The Superposition Game

I thought up what I call the superposition game back in 2004 as a gedanken device for exploring the foundations of quantum mechanics. The game is to sit down any physicist in the world and get him/her to list all the physical quantities he/she can think of. Then put the quantities into two groups: (i) those that can exist in a quantum superposition, and (ii) those that cannot.

Firstly, it's amazing how difficult it is to actually remember all the types of physical quantities—many physicists will get so focussed on obscure ones from particle physics that they'll forget to even list simple things like pressure or angular momentum. Secondly, nomatter how esteemed the physicist, I guarantee that his/her two columns will disagree with that of any other. What this all means is that this game is a great excercise, as it identifies open questions and may just trigger a useful insight.

I have played the game with a few people, and below is the list we've come up with so far. So this is the web version of the game, where we'll build up a concensus over time. Feel free to email me if you disagree with any items, and also if you can spot quantities we've missed out. In the comment box, I would like to ideally cite the paper where the first experimental observation (if any) was carried out on each type of superposition. Email me if you know of a correct reference.

Can exist in a quantum superposition Comment

postion

velocity

acceleration

momentum

energy For example, remember that electrons can be in a superpostion of atomic energy levels.

time Notice we are not talking about the direction of time.

angular momentum

angle

angular velocity

angular acceleration

electrical current

magnetic flux

magnetic field

electric field

phase

particle number This one surprises a few people.

spin direction

polarization

frequency

Cannot exist in a quantum superposition	Comment
temperature
pressure
entropy
charge
direction of time
parity
z component of spin
any quark flavour (charm, strangeness....)
colour triality

The Heisenberg Game

The idea here is to now exhaustively list conjugate variables for which a Heisenberg uncertainty relation exists. It is rather annoying that no text book does this. So it is a worthy exercise to do it here. If you see any we've missed, please email me.

Conjugate variables	Comment
momentum & position
energy & time
angular momentum & angle
particle number & phase
frequency & time

The Open Questions

The above two games now lead us to a set of open questions. If you can think of more questions or can provide an argument to close a question then email me.

1) Where does "mass" go? We don't seem to know whether it should be in the superposition column or not. If momentum and velocity are both in the superposition column, why is it that mass does not so easily follow? Can we say anything about inertial mass versus gravitational mass in this context, or must we say we just don't know?

2) Are there any other quantities like mass that cause trouble?

3) What should we say about "force"?

4) What general principle can we find in common between variables that satisfy the superposition and Heisenberg games? Is there something we can say regarding superselection rules? In terms of Hibert space, what about decomposition into diagonal form?

5) Are the quantities that satisfy the superposition game a subset of those that satisfy the Heisenberg game, or vice-versa? Or are the two sets of quantities exactly congruent? Given the answer to this question, what does it all mean?