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Elements of Classical Field Theory C6, HT 2016

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Elements of Classical Field Theory C6, HT 2016 Elements of Classical Field Theory C6, HT 2016 Uli Haischa aRudolf Peierls Centre for Theoretical Physics University of Oxford OX1 3PN Oxford, United Kingdom Please send corrections to [email protected]. 1 Classical Field Theory In this part of the lecture we will discuss various aspects of classical fields. We will cover only the bare minimum ground necessary before turning to the quantum theory, and will return to classical field theory at several later stages in the course when we need to introduce new concepts or ideas. 1.1 Lorentz Group The Lorentz group L is of fundamental importance for the construction of relativistic field theories, since L is associated to the symmetry of 4-dimensional space-time. Using the Minkowski metric η = diag (1; 1; 1; 1), the Lorentz group L consists of the real 4 4 matrices Λ that satisfy − − − × η = ΛT ηΛ ; (1.1) which written in components (µ, ν; ρ, σ = 0; 1; 2; 3) takes the form µν σρ µ ν η = η Λ ρ Λ σ : (1.2) It is readily seen that the transformations µ µ ν x Λ ν x ; (1.3) ! with Λ satisfying (1.1) leave the distance ds2 invariant. Setting for simplicity the speed of light c to 1, one has 2 µ ν µ ρ ν σ ρ σ 2 ds = ηµν dx dx ηµν Λ ρ dx Λ σ dx = ηρσ dx dx = ds : (1.4) ! The Lorentz transformations (LTs) (1.3) are therefore consistent with the postulate of special relativity that tells us that the speed of light is the same in all inertial frames. 1 4.1. SYMMETRIES 43 0 det(Λ) Λ 0 name contains given by +1 1 L↑ 1 L↑ ≥ + 4 + +1 1 L↓ PT PTL↑ ≤− + + 1 1 L↑ P PL+↑ − ≥ − 1 1 L↓ T TL+↑ − ≤− − Table 4.1: The fourFigure disconnected 1.1: The components four disconnected of the Lorentz components group. The of union the LorentzL+ = L group.+↑ L+↓ is also called the ∪ proper Lorentz group and L↑ = L+↑ L↑ is called the orthochronos Lorentz group (as it consists of transformations ∪ − preserving the direction of time). L+↑ is called the proper orthochronos Lorentz group. Let us examine the consequences of (1.1). First, we take its determinant O satisfying OT O = 1 and det(O)=1det.Writing (η) = detO =Λ1T +detiT (withη) det (purely (Λ) ; imaginary) generators T ,therelation(1.5) OT O = 1 implies T = T and, hence, that the Lie-algebra of SO(3) consists of 3 3 anti-symmetric matrices from which we deduce† that (multiplied by i). A basis for this Lie algebra is provided by the three matrices T defined× by det (Λ) = 1 : i (1.6) ± The case of det (Λ) = +1 ( 1) corresponds(Ti)jk = toi!properijk , (improper) LTs and the associated (4.16) − − subgroup is L+ (L−). This implies that parity or space-inversion P = diag (1; 1; 1; 1) as which satisfy the commutation relations − − − well as time-reversal T = diag ( 1; 1; 1; 1) are improper LTs and as such part of L−. Second, we look at the component η00, in− which[Ti,T casej]= onei!ijk findsTk . from (1.2) the relation (4.17) These are the same commutation relations as in Eq. (4.13) and,hence,theT form a three-dimensional (irreducible) 1 = ησρ Λ0 Λ0 = Λ0 2 Λii 2 : (1.7) representation of (the Lie algebra of) SU(2).Thisrepresentationmustfitintotheaboveclassificationoρ σ 0 0 f SU(2) − i=1;2;3 representations by an integer or half-integer number j and, simplyX on dimensional grounds, it has to be identified with theItj follows=1representation. that 0 Λ 0 1 : (1.8) j j ≥ 0 " 0 4.1.4When The ΛLorentz0 1 the group LT is said to be orthochronous and part of L , while Λ 0 1 gives a non-orthochronous≥ LT which belongs to L#. In consequence, the Lorentz group≤ consists − out The Lorentz group is of fundamentalimportance for the construction of field theories. It is the symmetry associated of four classes of LTs as illustrated in Figure 1.1 to four-dimensional Lorentz space-time and should be respected by field theories formulated in Lorentz space-time. Let us have a look at some simple example of LTs. A rotation by the angle θ about the Let us begin by formally defining the Lorentz group. With the Lorentz metric η = diag(1, 1, 1, 1) the Lorentz z-axis and a boost by v < 1 along the x-axis group L consists of real 4 4 matrices Λ satisfying − − − × 1 0 0 0 γ γv 0 0 ΛT ηΛ = η . − (4.18) µ 0 cos θ sin θ 0 µ γv γ 0 0 Λ ν = 0 − 1 ; Λ ν = 0− 1 ; (1.9) Special Lorentz transformationsare0 sin theθ identitycos θ14,parity0 P = diag(1, 10, 1, 01) 1,timeinversion 0 T = diag( 1, 1, 1, 1) B C B − − − C − and the product PT = 14.WenotethatthefourmatricesB0 0 0 1C 14,P,T,PTB 0form 0 a 0 finite 1C sub-group of the Lorentz − B C { B } C group. By taking the determinant@ of the defining relationA (4.18) we@ immediately learn thatA det(Λ)= 1 for all Lorentzwith transformations.