I isn't backwards compatible because it doesn't have the "Graphics Synthesizer" its the GPU that was in the ps2. It lacks all hardware essential for backwards compatibility
No it is impossible, to be backwards compatible it needs the Graphics Synthesizer, and either the Emotion Engine(EE) (the processor used in th ps2) or software emulation.
The 60GB and 20GB had both the EE and the Graphics Synthesizer
The 80GB had the Graphics Synthesizer but used Software Emulation instead of the EE
Some people say that Sony will provide Software Emulation in a future update.
It would be possible to put Software emulation in the 40GB, but since it doesn't have the Graphics Synthesizer, it would be useless.
The 40GB will never be able to play ps2 games, unless Sony decides to put the Graphics Synthesizer in the 40GB. But since that is hardware and not software, it won't be as simple as a firmware update.
EDIT:
Like the guy said above me, if someone creates a modchip it may be possible, but it would have to be very similar to the Graphics Synthesizer, then they would have to create the Emulation for it, so that is not likely