Consider two Cartesian coordinate systems rotated with respect to each other in the x-y plane as shown in Fig. 1.4.

Fig. 1.4.

(a) Show that the rotation matrix that takes the barred basis vectors to the unbarred basis vectors is

and show that the inverse of this rotation matrix is, indeed, its transpose, as it must be if this is to represent a rotation.

(b) Verify that the two coordinate systems are related by Eq. (1.13c).

(c) Let A_j be the components of the electromagnetic vector potential that lies in the x-y plane, so that A_z = 0. The two nonzero components A_x and A_y can be regarded as describing the two polarizations of an electromagnetic wave propagating in the z direction. Show that A_x̅ + iA_y̅ = (A_x + iA_y)e−i∅. One can show (cf. Sec. 27.3.3) that the factor e^−i∅ implies that the quantum particle associated with the wave—the photon—has spin one [i.e., spin angular momentum ℏ = (Planck’s constant)/2π].

(d) Let h_jk be the components of a symmetric tensor that is trace-free (its contraction h_jj vanishes) and is confined to the x-y plane (so h_zk = h_kz = 0 for all k). Then the only nonzero components of this tensor are h_xx = −h_yy and h_xy = h_yx. As we shall see in Sec. 27.3.1, this tensor can be regarded as describing the two polarizations of a gravitational wave propagating in the z direction. Show that h_x̅x̅ + ih_x̅y̅ = (h_xx + ih_xy)e^−2i∅. The factor e^−2i∅ implies that the quantum particle associated with the gravitational wave (the graviton) has spin two (spin angular momentum 2ℏ); cf. Eq. (27.31).

y X