Why is it appropriate that the story says of Montag "He felt his body divide itself..." in Fahrenheit 451?
Towards the beginning of the novel, Montag has an enlightening, significant conversation with Clarisse, who takes a dandelion and rubs it against his chin. Clarisse then examines Montag's chin and declares that he is not in love with anyone, which upsets him. Clarisse then begins to comment on the fact that Montag doesn't behave or act like the other firemen and to say that she believes there is something unique about him. After Clarisse mentions that being a fireman doesn't seem right for Montag, Bradbury writes,
"He felt his body divide itself into a hotness and a coldness, a softness and a hardness, a trembling and a not trembling, the two halves grinding one upon the other" (11).
This description of Montag's body and mind dividing itself into two different parts reveals his inner struggle between who he truly is and his current role in society. Clarisse awakens Montag's conscience, and he begins to question his life. Montag's sympathetic, compassionate, genuine personality is stirred after having an intimate conversation with Clarisse, and he immediately begins to question his life and its trajectory.
I think that you could argue that this is the point in the story where Montag is, figuratively, becoming two different people. If so, then this is a very appropriate thing for the story to say.
For most of his life, Guy has been one person -- he is a fireman. As such, he stands for the values of his society. But now, his experiences with Clarisse and the near-death of his wife Millie have started to make a new him. Because of these things, he has started to question the values of the society.
When he does, he starts to become two people. The two parts of him (just as this passage says) become different from one another and they start to be "two halves grinding" against one another. He is starting to have different parts of him that are coming in conflict with one another.