Supersymmetry transformations are carried out via generators, associated with unitary operators, acting on particle states, and carry spinor indices and hence are of fermionic type. These generators together with the ones of the Poncaré transformations satisfy a new algebra, referred to as the Super-Poincaré Algebra, involving commutators as well as anti-commutators. We consider theories with only one supersymmetry generator as a spinor with four components corresponding to ones referred to asN D1(simple) supersymmetric field theories.
The mere fact that the supercharge operator, as will be derived below,does not commute with the angular momentum operatornecessarily implies that the particle supermultiplets, associated with this algebra, involve particles withdifferent spins and hence the derivation of the commutator in question is of importance. The fermionic character of the supersymmetric generator, as witnessed by working, in particular, with Grassmann variables, shows also that the number of particles in a supermultiplet are finite in number. Below we derive the details of this algebra and investigate the nature of supermultiplets associated with these symmetry transformations. The reader is advised to review first the content of Sect.2.1on superspace as well the section on the Poincaré Algebra in Sect. 4.2 in Vol. I [26].
In superspace, coordinates are labeled by.x; a/, where is a Majorana spinor with anti-commuting components which commute withx. For a supersymmetry transformation, not involving Lorentz transformations and spatial rotations, the transformation rulex; !x0; 0, including translations, is defined by [see (2.1.3), (2.1.4)]
x0 D x C i
2 b; (2.5.1)
0 D C; (2.5.2)
and a Super-Poincaré Transformation, which involves Lorentz transformations and spatial rotations, is defined by [see (2.1.10), (2.1.11)]
x0 D xC i
2Kb; (2.5.3)
0 D KC; (2.5.4)
with both sets written in convenient matrix notations. K denotes the spinor- Dirac representation (Sect. 2.4 in Vol. I, Sect. 2.1) of a Lorentz group. In the transformation rules (2.5.1)–(2.1.4), the spinor .a/ is tied together with the spacetime coordinate .x/ in a way similar that x0 is tied together with xi in special relativity. The transformation rules are also linear in.x; a/. We consider infinitesimal transformations via a generatorGspelled out in (2.5.5), (2.5.15) below, which results from considering infinitesimal transformations around a closed path represented pictorially by
emphasizing the reversal of the transformations in the third and the fourth segments of the path, described by successive unitary transformationsU21U11U2U1 with UjD1CiGjfor infinitesimal transformations.
For additional clarity we consider the simple supersymmetry transformations in (2.5.1), (2.5.2) first before tackling the ones in (2.5.3), (2.5.4). To this end, the generator of the corresponding transformation on operators and particle states is defined by
GD•bPC•aQa; (2.5.5)
whereP is the energy-momentum operator and the Qa denote the components of the spinor generator of supersymmetry, and is defined to satisfyQ D CQ>, QD.Q/0. The latter as well as•are taken as Majorana spinors which guaranty, in particular, the Hermiticity of the generatorG on account of the easily derived property
•aQaDQa•a; (2.5.6)
where the parametersaand their variations anti-commute with the operatorsQa. As mentioned above, we consider infinitesimal transformations forming a closed path described by
.2;b2/1.1;b1/1.2;b2/.1;b1/D.;b/; (2.5.7)
where the group properties are given in Sect.2.1below (2.1.8), and use the general commutation rule involving the generatorsG1,G2,G
GD 1
i ŒG1;G2; (2.5.8)
as follows from the structureU21U11U2U1DU,UjD1CiGj,UD1CiG, with corresponding infinitesimal parameters (•1; •b1), (•2; •b2), (•; •b), respectively.
The group property of the parameters are spelled out below (2.1.8), leads to
•b D i•2•1Di•1a•2b.C/ab; (2.5.9)
• D 0; (2.5.10)
whereC is the charge conjugation matrix (cf. Sect.2.2).
For the subsequent analysis, we replace the expressions for the generators,G1, G2,G, by their corresponding expressions given in (2.5.5) into (2.5.8), withG, for example, defined in (2.5.5), use the parametric relations in (2.5.9), (2.5.10), and note that
Œ •1aQa; •2bQbD •1a•2bfQa;Qbg; (2.5.11) converting, in the process, a commutator to an anti-commutator. Upon comparing the coefficients of identical products of the parameters•1,•2,•b1,•b2 on both sides of the resulting expression for (2.5.8), we obtain
ŒP;P D0; ŒQa;P D0; fQa;Qbg D.C/abP: (2.5.12) To generalize the above algebra to the Super-Poincaré one corresponding to the transformations in (2.5.3), (2.5.4), we first recall the explicit structures of the following infinitesimal deviations from the identity elements
D •C•!; •! D •! ; (2.5.13)
Ka b D •a b C i
2•! .S /ab; S D i
4Œ; : (2.5.14) The generator for infinitesimal transformations on operators and particle states takes the form
GD•bPC1
2•! J C•aQa: (2.5.15)
The group property in Sect.2.1below (2.1.16), corresponding to a closed path, as before, now with group multiplications of elements .;K;;b/, leads to the parametric relations
•! D •!2 ˛•!1 ˛•!2 ˛•!1 ˛; (2.5.16)
•b D •!2˛•b˛1•!1˛•b˛2 Ci•1a•2b.C/ab; (2.5.17)
•a D i 2
•!2 .S •1/a•!1 .S •2/a
; (2.5.18)
where we have used, in the process,28the identities
DK1 K; 1DKK1; (2.5.19)
in matrix notations in spinor indices. The expression for•aalso leads to
•aD i 2
•!2 •1b•!1 •2b
.S /b a; (2.5.20)
where note that.S /0D0.S /, andS is defined (2.5.14).
