L. Brink We then introduce the"second-quantized"representation O in terms of the "first-quantized"representation o as 0=2id+xd+o(x)oo(x). We then find that the commutator between two of the generators JI and J2 is J1,J21=2i/dxa+o(x)Ui,jlo(r) We can understand that P- truly is the Hamiltonian using(5) P-=2/d+xa+9(x)3-p(r) Legendre transforming to the lagrangian using the field momenta from(12)we get the action S=/4x+5(x)-(x)+a+0()-5(x)-2)+5x)9(x) dxo+p(x)口φ(x It is remarkable that there is a unique form of the kinematic term for any spin-n field We should remember though that to specify the theory we have to give all Poincare generators, since the action via the Hamiltonian is just one of those generators. They will show what spin the field describes <e In this representation it is straightforward to try to add interaction terms to the miltonian. This was done in [12]. Every dynamical generator will have interac- tion terms. The procedure is very painstaking and there are as far as I know no other way than trial and error to find the non-linear representation. On the other hand, such a is found it represents a possible relativistically invariant nteracting field theory. The result is that for every integer n there exists a possible three-point interaction. For even, the unique solutions are s=/dxs()oo(x) +812(-1)(n)5(x)0D+(x)xx(x)+cc +O For A odd, the field o(x) must be in the adjoint representation of an external group oa()and we have to introduce the fully antisymmetric structure consta fabe in the interaction terms to find a possible term. (It is really by checking the four- point coupling that we find that the field has to be representation of a Lie group The results is
36 L. Brink We then introduce the “second-quantized” representation O in terms of the “first-quantized” representation o as O = 2i d4x∂+φ¯(x) o φ(x). We then find that the commutator between two of the generators J1 and J2 is [J1,J2] = 2i d4x∂+φ¯(x)[ j1, j2]φ(x). (13) We can understand that P− truly is the Hamiltonian using (5) P− = 2 d4x∂+φ¯(x) ∂ ¯ ∂ ∂+ φ(x). (14) Legendre transforming to the Lagrangian using the field momenta from (12) we get the action S = d4x ∂+φ¯(x)∂−φ(x) +∂+φ(x)∂−φ¯(x)−2∂+φ¯(x) ∂ ¯ ∂ ∂+ φ(x) = d4x∂+φ¯(x)✷φ(x). (15) It is remarkable that there is a unique form of the kinematic term for any spin-λ field. We should remember though that to specify the theory we have to give all Poincar´e generators, since the action via the Hamiltonian is just one of those generators. They will show what spin the field describes. In this representation it is straightforward to try to add interaction terms to the Hamiltonian. This was done in [12]. Every dynamical generator will have interaction terms. The procedure is very painstaking and there are as far as I know no other way than trial and error to find the non-linear representation. On the other hand, once such a representation is found it represents a possible relativistically invariant interacting field theory. The result is that for every integer λ there exists a possible three-point interaction. For λ even, the unique solutions are S = d4 x φ¯(x)✷φ(x) +g λ ∑ n=0 (−1) n λ n φ¯(x)∂+λ ( ¯ ∂λ−n ∂+λ−n φ(x) ¯ ∂λ ∂+λ φ(x)) +c.c. +O(g2 ). (16) For λ odd, the field φ(x) must be in the adjoint representation of an external group φa(x) and we have to introduce the fully antisymmetric structure constants f abc in the interaction terms to find a possible term. (It is really by checking the fourpoint coupling that we find that the field has to be representation of a Lie group.) The results is
Light-Cone Field Theory /1(=o (17) e note the non-locality in the interaction term in terms of inverses of d+.The easiest way to understand it is to Fourier transform to momentum space. In the calculations it is really defined by the rule dff(x+)=f(x+). When performing a calculation one has to specify exactly the situation of the pole in d*. In an sense this is a remainder of the gauge invariance. We can now check for special values of a The dimension of the coupling constant g is l(in mass units)and this is the usual o--theory. This theory is superrenormalizable but not physical since it does not have a stable vacuum having a potential with no minimum =1 The dimension of the coupling constant g is 0 and this theory is nothing but non-abelian gauge theory in a specific gauge. If we go on we know that we need a four-point coupling to fully close the algebra. Note that the action has no local ymmetry and the gauge group only appears as the external symmetry group The dimension of the coupling constant g is -l and this theory is the beginning series of