Chapter 3
A Concise Introduction to Quantum
Field Theory
Manuel Asorey
Departamento de F ́ısica Te ́orica, Universidad de Zaragoza and
Centro de Astopart ́ıculas y F ́ısica de Altas Energ ́ıas
email: [email protected]We review the basic principles of Quantum Field Theory in a brief but comprehensive intro-
duction to the foundations of Quantum Field Theory. The principles of Quantum Field Theory
are introduced in canonical and covariant formalisms. The problem of ultraviolet divergences
and its renormalization is analyzed in the canonical formalism. As an application, we review the
roots of Casimir effect. For simplicity, we focus on the scalar field theory but the generalization
for fermion fields is straightforward. However, the quantization of gauge fields requires extra
techniques which are beyond the scope of these lectures. The special cases of free field theories
and conformal invariant theories in lower space-time dimensions illustrate the relevance of the
foundations of the theory. Finally, a short introduction to functional integrals and perturbation
theory in the Euclidean formalism is included in the last section.
3.1 Introduction..............................
Quantum Field Theory (QFT) is the current paradigm of Fundamental Physics. It
emerges from the convolution of Quantum Physics and Relativity, the two major
theoretical revolutions of the 20th century physics. The search for a theory of
quantum fields started right after the discovery of Quantum Mechanics, but the
appearance of ultraviolet divergences postponed the formulation of a consistent
theory for two decades. The main problemwas solved in perturbation theory by
the renormalization of all physical quantities: vacuum energy, fundamental fields,
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