Unveiling the Connection Between SR and Multiparticle Picture in QM

[qsa]

Why is it that when we combine SR with QM we are lead directly to the multiparticle picture. I know about the standard textbooks, I need to know EXACTLY why? What is it in SR that produces the multiparticle picture.

 

[Meir Achuz]

The appearance of negative energy states requires a multiparticle picture.
Also the appearance of a second time derivative prevents |psi|^2 from being a probability, so psi must become an operator.

 

[qsa]

The appearance of negative energy states requires a multiparticle picture.
Also the appearance of a second time derivative prevents |psi|^2 from being a probability, so psi must become an operator.

Thanks. Let me ask the same question from a different angle. What is it about lorentz invariance implied by SR that leads to multiparticle picture.

 

[Fredrik]

Why is it that when we combine SR with QM we are lead directly to the multiparticle picture. I know about the standard textbooks, I need to know EXACTLY why? What is it in SR that produces the multiparticle picture.

I suspect that what you have in mind is the fact that the Klein-Gordon field can't be interpreted as a wavefunction, already mentioned by clem. If that field is promoted to an operator, by imposing commutation relations on its Fourier coefficients, the Fourier coefficients can be interpreted as operators that change the number of particles of the state they act on. This is a very weak argument at best, so you might as well forget about it. If you want to read about it anyway, I think it's explained in Mandl & Shaw. (Not 100% sure...it's been a long time since I read it).

To define "relativistic QM" properly, we need to incorporate the idea that spacetime is Minkowski spacetime into QM. This can be done e.g. by postulating that there must exist a group homomorphism from the Poincaré group into the group of automorphisms on the set of states. There are some complicated mathematical arguments that can translate this into "There exists a group homomorphism from the covering group of the Poincaré group into the group of unitary operators on a complex separable Hilbert space".

Such a homomorphism is called a unitary representation. A subspace M of the Hilbert space is said to be an invariant subspace for an operator T if T(M) is a subset of M. The representation is a map $g\mapsto U(g)$ where U(g) is a unitary operator. It's said to be irreducible if no U(g) has an invariant subspace, other than {0} and the entire Hilbert space.

Each particle species is identified with an irreducible representation. The Hilbert space of an irreducible representation is interpreted as the set of 1-particle states for the particle species identified by the representation.

The 1-particle Hilbert spaces can be used to construct Hilbert spaces of n-particle states, and the n-particle Hilbert spaces can be combined into a single Hilbert space called a Fock space. This is the Hilbert space that's appropriate for a theory of an arbitrary number of non-interacting particles.

So SR+QM doesn't give you a picture of multiple particles. It gives you many pictures of single particles. I don't know a better place to start reading about these things than chapter 2 of Weinberg's QFT book. If you're going to read it, you might want to read a few pages from this talk first, where he explains his thoughts about what QFT is.

When I wrote something similar in another thread, arkajad posted this comment:

Quantized fields are not irreducible representations. But one-particle subspaces, for stable particles, carry nearly irreducible representations of the Poincare group. Why only "nearly"? Because we have to leave the room for parities, charges, other internal degrees of freedom. So, irreducible representations of the Poincare group enter with some (usually finite) multiplicity.