a gravity theory. It is clear from the dimensions of the coupling constant that interaction terms to arbitrary order can be constructed without serious non localities. The four-point function related to Einsteins theory is known [14]. Going beyond the four-point coupling is probably too difficult, unless powerful computer methods could be devised. We expect several solutions, of course, since we know that the Hilbert action is but the simplest of all actions consistent with the equiv alence principle. Note that the action above, which is a fully gauge fixed Hilbert ction expanded in the fluctuations around the Minkowski metric, has no local sym- metry,no covariance and knows nothing about curved spaces. It is probably useless for discussions about global properties of space and time but can be useful in study of quantum corrections; to understand the finiteness properties of the quantum theory ·元>2 The dimension of the coupling constant g is <-l and these theories are theo- ries for higher spins. Again they are non-renormalizable in the naive sense like the
Light-Cone Field Theory 37 S = d4 x φ¯a(x)✷φa(x) +g f abc λ ∑ n=0 (−1) n λ n φ¯a(x)∂+λ ( ¯ ∂λ−n ∂+λ−n φb (x) ¯ ∂λ ∂+λ φc (x)) +c.c. +O(g2 ). (17) We note the non-locality in the interaction term in terms of inverses of ∂+. The easiest way to understand it is to Fourier transform to momentum space. In the calculations it is really defined by the rule 1 ∂+ ∂+ f(x+) = f(x+). When performing a calculation one has to specify exactly the situation of the pole in ∂+. In an sense this is a remainder of the gauge invariance. We can now check for special values of λ. • λ = 0 The dimension of the coupling constant g is 1 (in mass units) and this is the usual φ3-theory. This theory is superrenormalizable but not physical since it does not have a stable vacuum having a potential with no minimum. • λ = 1 The dimension of the coupling constant g is 0 and this theory is nothing but non-abelian gauge theory in a specific gauge. If we go on we know that we need a four-point coupling to fully close the algebra. Note that the action has no local symmetry and the gauge group only appears as the external symmetry group. • λ = 2 The dimension of the coupling constant g is −1 and this theory is the beginning series of a gravity theory. It is clear from the dimensions of the coupling constant that interaction terms to arbitrary order can be constructed without serious nonlocalities. The four-point function related to Einstein’s theory is known [14]. Going beyond the four-point coupling is probably too difficult, unless powerful computer methods could be devised. We expect several solutions, of course, since we know that the Hilbert action is but the simplest of all actions consistent with the equivalence principle. Note that the action above, which is a fully gauge fixed Hilbert action expanded in the fluctuations around the Minkowski metric, has no local symmetry, no covariance and knows nothing about curved spaces. It is probably useless for discussions about global properties of space and time but can be useful in the study of quantum corrections; to understand the finiteness properties of the quantum theory. • λ > 2 The dimension of the coupling constant g is <−1 and these theories are theories for higher spins. Again they are non-renormalizable in the naive sense like the
L. Brink spin-2 theory above. There are strong reason to believe that these theories cannot be Poincare invariant one by one when we go to higher orders in the coupling constant, but the result above is an indication that certain sums of such theories interacting with each other could possibly be invariant theories. We can also find interacting solutions for n half-integer. We can, of course, not have a three-point coupling. We will in fact not be able to find self-interacting theo- ries but have to consider the coupling of the half-integer spin field to an integer spin field. We then find that we can couple a spin-1/2 field to a spin-l or a spin-0 field to recover in the first case a non-abelian gauge field coupled to a spin-1/2 field y(x) in a representation characterized by i of the external group such that we can have a coupling viy oCia, with Cia the Clebsch-Gordan coefficient. It is interesting to note that it is only in the interacting theory that we can prove the spin-statistics the- orem [15]. The formalism demands the spin-1/2 field to be of odd Grassmann type and the integer spin fields to be even. Note that there is no spinor space. The spin- 1/2 field is a complex( Grassmann odd)field with no space-time index. Its equation f motion looks just