
1. Hamlet is madly in love with Ophelia and their separation is driving him mad,
2. Hamlet is simply putting on an act, or
3. Hamlet is actually insane.
Polonius believes that Hamlet is lusting for Ophelia and that their disrupted relationship is upsetting him. As Hamlet "long stayed" (II.i.103) holding on to Ophelia's wrist. This hold can be seen as intimate, and his prolonged hold can portray a longing for intimacy with Ophelia. His sigh that seemed to "shatter all his bulk" (II.i.107) implies that his strength and masculinity are shed away with his sigh, exposing his affections towards Ophelia and his "weakness" because of it. As he leaves her, he keeps his eyes fixed upon her, "to the last bended their light on [her]" (II.i.112), expressing longing for her and a reluctance to look away.
Because Ophelia agrees to her father's commands and refuses communication with Hamlet, Polonius believes that this "hath made him mad" (II.i.123). As a result, Polonius regrets his commands and believes that Hamlet is indeed faithful in his love for Ophelia. He fears that Hamlet's inability to express his love for Ophelia will cause more troubles than "hate to utter" (II.i.133).
After encountering his father's spirit and hearing the deeds of Claudius and Gertrude, Hamlet is left bewildered and frantic. He tells his peers that he will "put an antic disposition on" (I.v.192), warning them that they are to tell no one that he is simply acting. His putting on a disposition makes him "seem" rather than "is," which is portrayed when Ophelia describes his appearance during the night he visits her. During his visit, Ophelia happens to be sewing in her closet. She describes him with "his doublet all unbraced" (II.i.88), his socks dirty and fallen to his ankles. Her sewing and his appearance all provide the symbolism of "seeming" rather than "being," emphasizing the possibility that Hamlet is putting on an act.
Hamlet's sanity is in question after he holds a conversation with his father's spirit. Prior to this encounter, Marcellus and Horatio warn him that the ghost may "deprive [him] sovereignty of reason / And draw [him] into madness" (I.v.81-82), bringing him past the brink of insanity. Their fear of his delving into insanity is seen when they physically restrain him from following the spirit, holding him back and forcing him to threaten to "make a ghost of him that lets me" (I.v.95). Already at this point, Hamlet seems mad as he so enthusiastically wishes to follow after his father's ghost, threatening to kill any who holds him back.
During his visit to Ophelia's home, Hamlet is described to look as though "he had been loosèd out of hell" (II.i.93), which could refer back to his father's spirit and his peers' pleas not to go after it. With his "hand thus o'er his brow" (II.i.101), he gestures towards his mind, as he did after he meets his father's ghost. He makes mention of his "distracted globe" (I.v.104), with annotations that imply that he's perhaps gesturing towards his head.
Although Hamlet's current state of mind is unclear, his bizarre behavior could foreshadow that an unlikely event is about to occur.