While equation (1.53) is the differential energy balance (local energyconservation law), equation (1.54) consitutes the local momentum conservationlaw for an arbitrary system described by the Lagrangian L. This means wehave obtained four scalar quantities Pα which are conserved.

What is remarkable about this result is its generality. We have not usedany specific system property, we only used its Lagrangian given in terms ofarbitrary fields Φ and considered infinitesimal space-time translations. It isobvious to guess that Equations (1.55) constitute the conservation laws ofenergy and momentum of the field Φ. We can readily verify this hypothesisby explicitly computing the quantities Pα. Inserting the result (1.50) intoequation (1.55) we find

P 0 =

∫d3r

π(x)Φ(x)− L(x)

≡∫d3rH(x) = H, (1.58)

P k =

∫d3r π(x)∂kΦ(x), (1.59)

where in the last line we took into account that g0k = 0. The first line exactlycoincides with our result (1.31) for the Hamilton density of the field Φ, i.eP 0 is nothing but the Hamilton function of the system and its conservationis the energy conservation law for the field Φ. Thus we can now establishthe meaning of the individual components of the tensor T kαc and of the localconservation laws (1.53) and (1.54). The local energy balance (1.53) connectsthe time derivative of the Hamilton density T 00

c with the divergence of theenergy current density – the vector with the components (T 10

c , T20c T

30c ). On

the other hand, the momentum balance equation (1.54) relates the time changeof the momentum density vector T 0j

c with the divergence of the stress tensorT kjc . In other words the j−column of T kjc is the momentum current densitycorresponding to the j−component of the momentum density T 0j

c .


We have considered only the simplest kind of symmetry transformation:space-time translations and established that homogeneity of space and time isrelated to conservation of total energy and momentum, respectively. Amongother important symmetry transformation we mention rotations. One candemonstrate that isotropy of space-time is directly related to Lorentz invari-ance, see [WG93].

1.3 Field quantization

Let us know quantize the pair of canonically conjugate fields Φ(r, t) and π(r, t)just as it is done in quantum mechanics. To this end we replace the fields byoperators

Φ(r, t) −→ Φ(r, t), (1.60)

π(r, t) −→ π(r, t). (1.61)

As in quantum mechanis, the two operators do not commute (Heisenberg un-certainty principle), and here we generalize the fundamental commutation re-lation, [ri, pk] = i~δi,k, to the case of functions of continuous arguments

[Φ(r, t), π(r′, t)]∓ = i~ δ(r− r′) (1.62)

[Φ(r, t), Φ(r′, t)]∓ = 0, (1.63)

[π(r, t), π(r′, t)]∓ = 0. (1.64)

Thus identical fields always commute, while a field and its canonical momen-tum commute always, except for exactly the same space arguments in bothoperators. Also, the commutation relations hold only if both operators havethe same time arguments. Finally, the subscript ∓ indicates an extension (be-yond standard quantum mechanics) to bosonic (-) and fermionic (+) fieldswhere we defined the commutator (anti-commutator) by

[A, B]∓ ≡ AB ∓ BA. (1.65)

As in quantum mechanics we now apply the correspondence principle stat-ing that functions of the canonical variables retain the same functional form.For example, the Hamilton density field now becomes an operator function

1.3. FIELD QUANTIZATION 21

of the same form as in classical field theory, and the same is true for theLagrangian, the energy-momentum tensor and other functions.

L[Φ, Φ] −→ L[Φ, ˆΦ],

H[Φ, π] −→ H[Φ, π],

P 0 −→ P 0 =

∫d3r H(r), (1.66)

P k −→ P k =

∫d3r π(r)∂kΦ(r). (1.67)

Thus the equations of motion following from the Euler Lagrange equationswill not change their form and be valid also for the operator fields. Similarly,the conservation laws of classical field theory, Eq. (1.51), remain formally validfor the operator of the energy-momentum tensor

∂µTµαc (x) = 0 (1.68)

There is an additional property of the energy-momentum operator Pα

which we formulate as a theorem:

Theorem: The operator P k generates space translations in the direction rk, k =1, 2, 3, and the operator P 0 generates a time translation of the field Φ of thefollowing form

[P k(t), Φ(r, t)]− =~i

∂Φ(r, t)

∂rk, (1.69)

[P 0(t), Φ(r, t)]− =~i

∂Φ(r, t)

∂t. (1.70)

Since P 0 is nothing but the density of the Hamilton operator, equation (1.70) isa generalization of Heisenberg’s equation for time-dependent operators to thecase of field operators. Moreover, Eq. (1.69) further generalizes this equationto space derivatives.

Proof of relation (1.69):The commutator (1.69) is straightforwardly transformed using the definition(1.67)

[P k(t), Φ(r, t)]− =

∫d3r′

[π(r′, t)

∂Φ(r′, t)

∂r′k, Φ(r, t)

]−

.


Transforming the commutator of a product according to [AB,C] = A[B,C] +[A,C]B yields∫

d3r′

π(r′, t)

[∂Φ(r′, t)

∂r′k, Φ(r, t)

]−

+[π(r′, t), Φ(r, t)

]−

∂Φ(r′, t)

∂r′k

.

The first term is zero due to the commutation of Φ with itself, whereas thesecond term can be integrated using the commutation relation (1.62), and weimmediately obtain the result (1.69). In a similar way, relation (1.70) is proven(see problem 2).

We may now combine the two relations (1.70) and (1.69) into a singlefour-vector relation

[Pα(t), Φ(x)]− =~i∂αΦ(x). (1.71)

Furthermore, we may extend this property to an arbitrary function of the

fields, F [Φ, ˆΦ, Φ′], i.e.

[Pα(t), F ]− =~i∂αF (x). (1.72)

An interesting example is F → P µ. Then Eq. (1.72) yields

[Pα(t), P µ(t)]− =~i∂αP µ(x) = 0. (1.73)

This expression vanhishes because the energy-momentum P µ, Eqs. (1.58),(1.59) is a conserved quantity and space-independent. Equation (1.73) showsthat the energy-momentum P µ is part of the Poincare algebra, for more detailssee Ref. [WG93].

1.4 Phonons

As the first application of canonical field theory and field quantization weconsider a continuous elastic medium which can perform oscillations aroundits equilibrium state. This generalizes our previous models, such as the one-dimensional chain or string, cf. Sections 2.4.1 and 2.5.1.

The state of the system is described by the displacement field q(r, t) de-scribing the local perturbation of the medium around the equilibrium position.The Lagrange density is the three-dimensional generalization of our previous1d−result (2.84),

L[q(r, t),q′(r, t)] =ρ

2

3∑1=1

(∂qi(r, t)

∂t

)2

− c2

3∑j=1

(∂qi(r, t)

∂xj

)2

=3∑

1=1

Li

1.4. PHONONS 23

Here ρ is the mass density, c =√

σρ

and σ is the elastic tension. For the relation

of these quantities to the discrete chain we refer to Section 2.5.1. Here our goalis primarily to demonstrate the application of the methods of field theory, sowe will limit ourselves to the simplest example of elastic deformations: we willconsider only longitudinal deformations and assume that σ is isotropic whichleads to a direction independent velocity c. Extensions to anisotropic systemsand transverse excitations are straightforward.

1.4.1 Application of canonical field theory

Let us now apply our general field theory results to the present system.

1. We have to identify the general field variable Φ. Here the fields arereplaced by a vector field q or three independent scalar fields q1, q2, q3, i.e.Φl −→ qi, which we already indicated in the arguments of the Lagrangedensity (1.74).

2. Apply the general Euler-Lagrange equation (1.30) to the deformationfield and evaluate the partial derivatives. For a fixed value i = 1, 2, 3 wehave

0 =δLiδqi− d

dt

δLiδqi−

3∑k=1

∂

∂xkδLi

δ (∂xkqi)

= 0− ρ ddtqi + ρc2

N∑k,j=1

∂

∂xk∂qi∂xj

δj,k

= qi − c2∆qi, (1.74)

where, in the last line, we have canceled the common factor ρ. Thus wehave obtained from the Euler-Lagrange equations a 3D wave equation,separately for each displacement component qi.

3. We now calculate the canonical momentum, applying Eq. (1.28),

πi =δLiδqi

= ρqi, i = 1, 2, 3, (1.75)

where φi is the momentum density associated with the deformation qiwhich generalizes the mechanical momentum pi to continuous systemsby the simple replacement m −→ ρ.


4. Next, compute the Hamilton density, according to Eq. (1.32), where wehavo to eliminate qi by the momentum, Eq. (1.75),

H =3∑i=1

πiqi − Li

=ρ

2~q 2 +

3∑i,j=1

ρc2

2

(∂qi∂xj

)2

=~π 2

2ρ+σ

2

3∑i,j=1

(∂qi∂xj

)2

. (1.76)

Obviously, the first term is the kinetic energy density and the sum con-tains all contributions to the potential energy arising from elastic tensionsin the medium.

5. The energy-momentum tensor follows form the general definition (1.50),µ, α = 0, 1, 2, 3,

T µαc (x) = πµ(x)∂αΦ(x)− gµαL(x), (1.77)

In particular, the total energy and momentum of the displacement fieldare obtained from Eqs. (1.58) and (1.59)

H(t) =

∫d3r

3∑i=1

πi(r, t)qi(r, t)− Li(r, t) =

∫d3rH(r, t), (1.78)

P k(t) =

∫d3r

3∑i=1

πi(r, t)∂kqi(r, t), k = 1, 2, 3 (1.79)

These equations are the basis for the mechanics of elastic continuous media,including fluids and solids.

1.4.2 Expansion in terms of eigenfunctions

The solutions of the equation of motion (1.74) are oscillations or waves whichdepend on the initial and boundary conditions. Stationary solutions whichsolve Poisson’s equations are standing waves with wave vector k. We maymodel an infinite system by considering a cube of side length L with volumeV = L3 and using perdiodic boundary conditions. Then the solutions are givenby

uk(r) = ekeikr

L3/2, ek =

k

k, (1.80)

1.4. PHONONS 25

and, obviously, form a complete orthonormal system∫V

d3r uk(r)u∗k′(r) = δk,k′ . (1.81)

Here we have chosen longitudinal polarization of the oscillations, i.e. ek ∼ k.The system (1.80) forms a basis for arbitrary displacements,

q(r, t) =∑k

Bk bk(t)uk(r) + b∗k(t)u∗k(r) , (1.82)

where we included arbitrary time-dependent complex expansion coefficientsand added the complex conjugate of the modes to assure that the displacementis real. The real coefficients Bk are introduced in order to adjust lateron theamplitude of the functions bk(t) to one. Inserting the ansatz (1.82) into thewave equation (1.74)

0 =∑k

Bk

(bk(t) + c2k2bk(t)

)uk(r) +

(b∗k(t) + c2k2b∗k(t)

)u∗k(r)

,

yields a condition for the coefficients bk(t). Since the functions uk form anorthonormal system, this equation can only be fulfilled if the terms in allparantheses vanish simultaneously, i.e. for all k

bk(t) + c2k2bk(t) = 0, with the solution bk(t) = bk0 e−iωkt, (1.83)

and the dispersion relation ωk = ck. By properly choosing the Bk we canalways use |bk0| = 1 leaving open an arbitrary phase φ, i.e. bk0 = eiφk , whichallows to fulfill the initial condition. Thus the final result for the displacementis

q(r, t) =1

L3/2

∑k

ekBk

bk0e

−i(ωkt−kr) + b∗k0ei(ωkt−kr)

. (1.84)

We can immediately write down the corresponding normal mode expansion ofthe canonically adjoint field, the momentum, Eq. (1.75), by differentiating theexpansion (1.84) with respect to time,

π(r, t) = −i ρ

L3/2

∑k

ekBkωkbk0e

−i(ωkt−kr) − b∗k0ei(ωkt−kr)

. (1.85)

Note the sign change in front of the adjoint contribution.


1.4.3 Quantization of the displacement field

Let us now quantize the canonical fields qi and πi with i = x, y, z by introducingoperators. We replace qi −→ qi and πi −→ πi and require that the standardcanonical commutation relations are fulfilled,

[qi(r, t), πl(r′, t)] = i~ δi,l δ(r− r′), (1.86)

whereas the commutators of two identical fields are zero. Again, the two fieldsare taken at equal times. We now consider the normal mode representation(1.84) of the displacement field and the momentum, Eq. (1.85). Replacing herethe fields by operators requires to introduce two independent operators (foreach mode) also on the right hand sides. The only reasonable way of doingthis is to replace the phase factors by operators, bk0 −→ bk and b∗k0 −→ b†k.I.e. the normal mode expansion of the operators becomes

q(r, t) =1

L3/2

∑k

ekBk

bke−i(ωkt−kr) + b†ke

i(ωkt−kr), (1.87)

π(r, t) =−iρL3/2

∑k

ekBkωk

bke−i(ωkt−kr) − b†ke

i(ωkt−kr). (1.88)

What remains is to establish the commutation relation between these oper-ators. The criterion is, of course, that the original commutation relation (1.86)is satisfied. To this end we calculate the commutator of the i and l componentsof (1.87) and (1.88) and set it equal to the r.h.s. of (1.86),

i~ δi,l δ(r− r′) =−iρL3

∑kk′

kik′l

kk′BkBk′ωk′

[bke−i(ωkt−kr) + b†ke

i(ωkt−kr),

×bk′e−i(ωk′ t−k

′r′) − b†k′ei(ωk′ t−k′r′)

]. (1.89)

This equation can be satisfied by imposing, as an ansatz, the following bosoniccommutation relations

[bk, bk′ ] = [b†k, b†k′ ] = 0 (1.90)

[bk, b†k′ ] = δk,k′ . (1.91)

Then, from the four commutators in (1.89) only two involving b and b† remain(giving identical contributions, therefore, the factor 2) where k′ = k and we

1.4. PHONONS 27

cancel the time-dependent exponents

~ρδi,l δ(r− r′) =

2

V

∑kk′

kik′l

kk′BkBk′ωk′

[bk, b

†k

]×e−i(ωk−ωk′ )tei(kr−k′r′) (1.92)

=2

V

∑k

kiklk2

B2kωke

ik(r−r′)

−→ 2

∫d3k

kiklk2

B2kωke

ik(r−r′). (1.93)

Since we consider a macroscopic system with L −→ ∞ the wave vector spec-trum is quasicontinuous, and we could, in the last line, replace the sum V −1

∑k

by an integral over kx, ky, kz, from minus to plus infinity. This integral vanishesif i 6= l since the integrand of the ki and kl integrals is an odd function, whichis in agreement with the Kronecker symbol on the l.h.s. On the other hand,in an isotropic medium, we expect that Bk = Bk. Then, for i = l, the integralis independent of i, i.e. the integrals involving k2

x, k2y and k2

z are equal to eachother and equal the integral containing k2/3. Thus the k factors cancel andthe 3d integral over k yields a delta function δ(r−r′), again in agreement withthe l.h.s. What is left to satisfy Eq. (1.93) is to properly choose the amplitudesBk, with the result

Bk =

(~

2× 3ρωk

)1/2

=L3/2

61/2x0k, (1.94)

where we have used ρ = m/L3 and introduced the oscillator ground state wavefunction extension x0k of mode k, x0k = (~/mω)1/2. The last expression (1.94)shows that Bk has the dimension length to the power 5/2 as it should be toguarantee the correct dimension (length) of the displacement q, cf. Eqs. (1.84)and (1.87).

