Doing PhD in math/physics after obtaining a bachelor degree in an unrelated field I want to do a PhD in mathematics (Specializing in algebraic topology or algebraic geometry ) or mathematical physics (String theory). I have a bachelor degree in a totally different field (medicine). 
Here's my situation :

I have independently studied mathematics and physics. I have spent a lot of time with hatcher's textbook in algebraic topology and lang's complex analysis among with other topics in math and physics. These are proof-heavy textbooks and I usually do the exercises. 

Now , after having studied analysis ,differential topology ,geometry, abstract algebra , quantum mechanics and QFT .

Is this a good preparation for a PhD in math/physics ? 

Please note that I have studied from standard textbooks and solved a lot of exercises. 

How do I convince PhD committee to accept my application ? 

If I apply now , Apart from formal qualifications,my application will be missing one crucial component which is the letters of recommendations. I'm not sure where do I get them ? I don't know any physics or math professors who can evaluate my knowledge.
You should contact potential advisors who work in the areas in which you are interested.  Introduce yourself, explain any work you've done, and ask if they have specific feedback for moving on in your career.  You should contact several people.  If you live near a university, arranging a time to meet with some current faculty members is also a good way to go about this.