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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.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.
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