What is a quantum field?

Philosopher of Physics Meinard Kuhlmann has a brilliant article, What is real?, in the August 2013 edition of Scientific American.

Kuhlmann provides a clear account of why the particle and field ontologies provide equally inadequte interpretations of quantum field theory. Whilst most physicists tend to resort to the lazy claim that the particle and field concepts are somehow 'complementary', Kuhlmann points out that this doesn't help "because neither of these conceptions works even in those cases where we are supposed to see one or the other aspect in purity."

Kuhlmann's account of how a quantum field is mathematically defined is particularly striking: "A classical field is like a weather map that shows the temperature in various cities. The quantum version is like a weather map that does not show you '40 degrees', but '√'.

The article concludes with a nice explanation of two alternative ontologies: structural realism, and the bundle theory of properties.

If you want an insight into the philosophical problems of modern physics, this is an excellent introduction.