Springer Tracts in Modern Physics, Volume 60

236 Pages·1971·2.783 MB·English

by  Professor Dr. J. Wess (auth.)

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Contents
Conformal Invariance and the Energy-
Momentum Tensor
J. SSEW
Representations of the Local Current Algebra.
A Constructional Approach
R. V. SEDNEM and Y. NAME'EN 81
Chiral Symmetry. An Approach to the Study
of the Strong Interactions
M. NIETSNIEW 32
Dual Quark Models
K. ZTEID 74
High Energy Inclusive Processes
CHUNG-I NAT 19
Deep Inelastic Electron-Nucleon Scattering
J. sE?mD 701
Hyperon-Nucleon Interaction
J. J. ,TRAWSED M. M. ,SLEGAN
T. A. RIJKEN and P. A. NEVEOHREV 831
How Important are Regge Cuts ?
P. D. B. SNILLOC 204
Conformal Invariance dna the
Energy-Momentum Tensor*
J. WEss
Contents
I. The Conformal Group . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. Low Energy Theorem for Gravitons . . . . . . . . . . . . . . . . . . . 4
III. Conformal Invariance and Effective Lagrangians .............. 6
IV. Invariant Lagrangian for a Scalar Field . . . . . . . . . . . . . . . . . . 51
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
I. The Conformal Group
The conformal group in the four-dimensional, pseudoeuclidean space
is the group of coordinate transformations which leave the relation
ds 2 = (dx~ 2 - (dx1) 2 - (dx2) 2- (dx3) 2 = 0 (1)
invariant. The infinitesimal transformations can be parameterized as
follows 1:
Translations" x'u = x'. + e,
Lorentz Rotations x, ! = xu + a.~ x, v (a~ = - ~)
)2(
Scale Transformations: x. ! = x, + e x.
(Proper) Conformal Transformations: x~ = x, + ~. x 2 - 2 x. ~ x.
The proper conformal transformations can be generated by an inversion
(x'. = -x~'/x2), a translation and an inversion.
A'u(x' ) = Au(x ) - 8A,(x)
(3)
A' u (x') = A u (x) + 2 a x A u (x) + 2 (x~ a u - x~, o~) A ~ (x).
It can be seen that Maxwell's equations are invariant, under scale and
conformal transformations, if the electromagnetic potential transforms
like above. Therefore, the interest in this group is quite old 2. Despite
of this it has not yet lead to much physical insight. But there is some hope
that by the skill we have acquired in dealing with symmetries and with
* Based on a Lecture given at the International Summer School for Theoretical
Physics 1970, University of Heidelberg.
2 J. Wess:
broken symmetries, we might be able to explore fruitfully conformal
invariance.
The natural domain, where one would hope to learn something
from conformal invariance is at processes where all energy variables are
large compared to the masses involved. This is so because in a Lagrangian
with dimensionless coupling constants it is the mass term which breakes
the symmetry. To demonstrate this let us assign the following trans-
formation properties to fields:
Scalar Field:
)'x('Sq = )x(b~ - e ~b(x) qb'(x') = )x(~q + b~x~c2 (x). )4(
Spin 1/2 Field:
~'(x') = ~ (x)- (3/2) e o~ (x)
(5)
~'(x') = w (x) + {3 ~c x + (1/2) ce x '~ <, y'} o~ (x)
or, more concise:
"p( (x') = 1( - de) p~(x) )6(
;o~ (x') = 1( + 2 d a x) ~q ~(x) + 2c~.xtS~ ~q ,(x),
where S.t is the generator for Lorentz rotations and d = 1 for Bose
fields and d = 3/2 for Fermi fields.
We find that a Lagrangian is conformal invariant if it is composed
of the massless free Lagrangian and the following interactions:
~4, ~)l/~lp, ,~157~/,u~q
A"pTu~ p, ,P~57u?p~UA )7(
AU~Su(o , A u x At(SuAv - 8tA.), (A" x A t) (a. x A O.
For this reason one might be willing to give some serious considerations
to the conformal group.
The group is isomorphic to S O ,4( 2) and the commutation relations
between the generators are, in an obvious notation:
pu, W = 0
M ,~u M ~ = ~Ug Mt~ + g~ M ~'~ _ ~Ug Mt~ _ gtO M~'X
MUt, p~. = gtZ pu _ ZUg W
IS, P~ = P~
Is, M"q = o (s)
C", C t = 0
C ,u W = 2(M u" - 9"tS)
C ,~ M.t = 9~. Ct _ gZt C u
C ,~ S = C%
Conformal Invariance and the Energy-Momentum Tensor 3
It is easy to show that in a theory, where the generators of the con-
formal group exist as linear operators in Hilbert space, the only possible
discrete eigenvalue of p2= M 2 is zero. Moreover, the continuous spec-
trum of p2 cannot start at any finite value.
