On non-Hermitian quantum mechanics

Our recent post "On non-Hermitian quantum mechanics" arXiv:0804.2051 we critique some recent work in the field by Carl Bender et al [see for example Phys. Rev. Lett 98, 040403 (2007) and Rep. Prog. Phys. 70, 947-1018 (2007)]. Their work is of interest but, in our opinion, there is some misuse of terminology with respect to the label "non-Hermitian". To us it is more like they are doing conventional quantum mechanics with novel inner products.

There already exists in the literature comments on this topic (e.g. Ali Mostafazadeh arXiv:quant-ph/0310164 and arXiv:quant-ph/0407070 as well as David B. Fairlie and Jean Nuyts arXiv:hep-th/0412148). While these papers present valid arguments they miss the central point of our comment. Namely, that if one chooses such an inner product then the Hamiltonian in question is actually Hermitian (and the whole exercise is somewhat redundant). Our argument is based on an observation that there exists in the literature some confusion over the definition of Hermiticity and the imposition of Schrödinger evolution on the evolution of state vectors. [This argument is not quite the same as the one presented in L.D. Landau and E.M. Lifshitz, “Quantum Mechanics (Non-relativistic Theory)” which, as we point out, also contains a flaw].

Holiday Halo

Just back from a great holiday that started with a 7 night cruse from Luxor to Aswan with Abercrombie & Kent. Absolutely excellent, would recommend them to anyone. Then we to Mt Siani with some good friends of ours for new year. It was there that we saw this Halo above St Catherine's monastery. Makes a rather pleasing photo, feel free to use this image to support your physics teaching if you want to (please remember to cite this blog).

Categories, Logic and Physics

Ever thought that we might be fundamentally missing something in our approach to physics?

Maybe its time to try something new such as applying category theory, of all things, to physics (of particular interest to me - the application of topos theory to the foundations of physics). I stumbled across this incipient branch of mathematical physics quite by accident while checking out the "Quantum Information Scientists of the World Unite" group on facebook.

I have yet to get to grips with this subject but it does look highly promising and very exciting. Especially when you consider that "Briefly put, a topos is a category whose structure is so rich that it is capable of modelling any situation which can even be discussed in mathematical terms." Chris Hillman "A Categorical Primer" (1997). Surly this must find great application in physics.

So far I have mostly looked at the work of Bob Coecke and Andreas Doering, you may want to look at their web site: Categories, Logic and Physics

if you are really interested and a bit lost (like me) see: quant-ph/0510032, quant-ph/0506132 and quant-ph/0703060, 62, 64 and 66

Correspondence Principle

Just finished a paper on the "Correspondence principle and chaos for a macroscopic quantum device". The surprising feature here is that the application of a traditional expression of the correspondence principle is not possible. It needs to be rephrased to something along the lines of:

Consider \hbar fixed (it is) and scale the Hamiltonian so that when compared with the minimum area \hbar/2 in phase space:

1. the relative motion of the expectation values of the observable become large and
2. the state vector is localised.
   

Then, under these circumstances, expectation values of operators will behave like their classical counterparts.

Even then application of this statement is limited by circuit parameters.

I would welcome any observations or constructive criticisms on this result:

http://arxiv.org/abs/0712.3043

Bad Puns

For me this is a first trial as a blogger. There seems to be a growing number of people using blogs to stimulate discussion in science. Time to see if it also works for me.

Why is it so hard to resist bad puns?
Ontoso(phi) from Ontosophy and phi from notation used in quantum mechanics (reflecting my interest in foundations of quantum mechanics).