Dear Sir,
We had gone through your essay. You had based your deductions on some generally accepted principles. But if we analyze the foundations of these policies, we find many inconsistencies. Hence without undermining your attempts and arguments, we are bringing out these facts for a healthy debate can take place to establish the truth.
You have referred to the expression: δE.δt ≥ ħ/2, involving time and energy. Time is not an observable property of a system in the normal sense. It is a parameter used to mark the interval between an epoch marking the beginning of measurement process and another marking its termination. Some scientists say that there is no such limitation. They can measure the energy and look at their watch. Then they know both energy and time. However, other scientists differ from this view. According to them, the equation places a limit on the accuracy with which one can specify the amount of energy transferred together with the knowledge of the time at which the transfer took place.
We have discussed the Uncertainty relation in our essay and shown that it is not a fundamental law of Nature, but arises as a consequence of natural laws relating to observation that reveal a kind of granularity at certain levels of existence that is related to causality. We have also shown that the mathematical format of the Uncertainty relation is wrong. Heisenberg postulated this relationship which he thought as corresponding to, as he claimed, the "well-known" relation tE - Et = iħ. However, the status of time variable in his illustrations is not clear. He also formulated the inequality δw . δJ ≥ ħ, where w is the angle and J is the action based on the "well-known" relation wJ - Jw = iħ. However, these "well-known" relations are actually false if energy E and action J are to be positive operators (Jordan 1927). In that case, self-adjoint operators t and w do not exist and inequalities analogous to Δψp Δψq ≥ ħ/2 cannot be derived. Also, these inequalities do not hold for angle and angular momentum (Uffink 1990). These obstacles have led to a quite extensive literature on time-energy and angle-action uncertainty relations (Muga et al. 2002, Hilgevoord 2005).
Heisenberg summarized his findings in a general conclusion: all concepts used in classical mechanics are also well-defined in the realm of atomic processes. But, as a pure fact of experience ("rein erfahrungsgemäß"), experiments that serve to provide such a definition for one quantity are subject to particular indeterminacies, obeying the relations δp . δq ≥ ħ,
δt . δE ≥ ħ, and
δw . δJ ≥ ħ
which prohibit them from providing a simultaneous definition of two canonically conjugate quantities. It may be noted that in this formulation the emphasis has slightly shifted. He now speaks of a limit on the definition of concepts, i.e. not merely on what we can know, but what we can meaningfully say about a particle.
In his Como lecture, published in 1928, Bohr gave his own version of a derivation of the uncertainty relations between position and momentum and between time and energy. He started from the relations: E = hν and p = h/λ, which connects the notions of energy E and momentum p from the particle picture with those of frequency ν and wavelength λ from the wave picture. He noticed that a wave packet of limited extension in space and time can only be built up by the superposition of a number of elementary waves with a large range of wave numbers and frequencies. Denoting the spatial and temporal extensions of the wave packet by Δx and Δt, and the extensions in the wave number σ := 1/λ and frequency by Δσ and Δν, it follows from Fourier analysis that in the most favorable case Δx Δσ ≈ Δt Δν ≈ 1, and, using E = hν and p = h/λ, one obtains the relations:
Δt ΔE ≈ Δx Δp ≈ h.
It may be noted that Δx, Δσ, etc., are not standard deviations but unspecified measures of the size of a wave packet. These equations determine, according to Bohr: "the highest possible accuracy in the definition of the energy and momentum of the individuals associated with the wave field" (Bohr 1928, p. 571). He noted, "This circumstance may be regarded as a simple symbolic expression of the complementary nature of the space-time description and the claims of causality" (ibid).
Bohr does not refer to discontinuous changes in the relevant quantities during the measurement process. Rather, he emphasizes the possibility of defining these quantities. This view is markedly different from Heisenberg's. A draft version of the Como lecture is even more explicit on the difference between Bohr and Heisenberg: "These reciprocal uncertainty relations were given in a recent paper of Heisenberg as the expression of the statistical element which, due to the feature of discontinuity implied in the quantum postulate, characterizes any interpretation of observations by means of classical concepts. It must be remembered, however, that the uncertainty in question is not simply a consequence of a discontinuous change of energy and momentum say during an interaction between radiation and material particles employed in measuring the space-time coordinates of the individuals. According to the above considerations the question is rather that of the impossibility of defining rigorously such a change when the space-time coordination of the individuals is also considered" (Bohr, 1985 p. 93).
Indeed, Bohr not only rejected Heisenberg's argument that these relations are due to discontinuous disturbances implied by the act of measurement, but also emphasized his view that the measurement process creates a definite result: "The unaccustomed features of the situation with which we are confronted in quantum theory necessitate the greatest caution as regard all questions of terminology. Speaking, as it is often done of disturbing a phenomenon by observation, or even of creating physical attributes to objects by measuring processes is liable to be confusing, since all such sentences imply a departure from conventions of basic language which even though it can be practical for the sake of brevity, can never be unambiguous" (Bohr, 1939, p. 24).
