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.Theyarethe Maxwell equations for classical electromagnetism, the Klein-Gordon equa- tion and the Dirac equation. Maxwell’s equations govern the dynamics of a vector field, the vector potentials Aµ x = A0 x , A x ,whereastheKlein- Gordon equation describes excitations of a scalar field φ x and the Dirac equation governs the behavior of the( ) four-component( ( ) ( )) spinor field ψα x , α = 0, 1, 2, 3 .Eachoneofthesefieldstransformsinaverydefiniteway( ) under the group of Lorentz transformations, the Lorentz group. The Lorentz( ) (group is defined) as a group of linear transformations Λof Minkowski space- time M onto itself Λ ∶ M ↦ M such that the new coordinates are related to the old ones by a linear (Lorentz) transformation
′µ µ ν x = Λν x (2.1)
0 The space-time components of a Lorentz transformation, Λi ,arethe Lorentz boosts.Lorentzboostsrelateinertialreferenceframesmovingatrel- ative velocity v with respect to each other. Lorentz boosts along the x1-axis 2.1 Relativistic invariance 11 have the familiar form 0 1 ′ x + vx c x0 = 1 − v2 c2 1 0/ 1′ !x + vx c x = / 1 − v2 c2 ′ / x2 = x!2 ′ / x3 = x3 (2.2) where x0 = ct, x1 = x, x2 = y and x3 = z (note: the superscripts indicate − components, not powers!). If we use the notation γ = 1 − v2 c2 1 2 ≡ cosh α,wecanwritetheLorentzboostasamatrix: /
0′ 0 ( / ) x cosh α sinh α 00 x ′ x1 sinh α cosh α 00 x1 2′ = 2 (2.3) ⎛x ⎞ ⎛ 0010⎞ ⎛x ⎞ ⎜ 3′⎟ ⎜ 3⎟ ⎜x ⎟ ⎜ 0001⎟ ⎜x ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ i ⎜ ⎟ The space components⎝ of⎠ Λj⎝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 ∂ = ∂ and µ, ν = 0, 1, 2, 3. The infinitesimal generators L satisfy µ ∂xµ ( ) µν 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 Lie group SO 3, 1 .Actually,anyoperatorofthe form ( ) Mµν = Lµν + Sµν (2.6) where Sµν are 4 × 4matricessatisfyingthealgebraofEq.(2.5)arealso generators of SO 3, 1 .Belowwewilldiscussexplicitexamples. Lorentz transformations form a group,since(a)theproductoftwoLorentz transformations is( a) Lorentz transformation, (b) there exists an identity 12 Classical Field Theory transformation, and (c) Lorentz transformations are invertible. Notice, how- ever, that in general two transformations do not commute witheachother. Hence, the Lorentz group is non-Abelian. The Lorentz group has the defining property of leaving invariant the rel- ativistic interval 2 2 2 2 2 2 x ≡ x0 − x = c t − x (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 isdenotedby SO 3 ,andtheLorentzgroupisdenotedbySO 3, 1 .Thisnotationmakes manifest the fact that the signature of the metric has one + sign and three − signs.( ) ( ) The group SO 3, 1 of linear transformations is non-compact in the fol- lowing sense. Let us consider first the group of rotations in three-dimensional space, SO 3 .Thelineartransformationsin( ) SO 3 leave the Euclidean dis- 2 2 2 2 tance (squared) R = x1 + x2 + x3 invariant. The set of points with a fixed value of R(is) the two-dimensional surface of a sphere( ) of radius R in three di- mensions, that we will denote by S2.Theelementsofthegroupofrotations SO 3 are in one-to-one correspondence with the points on S2.Theareaof a2-sphereS2 of unit radius is 4π.Thenwewillsaythatthe“volume”of the( group) SO 3 is finite and equal to 4π.Agroupoflineartransformations with finite volume is said to be compact.Incontrast,theLorentzgroupisthe set of linear transformations,( ) denoted by SO 3, 1 ,thatleavetherelativis- µ tic interval xµx invariant, which is not positive definite. As we well know, Lorentz boosts which map points along hyperbolas( ) of Minkowski space time. In this sense, the Lorentz group is non-compact since its “volume” is infinite. We will adopt the following conventions and definitions:
1) Metric Tensor:Wewillusethestandard(“BjorkenandDrell”)metricfor Minkowski space-time in which the metric tensor gµν is 10 0 0 µν 0 −10 0 g = g = (2.8) µν − ⎛ 00 10⎞ ⎜ 00 0−1 ⎟ ⎜ ⎟ ⎜ ⎟ With this notation the infinitesimal⎝ relativistic interval⎠ is 2 µ µ ν 2 2 2 2 2 ds = dx dxµ = gµν dx dx = dx0 − dx = c dt − dx (2.9) 2) 4-vectors: i) xµ is a contravariant 4-vector, xµ = ct, x
( ) 2.1 Relativistic invariance 13
ii) xµ is a covariant 4-vector xµ = ct, −x iii) Covariant and contravariant vectors (and tensors) are related through the metric tensor gµν ( ) µ µν A = g Aν (2.10) iv) x is a vector in R3 2 pµ = E , p p pµ = E − p2 v) c is the energy-momentum 4-vector . Hence, µ c2 is a Lorentz scalar. 3) Scalar Product( ) : µ µν p ⋅ q = pµq = p0q0 − p ⋅ q ≡ pµqνg (2.11) ∂ µ ∂ 2 4) Gradients: ∂µ ≡ µ and ∂ ≡ .WedefinetheD’Alambertian∂ ∂x ∂xµ
2 µ 1 2 2 ∂ ≡ ∂ ∂µ ≡ ∂ −& (2.12) c2 t which is a Lorentz scalar. From now on we will use units of time T and = = = length L such that h̵ c 1. Thus, T L and we will use units like centimeters (or any other unit of length). [ ] 2 5) Interval[:TheintervalinMinkowskispaceis] [ ] x [, ] 2 µ 2 2 x = xµx = x0 − x (2.13) Time-like intervals have x2 > 0whilespace-likeintervalshavex2 < 0. Since a field is a function (or mapping) of Minkowski space ontosome other (properly chosen) space, it is natural to require that the fields should have simple transformation properties under Lorentz transformations. For example, the vector potential Aµ x transforms like 4-vector under Lorentz ′µ µ ν ′µ ′ µ ν transformations, i.e. if x = Λν x ,thenA x = Λν A x .Inotherwords, µ µ A transforms like x .Thus,itisavector.Allvectorfieldshavethisprop-( ) erty. A scalar field Φ x ,ontheotherhand,remains( ) invariant( ) under Lorentz transformations, ′ ′ ( ) Φ x = Φ x (2.14)
A4-spinorψα x transforms under Lorentz transformations. Namely, there ( ) ( ) exists an induced 4 × 4lineartransformationmatrixS Λ such that ( ) −1 −1 S Λ = S Λ (2.15) ( ) and ′ ( ) ( ) Ψ Λx = S Λ Ψ x (2.16) Below we will give an explicit expression for S Λ . ( ) ( ) ( ) ( ) 14 Classical Field Theory
!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ x0
time-like x2 > 0
space-like
x1 x2 < 0
Figure 2.1 The Minkowski space-time and its light cone. Events at a rel- 2 2 2 ativistic interval with x = x0 − x > 0 are time-like (and are causally 2 2 2 connected with the origin), while events with x = x0 − x < 0arespace- like and are not causally connected with the origin.
