From Classical Mechanics to Quantum Field Theory

228 From Classical Mechanics to Quantum Field Theory. A Tutorial

To distinguish between both cases one has to work out the perturbation theory
and to find a good prescription to renormalize all UV divergences. The best renor-
malization scheme is the BPHZ method, developed by Bogoliubov, Parasiuk, Hepp
and Zimmerman to provide a rigorous proof to the perturbative renormalization
program[ 14 ].
In a heuristic way, one can distinguish the renormalizable theory just by a
power counting algorithm. It proceeds by assigning a physical dimension to the
fields according to their scale transformations in a such a way that the kinetic term
of the action holds scale invariant (i.e. dimensionless). In the scalar theory this
means that the scalar fieldφhas dimensiondφ= 1, like the space-time derivative
∂xoperators. In that way the kinetic term of the action

1

2 ‖∇φ‖

2

becomes dimensionless. The theory is renormalizable by power counting if all the
terms of the action have non-positive dimensions. This constraint only allows
Poincar ́e invariant terms like

m^2

2

‖φ‖^2 ,

which has dimensiond=−2,

σ

3!

∫

R^4

d^4 xφ(x)^3 ,

which has dimensiond=−1, or

λ

4!

∫

R^4

d^4 xφ(x)^4 ,

which is dimensionless. No other selfinteracting terms give rise to a renormaliz-
able theory. This limitation became very important in model building because it
introduces very stringent constraints on renormalizable models. What is highly
remarkable is that Nature has chosen renormalizable models to build the theory
of fundamental interactions.
The only fundamental theory which does not satisfy the renormalizability cri-
terium is Einstein theory of Gravitation. In that theory due to diffeomorphism
invariance the Einstein term contains an infinity of terms with positive dimensions,
which generate an infinite number of new counterterms by renormalization.
One of the advantages of the covariant approach is that it does not require
the existence of a classical Lagrangian. It is enough to have a complete set of
Schwinger functions satisfying the fundamental properties E1-E5 of a QFT. This
opens the possibility of quantum systems which are not defined by quantization
of a classical system. There are few examples of that sort. But also it opens the