Explaining Rank 0, 1 and 2 Field Theory in Simple Terms

[actionintegral]

"Between the simplest rank 0 field theory and the simplest rank 2 field theory is the simplest rank 1 field theory."

I found this quote buried in a huge thread. It seems to be the central point of that thread but I don't know what it means.

Can someone explain what it means using really small words that will fit into my little brain?

 

[robphy]

I think that these are references to ways to describe the gravitational field. A "rank 0" theory is a "scalar" theory, like ordinary Newtonian gravitation with the gravitational potential (a scalar field) $$\phi$$ [which satisfies the Poisson Equation $$\nabla^2\phi=\rho$$]. A "rank 2" theory is a "tensor" theory, like the Einstein's General Relativity with a "rank 2" object, the metric tensor field $$g_{ab}$$, which must satisfy the Einstein Field Equations. A "rank 1" theory is a "vector" theory, like Maxwell's Electrodynamics with a vector potential $$A_a$$.

From Quantum Field Theory, these ranks are associated with the "spin" of the [massless] quanta of that theory.

I vaguely recall an argument from a Quantum Field Theory class that somehow rejects odd-spin theories for gravitation. (Does it have to do with the attractive property of gravity?)

 

[Claude Bile]

The only time I have heard the "rank 0, rank 1.." terminology is wrt to Tensors, with a rank 0 tensor being a scalar, rank 1 being a vector and rank 2 being a matrix.

I guess the statement is saying that vector field theory is harder than scalar field theory but easier than matrix field theory...which doesn't make a great deal of sense. Tensor fields of different ranks are all linked via their derivatives, i.e. the derivative of a scalar field is a vector field and so on.

Perhaps whoever posted it meant that vector calculus lies in between scalar calculus and tensor calculus in difficulty...

Without any context, I can't really add any more insight.

Claude.