2.2 The Lagrangian, the action, and the Least Action Principle
The evolution of any dynamical system is determined by its Lagrangian. In the Classical Mechanics of systems of particles described bythegeneralized coordinates q,theLagrangianL is a differentiable function of the coordi- nates q and their time derivatives. L must be differentiable since, otherwise, the equations of motion would not be local in time, i.e. could not be written in terms of differential equations. An argument `a-la Landau and Lifshitz (Landau and Lifshitz, 1959a) enables us to “derive” the Lagrangian. For ex- ample, for a particle in free space, the homogeneity, uniformity and isotropy of space and time require that L be only a function of the absolute value of the velocity v .Since v is not a differentiable function of v,theLa- grangian must be a function of v2.Thus,L = L v2 .Inprinciplethereis no reason to assume∣ ∣ that ∣L ∣cannot be a function of the acceleration a (or 2 rather a )orofitshigherderivatives.Experiment( tells) us that in Classical Mechanics it is sufficient to specify the initial position x 0 of a particle and its initial velocity v 0 in order to determine the time evolution of the ( ) ( ) 2.2 The Lagrangian, the action, and the Least Action Principle 15 system. Thus we have to choose 1 L v2 = const + mv2 (2.17) 2 The additive constant is irrelevant in classical physics. Naturally, the coef- 2 ( ) ficient of v is just one-half of the inertial mass. However, in Special Relativity, the natural invariant quantity to consider is not the Lagrangian but the action S.Forafreeparticletherelativistic invariant (i.e. Lorentz invariant ) action must involve the invariant interval, 2 proper length ds = c − v dt the 1 c2 .Henceonewritestheactionfora relativistic massive particle+ as
sf tf v2 S = −mc ds = −mc2 dt 1 − (2.18) " " 2 si ti , c The relativistic Lagrangian then is
v2 L = −mc2 1 − (2.19) , c2 As a power series expansion, it contains all powers of v2 c2.Itiselementary to see that, as expected, the canonical momentum p is ∂L mv / p = = (2.20) ∂v v2 1 − , c2 from which it follows that the Hamiltonian (or energy) is given by mc2 H = = p2c2 + m2c4, (2.21) v2 1 − ! , c2 as it should be. Once the Lagrangian is found, the classical equations of motion are de- termined by the Least Action Principle.Thus,weconstructtheactionS
S = dt L q,q˙ (2.22) " q = dq q t where ˙ dt ,anddemandthatthephysicaltrajectories( ) leave the action S stationary, i.e. δS = 0. The variation of S is tf ∂L ∂L ( ) δS = dt δq + δq˙ (2.23) " ti ∂q ∂q˙ - . 16 Classical Field Theory
q
i
t
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 trace represents a variation.
Integrating by parts, we get tf ∂L tf ∂L d ∂L δS = dt q˙ δq + dt δq − (2.24) " " ti ∂q˙ ti ∂q dt ∂q˙ Hence, we get - . / - .0
tf ∂L tf ∂L d ∂L δS = δq + dt δq − (2.25) " ∂q˙ ti ∂q dt ∂q˙ 1ti 1 1 If we assume that the variation1 δq is an/ arbitrary- function.0 of time that 1 vanishes at the initial and1 final times ti and tf ,i.e.δq ti = δq tf = 0, we find that δS = 0ifandonlyiftheintegrandofEq.(2.25)vanishesidentically. Thus, ( ) ( ) ∂L d ∂L − = 0(2.26) ∂q dt ∂q˙
These are the equations of motion or- Newton’s. equations.Ingeneralthe equation that determine the trajectories that leave the action stationary is called the Euler-Lagrange equation.
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 .TheactionS must be invariant under Lorentz transformations. Since we want to construct local ( ) 2.3 Scalar field theory 17 theories it is natural to assume that S is given in terms of a Lagrangian density L
S = d4x L (2.27) " where L is a local differentiable function of the field and its derivatives. These assumptions in turn guarantee that the resulting equations of motion have the form of partial differential equations. In other terms, thedynamicsdoes not allow for action-at-a-distance. Since the volume element of Minkowski space d4x is invariant under Lorentz transformations, the action S is invariant if L is a local, differ- entiable function of Lorentz invariants that can be constructed out of the field Φ x .SimpleinvariantsareΦx itself and all of its powers. The gra- dient ∂µΦ ≡ ∂Φ is not an invariant but the D’alambertian ∂2Φis.Bilinears ∂xµ µ such as( ∂)µΦ∂ ΦarealsoLorentzinvariantaswellasunderachangeofthe( ) sign of Φ. So, we can write the following simple expression for L:
1 µ L = ∂ Φ∂ Φ − V Φ (2.28) 2 µ where V Φ is some potential, which we can assume( ) is a polynomial function of the field Φ. Let us consider the simple choice ( ) 1 V Φ = m¯ 2Φ2 (2.29) 2 = wherem ¯ mc h̵ .Thus, ( ) 1 µ 1 L = ∂ Φ∂ Φ − m¯ 2Φ2 (2.30) / 2 µ 2 This is the Lagrangian density for a free scalar field.Wewilldiscusslater on in what sense this field is “free”. Notice, in passing, that we could have added a term like ∂2Φ. However this term, in addition of being odd under Φ → −Φ, is a total divergence and, as such, it has an effect only on the boundary conditions but it does not affect the equations of motion. In what follows, unless stated to the contrary, will will not consider surface terms. The Least Action Principle requires that S be stationary under arbitrary variations of the field Φand of its derivatives ∂µΦ. Thus, we get
4 δL δL δS = d x δΦ + δ∂µΦ (2.31) " δΦ δ∂µΦ Notice that since L is a functional/of Φ, we have to use functional0 derivatives, i.e. partial derivatives at each point of space-time. Upon integrating by parts, 18 Classical Field Theory we get
4 δL 4 δL δL δS = d x∂µ δΦ + d xδΦ − ∂µ (2.32) " δ∂µΦ " δΦ δ∂µΦ Instead of considering- initial and. final conditions,/ we now- have.0 to imagine that the field Φis contained inside some very large box of space-time. The term with the total divergence yields a surface contribution. We will consider field configurations such that δΦ = 0onthatsurface.Thus,theEuler- Lagrange equations are δL δL − ∂µ = 0(2.33) δΦ δ∂µΦ More explicitly, we find - . δL ∂V = − (2.34) δΦ ∂Φ and, since ∂L µ = ∂ Φ, (2.35) δ∂µΦ Then δL µ 2 ∂µ = ∂µ∂ Φ ≡ ∂ Φ(2.36) δ∂µΦ By direct substitution we get the equation of motion (or field equation) ∂V ∂2Φ + = 0(2.37) ∂Φ For the choice m¯ 2 ∂V V Φ = Φ2, = m¯ 2Φ(2.38) 2 ∂Φ the field equation is ( ) ∂2 + m¯ 2 Φ = 0(2.39)
2 ∂2 = 1 ∂ −&2 where c2 ∂t2 .Thus,wefindthattheequationofmotionforthe( ) free massive scalar field Φis 1 ∂2Φ −&2Φ + m¯ 2Φ = 0(2.40) c2 ∂t2 This is precisely the Klein-Gordon equation if the constant m¯ is identified with mc .Indeed,theplane-wavesolutionsoftheseequationsare h̵ − ⋅ i p0x0 p x h̵ Φ = Φ0e (2.41) ( )/ 2.4 Classical field theory in the canonical formalism 19 where p0 and p are related through the dispersion law 2 2 2 2 4 p0 = p c + m c (2.42) This means that, for each momentum p,therearetwosolutions,onewith positive frequency and one with negative frequency.Wewillseebelowthat, in the quantized theory, the energy of the excitation is indeed equal to p 1 = h̵ 0 .Noticethat m¯ mc has units of length and is equal to the Compton wavelength for a particle of mass m.Fromnowon(unlessitstatedthe = = = contrary)∣ ∣ I will use units in which h̵ c 1inwhichm m¯ .
