I'm following my first course in field theory and the professor began, like many books do, by introducing the scalar field. However, I am a bit hesitant about the physical idea of fields. My question is: what is the physical meaning of the fields? Why they are introduced? I read the introduction of the books of Peskin and Weinberg but I'm not satisfied.
Answer
Probably the most fundamental and simple idea of the field arises from heat equation. You have a heat source and heat diffusion through media. It is described by field of temperature. It is the simplest scalar field I can imagine. But it has nontrivial equation of motion - has it? From that simplest cases more complicated arises: force fields introduced without any complicated mathematics by well known genius Michael Faraday, who just draw lines of equal potential from one charge to another in order to understand how electrostatic and magnetic forces works. Of course You may imagine even more complicated fields - general tensor or spinor ones. Strict definition of fields is of course a function which for given point on the manifold assign tensor or spinor to it. Physical fields often has strong continuity and differential conditions on them and obeys complicated differential or integral equations, but the most fundamental idea is still the same: lines which was drawn by Faraday, temperature in continuous media, velocity field of the fluid passing through water canal. Try to understand that simple ideas and return to "quantum-compacted-26-dimensional-theories of everything".
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