Why is time evolution replaced by strongly continuous unitary representation of the Poincaré group in relativistic QM?

Question

One of the basic postulates in quantum mechanics is the one of time evolution: a state $\psi$ on a Hilbert space $\mathscr{H}$ at $t_{0}\ge 0$ will evolve to a state $\psi_{t} := e^{-i(t-t_{0})H}\psi$ at a time $t \ge t_{0}$.

In relativity, space and time become one physical entity called space-time. Since space and time are in equal footing, one has to change the time evolution postulate. It is natural to consider wave functions to live on $\mathbb{R}^{4}$ now (actually, they live in so-called mass shell). However, why is the time evolution postulate replaced by the the existence of a strongly continuous unitary representation of the Poincaré group? What is the physical meaning of this postulate?

Answer 1

There are a few concepts you are mixing up.

First, there isn't really such a thing as relativistic quantum mechanics of a single particle. This was tried in the 1920s, but ran into various pathologies such as an infinite tower of negative energy states, as explained in the intros to most quantum field theory textbooks. It was discovered that in order to have a consistent interacting quantum theory involving relativity, we need to move to quantum field theory, which allows us to consider processes where the number of particles changes.

Second, within quantum field theory, it is perfectly possible to formulate a Schrodinger picture, where the state of the system is described by a wavefunction(al), and the state evolves in time according to the (functional) Schrodinger equation. There is a cost relative to ordinary quantum mechanics, because the basic degrees of freedom in quantum field theory are fields, or functions of space. Therefore, the wavefunction in quantum mechanics (which assigns a probability amplitude to each particle position) gets generalized to a wavefunctional (which assigns a probability amplitude to each possible configuration of the field). As a result, the wavefunctional is a much more complicated mathematical object then the wavefunction, and it is usually not practical to use this approach for calculations.

Finally, in any quantum system that is invariant under a given continuous symmetry group, the states must form unitary (or anti-unitary) representations of that group. This is due to well-known arguments by Wigner. In the case of relativistic quantum theory, the underlying group of spacetime symmetries is the Poincaire group. The requirement that the states must form unitary representations of the Poincaire group leads to the conclusion that we can classify the states (at least asymptotically) as particles with a given mass and spin. The logic of looking for representations of an underlying underlying symmetry group is not special to relativistic quantum theory; similar arguments in the Coulomb potential with rotational invariance let you conclude that the states can be classified by angular momentum, plus additional quantum numbers. (And an even trickier argument using the hidden $SO(4)$ symmetry of the Coulumb potential in 3 spatial dimensions lets you classify all the states). The only special thing that happens in the relativistic case is that the group of symmetries is the Poincaire group, as opposed to other groups you might consider.