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3 Classical and Conservation Laws

We have used the existence of symmetries in a physical system as a guiding principle for the construction of their Lagrangians and functionals. We will show now that these symmetries imply the existence of conservation laws. There are different types of symmetries which, roughly, can be classi- fied into two classes: (a) symmetries and (b) internal symmetries. Some symmetries involve discrete operations, hence called discrete symme- tries, while others are continuous symmetries. Furthermore, in some theories these are global symmetries,whileinotherstheyarelocal symmetries.The latter class of symmetries go under the name of gauge symmetries.Wewill see that, in the fully quantized theory, global and local symmetries play different roles. are the most common examples of symmetries that are encountered in . They include invariance and invariance. If the system is isolated, then -translation is also a symme- try. A non-relativistic system is in general under Galilean trans- formations, while relativistic systems, are instead Lorentz invariant. Other spacetime symmetries include time-reversal (T ), (P )andcharge con- jugation (C). These symmetries are discrete. In classical , the existence of symmetries has important conse- quences. Thus, translation invariance,whichisaconsequenceofuniformity of , implies the conservation of the total P of the system. Similarly, implies the conservation of the total L and time translation invariance implies the conservation of the total energy E. All of these concepts have analogs in field theory. However, infieldtheory new symmetries will also appear which do not have an analog in the classi- 52 Classical Symmetries and Conservation Laws cal mechanics of particles. These are the internal symmetries,thatwillbe discussed below in detail.

3.1 Continuous symmetries and Noether’s We will show now that the existence of continuous symmetries has very profound implications, such as the existence of conservation laws.Oneim- portant feature of these conservation laws is the existence of locally conserved currents.Thisisthecontentofthefollowingtheorem,duetoEmmyNoether. Noether’s theorem: For every continuous global there exists aglobalconservationlaw. Before we prove this statement, let us discuss the connectionthatexists between locally conserved currents and constants of .Inparticular, let us show that for every locally there exist a globally , i.e. a . To this effect, let jµ be some locally conserved current, i.e. jµ x satisfies the local constraint ( ) µ = ∂µj x( ) 0(3.1)

( )

V T + ∆T T + ∆T

( )

time Ω

δΩ

T V T space ( )

Figure 3.1 A spacetime 4-.

Let Ωbe a bounded 4-volume of spacetime, with boundary ∂Ω. Then, the 3.2 Internal symmetries 53 (Gauss) theorem tells us that

4 µ µ 0 = d x∂µj x = dSµj x (3.2) !Ω "∂Ω where the r.h.s. is a ( on) the oriented closed( ) surface ∂Ω(a3- volume). Let us suppose that the 4-volume Ωextends all the waytoinfinity in space and has a finite extent in time ∆T . µ If there are no currents at spacial infinity, i.e. lim x →∞ j x,x0 = 0, then only the top (at time T + ∆T )andthebottom(attimeT )oftheboundary ∂Ω(showninFig.(3.1))willcontributetothesurface(bound∣ ∣ ( ary)) integral. Hence, the r.h.s. of Eq.(3.2) becomes

0 0 0 = dS0 j x,T + ∆T − dS0 j x,T (3.3) !V T +∆T !V T 3 ( ) ( ) Since dS0 ≡ d x(,theboundarycontributionsreducetotwooriented3-) ( ) volume

0 = d3xj0 x,T + ∆T − d3xj0 x,T (3.4) !V T +∆T !V T

Thus, the quantity( Q T) ( ) ( ) ( )

3 0 ( )Q T ≡ d xj x,T (3.5) !V T is a constant of motion,i.e.( ) ( ) ( )

Q T + ∆T = Q T ∀ ∆T (3.6)

Hence, the existence of a locally conserved current, satisfying ∂ jµ = 0, ( ) ( ) µ implies the existence of a globally conserved charge (or Noether charge) Q = d3xj0 x,T ,whichisaconstant of motion.Thus,theproofofthe ∫ Noether theorem reduces to proving the existence of a locallyconserved current. In the( following) sections we will prove Noether’s theorem for internal and spacetime symmetries.

3.2 Internal symmetries Let us begin, for simplicity, with the case of the complex field φ x ≠ ∗ φ x .Theargumentsthatfollowbelowareeasilygeneralizedtoother cases. ∗ ∗ Let L φ,∂µφ,φ ,∂µφ be the Lagrangian density. We will assume( that) the( Lagrangian) is invariant under the continuous global internal symmetry ( ) 54 Classical Symmetries and Conservation Laws transformation ′ φ x ↦φ x = eiαφ x ∗ ′∗ −iα ∗ φ x ↦φ x = e φ x (3.7) ( ) ( ) ( ) where α is an arbitrary real (not a !). The system is invariant ( ) ( ) ( ) under the transformation of Eq.(3.7) if the Lagrangian L satisfies ′ ′ ′∗ ′∗ ∗ ∗ L φ ,∂µφ ,φ ,∂µφ ≡ L φ,∂µφ,φ ,∂µφ (3.8) Then we say that the transformation shown in Eq.(3.7) is a global symmetry ( ) ( ) of the system. In particular, for an infinitesimal transformation we have ′ ′∗ ∗ ∗ φ x = φ x + δφ x + ..., φ x = φ x + δφ x + ... (3.9) where δφ x = iαφ x .SinceL is invariant,itsvariationmustbeidentically ( ) ( ) ( ) ( ) ( ) ( ) equal to zero. The variation δL is

( ) δ(L) δL δL ∗ δL ∗ δL = δφ + δ∂µφ + ∗ δφ + ∗ δ∂µφ (3.10) δφ δ∂µφ δφ δ∂µφ Using the equation of motion δL δL − ∂µ = 0(3.11) δφ δ∂µφ and its complex conjugate, we can write" the# variation δL in the form of a total divergence δL δL ∗ δL = ∂µ δφ + ∗ δφ (3.12) δ∂µφ δ∂µφ ∗ ∗ Thus, since δφ = iαφ and δφ =$ −iαφ ,weget %

δL δL ∗ δL = ∂µ i φ − ∗ φ α (3.13) δ∂µφ δ∂µφ Hence, since α is arbitrary, &δL"will vanish identically# ' if and only if the 4- vector jµ,definedby

µ δL δL ∗ j = i φ − ∗ φ (3.14) δ∂µφ δ∂µφ is locally conserved, i.e. ( ) µ δL = 0iff∂µj = 0(3.15) In particular, if L has the form ∗ µ 2 L = ∂µφ ∂ φ − V φ (3.16)

( ) ( ) (∣ ∣ ) 3.3 Global symmetries and representations 55 which is manifestly invariant under the symmetry transformation of Eq.(3.7), we see that the current jµ is given by

µ µ ∗ ∗ µ ∗←→ j = i ∂ φ φ − φ ∂ φ ≡ iφ ∂µφ (3.17) Thus, the presence of a continuous internal symmetry impliestheexistence * + of a locally conserved current. Furthermore, the conserved charge Q is given by

∗←→ Q = d3xj0 x,x = d3xiφ ∂ φ (3.18) ! 0 ! 0

In terms of the canonical momentum( Π) x ,thegloballyconservedchargeQ of the charged scalar field is

(∗ ) ∗ Q = d3xi φ Π − φΠ (3.19) !

( ) 3.3 Global symmetries and group representations Let us generalize the result of the last subsection. Let us consider a scalar field φa which transforms irreducibly under a certain representation of a G.InthecaseconsideredintheprevioussectionthegroupG is the group of complex of unit , the group U 1 .Theelementsof this group, g ∈ U 1 ,havetheformg = eiα. ( ) S ( )1 R

|z| =1 1

Figure 3.2 The U 1 group is isomorphic to the unit while the real numbers R are isomorphic to a . ( ) This of complex numbers forms a group in the sense that, 56 Classical Symmetries and Conservation Laws 1.Itisclosedundercomplexmultiplication,i.e.

iα ′ iβ ′ i α+β g = e ∈ U 1 and g = e ∈ U 1 ⇒ g ∗ g = e ∈ U 1 . ( ) (3.20) 2.Thereisanidentityelement,i.e.( ) g = 1.( ) ( ) 3.Foreveryelementg = eiα ∈ U 1 there is a unique inverse element − − g 1 = e iα ∈ U 1 . ( ) The elements of the group U 1 are in one-to-one correspondence with the ( ) points of the S1.Consequently,theparameterα that labels the transformation (or element of this( ) group) is defined modulo 2π,anditshould be restricted to the 0, 2π .Ontheotherhand,transformations infinitesimally close to the identity element, 1, lie essentially on the line tangent to the circle at 1 and( are isomorphic] to the group of real numbers R.ThegroupU 1 is compact, in the sense that the length of the natural parametrization of its elements is 2π,whichisfinite.Incontrast,thegroup R of real numbers( ) is not compact (see Fig. (3.2)). In the sequel,wewill almost always work with internal symmetries with a compact Lie group. For infinitesimal transformations the groups U 1 and R are essentially identical. There are, however, field configurations for whichtheyarenot. Atypicalcaseisthevortex configuration in two( .) For a vortex the phase of the field on a large circle of R → ∞ winds by 2π.Such configurations would not exist if the was R instead of U 1 (note that analyticity requires that φ → 0asx → 0.) ( ) iθ φ = φ0e

