Mathematical representation and its implication in physics.

[MathematicalPhysicist]

my post i really about discussing the difference that a mathematical representation does to physics.
for example, let's take for example a system which can be be represented by quaternions and by complex numbers.

if we can represent a physical theory by quaternions which aren't commutative as oppose to complex numbers, does it have an impact on the theory itself?
or in other words we can represent a physical theory by distinct mathematical appraoches which aren't the same and in someway even contrdict each other, then which representation should we approve?
does ockham's razor should put to the test here?

 

[HallsofIvy]

It had better not! A physical theory stands or falls by its experimental evidence. If using one mathematical model rather than another results in a change in experemental evidence then one model was wrong. If neither model suggests an experiment that the other would fail, then they have no differential effect on the theory and are equivalent models.

 

[selfAdjoint]

my post i really about discussing the difference that a mathematical representation does to physics.
for example, let's take for example a system which can be be represented by quaternions and by complex numbers.

if we can represent a physical theory by quaternions which aren't commutative as oppose to complex numbers, does it have an impact on the theory itself?
or in other words we can represent a physical theory by distinct mathematical appraoches which aren't the same and in someway even contrdict each other, then which representation should we approve?
does ockham's razor should put to the test here?

If the complex numbers, as such, are adequate to describe the physical system, then quaternions cannot be appropriate, precisely because the physics must obey the commutative law in order to be satisfactorally represented by the complex numbers, in the sense that calculations with them can reproduce what happens in experiments. The quaternions, with their non-commutativity, would predict a different set of outcomes.

Now this is complicated because you can represent non-commutative physics using either quaternions or matrices of complex numbers. But those matrices are non-commutative among themselves, and in fact there is a mathematical equivalence (isomorphism) between the quaternions and certain matrices of complex numbers, so no contradiction arises.