Can a scalar field exert a torque on a particle?
It is unclear how you can get a torque on a classical point particle ... however, can you think of a scalar field in nature that shows characteristics of torque (a change in angular momentum). i.e. the gravitational scalar potential can result in rotating motion - but it conserves angular momentum. You can get gravitational coupling - i.e. earth-moon - where angular momentum of each individual object changes: but these are not point-particles.
Often, the scalar field may not by itself related to motion but require further conditions, like hydraulic pressure in a bucket is a scalar but on the walls, it is coupled with the area vector and becomes a vector and could force and bend which could be observed with flexible walls.
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