#bbinclude#="MacBook:Users:jeffrey:Sites:NT:NT_text:aBBIncludes:front_end.txt" #title#="natural theology > VII notes > 25 january 2009" #metaDescription#="natural, religion, theology, notes, 25 january 2009" #metaKeywords#="natural, religion, theology, notes, notebooks, 25 January 2009" #volHead#="vol 7: Notes" #capHead#="2009" #pageHead#="Sunday 25 January" #topPath#="../../" #nextURL#="" #nextText#="" #previousURL#="" #previousText#="" --> natural theology > VII notes > 25 january 2009

natural theology

This site is part of the The natural religion project
dedicated to developing and promoting the art of peace.

Contact us: Click to email

Notes

[Sunday 25 January 2009 - Saturday 31 January 2009]

[Notebook: DB 65 Symmetric U]

[page 80]

Sunday 25 January 2009

What we call the laws of physics are in fact (?) the constraints imposed by the simplicity of its early stages, ie its lower layers.

[page 81]

Monday 26 January 2009

PERIODIC = BORING (ie repetitive like the boring old soldier who tells the same tales repeatedly). The more skilled storyteller, like the more interesting musician, brings in unpredictable details every now and then.

Tuesday 27 January 2009

Hughes page 77: Systems are indifferent to (symmetric with respect to) coordinates not included in their Lagrangian. Hughes (Emmy Noether Emmy Noether - Wikipedia ) As systems become more complex they become connected to more coordinates and their lagrangians become more complex until we arrive at something like the Lagrangian of the Standard Model (Veltman page 249 sqq. Veltman). If the layered network model is good, we should be able to order the elements of the standard model lagrangian into layers, thus giving ourselves a more digestible perspective on the whole thing.

Wednesday 28 January 2009

To say that a coordinate is missing from a Lagrangian is to say that it has no input into the energy of the system. In physics the possession of energy (ie the ability to communicate) is the touchstone of existence, as we say a particle exists when the corresponding field mode is excited (has energy). In human government terms, a policy can be said to exist when it is funded, ie has cashflow.

CLASSICAL - PAST (COMPLETE)
QUANTUM = PRESENT (INCOMPLETE).

[page 82]

Thursday 29 January 2009

Zurek: Decoherence, Einselection and the quantum origins of the classical: arXiv: quant-ph/0105127v3 19 June 2003. Zurek

Einselection = environment induced superselection

Envariance = environment assisted invariance

Zurek's idea seems completely consistent with the network model insofar as the choice of a common set of eigenfunctions between two communicating systems is equivalent to sharing a protocol or common language in order to communicate and so can be seen as a negotiated joint effort between the system of interest and the environment with which it is communicating.

Pointer states: = set of orthogonal eigenfunctions shared by both parties in a communication.

Continuous encodings are a tiny subset of all possible encodings and restrict our ability to apply Shannon's theorems to error correction (Shannon 1949 Shannon)

There are no quantum measurements. All are classical, only the explanations are quantum, ie quantum mechanics explains them.

Wheeler 1978, 1983) quoted in Zurek (page 2): 'No [quantum] phenomenon is a phenomenon until it is a recorded (observed) phenomenon'. Wheeler & Zurek

[page 83]

Zurek page 3: Many Worlds Interpretation (Everett, Everett III) 'MWI is incomplete. It does not explain what is effectively classical and why.'

Classical is the outcome of a computation. Nothing is worse than a computer that 'hangs' (refuses to halt and yield an output). Often the only way out is to restart and lose work in progress in the process.

Friday 30 January 2009

Zurek page 4: 'Our aim is to explain why does the quantum Universe appear classical.'

Maybe more to the point is why does the classical Universe have a quantum explanation? We have two opposite meanings of classical: 1. It obeys classical logic, as in a classical computer; and 2: It is continuous. 1 is good. 2 is a simplifying assumption which appears true macroscopically because Planck's quantum of action is so small, but if we consider larger events like reproduction or a hurricane, which are of themselves quantized, we see that the classical Universe is not continuous.