With this we can write down the final expression for the operator of anarbitrary displacement in terms of the eigenfunctions of the elastic mediumand their creation and annihilation operators,

q(r, t) =

(~

6ρL3

)1/2∑k

ek

ω1/2k

bke−i(ωkt−kr) + b†ke

i(ωkt−kr). (1.95)

Finally we compute the Hamilton operator corresponding to the Hamiltonfunction (1.76) and express it interms of the operators bk and b†k. Accordingto the correspondence principle, we rewrite Eq. (1.76) in terms of operators

H =

∫d3r

~π 2

2ρ+σ

2

3∑i,j=1

(∂qi∂xj

)2. (1.96)


Inserting now the normal mode expansions (1.87) and (1.88) we obtain (seeProblem 3)

H =∑k

~ωk(nk +

1

2

), (1.97)

where nk = b†kbk is the number operator corresponding to mode k. Withexpression (1.97) we have succeeded to diagonalize the hamiltonian of theelastic continuum and brought it to a very intuitive form: the hamiltonianis a superposition of independent quantized normal mode contributions, eachhaving the form of a 1d linear harmonic oscillator with an occupation numberoperator and a zero point energy. This allows for a clear interpretation ofthe operators b†k and bk: they create (annhilate) and elementary excitationcharacterized by the energy ~ωk and momentum ~k. These excitations arecalled phonons. The time and space dependence of the elastic deformationcorresponding to the phonon mode k is given by a monochromatic plane wave,cf. Eq. (1.95).

In the k−sum we encounter the same problem as in Sec. 2.5.2: if the numberof modes is infinite the sum diverges due to the infinite contribution of theground state contribution. Thus, a cut-off is necessary (renormalization).

Time-dependent creation and annihilation operators. The operators band b† which were introduced in Eqs. (1.87) and (1.88) were time-independent.We simply replaced the phase factors bk0 and b∗k0 by operators and left thetime-dependence in the classical expressions (1.84) and (1.85) unchanged. Letus now discuss an alternative approach which leads to the same hamiltonian(1.97) and is frequently used. We then use new operators b(t) and b†(t) wherethe time-dependence is still open. Then the expansion of the displacementoperator becomes

q(r, t) =

(~

2 · 3ρL3

)1/2∑k

ek

ω1/2k

bk(t) eikr + b†k(t) e−ikr

. (1.98)

We can find the time-dependence, as in the classical case, by inserting (1.98)into the wave equation (1.74) which yields bk(t) = bke

−iωkt and b†k(t) = b†keiωkt.

Alternatively, we can compute the time-derivative of the creation and annihi-lation operators by using the general Heisenberg equations of canonical field

1.4. PHONONS 29

theory, Eq. (1.70) with the hamiltonian (1.97):

i~dbkdt

=[bk, H

]=

∑k′

~ωk′[bk,

(nk′ +

1

2

)]=

∑k′

~ωk′[bk, b

†k′ bk′

]=

∑k′

~ωk′b†k′

[bk, bk′

]+[bk, b

†k′

]bk′

= ~ωkbk,

where, in the last line, we used the commutation relations (1.90), (1.91). Thesolution of this equation yields the previous result bk(t) = bke

−iωkt.


1.5 Quantization of the free electromagnetic

field. Photons

We now apply the formalism of canonical field theory to the electromag-netic field which is determined by Maxwell’s equations. We will quantize theMaxwell field and will obtain the elementary excitations of the electromagneticfield – the photons.

1.5.1 Maxwell’s equations. Electromagnetic potentials.Field tensor

Let us recall the basic quantities and equations of electrodynamics. The equa-tions of motion of the electromagnetic field coupled to charges and currentsare given by (2.10,2.11)

divE(r, t) = 4πρ(r, t), ∇× E(r, t) = −1

c

∂B(r, t)

∂t, (1.99)

divB(r, t) = 0, ∇×B(r, t) =4π

cj(r, t) +

1

c

∂E(r, t)

∂t. (1.100)

The charge and current densities ρ, j are not independent but are coupled viathe charge conservation law (continuity equation)

∂ρ(r, t)

∂t+ div j(r, t) = 0. (1.101)

All equations are local, i.e. all quantities appear with the same space and timearguments.

Two of Maxwell’s equations (1.99,1.100) are satifsfied exactly by introduc-ing the scalar and vector potentials, φ and A, according to

B = ∇×A, (1.102)

E = −∇φ− 1

cA, (1.103)

where Eq. (1.102) solves the third of Maxwell’s equations and Eq. (1.103) thesecond. In the remaining two equations the fields can be eliminated giving riseto

−∇2φ− 1

c∇A = 4πρ, (1.104)(

1

c2

∂2

∂t2−∆

)A =

4π

cj− 1

c∇(c divA + φ

), (1.105)

1.5. PHOTONS 31

where, in the second line, we used rot rot = −∆ + grad div. The definition ofthe potentials is not unique. The equations for the measurable quantities –the electric and magnetic field strength E and B remain unchanged under anygauge transformation χ(r, t) of the potentials

A −→ A +∇χ, (1.106)

φ −→ φ− χ. (1.107)

Below we will use this gauge freedom to simplify the field equations. In par-ticular, we will use the Coullomb gauge, ∇A = 0, and the Lorentz gauge. Theidea of the latter is to eliminate the second term on the r.h.s. of Eq. (1.105),which is achieved by the following relation between A and φ

c divA + φ = 0. (1.108)

This allows to transform the equations for the potentials into two decoupledwave equations, (

1

c2

∂2

∂t2−∆

)φ = 4πρ, (1.109)(

1

c2

∂2

∂t2−∆

)A =

4π

cj. (1.110)

This highly symmetric form is, of course, no conincidence but reflects rela-tivistic covariance of the equations of the electromagnetic fields. This propertycan be made explicit by combining the scalar equation (1.109) and the vectorequation (1.110) into a single equation by introducing the four-vector notation,as we did in section 1.1, but now for the electrical charge density and currentdensity and for the potentials,

jµ = (cρ, j), (1.111)

Aµ = (φ,A). (1.112)

Recalling the definition of the D’Alambert operator = ∂µ∂µ = 1

c2∂2

∂t2−∆ we

can rewrite the wave equations (1.109), (1.110)

Aµ =4π

cjµ (1.113)

which is a covariant form of Maxwell’s equations. Both sides are 4−vectorsshowing that this equation is Lorentz invariant. Analogously, we may intro-duce a 4−vector form for the continuity equation (1.101), the gauge transform


(1.106), (1.107) and the Lorentz gauge condition (1.108)

∂µjµ = 0, (1.114)

Aµ −→ Aµ − ∂µχ,∂µA

µ = 0, (1.115)

which are again Lorentz invariant.While the Lorentz gauge is mathematically appealing, due to the symmetry

between φ and A, it partly masks the physical nature of the electromagneticfield suggesting the existence of four independent wave modes. However, thisis not the true. As in the case of the displacement field, Sec. 1.4, there existonly three orthogonal wave excitations – one longitudinal (parallel to the wavevector k) and two transverse ones (perpendicular to k). This is also obviousfrom the Lorentz gauge (1.115) which couples the four components of Aµ, leav-ing only three of them independent. We will return to this question in Section1.5.4 when we quantize the two transverse components of the electromagneticfield.

While the covariant formulation of the potentials is given by the 4−vectorAµ, the corresponding representation of the electric and magnetic field strengthsis the field tensor (for more details see textbooks on electrodynamics, e.g.[LL62])

F µν = ∂µAν − ∂νAµ =

0 −E1 −E2 −E3

E1 0 −B3 B2

E2 B3 0 −B1

E3 −B2 B1 0

(1.116)

which is anti-symmetric and gauge-invariant and has the following properties(i, j, k = 1, 2, 3)

F µν = −F νµ, (1.117)

F k0 = Ek, (1.118)

F ij = −εijkBk, (1.119)

Bk = −1

2εkijF

ij, (1.120)

where εijk is the completely antisymmetric tensor 5.

5εijk = 0 unless i, j, k are different and εijk = 1 (εijk = −1) if i = 1, j = 2, k = 3(i = 1, j = 3, k = 2) and analogously for cyclic permutations. The tensor εijk has the sameproperties.

1.5. PHOTONS 33

For completeness we give also the tensor with the lower indices which fol-lows from the general tensor relations (1.6)

Fµν = gµµ′gνν′Fµ′ν′ =

0 E1 E2 E3

−E1 0 −B3 B2

−E2 B3 0 −B1

−E3 −B2 B1 0

, (1.121)

where gµµ′ is the metric tensor (1.4). Fµν is also antisymmetric and differsfrom F µν only by the signs of the electric field components. The completedescription of the electromagnetic field requires a second tensor, the so-calleddual tensor F µν given by

F µν =1

2εµναβFαβ =

0 −B1 −B2 −B3

B1 0 E3 −E2

B2 −E3 0 E1

B3 E2 −E1 0

(1.122)

which follows from F µν by the so-called dual transform where the electric andmagnetic fields are interchanged with the following sign change, (E,B) −→(B,−E). The antisymmetric tensor εµναβ is the generalization of εµνα. Verifi-cation of the components of the tensor F µν is subject of problem 4.

The two field tensors can be used to construct Lorentz invariants of the fieldwhich are of particular importance for the further analysis, in particular, forthe construction of the Lagrange density and the action of the electromagneticfield in Sec. 1.5.2. In fact, one can show (see problems 5,6) that there existtwo invariants,

F µνFµν = 2(B2 − E2

)(1.123)

F µνFµν = −4B · E. (1.124)

The first is a Lorentz scalar and the second a pseudo-scalar 6. Finally, weuse the tensors to give another co-variant formulation of Maxwell’s equationswhich follow from equation (1.113). In fact, differentiating the field tensor(1.116), we obtain

∂µFµν = ∂µ (∂µAν − ∂νAµ) (1.125)

= Aν − ∂ν∂µAµ

= Aν =4π

cjν ,

6it involves a 3d scalar product rather than a scalar product of two 4−vectors


where the second term in the second line vanishes due to the Lorentz gauge con-dition (1.115) and, in the last line, we have used Maxwell’s equations (1.113).These are four equations since ν = 0, 1, 2, 3. For ν = 0, inserting the 0−th col-umn of the tensor, F µ0, yields divE = 4πρ, whereas the equations for ν = 1, 2, 3yield ∇×B− E/c = 4πj/c, i.e. we recover two of Maxwell’s equations.

Note that another differentiation of the l.h.s. of (1.125) yields zero, i.e.∂ν∂µF

µν = 0 which is a consequence of definition (1.116) and of the Lorentzgauge. This requires that also the r.h.s. vanishes, i.e. ∂νj

ν = 0, showing thatfulfilment of the continuity equation (1.114) by the charge and current densityis a necessary condition for internal consistency of the electromagnetic fieldequations. The second pair of Maxwell’s equations is given by the dual tensor:

∂µFµν = 0. (1.126)

When inserting the components of the dual tensor into this equations, weobtain, for ν = 0, divB = 0 and, for ν = 1, 2, 3,∇×E+B/c = 0, i.e. we recoverthe remaining two equations. In contrast to the equations for F µν which coupleto the electrical current density jν these equations have a zero r.h.s. This is aconsequence of the incomplete symmetry between electric and magnetic field,arising from the nonexistence of magnetic monopoles (“magnetic charges”).

1.5.2 Lagrange density of the free electromagnetic field

Let us now discuss how to construct the Lagrange density of the electromag-netic field. The Lagrange density L has the dimension of an energy densitywhich, for the electromagnetic field, is well known and given by (E2 +B2)/8π.Furthermore, L has to be Lorentz invariant and gauge invariant. While gaugeinvariance would be satisfied if L depends only on E and B, Lorentz invari-ance is achieved if L is expressed by a combination of the two invariants (1.123)and (1.124). In fact, the first invariant (1.123) is sufficient and we rewrite itin terms of the field tensor

L =1

8π

(E2 −B2

)= − 1

16πF µνFµν . (1.127)

The prefactor is arbitrary but, together with the factor 2 in Eq. (1.123), wemay expect that 1/16π will yield the correct field energy. The minus sign is aguess which will lateron be confirmed by the correct sign of the field energy.

Our goal now is to express L in terms of the electromagnetic potentials Aµ,i.e. to find the functional form L[Aµ, ∂µAν ]. Using the definition of the field

1.5. PHOTONS 35

tensor (1.116) we obtain from (1.127) and (1.116)

−16πL = F µν (∂µAν − ∂νAµ)

= F µν∂µAν − F νµ∂µAν

= 2F µν∂µAν ,

where the second line was obtained by exchanging, in the second term, theindices ν and µ. 7 Thus, the final result is

L = − 1

8π(∂µAν − ∂νAµ) ∂µAν

showing that the Lagrange density does not depend on the potential, but onlythe derivative, L = L[∂µAν ].

We now derive the equations of motion of the electromagnetic field fromthe general Euler-Lagrange equations (1.18). Substituting Aν for the generalfield variable Φl, we obtain

0 =δLδAν− ∂µ

δLδ∂µAν

, ν = 0, . . . 3, (1.128)

which is a system of four equations. The first term vanishes because L isindependent of Aν , whereas the second is most easily evaluated starting fromexpression (1.127)

− δLδ∂µAν

= 2F µν

16π

δFµνδ∂µAν

=F µν

4π, (1.129)

so the Euler-Lagrange equations become

∂µFµν = 0, (1.130)

i.e. we recover Maxwell’s equations in the form (1.125). The zero of ther.h.s., i.e. missing current density, is due to the fact that we started fromthe Lagrange density of the free electromagnetic field. The generalization tononzero charge density will be considered below in Sec. 1.6.

After having derived the field equations from the Lagrange density we cannow use the Lagrange density to find the canonical momentum and the energy-momentum tensor of the electromagnetic field. Using the general definition(1.28) of the canonical momentum we can write, using the result (1.129),

π0ν =δL

δ∂0Aν= −F

0ν

4π, ν = 0, 1, 2, 3. (1.131)

7Recall that summation over the repeated indices µ and ν is implied, so interchangingthe indices does not affect the result.


