The Poincare superalgebra

2.2.1. Uniqueness of the N

=

1 Poincare superalgebra

Now all the facilities which we require to introduce a basic object of our book - the Poincare superalgebra, which presents an extension of the Poincare algebra [l/' - are at our disposal. The main motivation for the appearance of this object in theoretical physics was a very old theorists' dream to find a non-trivial extension (other than the direct sum) of the Poincare algebra, i.e. of the Lie algebra of the space-time symmetry group of any relativistic quantum field theory. The problem has proved to have no affirmative solution in the class of Lie algebras. The well-known theorem ofS. Coleman and J. Mandula states that, due to assumptions of the S-matrix approach, the most general Lie group of symmetries in a quantum field theory (whose spectrum contains massive particles) is a direct product of the Poincare group and some internal group,

nxG

G

=

Gl X [U(1)]" (2.2.1)

where G_l is a semi-simple compact group. The corresponding Lie algebra has the direct sum structure

gt'EB~ ~=~1E9~2 (2.2.2)

where ~l is the semi-simple Lie algebra of G_l and ~2 is an Abelian algebra.

Since elements of CS commute with elements of 9, the only generators with Lorentz indices present in the symmetry Lie algebra are the Poincare generators {Pa,jab} of space-time translations and Lorentz rotations. After suffering a setback in the class of Lie algebras, it was natural to look for success in the class of superalgebras (in fact, this is the way superalgebras were created). Fortunately, it turned out that it was possible to extend the Poincare algebra by several sets of fermionic generators with spinor indices thus obtaining superalgebras.

Let us show that there exists a unique superalgebra extension of the Poincare algebra by four a-type generators (q"" qeX), with q" carrying an undotted spinor index and

q"

a dotted spinor index . Recall, in accordance with the results of subsection 2.1.1, a-type elements of a superalgebra CS transform in some representation of the Lie algebra °CS. Since we have chosen the representations (t,O) and (0,

t)

to act on the subspaces generated by q",

where we have used the fact that space-time translations do not act on spinor or vector indices.

Now we are going to analyse anticommutation relations in the assumed superalgebra. From equation (2.l.11c), we can write

=

^{f a 1 f ab'}

140 Ideas and Methods ot'Supersymmetry and Supergravity

satisfy equation (2.1.18). The indices i,j and I from (2.1.18) are now Poincare indices, and the indices et,

f3

and 'Y from (2.1.18) are now spinor indices. The non-vanishing structure constants of the Poincare algebra are fa.

cl

^{= -} fed.}

and fah.

j' = -

fed. al, and their explicit values can be readily found from (1.5.5). Then, choosing in equation (2.1.18) i = a and I = b and using equation (2.2.3), one obtains

r uh

f

.ah -

f

ah 0

.I(t.{!

=

if! - (t.~

= .

The relations (2.1.18) tell us that the other structure constants

f!Y./,

^f~l and

.f~;(/ are invariant tensors of the Lorentz algebra. But the Lorentz group has no invariant tensors like

f(t,l

or

fit

Further, the only candidate for the role of j(t.;" is the invariant tensor (O'a)(t.~. As a result, the anticommutation relations (2.2.4) are simplified drastically:

{q(t" qf3}

=

0 {ih, qp} = 0

(2.2.5) { q(t"

q&J =

2k( O'a)CI.~pa

with k being some constant. It is a simple exercise to check that the second equation (2.1.19) for the structure constants is satisfied now identically. So, we have obtained a superalgebra.

Finally, it would be desirable to demand that in any unitary representation T of the obtained superalgebra (from the physical point of view, such representations are of the greatest importance) the generators 00:

=

T(qo:) and

ifJ" =

T(q,,) were Hermitian conjugate to each other,

0" =

(00:) +. (2.2.6)

In other words, we wish to treat the pair (qCl., q~) as a Majorana spinor. Then, the constant k in (2.2.5) should be real and positive. Indeed, equation (2.2.5) leads to

1 . _ 1 .

- kIPa =

:4

(aa){icx{ 00:, Op} =

:4

(aa)f3C1.{ OCt, (O{!) + } (2.2.7) where IPa = T(Pa)

= (- [, P)

is the (Hermitian) energy-momentum operator, hence k is real. Choosing here a

=

^{0 gives}

1 + 1 + 1 + 1 +

k[

=

-O\(Od

+

-0z(02)

+

-(0\) 0\

+

^-(02 ) O 2 , (2.2.8)

4 4 4 4

Since physically acceptable unitary Poincare representations are characterized by condition (1.5.16) (positivity of energy) and due to positive definiteness of the operator in the right-hand side of equation (2.2.8), we must choose k > O. Hence one can set k

=

1 by making a simple rescaling of qo: and

q&.

