vol 3: Development
chapter 3: Physics
page 6: Invariance with respect to complexity
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Invariance with respect to complexity
Cantor, describing his understanding of the transfinite numbers, wrote:
We shall show that the transfinite cardinal numbers can be arranged according to their magnitude, and, in this order, form, like the finite numbers, a "well-ordered aggregate' in the extended sense of the words. Out of aleph(0) proceeds, by a definite law, the next greater cardinal number aleph(1), out of this by the same law the next greater aleph(2) and so on. But even the unlimited sequence of cardinal numbers
aleph(0), aleph(1), aleph(2), . . . aleph(nu), . . . does not exhaust the conception of transfinite cardinal number. We will prove the existence of a cardinal number which we denote by aleph(omega) and which shows itself to be the next greater to all the numbers aleph(nu) ; out of it proceeds in the same way as aleph(1) out of aleph(0), a next greater aleph(omega + 1), and so on, without end. Cantor, page 109
We might call this construction the Cantor line or cantor Universe . Unlike Euclid's line, which lies evenly with the points on itself the Cantor line expands exponentially, its law of creation generating ever greater cardinal numbers. Heath Nevertheless, we can assign a scalar parameter to each of these lines, the cardinal number of a point, measured from some starting point. Both lines have a metric which is simple count of points.
The definite law described here operates in the same way no matter what the initial cardinality of the set upon which ot operates may be. We will call it the law of formal creation. We say it is invariant with respect to count that is cardinal number, complexity or entropy. It is all the same to this law whether the universe we apply it to is a one state initial singularity or a system comprising any finite or transfinite number of states.
Cantor proposed to prove this law, and we will accept that it is formally true. In other words its contradiction leads to contradiction. Cantor's diagonal argument - Wikipedia , We will imagine this proof to be the source of the Cantor force, that is the principle that points any consistent symbolic universe toward ever greater complexity.
Quantum mechanics, is also invariant with respect to complexity. Quantum mechanics, Hilbert spaces The fundamental mathematical mechanisms of quantum mechanics work the same way no matter what the cardinal of the Hilbert spaces, functions (or vectors) involved.
The universe is both one and many. This fact raises some of the oldest question in philosophy and science. Here we hope that the gap between one and many can be bridged with complexity invariant formalisms like those of quantum mechanics and the law of formal creation.
Unitarity
Our current standard explanation of our physical experiences of the world is expressed in two theories, general relativity and quantum field theory. Einstein, Peacock, Zee The latter is the connection of special relativity and quantum mechanics. General relativity has yet to be satisfactorily explained in quantum mechanics
From the point of view espoused here, we understand quantum mechanics to model of physical line (measured by time) as first glimpsed by Cantor.
Our study of physics is the study of timelines. We note first that nothing, appears to live forever. It has a birth and death. The only exception may be the universe as a whole, which appears to have been born from an initial singularity, but has no discernible end. In this is shares the structure of the Cantor line.
Quantum mechanics assumes that systems come into contact with the world only at birth and death. In between they are isolated independent universes and their internal evolution is unitary. Unitary evolution conserves complexity as measured by the number of dimensions of the Hilbert space in which the system is modelled.
For historical reason, quantum mechanics calls birth and death measurement. The effect of measurement is described by postulates (iv) and (v):
(iv) measurement outcomes are restricted to an orthonormal set { | sk > } of eigenstates of the measured observable'
(v) the probability of finding a given outcome is pk = |<sk || q >|2, where |q > is the preexisting state of the system.
Mathematically the requirement for unitary evolution between measurements is required by postulate (v), where it is assumed that the sum of the probabilities pk must be 1.
Unitarity is not maintained through birth and death. Instead, like the fall of a die, one possible outcome is chosen and the system is found in a definite state. The probability of this state depends upon the operator used to measure the living state and is given by postulate (v). This is the quantum jump or collapse of the wavefunction. Here we interpret it as the emission or reception of a signal by a node in the physical network.
Communication theory measures the information carried by a symbol by the entropy of the space from which it is drawn. This idea suggests that even though the collapse of the wave function is not unitary, the entropy of the superposition is nevertheless represented faithfully by the information obtained from the measurement.
The wider viewpoint
The complexity invariance of quantum mechanics means that it can deal with universe within universes. There is a collapse of the wave function from the point of view of the two systems observing one another, but if we consider the two of them to be an isolated system, unitary evolution is maintained across the collapse. Of course, we cannot observe this larger system without killing it, but it provides a theoretical window on what is happening.
