"You have heard that it was said,'You shall not commit adultery.' But I say to you that everyone who looks at a woman with lust has already committed adultery with her in his heart. If your right eye causes you to sin, tear it out and throw it away; it is better for you to lose one of your members than for your whole body to be thrown into hell. And if your right hand causes you to sin, cut it off and throw it away; it is better for you to lose one of your members than for your whole body to go into hell." (Matthew 5:27-30) (NRSV)
(One aside: Rami, would it be possible for you provide dates for some of the rabbinic sources you cite from time to time. Insofar as I know, the earliest date from the second and third centuries.)
Jesus cites Exodus 20:14, prohibiting adultery. Most American Christians probably understand the commandment to forbid all sex outside of marriage. In its ancient setting, however, the commandment referred to sexual relations between a married man and the wife of another man. Many commentators argue that adultery, therefore, had to do with property laws, in that a wife was considered to be her husband's possession. Officially, the offense required the death penalty, though in practice the penalty was seldom if ever exacted, at least insofar as we can determine from written accounts.
Whereas the commandment deals with a specific action, Jesus moves to deal with the heart (as in the imagination, will, desire, etc.). In effect, Jesus calls for a redefined relationship between men and women, one in which lust per se is taken off the table. Normal desire is not the issue here. Lust is. Lust is the feeling that reduces others to objects, even as it atrophies our own capacity for empathy.
Two things need to be said about the illustrations of eye and hand. First, they are a kind of hyperbole, designed to drive home how seriously Jesus takes the matter. Second, they make a theological/existential point: in the end unbridaled lust reduces the one lusting to an utterly self-centered person, the kind of person who cannot experience the genuine, interactive presence of any other living being, even God. Such isolation is among the classic definitions of hell.