Complementary Parameters

[Feeble Wonk]

I was thinking about the various complementary parameters in quantum physics. I'm curious if anyone is aware of a "complete" list of such measurements or qualities that has been compiled in one source, based on current knowledge.

 

[jfizzix]

I don't know of any table, but it is generally accepted that any pair of generalized positions and momenta will be complementary in the sense that resolving measurements of both is limited by the Heisenberg uncertainty principle. This includes ordinary positions/momenta, angular positions/momenta, and any observables associated to generalized positions and momenta in Hamiltonian/Lagrangian mechanics.

Alternatively, there are also pairs of observables that are complementary, but not conjugate. As an example, it is not possible to prepare a spin-1/2 particle in a state where you will be able to predict the measurement outcomes of all its spin components with accuracy.

As an interesting side note, the list of all kinds of sets of complementary observables is not complete yet, even for simple systems. For example, for quantum systems of dimension 6 (say, a pair of particles; one spin-1/2 and one spin-1) it is an unsolved problem to find a complete set of complementary observables (also called "mutually unbiased" observables).

 

[Vanadium 50]

There are an infinite number of such pairs, so nobody has considered it a good use of their time to write them all down.

 

[Feeble Wonk]

There are an infinite number of such pairs, so nobody has considered it a good use of their time to write them all down.

Fair enough. I was aware of the various forms position/momenta complimentarity. But I'd recently read a passage where Lee Smolin referred to aspects of time and space having a similar complementary relationship, and it got me thinking about the general concept.

 

[atyy]

In a sense, complementary observables are the basis of any quantum theory.

A quantum theory is specified by (1) Hilbert space (2) Observables (3) Hamiltonian.

In specifying (2) Observables, a very important part is their commutation relations, which is how "complementary observables" are formalized in the mathematics.

 

[Vanadium 50]

Atyy is absolutely right - I might even drop the "in a sense". Once you define the commutator algebra, you have defined the theory. That's both its power, and the reason you can't write it all down.

 

[Feeble Wonk]

Thanks to both of you. That's the input I was looking for. It gives me a different way of looking at the idea.

 

[atyy]

Atyy is absolutely right - I might even drop the "in a sense". Once you define the commutator algebra, you have defined the theory. That's both its power, and the reason you can't write it all down.

Yes, I put "in a sense" in at the last moment, knowing that this is PF and there will be all sorts of tricky questions, like whether an anti-commutation relation is also "complementary" :)