Mathematica Mathematical representation and its implication in physics.

AI Thread Summary
The discussion focuses on how different mathematical representations, such as quaternions and complex numbers, can influence physical theories. It questions whether the choice of representation affects experimental outcomes, emphasizing that a theory's validity relies on empirical evidence. If one model leads to different predictions than another, it may indicate that one is incorrect. The conversation also touches on the relationship between non-commutative mathematics and physical representation, noting that quaternions and certain matrices of complex numbers can be mathematically equivalent. Ultimately, the effectiveness of a mathematical model in physics is determined by its ability to accurately reflect experimental results.
MathematicalPhysicist
Science Advisor
Gold Member
Messages
4,662
Reaction score
372
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?
 
Physics news on Phys.org
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.
 
loop quantum gravity said:
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.
 
Back
Top