RE: the meaning of imaginary numbers & MDT & quantization
Hello George,
I just now saw your post from Dec. 4th, where you write, " George Ellis wrote on Dec. 4, 2008 @ 10:00 GMT
Dear Dr. E (The Real McCoy)
I agree with the spirit of what you do in your essay, which as you point out is going in the same direction as mine. The main point where I differ is in the use of the imaginary time coordinate. I prefer to see it all done with a real time coordinate (see e.g. my book with Ruth Williams entitled "Flat and Curved Spacetimes"). So a key element is how proper time relates to coordinate time as we move to the future, which is what your key equation establishes; I think one can do this without introducing the imaginary quantity "i"."
I contend that the "i" is there for a reason, and that rather than get rid of it, our job is to ponder what "i" *physically* means.
Here is a great passage from Max Born's book on relativity, which you will enjoy, regarding imaginary numbers--he agrees that they have real physical meaning:
http://books.google.com/books?id=Afeff9XNwgoC&printsec=frontcover&dq= Einstein%27s+Theory+of+ Relativity&ei=i7ZCSbaIKoqakQT5tenUDg#PPA306,M1
Born writes, "It is true that in the domain of ordinary number one cannot extract the square root of the negative quantity -c^2t^2; hence u has no elementary menaing. But mathematicians have long been accustomed to overcoming such diificulties. The imaginary quanity "i" has been firmly established in mathematics since the time of Gauss. We cannot here enter into the question of how the doctrines of imaginary numbers can be rigorously established. These numbers are essentially no more "imaginary" than a fraction such as 2/3., for numbers with which we number things or count properly comprise only the natural intergesrs 1,2,3,4. . . The number 2 is not dividsible by 3, so that 2/3 is an operation that can be carried out just as little as (-1)^(1/2). Fractions such as 2/3 signify an extension of the natural concept of numbers; however, they have become familiar through education and custom, and excite no feeling of strangeness. The introduction of imgainary numbers is a similar extension: all formulae that contain imaginary numbers have just as definite a meaning as those formed from ordinary "real" numbers, and the inferences drawn from them are just as convincing." --p. 306, Einstein's Relativity, Max Born
So it is that both time and the fourth dimension are very, very real. Time is a parameter, often measured on our watches, which emerges because the fourth dimesnion is expanding relative to the three spatial dimensions at the rate of c, or dx4/dt=ic.
Attached please find some figures illustrating the *physical* meaning of "i" in dx4/dt=ic (attachment#1).
The fourth dimension is expanding at the rate of c in a manner perpendicular to the three spatial dimensions, manifesting itself as a spherically-symmetric, nonlocal expanding wavefront--which perfectly describes the photon's motion!
That is why we see:
dx4/dt = ic (MDT's equation, underlying relativity)
and
xp - px = ih (fundamental to quantum mechanics)
Both equations contain "i". Both equations represent the fact that something is *physically* changing in a manner perpendicular to our three spatial dimensions.
dx4/dt=ic posits that the fourth dimension is expanding relative to the three spatial dimensions at the rate of c.
xp - px = ih
also shows that spacetime is changing--the fourth dimension is expanding in units of Planck's length--in a manner perpendicular to our three spatial dimensions, as both x and p are measured in our three spatial dimensions; and yet, when one considers xp-px, one sees a quantity lying beyond our three spatial dimensions!
MDT's new universal invariant--the expansion of the fourth dimension, at the rate of c in units of the Planck length, sets both h and c, while weaving change into the fundamental fabric of spacetime for the first time in the history of relativity; and also providing a deeper physical model from where Einstein's principle of relativity naturally emerges, and also accounting for time and all its arrows and assymetries, all quantum/wave/particle behavior, entropy, and quantum nonlocality and entanglement.
In the second attachment, please see how MDT accounts for the gravitational slowing of light and time, as well as the quantum nature of all measurement--the fourth dimension has a nonlocal, wavelike, quantized quality via its invariant expansion in units of the Planck wavelength--something that is attested to by every photon which 1) is timeless and ageless, 2) moves at the speed of light, 3) remains fixed in the fourth dimension, and 4) is described by a nonlocal, spherically-expanding wavefront.
More in this in the upcoming book! "HERO'S JOURNEY PHYSICS & MOVING DIMENSIONS THEORY: FROM HERACLITIS, TO PLATO, TO ARISTOTLE, TO COPERNICUS, TO BRUNO, TO KEPLER, TO GALILEO, TO NEWTON, TO PLANCK/EINSTEIN/BOHR/BORN--AND YET IT MOVES! Unifying relativity, quantum mechanics, entropy, and time's arrows and assymetries with a new universal invariant: dx4/dt=ic."
Best,
Dr. E :) (The Real McCoy)
MDT allows us to keep all time and change by weaving change into the fundamental fabric of spacetime with dx4/dt=ic. This means that come 2009, both time and progress in theoretical physics will be unfrozen, and we will be liberated from the block universe and granted free will!
Moving Dimensions Theory is bold, subtle, far-reaching, simple, and consistent. MDT agrees with all empirical evidence and all of relativity and quantum mechanics. MDT has come not to abolish the laws of the founding fathers of physics, but to unite them. MDT liberates us from the block universe and frozen time, while allowing us to keep spacet-time diagrams so as to sell coffee-table books on physics and time travel (although MDT shows that time travel into the past is impossible). MDT goes so far as to show that seeming paradoxes across all realms--from the paradoxes of Godel's block universe to the EPR paradox--can be expalined and accounted for with a novel universal invariant which underlies Einstein's Principle of Relativity, entropy, and emergent time and all its arrows and assymetries.
MDT resembles neither String Theory nor Loop Quantum Gravity, but this should not count against it.Attachment #1: 2_MDT_PERVADES_NATUREIMAGINARY_NUMBERS_IMPLY_PERPENDICULARITY.pdfAttachment #2: 1_5_MOVING_DIMENSIONS_THEORY_EXAMINES_THE_GRAVITATIONAL_REDSHIFT_SLOWING_OF_CLOCKS.pdf