Interpretation of an "interaction" term In QFT a polynomial (of degree >2) in the fields is said to be an interaction term, Ex.: $\lambda\phi^4$.
Question
Is it possible to give an interpretation to terms like $\frac{1}{\phi^n}$? (for $n\in\mathbb{N}$)
Cheers
 A: In principle, yes, but only if the expectation value of $\phi$ is nonzero, so $\phi$ would immediately be shifted. Moreover, the result would be badly nonrenormalizable, so nobody is using such terms.
An important case of a nonpolynomial interaction that received considerable attention in 1+1D is the interaction $\sin\phi(x)$ of the sine-Gordon model 
http://en.wikipedia.org/wiki/Sine-Gordon#Quantum_version
A: As Arnold Neumaier points out, the particle interpretation is not very good.  
Another problem:  The interaction $V(\phi,x) = 1/\phi^n(x)$ isn't stable if $n$ is odd. This energy isn't bounded below.  You can try to fix this by setting $V(\phi,x) = 1/|\phi|^n(x)$, but there's still a kind of stability problem.  The minimum of this potential is at $\phi(x) = \infty$, so if you start in the naive vacuum, you'll generate a huge expectation value.
If you put this theory on a lattice, you'll see the magic of effective field theory in action:  The expectation values of observables whose support is large relative to the lattice spacing will be governed by an effective field theory with polynomial interactions.  You'll find you could have computed these expectation values just as well by assuming that the lattice theory was renormalizable, with coefficients gotten by matching the input and output of renormalization flow.
