From Matthew D. Shwartz Quantum Field Theory textbook, he writes: "Most of the time, a symmetry of a classical theory is also a symmetry of the quantum theory based on the same Lagrangian. When it is not, the symmetry is said to be anomalous." A priori, how is this even possible? If a symmetry ...

The whole system is adiabatic, no heat exchange can take place, if the volume of the gas now doubles it should actually cool down. That's why I don't understand $dT=0$