Zurek (continues): 'This question can be motivated only in a Universe divided into systems [ie a discontinuous Universe] . . . In the absence of systems the Schrödinger equation dictates deterministic evolution and the problem of interpretation seems to disappear.

'There is no need for 'collapse' in a Universe with no systems. The division into systems is imperfect. As a consequence, the Universe is a collection of open (interacting) quantum systems.'

[page 84]

In other words a network of o[pen]-systems (oracle machines Turing, Oracle machine - Wikipedia).

All of the Universe appears classical to observers: all that is non-classical is the explanation of the classical appearances. Quantum mechanics currently does this with continuous functions, and the collapse hypothesis. The network model does it with Turing machines.

Zurek page 4: '. . . typically observers use environment as a "communication channel" and monitor it to find out about the system.'

Why do continuous functions work so well in physics? Because we are mainly concerned with 'material' measures like mass, velocity and so on which can be expressed in cardinal numbers without reference to any order which may exist within the set so measured. From a Newtonian dynamic point of view all we need to know is that a certain book has a mass of one kilogram. The meaning of the book, whether Bible of fashion magazine full of advertisements is irrelevant. Pure quantum mechanics has the same property, dealing with a conserved quantity we call probability. It is only when we begin to apply these models to particular cases that we begin to take into account the interaction of ordered parts, as a Newtonian investigation of vehicle suspension or the quantum mechanical examination of an atom. Quantum mechanics, because it works in infinite dimensional vector space, is better suited to examining complex systems, but it is still very restrained by its assumption of continuity compared to the ability of a Turing Machine

(Ross Ashby page 133 ['A common and very powerful constraint is that of continuity. It is a constraint because whereas the function that changes arbitrarily can undergo any change, the continuous function can change, at each step, only to a neighbouring value.] Ashby

Zurek p 6: 'The freedom is basis choice -- basis ambiguity -- is guaranteed by the principle of superposition.'

[page 85]

We believe in musical superposition because we can hear the different voices in an orchestra or choir. Quantum mechanical superpositions arise because the same differential equation may be satisfied by any frequencies that honour the boundary conditions (overtones).

Let us say that the layers of the cosmic network are parametrized by inverse time, that is energy. The aim of high energy physics is to get as near as possible to the physical layer of the network by generating higher and higher energies per particle. Ultimately we expect all the energy in the Universe to be concentrated in a two state system, the fundamental clock which is the dynamic hardware for the whole system. As the Universe complexifies, this clock is shared among all the processes in the Universe by a sort of time or frequency division multiplexing. So my life depends on my tiny fraction of time on the universal Central Processing Unit and all the other personalities in the Universe (users) similarly get their cut just like the users on a computer network with a single CPU.

In orthodox quantum mechanics the state vectors are never at rest but always moving like an orchestra with a superposition of orthogonal frequencies whose sum is the energy of the system of interest, and whose relative energies are encoded in the amplitudes associated with each possible state. In a classical computer on the other hand, the clock ticks discontinuously and the algorithms in the computer are executed in discrete steps, the transitions between steps being masked by the clock. The clock frequency represents the total energy of the process, which is shared by the different subroutines which make up the overall process, each being called and returning its value with a particular (and possibly variable) frequency as the

[page 86

computation progresses. In the Schrödinger equations of motion of a quantum system, the energy attributed to each frequency is encoded in the Hamiltonian matrix which also encodes the transition probabilities (frequencies) between each mode represented by a basis vector. In a classical computer the role of the Hamiltonian is played by the program, which is made up of layered subroutines, some used more frequently than others.

One section of why_quantized must be devoted to mapping a Turing machine into the Schrödinger equation, insofar as that is possible, and pointing out the differences between the two, particularly as concerns time ordering and the restrictions of continuity and linearity.

