Can Calculus Effectively Handle Discontinuous Fundamental Quantities?

[meaw]

i wondered at the idea that calculus works for continuous functions and in reality fundamental quantities are discontinuous. For example energy or an electron can't asume all values. Therefore isn't there a conflict when we work on a equation like dE/dt or something similar which involve calculus of discountinous things ?

 

[HallsofIvy]

Mathematics is not physics. No mathematical model can be expected to work perfectly. That is taken into account when developing a mathematical model for a physical situation.

 

[meaw]

"No mathematical model can be expected to work perfectly" - you mean it really ? Then why are we so bothered about mathematical inconsistencies when GR and QM are put together ? Unless you are hiniting at thephylosophical question of whether maths is invented or discovered, the general believe is that Maths is reality. Maybe I misread you. Can you clarify ?

 

[genneth]

It is certainly possible to do physics without ever introducing continuous quantities, but it's just that much more cumbersome. In a way, we use things like real numbers or calculus as shorthands for large structures which are well-regulated. The fact that this is often ignored is an unfortunate state of affairs, but textbooks have to draw a line somewhere. In practice, physicists will often do things like use integration when they are talking about discrete sums, as long as the results are reasonable.

 

[meaw]

right you are. I am wondering whether any attempts have been made to redefine calculus for discountinouous functions where the discontinuity is of the order of plank's constant which can be ignored for general physics but not so for quantum mech.

 

[genneth]

Inside differential geometry (which is basically just calculus on curved manifolds) there is a large combinatorial, simplicial structure. Google for "discrete differential forms". These have found good use in numerical simulations, where things have to be discretised for computation. Similarly, things like lattice QCD or Regge calculus in GR have similar backgrounds.

 

[Gib Z]

There are discrete analogs for the derivative and the integral: the finite difference operator and discrete summation. It's just that a lot of the time, its much easier to do the continuous version, and the results are reasonable.