2.4 Classical field theory in the canonical formalism In Classical Mechanics it is often convenient to use the canonical formulation in terms of a Hamiltonian instead of the Lagrangian approach.Forthecase of a system of particles, the canonical formalism proceeds asfollows.Given aLagrangianL q,q˙ ,acanonicalmomentump is defined to be ∂L = p (2.43) ( ) ∂q˙ The classical Hamiltonian H p, q is defined by the Legendre transformation H p, q = pq˙ − L q,q˙ (2.44) ( ) If the Lagrangian L is quadratic in the velocitiesq ˙ and separable, e.g. ( ) ( ) 1 L = mq˙2 − V q (2.45) 2 then, H pq˙ is simply given by ( ) mq˙2 p2 ( ) H p, q = pq˙ − − V q = + V q (2.46) 2 2m where ( ) 2 ( )3 ( ) ∂L p = = mq˙ (2.47) ∂q˙ The (conserved) quantity H is then identified with the total energy of the system. In this language, the Least Action Principle becomes
δS = δ Ldt= δ pq˙ − H p, q dt = 0(2.48) " "
Hence [ ( )] ∂H ∂H dt δp q˙ + pδq˙ − δp − δq = 0(2.49) " ∂p ∂q
- . 20 Classical Field Theory Upon an integration by parts we get ∂H ∂H dt δp q˙ − + δq − − p˙ = 0(2.50) " ∂p ∂q which can only be satisfied/ - for arbitrary. variations- δq.0t and δp t if ∂H ∂H q˙ = p˙ = − (2.51) ∂p ∂q ( ) ( ) 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{ ∂}A ∂B A, B ≡ − (2.52) qp ∂q ∂p ∂p ∂q
Let F q,p,t be some di{ fferentiable} function of q,p and t.Thenthetotal time variation of F is ( ) dF ∂F ∂F dq ∂F dp = + + (2.53) dt ∂t ∂q dt ∂p dt Using Hamilton’s Equations we get the result dF ∂F ∂F ∂H ∂F ∂H = + − (2.54) dt ∂t ∂q ∂p ∂p ∂q or, in terms of Poisson Brackets, dF ∂F = + F, H (2.55) dt ∂t qp
In particular, { } dq ∂H ∂q ∂H ∂q ∂H = = − = q,H (2.56) dt ∂p ∂q ∂p ∂p ∂q qp since { } ∂q ∂q = 0and = 1(2.57) ∂p ∂q Also the total rate of change of the canonical momentum p is dp ∂p ∂H ∂p ∂H ∂H = − ≡ − (2.58) dt ∂q ∂p ∂p ∂q ∂q ∂p = ∂p = since ∂q 0and ∂p 1. Thus, dp = p, H (2.59) dt qp
{ } 2.4 Classical field theory in the canonical formalism 21 Notice that, for an isolated system, H is time-independent. So, ∂H = 0(2.60) ∂t and dH ∂H = + H, H = 0(2.61) dt ∂t qp since { } H, H qp = 0(2.62) Therefore, H can be regarded as the generator of infinitesimal time transla- { } ∂H = tions. Since it is conserved for an isolated system, for which ∂t 0, we can indeed identify H with the total energy. In passing, let us also notice that the above definition of the Poisson Bracket implies that q and p satisfy
q,p qp = 1(2.63) 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 Φwith Lagrangian density L Φ,∂µ, Φ .Wewillchoose Φ x to be the (infinite) set of canonical coordinates.Thecanonical momentum Π x is defined by ( ) δL ( ) = Π x ( ) (2.64) δ∂0Φ x
If the Lagrangian is quadratic(in) ∂µΦ, the canonical momentumΠ x is simply given by ( ) ˙ Π x = ∂0Φ x ≡ Φ x ((2.65)) The Hamiltonian density H Φ, Π is a local function of Φ x and Π x given ( ) ( ) ( ) by
H Φ, Π (= Π )x ∂0Φ x − L Φ,∂0Φ ( ) ( )(2.66) If the Lagrangian density L has the simple form ( ) ( ) ( ) ( ) 1 L = ∂ Φ 2 − V Φ (2.67) 2 µ then, the Hamiltonian density H Φ, Π is ( ) ( ) 1 1 H = ΠΦ˙ − L Φ, Φ˙ ,∂ Φ ≡ Π2 x + !Φ x 2 + V Φ x (2.68) j 2 ( ) 2 which is explicitly a positive definite quantity. Thus, the energy of a plane ( ) ( ) ( ( )) ( ( )) wave solution of a massive scalar field theory, i.e. a solutionoftheKlein- Gordon equation, is always positive, no matter the sign of the frequency. 22 Classical Field Theory In fact, the lowest energy state is simply Φ = constant. A solution made of linear superpositions of plane waves (i.e. a wave packet) haspositiveenergy. Therefore, in field theory, the energy is always positive.Wewillseethat,in the quantized theory, in the case of a complex field, the negative frequency solutions are identified with antiparticle states and their existence do not signal a possible instability of the theory. The canonical field Φ x and the canonical momentum Π x satisfy the equal-time Poisson Bracket (PB) relations ( ) ( ) Φ x,x0 , Π y,x0 PB = δ x − y (2.69)
x where δ is the Dirac{ ( δ-function) ( and)} the Poisson( ) Bracket A, B PB is defined to be ( ) { } 3 δA δB δA δB A, B PB = d x − (2.70) " δΦ x,x0 δΠ x,x0 δΠ x,x0 δΦ x,x0 for{ any} two functionals/ A and B of Φ x and Π x .Thisapproachcan0 ( ) ( ) ( ) ( ) be extended to theories other than that of a scalar field without too much difficulty. We will come back to these issues( ) when we( ) consider the problem of quantization. Finally we should note that while Lorentz invariance is apparent in the Lagrangian formulation, it is not so in the Hamiltonian formulation 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 discuss the theory of spinors as a classical field theory. We will find that this theory is not consistent unless it is properly quantized as a quantumfieldtheoryof spinors. We will return to this problem in chapter 7. Let us rewrite the Dirac equation ∂Ψ hc ih = ̵ α ⋅ !Ψ + βmc2 Ψ ≡ H Ψ(2.71) ̵ ∂t i Dirac in a manner in which relativistic covariance is apparent. TheoperatorHDirac is the Dirac Hamiltonian. We first recall that the 4 × 4hermitianmatricesα and β should satisfy the algebra
2 2 αi,αj = 2δij I, αi,β = 0,αi = β = I (2.72) × where I is the{ 4 4identitymatrix.} { } 2.5 Field theory of the Dirac equation 23 Asimplerepresentationofthisalgebraarethe2×2block(Dirac)matrices
i i 0 σ I 0 α = β = (2.73) σi 0 0 −I
i 2 3 - . where the σ matrices are the three 2 × 2Paulimatrices 01 0 −i 10 σ1 = σ2 = σ3 = (2.74) 10 i 0 0 −1 and I is the 2-× 2identitymatrix.Thisisthe. - . Dirac- representation. of the Dirac algebra. It is now convenient to introduce the Dirac gamma matrices,
i i γ0 = βγ= βα (2.75)
The Dirac gamma matrices γµ have the block form
i I 0 i 0 σ γ0 = β = ,γ= (2.76) 0 −I −σi 0 an obey the Dirac algebra- . 2 3
µ ν µν γ ,γ = 2g I (2.77)
I × where is the 4 4identitymatrix.{ } In terms of the gamma matrices, the Dirac equation takes the much sim- pler, covariant, form µ mc iγ ∂µ − Ψ = 0(2.78) h̵ where Ψis a 4-spinor. It is also4 customary5 to introduce the notation (known as Feynman’s slash) µ a ≡ aµγ (2.79)
