Energy nonconservation as a link between $f(R,T)$ gravity and noncommutative quantum theory

Understand

$f(R,T)$ gravity was proposed as an extension of the $f(R)$ theories, containing not just geometrical correction terms to the General Relativity equations, but also material correction terms, dependent on the trace of the energy-momentum tensor $T$.

• These material extra terms prevent the energy-momentum tensor of the theory to be conserved, even in a flat background.
• Energy nonconservation is a prediction of quantum theory with time-space noncommutativity.
• If time is considered as an operator and there are compact spatial coordinates which do not commute with time, then the time evolution gets quantized and energy conservation can be violated.