=0 φ θ φ = φ0

Figure 3.3 A vortex

Another example is the N-component real scalar field φa x ,witha = 1,...,N.Inthiscasethesymmetryisthegroupofrotations in N-dimensional ( ) ′a ab b φ x = R φ x (3.21)

( ) ( ) 3.3 Global symmetries and group representations 57 The field φa is said to transform like the N-dimensional (vector) represen- tation of the O N . The elements of the orthogonal group, R ∈ O N ,satisfy ( ) 1.IfR1 ∈ O N and R2 ∈ O N ,thenR1R2 ∈ O N , 2.∃I ∈ O N such that ∀R ∈ O N then RI =( IR) = R, −1 −1 t 3.∀R ∈ O N( )∃R ∈ O N ( such) that R = R (, ) where Rt is( the) transpose of the ( ) R. Similarly,( if the) N-component( ) vector φa x is a complex field,ittransforms under the group of N × N Unitary transformations U ′a ab ( b ) φ x = U φ x (3.22) The complex N × N matrices U are elements of the U N ( ) ( ) and satisfy ( ) 1.U1 ∈ U N and U2 ∈ U N ,thenU1U2 ∈ U N , 2.∃I ∈ U N such that ∀U ∈ U N , UI = IU = U, −1 −1 † † t ∗ 3.∀U ∈ U( N) , ∃U ∈ U(N) such that U =( U),whereU = U . In the particular( ) case discussed( above) φa transforms like the fundamental () representation( ) of U( N) .Ifweimposethefurtherrestrictionthat( ) det U = 1, the group becomes SU N .Forinstance,ifN = 2, the group is SU 2 and ( ) ∣ ∣ ( φ) φ = 1 (3.23) ( ) φ2 it transforms like the spin-1 2representationof" # SU 2 . In general, for an arbitrary continuous Lie group G,thefieldtransforms like / ( ) ′ k k φ x = exp iλ θ φ x (3.24) a ab b where the vector θ is arbitrary and constant (i.e. independent of x). The k ( ) , - ./ ( ) matrices λ are a set of N × N linearly independent matrices which span the of the Lie group G.ForagivenLiegroupG,thenumberof such matrices is D G ,anditisindependentofthedimension N of the representation that was chosen. D G is called the of the group. The k matrices λab are the( generators) of the group in this representation. In general, from a symmetry ( ) of view, the field φ does not have to be a vector, as it can also be a or for the transform under any representation of the group. For simplicity, we will onlyconsiderthe case of the vector representation of O N ,andthefundamental(spinor) and adjoint (vector) (see below) representations of SU N . ( ) ( ) 58 Classical Symmetries and Conservation Laws

For an arbitrary compact Lie group G,thegenerators λj , j = 1,...,D G , † = = are a set of hermitean, λj λj,tracelessmatrices,trλj 0, which obey the commutation relations { } ( ) j k jkl l λ ,λ = if λ (3.25)

The numerical constants f jkl are known as the structure constants of the Lie [ ] group and are the same in all its representations. In addition, the generators have to be normalized. It is standard to require the normalization condition

a b 1 ab trλ λ = δ (3.26) 2 In the case considered above, the complex scalar field φ x ,thesymme- try group is the group of unit length complex numbers of the form eiα. This group is known as the group U 1 .Allitsrepresentationsareone-( ) dimensional, and has only one . AcommonlyusedgroupisSU 2 .Thisgroup,whichisfamiliarfromnon-( ) relativistic , has three generators J1,J2 and J3,thatobey the angular momentum algebra ( )

Ji,Jj = i(ijkJk (3.27) with [ ] 1 tr J J = δ and trJ = 0(3.28) i j 2 ij i

The representations of SU( 2 )are labelled by the angular momentum quan- tum number J.EachrepresentationJ is a 2J + 1-fold degenerate multiplet, i.e. the of the representation( ) is 2J + 1. The lowest non-trivial representation of SU 2 ,i.e. J ≠ 0, is the spinor J = 1 representation which has 2 and is two-dimensional. In this representa- tion, the field φa x is a two-component complex( ) spinor and the generators J ,J J × J = 1 σ 1 2 and 3 are given by the set of 2 2Paulimatrices j 2 j. The vector, or( spin) 1, representation is three-dimensional,andφa is a three-component vector. In this representation, the generators are very sim- ple

Jj kl = (jkl (3.29) Notice that the dimension of this representation (3) is the same as the rank ( ) (3) of the group SU 2 .Inthisrepresentation,knownastheadjoint repre- sentation, the matrix elements of the generators are the structure constants. This is a general property( ) of all Lie groups. In particular, for the group SU N ,whoserankisN 2 − 1, it has N 2 − 1infinitesimalgenerators,and

( ) 3.3 Global symmetries and group representations 59 the dimension of its adjoint (vector) representation is N 2 − 1. For instance, for SU 3 the number of generators is eight. Another important case is the group of of N-dimensional Eu- clidean( space,) O N .Inthiscase,thegrouphasN N − 1 2generators which can be labelled by the matrices Lij i, j = 1,...,N .Thefundamental (vector) representation( ) of O N is N-dimensional and,( in this)/ representa- tion, the generators are ( ) ij ( ) L kl = −i δikδj$ − δi$δjk (3.30)

Lij N It is easy to see that the ( ’s) generate( infinitesimal rotations) in -dimensional space. Quite generally, in a given representation an element of a Liegroupis labelled by a set of Euler denoted by θ.IftheEuleranglesθ are infinitesimal, then the representation matrix exp iλ ⋅ θ is close to the iden- tity, and can be expanded in powers of θ.Toleadingorderinθ the change a in φ is ( ) a ab b δφ x = i λ ⋅ θ φ x + ... (3.31) If φ is real, the conserved current jµ is a ( ) ( ) ( ) δL k = k jµ x a λab φb x (3.32) δ∂µφ x

k = ,...,D G ( ) λk ( ) where 1 .Here,thegenerators( ) are real hermitean matrices. In contrast, for a complex field φa,theconservedcurrentsare ( δ)L δL k = k k ∗ jµ x i a λab φb x − a ∗ λab φb x (3.33) δ∂µφ x δ∂µφ x

( ) " λk ( ) ( ) # Here the generators (are) hermitean matrices( (but) are not all real). Thus, we conclude that the number of conserved currents is equal to the number of generators of the group.Fortheparticularchoice ∗ µ ∗ L = ∂µφa ∂ φa − V φa φa (3.34) the conserved current is ( ) ( ) ( ) k k ∗←→ jµ = iλab φa ∂µφb (3.35) and the conserved charges are

k 3 k ∗←→ Q = d xiλab φa ∂0 φb (3.36) !V where V is the volume of space. 60 Classical Symmetries and Conservation Laws 3.4 Global and local symmetries: gauge invariance The existence of global symmetries assumes that, at least in principle, we can and change all of the components of a field φa x at all points x in space at the same time. Relativistic invariance tells us that, although the theory may posses this global symmetry, in principle this ex( periment) cannot be carried out. One is then led to consider theories which are invariant if the symmetry operations are performed locally.Namely,weshouldrequire that the Lagrangian be invariant under local transformations ′ k k φ x → φ x = exp iλ θ x φ x (3.37) a a ab b For instance, we can demand that the theory of a complex scalarfieldφ x ( ) ( ) , - ( )./ ( ) be invariant under local changes of phase ′ iθ x ( ) φ x → φ x = e φ x (3.38) ( ) The standard local Lagrangian L ( ) ( ) ( ) ∗ µ 2 L = ∂µφ ∂ φ − V φ (3.39) is invariant under global transformations with θ = constant, but it is not in- ( ) ( ) (∣ ∣ ) variant under arbitrary smooth local transformations θ x .Themainprob- lem is that since the of the field does not transformlikethefield itself, the is no longer invariant. Inde( )ed, under a local transformation, we find ′ iθ x iθ x ∂µφ x → ∂µφ x = ∂µ e φ x = e ∂µφ + iφ∂µθ (3.40) ( ) ( ) In order to make L locally invariant we must find a new derivative ( ) ( ) $ ( )% 0 1 Dµ,thecovariant derivative,whichtransformsinthesamewayasthefield φ x under local phase transformations, i.e. ′ ′ iθ x D φ → D φ = e D φ (3.41) ( ) µ µ µ ( ) From a “geometric” point of view we can picture the situation as follows. In order to define the phase of φ x locally, we have to define a local frame, or fiducial field, with respect to which the phase of the field is measured. Local gauge invariance is then the( ) statement that the physical properties of the system must be independent of the particular choice of frame. From this point of view, local gauge invariance is an extension of the to the case of internal symmetries. Now, if we wish to make phase transformations that differ from point to point in spacetime, we have to specify how the phase changes aswegofrom one point x in spacetime to another one y.Inotherwords,wehavetodefine 3.4 Global and local symmetries: gauge invariance 61 a connection that will tell us how to transport the phase of φ from x to y as we travel along some path Γ. Let us consider the situation inwhich x and y are arbitrarily close to each other, i.e. yµ = xµ + dxµ where dxµ is an infinitesimal 4-vector. The change in φ is φ x + dx − φ x = δφ x (3.42) If the transport of φ along some path going from x to x +dx is to correspond ( ) ( ) ( ) to a phase transformation,thenδφ must be proportional to φ.Soweareled to define µ δφ x = iAµ x dx φ x (3.43) where A x is a suitably chosen vector field.Clearly,thisimpliesthatthe µ ( ) ( ) ( ) Dµ must be defined to be ( ) Dµφ ≡ ∂µφ x − ieAµ x φ x ≡ ∂µ − ieAµ φ (3.44) where e is a parameter which we will give the physical interpretationofa ( ) ( ) ( ) * + . How should Aµ x transform? We must choose its transformation law in iθ such a way that Dµφ transforms like φ x itself. Thus, if φ → e φ we have ( ) ′ ′ ′ D φ = ∂ − ieA eiθφ ≡ eiθD φ µ µ ( ) µ (3.45) This requirement can be met if ( )( ) ′ i∂µθ − ieAµ = −ieAµ (3.46)

