Classical Field Theory - Eduardo Fradkin...

2

Classical Field Theory

In what follows we will consider rather general field theories. The only guid-ing principles that we will use in constructing these theories are (a) symme-tries and (b) a generalized Least Action Principle.

2.1 Relativistic Invariance

Before we saw three examples of relativistic wave equations. They are theMaxwell equations for classical electromagnetism, the Klein-Gordon equa-tion and the Dirac equation. Maxwell’s equations govern the dynamics ofa vector field, the vector potentials Aµ(x) = (A0(x),A(x)), whereas theKlein-Gordon equation describes excitations of a scalar field φ(x) and theDirac equation governs the behavior of the four-component spinor fieldψα(x)(α = 0, 1, 2, 3). Each one of these fields transforms in a very definiteway under the group of Lorentz transformations, the Lorentz group. TheLorentz group is defined as a group of linear transformations Λ of Minkowskispace-time M onto itself Λ : M !→ M such that the new coordinates arerelated to the old ones by a linear (Lorentz) transformation

x′µ = Λµνx

ν (2.1)

The space-time components of Λ0i are the Lorentz boosts which relate

inertial reference frames moving at relative velocity v from each other. Thus,

12 Classical Field Theory

Lorentz boosts along the x1-axis have the familiar form

x0′ =x0 + vx1/c√1− v2/c2

x1′ =x1 + vx0/c√1− v2/c2

x2′ = x2

x3′ = x3

(2.2)

where x0 = ct, x1 = x, x2 = y and x3 = z (note: these are components, notpowers!). If we use the notation γ = (1− v2/c2)−1/2 ≡ coshα, we can writethe Lorentz boost as a matrix:

⎛

⎜⎜⎝

x0′

x1′

x2′

x3′

⎞

⎟⎟⎠ =

⎛

⎜⎜⎝

coshα sinhα 0 0sinhα coshα 0 0

0 0 1 00 0 0 1

⎞

⎟⎟⎠

⎛

⎜⎜⎝

x0

x1

x2

x3

⎞

⎟⎟⎠ (2.3)

The space components of Λij are conventional rotations R of three-dimensional

Euclidean space.Infinitesimal Lorentz transformations are generated by the hermitian op-

erators

Lµν = i(xµ∂ν − xν∂µ) (2.4)

where ∂µ = ∂∂xµ and µ, ν = 0, 1, 2, 3. The infinitesimal generators Lµν satisfy

the algebra

[Lµν , Lρσ] = igνρLµσ − igµρLνσ − igνσLµρ + igµσLνρ (2.5)

where gµν is the metric tensor for flat Minkowski space-time (see below).This is the algebra of the group SO(3, 1). Actually any operator of the form

Mµν = Lµν + Sµν (2.6)

where Sµν are 4 × 4 matrices satisfying the algebra of Eq.(2.5) are alsogenerators of SO(3, 1). Below we will discuss explicit examples.

Lorentz transformations form a group, since (a) the product of two Lorentztransformations is a Lorentz transformation, (b) there exists an identitytransformation, and (c) Lorentz transformations are invertible. Notice, how-ever, that in general two transformations do not commute with each other.Hence, the Lorentz group is non-Abelian.

2.1 Relativistic Invariance 13

The Lorentz group has the defining property of leaving invariant the rel-ativistic interval

x2 ≡ x20 − x2 = c2t2 − x2 (2.7)

The group of Euclidean rotations leave invariant the Euclidean distance x2

and it is a subgroup of the Lorentz group. The rotation group is denoted bySO(3), and the Lorentz group is denoted by SO(3, 1). This notation makesmanifest the fact that the signature of the metric has one + sign and three− signs.

We will adopt the following conventions and definitions:

1 Metric Tensor: We will use the standard (“Bjorken and Drell”) metric forMinkowski space-time in which the metric tensor gµν is

gµν = gµν =

⎛

⎜⎜⎝

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

⎞

⎟⎟⎠ (2.8)

With this notation the infinitesimal relativistic interval is

ds2 = dxµdxµ = gµνdxµdxν = dx20 − dx2 = c2dt2 − dx2 (2.9)

2 4-vectors:

1 xµ is a contravariant 4-vector, xµ = (ct,x)2 xµ is a covariant 4-vector xµ = (ct,−x)3 Covariant and contravariant vectors (and tensors) are related through

the metric tensor gµν

Aµ = gµνAν (2.10)

4 x is a vector in R3

5 pµ =(Ec ,p

)is the energy-momentum 4-vector. Hence, pµpµ = E2

c2 − p2

is a Lorentz scalar.

3 Scalar Product:

p · q = pµqµ = p0q0 − p · q ≡ pµqνg

µν (2.11)

4 Gradients: ∂µ ≡ ∂∂xµ and ∂µ ≡ ∂

∂xµ. We define the D’Alambertian ∂2

∂2 ≡ ∂µ∂µ ≡1

c2∂2t −∇2 (2.12)

which is a Lorentz scalar. From now on we will use units of time [T ] andlength [L] such that ! = c = 1. Thus, [T ] = [L] and we will use units likecentimeters (or any other unit of length).


5 Interval: The interval in Minkowski space is x2,

x2 = xµxµ = x20 − x2 (2.13)

Time-like intervals have x2 > 0 while space-like intervals have x2 < 0.

!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]_abcdefghijklmnopqrstuvwxyz{|}~

time-like

space-like

x0

x1

x2

> 0

x2

< 0

Figure 2.1 The Minkowski space-time and its light cone. Events at a rel-ativistic interval with x2 = x2

0 − x2 > 0 are time-like (and are causallyconnected with the origin), while events with x2 = x2

0 − x2 < 0 are space-like and are not causally connected with the origin.

Since a field is a function (or mapping) of Minkowski space onto some other(properly chosen) space, it is natural to require that the fields should havesimple transformation properties under Lorentz transformations. For exam-ple, the vector potential Aµ(x) transforms like 4-vector under Lorentz trans-formations, i.e., if x′µ = Λµ

νxν , then A′µ(x′) = ΛµνAν(x). In other words, Aµ

transforms like xµ. Thus, it is a vector. All vector fields have this property.A scalar field Φ(x), on the other hand, remains invariant under Lorentztransformations,

Φ′(x′) = Φ(x) (2.14)

A 4-spinor ψα(x) transforms under Lorentz transformations. Namely, thereexists an induced 4× 4 linear transformation matrix S(Λ) such that

S(Λ−1) = S−1(Λ) (2.15)

2.2 The Lagrangian, the Action and and the Least Action Principle 15

and

Ψ′(Λx) = S(Λ)Ψ(x) (2.16)

Below we will give an explicit expression for S(Λ).