Upon comparing the coefficients of identical products of the parameters• !1,
• !2,•1,•2,•b1,•b2 on both sides of the resulting expression for (2.5.8), the Super-Poincaré Algebra now readily emerges:
fQa;Qbg D .C/abP; fQa;Qag D0; (2.5.21)
ŒQa;J D .S/abQb; (2.5.22)
ŒQa;P D0; ŒQa;P2D0; (2.5.23)
ŒP;P D 0; (2.5.24)
ŒP;J D i .PP/; (2.5.25)
ŒJ;J D i.JJCJJ/: (2.5.26) where the second anti-commutation relation in (2.5.21) holds with or without a sum overa. The derived non-commutativity property of the generatorQwith the angular momentum should be noted and is responsible for the fact that supermultiplets involve particles with different spins. This we consider next.
28See (2.1.9).
We recall the basic property of a Majorana spinor
QD Q>C1: (2.5.27)
Therefore upon multiplying (2.5.21) byCbc1we obtain fQa;Qcg D ./acP: Upon multiplying the latter by.0/cbgives
fQa;Qbg D .0/abP: (2.5.28) The commutation relation ofQa with the angular momentum operator Ji D ij kJj k=2follows from (2.5.22) to be in matrix notation
ŒQ;JD 1
2†Q; †D 0
0 : (2.5.29)
Its adjoint satisfies the equation
ŒJ;QD 1
2Q†: (2.5.30)
Also note, by now familiar, the commutation relations [see (2.5.25), (2.5.26)]
ŒP0;J D0; ŒJ3;J2 D0: (2.5.31) We first treat massive supermultiplets. Consider a massive particle of massm.
By going to the rest frame of the particle, states may be labeled by.m;j; /, with P D .m;0/. In the sequel, we suppress the dependence of the states onm. Upon setting
Aa D p1
mQa; AaD p1
mQa; (2.5.32)
in the process, and observing the simple structure of† in (2.5.29), (2.5.30), and using the fact that00D I, we obtain from (2.5.28)–(2.5.30), (2.5.32)
fAa;Abg D ıab; JAa DAb
Jıba1 2ab
; a;bD1; 2;
JAaDAb
JıbaC 1 2ba
; a;bD1; 2;
J3A1DA1
J31 2
; J3A2DA2
J3C1 2
;
J3A1DA1
J3C1 2
; J3A2DA2
J31 2
; (2.5.33)
together with the elementary identities that follow from (2.5.21)
.A]1/2D0; .A]2/2D0; (2.5.34) whereA]refers to the operator or its adjoint.
One can always define a normalized statej'isuch thatAa j'i D0,aD1; 2. For example, ifA1 j'i ¤ 0, then you may consider the normalized stateA1 j'iwhich is annihilated byA1and so on. Accordingly, givenAaj'i D0,a D 1; 2, consider the following. For a given unit vectorn, set
ei#nJAaei#nJDAaŒ#;n: (2.5.35) Then, from the second equation in the first line of (2.5.33), witha;b D 1; 2, its derivative with respect to#is given by
@
@ #AaŒ#;nD in 2
AŒ#;n
a; (2.5.36)
which leads from (2.5.35) to
ei#nJAaei#nJD ei#n2
a bAb: (2.5.37)
That is AaŒ#;n annihilatesj'ias well. Hence one may carry out the following expansion with the property
0DAaei#nJ j'i DX
j;
Aa jj; ihj; jei#nJ j'i; (2.5.38) which is true forall #,n. That is, the latter is true for all arbitrary coefficients aj;.#;n/D hj; jei#nJj'i, as we may vary over the infinite possible values that may be taken by.n; #/. Thus we may infer that one may set-up normalized spin statesjj; isuch thatAa jj; i D0, foraD1; 2.
Given the above normalized spin statesjj; i, one may use the third equation in (2.5.33), together with the identities in (2.5.34), to obtain
ŒJ;A1A2D 1 2A1A2
11C22
D0; (2.5.39)
where we have used the fact that
11C22
Tr D0. That is,
JA1A2DA1A2J: (2.5.40)
We may, therefore, introduce new normalized spin states
jj; i0DA1A2 jj; i; (2.5.41) orthogonal to the states jj; i as a consequence of the fact that the latter is annihilated byAa.