Using Feynman’s slash, we can write/ the Dirac equation in the form mc i∂ − Ψ = 0(2.80) h̵ From now on I will use units in which h = c = 1. In these units energy has − 4 / 5̵ units of (length) 1 and time has units of length. Notice that, if Ψsatisfies the Dirac equation, then it also satisfies
i∂ + m i∂ − m Ψ = 0(2.81)
( / )( / ) 24 Classical Field Theory Also, we find 1 1 ∂ ⋅ ∂ =∂ ∂ γµγν = ∂ ∂ γµ,γν + γµγν µ ν µ ν 2 2 = µν = 2 / / ∂µ∂ν g ∂ - { } [ ]. (2.82) where we used that the commutator γµ,γν is antisymmetric in the indices µ and ν.Asaresult,wefindthateachcomponentofthe4-spinorΨmust also satisfy the Klein-Gordon equation[ ] ∂2 + m2 Ψ = 0(2.83)
( ) 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 Ψmust be constant in space and all its space derivatives must vanish. The Dirac equation becomes ∂Ψ iγ0 = mΨ(2.84) ∂t where t = x0 (c = 1). Let us introduce the bispinors φ and χ φ Ψ = (2.85) χ
We find that the Dirac equation reduces- . to a simple system of two2× 2 equations ∂φ ∂χ i = +mφ, i = −mχ (2.86) ∂t ∂t The four linearly independent solutions are
−imt 1 −imt 0 φ = e φ = e (2.87) 1 0 2 1 and - . - . imt 1 imt 0 χ = e χ = e (2.88) 1 0 2 1
Thus, the upper component φ -represents. the solutions- . with positive energy +m,whileχ represents the solutions with negative energy −m.Thead- ditional two-fold degeneracy of the solutions is related to the spin of the particle, as we will see below. 2.5 Field theory of the Dirac equation 25 More generally, in terms of the bispinors φ and χ the Dirac Equation takes the form, ∂φ 1 i = mφ + σ ⋅ !χ (2.89) ∂t i ∂χ 1 i = −mχ + σ ⋅ !φ (2.90) ∂t i Furthermore, in the non-relativistic limit, taking formally c → ∞,itshould reduce to the Schr¨odinger-Pauli equation. To see this, we define the slowly varying amplitudes φ˜ andχ ˜ − φ = e imtφ˜ −imt χ = e χ˜ (2.91) The fieldχ ˜ is small and nearly static. We will now see that the field φ˜ describes solutions with positive energies close to +m.Intermsofφ˜ andχ ˜, the Dirac equation becomes ∂φ˜ 1 i = σ ⋅ !χ˜ (2.92) ∂t i ∂χ˜ 1 i = −2mχ˜ + σ ⋅ !φ˜ (2.93) ∂t i Indeed, in this limit, the l.h.s of Eq. (2.93) is much smaller than its r.h.s. Thus we can approximate 1 2mχ˜ ≈ σ ⋅ !φ˜ (2.94) i We can now eliminate the “small component”χ ˜ from Eq. (2.92) to find that φ˜ satisfies ∂φ˜ 1 i = − &2 φ˜ (2.95) ∂t 2m which is indeed the Schr¨odinger-Pauli equation.
2.5.2 Conserved current Let us introduce one last bit of useful notation. Let us define Ψ¯ by † Ψ¯ = Ψ γ0 (2.96) in terms of which we can write down the 4-vector jµ µ µ j = Ψ¯ γ Ψ(2.97) 26 Classical Field Theory which is conserved, i.e. µ ∂µj = 0(2.98) Notice that the time component of jµ is the density † j0 = Ψ¯ γ0Ψ ≡ Ψ Ψ(2.99) and that the space components of jµ are † † j = Ψ¯ γΨ = Ψ γ0γΨ = Ψ αΨ(2.100)
Thus the Dirac equation has an associated 4-vector field, jµ x ,whichis conserved and obeys a local continuity equation ! ( ) ∂0j0 + ⋅ j = 0(2.101)
However it is easy to see that in general the density j0 can be positive or negative. Hence this current cannot be associated with a probability current (as in non-relativistic quantum mechanics). Instead we willseethatthatit is associated with the charge density and current.
2.5.3 Relativistic covariance Let Λbe a Lorentz transformation. Let Ψ x be a spinor field in an inertial ′ ′ frame and Ψ x be the same Dirac spinor field in the transformed frame. The Dirac equation is covariant if the Lorentz( ) transformation ( ) ′ ν xµ = Λµxν (2.102) induces a linear transformation S Λ in spinor space ′ ′ Ψ x = S Λ Ψ x (2.103) α ( ) αβ β such that the transformed Dirac equation has the same form as the original ( ) ( ) ( ) equation in the original frame, i.e. we will require that if ∂ ∂ µ − = µ − ′ ′ = iγ µ m Ψβ x 0, then iγ ′µ m Ψβ x 0 ∂x αβ ∂x αβ (2.104) - . ( ) - . ( ) Notice two important facts: (1) both the field Ψand the coordinate x change under the action of the Lorentz transformation, and (2) the gamma matrices and the mass m do not change under a Lorentz transformation. Thus, the gamma matrices are independent of the choice of a reference frame. However, they do depend on the choice of the basis in spinor space. 2.5 Field theory of the Dirac equation 27 What properties should the representation matrices S Λ have? Let us ′µ µ ν first observe that if x = Λν x ,then ν ( ) ∂ ∂x ∂ − ν ∂ = ≡ Λ 1 (2.105) ∂x′µ ∂x′µ ∂xν µ ∂xν ∂ Thus, ∂xµ is a covariant vector. By substituting( ) this transformation law back into the Dirac equation, we find
µ ∂ ′ ′ µ − ν ∂ iγ Ψ x = iγ Λ 1 S Λ Ψ x (2.106) ∂x′µ µ ∂xν Thus, the Dirac equation( now) reads( ) ( ( ) ( )) µ − ν ∂Ψ iγ Λ 1 S Λ − mS Λ Ψ = 0(2.107) µ ∂xν Or, equivalently ( ) ( ) ( ) − µ − ν ∂Ψ S 1 Λ iγ Λ 1 S Λ − mΨ = 0(2.108) µ ∂xν Therefore, covariance holds provided S Λ satisfies the identity ( ) ( ) ( ) − µ − ν ν S 1 Λ γ S Λ Λ 1 = γ (2.109) ( ) µ Since the set of Lorentz transformations form a group, the representation ( ) ( )( ) matrices S Λ should also form a group. In particular, it must be true that the property − − ( ) S 1 Λ = S Λ 1 (2.110) 2 = µ holds. We now recall that the invariance( ) of( the) relativistic interval x xµx implies that Λmust obey ν λ λ λ Λ µΛν = gµ ≡ δµ (2.111) Hence, ν −1 µ Λµ = Λ ν (2.112) So we can rewrite Eq.(2.109) as ( ) µ −1 −1 µ ν S Λ γ S Λ = Λ ν γ (2.113) Eq.(2.113) shows that a Lorentz transformation induces a similarity trans- ( ) ( ) ( ) formation on the gamma matrices which is equivalent to (the inverse of) a Lorentz transformation. From this equation it follows that,forthecaseof Lorentz boosts, Eq.(2.113) shows that the matrices S Λ are hermitian.In- stead, for the subgroup SO 3 of rotations about a fixed origin, the matrices S Λ are unitary. These different properties follow from( ) the fact that the ( ) ( ) 28 Classical Field Theory matrices S Λ are representation of the Lorentz group SO 3, 1 which is a non-compact Lie group. We will now( ) findthe form of S Λ for an infinitesimal Lorentz( ) transforma- µ µ tion. Since the identity transformation is Λν = gν ,aLorentztransformation infinitesimally close to the identity( ) should have the form µ µ µ − µ µ µ Λ = g + ω , and Λ 1 = g − ω (2.114) ν ν ν ν ν ν where ωµν is infinitesimal and antisymmetric in its space-time indices ( ) µν νµ ω = −ω (2.115)
