From: Osher Doctorow osher@ix.netcom.com, Tues. July 11, 2000, 11:00AM Dear Colleagues: The "conversions" from x/y to 1 - x + y and from x/y to x + y - xy which I introduced here on July 8, 2000 not only can be used across probability-statistics and (fuzzy) multivalued logical categories and ring theory and number theory categories (Fermat's Last Theorem) as indicated in that paper, but they are especially useful in special relativity and quantum field theory as well as quantum mechanics as I shall show here. The categories in the latter two fields will be discussed in detail later, but here I would like to indicate what results are obtained. Applying the conversion from x/y to 1 - x + y to the Heisenberg Uncertainty Principle in the form xy > k where k is a positive constant (x, y uncertainties in position and momentum respectively, for example, where notice x and y are nonnegative in the standard deviation version of uncertainty) yields xy = (x/y)y^^2 (where ^^ is exponentiation) --> (1 - x + y)y^^2. For 1 - x + y < 0, which says x > 1 + y, the latter expression is negative, so the converted form reads: (1 - x + y)y^^2 = -/1 - x + y/y^^2 > k and therefore /1 - x + y/y^^2 < -k for k positive. Since y^^2 is always nonnegative, this conditional holds trivially (always) provided that x > 1 + y. Since x and y could be selected with exchanged physical roles (uncertainties in momentum and position respectively, for example), the condition that x > 1 + y is rather arbitrary and certainly will be fulfilled for one of the two orders in which x and y are defined. The conclusion which we must come to is that there are two phases: the phase in which the Heisenberg Uncertainty Principle is satisfied (which phase may for example correspond to an interaction between macroscopic observation and microscopic phenomena) and the phase in which the Heisenberg Uncertainty Principle is not satisfied (which phrase may correspond to two macroscopic or two microscopic observation/phenomena pairs or both or some other regime). A similar but even more explicit results holds when we make the conversion in special relativity in the beta or 1/beta Lorentz contraction factor sqrt(1 - v^^2/c^^2) where sqrt means square root of. We get 1 - v^^2/c^^2 = 1 - (v/c)^^2 = 1 - y/x for y = v^^2, x = c^^2 and this goes over to 1 - (1 - x + y = x - y. So beta or 1/beta (depending on what notation one uses) goes over to sqrt(x - y). However, it has been pointed out in earlier papers (and it will be pointed out again) that the closure law holds in most of the categories to which these conversions x/y --> 1 - x + y for example apply, and let us assume the same thing here (closure under subtraction means that if x and y are in the category or in the set part of the category other than the morphism, then x - y and y - x are also in the set part). Therefore, sqrt(y - x) is an alternate form of the result for beta or 1/beta, and since sqrt (y - x) is imaginary when y < x and real when y < x, it must be that the real and imaginary scales are themselves arbitrary. In more common terminology, they merely measure different phases in the sense of liquid-solid-gas etc. Thus, not only can the speed of light be exceeded (although the object exceeding it enters a different phase), but the Heisenberg Uncertainty Principle can be violated (but the objects violating it enter different phases). This is not surprising in view of the superluminal group velocity results obtained in 1997 and later experiments by Nimtz in Cologne/Koln and Berkeley and elsewhere (which according to Nimtz are also applicable to ordinary velocities). It is also not surprising in view of M. Jammer's comprehensive analysis (The Philosophy of Quantum Mechanics, Wiley: New York 1974) of the Heisenberg Uncertainty Principle in which it is found that the principle does not apply to individual measurements of physical objects but to statistical summaries of their uncertainties (Schrodinger in fact proved that on Hilbert Space self-adjoint operators obey this inequality for the product of their uncertainties, but Banach Space is far more general than Hilbert Space and gives enough room by far to define or contain a second phase sets of objects - which may not be operators in the usual sense). The implications for special relativity and for quantum field theory (which combines quantum mechanics with special relativity) are obviously serious, although not necessarily catastrophic. It just means that these theories are again approximations to one or two states or category of states of physical objects but not to more general or different categories of states of physical objects. Since light itself satisfies both categories (sqrt (1 -(c^^2/c^^2)) is 0 which can be regarded as both real and imaginary), there certainly is at least one non-empty element in each category. Osher Doctorow Doctorow Consultants Culver City, California USA