44 Group Theory The matrix D (/,) that represents the Lorentz transformation (.4) L = e i `J` i `K` (.5) is D (/,) (, )=exp( i / /). (.5) And so the generic D (/,) matrix is D (/,) (, )=e z / (.53) with =Rez and =Imz. It is nonunitary and of unit determinant; it is a member of the group SL(,C) of complex unimodular matrices. The (covering) group SL(,C) relates to the Lorentz group SO(3, ) as SU() relates to the rotation group SO(3). Example.3 (The Standard Left-Handed Boost ) For a particle of mass m>, the standard boost that takes the 4-vector k =(m, ) to p =(p, p), where p = p m + p, is a boost in the ˆp direction B(p) =R(ˆp) B 3 (p ) R (ˆp) =exp( ˆp B) (.54) in which cosh = p /m and sinh = p /m, as one may show by expanding the exponential (exercise.33). For = ˆp, one may show (exercise.34) that the matrix D (/,) (, ) is D (/,) (, ˆp) =e ˆp / = I cosh( /) ˆp sinh( /) = I p p (p + m)/(m) ˆp (p m)/(m) = p + m p p m(p + m) in the third line of which the identity matrix I is suppressed. (.55) Under D (/,), the vector ( I, ) transforms like a 4-vector. For tiny and, one may show (exercise.36) that the vector ( I, ) transforms as D (/,) (, )( I)D (/,) (, )= I + D (/,) (, ) D (/,) (, )= +( I) + ^ (.56) which is how the 4-vector (t, x) transforms (.4). Under a finite Lorentz transformation L the 4-vector S a ( I, ) becomes D (/,) (L) S a D (/,) (L) =L a b Sb. (.57)
.3 Two-Dimensional Representations of the Lorentz Group 45 A massless field (x) that responds to a unitary Lorentz transformation U(L) like U(L) (x) U (L) =D (/,) (L ) (Lx) (.58) is called a left-handed Weyl spinor. We will see in example.3 why the action density for such spinors is Lorentz covariant, that is L`(x) =i (x)(@ I r ) (x) (.59) U(L) L`(x) U (L) =L`(Lx). (.6) Example.3 (Why L` Is Lorentz Covariant) We first note that the derivatives @ b in L`(Lx) are with respect to x = Lx. Since the inverse matrix L takes x back to x = L x or in tensor notation x a = L a b xb, the derivative @ b is @ b = @ @xa = @xb @x b @ @x a = L a b @ @x a = @ a L a b. (.6) Now using the abbreviation @ I r @ a S a and the transformation laws (.57 &.58), we have U(L) L`(x) U (L) =i (Lx)D (/,) (L )( = i (Lx)( @ a L a b Sb ) (Lx) which shows that L` is Lorentz covariant. @ a S a )D (/,) (L ) (Lx) = i (Lx)( @ b Sb ) (Lx) =L`(Lx) (.6) Incidentally, the rule (.6) ensures, among other things, that the divergence @ a V a is invariant (@ a V a ) = @ av a = @ b L b a L a cv c = @ b b c V c = @ b V b. (.63) Example.33 (Why is Left Handed) The space-time integral S of the action density L` is stationary when (x) satisfies the wave equation or in momentum space (@ I r ) (x) = (.64) (E + p ) (p) =. (.65) Multiplying from the left by (E p ), we see that the energy of a particle created or annihilated by the field is the same as its momentum E = p in accord with the absence of a mass term in the action density L`. And
46 Group Theory because the spin of the particle is represented by the matrix J = /, the momentum-space relation (.65) says that (p) is an eigenvector of ˆp J ˆp J (p) = (p) (.66) with eigenvalue /. A particle whose spin is opposite to its momentum is said to have negative helicity or to be left handed. Nearly massless neutrinos are nearly left handed. One may add to this action density the Majorana mass term L M (x) = m (x) (x)+ T (x) (x) (.67) which is Lorentz covariant because the matrices and 3 anti-commute with which is antisymmetric (exercise.39). Since charge is conserved, only neutral fields like neutrinos can have Majorana mass terms. The generators of the representation D (,/) with j = and j =/ are given by (.47 &.49) with J + = and J = /; they are J = Thus matrix D (,/) (, (.4) and K = i. (.68) ) that represents the Lorentz transformation L = e i `J` i j K j (.69) is D (,/) (, )=exp( i /+ /) = D (/,) (, ) (.7) which di ers from D (/,) (, ) only by the sign of. The generic D (,/) matrix is the complex unimodular matrix D (,/) (, )=e z / (.7) with =Rez and =Imz. Example.34 (The Standard Right-Handed Boost ) For a particle of mass m>, the standard boost (.54) that transforms k =(m, ) to p =(p, p) isthe4 4 matrix B(p) =exp( ˆp B) in which cosh = p /m and sinh = p /m. This Lorentz transformation with = and = ˆp
.3 Two-Dimensional Representations of the Lorentz Group 47 is represented by the matrix (exercise.35) D (,/) (, ˆp) =e ˆp / = I cosh( /) + ˆp sinh( /) = I p p (p + m)/(m)+ˆp (p m)/(m) = p + m + p p m(p + m) (.7) in the third line of which the identity matrix I is suppressed. Under D (,/), the vector (I, ) transforms as a 4-vector; for tiny z D (,/) (, ) ID (,/) (, )=I + D (,/) (, ) D (,/) (, )= + I + ^ (.73) as in (.4). A massless field (x) that responds to a unitary Lorentz transformation U(L) as U(L) (x) U (L) =D (,/) (L ) (Lx) (.74) is called a right-handed Weyl spinor. One may show (exercise.38) that the action density is Lorentz covariant L r (x) =i (x)(@ I + r ) (x) (.75) U(L) L r (x) U (L) =L r (Lx). (.76) Example.35 (Why Is Right Handed) An argument like that of example (.33) shows that the field (x) satisfies the wave equation or in momentum space Thus, E = p, and (p) is an eigenvector of ˆp J (@ I + r ) (x) = (.77) (E p ) (p) =. (.78) ˆp J (p) = (p) (.79) with eigenvalue /. A particle whose spin is parallel to its momentum is said to have positive helicity or to be right handed. Nearly massless antineutrinos are nearly right handed.
