Quantum field theory (QFT) and quantum mechanics (QM) are both frameworks within physics that describe the behavior of particles at the quantum level. However, there are some fundamental differences between the two.
Quantum mechanics (QM) is a theory that deals with the behavior of individual particles, such as electrons or photons. It provides a mathematical framework for understanding the wave-particle duality and the probabilistic nature of quantum phenomena. QM uses wave functions to describe the state of a particle, and the Schrödinger equation or other similar equations to determine how the wave function evolves over time.
On the other hand, quantum field theory (QFT) extends the principles of quantum mechanics to fields rather than individual particles. In QFT, fields are pervasive throughout space and time and are associated with specific types of particles. These fields are described by operators, and particles are interpreted as excitations or quanta of these fields. QFT combines quantum mechanics with special relativity to account for relativistic effects and interactions between particles.
Now, regarding the conservation of particle number, in quantum mechanics, particle number is not strictly conserved. Quantum systems can undergo processes such as particle creation and annihilation, where particles can be transformed into other particles or appear and disappear altogether. This is known as particle non-conservation in quantum mechanics.
In quantum field theory, however, the conservation of particle number is a consequence of a fundamental symmetry known as gauge symmetry. Gauge symmetry requires that the total number of particles of a particular type (such as electrons or quarks) remains constant in a system, i.e., it is conserved. This conservation arises due to the mathematical structure of QFT and is deeply connected to the underlying symmetries and conservation laws of the theory.
It's important to note that in certain situations, such as high-energy particle interactions or in the presence of strong gravitational fields, particle number conservation can be violated or appear to be violated in both QFT and QM. These scenarios fall beyond the realm of everyday observations and require more specialized theoretical frameworks, such as quantum field theory in curved spacetime or quantum field theory at high energies (such as in particle accelerators or during the early universe).