Hence, Aµ should transform like

′ 1 → = + Aµ Aµ Aµ e∂µθ (3.47) But this is nothing but a gauge transformation! Indeed, if we define the gauge transformation Φ x

≡ 1 ( ) Φ x eθ x (3.48) we see that the vector field A transforms like the vector potential of Maxwell’s µ ( ) ( ) . We conclude that we can promote a global symmetry to a local (i.e. gauge) symmetry by replacing the derivative operator by the covariant derivative. Thus, we can make a system invariant under local gauge transformations at the price of introducing a vector field Aµ,thegaugefield,thatplaysthe role of a connection. From a physical point of view, this result means that the impossibility of making a comparison at a of the phase of the 62 Classical Symmetries and Conservation Laws

field φ x requires that a physical gauge field Aµ x must be . This procedure, that relates the matter and gauge fields through the covariant derivative,( ) is known as minimal coupling. ( ) There is a set of configurations of φ x that changes only because of the presence of the gauge field. These are the configurations φc x . They satisfy the equation ( ) ( ) Dµφc = ∂µ − ieAµ φc ≡ 0(3.49) which is equivalent to the (see Eq. (3.43)) ( ) ∂µφc = ieAµφc (3.50) Let us consider, for example, two points x and y in spacetime at the ends of a path Γ x, y .Foragiven path Γ x, y ,thesolutionofEq.(3.49)isthe path-ordered exponential of a ( ) − ( ) µ ie Γ x,y dzµA z φc x = e ∫ φc y (3.51) ( ) Indeed, under a gauge transformation, the line( ) integral transforms like ( ) ( ) 1 µ ↦ µ + µ e dzµA e dzµA e dzµ e∂ θ !Γ x,y !Γ x,y !Γ x,y µ ( ) = e ( )dzµA z + θ (y )− θ x (3.52) !Γ x,y

Hence, we get ( ) ( ) ( ) ( ) µ µ −ie Γ dzµA −ie Γ dzµA −iθ y iθ x φc y e ∫ ↦ φc y e ∫ e e iθ x ( ) ( ) ≡ e φc x (3.53) ( ) ( ) as it should be. ( ) ( ) However, we may now want to ask how the change of phase of φc depends Γ1 Γ2 on the choice of the path Γ. Thus, let φc y and φc y be solutions of the geodesic equations for two different pathsΓ 1 and Γ2 with the same end points, x and y.Clearly,wehavethatthechangeofphase( ) ( )∆γ given by

µ µ µ ∆γ = −e dzµA + e dzµA ≡ −e dzµA (3.54) ! ! " + Γ1 Γ2 Γ + Here Γ is the closed oriented path + + − Γ = Γ1 ∪ Γ2 (3.55) and

dzµAµ = − dzµAµ (3.56) ! − ! + Γ2 Γ2 3.4 Global and local symmetries: gauge invariance 63

Σ

Γ

Figure 3.4 A closed path Γis the boundary of the open surface Σ.

+ Using Stokes theorem we see that, if Σ is an oriented surface whose + + + boundary is the oriented closed path Γ , ∂Σ ≡ Γ (see Fig.(3.4), then ∆γ is given by the flux Φ Σ of the of the vector field Aµ through the + surface Σ ,i.e. ( ) e µν ∆γ = − dSµν F = −e Φ Σ (3.57) 2 !Σ+ where F µν is the field tensor ( )

µν µ ν ν µ F = ∂ A − ∂ A (3.58) dSµν is the oriented element, and Φ Σ is the flux through the sur- µν face Σ. Both F and dSµν are antisymmetric in their spacetime indices. µν In particular, F can also be written as( a) commutator of two covariant i µν = µ µ F e D ,D (3.59)

µν Thus, F measures the (infinitesimal)[ incompatibility] of displacements along two independent directions. In other words, F µν is a tensor. These results show very clearly that if F µν is non-zero in some region of spacetime, then the phase of φ cannot be uniquely determined: the phase of φc depends on the path Γalongwhichitismeasured. 64 Classical Symmetries and Conservation Laws 3.5 The Aharonov-Bohm effect

The path dependence of the phase of φc is closely related to Aharonov-Bohm Effect. This is a subtle effect, which was first discovered in the context of elementary quantum mechanics, and plays a fundamental role in (quantum) field theory as well. Consider a quantum mechanical particle of charge e and m mov- ing on a . The particle is coupled to an external electromagnetic field Aµ (here µ = 0, 1, 2only,sincethereisnomotionoutoftheplane).Let us consider the geometry shown in Fig.(3.5) in which an infinitesimally thin pierces the plane in the vicinity of some point r = 0. The Σ+

r

Γ2 Φ Γ1

r0 Γ+

Figure 3.5 Geometric setup of the Aharonov-Bohm effect: a magnetic flux + + Φ ≠ 0 is thread through the small hole in the punctured plane Σ .HereΓ + are the oriented outer and inner edges of Σ ;Γ1 and Γ2 are two inequivalent paths from r0 to r described in the text.

Schr¨odinger Equation for this problem is ∂Ψ HΨ = ih (3.60) ̵ ∂t where 1 h e 2 H = ̵ *+ A (3.61) 2m i c is the Hamiltonian. The magnetic field" B = Bzˆ#vanishes everywhere except at r = 0,

B = Φ0 δ r (3.62)

( ) 3.5 The Aharonov-Bohm effect 65 Using Stokes theorem we see that the flux of B through an arbitrary region + + Σ with boundary Γ is

Φ = dS ⋅ B = d$ ⋅ A (3.63) !Σ+ "Γ+ + Hence, Φ = Φ0 for all surfaces Σ that enclose the point r = 0, and it is equal to zero otherwise. Hence, although the magnetic field iszeroforr ≠ 0, the vector potential does not (and cannot) vanish. The function Ψ r can be calculated in a very simple way. Let us define ( ) iθ r Ψ r = e Ψ0 r (3.64) where Ψ r satisfies the Schr¨odinger( Equation) in the absence of the field, 0 ( ) ( ) i.e. ∂Ψ ( ) H Ψ = ih 0 (3.65) 0 0 ̵ ∂t with h2 H = − ̵ *2 (3.66) 0 2m

Since the Ψhas to be differentiable and Ψ0 is single valued, we must also demand the boundary condition that

lim Ψ0 r,t = 0(3.67) r→0 iθ The wave function Ψ = Ψ0 e looks( like) a gauge transformation. But we will discover that there is a subtlety here. Indeed, θ can be determined as follows. By direct substitution we get

h e iθ iθ e h ̵ *+ A e Ψ = e h *θ + A + ̵ * Ψ (3.68) i c 0 ̵ c i 0

Thus, in order" to succeed# ( in our) task, we" only have to require# that A and θ must obey the relation e * + ≡ h̵ θ cA 0(3.69) Or, equivalently, e *θ r = − A r (3.70) hc̵ However, if this relation holds, θ cannot be a smooth function of r.Infact, ( ) ( ) + the line integral of *θ on an arbitrary closed path Γ is given by

d$ ⋅*θ = ∆θ (3.71) !Γ+ 66 Classical Symmetries and Conservation Laws where ∆θ is the total change of θ in one full counterclockwise turn around the path Γ. It is the immediate to see that ∆θ is given by e ∆θ = − d$ ⋅ A (3.72) hc̵ "Γ+ We must conclude that, in general, θ r is a multivalued function of r which has a branch cut going from r = 0outtosomearbitrarypointatinfin- ity. The actual and of( the) branch cut is irrelevant, but the discontinuity ∆θ of θ across the cut is not irrelevant. Hence, Ψ0 is chosen to be a smooth, single valued, solution of the Schr¨odinger equation in the absence of the solenoid, satisfying the boundary condition of Eq. (3.67). Such wave functions are (almost) plane . Since the function θ r is multivalued and, hence, path-dependent, the wave function Ψis also multivalued and path-dependent. In particular, let r0 be some arbitrary point( ) on the plane and Γ r0, r is a path that begins in r0 and ends at r.Thephaseθ r is, for that choice of path, given by e ( ) θ r = θ r − dx ⋅ A x (3.73) 0 ( hc) ! ̵ Γ r0,r

