New Paths Towards Quantum Gravity (Lecture Notes in Physics, 807)

1 Notes on “Quantum Gravity” and Noncommutative Geometry 29The rule[]a αβ (k), a μν † (k′ ) = g αα g ββ t αβμν δ(k − k ′ )follows.The scalar field is now constructed in a way close to the standard one:∫ϕ(x) = (2π) −3/2 dμ(k)(e −ikx a(k) − e ikx a † (k)), (32)where the (not Lorentz covariant) operators a # satisfy[a(k), a † (k)] =4δ(k − k ′ ).The traceless sector is represented as∫)H αβ (x) = (2π) −3/2 dμ(k)(e −ikx a αβ (k) + g αα g ββ t αβμν e ikx a † αβ (k) .Now one can verify (12) painstakingly.The last task in this section is to identify finally the physical degrees of freedom.For that, let us choose and fix k μ = (ω, 0, 0,ω). One can verify that the only statesnot present in Q (that is, belonging to the kernel of [Q, Q † ] + )are(b 11 − b 22 ) † |0〉 and b † 12 |0〉 =b† 21 |0〉.They correspond to linear polarization states. Their complex combinations (circularpolarization states) may be represented by matriceswhich transform like⎛ ⎞0 0 0 0ε ± := ⎜0 1 ±i 0⎟⎝0 ±i −1 0⎠ ,0 0 0 0ε ′ ± = e±2iφ ε ±under a rotation of angle φ about the direction of propagation. The reader can verifythis by using the generator of rotations⎛ ⎞0 0 0 0⎜0 0 1 0⎟⎝0 −1 00⎠ .0 0 0 0