QUANTUM FIELD THEORY in Visual Studio .NET

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QUANTUM FIELD THEORY
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initial data. In particular, the observed dimensions of the fields (or other composite operators) will in general depend on the dynamics. Of special interest is the case where the origin is an ultraviolet fixed point. This situation is called asymptotic freedom. Logarithmic corrections to naive scaling will then emerge as a result of renormalization.
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13-2-1 Scale and Conformal Invariance
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If in a classical action all dimensional constants vanish we would expect the theory to be scale invariant. This could also be the case in a massive theory at short distance (typically m[x[ 1). If the configuration variables are scaled down
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Je -lX
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(13-37)
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the fields, noted generically as <p, would transform according to
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<p(x) -4 A<p(x) = U(Je)<p(Jex)
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(13-37a)
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with U(Je) standing for a finite dimensional representation of the abelian group of dilatations. We shall assume this representation to be fully reducible. This means that we may write
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U(Je) =
eDInJc
where the matrix D can be diagonalized. In infinitesimal form, when In Je = be, the transformation law reads
b<p = be (x.
:x +
D )<p
(13-38)
In a classical massless theory (13-37) or (13-38) lead to an invariance provided the eigenvalues of D are equal to 1(dj2 - 1) for Bose fields, and i(dj2 - i) for Fermi fields. The quantities in parentheses apply to the case of an arbitrary dimension d instead of four. We can also consider the effect of such transformations in a massive theory, obtaining therefore a Ward identity reflecting the breaking of scale invariance. In this sense we differ from pure dimensional analysis, since we consider the effect of a transformation of dynamical variables (the fields) but not of the dimensional parameters such as masses. If we were to do so we would relate two different physical situations. For our favorite examplei (13-39)
t To avoid confusion with the scale parameter .J., the throughout this chapter.
coupling constant
WIIl
be denoted g
ASYMPTOTIC BEHAVIOR
we find a variation
Hence, if D
[yft'
x ~+ 2(D + 1) -2- - 4Dg
aft'
(a<p)2
41- 2D T
m2 <p2
(13-40)
[yft' = (x.
~ + 4)ft' + m 2<p2
The integral Jd4x A4 ft'(AX) is independent of A (positive). From a differentiation at A = 1 it follows that
meaning that [x (a/ax) action I reads
d x (x.
:x +
4)ft'(X) = 0
+ 4] ft'(x)
is a divergence and that the variation of the
M = be
d4x m2<p2(x)
(13-41)
Obviously, when m vanishes we have classical scale in variance.
Let us show that conformal invariance is then a consequence of scale invariance. The conformal group is defined as the set of transformations leaving the angles invariant. This carries over to Minkowski space, where we deal with hyperbolic as weII as circular angles. This group is obtained by adjoining to the Poincare transformations an inversion with respect to an arbitrary point-the origin, for instance, (13-42) For this definition to be meaningful, the usual R4 space must be completed by a cone at infinity. Let us introduce a useful geometrical representation. We consider a six-dimensional space with a (2,4) metric, i.e., such that
The lines belonging to the isotropic cone Z2 = 0 are identified with the usual R4 space completed by a cone at infinity. This may be realized, for instance, by cutting the cone Z2 = 0 by the hyperplane Z_j = 1 and projecting stereographicaIIy on R4 the resulting one-sheeted hyperboloid from the point (0,0,0,0, -1) in R 5 (Fig. 13-3). The explicit formulas are
(13-43)
xp=--y~
1 + Y4
x2 = ---
Y4 - 1 Y4
The action of the four-sheeted pseudoorthogonal group 0(2, 4) is reflected on Minkowski space as conformal transformations. In particular dilatations correspond to hyperbolic rotations in the plane
QUANTUM FIELD THEORY
Figure 13-3 Projection of a (1,4) hyperboloid on Minkowski space.
, -0, x =e x=yp=
YP ,Y4cosh8+sinh8 ,Y4= cosh 8 + Y4 sinh 8 Y4 sinh 8 + cosh 8
Z4 = Z4 cosh 8 +
sinh 8,
Z'-l
cosh 8 + Z4 sinh 8
(13-44)
As an exercise, find the four other types of conformal transformations, completing therefore the number of generators to 15. Write the corresponding transformations in Minkowski space. Perform the similar construction in the case of an euclidean four-dimensional space. The conformal group is then 0(1,5) and R4, completed by a point at infinity, may be identified with the stereographic projection of a unit sphere in five-dimensional space. In order to prove the conformal invariance of the massless ((!4 theory it is therefore sufficient to study the effect of an inversion. In five-dimensional y space this transformation corresponds to a symmetry y ....... - y of the unit hyperboloid. We have to choose a transformation law for the field. Equation (13-37a), .l((!(x) = .l.((!(h), suggests the definition
((!(x)~ ((!'(x)
:2 ((!(:2)
(13-45)
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