Let us parameterize S Λ in terms of a 4 × 4matrixσµν which is also antisymmetric in its indices, i.e. σµν = −σνµ.Then,wecanwrite ( ) i µν S Λ = I − σ ω + ... 4 µν − i µν S 1 Λ = I + σ ω + ... ( ) 4 µν (2.116) ( ) where I stands for the 4 × 4identitymatrix.Ifwesubstitutebackinto Eq.(2.113), we get
i µν λ i αβ λ λ ν I − σ ω + ... γ I + σ ω + ... = γ − ω γ + ... (2.117) 4 µν 4 αβ ν Collecting all the terms linear in ω,weobtain ( ) ( ) i λ µν λ ν γ ,σ ω = ω γ (2.118) 4 µν ν Or, what is the same, the matrices σ must obey 6 7µν µ = µ µ γ ,σνλ 2i gν γλ − gλ γν (2.119) This matrix equation has the solution [ ] ( ) i σ = γ ,γ (2.120) µν 2 µ ν ′ = Under a finite Lorentz transformation[x Λ]x,the4-spinorstransformas ′ ′ Ψ x = S Λ Ψ(2.121) with ( ) ( ) i µν S Λ = exp − σ ω (2.122) 4 µν
The matrices σµν are the (generators) -of the group. of Lorentz transforma- tions in the spinor representation. From this solution we seethatthespace 2.5 Field theory of the Dirac equation 29 components σjk are hermitean matrices, while the space-time components σ0j are antihermitean.ThisfeatureistellingusthattheLorentzgroupis not a compact unitary group, since in that case all of its generators would be hermitian matrices. Instead, this result tells us that theLorentzgroup is isomorphic to the non-compact group SO 3, 1 .Thus,therepresentation matrices S Λ are unitary only under space rotations with fixed origin. The linear operator S Λ gives the field in( the )transformed frame in terms of the coordinates( ) of the transformed frame.However,wemayalsowishto ask 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 ′ ( ) −1 Ψ x = U Λ Ψ x = S Λ Ψ Λ x (2.123) ( ) For an infinitesimal Lorentz transformation, we seek a matrix U Λ of the ( ) ( ) ( ) ( ) ( ) form i µν U Λ = I − J ω + ... ( )(2.124) 2 µν and we wish to find an expression for Jµν .Wefind ( ) i µν i µν ρ ρ ν I − J ω + ... Ψ = I − σ ω + ... Ψ x − ω x + ... 2 µν 4 µν ν
i µν ρ ν - . ≅( I − σ ω + ...) 8Ψ − ∂ Ψ ω x +9... 4 µν ρ ν (2.125) - . 8 9 Hence, ′ i µν µν Ψ x ≅ I − σ ω + x ω ∂ + ... Ψ x (2.126) 4 µν µ ν
From this expression( ) we- see that Jµν is given by the operator. ( ) 1 J = σ + i x ∂ − x ∂ (2.127) µν 2 µν µ ν ν µ We easily recognize the second term as the orbital angular momentum op- ( ) erator (we will come back to this issue shortly). The first termistheninter- preted as the spin. In fact, let us consider purely spacial rotations, whose infinitesimal gen- erator are the space components of Jµν ,i.e. 1 J = i x ∂ − x ∂ + σ (2.128) jk j k k j 2 jk We can also define a three component vector J( as the 3-dimensional dual ( ) of Jjk
Jjk = -jklJ( (2.129) 30 Classical Field Theory where -ijk is the third-rank Levi-Civita tensor:
1ifijk is an even permutation of 123 ijk - = −1ifijk is an odd permutation of 123 (2.130) ⎧ ⎪ ( ) ( ) ⎪0otherwise ⎨⎪ ( ) ( ) ⎪ Thus, we get⎪ (after restoring the factors of h) ⎩⎪ ̵ ih h J = ̵ - x ∂ − x ∂ + ̵ - σ ( 2 (jk j k k j 4 (jk jk h 1 = i h- x ∂ + ̵ - σ ̵ (jk( j k 2 2) (jk jk h J ≡ x × pˆ + ̵ σ - . ( ( 2 ( (2.131) ( ) The first term is clearly the orbital angular momentum and the second term can be regarded as the spin. With this definition, it is straightforward to check that the spinors φ,χ ,whicharesolutionsoftheDiracequation, carry spin one-half. ( )
2.5.4 Transformation properties of field bilinears in the Dirac theory We will now consider the transformation properties of several physical ob- servables of the Dirac theory under Lorentz transformations. Let Λbe a general Lorentz transformation, and S Λ be the induced transformation for the Dirac spinors Ψa x (with a = 1,...,4): ( ) ′ ′ = ( Ψ) a x S Λ ab Ψb x (2.132)
Using the properties of the induced( ) Lorentz( ) transformation( ) S Λ and of the Dirac gamma matrices, is straightforward to verify that the following Dirac bilinears obey the following transformation laws: ( )
Scalar: ′ ′ ′ ′ Ψ¯ x Ψ x = Ψ¯ x Ψ x (2.133) which transforms as a scalar( . ) ( ) ( ) ( ) 2.5 Field theory of the Dirac equation 31
Pseudoscalar: Let let us define the Dirac matrix γ5 = iγ0γ1γ2γ3.Thenthe bilinear ′ ′ ′ ′ Ψ¯ x γ5 Ψ x = det Λ Ψ¯ x γ5 Ψ x (2.134) transforms as a pseudo-scalar( ) . ( ) ( ) ( )
Vector: Likewise
′ ′ µ ′ ′ µ ν Ψ¯ x γ Ψ x = Λν Ψ¯ x γ Ψ x (2.135) transforms as a vector(,and) ( ) ( ) ( )
Pseudovector:
′ ′ µ ′ ′ µ ν Ψ¯ x γ5γ Ψ x = det ΛΛν Ψ¯ x γ5γ Ψ x (2.136) transforms as( a pseudo-vector) ( ).Finally, ( ) ( )
Tensor: ′ ′ µν ′ ′ µ ν αβ Ψ¯ x σ Ψ x = Λα Λβ Ψ¯ x σ Ψ x (2.137) transforms as a tensor ( ) ( µ ) ( ) ( ) Above we have denoted by Λν aLorentztransformationanddetΛisits determinant. We have also used that
−1 S Λ γ5S Λ = det Λ γ5 (2.138)
Finally we note that the Dirac( ) algebra( ) provides for a natural basis of the space of 4 × 4matrices,whichwewilldenoteby
S V T A P Γ ≡ I, Γµ ≡ γµ, Γµν ≡ σµν , Γµ ≡ γ5γµ, Γ = γ5 (2.139) where S, V , T , A and P stand for scalar, vector, tensor, axial vector (or pseudo-vector) and parity respectively. For future reference we will note here the following useful trace identities obeyed by products of Dirac gamma matrices
trI = 4, trγµ = trγ5 = 0, trγµγν = 4gµν (2.140)
Also, if we denote by aµ and bµ two arbitrary 4-vectors, then µ µ ν a b = aµb − iσµν a b , and tr a b = 4a ⋅ b (2.141)
/ / 8/ /9 32 Classical Field Theory 2.5.5 The Dirac Lagrangian We now seek a Lagrangian density L for the Dirac theory. It should be a local differentiable Lorentz-invariant functional of the spinor field Ψ. Since the Dirac equation is first order in derivatives and it is Lorentz covariant, the Lagrangian should be Lorentz invariant and first order in derivatives. A simple choice is 1 ←→ L = Ψ¯ i∂ − m Ψ ≡ Ψ¯ i ∂ Ψ − mΨΨ¯ (2.142) 2 ←→ µ where Ψ¯ ∂ Ψ ≡ Ψ¯ ∂Ψ − ∂(µΨ¯/ γ Ψ). This choice/ satisfies all the requirements. The equations of motion are derived in the usual manner, i.e. by demand- ing that the/ action/S = d4x L be stationary ( ) (∫ ) 4 δL δL δS = 0 = d x δΨα + δ∂µΨα + Ψ ↔ Ψ¯ (2.143) " δΨα δ∂µΨα The equations of motion> are ( )? δL δL − ∂µ = 0 δΨα δ∂µΨα δL δL − ∂µ = 0(2.144) δΨ¯ α δ∂µΨ¯ α By direct substitution we find ←− i∂ − m Ψ = 0, and Ψ¯ i ∂ + m = 0(2.145) ←− Here, ∂ indicates( that/ the) derivatives are acting( / on) the left. Finally, we can also write down the Hamiltonian density that follows from the Lagrangian/ of Eq.(2.142). As usual we need to determine the canonical momentum conjugate to the field Ψ, i.e.
