Can I run a MacBook off a 45W MacBook Air PSU?

There most likely will not be enough current/amperage for the MacBook to power on without some added boost from the battery. Initially, when you press the power button, everything involved with the boot process runs full tilt until the boot is complete. Thus that is the time when the most power is needed. The MacBook will power on, run and charge fine with either the 65 watt or 85 watt PSU and no charge to the battery.

MacBook Pros require the 85 watt magsafe to power up without the battery. They will power on with the 65 watt PSU if the battery has 15-20% charge. Then run fine, along with the battery continuing to charge. It would only make sense that the same would hold true for the MacBook and a 45 Watt magsafe.