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
viceversa? Or are the two sets of quantities exactly congruent?
Given the answer to this question, what does it all mean?
