Fields transforming into Space

[DuckAmuck]

Are there any theories that have mixing between fields and space?
For instance, a theory with mixing between two fields might look like:
$$L_{int} = k \phi(x) \psi(x)$$
Where mixing between a field and space might look like:
$$L_{int} = k \phi(x) x$$
What are the consequences of something like this?

 

[Ben Niehoff]

Such a term will never be generally-covariant, so at the very least, it will pick out a preferred coordinate system.

In flat space, with Cartesian coords, you still have to make it Lorentz-invariant. So you'll need a vector-valued field: $\phi_\mu(x) x^\mu$.

 

[DuckAmuck]

Could you construct a model that turns fields into space and vice verse?

 

[Berlin]

Not fully sure whether this is the same, but when you look at Mannheim's conformal gravity paper on conformal gravity (http://arxiv.org/pdf/1506.01399.pdf), equation (92) it rephrases the Einstein equation as T (universe) = T (matter) + T (Space) = 0 (with T(space) defines as -G(Einstein)).

Reminds me of the +/- m solutions of Dirac, in this case the idea that matter creation goes hand in hand with equal negative energy of space as a dynamical process. Feels like there should be a kind of (Higgs/ Dilaton) doublet... with a two soft bosons proces to create them (I assume it is allowed to speculate!).

berlin