ABSTRACT
Let’s return to our observers M and P as they were at the beginning, located together, looking at empty space. In Chapter 1, we studied the consequences of their choosing different coordinate axes for the space around their common location, which led us to the consideration of the rotation group SO(3). In Chapter 2 we studied the universal double cover f : SU(2)→ SO(3), and found that the defining representation of SU(2) provides a model for electron spin. Later, in Chapters 7 and 8, we wondered about the consequences of M and P choosing different locations to plant their feet, and we were led to the translation action of (R3,+) and the Schrödinger representation of the Heisenberg group H3 on position space L2(R3). In all of these cases, we assumed that M and P were at rest with respect to each other. In this chapter, we ask about the consequences of relative, uniform motion in a straight line.
9.1 Galilean relativity Suppose that P has chosen an orthonormal basis for (V, 〈, 〉) as in Chap-
ter 1, thereby identifying the physical space around him with (R3, ·). In addition to having identical meter sticks and protractors (when compared to each other at rest), we now assume that M and P have identically constructed watches (a tacit assumption throughout this book). Just as in Chapter 1, M has chosen a different orthonormal basis, obtained from P’s by a rotation A ∈ SO(3). But in addition, we now assume that M is traveling with constant velocity v with respect to P, so that at time t, M is located at x(t) = tv (see Figure 9.1). If t′ denotes the time on M’s watch, we assume that t′ = 0 when t = 0 (i.e., when M and P are at the same location). Our question is simple: what is the relationship between P’s space-time coordinates (t,x) and M’s coordinates (t′,x′)? In the next section, we will briefly review Einstein’s radical answer to this question (in the form of special relativity) and investigate some immediate consequences for quantum mechanics. But in this section, we stick with the non-relativistic framework, in which the passage of time is assumed to be common to all observers, regardless of their states of motion, so t′ = t. Furthermore, space is assumed to be an immutable stage on which events transpire.