δL † Π x = = iΨ¯ x γ0 ≡ iΨ x (2.146) δ∂0Ψ x Thus the Hamiltonian( density) is ( ) ( ) ( ) ¯ 0 H =Π x ∂0Ψ x − L = iΨγ ∂0Ψ − L =Ψ¯ iγ ⋅ !Ψ + mΨΨ¯ († ) ( ) =Ψ iα ⋅ ! + mβ Ψ(2.147) Thus we find that the “one-particle” Dirac Hamiltonian H of Eq.(2.71) ( ) Dirac appears naturally in the field theory as well. Since the Hamiltonian of Eq.(2.147) is first order in derivatives, unlike its 2.6 Classical electromagnetism as a field theory 33 Klein-Gordon relative, it is not manifestly positive. Thus,thereisaquestion of the stability of this theory. We will see below that the proper quantization of this theory as a quantum field theory of fermions solves this problem. In other words, it will be necessary to impose the Pauli Principle for this theory to describe a stable system with an energy spectrum that is bounded from below. In this way we will see that there is natural connection between the spin of the field and the statistics.Thisconnection,whichactuallyisan axiom of Quantum Field Theory, 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 a set of sources. Let ρ x and j x represent the charge and current densities at a point x of space-time. Charge conservation requires that a continuity equation has to be obeyed,( ) ( ) ∂ρ + ! ⋅ j = 0(2.148) ∂t Given an initial condition, that specifies the values of the electric field E x and the magnetic field B x at some t0 in the past, the time evolution is governed by the Maxwell equations ( ) ( ) ! ⋅ E =ρ ! ⋅ B =0(2.149) 1 ∂E 1 ∂B ! × B − =j ! × E + =0(2.150) c ∂t c ∂t It is possible to reformulate classical electrodynamics in amannerinwhich (a) relativistic covariance is apparent and (b) that the Maxwell equations follow from a Least Action Principle. A convenient way to see the above is to define the electromagnetic field tensor F µν ,whichisthe(contravariant) antisymmetric real tensor whose components are given by
0 −E1 −E2 −E3 1 3 2 µν νµ E 0 −B B F = −F = (2.151) ⎛E2 B3 0 −B1⎞ ⎜ 3 2 1 ⎟ ⎜E −B B 0 ⎟ ⎜ ⎟ ⎜ ⎟ Or, equivalently ⎜ ⎟ ⎝ ⎠ F 0i = −F i0 = −Ei F ij = −F ji = -ijkBk (2.152) 34 Classical Field Theory
The dual tensor F$µν is defined as follows µν νµ 1 µνρσ F = −F = - F (2.153) $ $ 2 ρσ where -µνρσ is the fourth rank Levi-Civita tensor, defined similarly to the third rank Levi-Civita tensor of Eq.(2.130). In particular 0 −B1 −B2 −B3 1 3 2 µν B 0 E −E F = (2.154) $ ⎛ B2 −E3 0 E2 ⎞ ⎜ 3 2 − 2 ⎟ ⎜ B E E 0 ⎟ ⎜ ⎟ With these notations, we can⎜ rewrite the left column⎟ of Maxwell equations, ⎝ ⎠ Eq.(2.150), in the more compact and manifestly covariant form µν ν ∂µF = j (2.155) which we will interpret as the equations of motion of the electromagnetic field. The right column of the Maxwell equations, Eq.(2.150),becomethe constraint µν = ∂µF$ 0(2.156) which is known as the Bianchi identity. Then, consistency of the Maxwell equations requires that the continuity equation µ ∂µj = 0(2.157) be satisfied. By inspection we see that the field tensor F µν and the dual field tensor µν F$ map into each other upon exchanging the electric and magneticfields. This electro-magnetic duality would be an exact property of electrodynamics if in addition to the electric charge current jµ the Bianchi Identity Eq.(2.156) included a magnetic charge current (of magnetic monopoles). At this point it is convenient to introduce the vector potential Aµ whose contravariant components are A0 µ = A x c , A (2.158)
The current 4-vector jµ x ( ) 2 3 µ j x = ρc, j ≡ j0, j (2.159) ( ) The electric field strength E and the magnetic field B are defined to be ( ) ( ) ( ) 1 1 ∂A E = − !A0 − , B = ! × A (2.160) c c ∂t 2.6 Classical electromagnetism as a field theory 35 In a more compact, relativistically covariant, notation we write µν µ ν ν µ F = ∂ A − ∂ A (2.161) In terms of the vector potential Aµ,theMaxwellequationsremainun- changed under the (local) gauge transformation µ µ µ A x ↦ A x + ∂ Φ x (2.162) where Φ x is an arbitrary smooth function of the space-time coordinates µ ( ) ( ) ( ) x .Itiseasytocheckthat,underthetransformationofEq.(2.162), the field µν µν strength( remains) invariant i.e. F ↦ F . This property is called Gauge Invariance and it plays a fundamental role in modern physics. By directly substituting the definitions of the magnetic field B and the electric field E in terms of the 4-vector Aµ into the Maxwell equations, we obtain the wave equation. Indeed, the equation of motion µν ν ∂µF = j (2.163) yields the equation for the vector potential 2 ν ν µ ν ∂ A − ∂ ∂µA = j (2.164) which is the wave equation. ( ) We can now use gauge invariance to further restrict the vectorpotential Aµ,whichisnotcompletelydetermined.Theserestrictionsareknownasthe procedure of fixing a gauge.Thechoice µ ∂µA = 0(2.165) known as the Lorentz gauge, yields the simpler and standard form of the wave equation µ µ ∂2A = j (2.166) Notice that the Lorentz gauge preserves Lorentz covariance. Another popular choice is the radiation (or Coulomb) gauge ! ⋅ A = 0(2.167) which yields (in units with c = 1) 2 ν ν 0 ν ∂ A − ∂ ∂0A = j (2.168) which is not Lorentz covariant. In the absence of external sources, jν = 0, ( ) 0 we can further impose the restriction that A = 0. This choice reduces the set of three equations, one for each spacial component of A,whichsatisfy ∂2A = 0, provided ! ⋅ A = 0(2.169) 36 Classical Field Theory The solutions are, as we well know, plane waves of the form
i p x −p⋅x A x = A e 0 0 (2.170) ( ) 2 − 2 = ⋅ = which are only consistent if p(0 ) p 0andp A 0. For the last reason, this choice is also known as the transverse gauge. We can also regard the electromagnetic field as a dynamical system and construct a Lagrangian picture for it. Since the Maxwell equations are lo- cal, gauge invariant and Lorentz covariant, we should demandthattheLa- grangian density should be local, gauge invariant and Lorentz invariant. Since, in terms of the vector potential Aµ,theMaxwellequationsaresecond order in derivatives, we seek a local Lagrangian density which is also second order in derivatives of the vector potential. A simple choicethatsatistiesall the requirements is
