General Leslie Groves was the military commander in charge of the Manhattan Project.  According to experts, including Stanford Professor Barton Bernstein, no physical order can be produced documenting Truman's direct authorization for dropping the Hiroshima bomb. 
Here is a link to Prof. Bernstein's visit to Palo Alto HIgh School when he discussed this point.

General Leslie Groves

Click here for Article on Dr. Bernstein's Visit




National Security Files on Decision to Drop The Atomic Bomb