The overlap of two wave( ) functions( ) that( are) defined( by) two different paths Γ1 r0, r and Γ2 r0, r is (with r0 fixed)

∗ Γ Γ = d2r Ψ r Ψ r ( 1) 2 !( ) Γ1 Γ2 ie ⟨ ≡ ∣ d⟩2r Ψ r 2 exp( )+ ( ) d$ ⋅ A − d$ ⋅ A ! 0 hc ! ! ̵ Γ1 r0,r Γ2 r0,r (3.74) ∣ ( )∣ 4 " ( ) ( ) #5 If Γ1 and Γ2 are chosen in such a way that the (where the solenoid is piercing the plane) is always to the left of Γ1 but it is also always to the right of Γ2, the difference of the two line integrals is the circulation of A

d$ ⋅ A − d$ ⋅ A ≡ d$ ⋅ A (3.75) !Γ r ,r !Γ r ,r "Γ+ 1 0 2 0 r0

+ ( ) on the closed, positively( ) oriented, contour( ) Γ = Γ1 r0, r ∪ Γ2 r, r0 .Since this circulation is constant, and equal to the flux Φ, we find that the overlap Γ1 Γ2 is ( ) ( ) ie Γ1 Γ2 = exp Φ (3.76) ⟨ ∣ ⟩ hc̵ where we have taken Ψ0 to⟨ be∣ normalized⟩ 4 to unity.5 The result of Eq.(3.76) is known as the Aharonov-Bohm Effect. 3.6 Non-abelian gauge invariance 67 We find that the overlap is a pure phase factor which, in general, is dif- ferent from one. Notice that, although the wave function is always defined up to a constant arbitrary phase factor, phase changes are physical effects. In addition, for some special choices of Φthe wave function becomes single valued. These values correspond to the choice e Φ = 2πn (3.77) hc̵ where n is an arbitrary integer. This requirement amounts to a condition for the magnetic flux Φ, i.e. hc = ≡ Φ n e nΦ0 (3.78) = hc where Φ0 is the flux quantum,Φ0 " e .# In 1931 Dirac considered the effects of a monopole configurationofmag- netic fields on the quantum mechanical wave functions of charged particles (Dirac, 1931). In Dirac’s construction, a magnetic monopoleisrepresented as a long thin solenoid in three-dimensional space. The magnetic field near the end of the solenoid is the same as that of a magnetic charge m equal to the magnetic flux going through the solenoid. Dirac argued that for the solenoid (the “Dirac ”) to be unobservable, the wave function must be single-valued. This requirement leads to the Dirac quantization condition for the smallest magnetic charge, = me 2πhc̵ (3.79) which we recognize is the same as the flux quantization condition of Eq.(3.78).

3.6 Non-abelian gauge invariance Let us now consider systems with a non-abelian global symmetry. This means that the field φ transforms like some representation of a Lie group G, ′ φa x = Uabφb x (3.80) where U is a matrix that represents the of a group element. The local ( ) ( ) Lagrangian density ∗ µ a 2 L = ∂µφa ∂ φ − V φ (3.81) is invariant under global transformations. (∣ ∣ ) Suppose now that we want to promote this global symmetry to a local one. However, it is also correct for this general case as well that while the potential term V φ 2 is invariant even under local transformations U x ,

(∣ ∣ ) ( ) 68 Classical Symmetries and Conservation Laws the first term of the Lagrangian of Eq.(3.81) is not. Indeed, the of φ does not transform properly (i.e. covariantly) under the action of the Lie group G,

′ ∂µφ x =∂µ U x φ x

= ∂µU x φ x + U x ∂µφ x ( ) [ ( ) ( )] −1 =U x ∂µφ x + U x ∂µU x φ x (3.82) * ( )+ ( ) ( ) ( ) Hence ∂µφ does not transform( )[ in the( ) same way( ) as the( )φ(field.)] We can now follow the same approach that we used in the abelian case and define a covariant derivative operator Dµ which should have the property that Dµφ should obey the same transformation law as the field φ,i.e.

′ Dµφ x = U x Dµφ x (3.83)

It is clear that Dµ is now* both( )+ a differential( )* operator( )+ as well as a matrix acting on the field φ.Thus,Dµ depends on the representation that was chosen for the field φ.Wecannowproceedinanalogywithwhatwedid in the case of electrodynamics, and guess that the covariant derivative Dµ should be of the form

Dµ = I∂µ − igAµ x (3.84) × where g is a coupling constant, I is the N N( )identity matrix, and Aµ is a matrix-valued vector field. If φ has N components, the vector field Aµ x is an N ×N hermitian matrix which can be expanded in the of the group k generators λab (with k = 1,...,D G ,anda, b = 1,...,N)whichspanthe( ) algebra of the Lie group G, ( ) = k k Aµ x ab Aµ x λab (3.85)

Thus, the vector field Aµ x is parametrized by the D G -component 4- k * ( )+ ( ) vectors Aµ x .WewillchoosethetransformationpropertiesofAµ x in such a way that Dµφ transforms( ) covariantly under gauge( transformations,)

′ ′ ( ′) ′ ( ) Dµφ x ≡Dµ U x φ x = ∂µ − igAµ x Uφ x −1 −1 ′ =U x ∂µφ x + U x ∂µU x φ x − igU x Aµ x U x φ x ( ) ( ( ) ( )) , ( )/( ( )) ≡U x Dµφ x (3.86) ( )[ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )] This condition( ) is met( if) we require that −1 ′ −1 U x igAµ x U x = igAµ x + U x ∂µU x (3.87)

( ) ( ) ( ) ( ) ( ) ( ) 3.6 Non-abelian gauge invariance 69 or, equivalently, that Aµ obeys the transformation law i ′ = −1 − −1 Aµ x U x Aµ x U x g ∂µU x U x (3.88) U x Since the matrices( ) ( are) unitary( ) and( ) invertible,* ( we)+ have( ) − U 1 x U x = I (3.89) ( ) ′ we can equivalently write the transformed vector field A x in the form ( ) ( ) µ i ′ = −1 + −1 Aµ x U x Aµ x U x g U x ∂µU ( x) (3.90) This is the general form of a gauge transformation for a non-abelian Lie ( ) ( ) ( ) ( ) ( ), ( )/ group G. In the case of an abelian symmetry group, such as the group U 1 ,the iθ x matrix reduces a simple phase factor, U x = e ,andthefieldAµ x is arealnumber-valuedvectorfield.Itiseasytocheckthat,in( ) this case,( ) Aµ transforms as follows ( ) ( ) i ′ = iθ x −iθ x + iθ x −iθ x Aµ x e Aµ x e g e ∂µ e ( ) 1 ( ) ( ) ( ) ≡ + ( ) Aµ x (g)∂µθ x ( ) (3.91) which recovers the correct form for an abelian gauge transformation. ( ) ( ) Returning now to the non-abelian case, we see that under an infinitesimal transformation U x k k k k U x = exp iλ θ x ≅ δ + iλ θ x + ... (3.92) ab ( ) ab ab ab the scalar field φ x transforms as ( ( )) - , ( )/. ( ) k k δφ x ≅ iλ φ x θ x + ... (3.93) ( ) a ab b while the vector field Ak now transforms as µ ( ) ( ) ( ) 1 k ≅ ksj j s + k + δAµ x if Aµ x θ x g ∂µθ x ... (3.94) k Thus, Aµ x transforms( ) as a vector( ) in the( ) ( ) of the Lie group G since, in that representation, the matrix elements of the generators ksj k are the group( ) structure constants f .NoticethatAµ is always in the adjoint representation of the group G,regardlessoftherepresentationin which φ x happens to be in. From the discussion given above, it is clear that the field Aµ x can be interpreted( ) as a generalization of the vector potential of electromagnetism. Furthermore, Aµ provides for a natural connection which tell us( ) how the 70 Classical Symmetries and Conservation Laws “internal ,” in reference to which the field φ x is defined, changes from one point xµ of spacetime to a neighboring point xµ + dxµ. a In particular, the configurations φ x which are solutions of( the) geodesic equation

ab ( ) Dµ φb x = 0(3.95) correspond to the of(φ)from some point x to some point y. This equation can be written in the equivalent form

k k ∂µφa x = igAµ x λab φb x (3.96)