The Hilbert space of the combined system is the tensor product of the Hilbert spaces of the each of the two independent systems that have come into contact with one another. In other words, communication increases the size of the spaces of quantum mechanics. Insofar as we define the universe as that outside which there is nothing, the universe is a free particle and we assume that its overall evolution is unitary, although we see endless birth and death on our own path through life.
Quantum mechanics is invariant with respect to complexity so that it applies identically at all points on the Cantor line. Logically quantum mechanics owes its complexity invariance to the fact that it has no memory. It exists entirely in the present. Of itself, it has no past to constrain it and it places no constraint on the future. In this sense it is eternal, rather as we might expect the initial singularity to be.
Observation shows that unlike time, which marches along imperturbably, carrying us with it, space began small and has grown to the huge universe than we now inhabit. Here we identify space with memory. Without memory, the whole concept of evolution is meaningless since to an entity without memory, there is no past or future. Further, although we cannot move in time, we are free to move in space.
We turn to the creation of space on the next page.
Further reading
Books
| Cantor, Georg, Contributions to the Founding of the Theory of Transfinite Numbers (Translated, with Introduction and Notes by Philip E B Jourdain), Dover 1955 Jacket: 'One of the greatest mathematical classics of all time, this work established a new field of mathematics which was to be of incalculable importance in topology, number theory, analysis, theory of functions, etc, as well as the entire field of modern logic.' Amazon back |
| Einstein, Albert, and Robert W Lawson (translator) Roger Penrose (Introduction), Robert Geroch (Commentary), David C Cassidy (Historical Essay) , Relativity: The Special and General Theory, Pi Press 2005 Preface: 'The present book is intended, as far as possible, to give an exact insight into the theory of relativity to those readers who, from a general scientific and philosophical point of view, are interested in the theory, but who are not conversant with the mathematical apparatus of theoretical physics. ... The author has spared himself no pains in his endeavour to present the main ideas in the simplext and most intelligible form, and on the whole, in the sequence and connectionin which they actually originated.' page 3 Amazon back |
| Peacock, John A, Cosmological Physics, Cambridge University Press 1999 Nature Book Review: 'The intermingling of observational detail and fundamental theory has made cosmology an exceptionally rich, exciting and controversial science. Students in the field -- whether observers or particle theorists -- are expected to be acquainted with matters ranging from the Supernova Ia distance scale, Big Bang nucleosynthesis theory, scale-free quantum fluctuations during inflation, the galaxy two-point correlation function, particle theory candidates for the dark matter, and the star formation history of the Universe. Several general science books, conference proceedings and specialized monographs have addressed these issues. Peacock's Cosmological Physics ambitiously fills the void for introducing students with a strong undergraduate background in physics to the entire world of current physical cosmology. The majestic sweep of his discussion of this vast terrain is awesome, and is bound to capture the imagination of most students.' Ray Carlberg, Nature 399:322 Amazon back |
| Zee, Anthony, Quantum Field Theory in a Nutshell, Princeton University Press 2003 Amazon book description: 'An esteemed researcher and acclaimed popular author takes up the challenge of providing a clear, relatively brief, and fully up-to-date introduction to one of the most vital but notoriously difficult subjects in theoretical physics. A quantum field theory text for the twenty-first century, this book makes the essential tool of modern theoretical physics available to any student who has completed a course on quantum mechanics and is eager to go on. Quantum field theory was invented to deal simultaneously with special relativity and quantum mechanics, the two greatest discoveries of early twentieth-century physics, but it has become increasingly important to many areas of physics. These days, physicists turn to quantum field theory to describe a multitude of phenomena. Stressing critical ideas and insights, Zee uses numerous examples to lead students to a true conceptual understanding of quantum field theory--what it means and what it can do. He covers an unusually diverse range of topics, including various contemporary developments,while guiding readers through thoughtfully designed problems. In contrast to previous texts, Zee incorporates gravity from the outset and discusses the innovative use of quantum field theory in modern condensed matter theory. Without a solid understanding of quantum field theory, no student can claim to have mastered contemporary theoretical physics. Offering a remarkably accessible conceptual introduction, this text will be widely welcomed and used. Amazon back |
Links
| Cantor's diagonal argument - Wikipedia Cantor's diagonal argument - Wikipedia, the free encyclopedia 'Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.' back |
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