Hello,
This thread concerns a physical notion, but its origins go back to the
emergence of the theory of mathematical unification. In mathematics, two
theories are identical, if their corresponding topos (i.e. categories) are
equivalent.
I would like in this thread to discuss the unification of quantum mechanics,
and general relativity. What do we mean, mathematically speaking, by unifying
quantum mechanics and general relativity?
From a few readings that I had done several times in the past, the General
Theory of Relativity is a geometric theory. On the other hand, quantum
mechanics is an algebraic theory. So, finally, we seek to unify an algebraic
theory and a geometric theory. What would that mean?
Classically, I have seen how we unify an algebraic theory and a geometric
theory, via functors: differential geometry is identified in part with the
theory of commutative algebra,
because, the category of real differential varieties, are identified to a
subcategory of commutative algebras.
Is that, to unify the theory of general relativity, and the theory of quantum
mechanics consists in exhibiting a functor through which, we identify a
category of geometric objects within general relativity, and a category
of algebraic objects within quantum mechanics?
Thanks in advance for your help.