Two times or none?

Abstract: Attempts to treat time on an equivalent footing with space in quantum mechanics have been apparently dominated by timeless' approaches, such as the one of Page and Wootters, which allow meaningful discussion of atime operator'. However, there is an alternative, and significantly less studied approach, due to Bauer, which makes use of the pseudospin' extension of the state space, effectively adding a backwards-time degree of freedom. This two-time approach allows definition of atime operator' and moreover bears interesting relations with other time-symmetric formulations of quantum mechanics. We review and compare these approaches to quantum time, emphasizing that there is a subtle choice between the timeless framework and the two-time approach. Finally, we sketch a framework in which the timeless philosophy can be combined with two-time quantum mechanics.