Let us now write down the complete list of (anti)commutation relations of the superalgebra:

[Pa, Pb]

=

0 [jab' pcJ

=

ilJacPb - ilJbcPa [jab' id]

=

ilJajbd - ilJadhc

+

ilJbJac - ilJbJad

[jab' q,J

=

i(O'ab),/,q(1 [Pa, q,,]

=

0

(2.2.9) [j -"] ab' q

=

1 ab ^.(-)& 0' ~q ^{-(1 [} P a' q ^-"]

=

⁰

{q", q(l} = 0 {q", q~} = 0 {q",

q;J =

2(O'a),,&p a

This real superalgebra is known as the 'Poincare superalgebra'. It will be denoted by SfP. Its a-type generators qc< and

q",

are called 'supersymmetry generators'. Every element X of the Poincare superalgebra can be represented, in agreement with subsection 2.1. 7, as follows:

X

= i(

^-bapa

^{+ ~Kabjab) +}

Ji(K"q"

+

K"q,,)

(2.2.10) ab

=

ba, K

-

K^ba E IR K", K"

=

(K")* E C.

For later use, we rewrite the Poincare superalgebra in spin or notation, converting every vector index into a pair of dotted and undotted indices.

When applied to the Poincare generators, this operation leads to the change {Pa,jab} -+ {p"",L(I,J&~}. Then one finds

i i

[j "", _J" p" ( I .. ,]

= -

₂ 8""Pf~'"I I

+ -

2;8"flP,,'" I

. ]

= 2

^{i ( . . .}

[j _,,/3, J;,cl 8;',.,)/3"

+

8.,.(lI""

+

8,5,.,);'/3

+

8,5(l1),,, ^{. )}

i i (2.2.11 )

[j _,,/3, q" ]

= 2

8;,,,q(J

+ 2

8;'(Jq"

{q", q&} = 2p",,,.

Other (anti)commutators vanish or may be found by Hermitian conjugation.

2.2.2. Extended Poincare superalgebras

It was shown above that the Poincare superalgebra is the only possible superalgebra such that its even part coincides with [lJ and the odd part transforms in the real (Majorana) representation

d',

0) E9 (0,

i)

of the Lorentz group. However, one can consider a more general problem: to find all possible superalgebras A with 0 A being of the form (2.2.2) and lA being generated by a set of elements, each of which carries at least one spin or index. Then, since {I A, ^I ... it} c 0A, it follows from the Coleman-Mandula theorem that every generator of 1.# should carry exactly one spin or index (otherwise, °.If

These equations give very strong restrictions on the structure constants. An analysis similar to that of subsection 2.2.1 leads to the following

together with the Poincare commutation relations. Here {t;} are generators of the semi-simple Lie algebra ~!' {e AB

=

_eBA , CAB

=

-CBA} are generators that central charges may exist when N ~ 2. For a more detailed derivation of equations (2.2.12) and (2.2.13), see the book by J. Wess and J. Bagger.

The superalgebra (2.2.12) is known as an 'N -extended Poincare superalgebra', depending on the number of spinor generators. The Poincare superalgebra (2.2.9) is called the N

=

1 (or 'simple') Poincare superalgebra. It follows from the above consideration that the superalgebra (2.2.12) is the most general (finite-dimensional) extension of the Poincare algebra, consistent with axioms of quantum field theory. This assertion is known as the Haag, Lopuszanski and Sohnius theorem.

From now on, we shall study the N I Poincare superalgebra only; this case, being simple, is worked out in detail and contains the main ingredients of all supersymmetric theories.

2.2.3. Matrix realization of the Poincare superalgebra

The Poincare superalgebra has been introduced above as an abstract superalgebra. Now we give its realization in terms of matrices. Let us consider in M at(4, 1

I q

matrices {Pa, jab' qC(,

lie.}

defined as follows

o

_O

o

2 - lO'o

o

- t(fl4

₊

_Ys)Y.

o

o _O o

2

O

o

²

o

Po

= =

o o o o o o o o o o

0 0

- iO'ob

O

0 ² 0

-i~ab 0 0

0

O

² - iii"ob 0

jab

= =

o o o o ^o

I !

o o o o o

(2.2.14)

o

¹

- I

o

04

o

0⁴

o

o o

ql ^q2

=

o o o o o o o o o o

o o

4

o

0

o

0

o

⁴

o

o ^o

iE = qi =

o o

- I 0

I

⁰

o o o o

=

144 Ideas and Methods of Supersymmetry and Supergravity

where On means the zero n x n matrix. The matrices Ya, Lab and Ys were introduced in Section 1.4. One can readily check that the matrices (2.2.14) satisfy the (anti)commutation relations (2.2.9). Note that the superalgebra, generated by the matrices (2.2.14), is a subalgebra of sl(4, 11 IC).

2.2.4. Grassmann shell of the Poinc'are superalgebra

As we know, with every complex superalgebra ~(IC) one can relate a Berezin superalgebra ~(A:x:) (the Grassmann shell of ~(IC)) and a super Lie algebra o~(A~) (the even part of ~(A:x:)). We now construct these objects for the complex shell S.9'(1C) of the Poincare superalgebra. It is a complex superalgebra of dimension (l0

+

4) with a general element X E S.9'(1C) of the form

I ab' .~

x-X

=

^{X a} Pa

+ -

^X

2 ^Jab

+

^X q"

+

^X q"

(2.2.15) xa, xab

= _

xba, X", XX E C.