1 µν µ L = − F F − j A (2.171) 4 µν µ This Lagrangian density is manifestly Lorentz invariant. Gauge invariance µ is satisfied if and only if jµ is a conserved current, i.e. if ∂µj = 0, since, under a gauge transformation Aµ ↦ Aµ +∂µΦ x the field strength does not change. However, the source term changes as follows ( ) µ µ µ d4xj A ↦ d4x j A + j ∂ Φ " µ " µ µ µ µ µ = d4xj A + d4x∂ j Φ − d4x∂ j Φ(2.172) " >µ " ? µ " µ
If the sources vanish at infinity, lim x →∞ (jµ =)0, the surface term can be dropped. Thus the action S = d4xL is gauge-invariant if and only if the ∫ ∣ ∣ current jµ is locally conserved,
µ ∂µj = 0(2.173) which is the continuity equation. We can now derive the equations of motion by demanding that theaction S be stationary, i.e.
δL µ δL ν µ δS = d4x δA + δ∂ A = 0(2.174) " δAµ δ∂ν Aµ
Much as we did before, we can/ now proceed to integrate0 by parts to get
ν δL µ µ δL ν δL δS = d4x∂ δA + d4xδA − ∂ (2.175) " δ∂ν Aµ " δAµ δ∂ν Aµ
/ 0 / - .0 2.7 The Landau theory of phase transitions as a field theory 37 By demanding that at the surface the variation vanishes, δAµ = 0, we get
δL ν δL = ∂ (2.176) δAµ δ∂ν Aµ
Explicitly, we find - . δL δL µν = −j , and = F (2.177) δAµ µ δ∂ν Aµ Thus, we obtain µ µν j = −∂ν F (2.178) or, equivalently ν µν j = ∂µF (2.179) Therefore, the Least Action Principle implies the Maxwell 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. In order to be a little more specific, we will consider the simplest model of a ferromagnet: the classical Ising model. In this model, one considers an array of atoms on some lattice (say cubic). Each site is assumed to have a net spin magnetic moment S i .Fromelementaryquantummechanicsweknow that the simplest interaction among the spins is the Heisenberg exchange Hamiltonian ( ) H = − J S i ⋅ S j (2.180) % ij
! ! S(i) S(j) i j
Figure 2.3 Spins on a lattice. 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 prob- lems which in general are very di{ffi}cult. Landau first proposed to work not with the microscopic spins but with a set of coarse-grained configurations (Landau, 1937). One way to do this, using the more modern version of the argument due to Kadanoff(Kadanoff, 1966) and Wilson (Wilson, 1971), is to partition a large system of linear size L into regions or blocks of smaller linear size / such that a0 ≪ / ≪ L,wherea0 is the lattice spacing. Each one of these regions will be centered around a site, say x.Wewilldenotesuch aregionbyA x .Theideaisnowtoperformthesum,i.e.thepartition function Z,whilekeepingthetotalmagnetizationofeachregionA x fixed to the values ( ) ( ) 1 M x = σ y (2.183) N A % y∈A x ( ) ( ) ( ) where N A is the number of sites in[ A] x .Therestrictedpartitionfunction
[ ] ( ) 2.7 The Landau theory of phase transitions as a field theory 39 is now a functional of the coarse-grained local magnetizations M x ,
E σ 1 Z M = exp − δ M x − σ y ( )(2.184) % T & N A % σ x y∈A x [ ] ⎛ ⎞ [ ] @ A ⎜ ( ) A ( )⎟ The variables{ }M x have the property⎝ that, for( N) (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 = 0andthe system is{ in( a)} paramagnetic phase. On the other hand, at low temperatures{ } the average magnetization may be non-zero and the system⟨ ⟩ may be in a ferromagnetic phase. Thus, at high temperatures the partition function Z is dominated by configurations which have M = 0whileatverylowtemper- atures, the most frequent configurations have M ≠ 0. Landau proceeded to write down an approximate form for⟨ the⟩ partition function in terms of sums over smooth, continuous,configurationsM⟨ x⟩ which, formally, can be represented in the form ( ) E M x ,T Z ≈ DM x exp − (2.185) " T [ ( ) ] where DM x is an integration( measure) 2 which means “sum3 over all config- urations.” We will define this more properly later on, in chapter 15. We will assume( ) that for the dominant configurations in the sum (integral!) shown in Eq.(2.185) the local averages M x are smooth functions of x and are small.Then,theenergyfunctionalE M can be written as an expansion in powers of M x and of its space derivatives.( ) With these assumptions the free energy of the magnet in D dimensions[ ] can be approximated by the Ginzburg-Landau( ) form (Landau and Lifshitz, 1959b)
1 1 1 E M = dDx K T !M x 2 + a T M 2 x + b T M 4 x + ... " 2 2 4! (2.186) ( ) 2 ( )∣ ( )∣ ( ) ( ) ( ) ( ) 3 Ginzburg and Landau made the additional (and drastic) assumption that there is a single configuration M x that dominates the full partition func- tion. Under this assumption, which can be regarded as a mean-field approx- imation, the free energy F = −T (ln )Z takes the same form as Eq.(2.186). In a later chapter we will develop the theory of the effective action and we will return to this point. Thermodynamic stability requires that the stiffness K T and the coef- ficient b T of the quartic term must be positive. The second term has a ( ) ( ) 40 Classical Field Theory coefficient a T which can have either sign. A simple choice of these param- eters is ( ) K T ≃ K0,bT ≃ b0,aT ≃ a¯ T − Tc (2.187) where T is an approximate value of the critical temperature. c ( ) ( ) ( ) ( ) The free energy F M defines a Classical,orEuclidean,FieldTheory.In fact, by rescaling the field M x in the form ( ) Φ x = KM x (2.188) ( ) we can write the free energy as D ( ) ( ) d 1 F Φ = d x !Φ 2 + U Φ (2.189) " 2 where the potential U(Φ) is @ ( ) ( )A m¯ 2 λ ( U) Φ = Φ2 + Φ4 + ... (2.190) 2 4! m2 = a T λ = b where ¯ K and (K2).Exceptfortheabsenceoftheterminvolving 2 the canonical( momentum) Π x ,F Φ has a striking resemblance to the Hamiltonian of a scalar field in Minkowski space! We will see below that this is not an accident, and( that) the( ) Ginzburg-Landau theory is closely related to a quantum field theory of a scalar field with a Φ4 potential.
