Multidimensional Quartic Equations

Question

I know for the quadratic case (with $A$ an operator): $$ax^2 \Rightarrow x^T A x \Rightarrow \int xA[x]dx$$ Does any such analogy exist with $ax^4$ type functions? Either in the finite or infinite dimentional case? I'm thinking some sort of tensor notation could work, but not all operators are tensors so it could be confusing.