Why cant we use 64 bit OS on a 32 bit CPU?

"I'm assuming we're talking about standard PCs here.

Most 64-bit processors - especially the ones from Intel and AMD - have the full 32-bit instruction set as well as the 64-bit extensions, so they can run 32-bit software.

32-bit processers don't have a 64-bit instruction set though. Many were made even before the 64-bit instruction set was defined, so it's not possible they could implement it correctly, even emulated. They just don't understand that language."

"Why can't you run Windows on your old Atari? Incompatible hardware and processors.

x86-64 should be considered completely different from x86 because it is vastly different from the kernel and machine code point of view. The only thing they really share in common is the bootstrapping code(from the 16bit startup, to get to 64bit you must [usually] go to 32bit first) and the 32-bit compatability mode and the common CPU extensions like SSE. Other than that, the processors are really completely different as to how applications are made in machine code."

"A 64 bit OS relies on processor features of a 64 bit processor.

You can't tell a 32 bit process to perform an operation on the 64 bit register if it doesn't exist.

Whereas 64 bit processors have everything the 32 bit one has, so it can be done."