Question

Have Witten-type TQFT's nonconservation of energy and momentum in interactions?

Answer 1

Witten-type topological quantum field theories are based on cohomology theories. Every observable must lie in a cohomology class. May be a geometric field. Then every observable expectation value must be independent on variations in to satisfy the criterion that a theory is a topological field theory.
Now suppose I have the observable expectation value (normalization factor is involved in the Haar measure)
Then a physical observable is given by
, i.e. fields that lie in the cohomology class of the theory. Now one can decompose the general field into a cohomology class part (projection operator is ) and a "vertical" part . The Haar measure become
and it holds
Now one can go to from the position space to momentum space. There are also terms included with and the 4-momentum is the (transferred) momentum of . Since is not a physical observable, plays the role as an energy-momentum excess field..

Answer 2

Both conservation of energy and conservation of momentum always apply, but are not always equally convenient to use.