2.2 The Lagrangian, the Action and and the Least ActionPrinciple

The evolution of any dynamical system is determined by its Lagrangian. Inthe Classical Mechanics of systems of particles described by the generalizedcoordinates q, the Lagrangian L is a differentiable function of the coordinatesq and their time derivatives. L must be differentiable since, otherwise, theequations of motion would not be local in time, i.e. could not be writtenin terms of differential equations. An argument a-la Landau-Lifshitz enablesus to “derive” the Lagrangian. For example, for a particle in free space, thehomogeneity, uniformity and isotropy of space and time require that L beonly a function of the absolute value of the velocity |v|. Since |v| is not adifferentiable function of v, the Lagrangian must be a function of v2. Thus,L = L(v2). In principle there is no reason to assume that L cannot be afunction of the acceleration a (or rather a2) or of its higher derivatives.Experiment tells us that in Classical Mechanics it is sufficient to specify theinitial position x(0) of a particle and its initial velocity v(0) in order todetermine the time evolution of the system. Thus we have to choose

L(v2) = const +1

2mv2 (2.17)

The additive constant is irrelevant in classical physics. Naturally, the coef-ficient of v2 is just one-half of the inertial mass.

However, in Special Relativity, the natural invariant quantity to consideris not the Lagrangian but the action S. For a free particle the relativisticinvariant (i.e., Lorentz invariant ) action must involve the invariant interval,

the proper length ds = c√

1− v2

c2 dt. Hence one writes the action for arelativistic massive particle as

S = −mc

∫ sf

si

ds = −mc2∫ tf

ti

dt

√1− v2

c2(2.18)

The relativistic Lagrangian then is

L = −mc2√

1− v2

c2(2.19)


As a power series expansion, it contains all powers of v2/c2. It is elementaryto see that the canonical momentum p (as expected) is

p =∂L

∂v=

mv√1− v2

c2

(2.20)

from which it follows that the Hamiltonian (or energy) is given by

H =mc2√1− v2

c2

=√

p2c2 +m2c4, (2.21)

as it should be.

t

q

i

Figure 2.2 The Least Action Principle: the dark curve is the classical tra-jectory and extremizes the classical action. The curve with a broken tracerepresents a variation.

Once the Lagrangian is found, the classical equations of motion are de-termined by the Least Action Principle. Thus, we construct the action S

S =

∫dt L

(q,∂q

∂t

)(2.22)

and demand that the physical trajectories q(t) leave the action S station-ary,i.e., δS = 0. The variation of S is

δS =

∫ tf

ti

dt

(∂L

∂qδq +

∂L

∂ dqdt

δdq

dt

)(2.23)

2.3 Scalar Field Theory 17

Integrating by parts, we get

δS =

∫ tf

ti

dtd

dt

(∂L

∂ dqdt

δq

)+

∫ tf

ti

dt δq

[∂L

∂q− d

dt

(∂L

∂ dqdt

)](2.24)

Hence, we get

δS =∂L

∂ dqdt

δq∣∣∣tf

ti+

∫ tf

ti

dt δq

[∂L

∂q− d

dt

(∂L

∂ dqdt

)](2.25)

If we assume that the variation δq is an arbitrary function of time thatvanishes at the initial and final times ti and tf , i.e. δq(ti) = δq(tf ) = 0, wefind that δS = 0 if and only if the integrand of Eq.(2.25) vanishes identically.Thus,

∂L

∂q− d

dt

(∂L

∂ dqdt

)= 0 (2.26)

These are the equations of motion or Newton’s equations. In general theequations that determine the trajectories that leave the action stationaryare called the Euler-Lagrange equations.

2.3 Scalar Field Theory

For the case of a field theory, we can proceed very much in the same way.Let us consider first the case of a scalar field Φ(x). The action S must beinvariant under Lorentz transformations. Since we want to construct localtheories it is natural to assume that S is given in terms of a Lagrangiandensity L

S =

∫d4x L (2.27)

Since the volume element of Minkowski space d4x is invariant under Lorentztransformations, the action S is invariant if L is a local, differentiable func-tion of Lorentz invariants that can be constructed out of the field Φ(x). Sim-ple invariants are Φ(x) itself and all of its powers. The gradient ∂µΦ ≡ ∂Φ

∂xµ

is not an invariant but the D’alambertian ∂2Φ is. ∂µΦ∂µΦ is also an invari-ant under a change of the sign of Φ. So, we can write the following simpleexpression for L:

L =1

2∂µΦ∂

µΦ− V (Φ) (2.28)


where V (Φ) is some potential, which we can assume is a polynomial functionof the field Φ. Let us consider the simple choice

V (Φ) =1

2m2Φ2 (2.29)

where m = mc/!. Thus,

L =1

2∂µΦ∂

µΦ− 1

2m2Φ2 (2.30)

This is the Lagrangian density for a free scalar field. We will discuss lateron in what sense this field is “free”. Notice, in passing, that we could haveadded a term like ∂2Φ. However this term, in addition of being odd underΦ → −Φ, is a total divergence and, as such, it has an effect only on theboundary conditions but it does not affect the equations of motion. In whatfollows will will not consider surface terms.

The Least Action Principle requires that S be stationary under arbitraryvariations of the field Φ and of its derivatives ∂µΦ. Thus, we get

δS =

∫d4x

[δLδΦ

δΦ +δLδ∂µΦ

δ∂µΦ

](2.31)

Notice that since L is a functional of Φ, we have to use functional derivatives,i.e., partial derivatives at each point of space-time. Upon integrating byparts, we get

δS =

∫d4x ∂µ

(δLδ∂µΦ

δΦ

)+

∫d4x δΦ

[δLδΦ− ∂µ

(δLδ∂µΦ

)](2.32)

Instead of considering initial and final conditions, we now have to imaginethat the field Φ is contained inside some very large box of space-time. Theterm with the total divergence yields a surface contribution. We will considerfield configurations such that δΦ = 0 on that surface. Thus, the Euler-Lagrange equations are

δLδΦ− ∂µ

(δLδ∂µΦ

)= 0 (2.33)

More explicitly, we find

δLδΦ

= −∂V∂Φ

(2.34)

and, since

∂Lδ∂µΦ

= ∂µΦ, (2.35)

2.3 Scalar Field Theory 19

Then

∂µδLδ∂µΦ

= ∂µ∂µΦ ≡ ∂2Φ (2.36)

By direct substitution we get the equation of motion (or field equation)

∂2Φ+∂V

∂Φ= 0 (2.37)

For the choice

V (Φ) =m2

2Φ2,

∂V

∂Φ= m2Φ (2.38)

the field equation is(∂2 + m2

)Φ = 0 (2.39)

where ∂2 = 1c2

∂2

∂t2 −▽2. Thus, we find that the equation of motion for the

free massive scalar field Φ is

1

c2∂2Φ

∂t2−▽2Φ+ m2Φ = 0 (2.40)

This is precisely the Klein-Gordon equation if the constant m is identifiedwith mc

!. Indeed, the plane-wave solutions of these equations are

Φ = Φ0ei(p0x0−p·x)/! (2.41)

where p0 and p are related through the dispersion law

p20 = p2c2 +m2c4 (2.42)

which means that, for each momentum p, there are two solutions, one withpositive frequency and one with negative frequency. We will see below that,in the quantized theory, the energy of the excitation is indeed equal to|p0|. Notice that 1

m = !

mc has units of length and is equal to the Comptonwavelength for a particle of mass m. From now on (unless it stated thecontrary) I will use units in which ! = c = 1 in which m = m.

