Arguably, Wightman's greatest legacy is being the advisor and mentor to a selection of Princeton Ph.D students who went on to distinguished careers, mostly in mathematical physics.
There are some nice testimonials on the Princeton Physics web site. Reading them it struck me that Wightman would have measured rather poorly on today's common metrics [grant money, numbers of publications, journal impact factors, numbers of Ph.D students, citations]; yet, he had an incredible scientific impact!
Wightman was extremely helpful and generous to me when I was a beginning graduate student at Princeton in the mid 1980s. In particular, I had a paper from my undergraduate thesis that I was trying to publish. He gave me great encouragement, some helpful feedback, and arranged for it to be published in the Journal of Mathematical Physics. To quote the beautiful testimonial of John Preskill, "Though I did not sufficiently appreciate it at the time, Arthur was incredibly generous with his time."
The following observations by Jurg Frohlich are particularly poignant:
As Arthur Jaffe said, the disappearance of Arthur Wightman marks the end of an era. I fear it may also mark the gradual disappearance of an attitude and style among scientists that I associate directly with people like Arthur Wightman and Res Jost:
Focus on the central problems of your field – even if they may not be doable immediately
– generously share your time, insights and ideas with others, especially with young colleagues,
generously support the careers of young scientists,
maintain unerring intellectual honesty and integrity – in short, try to be a gentleman scientist!
More than his scientific oeuvre, I view the latter qualities as Arthur Wightman’s central legacy for which he will be remembered, and which, in a time when they are endangered, we should cherish!