Student colloquium - Frederik Nørfjand: Why quantum fields? From one particle path-integrals to quantum fields
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Supervisor: Dmitri Fedorov
You might have been introduced to quantum field theory or classical field theory. In that context you might have asked yourself, ‘’why do we need these fields?’’. At first, fields may seem like strange objects, in particular when coming from the single particle description in non-relativistic Schrödinger quantum mechanics. Why don’t we simply make a relativistic Schrödinger theory and leave the field all together? In this lecture, we will consider such a single-particle relativistic quantum theory by using Feynman path-integrals. This theory has aspects that at first glance seems to break with causality, but then turns out to motivate the field concept. We will see how the field theory ties it all together and gives a natural way to understand what seems more obscure in the path-integral formulation. For more details on the subject, please consider the reference .
In the picture the lefthand side is a path taken by a single particle. The particle moves both forwards and backwards in time before arriving at its endpoint. Such paths are allowed in the relativistic single particle pathintegral. On the righthand side, the same path is represented using multiple particles in a field theory approach. Here particles, a, and their antiparticles, b, are created and annihilated in order to make the path happen in a causal manner(moving only forwards in time).
 Padmanabhan, Quantum field theory: the why, what and how. 2016. Chapter 1.