The Hamiltonian for a classical field is found by a straightforward gener-alization of the Hamiltonian of a classical particle. Namely, one defines thecanonical momentum Π(x), conjugate to the field (the “coordinate ”) Φ(x),

Π(x) =δL

δΦ(x), with Φ(x) =

∂Φ

∂x0(2.43)

In Classical Mechanics the Hamiltonian H and the Lagrangian L are relatedby

H = pq − L (2.44)


where q is the coordinate and p the canonical momentum conjugate to q.Thus, for a scalar field theory the Hamiltonian density H is

H = Π(x)Φ(x)− L

=1

2Π2(x) +

1

2(∇Φ(x))2 + V (Φ(x))

(2.45)

Hence, for a free massive scalar field the Hamiltonian is

H =1

2Π2(x) +

1

2(∇Φ(x))2 +

m2

2Φ2(x) ≥ 0 (2.46)

which is always a positive definite quantity. Thus, the energy of a planewave solution of a massive scalar field theory (i.e., a solution of the Klein-Gordon equation) is always positive, no matter the sign of the frequency.In fact, the lowest energy state is simply Φ = constant. A solution made oflinear superpositions of plane waves (i.e., a wave packet) has positive energy.Therefore, in field theory, the energy is always positive. We will see that, inthe quantized theory, the negative frequency solutions are identified withantiparticle states and their existence do not signal a possible instability ofthe theory.

2.4 Classical Field Theory in the Canonical (Hamiltonian)Formalism

In Classical Mechanics it is often convenient to use the canonical formulationin terms of a Hamiltonian instead of the Lagrangian approach. For the caseof a system of particles, the canonical formalism proceeds as follows. Givena Lagrangian L(q, q), a canonical momentum p is defined to be

∂L

∂q= p (2.47)

The classical Hamiltonian H(p, q) is defined by the Legendre transformation

H(p, q) = pq − L(q, q) (2.48)

If the Lagrangian L is quadratic in the velocities q and separable, e.g.

L =1

2mq2 − V (q) (2.49)

then, H(pq) is simply given by

H(p, q) = pq −(mq2

2− V (q)

)=

p2

2m+ V (q) (2.50)

2.4 Classical Field Theory in the Canonical (Hamiltonian) Formalism 21

where

p =∂L

∂q= mq (2.51)

The (conserved) quantity H is then identified with the total energy of thesystem.

In this language, the Least Action Principle becomes

δS = δ

∫L dt = δ

∫[pq −H(p, q)] dt = 0 (2.52)

Hence ∫dt

(δp q + p δq − δp ∂H

∂p− δq ∂H

∂q

)= 0 (2.53)

Upon an integration by parts we get∫

dt

[δp

(q − ∂H

∂p

)+ δq

(−∂H∂q− p

)]= 0 (2.54)

which can only be satisfied for arbitrary variations δq(t) and δp(t) if

q =∂H

∂pp = −∂H

∂q(2.55)

These are Hamilton’s equations.Let us introduce the Poisson Bracket {A,B}qp of two functions A and B

of q and p by

{A,B}qp ≡∂A

∂q

∂B

∂p− ∂A

∂p

∂B

∂q(2.56)

Let F (q, p, t) be some differentiable function of q, p and t. Then the totaltime variation of F is

dF

dt=∂F

∂t+∂F

∂q

dq

dt+∂F

∂p

dp

dt(2.57)

Using Hamilton’s Equations we get the result

dF

dt=∂F

∂t+∂F

∂q

∂H

∂p− ∂F

∂p

∂H

∂q(2.58)

or, in terms of Poisson Brackets,

dF

dt=∂F

∂t+ {F,H, }qp (2.59)

In particular,

dq

dt=∂H

∂p=∂q

∂q

∂H

∂p− ∂q

∂p

∂H

∂q= {q,H}qp (2.60)


since∂q

∂p= 0 and

∂q

∂q= 1 (2.61)

Also the total rate of change of the canonical momentum p is

dp

dt=∂p

∂q

∂H

∂p− ∂p

∂p

∂H

∂q≡ −∂H

∂q(2.62)

since ∂p∂q = 0 and ∂p

∂p = 1. Thus,

dp

dt= {p,H}qp (2.63)

Notice that, for an isolated system, H is time-independent. So,

∂H

∂t= 0 (2.64)

anddH

dt=∂H

∂t+ {H,H}qp = 0 (2.65)

since

{H,H}qp = 0 (2.66)

Therefore, H can be regarded as the generator of infinitesimal time transla-tions. Since it is conserved for an isolated system, for which ∂H

∂t = 0, we canindeed identify H with the total energy. In passing, let us also notice thatthe above definition of the Poisson Bracket implies that q and p satisfy

{q, p}qp = 1 (2.67)

This relation is fundamental for the quantization of these systems.Much of this formulation can be generalized to the case of fields. Let us

first discuss the canonical formalism for the case of a scalar field Φ withLagrangian density L(Φ, ∂µ,Φ). We will choose Φ(x) to be the (infinite) setof canonical coordinates. The canonical momentum Π(x) is defined by

Π(x) =δL

δ∂0Φ(x)(2.68)

If the Lagrangian is quadratic in ∂µΦ, the canonical momentum Π(x) issimply given by

Π(x) = ∂0Φ(x) ≡ Φ(x) (2.69)

The Hamiltonian density H(Φ,Π) is a local function of Φ(x) and Π(x) givenby

H(Φ,Π) = Π(x) ∂0Φ(x)− L(Φ, ∂0Φ) (2.70)

2.5 Field Theory of the Dirac Equation 23

If the Lagrangian density L has the simple form

L =1

2(∂µΦ)

2 − V (Φ) (2.71)

then, the Hamiltonian density H(Φ,Π) is

H = ΠΦ− L(Φ, Φ, ∂jΦ) ≡1

2Π2(x) +

1

2(∇Φ(x))2 + V (Φ(x)) (2.72)

The canonical field Φ(x) and the canonical momentum Π(x) satisfy theequal-time Poisson Bracket (PB) relations

{Φ(x, x0),Π(y, x0)}PB = δ(x− y) (2.73)

where δ(x) is the Dirac δ-function and {A,B}PB is now

{A,B}PB =

∫d3x

[δA

δΦ(x, x0)

δB

δΠ(x, x0)− δA

δΠ(x, x0)

δB

δΦ(x, x0)

](2.74)

for any two functionals A and B of Φ(x) and Π(x). This approach can beextended to theories other than that of a scalar field without too muchdifficulty. We will come back to these issues when we consider the problemof quantization. Finally we should note that while Lorentz invariance isapparent in the Lagrangian formulation, it is not so in the Hamiltonianformulation of a classical field.

2.5 Field Theory of the Dirac Equation

We now turn to the problem of a field theory for spinors. We will discussthe theory of spinors as a classical fuel theory. We will find that this theoryis not consistent unless it is properly quantized as a quantum field theory ofspinors. We will return to this in a later chapter.

Let us rewrite the Dirac equation

i!∂Ψ

∂t=

!c

iα ·∇Ψ+ βmc2 Ψ ≡ HDiracΨ (2.75)

in a manner in which relativistic covariance is apparent. The operator HDirac

is the Dirac Hamiltonian.We first recall that the 4 × 4 hermitian matrices α and β should satisfy

the algebra

{αi,αj} = 2δij I, {αi,β} = 0, α2i = β2 = I (2.76)

where I is the 4× 4 identity matrix.