Inserting the 0−th row of the field tensor, we immediately obtain that π00 =F 00 = 0, i.e. there exists no canonical momentum for the zero component ofAν . In other words, A0 = φ is not a dynamical variable of the electromagneticfield. The remaining three components of the momentum are given by F 0k, k =1, 2, 3, with the result that the momentum vector which is canonically adjointto the vector potential A is given by the electric field:

~π0 =E

4π(1.132)

The special role of the scalar potential is obvious from the fact that by choosinga certain gauge, φ can be arbitrary.

Let us now derive the energy-momentum tensor of the electromagnetic field.We start from the general energy-momentum tensor of canonical field theory,Eq. (1.50),

T µαc (x) = πµ(x)∂αΦ(x)− gµαL(x).

Dropping the subscript “c”, switching to the corresponding expression withthe subscript α and replacing the scalar fields by four-vectors, Φ → Aν andπµ → πµν = −F µν/4π, generalizing the expression (1.131), we obtain 8

T µα = πµν∂αAν − gµαL = −Fµν

4π∂αAν − gµαL.

In the first term on the right we now use the definition (1.116) of the fieldtensor,

F µν∂αAν = F µν (Fαν + ∂νAα)

= F µνFαν + ∂ν (F µνAα)− (−)[∂νFνµ]Aα, (1.133)

where the last term vanishes due to Maxwell’s equations. Also, the second termon the right, being a full 4-divergence, has no influence on the conservationlaws and can be dropped. Then, using expression (1.127), we finally obtain

T µα = −Fµν

4πFαν +

1

16πgµαF

βνFβν

= −Fµν

4πFαν −

1

8πgµα(E2 −B2

), (1.134)

where, in the last line, we used the field invariant (1.123). This tensor issymmetric, gauge invariant and Lorentz invariant, as it should be.

8Note that, when changing from superscript α to subscript α, the tensor elements trans-form according to Eqs. (1.8), i.e. T 0

α = T 0α and T βα = −T βα for β 6= 0 and arbitraryα.

1.5. PHOTONS 37

As it follows from canonical field theory, the energy-momentum tensorobeys the following four conservation laws

0 = ∂µTµα , α = 0, 1, 2, 3, (1.135)

and directly yields the energy density (Hamilton density)

H = T 00 = −F

0ν

4πF0ν −

1

8πg0

0

(E2 −B2

),

= − 1

4π

(−E2

1 − E22 − E2

3

)− 1

8π

(E2 −B2

)=

1

8π

(E2 + B2

), (1.136)

and the momentum density of the field (k−component), see footnote 8,

Sk = −Sk = −T 0k =

F 0ν

4πFkν ,

where the second part of the tensor does not contribute. Using the propertiesof the tensor (1.118) and (1.119) we obtain

Sk =1

4π(−Eν)(−εkνjBj) =

1

4π(E×B)k .

From this the total energy of the electromagnetic field (Hamilton function)and the total momentum (Poynting vector) are obtained by integration overthe whole volume

H(t) =1

8π

∫d3rH(r, t) =

1

8π

∫d3r(E2(r, t) + B2(r, t)

), (1.137)

P(t) =1

4π

∫d3r c · S(r, t) =

c

4π

∫d3rE(r, t)×B(r, t). (1.138)

Using the results for the energy and momentum density, we can now explicitlywrite down the local conservation laws of energy and momentum (1.135), aftermultiplication with c

∂

∂tH + divcS = 0 (1.139)

∂

∂tSk +

∂

∂xjcT jk = 0, j, k = 1, 2, 3, (1.140)

4πT jk = −(EjEk +BjBk

)+

1

2δj,k(E2 + B2

), (1.141)

where we identified the pressure tensor of the electromagnetic field T jk.Problem 7: Verify the expression (1.141) by direct evaluation of the tensor

components (1.134).


1.5.3 Normal mode expansion of theelectromagnetic field

We now wish to analyze the eigenmodes of the free electromagnetic field. Theseare the solutions of Maxwell’s equations in vacuum with zero external chargesand currents. The formal structure of the equations of motion depends on thechosen gauge: in the Lorentz gauge there exists a wave equation for the four-dimensional vector potential Aµ, cf. Eq. (1.113), however the electromagneticfield in vacuum is transverse, i.e. electric and magnetic field vectors oscillateperpendicular to the wave vector k (and orthogonal to each other). This meansthere exist only two possible polarizations of the electromagnetic field alongthe two axes in the plane perpendicular to k. Furthermore, we have seenfrom the Lagrange formalism, cf. Sec. 1.5.2, that the scalar potential (thezero component of Aµ) is not a dynamical variable (there exists no canonicalmomentum associated with φ).

Thus, the Lorentz gauge masks the intrinsic properties of the field in vac-uum and it, therefore, is advantageous to use the Coulomb gauge given by

divA = 0, (1.142)

φ ≡ 0, (1.143)

where the last line follows from the continuity equation (1.101), which yieldsφ = 0 and, in the absence of charges, we may set the potential equal to zero.

We now perform an expansion of the fields in terms of a complete set ofeigenmodes. To this end we consider the case that the field is confined toa cube with side length L, i.e. the eigenmodes will be plane waves with twopossible transverse polarizations, λ = 1, 2, which form a complete orthonormalset of functions

uλ,k(r) =eikr

L3/2~eλ,k, (1.144)∑

λ=1,2

elλ,kejλ,k = δl,j −

klkjk2

, (1.145)∫d3r u∗λ,k(r)uλ′,k′(r) = δλ,λ′δk,k′ .

Eq. (1.145) is a compact mathematical form of the completeness and transver-sality condition which is easy to understand. Given the wave vector k we maydecompose any vector B into a parallel and a perpendicular component by

1.5. PHOTONS 39

applying the parallel and transverse projection operator, respectively

B = B|| + B⊥, (1.146)

B|| = P||B = kk ·Bk2

, (1.147)

B⊥ = P⊥B = (1− P||)B = B− kk ·Bk2

, (1.148)

which, in a Cartesian basis in wave number space9, are represented by thetensors

P ij|| =

kikjk2

, (1.149)

P ij⊥ = δij −

kikjk2

. (1.150)

The system (1.144) forms a basis for arbitrary realizations of the fields Aµ

and πµ = −Aµ/(4πc). We introduce time-dependent expansion coefficientsbλ,k(t) and have to ensure that the fields are real functions:

A(r, t) =∑k λ

Bλ,k

bλk(t)uλ,k(r) + b∗λ,k(t)u∗λ,k(r)

, (1.151)

The real coefficients Bλ,k are introduced in order to adjust lateron the ampli-tude of the functions bλ,k(t) to one. Inserting the ansatz (1.151) into the waveequation (1.113) with jµ = 0

0 =∑λ k

Bλ,k

(bλ,k(t) + c2k2bλ,k(t)

)uλ,k(r) +

(b∗λ,k(t) + c2k2b∗λ,k(t)

)u∗λ,k(r)

,

yields a condition for the coefficients bλ,k(t). Since the functions uλ,k forman orthonormal system, this equation can only be fulfilled if the terms in allparantheses vanish simultaneously, i.e. for all k and λ

bλ,k(t) + c2k2bλ,k(t) = 0, with the solution bλ,k(t) = bλ,k0 e−iωkt, (1.152)

and the dispersion relation, ωk = ck, which is independent of the polarizationλ. By properly choosing the Bλ,k we can always use |bλ,k0| = 1 leaving open anarbitrary phase φ, i.e. bλ,k0 = eiφλ,k , which allows to fulfill the initial condition.Thus the final result for the vector potential is

A(r, t) =1

L3/2

∑λ k

eλ,kBλ,k

bλ,k0e

−i(ωkt−kr) + b∗λ,k0ei(ωkt−kr)

. (1.153)

9For the representation in coordinate space, see Sec. 1.5.4


We can immediately write down the corresponding normal mode expansion ofthe canonically adjoint field, the momentum π, by differentiating the expansion(1.153) with respect to time,

π(r, t) = i1

L3/2

∑λ k

eλ,kBλ,kωk

4πc

bλ,k0e

−i(ωkt−kr) − b∗λ,k0ei(ωkt−kr)

. (1.154)

Note the sign change in front of the adjoint contribution.

1.5.4 Quantization of the electromagnetic field

In Sec. 1.5.2 we have derived the canonical conjugate field variables for thefree electromagnetic field – the 4-potential Aµ and the 4-momentum field. Wenow want to quantize these fields by replacing these quantities by operatorsand imposing the proper commutation relations. The quantization rules are

Aµ −→ Aµ, (1.155)

π0k =1

4πEk = − 1

4πcAk −→ πk = − 1

4πc

d

dtAk, k = 1, 2, 3, (1.156)[

Ak(r, t), πj(r′, t)]

= i~ δTj,k(r− r′), (1.157)

δTj,l(r) =

∫d3k

(2π)3P jl⊥ (k) eikr, (1.158)

where the 0 component of π is zero. The commutation rules (1.157) are anal-ogous to those of the displacement field, Sec. 1.310. However, since there existonly two independent field components which are orthogonal to k we havereplaced the three-dimensional delta function by the 2d transverse delta func-tion which is the Fourier transform of the transverse projection tensor (1.150).By inserting the definition (1.150) into relation (1.158) we find an alternativerepresentation,

δTj,k(r) = δj,kδ(r) +1

4π∂rj∂rk

1

r. (1.159)

We now apply the normal mode expansions (1.153) and (1.154) to thequantized fields. Quantization in these expressions is performed by replacing

10We use bosonic commutation rules. This choice will be discussed at the end of thissection

1.5. PHOTONS 41

the expansion coefficients by operators, bλ,k0 → bλ,k and b∗λ,k0 → b†λ,k, i.e.

~A(r, t) =1

L3/2

∑λ k

eλ,kBλ,k

bλ,ke

−i(ωkt−kr) + b†λ,kei(ωkt−kr)

, (1.160)

~π(r, t) =i

L3/2

∑λ k

eλ,kBλ,kωk

4πc

bλ,ke

−i(ωkt−kr) − b†λ,kei(ωkt−kr)

. (1.161)

The expansion coefficient b†λ,k (bλ,k) has the clear meaning of a creator (annihi-lator) of an elementary excitation of the electromagnetic field mode (λ,k, ωk),i.e. of a photon. What is left now is to obtain the commutation relations forthese photon operators. To this end, we use the commutation relation (1.157)and insert the expansions (1.160) and (1.161) and divide by i~∑

λ k λ′ k′

ejλ,kelλ′,k′

Bλ,kBλ′,k′ωk′

4πc~L3× (1.162)[

bλ,ke−i(ωkt−kr) + b†λ,ke

i(ωkt−kr),bλ′,k′e−i(ω

′kt−k

′r′) − b†λ′,k′ei(ω′

kt−k′r′)]

= δTj,l(r− r′).

This equation is fulfilled by the following bosonic commutation rules[bλ,k, bλ′,k′

]=

[b†λ,k, b

†λ′,k′

]= 0, (1.163)[

bλ,k, b†λ′,k′

]= δk,k′δλ,λ′ . (1.164)

Then Eq. (1.162) reduces to

δTj,l(r− r′) = (1.165)

−2∑

λ k λ′ k′

ejλ,kelλ′,k′

Bλ,kBλ′,k′ωk′

4πc~L3δk,k′δλ,λ′ × (1.166)

ei(ω′k−ωk)te−i(k

′r′−kr) =∑k

e−ik(r′−r)∑λ

ejλ,kelλ,k

−B2λ,kωk

2πc~L3. (1.167)

According to the completeness relation (1.145), the sum over λ yields thetransverse projector P jl

⊥ , provided there is no λ−dependent coefficient. Fi-

nally, the k-sum is just the Fourier transform of P jl⊥ which is nothing but the

transverse delta function, cf. Eq. (1.158), if there is no k−dependent coeffi-cient. The requirements on the coefficients are readily fulfilled by choosingBk = (−2π~cL3/ωk)

1/2.


Thus we have succeeded in finding the normal mode representation of thefield operators of the electromagnetic field which obey bosonic commutationrules. The final result for the field operators which follows from inserting thecoefficients into (1.160,1.161) is given by

~A(r, t) =√

2π~c∑λ k

eλ,k

ω1/2k

bλ,ke

−i(ωkt−kr) + b†λ,kei(ωkt−kr)

, (1.168)

~π(r, t) = i

√~

8πc

∑λ k

eλ,kω1/2k

bλ,ke

−i(ωkt−kr) − b†λ,kei(ωkt−kr)

. (1.169)

Finally, using the normal mode (photon) representation of the field opera-tors we can transform all second quantization operators to the photon picture.In particular, we obtain for the hamiltonian (1.137)

H(t) =1

8π

∫d3r(E2(r, t) + B2(r, t)

)=

∫d3r

2π~π

2(r, t) +

1

8π

(~∇× ~A

)2

(r, t)

=

∑λ k

~ωk(b†λ,kbλ,k +

1

2

). (1.170)

Thus, the hamiltonian of the free electromagnetic field is a superposition oflinear harmonic oscillator hamiltonians corresponding to the quantized normalmodes of the field. For details of the derivation of the final result, see problem8.

Let us briefly discuss our result.

i) We have quantized the free electromagnetic field which is transverse withtwo independent components (polarizations).

ii) Type and number of independent field modes (normal modes) is the same asin the classical description. In an infinite systems, the normal modes aremonochromatic plane waves, in other geometries the modes follow fromthe solution of the wave equation with the proper boundary conditions.

iii) Other examples would be a small radiation source which emittes in alldirections. In this case the normal modes would be spherical waves.Furthermore, in a finite system, the normal modes depend on the prop-erties of the boundaries such as the reflectivity of the walls or losses dueto dissipation. Special resonators are used to select a certain number ofmodes, thus reducing the k−sum (wave guides, lasers etc.).

1.5. PHOTONS 43

iv) The field consists of an integer number of elementary quanta (photons)with an elementary energy ~ωk and a momentum ~k. Experiments con-firm that each mode can be multiply excited, i.e. contain a large numberNk,λ of modes. This would not be possible if photons would obey Fermistatistics and, hence, the Pauli principle. Thus the choice of bosoniccommutation rules (1.157) is justified.

v) In second quantization the electromagnetic field is described by the fieldoperators A and π which are randomly fluctuating quantities. This trans-forms into randomly fluctuating mode amplitudes bk,λ, b

†k,λ or their com-

binations such as the number operator nk,λ = b†k,λbk,λ. Measurementswill yield the eigenvalues, i.e. the number of photons of a given mode,nk,λ = 0, 1, 2, . . .

vi) Other measurable quantities are expectation values of the operators Aand π or of operator products (correlation functions). To compute theexpectation values requires knowledge of the statistical properties of thefield. This can, for example, be a quantum mechanical state (pure state)which is being prepared in an experiment (such as in quantum optics)

vii) Alternatively, the field can be in a mixed state such as in thermodynamicequilibrium. Then the theory requires averaging of operator productswith a thermodynamic density operator which leads to a statistical oc-cupation of the photon modes with the mean occupations given by aBose distribution.

viii) The most general situation corresponds to a nonequilibrium state of thequantized field which can be caused by an external perturbation. Thenthe mean occupation of each mode may change in time, nk,λ = nk,λ(t),and the question arises how to theoretically describe this evolution. Themost general theory which solves this problem is the theory nonequilib-rium Green functions (here, the photon Green function), which considersexpectation values of operator products of the type 〈bk,λ(t)b†k′,λ′(t

′)〉, seee.g. Ref. [?].

