RQM vs QM ( relativistic quantum mechanics vs quantum mechanics)

[moriheru]

What are the advantages of RQM against QM? If RQM is more accurate that QM then why use QM (yet only if it is) or better said if it is more complete? I read Paul Diracs lectures 1-4 and he talked about relativistic quantum mechanics and quantization of flat and curved spaces in lectures 3-4, so I am no Expert in RQM but I very familar with QM.

 

[atyy]

There are two meanings of QM. The first meaning of QM is the general structure as a theory of observables and Hilbert spaces etc. In this sense, RQM is only a particular type of QM. In fact, RQM has some problems which can be cured by going to relativistic quantum field theory. In this first meaning of QM, relativistic quantum field theory is also a particular type of QM.

In the second sense, QM refers to non-relativistic quantum mechanics, which I shall abbreviate NRQM. This is only accurate in non-relativistic situations, and can be derived (in the physics sense) as a valid approximation from relativistic quantum field theory. In situations where the full relativistic quantum field theory is not needed, it is more convenient to use NRQM. NRQM accurately describes most condensed matter physics and quantum chemistry. In some cases in quantum chemistry, a relativistic treatment is needed, but it is still too inconvenient to use the full relativistic quantum field theory formalism, and RQM is used in such cases.

Also, my personal bias is that even relativistic quantum field theory can be thought of as ordinary NRQM by using lattice models. The main problem with this bias is that it is still not clear if chiral interactions can be properly treated in the lattice framework.

 

[moriheru]

Is RQM a branch of QM or a independent field?

 

[atyy]

Is RQM a branch of QM or a independent field?

RQM is a branch of QM in the sense of QM being a formalism about observables and Hilbert spaces.

 

[dextercioby]

There are consistency problems in a specially relativistic formulation of particle quantum mechanics - which is basically done with Klein-Gordon's and Dirac's wave equations - which force you to use relativistc quantum fields. I would venture to say that there's no theory of relativistic quantum mechanics.

 

[moriheru]

@atyy Ah, ok thanks. How would one transform NRQM into RQM using latice models?

 

[atyy]

@atyy Ah, ok thanks. How would one transform NRQM into RQM using latice models?

Roughly, in a lattice model, we consider a large but fixed number of "particles" stuck to a lattice. Since the lattice violates special relativity, it is non-relativistic. Since there are a fixed number of particles, it is just quantum mechanics, ie. NRQM.

From this NRQM lattice model, we wish to recover relativistic quantum field theory. To recover relativity, we make the lattice spacing fine enough so that the violation of special relativity is below current experimental constraints. To recover quantum field theory, write the quantum fields in gauge invariant form using Wilson loops, so we can see that the lattice model is a discrete version of the continuous quantum fields. In practice, lattice models are usually a discretization of the path integral, but to see that relativistic quantum field theory is recovered, it is easier to use Hamiltonian lattice gauge theory, eg. Kogut and Susskind's http://journals.aps.org/prd/abstract/10.1103/PhysRevD.11.395.

Since we can recover relativistic quantum field theory from a lattice model, we can also recover RQM and NRQM which are approximations to relativistic quantum field theory. As I said, I think the major unsolved problem with basing our thinking on lattice models is the problem of describing chiral interactions.

 

[moriheru]

I see, thanks again.

 

[atyy]

Incidentally, since lattice gauge theory is just a particular type of NRQM, there is the idea to use NRQM systems like cold atoms in optical lattices to simulate some aspects of lattice gauge theories, eg.
http://www.lattice2013.uni-mainz.de/presentations/Plenaries Tuesday/Reznik.pdf
http://arxiv.org/abs/1211.2704
http://arxiv.org/abs/1409.3085