A simple representation of this algebra are the 2×2 block (Dirac) matrices

αi =

(0 σi

σi 0

)β =

(I 00 −I

)(2.77)

where the σi matrices are the three 2× 2 Pauli matrices

σ1 =

(0 11 0

)σ2 =

(0 −ii 0

)σ3 =

(1 00 −1

)(2.78)

and I is the 2 × 2 identity matrix. This is the Dirac representation of theDirac algebra.

It is now convenient to introduce the Dirac γ-matrices, are defined by thefollowing relations:

γ0 = β γi = βαi (2.79)

The Dirac gamma matrices γµ have the block form

γ0 = β =

(I 00 −I

), γi =

(0 σi

−σi 0

)(2.80)

an obey the Dirac algebra

{γµ, γν} = 2gµν I (2.81)

where I is the 4× 4 identity matrix.In terms of the γ-matrices, the Dirac equation takes the much simpler

form

(iγµ∂µ −mc

!)Ψ = 0 (2.82)

where Ψ is a 4-spinor. It is also customary to introduce the notation (knownas Feynman’s slash)

/a ≡ aµγµ (2.83)

Using Feynman’s slash, we can write the Dirac equation in the form

(i/∂ − mc

!)Ψ = 0 (2.84)

From now on I will use units in which ! = c = 1. In these units energy hasunits of (length)−1 and time has units of length.

Notice that, if Ψ satisfies the Dirac equation, then it also satisfies

(i/∂ +m)(i/∂ −m)Ψ = 0 (2.85)


Also, we find

/∂ · /∂ = ∂µ∂νγµγν = ∂µ∂ν

(1

2{γµ, γν}+ 1

2[γµγν ]

)

= ∂µ∂νgµν = ∂2

(2.86)

where we used that the commutator [γµ, γν ] is antisymmetric in the indicesµ and ν. As a result, we find that the 4-spinor Ψ must also satisfy theKlein-Gordon equation

(∂2 +m2

)Ψ = 0 (2.87)

2.5.1 Solutions of the Dirac Equation

Let us briefly discuss the properties of the solutions of the Dirac equation.Let us first consider solutions representing particles at rest. Thus Ψ mustbe constant in space and all its space derivatives must vanish. The Diracequation becomes

iγ0∂Ψ

∂t= mΨ (2.88)

where t = x0 (c = 1). Let us introduce the bispinors φ and χ

Ψ =

(φχ

)(2.89)

We find that the Dirac equation reduces to a simple system of two 2 × 2equations

i∂φ

∂t= +mφ

i∂χ

∂t= −mχ

(2.90)

The four linearly independent solutions are

φ1 = e−imt

(10

)φ2 = e−imt

(01

)(2.91)

and

χ1 = eimt

(10

)χ2 = eimt

(01

)(2.92)


Thus, the upper component φ represents the solutions with positive energy+m, while χ represents the solutions with negative energy −m. The ad-ditional two-fold degeneracy of the solutions is related to the spin of theparticle, as we will see below.

More generally, in terms of the bispinors φ and χ the Dirac Equation takesthe form,

i∂φ

∂t= mφ+

1

iσ ·∇χ (2.93)

i∂χ

∂t= −mχ+

1

iσ ·∇φ (2.94)

In the limit c→∞, it reduces to the Schrodinger-Pauli equation. The slowlyvarying amplitudes φ and χ, defined by

φ = e−imtφ

χ = e−imtχ (2.95)

with χ small and nearly static, define positive energy solutions with energiesclose to +m. In terms of φ and χ, the Dirac equation becomes

i∂φ

∂t=

1

iσ ·∇χ (2.96)

i∂χ

∂t= −2mχ+

1

iσ ·∇φ (2.97)

Indeed, in this limit, the l. h. s. of Eq. (2.97) is much smaller than its r. h.s. Thus we can approximate

2mχ ≈ 1

iσ ·∇φ (2.98)

We can now eliminate the “small component” χ from Eq. (2.96) to find thatφ satisfies

i∂φ

∂t= − 1

2m∇2φ (2.99)

which is indeed the Schrodinger-Pauli equation.

2.5.2 Conserved Current

Let us introduce one last bit of useful notation. Let us define Ψ by

Ψ = Ψ† γ0 (2.100)


in terms of which we can write down the 4-vector jµ

jµ = ΨγµΨ (2.101)

which is conserved, i.e.,

∂µjµ = 0 (2.102)

Notice that the time component of jµ is the density

j0 = Ψγ0Ψ ≡ Ψ†Ψ (2.103)

and that the space components of jµ are

j = ΨγΨ = Ψ†γ0γΨ = Ψ†αΨ (2.104)

Thus the Dirac equation has an associated four vector field, jµ(x), which isconserved and hence obeys a local continuity equation

∂0j0 +∇ · j = 0 (2.105)

However it is easy to see that in general the density j0 can be positive ornegative. Hence this current cannot be associated with a probability current(as in non-relativistic quantum mechanics). Instead we will see that that itis associated with the charge density and current.

2.5.3 Relativistic Covariance

Let Λ be a Lorentz transformation. Let Ψ(x) a spinor field in an inertialframe and Ψ′(x′) be the same Dirac spinor field in the transformed frame.The Dirac equation is covariant if the Lorentz transformation

x′µ = Λνµxν (2.106)

induces a linear transformation S(Λ) in spinor space

Ψ′α(x

′) = S(Λ)αβΨβ(x) (2.107)

such that the transformed Dirac equation has the same form as the originalequation in the original frame, i.e. we will require that if(iγµ

∂

∂xµ−m

)

αβ

Ψβ(x) = 0, then

(iγµ

∂

∂x′µ−m

)

αβ

Ψ′β(x

′) = 0

(2.108)Notice two important facts: (1) both the field Ψ and the coordinate x changeunder the action of the Lorentz transformation, and (2) the γ-matrices and


the mass m do not change under a Lorentz transformation. Thus, the γ-matrices are independent of the choice of a reference frame. However, theydo depend on the choice of the set of basis states in spinor space.

What properties should the representation matrices S(Λ) have? Let usfirst observe that if x′µ = Λµ

νxν , then

∂

∂x′µ=∂xν

∂x′µ∂

∂xν≡(Λ−1

)νµ

∂

∂xν(2.109)

Thus, ∂∂xµ is a covariant vector. By substituting this transformation law back

into the Dirac equation, we find

iγµ∂

∂x′µΨ′(x′) = iγµ(Λ−1)νµ

∂

∂xν(S(Λ)Ψ(x)) (2.110)

Thus, the Dirac equation now reads

iγµ(Λ−1

)νµS(Λ)

∂Ψ

∂xν−mS (Λ)Ψ = 0 (2.111)

Or, equivalently

S−1 (Λ) iγµ(Λ−1

)νµS (Λ)

∂Ψ

∂xν−mΨ = 0 (2.112)

Therefore covariant holds provided that S (Λ) satisfies the identity

S−1 (Λ) γµS (Λ)(Λ−1

)νµ= γν (2.113)

Since the set of Lorentz transformations form a group, the representationmatrices S(Λ) should also form a group. In particular, it must be true thatthe property

S−1 (Λ) = S(Λ−1

)(2.114)

holds. We now recall that the invariance of the relativistic interval x2 = xµxµ

implies that Λ must obey

ΛνµΛλν = gλµ ≡ δλµ (2.115)

Hence,

Λνµ =(Λ−1

)µν

(2.116)

So we can rewrite Eq.(2.113) as

S (Λ) γµS (Λ)−1 =(Λ−1

)µνγν (2.117)

Eq.(2.117) shows that a Lorentz transformation induces a similarity trans-formation on the γ-matrices which is equivalent to (the inverse of) a Lorentztransformation. From this equation it follows that for the case of Lorentz


boosts, Eq.(2.117) shows that the matrices S(Λ) are hermitian. Instead, forthe subgroup SO(3) of rotations about a fixed origin, the matrices S(Λ)are unitary. These different properties follow from the fact that the ma-trices S(Λ) are representation of the Lorentz group SO(3, 1) which si anon-compact Lie group.