Recall that elements of the Poincare superalgebra have the form (2.2.10).

Now, introduce the Grassmann shell S.9'(A,,) of S2P(IC). It is a Berezin superalgebra of dimension (l0, 4) with a pure basis {Pa, Lb' q~, ih} such that

ZPa

=

Paz ZLb

=

jab

z

(2.2.16) zq"

= ( -

^{I y(Z)q~z} zq"

= ( -

I Y(Z)q"z

for any pure supernumber z E Aoc. Every element

X

^E S.9'(A",) can be represented as follows

X -

=

cap

+ -

¹ ):a ^h J'

+

P q

+

^{C"q-' '}

- a 2" ab "'" - "

(2.2.17)

~a,~ab = _~ba,~:"~"EA",.

U sing the matrix realization (2.2.14) of the Poincare superalgebra, it is not difficult to obtain a supermatrix realization of the Berezin superalgebra S.9'(A",).

Let us endow the supervector space S2P(A oc ) with an operation of complex conjugation according to the rule

(Pa)* == -Pa (jab)* == - jab (q,,)* ==

-q".

Then every real c-type supervector

X

E °S.9'(A"J is of the form - _ .( b a 1 ab' " - _,,) - _

*

X-I - Pa

+ 2.

^{K Jab}

+

E q~

+

E"q E"

=

(Ex)

(2.2.18) ab

ba, K

= -

^{Kba E} IRe eEiCa,

Due to the (anti)commutation relations (2.2.9), one easily finds that the subset oS.J'(k,) of real c-type supervectors forms a subalgebra of the super Lie algebra °S..J'(A,). So °S&R(A~) is a real super Lie algebra. It is called the '(N

=

I) super Poincare algebra'.

Recalling the spinor notation (1.4.3), the last two terms in (2.2.18) can be rewritten as follows:

eq"

+

EA~

=

Eq

+

Eq

=

qE

+

qE (2.2.19) where we have used equation (2.2.16).

2.2.5. The super Poincare group

A super Lie group corresponding to the super Poincare algebra is known as the (N = I) 'super Poincare group'. It is denoted by

sn.

Every element of

sn

is of the form

g( b, E, lO, K)

=

exp [ { - b P a a

+ ~

^Kabjab

⁺

^Eq

⁺

^{Eq) ] .}

'] (2.2.20)

=

exp

[{~b";;P";; ⁺

K"Pj"fl

+ K'x~h~ ⁺

^Eq

⁺

^Eq ) ^. So, points of the super Poincare group are parametrized by real c-number

a ab ba

variables b and K

= -

K as well as by a-number variables (E:x' E'x) forming a Majorana spinor. Similarly to the Poincare group, supergroup elements

gib)

=

exp (-ibapa) b^a EIRe ^(2.2.21) will be called 'space-timc translations' and elements

=

g(K) exp

(~Kabjab

^{) Kab EIRe (2.2.22)}

will be called 'Lorentz transformations'. Elements

g(E, E)

=

exp [i(Eq

+

Eq)] E"'EC_a (2.2.23) are said to be 'supersymmetry transformations'. The union of all elements (2.2.22), denoted by SO(3, l1IRY, forms a super Lie group, which represents a c-number shell of the Lorentz group SO(3, l)i. Ordinary translations and ordinary Lorentz transformations correspond to soulless parameters in (2.2.21) and (2.2.22). We will refer to SO(3, 1lIRe)i as the Lorentz group over IRe.

It should be noticed that, since

0,

p] - p and

0,

q] '" q, the set of elements

a ab

(2.2.20) with b EiRe' e ECa' but K EIR, forms a subgroup of the ab super Poincare group. Therefore, one may restrict the parameters K in (2.2.18) and (2.2.20) to be ordinary real numbers. However, since {q, q} ..., p,

146 Idl!{fs (fl1d Ml!thods o( Supl!rsymml!try and Supergravity

we have

[E1q

+

_{E1q, E2q}

+

_E2q]

=

_{2(E 1a}^a_{E2 - E}2a^aE₁)Pa and hence

exp [i(E1q

+

Elq)] exp [i(E2q

+

E₂q)]

=

exp [i{(El

+

E₂)q

+

(El

+

E )q 2

+

i(E la E^a2 ^- E₂a El^a )Pa}J (2.2.25) as a consequence of the Baker-Hausdorff formula, so the parameters b^a in (2.2.18) and (2.2.20) should be arbitrary real c-numbers.

An invariance in quantum field theory with respect to the super Poincare group is called 'N

=

1 (or simple) supersymmetry'.

Im Dokument Supersymmetry and Supergravity (Seite 159-167)