A Concise Introduction to Quantum Field Theory 229possibility of having different field theories with the same Schwinger functions.
In that case they are quantum-mechanically equivalent although their classical
theories are completely different.
In two space-time dimensions, theories which in addition to the fundamen-
tal properties E1-E5 are conformally invariant, have been analyzed and classified
without any reference to the corresponding classical systems[ 9 ]. In three dimen-
sions there has been a recent breakthrough which opens the possibility of having
similar results[ 20 ][ 22 ]. However, in four space-time dimensions the problem is far
from its solution.
In the early seventies Wilson developed an interpretation of the renormaliza-
tion method as a nonlinear representation of the one-dimensional group of dilations
[ 29 ]. In the Euclidean formalism using a space-time lattice regularization Wilson
mapped the quantum field theory system into a statistical mechanics model. Then,
using the properties of second order phase transitions he interpreted the renormal-
ization of a field theory as a limit process near a critical point of the renormaliza-
tion group associated to the second order phase transition. The Wilson method
provided a new non-perturbative approach to quantum field theory which allows a
numerical treatment and has been intensively used in quantum chromodynamics.
However, with Wilson’s approach it was also born the possibility of considering
QFT not as the ultimate theory of Nature. It can be considered just as a successful
approximation to the intimate structure of Nature. This approach, known as
effective field theory, considers that the range of validity of the quantum field
theory has an energy upper bound beyond which the theory does not hold. The
limit scale is sometimes associated to the Planck energy scale, but for some theories
might be smaller.
In the last three decades to solve the problem of quantizing gravity there have
been many attempts searching for theories which are beyond QFT. From one way
or another all these attempts consider the possibility of non-local interactions.
The most popular approach is superstring theory. The connection of all non-local
approaches with fundamental aspects of Nature has not yet been confirmed by
experiments.
Acknowledgments
Lecture notes of a minicourse given at the XXV International Fall Workshop on
Geometry and Physics, CSIC, Madrid (2016). I thank the organizers for their
invitation and hospitality during the Workshop. I also thank A. P. Balachan-
dran, F. Falceto, G. Marmo, G. Vidal and A. Wipf for many enlightening discus-
sions. This work has been partially supported by the Spanish MINECO/FEDER