We will now find the form of S (Λ) for an infinitesimal Lorentz transforma-tion. Since the identity transformation is Λµ

ν = gµν , a Lorentz transformationwhich is infinitesimally close to the identity should have the form

Λµν = gµν + ωµ

ν , and(Λ−1

)µν= gµν − ωµ

ν (2.118)

where ωµν is infinitesimal and antisymmetric in its space-time indices

ωµν = −ωνµ (2.119)

Let us parameterize S (Λ) in terms of a 4 × 4 matrix σµν which is alsoantisymmetric in its indices, i.e., σµν = −σνµ. Then, we can write

S (Λ) = I − i

4σµνω

µν + . . .

S−1 (Λ) = I +i

4σµνω

µν + . . .

(2.120)

where I stands for the 4 × 4 identity matrix. If we substitute back intoEq.(2.117), we get

(I − i

4σµνω

µν + . . .)γλ(I +i

4σαβω

αβ + . . .) = γλ − ωλνγν + . . . (2.121)

Collecting all the terms linear in ω, we obtain

i

4

[γλ,σµν

]ωµν = ωλνγ

ν (2.122)

Or, what is the same, the matrices σµν must obey

[γµ,σνλ] = 2i(gµν γλ − gµλγν) (2.123)

This matrix equation has the solution

σµν =i

2[γµ, γν ] (2.124)

Under a finite Lorentz transformation x′ = Λx, the 4-spinors transform as

Ψ′(x′) = S (Λ)Ψ (2.125)

with

S (Λ) = exp[− i

4σµνω

µν ] (2.126)


The matrices σµν are the generators of the group of Lorentz transforma-tions in the spinor representation. From this solution we see that the spacecomponents σjk are hermitian matrices, while the space-time componentsσ0j are antihermitian. This feature is telling us that the Lorentz group isnot a compact unitary group, since in that case all of its generators wouldbe hermitian matrices. Instead, this result tells us that the Lorentz groupis isomorphic to the non-compact group SO(3, 1). Thus, the representationmatrices S(Λ) are unitary only under space rotations with fixed origin.

The linear operator S (Λ) gives the field in the transformed frame in termsof the coordinates of the transformed frame. However, we may also wish toask for the transformation U (Λ) that compensates the effect of the coordi-nate transformation. In other words we seek for a matrix U (Λ) such that

Ψ′(x) = U (Λ)Ψ(x) = S (Λ)Ψ(Λ−1x) (2.127)

For an infinitesimal Lorentz transformation, we seek a matrix U (Λ) of theform

U (Λ) = I − i

2Jµνω

µν + . . . (2.128)

and we wish to find an expression for Jµν . We find(I − i

2Jµνω

µν + . . .

)Ψ = (I − i

4σµνω

µν + . . .)Ψ (xρ − ωρνxν + . . .)

∼=(I − i

4σµνω

µν + . . .

)(Ψ− ∂ρΨ ωρνx

ν + . . .)

(2.129)

Hence

Ψ′(x) ∼=(I − i

4σµνω

µν + xµωµν∂ν + . . .

)Ψ(x) (2.130)

From this expression we see that Jµν is given by the operator

Jµν =1

2σµν + i(xµ∂ν − xν∂µ) (2.131)

We easily recognize the second term as the orbital angular momentum op-erator (we will come back to this issue shortly). The first term is then in-terpreted as the spin. In fact, let us consider purely spacial rotations, whoseinfinitesimal generator are the space components of Jµν , i.e.,

Jjk = i(xj∂k − xk∂j) +1

2σjk (2.132)


We can also define a three component vector Jℓ as the 3-dimensional dualof Jjk

Jjk = ϵjklJℓ (2.133)

Thus, we get (after restoring the factors of !)

Jℓ =i!

2ϵℓjk(xj∂k − xk∂j) +

!

4ϵℓjkσjk

= i ! ϵℓjk xj∂k +!

2

(1

2ϵℓjkσjk

)

Jℓ ≡ (x× p)ℓ +!

2σℓ

(2.134)

The first term is clearly the orbital angular momentum and the second termcan be regarded as the spin. With this definition, it is straightforward to

check that the spinors

(φχ

)which are solutions of the Dirac equation, carry

spin one-half.

2.5.4 Transformation Properties of Field Bilinears in the DiracTheory

We will now consider the transformation properties of a number of physicalobservables of the Dirac theory under Lorentz transformations. Let Λ be ageneral Lorentz transformation, and S(Λ) be the induced transformation forthe Dirac spinors Ψa(x) (with a = 1, . . . , 4):

Ψ′a(x

′) = S(Λ)ab Ψb(x) (2.135)

Using the properties of the induced Lorentz transformation S(Λ) and ofthe Dirac γ-matrices, is straightforward to verify that the following Diracbilinears obey the following transformation laws:

Scalar :

Ψ′(x′ ) Ψ′(x′ ) = Ψ(x) Ψ(x) (2.136)

which transforms as a scalar.

Pseudoscalar : Let let us define the Dirac matrix γ5 = iγ0γ1γ2γ3. Then thebilinear

Ψ′(x′ ) γ5 Ψ′(x′ ) = detΛ Ψ(x) γ5 Ψ(x) (2.137)

transforms as a pseudo-scalar.