Finally, we note that, with the second quantization of the electromagneticfield and the introduction of photons, we have found a consistent conceptfor the description of the field – in a sense, this is the modern picture ofthe particle-field duality, originally introduced by Max Planck the founder ofquantum theory. Indeed, this theory contains wave properties – here in termsof the normal modes - and particles. Only the latter are not to be understoodas particles in a mechanical sense but as elementary excitations of the field


modes – the photons – which carry an energy quantum ~ωk and a momentumquantum ~k11.

1.6 Quantization of the electromagnetic field

coupled to charged particles

So far we have considered the free electromagnetic field and its quantization.Let us now generalize these results to the case that the field is coupled tocharged particles. We will generalize the Lagrange formalism to the full field-matter interaction and find the corresponding Euler-Lagrange equations. Thefirst question we have to solve is how to incroporate the coupling betweenparticles and fields into the minimal action principle. As with the choice ofthe Lagrange function, cf. Sec. 1.1 there is, in principle, large freedom inconstructing the action for particles and fields. We will again follow the ruleto prefer the simplest choice possible, see also the discussion in Ref. [LL62].

The total action of the coupled particle-field system can always be writtenas the sum of three terms

S = SF + SM + SMF (1.171)

where we denote the field (matter) term which contains only field (matter)variables by “F” (“M”) and the coupling term by “MF”. The first term is theaction of the free electromagnetic field which was studied in Sec. 1.5.2. We arenow going to analyze the second term.

1.6.1 Lagrangian of a classical relativistic particle

The total action of a system of N charged particles is additive SM =∑N

i=1 SMi.This is, because the Coulomb interaction of the particles is mediated by theelectromagnetic field, and this effect will be included in the action SMF .

Thus, let us start by consdiering a single classical free non-relativistic parti-cle “i” with mass mi and recall its action. Since the only energy contribution isdue to kinetic energy T , the Lagrange function coincides with T , cf. Sec. 1.1.1,L0i = miv

2/2. Obviously, this expression is not Lorentz invariant. To generalize

11Interestingly, Planck himself maintained serious doubts about the consistency of his owntheory [Pla]. Considering, as an example, a spherical wave with an intensity decreasing asr−2 where r is the distance from the source he could not imagine how to deal with thesituation when the intensity falls below that corresponding to a single photon. This isnaturally solved in the standard probabilistic interpretation of quantum mechanics.

1.6. EMF QUANTIZATION IN MATTER 45

L0i to a relativistic particle we need to start from Lorentz scalars. The sim-

plest mechanical Lorentz scalar is the space time interval, the square of whichis (dsi)

2 = c2(dt)2 − (dri)2, cf. Sec. 1.1. Then, the action for a macroscopic

physical process between points “a” and “b” involving particle “i” is

SMi = αi

∫ b

a

ds = αic

∫ tb

ta

√1− v2

i

c2dt

≡∫ tb

ta

dt LMi, (1.172)

where we used vi = dri/dt, and the constant αi is yet to be determined. In thelast line we used the general definition of the Lagrange function which yields,

for a relativistic point particle, LMi = αic

√1− v2

i

c2. The constant is readily

determined by requiring that the known non-relativistic limit is recovered, i.e.

L0i (vi) =

mi

2v2i = lim

vi/c→0LMi(vi) = αic

(1− 1

2

v2i

c2

).

Comparison of the left and right sides and dropping the velocity independentfirst term yields the constant to be αi = −mic with the result for the Lagrangefunction of a relativistic classical point particle

LMi(vi) = −mic2

√1− v2

i

c2(1.173)

Applying the standard formulas of the Lagrange formalism, from this we im-mediately recover the momentum

pi =∂LMi

∂vi=

mivi√1− v2

i

c2

, (1.174)

and the total (kinetic) energy

Ei = pivi − LMi =mic

2√1− v2

i

c2

. (1.175)

Now, the hamiltonian follows from Ei, as usual, by eliminating the velocity.Squaring (1.175), we obtain

E2i

c2=

m2i c

2

1− v2ic2

=m2iv

2i

1− v2i

c2

+m2i (c

2 − v2i )

1− v2i

c2

= p2i +mic

2,


which yields the familiar relativistic energy-momentum dispersion

HMi(pi) =√c2p2

i +m2i c

4 (1.176)

Finally, the Lagrange formalism immediately yields the (trivial) equation ofmotion (the Euler-Lagrange equation) of a free particle:

0 = −∂LMi

∂qi+d

dt

∂LMi

∂vi=

d

dtpi. (1.177)

To shorten the notation, below we will denote the relativistic square root byγi = (1− v2

i /c2)−1/2.

1.6.2 Relativistic particle coupled tothe electromagnetic field

Let us now analyze how the equation of motion of the particle changes if itinteracts with an electromagnetic field. We then have to re-derive the Euler-Lagrange equation starting from the total action S = S[qi,vi;A

µ], Eq. (1.171),and performing the variation with respect to the particle variables. Due tothe linear relation between action and Lagrange function, the latter will alsoconsist of three parts from which the field part is, by definition, independentof the particle coordinates. Therefore, the equation of motion of particle “i”follows from the Lagrange function LMi + LMi,F generalizing Eq. (1.177)

0 = −∂LMi + LMi,F

∂qi+d

dt

∂LMi + LMi,F

∂vi

= −∂LMi,F

∂qi+d

dtpi +

d

dt

∂LMi,F

∂vi(1.178)

Inserting the result for the relativistic momentum, Eq. (1.174), we obtainthe relativistic generalization of Newton’s equation

d

dt

mivi√1− v2

i

c2

=∂LMi,F

∂qi− d

dt

∂LMi,F

∂vi= FLi (1.179)

where the r.h.s. is the relativistic Lorentz force which is determined by thefield-matter interaction. While the explicit form of the interaction LagrangianLMi,F is still open it is clear that, in the equation of motion (1.179) for ourclassical point particle, the r.h.s. has to be taken (after differentiation) at thecurrent phase space point qi(t),vi(t) of the particle.


1.6.3 Lagrangian of charged particles in an EM field

Let us now consider how to describe, in the Lagrange formalism, the interactionbetween a charged particle with the electromagnetic field. Again, we choosethe simplest possible form. The key quantities describing the field matterinteraction are the particle charge (and eventuall the charge current density)whereas the field is represented by the potential Aµ. The simplest way tocouple both is to include both factors in first order. To do this in a Lorentzinvariant way we need a scalar product of two 4−vectors which we form fromAµ and xµ. The proper choice of the action is (the pre-factor is chosen tocorrectly reproduce the interaction energy)

SMi,F = −∫ b

a

eicAµdx

µ = −eic

∫ b

a

cφdt−Adr (1.180)

= −eic

∫ tb

ta

cφ−Avi dt,

which yields the following result for the field-matter Lagrangian

LMi,F [vi(t);Aµ(qi, t)] =

eic

Avi − eiφ (1.181)

where the potentials have to be taken at the current position of the particle.With this result we can derive the canonical momentum and the Hamilton

function of the particle in the electromagnetic field replacing, in our previousexpressions for the free particle, LMi → LMi +LMi,F ≡ Li. The momentum isthen

Pi =∂Li∂vi

= pi +eic

A(qi) = miviγi +eic

A(qi), (1.182)

where pi is given by Eq. (1.174). Analogously, we obtain the energy

Ei = pivi − Li = mic2γi + eiφ(qi). (1.183)

To obtain the Hamilton function we again eliminate the velocity, as was donewith Eq. (1.175),

(Ei − eiφ)2

c2= p2

i +mic2.

The hamiltonian of a relativistic particle should be written as a function ofthe canonical momentum which now is given by Eq. (1.182). Thus solving forEi and expressing pi by Pi we obtain

Hi(qi,Pi) =

√c2(Pi −

eic

A(qi))2

+m2i c

4 + eiφ(qi) (1.184)


For completeness, we consider the non-relativistic limit of the above results.Then LMi → L0

i but LMi,F remains unchanged, i.e. Li → mi2

v2i + ei

cAvi − eiφ.

Consequently, Pi = ∂Li∂vi

= mivi − eicA, and the non-relativistic hamiltonian

reduces to

Hi(qi,Pi) =1

2mi

(Pi −

eic

A(qi))2

+ eiφ(qi).

With the result for the field-matter interaction Lagrangian we can nowevaluate the Lorentz force (1.179) on a relativistic particle. The result is thesame as in the non-relativistic case, familiar from electrodynamics,

FLi(t) = eiE(qi, t) +eic

vi ×B(qi, t) (1.185)

Proof: To evaluate the expression

FLi =∂LMi,F

∂qi− d

dt

∂LMi,F

∂vi, (1.186)

we begin with transforming the second term taking into account that, for acontinuous vector field A(r, t), the total time derivative is d

dtA = ∂

∂t+(v~∇)A.

Recalling further the relation between electric field and vector potential, E =−~∇φ− 1

c∂∂t

A, we can transform

− d

dt

∂LMi,F

∂vi= −ei

c

d

dtA = −ei

c

∂

∂t+ (vi~∇)

A

= ei(E + ~∇φ)− eic

(vi~∇)A. (1.187)

For the transformation of the first term in (1.186) we recall the vector identitiesfor an arbitrary pair of vectors

~∇(C ·D) = (C · ~∇)D + (D · ~∇)C +

+ D× (~∇×C) + C× (~∇×D),

which allows to transform

∂

∂qi[A(qi)vi] =

(vi ·

∂

∂qi

)A(qi) + vi ×

(∂

∂qi×A(qi)

),

where we took into account that vi does not depend on qi. Now we cantransform

∂LMi,F

∂qi=

eic

∂

∂qi[A(qi)vi]− ei

∂

∂qiφ(qi) (1.188)

=eic

(vi ·

∂

∂qi

)A(qi) +

eic

vi ×(

∂

∂qi×A(qi)

)− ei

∂

∂qiφ(qi).


Now we can add up the two terms (1.187) and (1.188) and obtain, after can-cellations,

− d

dt

∂LMi,F

∂vi+∂LMi,F

∂qi= eiE(qi) +

eic

vi ×(

∂

∂qi×A(qi)

), (1.189)

which is just the Lorentz force (1.186) acting on point particle “i”.With this the treatment of the particle dynamics is completed. Using the

Lagrange formalism with a Lagrangian for a relativistic point particle andan additional field-matter interaction contribution we have derived the Euler-Lagrange equation for the particle which is just the relativistic generalizationof Newton’s equation. The force on the particle is the Lorentz force which isrelativistically invariant due to the covariance of Maxwell’s equations.

What is left now is to consider the influence of the field-matter interactionon the electromagnetic fields. We expect that this will give rise to modifiedMaxwell’s equations compared to the vacuum case - the equations will containa coupling to the particles via charge and current densities. Since this couplingis, in general, not created by point particles, we first study how to generalizethe field-matter interaction Lagrangian to arbitrary spatially extended chargedistributions ρ(r).

Generalization to a delocalized charge distribution

To study the interaction of the field with a continuous charge distributionwe will first generalize our result to an ensemble of N point particles withtotal charge Q =

∑Ni=1 ei and then let the individual charges go to zero and

increasing N such that Q remains finite. Then we can change from summationto integration over the volume,

Q =N∑i=1

ei −→∫V

d3r ρ(r),

and write the total interaction contribution to the action, generalizing Eq. (1.180)according to

SMF =N∑i=1

SMi,F = −1

c

∫ b

a

Aµ

N∑i=1

eidxµ

−→ −1

c

∫ b

a

∫V

d3r ρ(r)Aµ dxµ

= −1

c

∫ tb

ta

∫V

d3r ρ(r)Aµdxµ

dtdt ≡

∫ tb

ta

dt

∫V

d3rLMF (r, t). (1.190)


The last line allows us to identify the Lagrange density which can be expressedby the current 4 vector using the definition jµ = ρdx

µ

dt= (cρ, j), cf. Eq. (1.111),

LMF = −1

cAµj

µ (1.191)

which is an impressively simple result considering the scope of physical pro-cesses it describes. This is the simplest linear coupling of the electromagneticfield to charged matter where the latter is fully described by the current vectorjµ. This result is fully applicable to classical systems: in this case we haveto insert the charge current density and charge density which contain deltafunctions which brings us back to the first expression for the action involvinga sum over point charge contributions. At the same time, the result (1.191)also applies to quantum systems. Then the charge and current density aredetermined by the wave function,

ρ = e ψψ∗,

j =e~

2miψ∗∇ψ − ψ∇ψ∗ ,

if the system is in the (pure) quantum state described by the wave functionψ(r, t). Thus, in the quantum case, the current entering the Lagrange densityis to be understood as an expectation value. Therefore, any technical toolto compute averages can be applied here. If the quantum system is in amixed state, e.g. in thermodynamic equilibrium, the expectation value can becomputed e.g. via the trace with the density operator, a distribution functionor Green’s function. Finally, the coupling (1.191) has proved very successful inthe description of other field-matter interactions far beyond electrodynamics,including quantum chromodynamics, see e.g. Ref. [qcd].

1.6.4 Quantization of the electromagnetic field coupledto charges

After using the action of the field-matter system to derive the equations ofmotion of the particles we now have to do the same for the electromagneticfield. Since we will perform the variations with respect to the field variableAµ, from the three contributions to the action (1.171) only two will have aneffect; the particle term SM does not depend on the field and can be skipped.Thus the Lagrange density LF of the field in vacuum, cf. Sec. 1.5.2 has to begeneralized to LF −→ LF + LMF ≡ Lfield which reads

Lfield[Aµ, ∂νAµ, jµ] = − 1

8π(∂µAν − ∂νAµ) ∂µAν −

1

cAµj

µ (1.192)


In contrast to the vacuum case the Lagrangian now also depends on Aµ. Theequations of motion are given by the Euler-Lagrange equations (1.128) in whichonly the first term is new,

0 =δLfield

δAν− ∂µ

δLfield

δ∂µAν, (1.193)

= −1

cjµ + ∂µ

F µν

4π,

and we immediately obtain Maxwell’s equations for the field tensor includingthe coupling to matter. The second pair of equations for the dual tensor doesnot change compared to the vacuum case since it does not involve charge andcurrent densities.

