224 From Classical Mechanics to Quantum Field Theory. A Tutorial
wherekis a positive exponent. For instance, for a free massless bosonic scalar
k= 2 and the 2-point kernel is
Δ−^1 (x−y)=a^2
|x−y|^2.The power like behavior of the 2-point kernel is a consequence of conformal in-
variance. The Schwinger functions of a conformal invariant theory cannot contain
any physical dimensionful parameter up to a unique universal space-time scalea.
Conformal invariant theories are associated to second order phase transitions.
In 2+1 dimensional space times the free massless bosonic scalar and the 2-point
kernel is
Δ−^1 (x−y)=a
|x−y|.
However, in 1+1 space times the corresponding 2-point kernel is not scale
covariant,
Δ−^1 (x−y)=−1
2 πlog|x−y|
a.
Such an anomalous behavior means that the theory is pathological. In fact, the
Schwinger functionS 2 (f 1 ,f 2 ) does not satisfies the Osterwalder-Schrader positivity
condition (3.51) because
Δ−^1 (θx−x)=−1
2 πlog2 x 0
a,
which is not positive forx 0 >a 2.
In fact what happens is simply that the theory has not a normalizable vacuum
state. Indeed, the vacuum state measure (3.44) does not define a good Gaussian
measure in this case due to infrared problems.
The result agrees with the Coleman-Mermin-Wagner theorem which states that
a continuous global symmetry cannot be spontaneously broken in 1+1 dimensions.
According to Goldstone theorem such a breaking would imply the existence of
massless scalar fields which as we have pointed out are pathological.
On the other hand in 1+1 dimensions, the conformal group is also different.
In fact it is an infinite dimensional group of space transformations.
Thus, in 1+1 dimensions the conformal symmetry provides much more con-
straints than in higher dimensions. However, there is an infinity of consistent field
theories which are conformal invariant[2; 9].
The simplest case is theO(2) sigma model. It can be obtained (in the spin
wave regime) from the massless scalar by the following field transformation
Φ(x)=eφ(x). (3.53)