I Infinite number of spatial dimensions (maybe also time dimensions)

[mad mathematician]

How would one build mathematically an infinite number of spatial dimensions theory?

I can concieve mathematically an n-th vector or $\mathbb{R}^{\infty}$, I had done so in my Topology course back then.
But obviously it's not empirically possible to test.

But is a theory of everything ought to be "finite" and empirical?
I mean obviously if there are only 4 interactions (currently known); but then again there could be more interactions around the corner.
So to encompass it all seems to me quite impossible by definition of "a theory of everything".
This post is a bit philosophical.

 

[anuttarasammyak]

We are familiar with countable infinite dimension Hilbert space in QM. You may be able to get some hint for your problem from it.

 

[mad mathematician]

We are familiar with countable infinite dimension Hilbert space in QM. You may be able to get some hint for your problem from it.

but spacetime in QM isn't inifinite dimensional in spatial coordinates.

 

[anuttarasammyak]

I do not expect that learnig Hilbert space give you a direct answer to your question, but you may become familiar with mathematics of infinite dimension space with it.

but spacetime in QM isn't inifinite dimensional in spatial coordinates.

Though I have no objection, In a view-point, 1 dimension space {x} corresponds to uncountable infinite dimension space in QM the orthogonal relation of which is
$<x|x'>=\delta(x-x')$

 

[mad mathematician]

Hi @anuttarasammyak I know what is a Hilbert space.
x in you case is spatial coordinate, so if I want inifinite number of coordinates then I would get a product of infinite number of dirac deltas.

 

[Hornbein]

Infinite d non-Hilbert space is now being discussed at https://hi.gher.space/forum/viewtopic.php?f=27&t=2698.

 

[fresh_42]

There are significant mathematical differences between finite- and infinite-dimensional vector spaces. One leads to linear algebra, the other to topological algebra. However, any vector space used in physics is only the playground, the background for the scene. It is only determined by the model we need to describe physics. As such, it is a property of our model, not a physical reality. The question is, therefore, how far our models can be taken to represent real phenomena. This is indeed a philosophical question.

 

[Fra]

But is a theory of everything ought to be "finite" and empirical?

I would want it to be computable for any given confidence level, but a given "computer". So if you imagine an embedding of inifite dimensions, it would have to be constructed as some limit of computable models to be of any use.

So I agree that to entertain that the "solution" lies in some infinite dimensional structures, is going to be at least useless, that if anything gets us mathematical objects but which from perspective of physics is philosophical useless constructs. Unfortunately this is all over the place in current theories.

The similar case in hilbert space, any actual computation of a quantum mechanical problem truncates this, by an energy cutoff. So we get an effective theory that is computable, as it is indeed impossible to make finite computation otherwise.

So the practical matter of computability, is important, and not just philosophical.

/Fredrik