Let us now discuss how the presence of charges alters the quantization ofthe electromagnetic field. The first thing we need is to compute the canonicalmomentum of the field by functionally differentiating Lfield with respect to∂tAν , cf. Eq. (1.131). Since LMF is independent of ∂tAν the result is the sameas in vacuum, π0 ≡ 0 and ~π = E/4π. This means, the quantization can beperformed as in vacuum, by replacing the fields Aµ and ~π by operators withthe commutation relations[

Ak(r, t), πj(r′, t)]

= i~ δj,kδ(r− r′). (1.194)

Note that we did not use the transverse delta function here because we cannotguarantee that there exist only two transverse modes as in vacuum electrody-namics since the fields are subject to external currents,

Ak =4π

cjk, k = 1, 2, 3. (1.195)

Here we have excluded the 0−component, i.e. the equation for φ because it isnot independent of the dynamics of A. For example, in the Lorentz gauge, theconnection is given by Eq. (1.108). The general solution of this equation is

Ak = CkAkfree + Akext,

where Afree is the solution of the homogeneous equation (consisting of twopurely transverse components) and Aext is the particular solution of the inho-mogeneous equation

Aext(r, t) =4π

c

∫dt

∫d3r G(r, t; r, t) j(r, t),

where G is the Green function of Eq. (1.195).


The explicit form of G and of all solutions of Eq. (1.195) depends on thegeometry of the system, the boundary values of j and the associated initial andboundary conditions for A. These solutions can again be used to construct acomplete set of normal modes uλ,k(r, t; [j]) which, due to linearity of the waveequation, can be used as a basis for the canonically conjugate fields A andπ. The same basis can be used for the field operators A and π, following theprocedure outlined in Sec. 1.5.4. Each of the modes will again be associatedwith a pair of photon operators bλ,k(r, t) and b†λ,k(r, t). With this, the secondquantization procedure has been extended to the case of fields coupled toexternal charges.

1.6.5 Quantization of the EM field in a dielectric mediumor plasma

So far we have considered the interaction of the field with external charges andcurrents and derived the coupled quations of charges and field in various forms,including a quantized description. This is a correct microscopic descriptionHowever, if the field interacts with a macroscopic number of particles, forexample with a plasma, a gas or a polarizable medium, it is advantageous touse a statistical (continuum) approach. The idea is not to resolve the individualcharged particle dynamics but to average all quantitities over a finite volume.

This medium itself contains charges and currents which are “induced” bythe field giving rise to a total charge and current density ρ = ρext + ρind andj = jext + jind which appear in Maxwell’s equations. This leads to a modified“medium” electrodynamics, see e.g. the text books [ABA84, Jac75, LL80, ?].

Here we only outline the basic procedure. The common approach is torestore equations of the same structure as those of vacuum electrodynamics,cf. Sec. 1.5.1,

divD(r, t) = 4πρext(r, t), ∇× E(r, t) = −1

c

∂B(r, t)

∂t, (1.196)

divB(r, t) = 0, ∇×H(r, t) =4π

cjext(r, t) +

1

c

∂D(r, t)

∂t. (1.197)

keeping only the external charge and currents and absorbing the induced con-tributions into modified fields D and H. The relation between D and E on theone hand, and H and B, on the other is realized via additional functions – thedielectric function ε and the magnetic permeability µ, respectively which con-tain the complete information about the particular medium. In the simplest


case, the relation in Fourier space reads (i, j = 1, 2, 3),

Di(k, ω) = εij(k, ω)Ej(k, ω),

Bi(k, ω) = µij(k, ω)Hj(k, ω),

where summation over repeated indices is implied.

Now, the eigenvalue problem of the field in the presence of a polarizablemedium differs compared to vacuum. As a result, for the case of an unmagne-tized medium (µ = 1), the eigenmodes of the electromagnetic field are obtainedform the condition that the following determinant vanishes, i.e.

∣∣∣∣k2δij − kikj −ω2

c2εij(k, ω)

∣∣∣∣ = 0.

The result is a spectrum of modes with the dispersion ωs(k), s = 1, 2, . . . ,which are essentially influenced by the medium, for details see [ABA84, Bon, ?].As in the case of the electromagnetic field in vacuum, the details of the modesdepend on the geometry of the system. In case that the field is enclosed in acube, cf. Sec. 1.5.3, the solutions will again be monochromatic plane wavesbut with a modified frequency dispersion ωs(k). After diagonalization, thesemodes can again be used to construct a complete orthonormal set suitable forexpansion of any field excitation. In particular, the canonical fields and thefield operators can be expanded in terms of these normal modes.

This is, however, only part of the story. The procedure outlined before anddiscussed so far leads to the expansion of the field operators A and π in termsof normal modes where the coefficients are creation and annihilation operatorsb† and b of elemenatry excitations of a given normal mode. However, theseare randomly fluctuating operators and, therefore, also A and π are randomquantities. The only quantities of practical use which can be related to anexperiment are suitable expectation values such as 〈Aµ〉. Since in many casesthis mean value is zero the first non-trivial expectation value is that of aproduct 〈AµAν〉 which is closely related to the Green’s functions of many-body theory. Thus the task of a many-body theory of field and matter is toderive equations of motion for the Green’s functions of the electromagnetic field(photon Green’s function) coupled to the Green’s functions of charge particles.This is described in detail in Ref. [Bon].


1.7 Quantization of the Schrodinger field

1.8 Quantization of the Klein-Gordon field

1.9 Coupled equations for the Schrodinger and

Maxwell fields

1.10 Problems to Chapter 1

1. Analyze the second variation of the action. Are there cases when theEuler-Lagrange equations (1.18) lead to a maximum of the action Srather than to a minimum?

2. Proof relation (1.70).

3. Derive the hamiltonian (1.97) from the expression (1.96).

4. Verify the components of the tensor F µν by direct evaluation of thedefinition, Eq. (1.122).

5. Verify Eq. (1.123) by direct evaluation of the tensor product.

6. Verify Eq. (1.124) by direct evaluation of the tensor product.

7. Derive the second quantization representation of the hamiltonian, Eq. (1.170).

8. Derive the second quantization representation of the Poynting vector,Eq. (1.138).

Chapter 2

Introduction to secondquantization

Now we turn to a different approach of treating quantum N -particle systems.Instead of first constructing a classical field theory with two canonically con-jugate fields, that are quantized, in a second step, we now proceed differently.We will start from a quantum mechanical description of the state many par-ticles which is first symmetrized (anti-symmetrized), for the case of bosons(fermions). We then switch to occupation number representation for whichthe quantization is rather trivial, based on the introduction of creation andannihilation operators. This scheme will be realized for fermions and bosonsin chapter ??. But before that, it is very instructive to consider the analo-gous many-body problem in the limit of classical particles and to perform a“second quantization in phase space”. This is achieved by introducing the mi-croscopic phase space density which is due to Klimontovich, cf. Eq. (2.13). Wewill observe that this quantity obeys an equation of motion that is completelyanalogous to the equations of motion of the field operators and fermions andbosons which allows for valuable insights into the common statistical conceptsof second quantization.

2.1 Second quantization in phase space

2.1.1 Classical dynamics in terms of point particles

We consider systems of a large number N of identical particles which interactvia pair potentials V and may be subject to an external field U . The system

55

56 CHAPTER 2. SECOND QUANTIZATION

is described by the Hamilton function

H(p, q) =N∑i=1

p2i

2m+

N∑i=1

U(ri) +∑

1≤i<j≤N

V (ri − rj) (2.1)

where p and q are 3N -dimensional vectors of all particle momenta and coordi-nates, p ≡ p1,p2, . . .pN and q ≡ r1, r2, . . . rN. Examples of the externalpotentials can be the electrostatic potential of a capacitor, the potential ofan atomic nucleus or the potential an electron feels at a solid surface. Theinteraction potentials can arise from gravitational fields, from the Coulombinteraction of charged particles, the magnetic interaction of currents and soon. From the hamiltonian (2.1) the equations of motion follow by applyingHamilton’s equations1,

qi =∂H

∂pi=pim, (2.2)

pi = −∂H∂qi

= −∂U∂qi−∑j 6=i

∂V

∂qi, (2.3)

which is to be understood as two systems of 3N scalar equations for x1, y1, . . . zNand px1, py1, . . . pzN where ∂/∂q ≡ ∂/∂r1, . . . ∂/∂rN, and the last equalitiesare obtained by inserting the hamiltonian (2.1). The system (2.3) is nothingbut Newton’s equations containing the forces arising from the gradient of theexternal potential and the gradient from all pair interactions involving thegiven particle, i.e. for any particle i = 1 . . . N

pi = −∂U(r1, . . . rN)

∂ri−∑j 6=i

∂V (ri − rj)

∂ri. (2.4)

Consider, as an example, a system of identical charged particles with chargeei which may be subject to an external electrostatic potential φext and interactwith each other via the Coulomb potential Vc =

eiej|ri−rj | . Then Newton’s equa-

tion (2.4) for particle i contains, on the r.h.s., the gradients of the potentialU = eiφext and of the N − 1 Coulomb potentials involving all other particles.

pi = −∂eiφext(ri)

∂ri−∑j 6=i

∂

∂ri

eiej|ri − rj|

, (2.5)

1Generalized equations of motion can, of course, also be obtained for non-Hamiltonian(dissipative) systems

2.1. SECOND QUANTIZATION IN PHASE SPACE 57

2.1.2 Point particles coupled via classical fields

An alternative way of writing Eq. (2.5) is to describe the particle interactionnot by all pair interactions but to compute the total electric field, E(r, t),all particles produce in the whole space. The force particle “i” experiencesis then just the Lorentz force, eiE, which is minus the gradient of the totalelectrostatic potential φ which is readily identified from the r.h.s. of Eq. (2.5)

pi = eiE(ri, t) = −ei∂φ(r, t)

∂r

∣∣∣∣r=ri

, (2.6)

φ(r, t) = φext(r) +N∑j=1

ej|r− rj(t)|

. (2.7)

In this case we explicitly know the form of the potential (2.7) but we can alsorewrite this in terms of the Poisson equation which is solved by the potential(2.7)

∆φ(r, t) = −4πN∑j=1

ejδ[r− rj(t)] = −4πρ(r, t). (2.8)

φ contains the external potential and the potentials induced by all particles ata given space point r at time t. When computing the force on a given particlei the potential has to be taken at r = ri(t) and the contribution of particle “i”to the sum over the particles in Eqs. (2.7, 2.8) has to be excluded (to avoidselfinteraction). Also, on the r.h.s. we have introduced the charge density ρ ofthe system of N point particles.

Considering the formal structure of Eq. (2.6) we notice that the Coulombforces between discrete particles have been completely eliminated in favor ofa space-dependent function – the electric field. Obviously, this descriptionis readily generalized to the case of time-dependent external potentials andmagnetic fields which yields a coupled set of Newton’s and Maxwell’s equations,

pi = ei

E(ri, t) +

1

cvi ×B(ri, t)

, (2.9)

divE(r, t) = 4πρ(r, t), ∇× E(r, t) = −1

c

∂B(r, t)

∂t, (2.10)

divB(r, t) = 0, ∇×B(r, t) =4π

cj(r, t) +

1

c

∂E(r, t)

∂t, (2.11)

where we introduced the current density j which, for a system of point parti-cles, is given by j(r, t) =

∑Nj=1 ejvj(t)δ[r− rj(t)]. Charge and current density


are determined by the instantaneous phase space trajectories q(t), p(t) of allparticles.

The two sets of equations (2.6, 2.8) and (2.9, 2.10, 2.11) form closed sys-tems coupling the dynamics of classical charged point particles and a classi-cal electromagnetic field. This coupling occurs, in the particle equation–viathe Lorentz force–and, in the field equations–via the charge and current den-sity. The classical description, therefore, requires knowledge of the N dis-crete particle trajectories q(t), p(t) and of the two continuous vector fieldsE(r, t),B(r, t).

Using the electric and magnetic fields, we may rewrite the Hamilton func-tion (2.1) corresponding to the full system (2.6, 2.10, 2.11)

H =N∑i=1

p2i

2m+

N∑i=1

U(ri) +1

8π

∫d3rE2(r, t) + B2(r, t)

(2.12)

where the integral contains the energy of the electromagnetic field familiarfrom standard electrodynamics.

2.1.3 Classical dynamics via particle and Maxwell fields

We have now found two alternative descriptions of the dynamics of interactingpoint charges:

1. via the system (2.5), involving only discrete point particles, and

2. via the system (2.9, 2.10, 2.11) which gives a hybrid description in whichthe particles are discrete but the fields (or the particle interaction) con-tinuous.

One may ask if there is third form which contains only continuous field-typequantities. This would require to represent also the particles by fields, also inthe classical case. This is, indeed possible, as we show in this section.

In fact, the right hand sides of Maxwell’s equations already do contain(formally) continuous quantities, ρ and j, representing the particles. However,they contain (via the delta functions) only the particle coordinates. It is, there-fore, tempting to consider a symmetric, with respect to q and p, quantity – themicroscopic phase space density which was introduced by Yuri Klimontovichin the 1950s [Kli57]

N(r,p, t) =N∑i=1

δ[r− ri(t)]δ[p− pi(t)] ≡N∑i=1

δ[x− xi(t)] (2.13)


where we introduced the short notations x ≡ r,p and xi(t) ≡ ri(t),pi(t).2The function N is related to the particle density n(r, t), and it obeys a nor-malization condition, ∫

d3pN(r,p, t) = n(r, t), (2.14)∫d6xN(r,p, t) = N(t). (2.15)

If there are no particle sources or sinks, N(t) =const, and there exists a localconservation law, d

dtN = 0. From this we obtain the equation of motion of

N(r,p, t):

dN(r,p, t)

dt=∂N(r,p, t)

∂t+

N∑i=1

δ[p− pi(t)]∂

∂rδ[r− ri(t)]

∂ri∂t

+

N∑i=1

δ[r− ri(t)]∂

∂pδ[p− pi(t)]

∂pi∂t

.

Using Newton’s equations (2.5) the time derivatives can be computed afterwhich the delta functions allow to replace ri → r and pi → p which can betaken out of the sum. As a result we obtain3

∂

∂t+ v

∂

∂r+ e

[E(r, t) +

1

cv ×B(r, t)

]∂

∂p

N(r,p, t) = 0. (2.16)

Thus we have obtained a field description of the particles via the function Nand eliminated (formally) all discrete particle information. The field N alsoreplaces the charge and current density in Maxwell’s equations (2.10, 2.11) viathe relations [cf. Eq. (2.14)]

ρ(r, t) = e

∫d3pN(r,p, t), (2.17)

j(r, t) = e

∫d3pvN(r,p, t). (2.18)

A particularly simple form is obtained in the absence of a magnetic field,in the case of particles interacting by Coulomb potentials, cf. Eqs. (2.6, 2.7)

2Note that for a vector y = y1, y2, y3, δ[y] ≡ δ[y1]δ[y2]δ[y3], so N contains a productof six scalar delta functions.

