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There is No Free Will In a Michelson Interferometer!


LIGO's intention is to use "squeezed" light so that the phase shift of the recombined beam can be measured more precisely. But, because of Heisenberg's uncertainty principle, the conjugate variable--the phase--will experience fluctuations that will balloon, "making it all the more important that the mirror mass be made as heavy as possible."


Appeal is made to virtual particles which are putatively involved in "borrowing" energy so that there can be "virtual" transitions that violate the conservation of energy. Such particles are said to be "off shell", meaning that for short times they can get away with breaking the bank. However, virtual particles are "virtual", and are in no way visible. Hence, they are not detectable, and rely on concocted perturbation schemes like that of Feynman diagrams. What can't be measured, doesn't exist and virtual particles are a figment of the imagination, or better an artifact of perturbation theory.


Optics, unlike mechanics, has the special added feature that the spectrum of light generated from two point sources depends not only on their individual spectra, but, also, on the correlations between the fluctuations of their strengths. This means that 1+1=2,0,4 all at the same time. The "2" implies that the two fields are independent and their intensities are the sum of the two. The "0" implies that they interfere destructively, while the "4" indicates constructive interference. It all depends upon the cosine of the angle between the two fields. If the angle is 90 degrees, there is no interference and the beams are said to be in quadrature. If the angle is 0 or 180 degrees, there is constructive or destructive interference.


This has an impact upon Einstein's calculation of his spectroscopic coefficients for spontaneous, stimulated, and absorption. It has been claimed that if spontaneous and stimulated emission had not occurred with the same phase difference of 90 degrees with respect to the total field, Einstein would have had to considered phase shifts in his calculation of his spectroscopic coefficients. In fact, Pollnau, in his article Phase aspect of photon emission and absorption (optica 5 (2018) 465) claims that if "spontaneous emission occurred with an arbitrary phase difference with respect to an existing electromagnetic field, his derivation would have had to take interference into account." However, Einstein used a condition of dynamical equilibrium in order to derive Planck's law, and that is tantamount to energy conservation, i.e., a "balancing" of the books. And said law implies that both spontaneous and stimulated emission must be 90 out of phase with respect to the external field. Pollnau call this a "semi-classical" finding; however, it must also hold quantum mechanically if energy conservation is not to be violated.


Energy conservation tells you how the waves have to behave. Consider two beams at an interface separating two media of different indices of refraction. Could both exit beams interfere destructively or constructively so that the conservation of energy is violated? One of the beams is in the denser medium so that reflection involves a phase shift of 180 degrees. The other beam is in the less dense medium so that reflection involves no phase shift at all. Now, if the transmitted and reflected beams on the denser side is destructive, the exit beam on the other side is constructive. Hence, the intensity of the exit beam on the destructive side is zero, but the intensity on the constructive side is twice the intensity of each incident beam.


The result that the energy is always equal to the intensities of the two incidents beams holds no matter what their relative phase difference is, no matter what their amplitudes and initial intensities are, and no matter if one exit beam will have zero intensity or not. It is like Einstein's spooky action at a distance: the beams "know" how to combine so that energy conservation is upheld.


Now consider a beam splitter, like that in a Michelson interferometer. Two beams at 90 degrees with respect to one another shine light on a silver coated mirror oriented at a 45 degree angle. Half the beams are reflected and half are transmitted. Here, there is a single index of refraction, that of the mirror. There will no longer be a 180 degree phase difference between the reflected beams. Each reflected beam will be in phase with its corresponding incident beam, but the transmitted beam with be 90 degrees out of phase from the incident beam. The cosine of 90 degrees is zero and the total intensity will be sum of the individual, incident, intensities. Energy is conserved!


In general, there will be a redistribution of energy. If it occurs in space, the wavelength is involved, like in the Michelson interferometer. If it occurs in time, the frequency is involved. Two light beams of different frequencies will interfere with one another. The effect is the same as two sound waves combining to form "beats". Moreover, the initial beams do not have to be at different frequencies, as Wolf showed. If two light beams of the same frequencies, whose sources are at different positions in space, are scattered off of random media, the effect is similar to the Doppler effect. However, no relative motion is implicated. The relative frequency shift is independent of the frequency of the beams, and so it imitates the Doppler shift. The Wolf effect has also been implicated as a non-cosmological redshift. No energy loss is involved, just a redistribution in time.




A very clear article on energy conservation can be found in C.-H. Hu, "The wave nature of light--interference."

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