This linear partial differential( ) equation( ) can be( solved) as follows. Let xµ and yµ be two arbitrary points in spacetime and Γ x, y afixedpathwith endpoints at x and y.Thispathisparametrizedbyamappingzµ from the real interval 0, 1 to M (or any other( space),) zµ ∶ 0, 1 ↦ M,oftheform [ ] [ ] zµ = zµ t ,t∈ 0, 1 (3.97) with the boundary conditions ( ) [ ]

zµ 0 = xµ and zµ 1 = yµ (3.98)

By integrating the geodesic( ) equation Eq.(3.95)( ) along the path Γwe obtain

∂φ z a = µ dzµ ig dzµ Aab z φb z (3.99) !Γ x,y ∂zµ !Γ x,y ( ) Hence, we find that( )φ x must be the solution( ) of the( integral) ( ) equation

µ φ y( =) φ x + ig dzµ A z φ z (3.100) !Γ x,y where we have omitted( ) all( the) indices( to) simplify( the) ( notatio) n. In terms of the parametrization zµ t of the path Γ x, y we can write

1 ( ) dz(µ µ) φ y = φ x + ig dt A z t φ z t (3.101) !0 dt

We will solve this( ) equation( ) by means of an iteration( ( )) ( procedur( )) e, similar to what it is used for the evolution operator in quantum theory. By substituting 3.6 Non-abelian gauge invariance 71 repeatedly the l.h.s. of this equation into its r. h. s. we get the series

1 dzµ t φ y = φ x + ig dt Aµ z t φ x !0 dt 1 t1 ( ) 2 dzµ1 t1 dzµ2 t2 µ1 µ2 +( )ig ( )dt1 dt2 ( ( ))A( )z t1 A z t2 φ x !0 !0 dt1 dt2 n 1 t1 ( ) tn−1( ) dzµ tj + (...)+ ig n dt dt ... dt ( ( ))j A(µj( z))t ( )φ x ! 1 ! 2 ! n % dt j 0 0 0 j=1 j ( ) + ... ( ) 6 * ( )+7(3.102)( )

Here we need to keep in mind that the Aµ’s are matrix-valued fields which are ordered from left to right!. The nested integrals of Eq.(3.102) can be written in the form

1 t t − n 1 n 1 In = ig dt1 dt2 ... dtn F t1 ⋯F tn !0 !0 !0 ig n 1 n ≡ ( )P dt F t ( ) (3.103)( ) n! & !0 ( ) &" ( )# ' where the F ’s are matrices and the operator P& means the path-ordered prod- uct of the objects sitting to its right. If we formally define the exponential of an operator to be equal to its power series expansion,

∞ A 1 n e ≡ A (3.104) ' n! n=0 where A is some arbitrary matrix, we see that the geodesic equation has the formal solution

1 dzµ µ φ y = P exp +ig dt A z t φ x (3.105) & !0 dt or, what is the( ) same& " ( ( ))#' ( )

µ φ y = P exp ig dzµ A z φ x (3.106) & !Γ x,y

Thus, φ y is given( ) by an& operator," the( path-ordered) ( )#'exponential( ) of the line integral of the vector potential Aµ,actingonφ x . By expanding( ) the exponential in a power series, it is easy to check that, under an arbitrary local gauge transformation(U)z ,thepathorderedex-

( ) 72 Classical Symmetries and Conservation Laws ponential transforms as follows

µ ′ P exp ig dz Aµ z t & !Γ x,y µ −1 $ (≡ U y( Pˆ) exp ig( ( )))% dz Aµ z U x (3.107) !Γ x,y

In particular we can( ) consider& " the case( of) a closed( )#' path Γ (x,) x ,wherex is an arbitrary point on Γ. The path-ordered exponential W&Γ x,x ( ) µ ( ) WΓ x,x = P exp ig dz Aµ z (3.108) & & !Γ x,x ( ) is not gauge invariant since, under& " a gauge( transformation) ( )#' it transforms as

′ ˆ µ ′ WΓ x,x = P exp ig dz Aµ z & !Γ x,x ( ) µ −1 = U &x Pˆ " exp (ig ) dz( )A#'µ t U x (3.109) !Γ x,x ( ) & " ( )#' ( ) Therefore W&Γ x,x transforms like a group( ) element, = −1 ( ) W&Γ x,x U x W&Γ x,x U x (3.110)

However, the of W ( x,) x ,whichwedenoteby( ) &Γ ( ) ( ) µ WΓ = tr WΓ x,x ≡( tr P) exp ig dz Aµ z (3.111) & & !Γ x,x ( ) not only is gauge-invariant but& it is" also independent( ) ( of)#' the choice of the point x.However,itisafunctionalofthepathΓ.ThequantityWΓ,which is known as the Wilson loop,playsacrucialroleingaugetheories.Inthe quantum theory this object will become the Wilson loop operator. Let us now consider the case of a small closed path Γ.IfΓis small, then the minimal area a Γ enclosed by Γand its length + Γ are both infinitesimal. In this case, we can expand the exponential in powers and retain only the leading terms. We( ) get ( ) 2 2 µ ig µ WΓ ≈ I + igP dz Aµ z + P dz Aµ z +⋯ (3.112) & &"Γ 2! & "Γ ( ) Stokes theorem says that the( first) integral," the circulation( )# of the vector field Aµ on the closed path Γ, is given by

µ µ ν 1 dzµA z = dx ∧ dx ∂µAν − ∂ν Aµ (3.113) "Γ !!Σ 2 ( ) ( ) 3.6 Non-abelian gauge invariance 73 where ∂Σ = ΓistheinfinitesimalareaelementboundedbyΓ,anddxµ ∧dxν is the oriented infinitesimal area element. Furthermore, thequadraticterm in Eq.(3.112) can be expressed as follows

2 1 µ 1 µ ν P dz Aµ z ≡ dx ∧ dx − Aµ,Aν + ... (3.114) 2! & "Γ 2 !!Σ Therefore," for an infinitesimally( )# small loop, we( get[ ])

ig µ ν 2 WΓ x,x′ ≈ I + dx ∧ dx Fµν + O a Σ (3.115) & 2 !!Σ ( ) where Fµν is the field tensor, defined by ( ( ) )

Fµν ≡ ∂µAν − ∂ν Aµ − ig Aµ,Aν = i Dµ,Dν (3.116) Keep in mind that since the fields A are matrices, the field tensor F is µ [ ] 0 1 µν also a matrix. Notice also that now Fµν is not gauge invariant. Indeed, under a local gauge transformation U x ,Fµν transforms as a transformation ′ − F x = U x F x U 1 x (µν) µν (3.117)

This property follows from the transformation properties of Aµ.However,al- ( ) ( ) ( ) ( ) µν though Fµν itself is not gauge invariant, other quantities such as tr Fµν F are gauge invariant. Let us finally note the form of Fµν in components. By expanding( Fµν in) the basis of the group generators λk (hence, in the algebra of the gauge group), k k Fµν = Fµν λ (3.118)

k we find that the components, Fµν are

k k k k$m $ m Fµν = ∂µAν − ∂ν Aµ + gf AµAν (3.119) The natural local, gauge-invariant, theory for a non-abelian gauge group is the Yang-Mills Lagrangian

1 µν L = − trF F (3.120) 4 µν for a general compact Lie group G,thatwewillcallthegauge group.Notice that the apparent similarity with the Maxwell Lagrangian is only superficial, since in this theory it is not quadratic in the vector potentials. We will see in later chapters that other Lagrangians are possible in other dimensions if some symmetry, e.g. time-reversal, is violated. 74 Classical Symmetries and Conservation Laws 3.7 Gauge invariance and minimal coupling We are now in a position to give a general prescription for the coupling of matter and gauge fields. Since the issue here is local gauge invariance, this prescription is valid for both relativistic and non-relativistic theories. So far, we have considered two cases: (a) fields that describe the of matter and (b) gauge fields that describe electromagnetismandchromo- dynamics. In our description of Maxwell’s electrodynamics we saw that, if the Lagrangian is required to respect local gauge invariance, then only con- served currents can couple to the gauge field. However, we havealsoseen that the presence of a global symmetry is a sufficient conditionfortheexis- tence of a locally conserved current. This is not only a necessary condition since a local symmetry also requires the existence of a conserved current. We will now consider more general Lagrangians that will include both matter and gauge fields. In the last sections we saw that if a system with Lagrangian L φ,∂µφ has a global symmetry φ → Uφ,thenbyreplacing all derivatives by covariant derivatives we promote a globalsymmetryinto alocal(orgauge)symmetry.Wewillproceedwithourgeneral( ) philosophy and write down gauge-invariant Lagrangians for systems which contain both matter and gauge fields. I will give a few explicit examples