3To simplify the notation, in the following we consider a one-component system withidentical charges, e1 = . . . eN = e. An extension to multi-component systems is straightfor-wardly done by introducing a separate function Na, for each component.


above. Then, we may use the solution of Poisson’s equation, expressing ρ viaN(r,p, t),

φ(r, t) = φext(r, t) +

∫dr′

ρ(r′, t)

|r− r′|

= φext(r, t) +

∫dr′∫dp′

eN(r′,p′, t)

|r− r′|. (2.19)

With this expression the electrostatic field has been eliminated, and the par-ticle dynamics (2.16) become a closed equation for N(r,p, t):

∂

∂t+ v

∂

∂r− e ∂

∂r

[φext +

∫d6x′

eN(r′,p′, t)

|r− r′|

]∂

∂p

N(r,p, t) = 0 (2.20)

Using the phase space density, all observables of the system can be expressedin terms of fields. For example, the hamilton function now becomes

H =

∫d6x

p2

2mN(r,p, t) +

∫d6xU(r)N(r,p, t) +

+1

8π

∫d3rE2(r, t) + B2(r, t)

, (2.21)

We underline that this is an exact equation (as long as a classical descriptionis valid) – no assumptions with respect to the interactions have been made.It is fully equivalent to Newton’s equations (2.6). The discrete nature of theparticles has now vanished – it is hidden in the highly singular phase spacefield N . As Newton’s equation, Eq. (2.20) allows for an exact solution of theparticle dynamics, once the initial conditions are precisely known.

2.1.4 Discussion: ensemble averages, fluctuations, quan-tum effects

The possibility to find an exact solution is, of course, restricted to few particles.In contrast, if our goal is to describe the behavior of a macroscopic particleensemble any particular initial condition, and the resulting dynamics have tobe regarded as random. However, we will be interested in statistically reliablepredictions, so we need to average over a certain statistical ensemble, e.g. overall possible initial conditions. An ensemble average of N yields directly thesingle-particle phase space distribution 〈N(r,p, t)〉 = f(r,p, t), and Eq. (2.20)turns into an equation for the distribution function f , i.e. a kinetic equation.

It is easy to see that application of such an ensemble average does notlead to a closed equation for f(r,p, t). This is because Eq. (2.20) contains a


product of N functions for which, of course, 〈NN〉 6= 〈N〉〈N〉. Instead one hasto introduce the fluctuations δN , by writing N = f + δN . Then one obtains

〈N ·N〉 = f · f + 〈δN · δN〉. (2.22)

The last term appears additionally in the ensemble averaged version of Eq. (2.20)and gives rise to collision and correlation effects. In a similar manner, also thefields have to be decomposed into an average and a fluctuating part,

E(r, t) = 〈E(r, t)〉+ δE(r, t)

B(r, t) = 〈B(r, t)〉+ δB(r, t) .

This expansion has to be inserted into Maxwell’s equations. Obviously, theexpectation values of the fields then obey equations that are driven by theexpectation values of the charge density and the charge current density, re-spectively. On the other hand, the fluctuations, δN , will give rise to fluctua-tions δρ and δj and, via Maxwell’s equations, to the fluctuating electric andmagnetic fields. Similarly, in the electrostatic case, the fluctation δN createsa fluctuation of the induced potential, δφ.

The use of the phase space density N and the concept of fluctuations asstarting point for the derivation of a kinetic theory of gases and plasmas hasbeen successfully demonstrated by Klimontovich, for details see his text books[Kli75, Kli80].

With Eqs. (2.16) and (2.20) we have realized the third picture of coupledparticle-electromagnetic field dynamics – in terms of the particle field, N ,and the electric and magnetic fields. While we have concentrated on chargedparticles and Coulomb interaction, the approach may be equally applied toother interactions, e.g. electrons interaction with lattice vibrations of a soliddescribed by the displacement field, see Sec. 2.5.1. Thus the basis for a classicalfield theory has been achieved.

Of course, this picture is based on classical physics, i.e. on Newton’s equa-tions for point particles and Maxwell’s equations for the electromagnetic field.No quantum effects appear, neither in the description of the particles nor thefield. The classical picture has been questioned only at the end of the 19thcentury where the experiments on black body radiation could not be correctlyexplained by Maxwell’s theory of the electromagnetic field. The classical ex-pression for the field energy of an electromagnetic wave which only dependson the field amplitudes E0,B0 and, therefore, is a continuous function, cf.hamiltonian (2.21), did not reproduce the measured spectral energy density.The solution which was found by Max Planck indicated that the field energy


cannot depend on the field amplitudes alone. The energy exchange betweenelectromagnetic field and matter is even entirely independent of E0,B0, in-stead it depends on the frequency ω of the wave. Thus the total energy of anelectromagnetic wave of frequency ω is an integer multiple of an elementaryenergy, Wfield(ω) = N~ω, where ~ is Planck’s constant, i.e. the energy is quan-tized. For the (“canonical”) quantization of the electromagnetic field, we referto Chapter 1.

What Planck had discovered in 1900 was the quantization of the electro-magnetic field 4. This concept is very different from the quantum mechanicaldescription of the electron dynamics from which it is, therefore, clearly distin-guished by the now common notion of “second quantization”. Interestingly,however, the “first quantization” of the motion of microparticles was intro-duced only a quarter century later when quantum mechanics was discovered.

Problem 1: Perform the ensemble average of Eqs. (2.16) and (2.20) andfind an explicit expression for the additional term arising from the correlationfunction of the fluctuations.

2.2 Quantum mechanics and first quantization

2.2.1 Reminder: State vectors and operators in Hilbertspace

Let us briefly recall the main ideas of quantum theory. The essence of quantummechanics or “first” quantization is to replace functions by operators, startingfrom the coordinate and momentum (here we use the momentum representa-tion),

r → r = r,

p → p =~i

∂

∂r,

where the last equalities refer to the coordinate representation. These opera-tors are hermitean, r† = r and p† = p, and do not commute

[xi, pj] = i~δi,j, (2.23)

4Planck himself regarded the introduction of the energy quantum ~ω only as a formalmathematical trick and did not question the validity of Maxwell’s field theory. Only halfa century later, when field quantization was systematically derived, the coexistence of theconcepts of energy quanta and electromagnetic waves became understandable, see Sec. 1.5

2.2. QUANTUM MECHANICS AND FIRST QUANTIZATION 63

which means that coordinate and momentum (the same components) cannotbe measured simultaneously. The minimal uncertainty of such a simultaneousmeasurement is given by the Heisenberg relation

∆xi∆pi ≥~2, (2.24)

where the standard deviation (“uncertainty”) of an operator A is definied as

∆A =

√⟨(A− 〈A〉

)2⟩, (2.25)

and the average is computed in a given state |ψ〉, i.e. 〈A〉 = 〈ψ|A|ψ〉. Thegeneral formulation of quantum mechanics describes an arbitrary quantumsystem in terms of abstract states |ψ〉 that belong to a Hilbert space (Dirac’snotation), and operators act on these state returning another Hilbert spacestate, A|ψ〉 = |φ〉.

The central quantity of classical mechanics – the hamilton function – retainsits functional dependence on coordinate and momentum in quantum mechan-ics as well (correspondence principle) but becomes an operator depending onoperators, H(r,p) → H(r, p). The classical equations of motion – Hamil-ton’s equations or Newton’s equation (2.4) – are now replaced by a partialdifferential equation for the Hamilton operator, the Schrodinger equation

i~∂

∂t|ψ(t)〉 = H|ψ(t)〉. (2.26)

Stationary properties are governed by the stationary Schrodinger equation thatfollows from the ansatz5

|ψ(t)〉 = e−i~ Ht|ψ〉 ,

H|ψ〉 = E|ψ〉. (2.27)

The latter is an eigenvalue equation for the Hamilton operator with the eigen-functions |ψ〉 and corresponding eigenvalues E.

2.2.2 Probabilistic character of “First” quantization.Comparison to experiments

Experiments in quantum mechanics never directly yield the wave functionor the probability distribution. Individual (random) realizations of possibleconfigurations and their dynamics.

Examples:

5Here we assume a time-independent hamiltonian.


1. double slit experiment with electrons of Tonomura

2. photons on photo plate,

3. many-body dynamics in ultracold atom experiments in optical lattices

Possible configurations are particularly evident in the case of fermionic atoms(assuming spin s = 1/2, for simplicity) in optical lattices. Due to the Pauliprinciple each lattice site can be occupied only by zero, one or two atoms – inthe latter case they have to have different spin projections. If an initial config-uration of atoms is excited (e.g. by a confinement quench), a dynamical evo-lution will start. This will, of course, not be described by the time-dependentSchrodinger equation for the many-atom wave function! This equation onlydescribes the average dynamics that follow from averaging over the dynamicsthat start from many independent realizations. This has been very successfulbut, on the other hand, reproduces only part of the information. For exam-ple, it completely misses the fluctuations of the numbers of atoms around theaverage.6

“First” quantization is evident in the case of particle motion in a confiningpotential U(r), such as an oscillator potential: classical bounded motion trans-forms, in quantum mechanics, into a set of eigenstates |ψn〉 (that are localizedas well) that exist only for a sequence of discrete (quantized) energies En. Thisexample is discussed more in detail below.

2.3 The linear harmonic oscillator and the lad-

der operators

Let us now recall the simplest example of quantum mechanics: one particle ina one-dimensional harmonic potential U(x) = m

2ω2x2, i.e. in Eq. (2.1), N = 1

and the interaction potentials vanish. We will use this example to introducethe basic idea of “second quantization”. In writing the potential U(x) weswitched to the coordinate representation where states |ψn〉 are represented byfunctions of the coordinate, ψn(x). At the end we will return to the abstractnotation in terms of Dirac states.

6This section is not complete yet.

2.3. THE LADDER OPERATORS 65

2.3.1 One-dimensional harmonic oscillator

The stationary properties of the harmonic oscillator follow from the stationarySchrodinger equation (2.27) which now becomes, in coordinate representation

H(x, p)ψn(x) =

p2

2m+mω2

2x2

ψn(x) = Enψn(x), (2.28)

where x = x and p = ~iddx

. We may bring the Hamilton operator to a moresymmetric form by introducing the dimensionless coordinate ξ = x/x0 withthe length scale x0 = [~/mω]1/2, whereas energies will be measured in units of~ω. Then we can replace d

dx= 1

x0ddξ

and obtain

H

~ω=

1

2

− ∂2

∂ξ2+ ξ2

. (2.29)

This quadratic form can be rewritten in terms of a product of two first orderoperators a, a†, the “ladder operators”,

a =1√2

(∂

∂ξ+ ξ

), (2.30)

a† =1√2

(− ∂

∂ξ+ ξ

). (2.31)

Indeed, computing the product

N = a†a =1√2

(∂

∂ξ+ ξ

)1√2

(− ∂

∂ξ+ ξ

)(2.32)

=1

2

− ∂2

∂ξ2+ ξ2 − 1

,

the hamiltonian (2.29) can be written as

H

~ω= N +

1

2. (2.33)

It is obvious from (2.33) that N commutes with the hamiltonian,

[H, N ] = 0, (2.34)

and thus the two have common eigenstates. This way we have transformed thehamiltonian from a function of the two non-commuting hermitean operators


x and p into a function of the two operators a and a† which are also non-commuting7, but not hermitean, instead they are the hermitean conjugate ofeach other,

[a, a†] = 1, (2.35)

(a)† = a†, (2.36)

which is easily verified.The advantage of the ladder operators is that they allow for a straight-

forward computation of the energy spectrum of H, using only the properties(2.32) and (2.35), without need to solve the Schrodinger equation, i.e. avoid-ing explicit computation of the eigenfunctions ψn(ξ) 8. This allows us to re-turn to a representation-independent notation for the eigenstates, ψn → |n〉.The only thing we require is that these states are complete and orthonormal,1 =

∑n |n〉〈n| and 〈n|n′〉 = δn,n′ .

Now, acting with N on an eigenstate, using Eq. (2.33), we obtain

N |n〉 = a†a |n〉 =

(En~ω− 1

2

)|n〉 = n|n〉, ∀n, (2.37)

n =En~ω− 1

2,

where the last line relates the eigenvalues of N and n that correspond to thecommon eigenstate |n〉. Let us now introduce two new states that are createdby the action of the ladder operators,

a|n〉 = |n〉,a†|n〉 = |n〉,

where this action is easily computed. In fact, multiplying Eq. (2.37) from theleft by a, we obtain

aa† |n〉 =

(En~ω− 1

2

)|n〉.

Using the commutation relation (2.35) this expression becomes

a†a |n〉 =

(En~ω− 3

2

)|n〉 = (n− 1)|n〉,

7The appearance of the standard commutator indicates that these operators describebosonic excitations.

8We will use the previous notation ψ, which means that the normalization isx0∫dξ|ψ(ξ)|2=1


which means the state |n〉 is an eigenstate of N [and, therefore, of H] and hasan energy lower than |n〉 by ~ω whereas the eigenvalue of N is n = n−1. Thus,the action of the operator a is to switch from an eigen state with eigenvaluen to one with eigenvalue n − 1. Obviously, this is impossible for the groundstate, i.e. when a acts on |0〉, so we have to require

|0〉 = a|0〉 ≡ 0. (2.38)

When we use this result in Eq. (2.37) for n = 0, the l.h.s. is zero with the con-sequence that the term in parantheses must vanish. This immediately leads tothe well-known result for the ground state energy: E0 = ~ω/2, correspondingto the eigenvalue 0 of N .

From this we now obtain the energy spectrum of the excited states: actingwith a† from the left on Eq. (2.37) and using the commutation relation (2.35),we obtain

N |n〉 =

(En~ω− 1

2+ 1

)|n〉 = n|n〉.

Thus, n is again an eigenstate of N and H. Further, if the eigenstate |n〉has an energy En, cf. Eq. (2.37), then n has an energy En + ~ω, whereasthe associated eigenvalue of N is n = n + 1. Starting from the ground stateand acting repeatedly with a† we construct the whole spectrum, En, and mayexpress all eigenfunctions via ψ0:

En = ~ω(n+

1

2

), n = 0, 1, 2, . . . (2.39)

|n〉 = Cn(a†)n |0〉. (2.40)

Cn =1√n!, (2.41)

where the normalization constant Cn will be verified from the properties of a†

below. The above result shows that the eigenvalue of the operator N is justthe quantum number n of the eigenfunction |n〉. In other words, since |n〉 isobtained by applying a† to the ground state function n times or by “n-foldexcitation”, the operator N is the number operator counting the number ofexcitations (above the ground state). Therefore, if we are not interested in theanalytical details of the eigenstates we may use the operator N to count thenumber of excitations “contained” in the system. For this reason, the commonnotion for the operator a (a†) is “annihilation” (“creation”) operator of anexcitation. For an illustration, see Fig. 2.3.1.


