Is quantum field theory derived from quantum mechanics or vice-versa?
Quantum Field Theory grew out of Quantum Mechanics, specifically, the need to make QM compliant with Special Relativity. In SR, space and time are on equal footing, and “mix together” when you switch reference frames. But, in vanilla QM, time is treated as a normal number (basically just a label), while the position in space is treated as a quantum operator! You can’t really “mix” those things. The solution that the physics community settled on was to demote the position to being just a label as well and to use these spacetime labels for operator-valued fields. Thus, “Quantum Field Theory”. So, QFT was “derived from” QM in the sense that QM defined the behavior that QFT needed to reproduce, and was somewhat of a guide.
On the other hand, QM can be “derived from” QFT more or less by sending c→∞c→∞ and then throwing away / simplifying the bits that are no longer relevant. It’s a bit like how Special Relativity basically simplifies to Galilean Relativity for sufficiently slow speeds. QM can somewhat be considered a “special case” of QFT, in which particles can’t be created or destroyed, and all speeds are far below cc. The representation of the two theories may still look very different, but they become empirically equivalent.
On the other hand, QM can be “derived from” QFT more or less by sending c→∞c→∞ and then throwing away / simplifying the bits that are no longer relevant. It’s a bit like how Special Relativity basically simplifies to Galilean Relativity for sufficiently slow speeds. QM can somewhat be considered a “special case” of QFT, in which particles can’t be created or destroyed, and all speeds are far below cc. The representation of the two theories may still look very different, but they become empirically equivalent.