To prove this let us labte the eigenvalues of pZ by #, or K, depending
if they are discrete or in the continuum. We normalize the states as
follows:
(Pi#j) = 6u, (g, K') = 6(K-K'), (p~, K) = 0. (9)
If S is a linear operator and if the eigenstates of p2 form a basis and are
in the domain of definition of S we can expand:
S#~) = ,_~ Ci jl#j) + . d K C~(K) K)
J (10)
SK) = ~ C,(K) #,) + ~ d K' C (K, K') K') .
i
From the commutator IS, p2 = 2p2 follows:
2#rSrs = C,s(#r-- #s)
(11)
2 K 6 (K - K') = (K - K') C (K, K').
Taking r = s in the first equation yields #, = 0, i.e. the only possible dis-
crete eigenvalue of pZ is zero.
The second equation has the solution
C(K, K')= - 2K S'(K - K'). (12)
For K > 0, K' > 0 we find that C (K, K') = - C (K', K) + 2 6 (K- K'), as a
consequence, the continuous spectrum cannot start at any finite value
of K.
Notice that in this proof we have used only the property that S is
defined on the eigenstates of p2, it was not necessary to assume that S 2
or expi2S exists.
This result, valid for a strictly conformal invariant, conventional
theory is not very encouraging. To approximate any realistic theory
by a conformal invariant theory does not seem to be realistic because
the vanishing masses would give rise to severe infrared problems.
Fortunately, we have learned from chiral invariance and PCAC how
to deal with approximate symmetries without really having to start
from the symmetric limit. In order to apply this techniques we are now
going to construct the relevant currents by Noether's theorem. These
currents might be meaningful, even when the corresponding charges
(like S) do not exist.
4 J. Wess:
We find 3:
s,,= o.vv-O/2) ~ ~r F.
ralacs sdleif
C,v = Ou~(gX~x 2- 2xXx~) )31(
+ 2 vX( eu ~2 -- guy 42) q- 2X~ Fu
ralacs sdleif
where L= ~, 6 o(~0~cZat dgu~q~+Zuvq) and ~uO is the symmetric
lla sdleif
energy momentum tensor. If Fu--0,A, the definition of the energy
momentum tensor can be changed 4:
O.v = O.~ - (1/3) (G~ - 0.0~)(A - 42/2). )41(
This new tensor has all the properties required for Poincar~ invariance
and so have the new currents:
~ = O,v~ x, v C#v = O,(g~vx 2 2 - 2x2xv) )51(
for scale and conformal invariance. They differ from the original currents
S, and C,~ only by terms which do not contribute to the charges. In
this very interesting case we find:
0 u d V" = - 2x~ O"Su = Of )61(
i.e., the breaking of the conformal and scale invariance are both charac-
terized by the trace of the energy momentum tensor.
II. Low Energy Theorem for Gravitons
The connection between conformal invariance and the energy momentum
tensor was established in the preceeding lecture. The energy momentum
tensor, on the other hand, is thought to be the source for gravitons.
Therefore, we might hope to learn something about conformal invariance
by studiing the emission of gravitons.
We consider the matrix element:
M ~" = (~10u~ Ifl) )71(
of the energy momentum tensor between a state fl) with an arbitrary
number of incoming particles and the state @ with an arbitrary number
of outgoing particles. The process fl-+ e is described by the amplitude T:
53/1~( = 6 )k( (~l Tiff) )81(
lamrofnoC Invariance dna eht mutnemoM-ygrenE Tensor 5
where k = ~P - Pp is the difference between the total momentum of the
outgoing particles (P.) and the total momentum of the incoming parti-
cles (Pp).
From Lorentz invariance follows:
Uk M~,v = k~ M,,v = Uk V.k ~uI,if = 0. (19)
If ~ and fi are not one-particle states the fourvector k" will, in general,
have four independent components. In this case we can differentiate
Eq. (19) with respect to k, and we obtain, using also the symmetry of Mu~:
M0, = (1/2) k,k,(c3/~3 k )o (O/~ k') M ~" . (20)
Eq. (20) shows that M,v is either singular as k--+0 or of order k"kL Thus,
if we are able to determine the singular part of M,~, (M,~), we know
M,~ up to second order in k.
Me. = (i/2) G (O/a k k~) (a/a )"k M'"" + 0 k2 . (21)
From perturbation theory we learn that a singular contribution to M "v
can only arise from a process where the graviton is emmitted from an
external line. This contribution can be computed in terms of the gravita-
tional form factors of the relevant particles and the amplitude for the
process fi--+ a.
