General relativity vs quantum field theory

[Urkel]

Hi,
I am curious about the following and I aim these questions to the people who do general relativity and uantum field theory over there.
What is the difference between field theory of general relativity and field theory of quantum field theory? Is the former only for study of gravitation while the latter covers broader scope, including quantum theory of gravitation?
Are mathematics subjects like differential geometry, manifold, topology and the likes are that needed and useful for quantum field theory (while to my knowledge these mathematics are mandatory to study GR)?
In particular, do condensed matter theorists indeed need these mathematics to study high level and rather abstract "types" of condensed matter physics like fractional statistics, anyon, fractional quantum Hall effects,etc?
I am quite surprised to hear that many people say that quantum field theory unifies relativity and quantum mechanics, but which relativity? If it is the special one, then I guess quantum field theorists need no bother about general relativity, which I can't believe!
Conversely, since general relativity, if I'm not mistaken, is still classical physics stuff, then experts on these need not deal with quantum field theory (since the former deals more with theory of space of time in gravitation than field theory in general as the latter does)?
Please enlighten me. Thanks.

regards

 

[cristo]

Quantum field theory unifies special relativity with quantum mechanics, however does not deal with gravity. This isn't to say that some people feel there is no need to bother with gravity-- it is just that we have not found a way to unify gravity with quantum theory. This, the search for a unified theory of gravity, is an active area of research.

 

[asluder]

Here is a comparison between the fields in GR and QFT:
GR
1.) The field is the metric (a second rank tensor field)
2.) The field determines the geometry of spacetime
3.) There is a relationship between the metric and the stress energy tensor given by einstein's equation.
4.) The field is NOT an operator.
5.) The field never obeys the laws of QM but obeys the laws of SR at freely falling points.
6.) This is a theory of space, time, and gravity. It does not tell us anything about the fundamental nature of leptons, quarks and the strong, electromagnetic, and weak interactions.
QFT
1.) There are scalar, spinor, and vector fields corresponding to spin 0, 1/2 and 1 fields.
2.) The field is defined over fixed Minskowski geometry. Particles arise as exctied states of the fields.
3.) The relationship between fields is given by the interaction terms in the Lagrangian of the fields.
4.) The field is an operator on a Fock space. The field creates and destoys particles when it acts on elements of a fock space.
5.) The field obeys the laws of QM and SR everywhere.
6.) This is a theory of the fundamental nature of leptons, quarks and the strong, electromagnetic, and weak interactions. It does not tell us anything about space, time, and gravity.


It would be nice to have a single theory of space, time, all 4 forces, leptons and quarks. But no such theory exists yet.