Figure 2.1: Left: oscillator potential and energy spectrum. The action of theoperators a and a† is illustrated. Right: alternative interpretation: the op-erators transform between “many-particle” states containing different numberof elementary excitations.

From the eigenvalue problem of N , Eq. (2.37) we may also obtain theexplicit action of the two operators a and a†. Since the operator a transformsa state into one with quantum number n lower by 1 we have

a|n〉 =√n|n− 1〉, n = 0, 1, 2, . . . (2.42)

where the prefactor may be understood as an ansatz9. The correctness isproven by deriving, from Eq. (2.42), the action of a† and then verifying thatwe recover the eigenvalue problem of N , Eq. (2.37). The action of the creationoperator is readily obtained using the property (2.36):

a†|n〉 =∑n

|n〉〈n|a†|n〉 =∑n

|n〉 a[〈n|] |n〉

=∑n

|n〉√n 〈n− 1 |n〉 =

√n+ 1 |n+ 1〉. (2.43)

Inserting these explicit results for a and a† into Eq. (2.37), we immediatelyverify the consistency of the choice (2.42). Obviously the oscillator eigenstates|n〉 are no eigenstates of the creation and annihilation operators 10.

Problems:

1. Calculate the explicit form of the ground state wave function by usingEq. (2.38).

2. Show that the matrix elements of a† are given by⟨n+ 1|a†|n

⟩=√n+ 1,

where n = 0, 1, . . . , and are zero otherwise.

3. Show that the matrix elements of a are given by 〈n− 1|a|n〉 =√n, where

n = 0, 1, . . . , and are zero otherwise.

4. Proof relation (2.41).

9This expression is valid also for n = 0 where the prefactor assures that application of ato the ground state does not lead to a contradiction.

10A particular case are Glauber states (coherent states) that are a special superpositionof the oscillator states which are the eigenstate of the operator a.


2.3.2 Generalization to several uncoupled oscillators

The previous results are directly generalized to a three-dimensional harmonicoscillator with frequencies ωi, i = 1, 2, 3, which is described by the hamiltonian

H =3∑i=1

H(xi, pi), (2.44)

which is the sum of three one-dimensional hamiltonians (2.28) with the po-tential energy U(x1, x2, x3) = m

2(ω2

1x21 + ω2

2x22 + ω2

3x23). Since [pi, xk] ∼ δk,i

all three hamiltonians commute and have joint eigenfunction (product states).The problem reduces to a superposition of three independent one-dimensionaloscillators. Thus we may introduce ladder operators for each component inde-pendently as in the 1d case before,

ai =1√2

(∂

∂ξi+ ξi

), (2.45)

a†i =1√2

(− ∂

∂ξi+ ξi

), [ai, a

†k] = δi,k. (2.46)

Thus the hamiltonian and its eigenfunctions and eigenvalues can be written as

H =3∑i=1

~ωi(a†iai +

1

2

)ai|0〉 = 0, i = 1, 2, 3

ψn1,n2,n3 = |n1n2n3〉 =1√

n1!n2!n3!(a†1)n1(a†2)n2(a†3)n3|0〉 (2.47)

E =3∑i=1

~ωi(ni +

1

2

).

Here |0〉 ≡ |000〉 = |0〉|0〉|0〉 denotes the ground state and a general state|n1n2n3〉 = |n1〉|n2〉|n3〉 contains ni elementary excitations in direction i, cre-ated by ni times applying operator a†i to the ground state.

Finally, we may consider a more general situation of any number M ofcoupled independent linear oscillators and generalize all results by replacingthe dimension 3→M .


Figure 2.2: Illustration of the one-dimensional chain with nearest neighborinteraction. The chain is made infinite by connecting particle N + 1 withparticle 1 (periodic boundary conditions).

2.4 Generalization to interacting particles.

Normal modes

The previous examples of independent linear harmonic oscillators are of coursethe simplest situations which, however, are of limited interest. In most prob-lems of many-particle physics the interaction between the particles which wasneglected so far, is of crucial importance. We now discuss how to apply theformalism of the creation and annihilation operators to interacting systems.

2.4.1 One-dimensional chain and its normal modes

We consider the simplest case of an interacting many-particle system: N iden-tical classical particles arranged in a linear chain and interacting with theirleft and right neighbor via springs with constant k.11, see Fig. 2.2.

This is the simplest model of interacting particles because each particle isassumed to be fixed around a certain position xi in space around which it canperform oscillations with the displacement qi and the associated momentumpi.

12 Then the hamiltonian (2.1) becomes

H(p, q) =N∑j=1

p2j

2m+k

2(qj − qj+1)2

. (2.48)

Applying Hamilton’s equations we obtain the system of equations of motion(2.4)

mqj = k (qj+1 − 2qj + qj−1) , j = 1 . . . N (2.49)

which have to be supplemented with boundary and initial conditions. In thefollowing we consider a macroscopic system and will not be interested in ef-fects of the left and right boundary. This can be achieved by using “peri-odic” boundary conditions, i.e. periodically repeating the system according to

11Here we follow the discussion of Huang [Hua98].12Such “lattice” models are very popular in theoretical physics because they allow to study

many-body effects in the most simple way. Examples include the Ising model, the Andersonmodel or the Hubbard model of condensed matter physics.

2.4. INTERACTING PARTICLES 71

Figure 2.3: Dispersion of the normal modes, Eq. (2.52), of the 1d chain withperiodic boundary conditions.

qj+N(t) = qj(t) for all j [for solutions for the case of a finite system, see Prob-lem 5]. We start with looking for particular (real) solutions of the followingform13

qj(t) = ei(−ωt+jl) + c.c., (2.50)

which, inserted into the equation of motion, yield for any j

−mω2(qj + q∗j

)= k

(eil − 2 + e−il

) (qj + q∗j

), (2.51)

resulting in the following relation between ω and k (dispersion relation)14:

ω2(l) = ω20 sin2 l

2, ω2

0 = 4k

m. (2.52)

Here ω0 is just the eigenfrequency of a spring with constant k, and the prefactor2 arises from the fact that each particle interacts with two neighbors. Whilethe condition (2.52) is independent of the amplitudes q0

j , i.e. of the initialconditions, we still need to account for the boundary (periodicity) condition.Inserting it into the solution (2.50) gives the following condition for l, indepen-dently of ω: l → ln = n

N2π, where n = 0,±1,±2, · · · ± N

2. Thus there exists

a discrete spectrum of N frequencies of modes which can propagate along thechain (we have to exclude n = 0 since this corresponds to a time-independenttrivial constant displacement),

ω2n = 4

k

msin2 nπ

N, n = ±1,±2, · · · ± N

2. (2.53)

This spectrum is shown in Fig. 2.3. These N solutions are the complete set ofnormal modes of the system (2.48), corresponding to its N degrees of freedom.These are collective modes in which all particles participate, all oscillate withthe same frequency but with a well defined phase which depends on the particlenumber. These normal modes are waves running along the chain with a phasevelocity15 cn ∼ ωn/ln.

13In principle, we could use a prefactor q0j = q0 different from one, but by rescaling ofq it can always be eliminated. The key is that the amplitudes of all particles are strictlycoupled.

14We use the relation 1− cosx = 2 sin2 x2 .

15The actual phase velocity is ωn/kn, where the wave number kn = ln/a involves a lengthscale a which does not appear in the present discrete model.


Due to the completeness of the system of normal modes, we can expandany excitation of particle j and the corresponding momentum pj(t) = mqj(t)into a supersposition of normal mode contributions (n 6= 0)

qj(t) =1√N

N2∑

n=−N2

Q0n e

i(−ωnt+2π nNj) =

1√N

N2∑

n=−N2

e−iωntQn(j) (2.54)

pj(t) =1√N

N2∑

n=−N2

P 0n e

i(−ωnt+2π nNj) =

1√N

N2∑

n=−N2

e−iωntPn(j), (2.55)

where P 0n = −imωnQ0

n. Note that the complex conjugate contribution to moden is contained in the sum (term −n). Also, qj(t) and pj(t) are real functions.By computing the complex conjugate q∗j and equating the result to qj we obtainthe conditions (Q0

n)∗ = Q0−n and ω−n = −ω−n. Analogously we obtain for the

momenta (P 0n)∗ = P 0

−n. To make the notation more compact we introduced

the N -dimensional complex vectors ~Qn and ~Pn with the component j beingequal to Qn(j) = Q0

nei2πnj/N and Pn(j) = P 0

nei2πnj/N . One readily proofs16

that these vectors form an orthogonal system by computing the scalar product(see problem 5)

~Qn~Qm = Q0

nQ0m

N∑j=1

ei2πn+mN

j = NQ0nQ

0mδn,−m. (2.56)

Using this property it is now straightforward to compute the hamiltonfunction in normal mode representation. Consider first the momentum contri-bution,

N∑j=1

p2j(t) =

1

N

N2∑

n=−N2

N2∑

m=−N2

~Pn ~Pm e−i(ωn+ωm)t, (2.57)

where the sum over j has been absorbed in the scalar product. Using now theorthogonality condition (2.56) we immediately simplify

N∑j=1

p2j(t) =

∑n

|P 0n |2. (2.58)

16See problem 5


Analogously, we compute the potential energy

U =k

2

N∑j=1

[qj(t)− qj+1(t)]2 =k

2N

N2∑

n=−N2

N2∑

m=−N2

e−i(ωn+ωm)t

×N∑j=1

Q0nQ

0m

ei2π

nNj − ei2π

nN

(j+1)

ei2πmNj − ei2π

mN

(j+1).

The sum over j can again be simplified using the orthogonality condition (2.56),which allows to replace m by −n,

1

N

N∑j=1

Q0nQ

0m

ei2π

nNj − ei2π

nN

(j+1)

ei2πmNj − ei2π

mN

(j+1)

=

=(1− ei2π

nN

) (1− ei2π

mN

)~Qn~Qm =

= 2

[1− cos

2πn

N

]δn,−mQ

0nQ

0m = 4

ω2n

ω20

δn,−m|Q0n|2,

where we have used Eq. (2.52) and the relation 1−cosx = 2 sin2 x2. This yields

for the potential energy

U =k

2

∑n

mω2n

k

and for the total hamilton function

H(P,Q) =

N2∑

n=−N2

1

2m|P 0n |2 +

m

2ω2n|Q0

n|2. (2.59)

Problem 5: Proof the orthogonality relation (2.56).

2.4.2 Quantization of the 1d chain

We now quantize the interacting system (2.48) by replacing coordinates andmomenta of all particles by operators

(qi, pi) → (qi, pi) , i = 1, . . . N,

with q†i = qj, p†i = pi, [qi, pj] = i~δij. (2.60)


The Hamilton function (2.48) now becomes an operator of the same functionalform (correspondence principle),

H(p, q) =N∑j=1

p2j

2m+k

2(qj − qj+1)2

,

and we still use the periodic boundary conditions qN+i = qi. The normalmodes of the classical system remain normal modes in the quantum case aswell, only the amplitudes Q0

n and P 0n become operators

qj(t) =1√N

N2∑

n=−N2

e−iωntQn(j) (2.61)

pj(t) =1√N

N2∑

n=−N2

e−iωntPn(j), (2.62)

where Qn(j) = Q0n expi2πnj/N, Pn(j) = P 0

n expi2πnj/N and P 0n =

−imωnQ0n.

What remains is to impose the necessary restrictions on the operators Q0n

and P 0n such that they guarantee the properties (2.60). One readily verifies

that hermiticity of the operators is fulfilled if (Q0)†n = Q0−n, (P 0)†n = P 0

−n andω−n = −ωn. Next, consider the commutator of qi and pj and use the normalmode representations (2.61, 2.62),

[qk, pj] =1

N

∑n

∑m

[Q0n, P

0m]e−i(ωn+ωm)tei

2πN

(kn+jm). (2.63)

A sufficient condition for this expression to be equal i~δk,j is evidently [Q0n, P

0m] =

i~δn,−m which is verified separately for the cases k = j and k 6= j. In otherwords, the normal mode operators obey the commutation relation[

Q0n, (P

0m)†]

= i~δn,m, (2.64)

and the hamiltonian becomes, in normal mode representation,

H(P , Q) =

N2∑

n=−N2

1

2m|P 0n |2 +

m

2ω2n|Q0

n|2. (2.65)

This is a superposition of N independent linear harmonic oscillators with thefrequencies ωn given by Eq. (2.53). Applying the results for the superposition


of oscillators, Sec. 2.3.2, we readily can perform the second quantization bydefining dimensionless coordinates, ξn =

√mωn~ Qn, n = −N

2, . . . N

2, n 6= 0,

and introducing the creation and annihilation operators,

an =1√2

(∂

∂ξn+ ξn

), (2.66)

a†n =1√2

(− ∂

∂ξn+ ξn

), [an, a

†k] = δn,k. (2.67)


H =

N2∑

n=−N2

~ωn(a†nan +

1

2

)

an|0〉 = 0, n = −N2, . . .

N

2

ψm1,...mN = |m1 . . .mN〉 =1√

m1! . . .mN !

(a†−N

2

)m1

. . .(a†N

2

)mN|0〉

E =

N2∑

n=−N2

~ωn(mn +

1

2

).

Here |0〉 ≡ |0 . . . 0〉 = |0〉 . . . |0〉 [N factors] denotes the ground state and ageneral state |m−N/2 . . .mN/2〉 = |m−N/2〉 . . . |mN/2〉 contains mn elementaryexcitations of the normal mode n, created by mn times applying operator a†nto the ground state.

Problem 6: The commutation relation (2.64) which was derived to satisfythe commutation relations of coordinates and momenta is that of bosons. Thisresult was independent of whether the particles in the chain are fermions orbosons. Discuss this seeming contradiction.