like the one for a bosonic field. (Remember the free equation the follows from(5). )However, the dimension of the field y(r)is different from the one of the bosonic field. so the free action is S=/xn+()v( The fact that we do not need to use spinors is very special for d= 4, since the transverse symmetry which is covariantly realized is SO(2)AU(1), which does not distinguish spinor representations We have hence seen that we can find all known unitary relativistic field theories as representations of the Poincare algebra, and we see their uniqueness and also what kind of possibilities there are for higher spin fields. In a gauge invariant formulation one can attempt to add in new terms that are gauge invariant. Invariably they lead to problems with unitarity. We do not see those terms here since the theories are unitary by construction. It should be said here that we could have derived the expressions above by starting with a gauge-covariant action and implement the light-cone gauge by hoosing the A-component of the vector field to be zero and then solve for the 3 Light-Frame Formulation of Supersymmetric Field Theories The known extension of the Poincare algebra is to make it into a supersymmetry algebra. This will lead to a restriction on relativistic dynamics. It is true that the at some stage supersymmetry is indeed a symmetry of the world pothesis is that world does not look supersymmetric as such, but a good working hy
38 L. Brink spin-2 theory above. There are strong reason to believe that these theories cannot be Poincar´e invariant one by one when we go to higher orders in the coupling constant, but the result above is an indication that certain sums of such theories interacting with each other could possibly be invariant theories. We can also find interacting solutions for λ half-integer. We can, of course, not have a three-point coupling. We will in fact not be able to find self-interacting theories but have to consider the coupling of the half-integer spin field to an integer spin field. We then find that we can couple a spin-1/2 field to a spin-1 or a spin-0 field to recover in the first case a non-abelian gauge field coupled to a spin-1/2 field ψi (x) in a representation characterized by i of the external group such that we can have a coupling ψ¯iψj φaCi ja, with Ci ja the Clebsch–Gordan coefficient. It is interesting to note that it is only in the interacting theory that we can prove the spin-statistics theorem [15]. The formalism demands the spin-1/2 field to be of odd Grassmann type and the integer spin fields to be even. Note that there is no spinor space. The spin- 1/2 field is a complex (Grassmann odd) field with no space–time index. Its equation of motion looks just like the one for a bosonic field. (Remember the free equation the follows from (5).) However, the dimension of the field ψ(x) is different from the one of the bosonic field, so the free action is S = d4x∂+ψ¯(x) ✷ ∂+ ψ(x). (18) The fact that we do not need to use spinors is very special for d = 4, since the transverse symmetry which is covariantly realized is SO(2) ≈ U(1), which does not distinguish spinor representations. We have hence seen that we can find all known unitary relativistic field theories as representations of the Poincar´e algebra, and we see their uniqueness and also what kind of possibilities there are for higher spin fields. In a gauge invariant formulation one can attempt to add in new terms that are gauge invariant. Invariably they lead to problems with unitarity. We do not see those terms here since the theories are unitary by construction. It should be said here that we could have derived the expressions above by starting with a gauge-covariant action and implement the light-cone gauge by choosing the A+-component of the vector field to be zero and then solve for the A−-component. 3 Light-Frame Formulation of Supersymmetric Field Theories The known extension of the Poincar´e algebra is to make it into a supersymmetry algebra. This will lead to a restriction on relativistic dynamics. It is true that the world does not look supersymmetric as such, but a good working hypothesis is that at some stage supersymmetry is indeed a symmetry of the world
