How many uncertainty relationships are there between physical variables? For example the well-known ones are: [momentum,position], [energy,time], [particle number, phase], [angular momentum, angle].
Are there any others? How do we know when we have a complete set?
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For any two quantum mechanical observables A,B, there is the uncertainty relation s(A,f) s(B,f) >= 1/2 |E_f([A,B])|, where s(A,f) denotes the standard deviation of A in the state f (analogously for B), and |E_f([A,B])| the absolute value of the expectation value of the commutator [A,B] in f. In case A and B commute, the relation is trivial, whereas in all other cases, there exists a non-trivial lower bound on the product of the uncertainties. Thus, for typical systems, there exist infinitely many uncertainty relations.
Basically, there are as many such relations as there are pairs of so-called complementary observables in the system you are considering. Mathematically, they are pairs of conjugated variables, such as any derivative of action and the quantity with respect to which has been differentated. In the parlance of quantum mechanics, the operators corresponding to such pairs of quantities never commutate.
@Gandalf, excellent response, and certainly correct: but I wonder if there's a way to understand/reinterpret this question is a less trivial way.
Let me clarify what I mean.
If we take a simple spin 1/2 system, we know that we can always construct an observable at any angle to the z axis, and hence find a maximally non-commuting variable, and therefore there must be an (non-countably) infinite number of commutation relations. But this seems like a bit of a fudge, really, as all of these sets of non-commuting operators are linked by a trivial rotation. So in some sense, all such angular momentum commutation relations are the same relation.
So in the spirit of the clarification to the question, is there a way to enumerate the number of distinct classes of commutation relations for a given system, where I still haven't defined the distinction (as I'm not sure how to) but I guess that it should be something like a relationship between the commutation relations that is not just a simple rotation of similar.
As can be seen, I neither know the answer nor even know how to pose it properly. Nevertheless, my intuition tells me that the answer for 'all classes of systems' must be uncountably infinite (and here, I'm thinking that fractional quantum Hall systems are likely to be the clearest exemplar). But for a finite system, then the answer is likely to be related to questions such as: how many distinct classes of entanglement there are, which is known to grow very rapidly ;).
Other than the four uncertainty relationships mentioned in my question, can anyone list anymore canonical ones?
It would be better to rephrase the question I think. One of the things about complementary observables is that knowing information about one should not give you any information about the other. This is ensured by having mutually unbiased eigenbases for each of the pair of observables. For example, the innerproduct of a position eigenstate with a momentum eigenstate is a constant (independent of the position or momentum eigenvalues). So if you know the system is in a position eigenstate, then you don't know what the momentum is, and vice versa. This is what is means by being mutually unbiased - the innerproduct between one state of each basis is the same constant regardless of the states you choose from each basis.
All the pairs of variables mentioned in the question have mutually unbiased bases. So the rephrased question is, how many mutually unbiased bases are there for a given state space? The 2D state space of a spin 1/2 has 3 mutually unbiased bases corresponding to the eigenbases of the 3 components of spin angular momentum. You can look up the latest research on mutually unbiased bases to find out more. I think that the number of mutually unbiased bases for dimension 6 and above is an unsolved problem. But this is from memory.
By canonical I am taking that you mean of the kind where the commutator is a constant. There is a general way for deriving them using the limiting procedure of the Pegg-Barnett phase formalism. See the paper below:
Article Quantum-optical Phase and Canonical Conjugation
Two more answers:
A. You can also define uncertainty relations using entropy as the measure of uncertainty rather than variance. See the paper below on Entropic uncertainty relations.
B. And further, you can use the amount of information that can be encoded in a state as a measure of how much your know about the system. To distinguish between two ways of encoding the information, you can use symmetry for one and asymmetry for the other. See the paper on Particle-wave duality.
Article Entropic uncertainty relations for canonically conjugate operators
Article Particle-wave duality: A dichotomy between symmetry and asymmetry
You can exclude [energy,time]. It is weird and requires many words to hold.
[angular momentum, angle] is actually the consequence of [momentum,position], so they fall in one class.
[particle number, phase] is from the second quantization. In some cases it is the consequence of [momentum,position].
You have omitted variables that have only matrix representation, such as spins, which are 2x2 matrices (Pauli) in the simplest case.
There are entities described by matrix operators of higher rank. Even spin 1 matrices are 3x3 and so on. The number of uncertainties is infinite because the matrices of any rank can feature in the theory.
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