i would like an answer from experts in string theory, quantum physics...
Well there are some,depending on the properties you wish to specify.I'm not an excellent mathematician, but I think the most important geometrical properties of spaces for physicists are the commutation relations that become important especially when one wants to quantize phenomena,be it particles,fields or strings.As you get deeper and deeper from first quantization in non relativisic quantum mechanics to second quantization of quantum fields and finally to string quantization you will use different algebras.For non-relativistic quantum mechanics you will invoke fairly standard algebra,for quantum field theory you will use dirac matrices to mathematically describe the commutation between a field and its conjugate momentum field.
Finally in string theory you will use some 'exotic ' algebras called virasoro algebras to understand the commutation relations between fields.These effectively play an important role in string quantization.You may wonder why I have insisted on the definition of commutation algebras to properly understand geometry.If for example we have a rectangle length 'a' and 'b'.The property of area in normal vector space is just ab=ba.But in other generalised cases a*b will not in general to b*a!Hence the need to understand the commutation relations which are basically statements of how much a*b is different from b*a or vice versa.
So i'd start from commutation relations if I were to look for a formula...in fact the commutator is the most general formula I know of.
Dear Jayanth,
I just want to add that since you have specified String Theory and Physics and you wish to specify the dynamics of the geometry you have to have the physical intrepretation of how space and time interacts in your problem to give rise to the particular geometry say of distribution of particles. Otherwise in Quantum Physics you cannot draw a geometry of forces. This is a result of the Law of General Relativity and of Heisenberg's Uncertainty Principle. At least not in String Theoretic Econophysics. You can take a look at some of my papers on my RG site particularly the ones which have graphs. Earl Prof. Dr. SKM QC EPS Fellow (In) MES MRES MAICTE
In Quantum Physics you cannot draw a geometry of forces, unless you specify the dynamics of the geometry and the physical interpretation of how space and time interacts in your problem to give rise to the particular geometry say of distribution of particles.
This Is a good question with many possible answers.
Let A be an n-dimensional geometrical figure represented by.a set
of points on an n-sphere S^n (hypersphere). And let
f: S^n --> R^n be a continuous mapping such that f(A) is a
feature vector that describes A, e.g.,
f(A) = (area(A), diameter(A), ...) in R^n
Then represents (describes) A.
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