Vector : Likewise

Ψ′(x′ ) γµ Ψ′(x′ ) = Λµν Ψ(x) γν Ψ(x) (2.138)

transforms as a vector, and

Pseudovector :

Ψ′(x′ ) γ5γµ Ψ′(x′ ) = detΛ Λµ

ν Ψ(x) γ5γν Ψ(x) (2.139)

transforms as a pseudo-vector. Finally,

Tensor :

Ψ′(x′ ) σµν Ψ′(x′ ) = Λµα Λνβ Ψ(x) σαβ Ψ(x) (2.140)

transforms as a tensorAbove we have denoted by Λµ

ν a Lorentz transformation and detΛ is itsdeterminant. We have also used that

S−1(Λ)γ5S(Λ) = detΛ γ5 (2.141)

Finally we note that the Dirac algebra provides for a natural basis of thespace of 4× 4 matrices, which we will denote by

ΓS ≡ I, ΓVµ ≡ γµ, ΓT

µν ≡ σµν , ΓAµ ≡ γ5γµ, ΓP = γ5 (2.142)

where S, V , T , A and P stand for scalar, vector, tensor, axial vector (orpseudo-vector) and parity respectively. For future reference we will note herethe following useful trace identities obeyed by products of Dirac γ-matrices

trI = 4, trγµ = trγ5 = 0, trγµγν = 4gµν (2.143)

Also, if we denote by aµ and bµ two arbitrary 4-vectors, then

/a /b = aµbµ − iσµν aµbν , and tr

(/a /b)= 4a · b (2.144)

2.5.5 Lagrangian for the Dirac Equation

We now seek a Lagrangian density L for the Dirac theory. It should be alocal differentiable Lorentz-invariant functional of the spinor field Ψ. Sincethe Dirac equation is first order in derivatives and it is Lorentz covariant,the Lagrangian should be Lorentz invariant and first order in derivatives. Asimple choice is

L = Ψ(i/∂ −m)Ψ ≡ 1

2Ψi/∂

↔Ψ−mΨΨ (2.145)


where Ψ/∂Ψ ≡ Ψ(/∂Ψ)− (∂µΨ)γµΨ. This choice satisfies all the requirements.The equations of motion are derived in the usual manner, i.e., by demandingthat the action S =

∫d4x L be stationary

δS = 0 =

∫d4x

[ δLδΨα

δΨα +δL

δ∂µΨαδ∂µΨα + (Ψ↔ Ψ)

](2.146)

The equations of motion are

δLδΨα

− ∂µδL

δ∂µΨα= 0

δLδΨα

− ∂µδL

δ∂µΨα= 0 (2.147)

By direct substitution we find

(i/∂ −m)Ψ = 0, and Ψ(i←−/∂ +m) = 0 (2.148)

Here,←−/∂ indicates that the derivatives are acting on the left.

Finally, we can also write down the Hamiltonian density that follows fromthe Lagrangian of Eq.(2.145). As usual we need to determine the canonicalmomentum conjugate to the field Ψ, i.e.,

Π(x) =δL

δ∂0Ψ(x)= iΨ(x)γ0 ≡ iΨ†(x) (2.149)

Thus the Hamiltonian density is

H = Π(x)∂0Ψ(x)− L = iΨγ0∂0Ψ− L= Ψiγ ·∇Ψ+mΨΨ

= Ψ† (iα ·∇+mβ)Ψ

(2.150)

Thus we find that the “one-particle” Dirac Hamiltonian HDirac of Eq.(2.75)appears naturally in the field theory as well. Since this Hamiltonian is firstorder in derivatives (i.e., in the “momentum), unlike its Klein-Gordon rel-ative, it is not manifestly positive. Thus there is a question of the stabilityof this theory. We will see below that the proper quantization of this theoryas a quantum field theory of fermions solves this problem. In other words, itwill be necessary to impose the Pauli Principle for this theory to describe astable system with an energy spectrum that is bounded from below. In thisway we will see that there is natural connection between the spin of the fieldand the statistics. This connection is known as the Spin-Statistics Theorem.


2.6 Classical Electromagnetism as a field theory

We now turn to the problem of the electromagnetic field generated by aset of sources. Let ρ(x) and j(x) represent the charge density and currentat a point x of space-time. Charge conservation requires that a continuityequation has to be obeyed.

∂ρ

∂t+∇ · j = 0 (2.151)

Given an initial condition, i.e., the values of the electric field E(x) and themagnetic field B(x) at some t0 in the past, the time evolution is governedby Maxwell’s equations

∇ ·E =ρ ∇ ·B =0 (2.152)

∇×B − 1

c

∂E

∂t=j ∇×E +

1

c

∂B

∂t=0 (2.153)

It is possible to recast these statements in a manner in which (a) the rel-ativistic covariance is apparent and (b) the equations follow from a LeastAction Principle. A convenient way to see the above is to consider the elec-tromagnetic field tensor Fµν which is the (contravariant) antisymmetric realtensor

Fµν = −F νµ =

⎛

⎜⎜⎝

0 −E1 −E2 −E3

E1 0 −B3 B2

E2 B3 0 −B1

E3 −B2 B1 0

⎞

⎟⎟⎠ (2.154)

In other words

F 0i = −F i0 = −Ei

F ij = −F ji = ϵijkBk

(2.155)

where ϵijk is the third-rank Levi-Civita tensor:

ϵijk =

⎧⎨

⎩

1 if (ijk) is an even permutation of (123)−1 if (ijk) is an odd permutation of (123)0 otherwise

(2.156)

The dual tensor Fµν is defined by

Fµν = −F νµ =1

2ϵµνρσFρσ (2.157)

2.6 Classical Electromagnetism as a field theory 35

where ϵµνρσ is the fourth rank Levi-Civita tensor. In particular

Fµν =

⎛

⎜⎜⎝

0 −B1 −B2 −B3

B1 0 E3 −E2

B2 −E3 0 E2

B3 E2 −E2 0

⎞

⎟⎟⎠ (2.158)

With these notations, we can rewrite Maxwell’s equations in the mani-festly covariant form

∂µFµν =jν (Equation of Motion) (2.159)

∂µFµν =0 (Bianchi Identity) (2.160)

∂µjµ =0 (Continuity Equation) (2.161)

By inspection we see that the field tensor Fµν and the dual field tensor Fµν

map into each other by exchanging the electric and magnetic fields with eachothers. This electro-magnetic duality would be an exact property of electro-dynamics if in addition to the electric charge current jµ the Bianchi IdentityEq.(2.160) included a magnetic charge current (of magnetic monopoles).

At this point it is convenient to introduce the vector potential Aµ whosecontravariant components are

Aµ(x) =

(A0

c,A

)(2.162)

The current 4-vector jµ(x)

jµ(x) = (ρc, j) ≡ (j0, j) (2.163)

The electric field strength E and the magnetic field B are defined to be

E = −1

c∇A0 − 1

c

∂A

∂tB = ∇×A

(2.164)

In a more compact, relativistically covariant, notation we write

Fµν = ∂µAν − ∂νAµ (2.165)

In terms of the vector potential Aµ, Maxwell’s equations have the followingadditional local (or gauge) symmetry

Aµ(x) !→ Aµ(x) + ∂µΦ(x) (2.166)

where Φ(x) is an arbitrary smooth function of the space-time coordinatesxµ. It is easy to check that, under the transformation of Eq.(2.166), the field


strength remains invariant i.e., Fµν !→ Fµν . This property is called GaugeInvariance and it plays a fundamental role in modern physics.