3.7.1 Quantum Electrodynamics (QED) is a theory of and . The electrons are described by fields ψα x .Thereasonforthis choice will become clear when we discuss the quantum theory and the Spin- Statistics theorem. Photons are described by a U( 1) gauge field Aµ.The Lagrangian for free electrons is just the free Dirac Lagrangian, LDirac ¯ ¯ ( ) LDirac ψ, ψ = ψ i∂ − m ψ (3.121) The Lagrangian for the gauge field is the free Maxwell Lagrangian L A ( ) ( / ) gauge 1 µν 1 L A = − F F ≡ − F 2 (3.122) gauge 4 µν 4 ( ) The prescription that we will adopt, known as minimal coupling,consistsin ( ) requiring that the total Lagrangian be invariant under local gauge transfor- mations. The free Dirac Lagrangian is invariant under the global phase transforma- tion (i.e. with the same phase factor for all the Dirac components)

′ iθ ψα x → ψα x = e ψα x (3.123)

( ) ( ) ( ) 3.7 Gauge invariance and minimal coupling 75 if θ is a constant, arbitrary phase, but it is not invariant not under the local phase transformation

′ iθ x ψα x → ψα x = e ψα x (3.124) ( ) As we saw before, the matter part of the Lagrangian can be made invariant ( ) ( ) ( ) under the local transformations ′ iθ x ψα x → ψα x = e ψα x ′ ( ) 1 A x → A x = A x + ∂ θ x (3.125) µ ( ) µ( ) µ ( e) µ if the derivative ∂ ψ is replaced by the covariant derivative D µ ( ) ( ) ( ) ( ) µ

Dµ = ∂µ − ieAµ x (3.126) The total Lagrangian is now given by the sum of two terms ( ) ¯ L = Lmatter ψ, ψ,A + Lgauge A (3.127) where L ψ, ψ¯,A is the gauge-invariant extension of the Dirac La- matter ( ) ( ) grangian, i.e. ( ) ¯ ¯ Lmatter ψ, ψ,A = ψ iD − m ψ µ = ψ¯ i∂ − m ψ + eψγ¯ µψA (3.128) ( ) ( / ) L A is the usual Maxwell term and D is a shorthand for D γµ.Thus, gauge * / + µ the total Lagrangian for QED is ( ) / 1 L = ψ¯ iD − m ψ − F 2 (3.129) QED 4 Notice that now both matter and gauge fields are dynamical degrees of ( / ) freedom. The QED Lagrangian has a local gauge invariance. Hence, it also has a locally conserved current. In fact the argument that we used above to show that there are conserved (Noether) currents if there is a continuous global symmetry, is also applicable to gauge invariant Lagrangians. As a matter of fact, under an arbitrary infinitesimal gauge transformation = ¯ = − ¯ = 1 δψ iθψ δψ iθψδAµ e ∂µθ (3.130) the QED Lagrangian remains invariant, i.e. δL = 0. An arbitrary variation of L is δL δL δL δL δL = δψ + δ∂µψ + ψ ↔ ψ¯ + δ∂µAν + δAµ (3.131) δψ δ∂µψ δ∂µAν δAµ ( ) 76 Classical Symmetries and Conservation Laws After using the and the form of the gauge transforma- tion, δL can be written in the form

µ 1 µν δL 1 δL = ∂µ j x θ x − F x ∂µ∂νθ x + ∂µθ x (3.132) e δAµ e µ where j x is[ the( ) ( )] number( current) ( ) ( )

µ ∂L δL ( ) j = i ψ − ψ¯ (3.133) δ∂µψ δ∂µψ¯ µν For smooth gauge transformations6 θ x ,theterm7 F ∂µ∂ν θ vanishes be- cause of the antisymmetry of the field tensor Fµν .Hencewecanwrite ( ) µ µ 1 δL δL = θ x ∂µj x + ∂µθ x j x + (3.134) e δAµ x The first term tells( ) us that( since) the( infinitesimal) & ( ) gauge tran' sformation θ x µ ( ) µ is arbitrary, the Dirac current j x locally is conserved,i.e.∂µj = 0. µ Let us define the charge (or gauge)currentJ x by the relation ( ) ( ) µ δL J x ≡ ( ) (3.135) δAµ x which is the current that enters( in) the Equation of Motion for the gauge ( ) field Aµ,i.e.theMaxwellequations.ThevanishingofthesecondtermofEq. (3.134), required since the changes of the infinitesimal gauge transformations are also arbitrary, tells us that the charge current and the number current are related by

Jµ x = −ejµ x = −e ψγ¯ µψ (3.136) This relation tells us that since jµ x is locally conserved, then the global ( ) ( ) conservation of Q0

( ) † Q = d3xj x ≡ d3xψ x ψ x (3.137) 0 ! 0 ! implies the global conservation of( the) ( ) (Q) † Q ≡ −eQ = −e d3xψ x ψ x (3.138) 0 !

This property justifies the interpretation of the( )coupling( ) constant e as the electric charge.InparticularthegaugetransformationofEq.(3.125),tells us that the matter field ψ x represents excitations that carry the unit of charge, ±e .Fromthispointofview,theelectricchargecanberegardedas a .Thispointofviewbecomesveryusefulinthequantum( ) 3.7 Gauge invariance and minimal coupling 77 theory in the strong coupling limit. In this case, under special circumstances, the excitations may acquire unusual quantum numbers. This is not the case of Quantum Electrodynamics, but it is the case of a number of theories in one and two space dimensions, with applications in Condensed Matter systems such as polyacetylene, or the two-dimensional electron in high magnetic fields, i.e. the fractional quantum Hall effect, or in gauge theories with magnetic monopoles).

3.7.2 Quantum Chromodynamics (QCD) is the gauge field theory of strong in- teractions in hadron physics. In this theory the elementary constituents of i hadrons, the ,arerepresentedbytheDiracspinorfieldψα x .The a theory also contains a set of gauge fields Aµ x that represent the gluons. The fields have both Dirac indices α = 1,...,4andcolor( indices) i = 1,...,Nc,whereNc is the number of colors.( ) In the of weak, strong interactions, and electromagnetic in particlephysics,andin QCD, there are in addition Nf = 6flavorsofquarks,groupedintothree generations, each labeled by a flavor index, and six flavors of leptons, also grouped into three generations. The flavor symmetry is a global symmetry of the theory. Quarks are assumed to transform under the fundamental representation of the gauge (or color) group G,saySU Nc .Thetheoryisinvariantun- der the group of gauge transformations. In QCD, the color group is SU 3 and so Nc = 3. The color symmetry is a non-abelian( ) gauge symmetry. The gauge field Aµ is needed in order to enforce local gauge invariance. In com-( ) a a a ponents, we get Aµ = Aµλ ,whereλ are the generators of SU Nc .Thus, 2 2 a = 1,...,D SU Nc ,andD SU Nc = Nc − 1. Thus, Aµ is an Nc − 1 2 dimensional vector in the adjoint representation of G.ForSU 3 (,Nc)−1 = 8 and there are( eight( generators.)) ( ( ) The gauge-invariant matter term of the Lagrangian, Lmatter( is) ¯ ¯ Lmatter ψ, ψ,A = ψ iD − m ψ (3.139) where D = ∂ − igA ≡ ∂ − igAaλa is the covariant derivative. The gauge field ( ) ( / ) term of the Lagrangian Lgauge / / / / / 1 µν 1 a µν L A = − trF F ≡ − F F (3.140) gauge 4 µν 4 µν a is the Yang-Mills Lagrangian. The total Lagrangian for QCD is L ( ) QCD ¯ LQCD = Lmatter ψ, ψ,A + Lgauge A (3.141)

( ) ( ) 78 Classical Symmetries and Conservation Laws Can we define a ?Sincethecolorgroupisnon-abelianit has more than one generator. We showed before that there are asmany conserved currents as generators are in the group. Now, in general, the group generators do not commute with each other. For instance, in SU 2 there is only one generator, J3,whileinSU 3 there are only two diagonal a generators, etc. Can all the global charges Q ( )

a † a( ) Q ≡ d3xψ x λ ψ x (3.142) ! be defined simultaneously? It is straightforward( ) ( ) to show thatthePoisson Brackets of any pair of charges are, in general, different from zero. We will see below, when we quantize the theory, that the charges Qa obey the same commutation relations as the group generators themselves do. So, in the quantum theory, the only charges that can be assigned to states are precisely the same as the quantum numbers that label the representations. Thus, if the group is SU 2 ,wecanonlyassigntothestatesthevaluesofthequadratic 2 Casimir operator J and of the J3.Similarrestrictionsapplyto the case of SU( ) 3 and to other Lie groups.