Nor did Bohr approve of an epistemological formulation or one in terms of experimental inaccuracies: "...a sentence like 'we cannot know both the momentum and the position of an atomic object' raises at once questions as to the physical reality of two such attributes of the object, which can be answered only by referring to the mutual exclusive conditions for an unambiguous use of space-time concepts, on the one hand, and dynamical conservation laws on the other hand" (Bohr, 1948, p. 315; also Bohr 1949, p. 211). It would in particular not be out of place in this connection to warn against a misunderstanding likely to arise when one tries to express the content of Heisenberg's well-known indeterminacy relation by such a statement as "the position and momentum of a particle cannot simultaneously be measured with arbitrary accuracy". According to such a formulation it would appear as though we had to do with some arbitrary renunciation of the measurement of either the one or the other of two well-defined attributes of the object, which would not preclude the possibility of a future theory taking both attributes into account on the lines of the classical physics. (Bohr 1937, p. 292)
Instead, Bohr always stressed that the uncertainty relations are first and foremost an expression of complementarity. This may seem odd since complementarity is a dichotomic relation between two types of description whereas the uncertainty relations allow for intermediate situations between two extremes. They "express" the dichotomy in the sense that if we take the energy and momentum to be perfectly well-defined, symbolically ΔE = Δp = 0, the position and the time variables are completely undefined, Δx = Δt = ∞, and vice versa. But they also allow intermediate situations in which the mentioned uncertainties are all non-zero and finite. It must here be remembered that even in the indeterminacy relation (Δq Δp ≈ h) we are dealing with an implication of the formalism which defies unambiguous expression in words suited to describe classical pictures. Thus a sentence like "we cannot know both the momentum and the position of an atomic object" raises at once questions as to the physical reality of two such attributes of the object, which can be answered only by referring to the conditions for an unambiguous use of space-time concepts, on the one hand, and dynamical conservation laws on the other hand (Bohr, 1949, p. 211).
The above expression means, if the change in energy is zero, momentum, which involves velocity that requires energy, also becomes zero. This is an idealistic situation, which is "un-physical", as nothing in this universe is ever stationary. Some may argue that even in such a situation, the particle may move due to inertia. But that will lead to interaction with at least the field, which will lead to non-zero energy exchange. In an idealistic situation, there is no movement. Thus, the concept of space and time are not applicable and become indeterminate, as perception is possible only during transition from one state to another and time is the interval between two perceptible events.
On a more formal level, it may be noted that Bohr's derivation does not rely on the commutation relations qp - pq = iħ and tE - Et = iħ, but on Fourier analysis. As far as the relationship between position and momentum is concerned, these two approaches are equivalent. But since most physical systems do not have a time operator, this is not so for time and energy. Indeed, in his discussion with Einstein (Bohr, 1949), Bohr considered time as a simple classical variable. This even holds for his famous discussion of the "clock-in-the-box" thought-experiment where the time, as defined by the clock in the box, is treated from the point of view of classical general relativity. Thus, in an approach based on commutation relations, the position-momentum and time-energy uncertainty relations are not on equal footing, which is contrary to Bohr's approach in terms of Fourier analysis (Hilgevoord 1996 and 1998).
There is also another interpretation of the said equation δt.δE ≥ ħ. According to the quantum mechanical dogma, the above equation implies that the so-called empty space is not actually empty, but is full of virtual particles. These virtual particles with opposite charge are postulated to have been created in pairs drawing energy ΔE at a point over a very short period of time Δt, which are then immediately annihilated. The apparently empty space is thus said to be capable of producing particles. This state is described by a quantum state with the lowest possible energy: thus called the zero-point energy state. This implies that there is an underlying "veiled reality" layer present, determining the quantum states of the system even when apparently there are no particles. However, the layer is completely undetectable to our sense organs and measuring instruments (we have to accept the words of the scientists blindly) - all of which are made up of particles. This is said to "prove" the probabilistic nature of the wave-function! It has been suggested that at the Planck scale, i.e., 10^-35 m, quantum fluctuations become powerful enough to twist and turn the geometry of the Universe. Space and time break down to quantum foam. Like the non-existent Higgs boson that has misled the scientific community for close to half a century, this is also another red herring.
Regarding the time cone and event horizon, we have separately shown that these are also wrong and misleading concepts. The light cone is said to be an imaginary surface associated with a point in space-time comprising the paths of all possible light rays that pass through that point. If this description is right, then there cannot be any "cone". If light moves in straight lines through one point, then it will comprise rays from "all" directions and not select directions to prove the theory right (unless someone claims that Nature does not follow its own law, but follows his laws).
The other explanation of drawing the world lines is also wrong. The trick is first to take two spatial dimensions and one time dimension and show the evolution of the light pulse as a conic section. Then the third spatial dimension is added to show the picture of the light cone. It is surprising that till date no scientist has challenged this. Light moves in straight lines (unless it is subjected to other effects). Thus, a photon will move in time in a straight line only and a light pulse will evolve in time spherically with the starting point as the origin. In both cases there cannot be any "cone". If two space dimensions are taken, it would be a chain of concentric circles. If the third spatial dimension is added, then it will be a chain of concentric circles. If the concept of light cone and event horizon are wrong, then the entire edifice built upon such wrong foundation is also wrong. It is surprising that till now, this wrong concept has gone unchallenged.
Regards.
basudeba
