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Witten-type topological quantum field theories are based on cohomology theories. Every observable must lie in a cohomology class. May be $G$ a geometric field. Then every observable expectation value must be independent on variations in $G$ 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)

$$<O>=\int d[G] Oe^{iS(G)}.$$

Then a physical observable is given by $O = G|_{coh}$, i.e. fields that lie in the cohomology class of the theory. Now one can decompose the general field $G$ into a cohomology class part $G|_{coh} = P_{coh}G$ (projection operator is $P_{coh}$) and a "vertical" part $H$. The Haar measure become

$$d[G] = d[G|_{coh}]d[H]$$

and it holds $$S(G) = S(G|_{coh},H). $$

Now one can go to from the position space to momentum space. There are also terms included with $H_k$ and the 4-momentum $k$ is the (transferred) momentum of $H$. Since $H$ is not a physical observable, $H$ plays the role as an energy-momentum excess field.

Maybe I am wrong on a certain statement, but I am asking:

Is there really nonconservation of energy and momentum in a general topological quantum field theory?

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  • $\begingroup$ I don't understand this question. The stress-energy tensor of topological field theories is zero rather by definition (since it is the variation of the action w.r.t. to the metric, but the action doesn't depend on the metric if it is topological), hence it (and energy and momentum) are trivally conserved (as being zero). In theories where this tensor is not naively zero it turns out it is just a gauge variation, and hence can be gauged to zero, anyway. $\endgroup$ – ACuriousMind Sep 20 '15 at 13:38
  • $\begingroup$ Tip: Consider adding references in order to get useful and focused answers. $\endgroup$ – Qmechanic Sep 29 '15 at 13:43

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