As discussed in the introduction to the chapter, to describe transformations between particles of half-odd integer spins, the fermions, and particles of integer spins, the bosons, one introduces a generator (generators) which necessarily carry a spinorial
18See [2, p. 74]), [41, p. 124]), [3, p. 92], [41], Weinberg in his monumental work on supersymme- try, on p. 124, line 13 from below, states that it is rather surprising that the need of a spinor field arises in such a case.
19See, e.g., [8,9,21,36,37]. Here we are referring to the case ofD D4dimensions with one (N D1) supersymmetry charge operator in the theory. For supersymmetric extensions (i.e., for 1N 8) and arbitrary dimensions (4 D11) of the theory, see these references on the renormalizability problems of such extensions.
index. Accordingly, to accommodate such a symmetry, in a spacetime description as an extension of the Minkowski spacetime, one may introduce a four component spinor .a/, which commutes with the spacetime variable x, as an additional coordinate tied together with x in a way similar the time variable x0 is tied together with those of the space ones x in special relativity. The resulting space with coordinates labels.x; a/is referred to assuperspace. Supersymmetry is a spacetime symmetry. By the extension of Minkowski spacetime to superspace, we will, in later sections, develop supersymmetric field theories.
Before discussing on how this spinor is tied together with the spacetime coordinatex, we note that the spinor is chosen as a Majorana spinor with anti- commuting components and, as mentioned above, that commute withx:
fa; bg D0; aaD0; Πa;xD0: (2.1.1) Here we recall the definition of a Majorana spinor
DC>; D >C1; D0;
where the charge conjugation matrixC is defined byC Di20and satisfies:20 C1C D ./>; CDC>D C:
Also for two anti-commuting Majorana spinors,, it is easily verified that21 D .Hermiticity condition/; D : (2.1.2) We introduce a supersymmetry transformation as a shift of by a Majorana spinor, with anti-commuting components, and anti-commuting with the compo- nents of:fa;bg D0as well. In order to tiea with the spacetime coordinates x, we also have to shiftxby a vector depending on,which must be Hermitian.
As in special relativity the transformations are taken to be linear, now inxand, given by the simple supersymmetry transformation on the coordinates.x; a/in an 8-dimensional superspace:
x0 D xC i
2b; (2.1.3)
0a D aCa: (2.1.4)
Here we have also introduced an ordinary spacetime translation ofx specified by the vector componentsb. The i factor in (2.1.3) is necessary to ensure Hermiticity
20For details on the gamma matrices see AppendixIat the end of this volume. See also AppendixII to see how the charge conjugation matrixC arises.
21For additional such identities see (2.2.8)–(2.2.10).
according to the rule (second equality) in (2.1.2). Below we will generalize the transformations in (2.1.3), (2.1.4) to include Lorentz boosts and spacial rotations.
For a subsequent transformationx0!x00,0!00, x00 D x0C i
200b0; (2.1.5)
00a D 0aC0a; (2.1.6)
from which we may infer that x00 D xC i
2.C0/
b Cb0 i 20
; (2.1.7)
00a D aC.aC0a/: (2.1.8)
That is, the supersymmetry transformations may be specified by the pair.;b/
satisfying the group properties:
1. Group multiplication:.0;b0/.;b/D.C0;bCb0 2i0/.
2. Identity.0; 0/W.0; 0/.;b/D.;b/.
3. Inverse.;b/1D.;b/W.;b/.;b/D.0; 0/.
4. Associativity Rule:.3;b3/Œ.2;b2/.1;b1/DŒ.3;b3/.2;b2/.1;b1/.
In writing the last equality in property 3, we have used the fact that, according to the second equality in (2.1.2),D0.
We now extend the transformation in (2.1.3), (2.1.4) to include Lorentz boosts and spacial rotations. We recall that under a homogeneous Lorentz transformation, xis transformed via the matrix, of special relativity, while the four component spinor transforms via a matrixŒKa b, ! K, under a Lorentz transformation, whose explicit structure is not needed here, satisfying the properties:22
DK1K; K0D0K1: (2.1.9) We define the supersymmetric transformations in superspace, known as Super- Poincaré Transformations, or Super-Inhomogeneous Lorentz Transformations, by
x0 D xC i
2Kb; (2.1.10)
0 D KC; (2.1.11)
in a convenient matrix notation. These include supersymmetry transformations, translations as well as Lorentz boosts and spatial rotations. Consistency of such a transformation requires that its structure remains the same under subsequent
22See AppendixIIat the end of this volume for such transformations.
transformations leading to group properties spelled out below as done for the pure supersymmetry ones above.
Under a subsequent transformation.x0; 0/!.x00; 00/one has x00 D 0x0C i
20K00b0; (2.1.12)
00 D K00C0; (2.1.13)
from which we may infer that x00 D Π0xC i
2
0CK01
ŒK0K
0bCb0 i
20K0
; (2.1.14) 00 D ŒK0KC
0CK0
: (2.1.15)
where in writing the second term on the right-hand side of (2.1.14), we have used the first identity in (2.1.9) to write
0KDK01K0K: (2.1.16)
That is, a Super-Poincaré Transformation, or a Super-Inhomogeneous Lorentz Transformation, may be specified by the quadruplet.;K;;b/, and upon com- parison of (2.1.14)/(2.1.15) with (2.1.10)/(2.1.11) we learn that upon a subsequent transformation:
!Œ 0 ; K!ŒK0K; !Œ0CK; b!ŒbCb0 i
20K0: That is we have the following group properties of the transformations:
1. Group Multiplication:
.0;K0;0;b0/.;K;;b/D.0;K0K;0CK0; 0bCb02i0K0/.
2. Identity.I;I; 0; 0/:
.I;I; 0; 0/.;K;;b/D.;K;;b/.
3. Inverse.;K;;b/1D.1;K1;K1;1b/:
.1;K1;K1;1b/.;K;;b/D.I;I; 0; 0/.
4. Associativity Rule:
.3;K3;3;b3/Œ.2;K2;2;b2/.1;K1;1;b1/ DŒ.3;K3;3;b3/.2;K2;2;b2/.1;K1;1;b1/:
Here we note that
.0CK0/0D0CK01;
as a result of the second identity in (2.1.9), relevant to the third entry on the right- hand side of the equality in property 1. Also that in writing the last equality in property 3, we have used the fact that according to (2.1.2) and the second identity in (2.1.9),
KK1D.K1/ .K1/D0:
In Sect.2.6, we will use these group properties to develop the Super-Poincaré Algebra satisfied by the generators of the corresponding unitary operators acting on the so-called superfields and on particle supermultiplets. Next we derive basic properties involving products of components of the spinorand related summation formulae needed in developing supersymmetric field theories.