Noise in gravitational-wave detectors and other classical-force measurements is not influenced by test-mass quantization

It is shown that photon shot noise and radiation-pressure back-action noise are the sole forms of quantum noise in interferometric gravitational wave detectors that operate near or below the standard quantum limit, if one filters the interferometer output appropriately. No additional noise arises from the test masses' initial quantum state or from reduction of the test-mass state due to measurement of the interferometer output or from the uncertainty principle associated with the test-mass state. Two features of interferometers are central to these conclusions: (i) The interferometer output [the photon number flux (N) over cap (t) entering the final photodetector] commutes with itself at different times in the Heisenberg picture, [(N) over cap (t),(N) over cap (t')]=0 and thus can be regarded as classical. (ii) This number flux is linear to high accuracy in the test-mass initial position and momentum operators (x) over cap (o) and (p) over cap (o), and those operators influence the measured photon flux (N) over cap (t) in manners that can easily be removed by filtering. For example, in most interferometers (x) over cap (o) and (p) over cap (o) appear in (N) over cap (t) only at the test masses' similar to1 Hz pendular swinging frequency and their influence is removed when the output data are high-pass filtered to get rid of noise below similar to10 Hz. The test-mass operators (x) over cap (o) and (p) over cap (o) contained in the unfiltered output (N) over cap (t) make a nonzero contribution to the commutator [(N) over cap (t),(N) over cap (t')]. That contribution is precisely canceled by a nonzero commutation of the photon shot noise and radiation-pressure noise, which also are contained in (N) over cap (t). This cancellation of commutators is responsible for the fact that it is possible to derive an interferometer's standard quantum limit from test-mass considerations, and independently from photon-noise considerations, and get identically the same result. These conclusions are all true for a far wider class of measurements than just gravitational-wave interferometers. To elucidate them, this paper presents a series of idealized thought experiments that are free from the complexities of real measuring systems.