The schematic shows the transceiver is energized both when the key is both in the start and run positions. I know it communicates with the chip to start your car, but does the transceiver and the chip constantly have to be communicating in order to keep the engine running? Or is it only for starting the vehicle? If it's just for starting the vehicle then I would think the transceiver would only need to be energized when the key is on the start position only and wouldn't be needed with the key in the run position.
The reasoning is that my engine dies all of a sudden and what is very very odd is that the starter will not turn over or crank after the engine dies. then after a while the starter will work ok and get the engine running again. Is the transceiver faulty?

 

You don't mention what year your Explorer is, but in several model years, the PATS system only needs to see the transceiver to start the car. It does not require the transceiver to be present once the engine is running.

However, also on past models that I'm familiar with, a key without a transceiver, or with an unprogrammed transceiver, will not prevent the starter from engaging. It only shuts off the fuel and maybe the spark. Also, the Theft or Security light would rapidly flash if trying to start the engine with an unrecognized key.

It sounds to me like you have a loose electrical connection. Do the lights in the instrument cluster perform the bulb test in the RUN position when the engine won't start, specifically the Check Engine Light?

-Rod