U(Φ)
T>T0
T −Φ0 Φ0 Φ Figure 2.4 The Landau free energy for the order parameter field Φ: for T > T0 the free energy has a unique minimum at Φ = 0 while for T < T0 there are two minima at Φ = ±Φ0 2.7 The Landau theory of phase transitions as a field theory 41 Let us now ask the following question: is there a configurationΦc x that gives the dominant contribution to the partition function Z?Ifso,weshould be able to approximate ( ) Z = DΦexp−F Φ ≈ exp −F Φ 1 +⋯ (2.191) " c This statement is usually{ called( )} the Mean{ Field( )}{ Approximation} .Sincethe integrand is an exponential, the dominant configuration Φc must be such that F has a (local) minimum at Φc.Thus,configurationsΦc which leave F Φ stationary are good candidates (we actually need local minima!). The problem of finding extrema is simply the condition δF = 0. This is the same problem( ) we solved for classical field theory in Minkowski space-time. Notice that in the derivation of F we have invoked essentially the same type of arguments as before: (a) invariance and (b) locality (differentiability). The Euler-Lagrange equations can be derived by using the sameargu- ments that we employed in the context of a scalar field theory. In the case at hand they become δF δF − + ∂j = 0(2.192) δΦ x δ∂jΦ x For the case of the Landau theory the- Euler-Lagrange. Equationbecomes the Ginzburg-Landau Equation( ) ( ) λ 0 = −&2 Φ x + m¯ 2Φ x + Φ3 x (2.193) c c 3! c x The solution Φc that minimizes( ) the energy( ) is uniform( ) in space and thus has ∂jΦc = 0. Hence, Φc is the solution of the very simple equation ( ) λ m¯ 2Φ + Φ3 = 0(2.194) c 3! c Since λ is positive andm ¯ 2 may have either sign, depending on whether T > Tc or T < Tc,wehavetoexplorebothcases. 2 For T > Tc, m¯ is also positive and the only real solution is Φc = 0. This 2 is the paramagnetic phase. But, for T < Tc, m¯ is negative and two new solutions are available, namely 6 m¯ 2 Φ = ± (2.195) c , λ These are the solutions with lowest energy∣ and∣ they are degenerate.They both represent the magnetized (or ferromagnetic)) phase. We now must ask if this procedure is correct or, rather, when can we 42 Classical Field Theory expect this approximation to work. The answer to this question is the central problem of the theory of phase transitions which describes the behavior of statistical systems in the vicinity of a continuous (or second order) phase transition. It turns out that this problem is also connected with a central problem of Quantum Field theory, namely when and how is it possible to remove the singular behavior of perturbation theory, and in the process remove all dependence on the short distance (or high energy) cutofffrom physical observables. In Quantum Field Theory this procedure amounts to a definition of the continuum limit. The answer to these questions motivated the development of the Renormalization Group which solved both problems simultaneously. 2.8 Field theory and Statistical Mechanics We are now going to discuss a mathematical procedure that willallowusto connect quantum field theory to classical statistical mechanics. We will use amappingcalledaWickrotation. Let us go back to the action for a real scalar field Φ x in D = d + 1 space-time dimensions ( ) D S = d x L Φ,∂ Ψ (2.196) " µ where dDx is ( ) D d d x ≡ dx0d x (2.197) Let us formally carry out the analytic continuation of the time component x0 of xµ from real to imaginary time xD x0 ↦ −ixD (2.198) under which Φ x0, x ↦ Φ x,xD ≡ Φ x (2.199) where x = x,x .Underthistransformation,theaction(orratheri times D ( ) ( ) ( ) the action) becomes ( ) d D iS ≡ i dx d x L Φ,∂ Ψ,∂ Φ ↦ d x L Φ, −i∂ Φ,∂ Φ (2.200) " 0 0 j " D j If L has the form ( ) ( ) 1 1 1 L = ∂ Φ 2 − V Φ ≡ ∂ Φ 2 − !Φ 2 − V Φ (2.201) 2 µ 2 0 2 ( ) ( ) ( ) ( ) ( ) 2.8 Field theory and Statistical Mechanics 43 then the analytic continuation yields 1 1 L Φ, −i∂ Ψ, &Φ = − ∂ Φ 2 − !Φ 2 − V Φ (2.202) D 2 D 2 Then we can write ( ) ( ) ( ) ( ) D 1 2 1 2 iS Φ,∂µΦ −−−−−−−−→ − d x ∂DΦ + !Φ + V Φ (2.203) x0 → −ixD " 2 2 This( expression) has the same form/ as( (minus)) the( potential) en( ergy)0 E Φ for a classical field Φin D = d + 1 space dimensions. However it is also the same as the energy for a classical statistical mechanics problem in( the) same number of dimensions i.e. the Landau-Ginzburg free energy of the last section. In Classical Statistical Mechanics, the equilibrium properties of a system are determined by the partition function.ForthecaseoftheLandautheory of phase transitions the partition function is −E Φ T Z = DΦ e (2.204) " where the symbol “ DΦ” means sum over( all)/ configurations.Wewilldiscuss ∫ the meaning of this expression and the definition of the “measure” DΦlater on. If we choose for energy functional E Φ the expression D 1 E Φ = d x ∂Φ 2 + V Φ (2.205) " 2 ( ) where ( ) / ( ) ( )0 2 2 2 ∂Φ ≡ ∂DΦ + !Φ (2.206) we see that the partition function Z is formally the analytic continuation of ( ) ( ) ( ) iS Φ,∂ Φ h Z = DΦ e µ ̵ (2.207) " ( )/ where we have used h̵ which has units of action (instead of the temperature). What is the physical meaning of Z?ThisexpressionsuggeststhatZ 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 factor i exp h S Φ,∂µΦ .WewilldiscoverlateronthatifT is formally( identified) ̵ Z with the Planck constant h̵ ,then represents the path-integral quantization → of the{ field( theory!)} Notice that the semiclassical limit h̵ 0isformally equivalent to the low temperature limit of the statistical mechanical system. The analytic continuation procedure that we just discussed is called a Wick rotation.ItamountstoapassagefromD = d+1-dimensional Minkowski 44 Classical Field Theory space to a D-dimensional Euclidean space. We will find that this analytic continuation 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 given precise meaning by a deformation of the integration contoursfromthereal time (or frequency) axis to the imaginary time (or frequency)axis.Thede- formation of the contour amounts to a definition of the theory in Euclidean rather than in Minkowski space. It is an underlying assumption that the analytic continuation can be carried out without problems. Namely, the as- sumption is that the result of this procedure is unique and that, whatever singularities may be present in the complex plane, they do not affect the result. It is important to stress that the success of this procedure is not guaranteed. 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 the theory of Quantum Gravity (which we will not discuss in this book).