Some years ago, while he was still bishop of Denver, Archbishop Chaput wrote a public note about holding hands during the Our Father. His comment was that there is no rubric for how one is to position one’s hands during the Our Father. He noted that as there was no rubric, no one had the right to say that it was wrong, or not permitted, or a violation of the rubrics.
He further said that no one had the right to force others to hold hands.
His note seemed to express some aggravation with the matter; and it was his intent to bring the matter to a close. He said that those who wished to hold hands could do so, and that no one else had the right to criticize them; and the reverse also applied; those who wished to not hold hands had the right to not do so, and no one had the right to criticize them.
Since moving to Philadelphia, he has addressed the issue at least once again, with the same conclusion, but with less commentary.
Hand holding during the Our Father started at least by 1966 if not by 1964. There have been at least three editions of the GIRM a well as specific instructions such as Summorum Pontificum (which spoke to specific acts noted to be contrary to the GIRM) and Rome has remained resoundingly silent on the matter. There is no doubt that Rome has known about the issue for a long time; and they have repeatedly chosen to ignore it.
At least one bishop, as chief liturgist in his diocese, has forbidden it. As it is solely within the purview of the bishop to regulate it, it is not the option of the parish priest to either encourage it or forbid it; it is outside their authority.
Any number of people have waded in on the matter in the last 40 to 50 years; as they are not liturgists, it is solely their opinion.
The bottom line, as Archbishop Chaput noted, is that no one should be forced to do it; and those who do not like it should keep their peace.
As there is no instruction as to one’s hands, the comment that “one is introducing” something into the liturgy by holding hands, by logic, holding one’s hands together would also be introducing something. And if one wants to argue that it is “traditional”, then after 48 years it would appear that holding hands is also a tradition. That may be a point for Canon lawyers, and they have disagreed on the matter.