Hi,

i have some questions about the mathematical, physical and philosphical consequences of quantized spacetime ..

But first a disclaimer: im neither a physicist not mathematician, but a programmer.

However, until i started to study computer science, math and physics were my favourite classes in school. That means i know about differential and integral calculus, limits and how they are applied in classical physics. Because of my interest in astronomy, i have always been interested in contemporary physics (special and general relativity (SR/GR), quantum physics, cosmology, string/brane theory, loop quantum gravity (LQG) and lately even non mainstream stuff like Heim theory (HT)). I think i have a shadow of a clue what they are about, but i simply lack the mathematical theory and practice to calculate tensors and riemann geometry and anything beyond by myself.

(read the next sentence with an Ali G voice)
So in your answers type slowly and in uppercase, coz i is blond ..
Just kidding, please no uppercase ..

As a progammer, i know the difference between integer numbers and floating point numbers and real numbers. On a computer, integer math is exact, as long as you stay inside the bounds, but i know how to get around those bounds by programming integer math with arbitrary but still limited precision. Physics use real numbers with unlimited precision. On a computer, real numbers are approxmated by floating point numbers, but hey are a limited subset only (my fav joke about that: between any two consequtive floating point numbers is an inifinite number of missing values) ..

OK, after that lets get back to physics. In classical newtonian physics, space and time are the "stage" where physics happens. Einsteins SR (special relativity) unifies space and time to spacetime (Lorentz, Minkowski), and with GR (general relativity), spacetime is not just the stage anymore but becomes part of the play (mass warps spacetime ..). No problems so far, except that i cant sit down and calculate stuff - which stops the real deep understanding somehow ..

Now my questions (thanks to those still here and awake ..) :

Both LQG (loop quantum gravity) and HT (Heim theory) assume or derive a quantized spacetime. As far as i understand, that means that space and time coordinates are discrete, non continous values, and these values are integral multiples of a smallest length / timespan. I think for my questions it doesnt really matter whether that smallest length or timespan is the Planck length / time or something bigger / smaller (e.g. Heims metron or whatever).

A: Am i right, that if we assume that spacetime is actually quantizised, any physical objects coordinates always have to be integral multiples of that smallest spacetime unit ?
Does that mean a physical object is either here (x = 2783) or there (x = 2784) but never in between (x = 2783,4286409453) - same for y,z,t ...
Or am i oversimplifying ?

B: What about physical mathematics ? Does limit theory as the mathematical foundation of calculus break down ? You cant go from x=1 to x=0 in smaller and smaller steps if there are no smaller steps ..
Isnt it necessary in physics to "throw away" real number math and replace it by integral math, cause real number math gives only an approximation ?

C: Are there still singularities in quantizised spacetime ? Doesnt the limitation to integral multiples of the smallest unit exclude lots of interesting and weird solutions in continuos real math, e.g. black holes, big bang, ...

D: Heisenbergs uncertainty principle (UP) claims that we cant measure position and momentum with arbitrary precision. Not just because of bad equipment and/or dumbness, but in principle. While that sounds weird in a continous world, it sounds pretty trivial in a non continous world. Is the UP an extremly trivial consequence of quantizised spacetime ?

E: The UP is a cornerstone of quantum mechanics/physics (QM), which has a bunch of "weird" philosophical consequences (Bohr: If QM doesnt look weird, you havent understood it yet). As far as i know, physicists still argue which interpretation (Kopenhagen, Everett, ..) is more convincing ..
E.g. if some physical object can only be either here (x=0) or there (x=1) and only now (t=0) or then (t=1), but never in between (x=0.4, t=0.6), i can come up with an probability interpretation saying that the probability of the object being here now (x=0,t=0) goes from 1 to 0 while its probability of being there then (x=1,t=1) goes from 0 to 1. But doesnt Occams razor cut that interpretations throat if spacetime is quantizised ?

F: Has anyone (except Heim) investigated the consequences of quantizised spacetime for all the models, ideas and theories of contemporary physics ?
To me it looks as if the whole branch of physics still happily uses the calculus tools of Newton and Leibniz, without even thinking whether they are still the right hammer for the nails ?
Of course, its fine for pretty much every application, but maybe not for every theoretical question ..

Ok, lots of dumb questions, but my physics/math teacher always said there are no dumb questions, just dumb answers ..

millka

I confess to wondering much the same as you ie finite mathematics of some description might be a better way.

Mainly because the GR equations (based on continuous notions) produce odd results "at both ends" so to speak (although they seem fine everywhere else). Time has been "baked in" and provides the link to analysis.

Just cannot help thinking that something is wrong somewhere when we cannot establish the link between quantum and the rest.

A No, that's pretty much right

B In a literal sense, the continuum nature of calculus breaks down, but this doesn't cause too much of a problem for large scale objects. For instance, fluid mechanics is based upon modelling fluids as a continuum, yet we've been aware for 100 years that fluids are actually a HUGE collection of particles, not a continuum, yet fluid mechanics is very much valid for scales larger than a few microns, because the discrete nature of the medium is not important, it behaves so much like a continous medium it's okay to model it as such.

When you get into how tiny amounts of liquids behave, a few micrograms, then you're going to need to consider atomic forces and possibly even quantum effects.

C It's hoped that by quantising gravity singularities like those found in the centre of a black hole will no longer be part of the theory. Sure, there'll still be an enormously dense blob of matter, but not a zero sized one.

The big bang and normal black holes are theories about objects which are so huge compared to the scale of the quantised distance of gravity (ie planck length) that relativity is a perfectly valid theory to use, because the discrete nature of space-time limits to a continuum when you're talking about distances of kilometres and certainly so when you're talking about light years.

For primordial black holes (ie ones perhaps created by the BB and are only a few billion tons in mass) a quantum theory of gravity is needed because they are proton sized or smaller, but stellar black holes are modelled by relativity to a very acceptable level of accuracy (parts in trillions or more).

D HUP is a consequence of quantum theory, regardless of gravity. It's derivation can be done in quantum mechanics (ie non-relativistic) without even considering the nature of space-time.

E While Occams razor is a powerful tool, it should be used with precision, not wealded as a blunt instrument. There are cases where the more complex theory is the better one, though usually you'd require experimental evidence to accept that. After all, by Occam's razor 'God did it' is the simplest explaination for everything, but then it answers no questions meaningfully.

F You can still use calculus in quantised theory. Quantising doesn't mean all your differential equations suddenly turn into difference equations. I don't know about Heim's theory or things like quantum loop gravity, but in more standard quantum field theory and string theory quantising means working out (anti)commutator relations for wave modes. The waves are still waves, but they interact with one another is a slightly different fashion. It's hard to explain without showing some maths. Chapter 2 of Peskin and Schroder's An introduction to Quantum Field Theory has a good explaination for the Klein-Gordon field and Chapter 5 does the Dirac field.