γ = (1 v2) Further,−1=2 are if part we write of the out Lorentz Eq. (4.18 group.)withindices In fact, any 3-dimensional rotation ± − µ ν ηµνµΛ ρΛ 1σ = 0ηρσ (4.19) Λ ν = ; (1.10) 0 O !0 2 i 2 0 and focus on the component ρ = σ =0we conclude that (Λ 0) =1+ i(Λ 0) 1,soeitherΛ 0 1 0 0 ≥ ≥ or Λ 0 leaves1.Thissignchoicefords2 invariant, since oneΛ has0 combinedOT O = withOOT the= choice1 by definition, for det(Λ) ifleadsO is to an four element classes of of the Lorentz ≤− 3 ! transformations3-dimensional which rotation are summarised group SO in Table(3).1 4.1. Also notethattheLorentzgroupcontainsthree-dimensional rotations since matrices of the form 1The group of 3-dimensional orthogonal matrices is10 denoted by O(3), while its subgroup of matrices with Λ = (4.20) determinant +1 is called the special orthogonal group0SOO(3). " # T satisfy the relation (4.18) and are hence special Lorentz tr2ansformations as long as O satisfies O O = 13. To find the Lie algebraof the Lorentzgroupwe write Λ = 14 +iT +... with purely imaginary 4 4 generators T .Thedefiningrelation(4.18)thenimpliesforthegeneratorsthatT = ηT T η,soT must be anti-symmetric× in the space-space components and symmetric in the space-time components.− The space of such matrices is six- dimensional and spanned by 0 i 00 00i 0 000i 00 i 000 0000 0000 J = ,K = ,K = ,K = , (4.21) i 0 T 1 0000 2 i 000 3 0000 " i # 0000 0000 i 000 44 CHAPTER 4. CLASSICAL FIELD THEORY (j+,j ) dimension name symbol − (0, 0) 1 scalar φ (1/2, 0) 2 left-handed Weyl spinor χL (0, 1/2) 2 right-handed Weyl spinor χR (1/2, 0) (0, 1/2) 4 Dirac spinor ψ ⊕ (1/2, 1/2) 4 vector Aµ FigureTable 1.2: 4.2: Low-dimensional representations representations of of the the Lorentz Lorentzgroup. group. where Ti areAny the LT generators can be (4.16) decomposed of the rotation as the group.product Given of a the rotation, embedding a boost, (4.20) space-inversion of the rotationP group, into the Lorentzand group time-reversal the appearanceT . Let of us the concentrateTi should not on come the continuous as a surprise. transformations. It is straightforward Since to there work out the commutationare three relations rotations and three boosts, one for each space direction, the continuous LTs are described in terms of six parameters. To find the corresponding six generators, i.e., a basis of [J ,J ]=i$ J , [K ,K ]= i$ J , [J ,K ]=i$ K . (4.22) transformation matricesi j thatijk describesk i infinitesimalj − ijk rotationsk i andj boosts,ijk wek write The above matrices can also be written in a four-dimensional covariant form by introducing six 4 4 matrices σ , Λ = 14 + iT ; (1.11) µν labelled by two anti-symmetric four-indices and defined by × where T are purely imaginary 4 4 matrices. Inserting this linearized LT into the defining × ρ ρ ρ relation (1.1), implies (σµν ) σ = i(ηµηνσ ηµσην ) . (4.23) T = ηT T η− : (1.12) By explicit computation one finds that J = 1 $ σ and− K = σ .Introducingsixindependentparameters$µν , This tells us that the generatorsi T must2 ijk bejk anti-symmetrici 0i in the space-space components, but labelledsymmetric by an anti-symmetric in the space-time pair of indices, components. a Lorentz The tra spacensformation of such close matrices to the has identity indeed can dimension be written as six and is spanned by the set (i = 1; 2; 3) ρ ρ i µν ρ ρ ρ Λ σ δ σ $ (σµν ) σ = δ σ + $ σ; . (4.24) $ − 2 0 0 Ji = ; The commutation relations (4.22) for the Lorentz group 0 T arei! very close to the ones for SU(2) in Eq. (4.13). This analogy can be made even more explicit by introducing a new basis of generators 0 i 0 0 01 0 i 0 0 0 0 i (1.13) J ± = (J iK ) . (4.25) i 0 0 0 i 02 0 0i ± 0 i 0 0 0 0 K1 = 0 1 ;K2 = 0 1 ;K3 = 0 1 : 0 0 0 0 i 0 0 0 0 0 0 0 In terms of these generators,B the algebraC (4.22) takesB the formC B C B0 0 0 0C B0 0 0 0C Bi 0 0 0C B C B C B C @ A @ A + @ A [Ji±,Jj±]=i$ijkJk± , [Ji ,Jj−]=0, (4.26) Here (Ti)jk = iijk with ijk the fully anti-symmetric Levi-Civita tensor (123 = +1) are the − that is, preciselygenerators the of formSO of(3).
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