From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1565 Path: news.gmane.org!not-for-mail From: "Osher Doctorow" Newsgroups: gmane.science.mathematics.categories Subject: Re: Cross-category "conversions" of some interest Date: Tue, 11 Jul 2000 11:27:52 -0700 Message-ID: <000201bfeb66$d41068e0$edf46ed1@osherphd> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: multipart/alternative; boundary="----=_NextPart_000_0007_01BFEB2B.0D635D00" X-Trace: ger.gmane.org 1241017928 31839 80.91.229.2 (29 Apr 2009 15:12:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:12:08 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Wed Jul 12 10:29:31 2000 -0300 Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.9.3/8.9.3) id KAA13924 for categories-list; Wed, 12 Jul 2000 10:25:47 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.00.2615.200 X-MimeOLE: Produced By Microsoft MimeOLE V5.00.2615.200 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 7 Original-Lines: 259 Xref: news.gmane.org gmane.science.mathematics.categories:1565 Archived-At: This is a multi-part message in MIME format. ------=_NextPart_000_0007_01BFEB2B.0D635D00 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable 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 =3D (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 =3D = -/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 =3D 1 - (v/c)^^2 =3D 1 - y/x for y =3D v^^2, x =3D = c^^2 and this goes over to 1 - (1 - x + y =3D 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 =20 ------=_NextPart_000_0007_01BFEB2B.0D635D00 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable
From: Osher Doctorow osher@ix.netcom.com, Tues. July = 11, 2000,=20 11:00AM
 
Dear Colleagues:
 
The "conversions" from x/y to 1 - x + y = and from=20 x/y to x + y - xy which I introduced here on July 8, 2000 not only can = be used=20 across probability-statistics and (fuzzy) multivalued logical categories = and=20 ring theory and number theory categories (Fermat's Last Theorem) as = indicated in that paper, but they are especially useful in special = relativity=20 and quantum field theory as well as quantum mechanics as I shall = show=20 here.  The categories in the latter two fields will be discussed in = detail=20 later, but here I would like to indicate what results are = obtained.
 
Applying the conversion from x/y = to 1 - x + y=20 to the Heisenberg Uncertainty Principle in the form xy > k where = k is a=20 positive constant (x, y uncertainties in position and momentum=20 respectively, for example, where notice x and y are nonnegative in the = standard=20 deviation version of uncertainty) yields xy =3D (x/y)y^^2 (where ^^ is=20 exponentiation) --> (1 - x + y)y^^2.  For 1 - x + y = < 0,=20 which says x > 1 + y, the latter expression is negative, so the=20 converted form reads: (1 - x + y)y^^2 =3D -/1 - x + y/y^^2 > k = and=20 therefore /1 - x + y/y^^2 < -k for k positive.  Since y^^2 is = always=20 nonnegative, this conditional holds trivially (always) provided = that x >=20 1 + y.   Since x and y could be selected with = exchanged=20 physical roles (uncertainties in momentum and position respectively, for = example), the condition that x > 1 + y is rather arbitrary and = certainly will=20 be fulfilled for one of the two orders in which x and y are=20 defined.
 
The conclusion which we must come to is = that there=20 are two phases: the phase in which the Heisenberg Uncertainty Principle = is=20 satisfied (which phase may for example correspond to an interaction = between=20 macroscopic observation and microscopic phenomena) and the phase in = which the=20 Heisenberg Uncertainty Principle is not satisfied (which phrase may = correspond=20 to two macroscopic or two microscopic observation/phenomena pairs or = both or=20 some other regime).
 
A similar but even more explicit = results holds when=20 we make the conversion in special relativity in the beta or 1/beta = Lorentz=20 contraction factor sqrt(1 - v^^2/c^^2) where sqrt means square root = of.  We=20 get 1 - v^^2/c^^2 =3D 1 - (v/c)^^2 =3D 1 - y/x for y =3D v^^2, = x =3D c^^2 and=20 this goes over to 1 - (1 - x + y =3D x - y.   So = beta or=20 1/beta (depending on what notation one uses) goes over to sqrt(x - = y). =20 However, it has been pointed out in earlier papers (and it = will be=20 pointed out again) that the closure law holds in most of the = categories to=20 which these conversions x/y --> 1 - x + y for example apply, and let = us=20 assume the same thing here (closure under subtraction means that if x = and y are=20 in the category or in the set part of the category other than the = morphism, then=20 x - y and y - x are also in the set part).   Therefore, sqrt(y = - x) is=20 an alternate form of the result for beta or 1/beta, and since sqrt (y - = x) is=20 imaginary when y < x and real when y < x, it must be that the = real=20 and imaginary scales are themselves arbitrary.  In more common = terminology,=20 they merely measure different phases in the sense = of liquid-solid-gas=20 etc.
 
Thus, not only can the speed of = light be=20 exceeded (although the object exceeding it enters a different phase), = but the=20 Heisenberg Uncertainty Principle can be violated (but the objects = violating it=20 enter different phases).    This is not surprising = in view=20 of the superluminal group velocity results obtained in 1997 and later=20 experiments by Nimtz in Cologne/Koln and Berkeley and elsewhere (which = according=20 to Nimtz are also applicable to ordinary velocities).  It is also = not=20 surprising in view of M. Jammer's comprehensive analysis (The Philosophy = of=20 Quantum Mechanics, Wiley: New York 1974) of the Heisenberg = Uncertainty=20 Principle in which it is found that the principle does not apply to = individual=20 measurements of physical objects but to statistical summaries of their=20 uncertainties (Schrodinger in fact proved that on Hilbert Space = self-adjoint=20 operators obey this inequality for the product of their uncertainties, = but=20 Banach Space is far more general than Hilbert Space and gives enough = room by far=20 to define or contain a second phase sets of objects - which may not be = operators=20 in the usual sense).
 
The implications for special relativity = and for=20 quantum field theory (which combines quantum mechanics with special = relativity)=20 are obviously serious, although not necessarily = catastrophic.   It=20 just means that these theories are again approximations to one or two = states or=20 category of states of physical objects but not to more general or = different=20 categories of states of physical objects.  Since light=20 itself satisfies both categories (sqrt (1 -(c^^2/c^^2)) is 0 which = can be=20 regarded as both real and imaginary), there certainly is at least one = non-empty=20 element in each category.
 
Osher Doctorow
Doctorow Consultants
Culver City, California=20 USA   
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