By directly substituting the definitions of the magnetic field B and theelectric field E in terms of the 4-vector Aµ into Maxwell’s equations, weobtain the wave equation. Indeed, the equation of motion

∂µFµν = jν (2.167)

yields the equation for the vector potential

∂2Aν − ∂ν(∂µAµ) = jν (2.168)

This is the wave equation.We can now use the gauge invariance to further restrict the vector poten-

tial Aµ, without which Aµ is not completely determined. These restrictionsare known as the procedure of fixing a gauge. The choice

∂µAµ = 0 (2.169)

known as the Lorentz gauge, yields the simpler (and standard) wave equation

∂2Aµ = jµ (2.170)

Notice that the Lorentz gauge preserves Lorentz covariance.Another “popular” choice is the radiation (or Coulomb) gauge

∇ ·A = 0 (2.171)

which yields (in units with c = 1)

∂2Aν − ∂ν(∂0A0) = jν (2.172)

which is not Lorentz covariant. In the absence of external sources, jν = 0,we can further make the choice A0 = 0. This choice reduces the set of threeequations, one for each spacial component of A, which satisfy

∂2A = 0, provided ∇ ·A = 0 (2.173)

The solutions are, as we well know, plane waves of the form

A(x) = A ei(p0x0−p·x) (2.174)

which are only consistent if p20 − p2 = 0 and p ·A = 0. This choice is alsoknown as the transverse gauge.

We can also regard the electromagnetic field as a dynamical system and

2.6 Classical Electromagnetism as a field theory 37

construct a Lagrangian picture for it. Since the Maxwell equations are lo-cal, gauge invariant and Lorentz covariant, we should demand that the La-grangian density should be local, gauge invariant and Lorentz invariant. Asimple choice is

L = −1

4FµνF

µν − jµAµ (2.175)

This Lagrangian density is manifestly Lorentz invariant. Gauge invarianceis satisfied if and only if jµ is a conserved current, i.e. if ∂µjµ = 0, sinceunder a gauge transformation Aµ !→ Aµ + ∂µΦ(x) the field strength doesnot change but the source term does. Hence,∫

d4x jµAµ !→

∫d4x

[jµA

µ + jµ∂µΦ]

=

∫d4x jµA

µ +

∫d4x ∂µ(jµΦ)−

∫d4x ∂µjµ Φ (2.176)

If the sources vanish at infinity, lim|x|→∞ jµ = 0, the surface term can bedropped. Thus the action S =

∫d4xL is gauge-invariant if and only if the

current jµ is locally conserved,

∂µjµ = 0 (2.177)

We can now derive the equations of motion by demanding that the actionS be stationary, i.e.,

δS =

∫d4x

[δLδAµ

δAµ +δL

δ∂µAµδ∂µAµ

]= 0 (2.178)

Once again, we can integrate by parts to get

δS =

∫d4x ∂ν

[∂L

δ∂µAµδAµ

]+

∫d4x δAµ

[δLδAµ

− ∂ν(

δLδ∂νAµ

)](2.179)

If we demand that at the surface δAµ = 0, we get

δLδAµ

= ∂ν(

δLδ∂νAµ

)(2.180)

Explicitly, we find

δLδAµ

= −jµ, andδL

δ∂νAµ= Fµν (2.181)

Thus, we obtain

jµ = −∂νFµν (2.182)

or, equivalently

jν = ∂µFµν (2.183)


Therefore, the Least Action Principle implies Maxwell’s equations.

2.7 The Landau Theory of Phase Transitions as a Field Theory

We now turn to the problem of the statistical mechanics of a magnet. Inorder to be a little more specific, we will consider the simplest model of aferromagnet: the classical Ising model. In this model, one considers an arrayof atoms on some lattice (say cubic). Each site is assumed to have a netspin magnetic moment S(i). From elementary quantum mechanics we knowthat the simplest interaction among the spins is the Heisenberg exchangeHamiltonian

H = −∑

<i,j>

JijS(i) · S(j) (2.184)

where < i, j > are nearest neighboring sites on the lattice. In many situa-

i j

S(j)S(i)

Figure 2.3 Spins on a lattice.

tions, in which there is magnetic anisotropy, only the z-component of thespin operators plays a role. The Hamiltonian now reduces to that of the

2.7 The Landau Theory of Phase Transitions as a Field Theory 39

Ising model HI

HI = −J∑

<ij>

σ(i)σ(j) ≡ E[σ] (2.185)

where [σ] denotes a configuration of spins with σ(i) being the z-projectionof the spin at each site i.

The equilibrium properties of the system are determined by the parti-tion function Z which is the sums over all spin configurations [σ] of theBoltzmann weight for each state,

Z =∑

{σ}

exp

(−E[σ]

T

)(2.186)

where T is the temperature and {σ} is the set of all spin configurations.In the 1950’s, Landau developed a picture to study these type of problems

which in general, are very difficult. Landau first proposed to work not withthe microscopic spins but with a set of coarse-grained configurations. Oneway to do this (using the more modern version due to Kadanoff and Wilson)is to partition a large system of linear size L into regions or blocks of smallerlinear size ℓ such that a0 << ℓ << L, where a0 is the lattice spacing. Eachone of these regions will be centered around a site, say x. We will denote sucha region by A(x). The idea is now to perform the sum, i.e., the partitionfunction Z, while keeping the total magnetization of each region M(x) fixed

M(x) =1

N [A]

∑

y∈A(x)

σ(y) (2.187)

where N [A] is the number of sites in A(x). The restricted partition functionis now a functional of the coarse-grained local magnetizations M(x),

Z[M ] =∑

{σ}

exp

{−E[σ]

T

} ∏

x

δ

⎛

⎝M(x)− 1

N(A)

∑

y∈A(x)

σ(y)

⎞

⎠ (2.188)

The variables M(x) have the property that, for N(A) very large, they ef-fectively take values on the real numbers. Also, the coarse-grained configu-rations {M(x)} are smoother than the microscopic configurations {σ}.

At very high temperatures the average magnetization ⟨M⟩ = 0 since thesystem is in a paramagnetic phase. On the other hand, at low temperaturesthe average magnetization may be non-zero since the system may now bein a ferromagnetic phase. Thus, at high temperatures the partition func-tion Z is dominated by configurations which have ⟨M⟩ = 0 while at verylow temperatures, the most frequent configurations have ⟨M⟩ = 0. Landau


proceeded to write down an approximate form for the partition function interms of sums over smooth, continuous, configurations M(x) which can berepresented in the form

Z ≈∫

DM(x) exp

[−E [M(x), T ]

T

](2.189)

where DM(x) is a measure which means “sum over all configurations.” If forthe relevant configurations M(x) is smooth and small, the energy functionalE[M ] can be written as an expansion in powers of M(x) and of its spacederivatives. With these assumptions the free energy of the magnet can beapproximated by the Landau-Ginzburg form

F (M) ≡

=

∫ddx

[1

2K(T )|∇M(x)|2 + 1

2a(T )M2(x) +

1

4!b(T )M4(x) + . . .

]

(2.190)

Thermodynamic stability requires that the stiffness K(T ) and the non-linearity coefficient b(T ) must be positive. The second term has a coefficienta(T ) with can have either sign. A simple choice of parameters is

K(T ) ≃ K0, b(T ) ≃ b0, a(T ) ≃ a (T − Tc) (2.191)

where Tc is an approximation to the critical temperature.The free energy F (M) defines a Classical, or Euclidean, Field Theory. In

fact, by rescaling the field M(x) in the form

Φ(x) =√KM(x) (2.192)

we can write the free energy as

F (Φ) =

∫ddx

{1

2(∇Φ)2 + U(Φ)

}(2.193)

where the potential U(Φ) is

U(Φ) =m2

2Φ2 +

λ

4!Φ4 + . . . (2.194)

where m2 = a(T )K and λ = b

K2 . Except for the absence of the term involvingthe canonical momentum Π2(x), F (Φ) has a striking resemblance to theHamiltonian of a scalar field in Minkowski space! We will see below thatthis is not an accident, and that the Landau-Ginzburg theory is closelyrelated to a quantum field theory of a scalar field with a Φ4 potential.