2.4.3 Generalization to arbitrary interaction

Of course, the simple 1d chain is a model with a limited range of applicability.A real system of N interacting particles in 1d will be more difficult, at least bythree issues: first, the pair interaction potential V may have any form. Second,the interaction, in general, involves not only nearest neighbors, and third, theeffect of the full 3d geometry may be relevant. We, therefore, now return to


the general 3d system of N classical particles (2.1) with the total potentialenergy17

Utot(q) =N∑i=1

U(ri) +∑

1≤i<j≤N

V (ri − rj), (2.68)

leading to Newton’s equations

mri = − ∂

∂riUtot(q), i = 1, . . . N. (2.69)

Let us consider stationary solutions, where the time derivatives on the l.h.s.vanish. The system will then be in a stationary state “s′′ corresponding toa minimum q

(0)s of Utot of depth U

(0)s = Utot(q

(0)s ) [the classical ground state

corresponds to the deepest minimum]. In the case of weak excitations from the

minimum, q = q(0)s + ξ, with |ξ| << q

(0)s , the potential energy can be expanded

in a Taylor series18

Utot(q) = U (0)s +

∂

∂qUtot(q = q(0)

s )ξ +1

2ξTH(s)ξ + ... (2.70)

where all first derivatives are zero, and we limit ourselves to the second order(harmonic approximation). Here we introduced the 3N × 3N Hesse matrix

H(s)ij = ∂2

∂xi∂xjUtot(q = q

(0)s ), where xi, xj = x1, y1, . . . zN , and ξT is the trans-

posed vector (row) of ξ. Thus, for weak excitations, the potential energy

change ∆Utot = Utot(q) − U (0)s is reduced to an expression which is quadratic

in the displacements ξ, i.e. we are dealing with a system of coupled harmonicoscillators19

We can easily transform this to a system of uncoupled oscillators by di-agonalizing the Hesse matrix which can be achieved by solving the eigenvalueproblem (we take the mass out for dimensional reasons)

λnmQn = HQn, n = 1, . . . 3N. (2.71)

Since H is real, symmetric and positive definite20 the eigenvalues are realand positive corresponding to the normal mode frequencies ωn =

√λn. Fur-

thermore, as a result of the diagonalization, the 3N -dimensional eigenvectors

17Here we follow the discussion of Ref. [HKL+09]18Recall that q, q

(0)s and ξ are 3N -dimensional vectors in configuration space.

19Strictly speaking, from the 3N degrees of freedom, up to three [depending on the sym-metry of U ] may correspond to rotations of the whole system (around one of the threecoordinate axes, these are center of mass excitations which do not change the particle dis-tance), and the remaining are oscillations.

20q(0)s corresponds to a mininmum, so the local curvature of Utot is positive in all directions


form a complete orthogonal system Qn with the scalar product QnQm ≡∑3Ni=1Qn(i)Qm(i) ∼ δm,n which means that any excitation can be expanded

into a superposition of the eigenvectors (normal modes),

q(t) = q(0)s +

3N∑n=1

cn(t)Qn. (2.72)

The expansion coefficients cn(t) (scalar functions) are the normal coordinates.Their equation of motion is readily obtained by inserting a Taylor expansionof the gradient of Utot [analogous to (2.70)] into (2.69),

0 = mq +∂Utot∂q

= mq +H · ξ, (2.73)

and, using Eq. (2.72) for q and eliminating H with the help of (2.71),

0 = m3N∑n=1

cn(t) + cn(t)ω2

n

Qn. (2.74)

Due to the orthogonality of the Qn which are non-zero, the solution of thisequation implies that the terms in the parantheses vanish simultaneously forevery n, leading to an equation for a harmonic oscillator with the solution

cn(t) = An cosωnt+Bn, n = 1, . . . 3N, (2.75)

where the coefficients An and Bn depend on the initial conditions. Thus, thenormal coordinates behave as independent linear 1d harmonic oscillators.

In analogy to the coordinates, also the particle momenta, correspondingto some excitation q(t), can be expanded in terms of normal modes by usingp(t) = mq(t). Using the result for cn(t), Eq. (2.75), we have the followinggeneral expansion

q(t)− q(0)s =

3N∑n=1

An cosωnt+BnQn ≡3N∑n=1

Qn(t) (2.76)

p(t) =3N∑n=1

An sinωnt+BnPn ≡3N∑n=1

Pn(t), (2.77)

where the momentum amplitude vector is Pn = −mωnQn. Finally, we cantransform the Hamilton function into normal mode representation, using theharmonic expansion (2.70) of the potential energy

H(p, q) =p2

2m+ Utot(q) = U (0)

s +N∑i=1

p2i

2m+

1

2

∑i 6=j

ξT (i)H(s)ij ξ(j). (2.78)


Eliminating the Hesse matrix with the help of (2.71) and inserting the expan-sions (2.76) and (2.77) we obtain

H(p, q)− U (0)s =

3N∑n=1

3N∑n′=1

Pn(t)Pn′(t)

2m+m

2ω2nδn,n′Qn(t)Qn′(t)

=3N∑n=1

P 2n(t)

2m+m

2ω2nQ

2n(t)

≡ H(P,Q), (2.79)

where, in the last line, the orthogonality of the eigenvectors has been used.Thus we have succeeded to diagonalize the hamiltonian of the N -particle

system with arbitrary interaction. Assuming weak excitations from a station-ary state the hamiltonian can be written as a superposition of 3N normalmodes. This means, we can again apply the results from the case of uncou-pled harmonic oscillators, Sec. 2.3.2, and immediately perform the “first” and“second” quantization.

2.4.4 Quantization of the N-particle system

For the first quantization we have to replace the normal mode coordinates andmomenta by operators,

Qn(t) → Qn(t) = An cosωnt+BnQn

Pn(t) → Pn(t) = An sinωnt+BnPn, (2.80)

leaving the time-dependence of the classical system unchanged. Further wehave to make sure that the standard commutation relations are fulfilled, i.e.[Qn, Pm] = i~δn,m. This should follow from the commutation relations ofthe original particle coordinates and momenta, [xiα, pjβ] = i~δi,jδα,β, whereα, β = 1, 2, 3 and i, j = 1, . . . N , see Problem 7. Then, the Hamilton operatorbecomes, in normal mode representation

H(P , Q) =3N∑n=1

P 2n(t)

2m+m

2ω2nQ

2n(t)

, (2.81)

which allows us to directly introduce the creation and annihilation operatorsby introducing ξn =

√mωn~ Qn, n = 1, . . . 3N)

an =1√2

(∂

∂ξn+ ξn

), (2.82)

a†n =1√2

(− ∂

∂ξn+ ξn

), [an, a

†k] = δn,k. (2.83)



H =3N∑n=1

~ωn(a†nan +

1

2

)an|0〉 = 0, n = 1, . . . 3N

ψn1,...n3N= |n1 . . . n3N〉 =

1√n1! . . . n3N !

(a†1)n1 . . . (a†3N)n3N |0〉

E =3N∑n=1

~ωn(nn +

1

2

).

Here |0〉 ≡ |0 . . . 0〉 = |0〉 . . . |0〉 [3N factors] denotes the ground state and ageneral state |n1 . . . n3N〉 = |n1〉 . . . |n3N〉 contains nn elementary excitations ofthe normal mode n, created by nn times applying operator a†n to the groundstate.

In summary, in finding the normal modes of the interacting N -particle sys-tem the description is reduced to a superposition of independent contributionsfrom 3N degrees of freedom. Depending on the system dimensionality, theseinclude (for a three-dimsional system) 3 translations of the center of mass and 3rotations of the system as a whole around the coordinate axes. The remainingnormal modes correspond to excitations where the particle distances change.Due to the stability of the stationary state with respect to weak excitations,these relative excitations are harmonic oscillations which have been quantized.In other words, we have 3N−6 phonon modes associated with the correspond-ing creation and annihilation operators and energy quanta. The frequencies ofthe modes are determined by the local curvature of the total potential energy(the diagonal elements of the Hesse matrix).

Problem 7: Proof the commutation relation [Qn, Pm] = i~δn,m.Problem 8: Apply the concept of the eigenvalue problem of the Hesse matrixto the solution of the normal modes of the 1d chain. Rederive the normal moderepresentation of the hamiltonian and check if the time dependencies vanish.


2.5 Continuous systems

2.5.1 Continuum limit of 1d chain

So far we have considered discrete systems containing N point particles. If thenumber of particles grows and their spacing becomes small we will eventuallyreach a continuous system – the 1d chain becomes a 1d string. We start withassigning particle i a coordinate xj = ja where j = 0, . . . N , a is the constantinterparticle distance and the total length of the system is l = Na, see Fig.

We again consider a macroscopic system which is now periodically repeatedafter length l, i.e. points x = 0 and x = l are identical21. In the discretesystem we have an equally spaced distribution of masses m of point particleswith a linear mass density ρ = m/a. The interaction between the massesis characterized by an elastic tension σ = κa where we relabeled the springconstant by κ. The continuum limit is now performed by simulataneouslyincreasing the particle number and reducing a but requiring that the densityand the tension remain unchanged,

a,m −→ 0

N, κ −→ ∞l, ρ, σ = const.

We now consider the central quantity, the displacement of the individual par-ticles qi(t) which now transforms into a continuous displacement field q(x, t).Further, with the continuum limit, differences become derivatives and the sumover the particles is replaced by an integral according to

qj(t) −→ q(x, t)

qj+1 − qj −→ a∂q

∂x∑j

−→ 1

a

∫ l

0

dx.

Instead of the Hamilton function (2.48) we now consider the Lagrangefunction which is the difference of kinetic and potential energy, L = T − V ,

21Thus we have formally introduced N + 1 lattice points but only N are different.

2.5. CONTINUOUS SYSTEMS 81

Figure 2.4: Illustration of the minimal action principle: the physical equa-tion of motion corresponds to the tractory q(x, t) which minimizes the action,Eq. (2.86) at fixed initial and final points (ti, 0) and (tf , l).

which in the continuum limit transforms to

L(q, q) =N∑j=1

m2

(qj)2 − κ

2(qj − qj+1)2

−→ 1

2

∫ l

0

dx

ρ

(∂q(x, t)

∂t

)2

− σ(∂q(x, t)

∂x

)2

(2.84)

The advantage of using the Lagrange function which now is a functional of thedisplacement field, L = L[q(x, t)], is that there exists a very general method offinding the corresponding equations of motion – the minimal action principle.

2.5.2 Equation of motion of the 1d string

We now define the one-dimensional Lagrange density L

L =

∫ l

0

dxL[q(x, t), q′(x, t)], (2.85)

where Eq. (2.84) shows that Lagrange density of the spring depends only ontwo fields – the time derivative q and space derivative q′ of the displacementfield. The action is defined as the time integral of the Lagrange functionbetween a fixed initial time ti and final time tf

S =

∫ tf

ti

dtL =

∫ tf

ti

dt

∫ l

0

dxL[q(x, t), q′(x, t)]. (2.86)

The equation of motion of the 1d string follows from minimizing the actionwith respect to the independent variables of L [this “minimal action principle”has been discussed in detail in Chapter 1, Sec. 1.1], for illustration, see Fig. 2.4,

0 = δS =

∫ tf

ti

dt

∫ l

0

dx

δLδqδq +

δLδq′

δq′

=

∫ tf

ti

dt

∫ l

0

dx ρq δq − σq′ δq′ . (2.87)


We now change the order of differentiation and variation, δq = ∂∂tδq and δq′ =

∂∂xδq and perform partial integrations with respect to t in the first term and x

in the second term of (2.87)

0 = −∫ tf

ti

dt

∫ l

0

dx ρq − σq′′ δq, (2.88)

where the boundary values vanish because one requires that the variationδq(x, t) are zero at the border of the integration region, δq(0, t) = δq(l, t) ≡ 0.Since this equation has to be fulfilled for any fluctuation δq(x, t) the term inthe parantheses has to vanish which yields the equation of motion of the 1dstring

∂2q(x, t)

∂t2− c2∂

2q(x, t)

∂x2= 0, with c =

√σ

ρ= a

√κ

m. (2.89)

This is a linear wave equation for the displacement field, and we introducedthe phase velocity, i.e. the sound speed c. The solution of this equation canbe written as

q(x, t) = q0ei(kx−ωt) + c.c., (2.90)

which, inserted into Eq. (2.89), yields the dispersion relation

ω(k) = c · k, (2.91)

i.e., the displacement of the string performs a wave motion with linear disper-sion – we observe an acoustic wave where the wave number k is continuous.

It is now interesting to compare this result with the behavior of the originaldiscrete N−particle system. There the oscillation frequencies ωn were givenby Eq. (2.53), and the wave numbers are discrete22 kn = 2πn/Na with n =±1, · · · ±N/2, and the maximum wave number is kmax = π/a. Obviously, thediscrete system does not have a linear dispersion, but we may consider thesmall k limit and expand the sin to first order:

ω2n ≈ 4

κ

m

(πnN

)2

= 4c2

a2

(akn2

)2

= ckn, (2.92)

i.e. for small k the discrete system has exactly the same dispersion as the con-tinuous system. The comparison with the discrete system also gives a hint atthe existence of an upper limit for the wave number in the continuous system.In fact, k cannot be larger than π/amin where amin is the minimal distanceof neighboring particles in the “continuous medium”. The two dispersions areshown in Fig. 2.5.

22The wave number follows from the mode numbers ln by dividing by a

2.5. CONTINUOUS SYSTEMS 83

Figure 2.5: Dispersion of the normal modes of the discrete 1D chain and ofthe associated continuous system – the 1D string. The dispersions agree forsmall k up to a kmax=π/a.

One may, of course, ask whether a continuum model has its own right of ex-istence, without being a limit of a discrete system. In other words, this wouldcorrespond to a system with an infinite particle number and, correspondingly,an infinite number M of normal modes. While we have not yet discussed howto quantize continuum systems it is immediately clear that there should beproblems if the number of modes is unlimited. In fact, the total energy con-tains a zero point contribution for each mode which, with M going to infinity,will diverge. This problem does not occur for any realistic system because theparticle number is always finite (though, possibly large). But a pure contin-uum model will be only physically relevant if such divergencies are avoided.The solution is found by co-called “renormalization” procedures where a max-imum k-value (a cut-off) is introduced. This maybe not easy to derive for anyspecific field theory, however, based on the information from discrete systems,such a cut-off can always be motivated by choosing a physically relevant par-ticle number, as we have seen in this chapter.

Thus we have succeeded to perform the continuum limit of the 1d chain– the 1d string and derive and solve its equation of motion. The solution isa continuum of acoustic waves which are the normal modes of the mediumwhich replace the discrete normal modes of the linear chain. Now the ques-tion remains how to perform a quantization of the continuous system, how tointroduce creation and annihilation operators. To this end we have to developa more general formalism which is called canonical quatization and which willbe discussed in the next chapter.


2.6 Solutions of Problems

1. A simple equation for ψ0 is readily obtained by inserting the definitionof a into Eq. (2.38),

0 = ψ′0(ξ) + ξψ0(ξ), (2.93)

with the solution ψ0(ξ) = C0e−ξ2/2, where C0 follows from the normal-

ization x0

∫∞−∞ dξψ

20 = 1, with the result C0 = (π1/2/x0)−1/2, where the

phase is arbitrary and chosen to be zero.

2. Proof: Using 〈ψ|a† = a|ψ〉 and Eq. (2.41), direct computation yields⟨ψn+1|a†|ψn

⟩=

1√n!(n+ 1)!

⟨ψ0|an+1a†(a†)n|ψ0

⟩.

The final result√n+ 1 is obtained by induction, starting with n = 0.

3. This problem reduces to the previous one by applying hermitean conju-gation

〈ψn−1|a|ψn〉 =⟨ψn|a†|ψn−1

⟩∗=√n

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[qcd] Quantum Chromodynamics.

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