( ) 3.8 Spacetime symmetries and the energy-momentum tensor Until now we have considered only the role of internal symmetries. We now turn to spacetime symmetries, and consider the role of coordinate transfor- mations. In this more general setting we will have to require the invariance of the action rather than only of the Lagrangian,aswedidforinternal symmetries. There are three continuous spacetime symmetries that will beimportant to us: a) translation invariance, b) rotation invariance andc) of time. While rotations are a subgroup of , space and time translations are examples of inhomogeneous Lorentztransforma- tions (in the relativistic case) and of Galilean transformations (in the non- relativistic case). Inhomogeneous Lorentz transformations also form a group, known as the Poincar´egroup. Note that the transformations discussed above are particular cases of more general coordinate transformations. However, it is important to keep in mind that, in most cases, general coordinate transfor- mations are not symmetries of an arbitrary system. They are the symmetries of . In what follows we are going to consider the response of a system to infinitesimal coordinate transformations of the form ′ xµ = xµ + δxµ (3.143) 3.8 Spacetime symmetries and the energy-momentum tensor 79 where δxµ may be a function of the spacetime point xµ.Underacoordinate transformation the fields change as

′ ′ µ φ x → φ x = φ x + δφ x + ∂µφδx (3.144) where δφ is the variation( ) of(φ in) the(absence) (of) a change of coordinates , i.e. afunctionalchange.Inthisnotation,auniforminfinitesimal translation by aconstantvectoraµ has δxµ = aµ and an infinitesimal rotation of the space axes is δx0 = 0andδxi = (ijkθjxk. In general, the action of the system is not invariant under arbitrary changes in both coordinates and fields. Indeed, under an arbitrary change ′ 4 of coordinates xµ → xµ xµ ,thevolumeelementd x is not invariant and changes by a multiplicative factor of the form

( ) ′ d4x = d4xJ (3.145) where J is the Jacobian of the coordinate transformation ′ ′ ′ ∂x ⋯x ∂xµ J = 1 4 ≡ det (3.146) ∂x1⋯x4 ∂xj 8 8 8 8 ′ 8= 6 78 For an infinitesimal transformation, xµ 8 xµ + δxµ 8x ,weget 8 8 ′ ∂xµ ν ν ( ) = gµ + ∂ δxµ (3.147) ∂xν

Since δxµ is small, the Jacobian can be approximated by

′ ∂xµ ν ν ν J = det = det g + ∂ δx ≈ 1 + tr ∂ δx + O δx2 (3.148) ∂x µ µ µ 8 ν 8 8 8 8 8 Thus 8 6 78 9 * +9 ( ) ( ) 8 8 8 8 µ 2 J ≈ 1 + ∂ δxµ + O δx (3.149)

The Lagrangian itself is in general not invariant.( ) For instance, even though we will always be interested in systems whose Lagrangians arenotanex- plicit function of x,stilltheyarenotingeneralinvariantunderthegiven transformation of coordinates. Also, under a coordinate change the fields may also transform. Thus, in general, δL does not vanish. The most general variation of L is

µ δL δL δL = ∂µL δx + δφ + δ∂µφ (3.150) δφ δ∂µφ 80 Classical Symmetries and Conservation Laws If φ obeys the equations of motion, δL δL = ∂µ (3.151) δφ δ∂µφ then, the general change δL obeyed by the solutions of the equations of motion is µ δL δL = δx ∂µL + ∂µ δφ (3.152) δ∂µφ The total change in the action is a sum" of two terms#

δS = δ d4x L = δd4x L + d4xδL (3.153) ! ! ! where the change in the integration measure is due to the Jacobian factor,

4 4 µ δd x = d x∂µδx (3.154)

Hence, δS is given by

4 µ µ δL δS = d x ∂µδx L + δx ∂µL + ∂µ δφ (3.155) ! δ∂µφ

Since the total variation&( of)φ, δT φ,isthesumofthefunctionalchangeof" #' the fields plus the changes in the fields caused by the coordinate transfor- mation, µ δT φ ≡ δφ + ∂µφδx (3.156) we can write δS as a sum of two contributions: one due to change of coor- dinates and another due to functional changes of the fields:

4 µ µ δL ν δS = d x ∂µδx L + δx ∂µL + ∂µ δT φ − ∂νφδx (3.157) ! δ∂µφ Therefore, for& the( change) of the action we" get ( )#'

4 µ δL ν δL δS = d x ∂µ gν L − ∂νφ δx + ∂µ δT φ (3.158) ! δ∂µφ δ∂µφ We have already4 encountered&" the # term' when& we discusse'5 dthecase of internal symmetries. The first term represents the change of the action S as a result of a change of coordinates. We will now consider a few explicit examples. To simplify we will consider the effects of coordinate transformations alone. Forsimplicity,here we will restrict our discussion to the case of the scalar field. 3.8 Spacetime symmetries and the energy-momentum tensor 81 3.8.1 Spacetime translations

Under a uniform infinitesimal translation δxµ = aµ,thefieldφ does not change

δT φ = 0(3.159) The change of the action now is

4 µν δL ν ν δS = d x∂µ g L − ∂ φ a (3.160) ! δ∂µφ For a system which is isolated and"translationally invariant# the action must not change under a redefinition of the origin of the coordinatesystem.Thus, µν δS = 0. Since aµ is arbitrary,itfollowsthatthetensorT

µν µν δL ν T ≡ −g L + ∂ φ (3.161) δ∂µφ is conserved, µν ∂µT = 0(3.162)

The tensor T µν is known as the energy-momentum tensor.Thereasonfor this name is the following. Given that T µν is locally conserved, by Noether’s theorem we can define the 4-vector P ν

ν ν P = d3xT0 x,x (3.163) ! 0 which is a constant of motion.Inparticular,( P 0 )is given by

0 3 00 3 δL 0 P = d xT x,x0 ≡ d x −L + ∂ φ (3.164) ! ! δ∂0φ L But δ is just the canonical( momentum) & Π x , ' δ∂0φ δL Π x = ( ) (3.165) δ∂0φ Then we can easily recognize that( the) quantity in brackets in Eq.(3.164) in the definition of P 0 is just the Hamiltonian density H

H = Π ∂0φ − L (3.166)

Therefore, the time component of P ν is just the total energy of the system

P 0 = d3x H (3.167) ! 82 Classical Symmetries and Conservation Laws

The space components P j are

j j j δL j P = d3xT0 = d3x −g0 L + ∂ φ (3.168) ! ! δ∂j φ

Thus, since g0j = 0, we get & '

P = d3x Π x ∂φ x (3.169) !

The vector P is identified with the total( ) linear( momentum) since (a) it is a constant of motion, and (b) it is the generator of infinitesimal space trans- lations. For the same reasons we will denote the component T 0j x with the linear momentum density Pj x .Itisimportanttostressthatthecanonical j momentum Π x and the total linear momentum density P are( ) obviously completely different physical( quantities.) While the canonical momentum is afieldwhichiscanonicallyconjugatetothefield( ) φ,thetotalmomentumis the linear momentum stored in the field, i.e. the linear momentum of the .

3.8.2 Rotations If the action is invariant under global infinitesimal Lorentztransformations, of which spacial rotations are a particular case, ν δxµ = ωµ xν (3.170) where ωµν is infinitesimal, and antisymmetric, then the variation of the action is zero, δS = 0. If φ is a scalar field,thenδT φ is also zero. This is not the case for spinor or vector fields which transforms under Lorentz transformations. Because of their transformation properties of these fields, the angular momentum tensor that we define below will be missing the contribution representing the spin of the field (which vanishes for a scalar field). Here we will consider only the case of scalars. Then, since for a scalar field δT φ = 0, we find

4 µν δL ν νρ δS = 0 = d x∂µ g L − ∂ φ ω xρ (3.171) ! δ∂µφ Since ωνρ is constant and arbitrary,&" the # in brackets' must also define a conserved current. Let M µνρ be the tensor defined by µνρ µν ρ µρ ν M ≡ T x − T x (3.172) 3.8 Spacetime symmetries and the energy-momentum tensor 83 1 ω M µνρ in terms of which, the quantity in brackets in Eq.(3.171) becomes 2 νρ . Thus, for arbitrary constant ω,wefindthatthetensorM µνρ is locally con- served, µνρ ∂µM = 0(3.173) In particular, the transformation

δx0 = 0,δxj = ωjkxk (3.174) represents an infinitesimal rotation of the spatial axes with

ωjk = (jklθl (3.175) µνρ where θl are three infinitesimal . Thus, we suspect that M must be related with the total angular momentum. Indeed, the local conser- vation of the current M µνρ leads to the global conservation of the tensorial quantity Lνρ νρ νρ L ≡ d3xM0 x,x (3.176) ! 0

In particular, the space components of Lνρ (are )

L = d3x T x x − T x x jk ! 0j k 0k j = d3x P x x − P x x (3.177) ! ( j ( ) j k ( ) j )

If we denote by Lj the (pseudo)( vector( ) ( ) ) 1 L ≡ ( L (3.178) j 2 jkl kl we get L ≡ d3x( x P x ≡ d3x+ x (3.179) j ! jkl k l ! j