Let us now ask the following question: is there a configuration Φc(x) that

2.7 The Landau Theory of Phase Transitions as a Field Theory 41

Φ

U(Φ)

Φ0

T > T0

T < T0

−Φ0

Figure 2.4 The Landau free energy for the order parameter field Φ: forT > T0 the free energy has a unique minimum at Φ = 0 while for T < T0

there are two minima at Φ = ±Φ0

gives the dominant contribution to the partition function Z? If so, we shouldbe able to approximate

Z =

∫DΦ exp{−F (Φ)} ≈ exp{−F (Φc)}{1 + · · · } (2.195)

This statement is usually called the Mean Field Approximation. Since theintegrand is an exponential, the dominant configuration Φc must be suchthat F has a (local) minimum at Φc. Thus, configurations Φc which leaveF (Φ) stationary are good candidates (we actually need local minima!). Theproblem of finding extrema is simply the condition δF = 0. This is the sameproblem we solved for classical field theory in Minkowski space-time. Noticethat in the derivation of F we have invoked essentially the same type ofarguments: (a) invariance and (b) locality (differentiability).

The Euler-Lagrange equations can be derived by using the same argu-ments that we employed in the context of a scalar field theory. In the caseat hand they become

− δF

δΦ(x)+ ∂j

(δF

δ∂jΦ(x)

)= 0 (2.196)

For the case of the Landau theory the Euler-Lagrange Equation becomesthe Landau-Ginzburg Equation

0 = −∇2Φc(x) + m2Φc(x) +λ

3!Φ3c(x) (2.197)


The solution Φc(x) that minimizes the energy is uniform in space and thushas ∂jΦc = 0. Hence, Φc is the solution of the very simple equation

m2Φc +λ

3!Φ3c = 0 (2.198)

Since λ is positive and m2 may have either sign, depending on whetherT > Tc or T < Tc, we have to explore both cases.

For T > Tc, m2 is also positive and the only real solution is Φc = 0. Thisis the paramagnetic phase. But, for T < Tc, m2 is negative and two newsolutions are available, namely

Φc = ±√

6 | m2 |λ

(2.199)

These are the solutions with lowest energy and they are degenerate. Theyboth represent the magnetized (or ferromagnetic)) phase.

We now must ask if this procedure is correct or, rather, when can weexpect this approximation to work. The answer to this question is the centralproblem of the theory of phase transitions which describes the behavior ofstatistical systems in the vicinity of a continuous (or second order) phasetransition. It turns out that this problem is also connected with a centralproblem of Quantum Field theory, namely when and how is it possible toremove the singular behavior of perturbation theory, and in the processremove all dependence on the short distance (or high energy) cutoff fromphysical observables. In Quantum Field Theory this procedure amounts to adefinition of the continuum limit. The answer to these questions motivatedthe development of the Renormalization Group which solved both problemssimultaneously.

2.8 Field Theory and Classical Statistical Mechanics

We are now going to discuss a mathematical “trick” which will allow us toconnect field theory with classical statistical mechanics. Let us go back tothe action for a real scalar field Φ(x) in D = d+ 1 space-time dimensions

S =

∫dDx L(Φ, ∂µΨ) (2.200)

where dDx is

dDx ≡ dx0ddx (2.201)

2.8 Field Theory and Classical Statistical Mechanics 43

Let us formally carry out the analytic continuation of the time componentx0 of xµ from real to imaginary time xD

x0 !→ −ixD (2.202)

under which

Φ(x0,x) !→ Φ(x, xD) ≡ Φ(x) (2.203)

where x = (x, xD). Under this transformation, the action (or rather i timesthe action) becomes

iS ≡ i

∫dx0 d

dx L(Φ, ∂0Ψ, ∂jΦ) !→∫

dDx L(Φ,−i∂DΦ, ∂jΦ) (2.204)

If L has the form

L =1

2(∂µΦ)

2 − V (Φ) ≡ 1

2(∂0Φ)

2 − 1

2(∇Φ)2 − V (Φ) (2.205)

then the analytic continuation yields

L(Φ,−i∂DΨ,∇Φ) = −1

2(∂DΦ)

2 − 1

2(∇Φ)2 − V (Φ) (2.206)

Then we can write

iS(Φ, ∂µΦ) −−−−−−−−→x0 → −ixD

−∫

dDx

[1

2(∂DΦ)

2 +1

2(∇Φ)2 + V (Φ)

](2.207)

This expression has the same form as (minus) the potential energy E(Φ)for a classical field Φ in D = d + 1 space dimensions. However it is alsothe same as the energy for a classical statistical mechanics problem in thesame number of dimensions i.e., the Landau-Ginzburg free energy of thelast section.

In Classical Statistical Mechanics, the equilibrium properties of a systemare determined by the partition function. For the case of the Landau theoryof phase transitions the partition function is

Z =

∫DΦ e−E(Φ)/T (2.208)

where the symbol “∫DΦ” means sum over all configurations. (We will dis-

cuss the definition of the “measure” DΦ later on). If we choose for energyfunctional E(Φ) the expression

E(Φ) =

∫dDx

[1

2(∂Φ)2 + V (Φ)

](2.209)

where

(∂Φ)2 ≡ (∂DΦ)2 + (∇Φ)2 (2.210)


we see that the partition function Z is formally the analytic continuation of

Z =

∫DΦ eiS(Φ, ∂µΦ)/! (2.211)

where we have used ! which has units of action (instead of the temperature).What is the physical meaning of Z? This expression suggests that Z

should have the interpretation of a sum of all possible functions Φ(x, t) (i.e.,the histories of the configurations of the field Φ) weighed by the phase factorexp{ i

!S(Φ, ∂µΦ)}. We will discover later on that if T is formally identified

with the Planck constant !, then Z represents the path-integral quantizationof the field theory! Notice that the semiclassical limit ! → 0 is formallyequivalent to the low temperature limit of the statistical mechanical system.

The analytic continuation procedure that we just discussed is called aWick rotation. It amounts to a passage fromD = d+1-dimensional Minkowskispace to a D-dimensional Euclidean space. We will find that this analyticcontinuation is a very powerful tool. As we will see below, a number of dif-ficulties will arise when the theory is defined directly in Minkowski space.Primarily, the problem is the presence of ill-defined integrals which are givenprecise meaning by a deformation of the integration contours from the realtime (or frequency) axis to the imaginary time (or frequency) axis. The de-formation of the contour amounts to a definition of the theory in Euclideanrather than in Minkowski space. It is an underlying assumption that theanalytic continuation can be carried out without difficulty. Namely, the as-sumption is that the result of this procedure is unique and that, whateversingularities may be present in the complex plane, they do not affect theresult. It is important to stress that the success of this procedure is notguaranteed. However, in almost all the theories that we know of this as-sumption holds. The only case in which problems are known to exist is thetheory of Quantum Gravity (which we will not discuss here).

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