With 120 Vac connected to the outside pins as intended, the center pin will have approx 60 Vac with respect to whichever outside pin is the neutral. BUT, this voltage will be at an extremely high source impedance because the internal contact surface area is so much smaller than the outside pins with their "electrodes". If you clip and filter this, you will have a DC level that can drive a MOSFET that can drive a DC relay coil if you have a DC voltage source referenced to the AC neutral. Or, two MOSFETS back-to-back and no filter to drive an AC relay coil.
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