48 Group Theory The Majorana mass term L M (x) = m (x) (x)+ T (x) (x) (.8) like (.67) is Lorentz covariant..33 The Dirac Representation of the Lorentz Group Dirac s representation of SO(3, ) is the direct sum D (/,) D (,/) of D (/,) and D (,/). Its generators are the 4 4 matrices J = and K = i. (.8) Dirac s representation uses the Cli ord algebra of the gamma matrices which satisfy the anticommutation relation a { a, b } a b + b a = ab I (.8) in which is the 4 4 diagonal matrix (.3) with = and jj = for j =,, and 3, and I is the 4 4 identity matrix. Remarkably, the generators of the Lorentz group J ij = ijk J k and J j = K j (.83) may be represented as commutators of gamma matrices J ab = i 4 [ a, They transform the gamma matrices as a 4-vector b ]. (.84) [J ab, c ]= i a bc + i b ac (.85) (exercise.4) and satisfy the commutation relations i[j ab,j cd ]= bc J ad ac J bd da J cb + db J ca (.86) of the Lorentz group (Weinberg, 995, p. 3 7) (exercise.4). The gamma matrices a are not unique; if S is any 4 4 matrix with an inverse, then the matrices a S a S also satisfy the definition (.8). The choice = i and = i (.87)
.33 The Dirac Representation of the Lorentz Group 49 makes J and K block diagonal (.8) and lets us assemble a left-handed spinor and a right-handed spinor neatly into a 4-component spinor =. (.88) Dirac s action density for a 4-spinor is L = ( a @ a + m) (6@ + m) (.89) in which i = =. (.9) The kinetic part is the sum of the left-handed L` and right-handed L r action densities (.59 &.75) a @ a = i (@ I r ) + i (@ I + r ). (.9) If is a left-handed spinor transforming as (.58), then the spinor! = i (.9) i transforms as a right-handed spinor (.74), that is (exercise.4) e z / = e z /. (.93) Similarly, if is right handed, then = is left handed. The simplest 4-spinor is the Majorana spinor M = = = i M (.94) whose particles are the same as its antiparticles. If two Majorana spinors () () M and M have the same mass, then one may combine them into a Dirac spinor D = p () M + i () M = p () + i () D () + i () =. (.95) D The Dirac mass term m D D = m D D + D D (.96)
43 Group Theory conserves charge. For a Majorana field, it reduces to m M M = m + = m + T (.97) = m + T a Majorana mass term (.67 or.8)..34 The Poincaré Group The elements of the Poincaré group are products of Lorentz transformations and translations in space and time. The Lie algebra of the Poincaré group therefore includes the generators J and K of the Lorentz group as well as the hamiltonian H and the momentum operator P which respectively generate translations in time and space. Suppose T (y) is a translation that takes a 4-vector x to x + y and T (z) is a translation that takes a 4-vector x to x + z. ThenT (z)t (y) and T (y)t (z) both take x to x + y + z. So if a translation T (y) =T (t, y) isrepresented by a unitary operator U(t, y) =exp(iht ip y), then the hamiltonian H and the momentum operator P commute with each other [H, P j ] = and [P i,p j ]=. (.98) We can figure out the commutation relations of H and P with the angularmomentum J and boost K operators by realizing that P a =(H, P )isa 4-vector. Let U(, )=e i J i K (.99) be the (infinite-dimensional) unitary operator that represents (in Hilbert space) the infinitesimal Lorentz transformation L = I + R + B (.3) where R and B are the six 4 4 matrices (.3 &.33). Then because P is a 4-vector under Lorentz transformations, we have U (, )PU(, )=e +i J+i K Pe i J i K =(I + R + B) P (.3) or using (.73) (I + i J + i K) H (I i J i K) =H + P (.3) (I + i J + i K) P (I i J i K) =P + H + ^ P.