Author: NickGolub Format: TextDear community, I would be grateful to hear any suggestions concerning the following issue.
In my master thesis I have stumbled upon some issue which makes me go crazy. I want to state the problem but I should discuss it so that it would not mislead me.
There is a model at hand, a field theory which interacts with gravity non minimally via metric in the lagrangian. This field theory is nontrivial topologically and prequantizes. Nonetheless, it has a prequantization when we fix the metric but it seems that it has no canonical prequantization as a field theory on the bundle of fields + metrics (satisfying certain conditions). Moreover, there is an obstruction to prequantization of such a field theory, since if we compute metric energy-momentum tensor locally it shall not glue properly to a global one.
Initially I intended to find out how gravity interacts with nontrivial topologically field theories so as to conclude inconsistencies with gravity even at the level of prequantization.
It seems that given a prequantum field theory which has interaction with gravity (via energy-momentum tensor) we must deform it (canonically some way, so that it could be functorial, may be additive…) so as to extend the prequantization to the bundle of fields + metrics.
Why this idea? I guess the answer is that we already do this with even locally-defined field theories. An obstruction to a proper nonminimal interaction (via Energy-Momentum tensor) of a field theory with gravity is that the Energy-Momentum tensor must be divergence-free. When we deform a given field theory we usually fall to such field configurations that satisfy divergence-freeness or even fall on-shell of the theory.
So we had:
obstruction to nonminimal interaction - divergence-freeness of EMT (analytical condition). We deform a bundle of a theory but not lagrangians. This deformation is “canonical”.
Now we have in addition:
obstruction to nonminimal interaction - EMT should be a globally defined tensor on a manifold (topological condition). We deform a field theory but I guess here we change the Lagrangians so that they could prequantize on a bundle of fields + metrics.
What do you think I could do in this case?
Dear community, I would be grateful to hear any suggestions concerning the following issue.
In my master thesis I have stumbled upon some issue which makes me go crazy. I want to state the problem but I should discuss it so that it would not mislead me.
There is a model at hand, a field theory which interacts with gravity non minimally via metric in the lagrangian. This field theory is nontrivial topologically and prequantizes. Nonetheless, it has a prequantization when we fix the metric but it seems that it has no canonical prequantization as a field theory on the bundle of fields + metrics (satisfying certain conditions). Moreover, there is an obstruction to prequantization of such a field theory, since if we compute metric energy-momentum tensor locally it shall not glue properly to a global one.
Initially I intended to find out how gravity interacts with nontrivial topologically field theories so as to conclude inconsistencies with gravity even at the level of prequantization.
It seems that given a prequantum field theory which has interaction with gravity (via energy-momentum tensor) we must deform it (canonically some way, so that it could be functorial, may be additive…) so as to extend the prequantization to the bundle of fields + metrics.
Why this idea? I guess the answer is that we already do this with even locally-defined field theories. An obstruction to a proper nonminimal interaction (via Energy-Momentum tensor) of a field theory with gravity is that the Energy-Momentum tensor must be divergence-free. When we deform a given field theory we usually fall to such field configurations that satisfy divergence-freeness or even fall on-shell of the theory.
So we had: obstruction to nonminimal interaction - divergence-freeness of EMT (analytical condition). We deform a bundle of a theory but not lagrangians. This deformation is “canonical”.
Now we have in addition: obstruction to nonminimal interaction - EMT should be a globally defined tensor on a manifold (topological condition). We deform a field theory but I guess here we change the Lagrangians so that they could prequantize on a bundle of fields + metrics.