The gravitational form factors for a scalar particle are:
11@ O,,, P2> = (1/2) P,P~FI(k )2 + (#,~k 2 - k,k~) f2(k )2
(22)
P =pl +P2, k=p,-P2.
From the definition of the energy H = 5 800 d3x follows F a (0)= .1
For a spin 1/2 field the form factors are:
<PI I O,~ P2> = ~(Pl) {(1/2) (?,Pv + 7~P,) ~G (k )2
(23)
-1- (1/2)p# vP 2G (k2) -~ (g/iv k2 -- ~/k )vk 6 3 (k2)} u (P2).
Again, from the definition of the energy and the angular momentum:
GI(0 ) ,1 G2(0)= 0.
=
The singular part of the matrix element (17) can now be computed.
It arises from the expression:
~. <pdo.vlp,+k> 7r(P,+k)
_ (p,+_k)2 m 2 A(P,+_-k, ...) (24)
i: ~n t particles
where A is the off-mass-shell amplitude for the process ~fl, rr is a
polynomial in Pi +- k, due to the propagators. The sign of k depends on
whether the graviton is emitted by an incomming or an outgoing particle.
We insert (24) into (20) and obtain M,~, the singular part of Mu~. We
6 J. :sseW
expand A (Pi -+ k, ...) in powers ofk and neglect terms of order k .2 Contrary
to what one might expect, there is no off-mass shell dependence to this
order in k.
Finally, we take the trace of M,~, we omit terms proportional to the
masses of the particles and obtain:
1=( o." >BI
= ~'~ {(di + Pi" 8/~ P,)
i: out, particles
+ k"Ed,. O/~p~ + P'. ~/OP,. O/OPg - (1/2) P,, ~2/8p2 )52(
- i Z~;~ 0/8 P/'} A
+ 0 k( )2
1~ scalar
where di = 3/2 spin 1/2 and the S~ are the generators for Lorentz
rotations. We would like to emphasize that, up to now, we have made
use of Lorentz invariance only. Scale and conformal invariance or
approximate invariance would tell us something about the trace of the
energy momentum tensor independent of Eq. (25). Invariance for ex-
ample would tell us:
(~10~ Ifl) = 0 + 0 Ek23. (26)
Approximate invariance could mean that
1~( Of >lfI = M2f(Pi .. .) + 0 k 2
and that in a domain, where all energy variables are big we can neglect
the term proportional to M ,2 M 2 being some fixed, but finite mass,
characteristic for the system.
It is very interesting to note that at the right hand side of Eq. (25),
the coefficient of the zero'th order in k is just the differential operator
which we would obtain from the Ward identities due to scale invariance,
assuming that the fields have their natural dimension, i.e. transform
according to (6). The coefficient of the first order in k is related to con-
formal invariance in the same way.
III. Conformal Invariance and Effective Lagrangians
In this lecture we are going to describe the interaction of an external
gravitational field with matter fields through an effective Lagrangian.
This means we have to use the formalism of the theory of general re-
lativity. Such a Lagrangian has to be invariant under general coordinate
Conformal Invariance and the Energy-Momentum Tensor 7
transformations:
x,~, = x'U {x ~ ... x }3
A'U(x ') = (0 x'U/O x ~ A ~ (x) (27)
A'~,(x') = (~ x~ x 'u) A o (x).
Because we consider our Lagrangian as an effective one and not as a
basic one we should be able to deal with fields of arbitrary spin. This
is best done if one uses the concept of a Vierbein 5, which we are going
to explain in a moment.
In the preceeding lectures we have developed some interest in the
trace of the energy momentum tensor. Therefore, it is advantageous to
write the Lagrangian in such a way that the trace of the energy momentum
tensor can be obtained by a simple variation. This can be achieved by
the conformal variation 6:
)x(22=~Ug6~o=,x6 ,~"g 64)=-d2(x).4~
(28)
6gu, = - 22(x) ~ug
where d is the appropriate dimension of the respective field.
We vary the action accordingly:
aa=aI l/Tad'x
)92(
= ,j { 6(~ /Tg) g.~22 6 (s /-~) }
ag ~" 5q6 ~bd2 d4x.
It is well known that the energy momentum tensor can be defined through:
6 ac~ ~ _ (1/2) 0~ /~g. (30)
6gU ~
The equations of motion say:
6 _ o. (31)
4a
Therefore, we obtain the desired result:
aA = - S d4x O.,g (32)
If Y' is invariant under the transformation (28) we find that the trace
of the energy momentum tensor is zero. Our aim, as customary, is to
construct the main part of the Lagrangian invariant under (28) such that
the trace of the energy momentum tensor can be computed by the varia-
tion of a simple term in the Lagrangian which breakes the symmetry.
For this purpose we are going to show that the free, massless field equa-