Clearly, the non-vanishing statesA1jj; i,A2jj; i,jj; i,jj; i0, are mutually orthogonal. On the other hand, by referring to the third equation in (2.5.33), involving the operatorsA1;A2, we note thatJC=2represents the addition of two angular momenta one of which corresponds to spin 1/2. Hence we may conclude that forj> 0, a massive supermultiplet contains also states of spinj˙1=2with all of the corresponding particles having thesamemass. Due to the Grassmann property of the spinor generators.A]a/2 D 0, and the anti-commutation relations in (2.5.34), no other states may be constructed by the applications of these generators and their products on the already obtained states above as they would either be zero or proportional to the pre-existing ones.
In particular, for a givenj > 0 we have learnt that there are two sates with spinj. These states will correspond to bosonic or fermionic degrees of freedom, depending whetherj is an integer or a half-odd integer, respectively. An elegant way of determining the number of fermionic and bosonic degrees of freedom in a supermultiplet is obtained in the following manner.
Since from (2.5.23),ŒP;QaD0, every particle in a supermultiplet has the same momentum, say,p, and same mass, say,m. Suppose that p D jpj.0; 0; 1/. Also from (2.5.25), (2.5.28)
fQ1;Q1g D P0; ŒJ3;P3D0;
ŒJ2;J3 D 0; ŒJ2;P3D2i.J2P1J1P2/; (2.5.42) and we may define statesjp;j; i, satisfying
J3jp;j; i D jp;j; i; pD jpj.0; 0; 1/;
ei2 J3jp;j; i D.1/2jp;j; i .1/2jjp;j; i: (2.5.43) On the other hand, from (2.5.37), we may infer that
ei2 J3Q1ei2 J3 D.ei 3/1bQbD Q1: (2.5.44) Now consider the following trace multiplied byEDp
p2Cm2, EX
j;
hp;j; jei2 J3jp;j; i DX
j;
hp;j; jei2 J3P0jp;j; i
D X
j;
hp;j; jei2 J3
Q1Q1CQ1Q1
jp;j; i
D X
j;
hp;j; j
Q1ei2 J3Q1Cei2 J3Q1Q1
jp;j; i D0; (2.5.45)
where in writing the second equality we have used the first equation in (2.5.42), and in writing the third equality we have used (2.5.44), and in writing the last one we have used the fact that TrŒABDTrŒB A. That is
0DTr ei2 J3
DX
j0
.1/2jNj.2jC1/Dnbnf; (2.5.46)
whereNj denotes number of particles with spinj, each with.2jC1/spin degrees of freedom, andnb=f denote the number of bosonic/fermionic degrees of freedom, respectively. Hence we learn that the number bosonic degrees of freedom is equal to the fermionic ones. For example for a givenj, sayj D 1=2, the above analysis shows that we have a supermultiplet with two1=2spins with2
2.1=2/C1 D4 fermionic degrees of freedom, one spin1of3 degrees of freedom and one spin0 of1degree of freedom. All in all we have,4fermionic degrees of freedom and4 bosonic ones.
Forj D0, we clearly have a massive supermuliplet involving two spin 0 states, and one of spin 1/2 with two degrees of freedom.
For massless supermultiplets, consider a particle with energy-momentumP D .E; 0; 0;E/,E > 0, i.e., which is moving along the 3-axis. For massless particles, working in the chiral representation of the Dirac matrices, (2.5.28) leads to
fB1;B1g D1; BaDQa=p
2E; BaDQa=p
2E; (2.5.47)
and
fB2;B2g D0; fB1;B2g D0; fB2;B1g D0; .B1/2D0; .B1/2D0:
(2.5.48) As in (2.5.33), we also have
J3B1DB1.J3C1=2/; J3B1DB1.J31=2/: (2.5.49) That is, ifN denotes the largest helicity in a supermultiplet, associated with some particle, then
B1 ˇˇE;N˛
D0: (2.5.50)
Hence another state in the supermultiplet would be given byB1jE;Ni D jE;N 1=2i. Also, .B1/2 D 0, implies that B1jE;N 1=2i D 0. That is, a massless supermuliplet contains just the two states ˇˇE;N˛
and ˇˇE;N1=2˛
. On the other hand, CPT invariance implies the existence also of its anti-supermultiplet consisting just of the statesˇˇE;N˛
andˇˇE;NC1=2˛
of opposite helicities. For example the superpartner of the photon, called the photino, is a massless fermion, of helicities
˙1=2, and that of the graviton, the superpartner, called the gravitino, is a fermion, a Rarita-Schwinger massless particle, of helicities ˙3=2. The photon and the graviton being so-called gauge fields, their superpartners are also each referred to as gauginos. In general supermultiplets withN D 1=2; 1are referred to as chiral and vector supermultiplets, respectively.
It is important to reiterate, in closing, that the second commutation rule in (2.5.23) implies that all the particles within a supermultiplet have the same masses, for both cases with massive or massless particles.