The description of the Antimagic field states (emphasis mine):
A 10-foot-radius invisible sphere of antimagic surrounds you. This area is divorced from the magical energy that suffuses the multiverse. Within the sphere, spells can't be cast, summoned creatures disappear, and even magic items become mundane.
The emphasized part suggests that magic of any kind cannot exist inside this area: any connection between the weapon and the caster is interrupted. The Spells entry of the Antimagic Fields states (emphases mine):
Any active spell or other magical effect on a creature or an object in the sphere is suppressed while the creature or object is in it.
The question is: is Spiritual Weapon active on the caster? The spell's description says (emphases mine):
You create a floating, spectral weapon within range that lasts for the duration or until you cast this spell again. When you cast the spell, you can make a melee spell attack against a creature within 5 feet of the weapon. On a hit, the target takes force damage equal to 1d8 + your spellcasting ability modifier.
As a bonus action on your turn, you can move the weapon up to 20 feet and repeat the attack against a creature within 5 feet of it.
There is a clear connection between the caster and the weapon; for the duration the caster can command the weapon via the magical link made by the spell: this means that the spell is active on the caster.
Therefore, the caster cannot use their bonus action to make the weapon move or attack while they are inside the AoE of the Antimagic Field: as soon as they exit from the dead magic area, however, they can resume command of the Spiritual Weapon (if its duration has not yet expired).