The Space Eater (1982) by David Langford?
DEATH WILL NOT RELEASE YOU Ken Jacklin is a soldier who has died many times. Rossa Corman has a communications talent that depends on pain. Together they must make an excruciating journey via makeshift matter transmitter and persuade a distrustful, war-torn planetary colony to halt research into Anomalous Physics -- the maverick science whose side effects include continent-busting nullbombs, exploding suns, and a mysterious final weapon called the Devourer. Gradually Jacklin and Corman realize that what Earth has told them about their own mission is a labyrinth of bluff and lies.... A high-tension hard SF thriller.
This review mentions the two (well, 1.933) centimeters gates and the main character dying over and over again:
If you go to my “About Paradox and Spike” page, you’ll see my interest in “hard science” sf; so why am I remembering and recommending a story about a “Force zombie killer?” Because Forceman Ken Jacklin, fighting to keep order in a war-torn London, has died 46 times and has gotten used to dying in combat, although he doesn’t have an interest in anything else any more … [...]
What remains of the government picks Jacklin to go to the one stellar colony to prevent them from making AP/Anomalous Physics experiments themselves. To get there, he and the tele(pathic)com officer will go through an AP gate – 1.933 centimeters wide …
Found with the Google query
scifi book transfer gate "centimeters wide" which returned the above review.