What evidence can you find to explain Leontes' jealousy in Act 1 of The Winter's Tale?
Some more 'tangible' proofs that Shakespeare might have given could be seen in the way the focus is on Leontes mostly in the first three acts. As we only see what Leontes is seeing- for instance, Hermione taking the hand of Polixenes or their lips so close as if on the point of kissing- we could say that Leontes's view might be biased for fear of being cuckolded. Further, though inadvertently, Hermione might be really making her king feel jealous when she "mingle[s] friendship too far."
Or, a far-fetched idea would be that Leontes is a masochist seeking for his own tragedy as his life seems too idyllic, with a best friend since his childhood, a perfect wife and son and a daughter making her way.
This is a very interesting question, especially because the eruption of Leontes' jealousy happens so fast. However, Shakespeare’s gives us some evidence that justifies Leontes’ abrupt and uncontrolled feelings.
In act 1.2, we learn that Polixenes, having been stayed long enough at Sicilia, wishes to return to Bohemia. Leontes, in vain, tries to persuade him to stay one week longer, and then, he asks Hermione to convince his friend, instead:
Hermione “I had thought, sir, to have held my peace until
You had drawn oaths from him not to stay. You, sir,
Charge him too coldly. Tell him you are sure
All in Bohemia’s well…” (1.2 28-32).
Nonetheless, the lines exchanged between Hermione, and Polixenes become more and more ambiguous making us to ponder about an eventual flirt between Leontes’ wife and the King of Bohemia:
Hermione “…Force me to keep you as a prisoner,
Not like a guest: so you shall pay your fees
When you depart, and save your thanks. How say you?
My prisoner? or my guest?...” (1.2 51-54).
Thus, when finally Polixenes agrees to extend his stay in Sicilia, this decision is enough to produce in Leontes an intense suspicion about his wife’s fidelity.
According to some scholars, in spite of being apparently irrational, Leontes' jealousy is substantiated by the kind of Patriarchal society existent in Jacobean England. Then, men were undoubtedly heads of families, but, at the same time they were dependent on women’s reproductive capacities to produce legitimate heirs. We see that in the play Hermione is nine months pregnant. and that Polixene has been stayed in Sicilia for “nine changes of the wat’ry star”. This fact is enough to raise deep suspicions in Leontes.
Lastly, we have to consider the nature of Leontes and Polixenes’ friendship. As children they had been close friends, but the loss of innocence had caused a change in this friendship- now, both men are married and have children. Then, in adulthood, competitiveness might have tarnished their friendship.