This question came up while reading the Peskin's An Introduction to Quantum Field Theory first lines, where, in order to explain the need for the field theory approach, he says relativistic mechanics imply that there will be more then one particles involved "since the Einstein relation $E=mc^2$ allows for the creation of particle-antiparticle pairs." Could anyone why this is so?
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2$\begingroup$ It's possible to write down theories which are relativistic and for which particles aren't created or destroyed --- these are called free theories. That is to say, $E = mc^2$ doesn't imply particle-antiparticle creation. However, in the real world we observe particles coming into and out of existence --- for instance, we see excited atoms spontaneously emit photons. Hence we observe that energy can be converted into mass, and particle number can change, and hence we need a theory that can accommodate this. This is quantum field theory. $\endgroup$– gj255Oct 4, 2016 at 16:18
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A simple answer could be as follows. The origin of the Einstein relation, in fact, is the energy-momentum relation $$E^2+(pc)^2=(m_0c^2)^2$$
(for more detail you may see:Energy-momentum_relation)
From the energy-momentum relation one could formally take the relativistic expression for the energy $$E=±(m_0^2c^4-p^2c^2)^{1/2}$$
Dirac by deriving his equation from this relation showed that the minus sign in this relation implies the existence of anti-particles.
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$\begingroup$ This is misleading. Dirac got into a lot of trouble because he was trying to explain what cannot be explained using a theory of a definite number of particles. He kind of tried to resolve it with considering infinite seas and such but that is not the modern approach. $\endgroup$– user87745May 21, 2020 at 10:28