- #1

- 241

- 43

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter Feeble Wonk
- Start date

- #1

- 241

- 43

- #2

- 757

- 355

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).

- #3

Vanadium 50

Staff Emeritus

Science Advisor

Education Advisor

2021 Award

- 28,056

- 12,592

- #4

- 241

- 43

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.

- #5

atyy

Science Advisor

- 14,751

- 3,257

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.

- #6

Vanadium 50

Staff Emeritus

Science Advisor

Education Advisor

2021 Award

- 28,056

- 12,592

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.

Last edited:

- #7

- 241

- 43

- #8

atyy

Science Advisor

- 14,751

- 3,257

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" :)

Share: