Quantum Field Theory I

Chapter 1
Introductions
Quantum field theory is
• a theory of particles
• mathematically ill-defined
• the most precise theory mankind ever had
• conceptually and technically quite demanding
Mainly because of the last feature, it seems reasonable to spend enough time
with introductions. The reason for plural is that we shall try to introduce the
subject in couple of different ways.
Our first introduction is in fact a summary. We try to show how QFT is
used in practical calculations, without any attempt to understand why it is used
in this way. The reason for this strange maneuver is that, surprisingly enough,
it is much easier to grasp the bulk of QFT on this operational level than to
really understand it. We believe that even a superficial knowledge of how QFT
is usually used can be quite helpful in a subsequent, more serious, study of the
subject.
The second introduction is a brief exposition of the nonrelativistic manyparticle
quantum mechanics. This enables a natural introduction of many basic
ingredients of QFT (the Fock space, creation and annihilation operators, calculation
of vacuum expectation values, etc.) and simultaneously to avoid the
difficult question of merging relativity and quantum theory.
It is the third introduction, which sketches that difficult question (i.e. merging
relativity and quantum theory) and this is done in the spirit of the Weinberg’s
book. Without going into technical details we try to describe how the
notionofarelativisticquantumfieldentersthegamein anaturalway. Themain
goal of this third introduction is to clearly formulate the question, to which the
canonical quantization provides an answer.
Only then, after these three introductions, we shall try to develop QFT
systematically. Initially, the development will concern only the scalar fields
(spinless particles). More realistic theories for particles with spin 1/2 and 1 are
postponed to later chapters.
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