192 From Classical Mechanics to Quantum Field Theory. A Tutorial
Another essential characteristic of relativistic field theories is that when the
field theory admits particle states they are accompanied by antiparticle states, i.e.
the theory requires the existence of antiparticles. This interesting property is also
a source of the ultraviolet problems of the theory.
(2)What are the mathematical tools of quantum field theory?As Gamow remarked, the first two revolutions had at their disposal the required
mathematical tools[ 11 ]:In their efforts to solve the riddles of Nature, physicists
often looked for the help of pure mathematics, and in many cases obtained it.
When Einstein wanted to interpret gravity as the curvature of four-dimensional,
continuum space-time, he found waiting for him Riemann’s theory of curved multi-
dimensional space. When Heisenberg looked for some unusual kind of mathematics
to describe the motion of electrons inside an atom, noncommutative algebra was
ready for him. However, the revolution of QFT was lacking an appropriate math-
ematical tool. The required mathematical theory to deal with UV singularities in
a rigorous way, the theory of distributions, was only formulated in the late 1940s
by L. Schwartz.
The fact that the quantum fields involve distributions is behind the existence
of UV divergences which in the quantum field theory requires a renormalization
program.
The goal of these lectures is to summarize the foundations of QFT and provide
some physical and mathematical insights to a reader with a solid mathematical
background. However, due to the space limitations, the level of mathematical rigor
will be softened. We will follow a path between the standards levels fixed by von
Neumann and Dirac in their approaches to quantum mechanics^1.
In Section 2, we summarize the basic elements of quantum mechanics and spe-
cial relativity. The principles of QFT in the canonical approach are introduced
in Section 3. The first problems of the canonical quantization approach appear
with the ultraviolet divergences of the vacuum energy. The renormalization of
the vacuum energy and the quantum Hamiltonian and their implications in the
Casimir effect are analyzed in Section 4. The relations between quantum fields and
particles are analyzed in the context of free field theories in Section 5, whereas the
introduction of interactions is postponed to Section 6. The covariant formulation
of QFT is analyzed in Section 7 and the functional integral approach to the quan-
tization of classical field theories in Section 8. We conclude with a brief outlook
of the main topics not covered by this review. Finally in three appendices we ad-
(^1) There are excellent textbooks in QFT, e.g.[ 1 ]–[ 13 ]. Our approach will be close that of Simon
and Glimm-Jaffe books[ 23 ],[ 13 ].