Light-Cone Field Theory The standard covariant supersymmetry generator @a is a spinor with the anti Qa,2B)=TapPu (19) The spinor @a is four-component. It satisfies the Majorana condition which makes it real in a certain representation of the r-matrices. In the light-cone frame the spinor plits up into two two-component spinor that can be rewritten as two complex op- rators, which we call Q x4+1-0 and Q-=-51-14Q. From the Clifford algebra r,y) g(-1, 1, 1, 1)we see that Q=Q++0-, and that the products -1+y-and -1-7+ are projection operators. We can linearly combine the two components of the spinors into complex entities with no indices. We can also augment by letting the @'s transform as the representation N under SU(N). The light-cone supersymmetry algebra is then {Q,+n}=-√2mP+ (20) {Qm,Q-n}=-√26mP- (21) {Q,a-n}=-√26mP (22) where all other anticommutators are zero, except for the complex conjugate of the last one. The indices m n run from 1 to n The superPoincare algebra can now be represented on a superspace with coor- dinates x± 8,Bn, where the coordinates 8 and An are complex conjugates, Grassmann odd and transform as N and N under SU(N). We will denote their derivatives as (23) The @s are then represented as(We use the notation with lower case letters for operators that act on the field. and the dynamical ones as On this space we can also represent"chiral"derivatives anticommuting with the supercharges Q dm=-am--0ma+ dn= n +-=8,a
Light-Cone Field Theory 39 The standard covariant supersymmetry generator Qα is a spinor with the anticommutator {Qα,Q¯β } = γ μ αβPμ. (19) The spinor Qα is four-component. It satisfies the Majorana condition which makes it real in a certain representation of the γ-matrices. In the light-cone frame the spinor splits up into two two-component spinor that can be rewritten as two complex operators, which we call Q+ = −1 2 γ+γ−Q and Q− = −1 2 γ−γ+Q. From the Clifford algebra {γμ , γν } = 2ημν with η = diag(−1,1,1,1) we see that Q = Q+ +Q−, and that the products −1 2 γ+γ− and −1 2 γ−γ+ are projection operators. We can linearly combine the two components of the spinors into complex entities with no indices. We can also augment by letting the Q’s transform as the representation N under SU(N). The light-cone supersymmetry algebra is then {Qm +,Q¯+n} = − √ 2δm n P+ (20) {Qm −,Q¯−n} = − √ 2δm n P− (21) {Qm +,Q¯−n} = − √ 2δm n P, (22) where all other anticommutators are zero, except for the complex conjugate of the last one. The indices m,n run from 1 to N. The superPoincar´e algebra can now be represented on a superspace with coordinates x±,x,x¯,θm,θ¯ n, where the coordinates θm and θ¯ n are complex conjugates, Grassmann odd and transform as N and N¯ under SU(N). We will denote their derivatives as ¯ ∂m ≡ ∂ ∂ θm ; ∂m ≡ ∂ ∂ θ¯ m . (23) The Q’s are then represented as (We use the notation with lower case letters for operators that act on the field.) qm + = −∂m + i √ 2 θm ∂+; ¯q+n = ¯ ∂n − i √ 2 θ¯ n ∂+, (24) and the dynamical ones as qm − = ¯ ∂ ∂+ qm +, q¯−m = ∂ ∂+ q¯+m. (25) On this space we can also represent “chiral” derivatives anticommuting with the supercharges Q. d m = −∂m − i √ 2 θm ∂+; d¯ n = ¯ ∂n + i √2 θ¯ n ∂+, (26)
which satisfy the anticommutation relations dn)=-iv28mna To find an irreducible representation we have to impose the chiral constraints dmo=0; dmo=0 on a complex superfield o(r, x, em, n). The solution is then that emA (29) We now have to add in B-terms into the Lorentz generators to complete the rep- resentation of the free algebra. The result is for 2=0 where the little group helicity generator is (gdp -pd) (ddp-dpdp) It ensures that the chirality constraints are preserved Lj, dm]=[j, dmI The other kinematical generators are The rest of the generators must be specified separately for chiral and antichiral fields Acting on o (6an+6n0°), chosen so as to preserve the chiral combination [j-,y-] and such that its commutators with the chiral derivatives
40 L. Brink which satisfy the anticommutation relations {dm, d¯ n } = −i √ 2δmn ∂+. (27) To find an irreducible representation we have to impose the chiral constraints d m φ = 0; d¯ m φ¯ = 0, (28) on a complex superfield φ(x±,x,x¯,θm,θ¯ n). The solution is then that φ = φ(x+,y− = x− − i √ 2 θm θ¯ m,x, x¯, θm). (29) We now have to add in θ-terms into the Lorentz generators to complete the representation of the free algebra. The result is for λ = 0 j = x ¯ ∂ −x¯∂ +S12, (30) where the little group helicity generator is S12 = 1 2 (θp ¯ ∂p − θ¯ p ∂ p ) − i 4 √ 2∂+ (dp d¯ p −d¯ p dp ). (31) It ensures that the chirality constraints are preserved [ j, dm ]=[ j, d¯ m ] = 0. (32) The other kinematical generators are j + = ix∂+, ¯j + = ix¯∂+. (33) The rest of the generators must be specified separately for chiral and antichiral fields. Acting on φ, we have j +− = ix− ∂+ − i 2 (θp ¯ ∂ p +θ¯ p ∂ p ), (34) chosen so as to preserve the chiral combination [ j +−, y− ] = −iy−, (35) and such that its commutators with the chiral derivatives [ j +−, dm ] = i 2 dm, [ j +−, d¯ m ] = i 2 d¯ m, (36)