I've read that one of the primary motivations for the need for QFT is that quantum mechanics cannot account for particle creation/annihilation, however special relativity "predicts" that such phenomena are possible (clearly they have been observed experimentally, but I'm going for a heuristic explanation for motivating QFT here). Hence, if one wishes to construct a fully consistent relativistic quantum theory, one must account for such phenomena, which naturally leads to the construction of QFT.(adsbygoogle = window.adsbygoogle || []).push({});

If this is correct, then there are a couple of things that I'm slightly unsure about.

Firstly, is the reason why quantum mechanics cannot treat scenarios in which there are a variable number of particles because of the the continuity equation: $$\frac{\partial}{\partial t}(\langle\psi\vert\psi\rangle) +\nabla\cdot\mathbb{j}=0$$ which implies conservation of particle number (I think it's correct to say that it is derived by assuming that no particles can be created or annihilated)?

Secondly, in what sense does special relativity "predict" that particle creation/annihilation should be possible? Is it implied by the energy dispersion relation $$E^2=m^2c^4+p^2c^2$$ so if the system has enough energy it can create a particle of mass ##m## from the vacuum?! Does it also come from Dirac's construction of a relativistic equation of motion (the Dirac equation) which predicts the existence of the positron which itself can annihilate with the electron?

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# I Particle number conservation and motivations for QFT

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