Rank 2 and rank 1 field theories

[kurious]

Can a rank 2 field be considered, in principle, to be the dot product of two rank 1 vector fields?

 

[AKG]

I don't know what you're talking about really, but I would guess "cross product" or "Cartesian product" instead of dot product. The reals (1-space) form a vector space and a field, and the cartestian product R x R gives R² (2-space), which is vector space (and would be a field, I imagine). So guessing as to what you mean, I would think Cartesian product is your answer.

 

[vanesch]

Can a rank 2 field be considered, in principle, to be the dot product of two rank 1 vector fields?

You should specify what you mean. I guess you're talking about tensorial rank. In that case, indeed, the tensor product of two rank 1 tensor fields yields a tensor field of rank 2. In a quantum field theory setting, tensorial rank is related (although not completely equivalent) with spin, and the tensor product is associated with combining two systems. So the quantum version is that the sum of two spin-1 systems contains (but is not equal to) a spin-2 system.
The difference is in what is called irreducible representations.
A tensor field of rank 1 is an irreducible representation, and hence fully maps onto a spin-1 field. However, a tensor field of rank 2 is not irreducible ; in fact it corresponds to a sum of a spin-2 field, a spin-1 field and a scalar (spin 0) field. This comes down to the property that combining 2 spin-1 systems gives you a total spin which can be 0, 1 or 2.

cheers,
Patrick.

 

[kurious]

Vanesch:
This comes down to the property that combining 2 spin-1 systems gives you a total spin which can be 0, 1 or 2.

Kurious:

Supposing I said that the gravitational force carrier was
made of two spin 1 particles coupled together.What would make the carrier just a spin 2 particle in total? I am thinking here of trying to reduce gravity
to being another case of electromagnetism.
My basic idea is this:
mass could be caused by electric charges in space interacting with protons and electrons.So a massless force carrier could just be an electrically neutral phenomenon moving through the
mass-causing charges in space.

 

[jtolliver]

Supposing I said that the gravitational force carrier was
made of two spin 1 particles coupled together.What would make the carrier just a spin 2 particle in total?

All you have to do is make sure the lagrangian depends only on the traceless symmetric product,
$$\frac{1}{2} (A_\mu B_\nu + B_\mu A_\nu) - \eta^{\mu \nu}(A_\rho B^\rho})$$.
Also you should be aware that not every spin 2 field can be constructed this way.