The vector Lj is the generator of infinitesimal( ) rotations( and) is thus identified with the total angular momentum,whereas+j x is the corresponding (spa- cial) angular momentum density. Notice that, since we are dealing with a scalar field, there is no spin contribution to the( angular) momentum density. The generalized angular momentum tensor Lνρ of Eq.(3.176) is not trans- lationally invariant since under a displacement of the origin of the coordinate system by aµ, Lνρ changes by an amount aνP ρ − aρP ν.Atrulyintrinsican- gular momentum is given by the Pauli-Lubanski vector W µ, νλ ρ µ 1 µνλρ L P W = − ( (3.180) 2 P 2 : 84 Classical Symmetries and Conservation Laws which, in the rest frame P = 0, reduces to the angular momentum. Finally, we find that if the angular momentum tensor M µνλ has the form µνλ µν λ µλ ν M = T x − T x (3.181)

Then, the conservation of the energy-momentum tensor T µν and of the angu- lar momentum tensor M µνλ together lead to the condition that the energy- momentum tensor should be a symmetric second rank tensor, µν µν T = T (3.182)

Thus, we conclude that the conservation of angular momentum requires that the energy momentum tensor T µν for a scalar field be symmetric. The expression for T µν that we derived in Eq.(3.161) is not manifestly µν µν symmetric. However, if T is conserved, then the “improved” tensor T̃ µν = µν + µνλ T̃ T ∂λK (3.183) is also conserved, provided the tensor Kµνλ is antisymmetric in µ, λ and µνλ µν ν,λ .ItisalwayspossibletofindsuchatensorK to make T̃ sym- . The improved, symmetric, energy-momentum tensor isknownasthe( ) (Belinfante) energy-momentum tensor. In particular, for the scalar field φ x whose Lagrangian density L is 1 L = ∂ φ 2 − V φ (3.184) 2 µ( ) µν the locally conserved energy-momentum( ) tensor( )T is

µν µν δL ν µν µ ν T = −g L + ∂ φ ≡ −g L + ∂ φ∂ φ (3.185) δ∂µφ which is symmetric. The conserved energy-momentum 4-vectoris

µ µ µ P = d3x −g0 L + ∂0φ∂ φ (3.186) !

Thus, we find that ( ) 1 1 P 0 = d3x Π ∂ φ − L = d3x Π2 + *φ 2 + V φ (3.187) ! 0 ! 2 2 is the total energy( of the field,) and [ ( ) ( )]

P = d3x Π x *φ x (3.188) ! is the linear momentum P of the field.( Both) are( ) constants of motion. 3.9 The energy-momentum tensor for the electromagnetic field 85 3.9 The energy-momentum tensor for the electromagnetic field

For the case of the Maxwell field Aµ,astraightforwardapplicationofthese methods yields an energy-momentum tensor T µν of the form

µν µν δL µ T = −g L + ∂ Aλ δ∂ν Aλ 1 µν αβ νλ µ = g F F − F ∂ A (3.189) 4 αβ λ µν It obeys ∂µT = 0and,hence,islocallyconserved.However,thistensor is not gauge invariant. We can construct a gauge-invariant and conserved energy momentum tensor by exploiting the ambiguity in the definition of µν T .Thus,ifwechooseKµνλ = FνλAµ,whichisanti-symmetricinthe indices ν and λ,wecanconstructtherequiredgauge-invariantandconserved energy momentum-tensor

µν 1 µν ν µλ T = g F 2 − F F (3.190) ̃ 4 λ where we used the equation of motion of the free electromagnetic field, λ ∂ Fνλ = 0. Notice that this “improved” energy-momentum tensor is both gauge-invariant and symmetric. From here we find that the 4-vector µ µ P = d3x T 0 (3.191) ! ̃ x0 fixed is a constant of motion. Thus, we identify 1 P 0 = d3x T 00 = d3x E2 + B2 (3.192) ! ̃ ! x0 fixed x0 fixed 2 with the Hamiltonian, and , / i i P = d3x T 0 = d3x E × B (3.193) ! ̃ ! i x0 fixed x0 fixed with the linear momentum (or Poynting vector) of the( electro) magnetic field.

3.10 The energy-momentum tensor and changes in the geometry The energy momentum tensor T µν appears in classical field theory as a result of the translation invariance, in both space and time, of the physical system. We have seen in the previous section that for a scalar field T µν is a as a consequence of the conservation of angular momentum. The definition that we found does not require that T µν should have any definite symmetry. However we found that it is always possible to modify T µν by 86 Classical Symmetries and Conservation Laws adding a suitably chosen antisymmetric conserved tensor to find a symmetric version of T µν .Giventhisfact,itisnaturaltoaskifthereisawaytodefine the energy-momentum tensor in a such a way that it is always symmetric. This issue becomes important if we want to consider theories for systems which contain fields which are not scalars. It turns out that it is possible to regard T µν as the change in the action due to a change of the geometry in which the system lives. From classi- cal physics, we are familiar with the fact that when a body is distorted in some manner, in general its energy increases since we have to perform some work against the body in order to deform it. A deformation of a body is a change of the geometry in which its component parts evolve. Examples of such changes of geometry are shear distortions, dilatations, bendings, and twists. On the other hand, there are changes that do not cost any energy since they are symmetry operations. Examples of symmetry operations are translations and rotations. These symmetry operations can be viewed as simple changes in the coordinates of the parts of the body which do not change its geometrical properties, i.e. the and angles of different points. Thus, coordinate transformations do not alter the energy of the sys- tem. The same type of arguments apply to any . In the most general case, we have to consider transformations whichleavetheac- tion invariant. This leads us to consider how changes in the geometry of the spacetime affect the action of a dynamical system. The information about the geometry in which a system evolves is encoded in the of the space (and spacetime). The metric tensor is a symmetric tensor that specifies how to measure the distance ds between a pair of nearby points x and x + dx,namely 2 µν ∣ ∣ ds = g x dxµ dxν (3.194) Under an arbitrary local change of coordinates x → x + δx ,themetric ( ) µ µ µ tensor changes as follows λ ρ ′ ′ ∂x ∂x g x = g x (3.195) µν λρ ∂x′µ ∂x′ν For an infinitesimal change,( this) means( ) that the functional change in the metric tensor δgµν is

1 ν µ λ δg = − g ∂ δx + g ∂ δx + ∂ g δx (3.196) µν 2 µλ λ λν λ µν λ 4 The , invariant, under coordinate transformations,/ is d x g, where g is the of the metric tensor. : 3.10 The energy-momentum tensor and changes in the geometry 87 Coordinate transformations change the metric of spacetime but do not change the action. Physical or geometric changes, are changes in the met- ric tensor which are not due to coordinate transformations. For a system in a space with metric tensor gµν x ,notnecessarilytheMinkowski(or Euclidean) metric, the change of the action is a linear function of the in- µν finitesimal change in the metric δg ( x) (i.e. “Hooke’s Law”). Thus, we can write the change of the action due to an arbitrary infinitesimal change of the metric in the form ( )

µν δS = d4x gT x δg x (3.197) ! µν : Below we will identify the proportionality constant,( ) ( the) tensor T µν ,withthe conserved energy-momentum tensor. This definition implies that T µν can be regarded as the derivative of the action with respect to the metric

µν δS T x ≡ (3.198) δgµν x Since the metric tensor is symmetric,( ) this definition always yields a sym- ( ) metric energy momentum tensor. In order to prove that this definition of the energy-momentum tensor agrees with the one we obtained before (which was not unique!)wehaveto prove that this form of T µν that we have just defined is a conserved current for coordinate transformations. Under an arbitrary local change of coordi- nates, which leave the distance ds unchanged, the metric tensor changes by the δgµν given above. The change of the action δS must be zero for this case. We see that if we substitute the expression for δgµν in δS,thenan integration by parts will yield a . Indeed, for the particular case of a flat metric, such as the Minkowski or Euclidean metrics, the change δS is 1 µν λ λ δS = − d4xT x g ∂ δx + g ∂ δx (3.199) 2 ! µλ ν λν µ since for a global coordinate transformation( ), the Jacobian/ factor g and the measure of spacetime d4x are constant. Thus, if δS is to vanish for : an arbitrary change δx,thetensorT µν x has to be a locally conserved µν current, i.e. ∂µT x = 0. This definition can be extended to the case of more general . We should note, however,( ) that the energy-momentum tensor can be made( symmetric) only if the space does not have a property known as torsion. Finally, let us remark that this definition of the energy-momentum tensor 88 Classical Symmetries and Conservation Laws also allows us to identify the spacial components of T µν with the - energy tensor of the system. Indeed, for a change in geometry which is does not vary with time, the change of the action reduces to a change of the total energy of the system. Hence, the space components oftheenergy momentum tensor tell us how much does the total energy change for a specific deformation of the geometry. But this is precisely what the stress energy tensor is!