N.T. Wright's 2003 study, The Resurrection of the Son of God, argues that the combination of an empty tomb and appearance experiences is needed to explain the rise of a very specific and, in Wright's reading, historically unprecedented belief: no strand of contemporary Jewish thought expected a single individual to be bodily raised in the middle of history, ahead of the general resurrection expected at the end of time, so the early Christians' claim needs some real event behind it to explain why they said what they said rather than something more conventional, such as Jesus's spirit living on or his cause continuing.
Gerd Ludemann, arguing from the same historical-critical method to a different conclusion, holds that the appearances are best explained as genuine but visionary experiences, arising out of grief in Peter's case and guilt in Paul's, which the earliest communities then interpreted, collectively and over time, as encounters with a bodily risen Jesus. Both Wright and Ludemann set aside the older swoon theory, that Jesus merely survived the crucifixion, and the stolen body theory, already answered polemically inside Matthew's own gospel, as too weak medically and logistically to explain the sources; the live disagreement among historians is between a real resurrection and a visionary origin for the appearance reports, not between those two older alternatives.