Wigner, Symmetries page 155: 'The Orthodox view: The possible states of a system can be characterized according to quantum mechanical theory by state vectors. These state vectors -- and this is an almost verbatim quotation of von Neumann - change in two ways. As a result of the passage of time they change continuously according to Schrödinger 's time dependent equation -- this equation will be called the equation of motion of quantum mechanics. The state vector also changes discontinuously, according to probability laws, if a measurement is carried out on the system. This second type of change is often called the reduction of the wavefunction. It is this reduction of the state vector which is unacceptable to many of our colleagues. Wigner, von Neumann

page 166: '. . . the state vector is a shorthand expression of that part of our information concerning the past of the system which is relevant for predicting (as far as possible) the future behaviour thereof.'

[page 87]

Shannon's theory tells us that we can construct orthogonal vectors in message space, ie vectors with no overlap which are sharply distinct from one another, just like orthogonal basis states in Hilbert space.

Shannon (1949) 'III Geometrical Representation of Signals. . . . Essentially, we have replaced a complex entity (say a television signal) in a simple environment [the signal requires only a plane for its representation as f (t)] by a simple entity (a point) in a complex environment (2TW dimensional space [where T is the duration of the signal and W its bandwidth]).

'If noise is added to the signal in transmission, it means that the point corresponding to the signal has been moved a certain distance in the space proportional to the rms value of the noise. This noise produces a small region of uncertainty about each point in the space.

'Mathematically the simplest types of mappings are those in which [the signal space and the message space] have the same number of dimensions.

It seems that this is the case in quantum mechanics. [but what about projectors?]

'It is not possible to map the message space onto the signal space in a one-to-one continuous manner (this is known mathematically as a topological mapping) unless the two spaces have the same dimensionality.'

[page 88]

However, there is no good reason to confine ourselves to topological mappings apart from the fact that non-topological mappings may be complex and non-linear, although well within the powers of a suitably powerful computer.

Theorem: Let P be the average transmitter power and suppose the noise is white thermal noise of power N in band W. By sufficiently complicated encoding systems it is possible to transmit binary digits at the rate

C = W log2 (P + N )/N

with as small a frequency of errors as desired. It is not possible by any encoding method to send at a higher rate and have an arbitrarily low frequency of errors.'

Here enters the velocity of light via coding delay.

'. . . we can send at the rate C but reduce errors by using more involved coding and longer delays at the transmitter and receiver. The transmitter will take long sequences of binary digits and represent the entire sequence by a particular signal function of long duration. The delay is required because the transmitter must wait for the full sequence before the signal can be determined Similarly the receiver must wait for the full signal function before decoding the binary digits.

'VIII Discussion. We will call a system that transmits without errors at the rate C an ideal system. Such a system cannot be achieved with any finite encoding process, but can be approximated as closely as desired. As we approximate

[page 89]

more closely to the ideal the following effects occur.

1. The rate of transmission . . . approaches C
2. The rate of errors approaches zero
3. The transmitted signal approaches white noise . . .
4. The threshold effect becomes very sharp. If the noise is increased over the value for which the system was designed, the frequency of errors increases very rapidly
5. The required delays at transmitter and receiver increase indefinitely . . . '

Khinchin Khinchin

Saturday 31 January 2009

Wigner: Relativistic Invariance and Quantum Phenomena

page 51: 'The principal theme of this discourse is the great difference between the relation of special relativity and quantum theory on the one hand and general relativity and quantum theory on the other.

'The difference between the two relations is, briefly, that while there are no conceptual problems to separate the theory of special relativity from quantum theory, there is hardly any common ground between the general theory of relativity and quantum mechanics. '

Quantum Electrodynamics: Tomonaga, Schwinger, Feynman and Dyson. Quantum electrodynamics - Wikipedia

'What is meant is . . . that the concepts which are used in quantum mechanics, measurements of positions, momenta and the like are the same concepts in terms of which the special relativistic postulate is formulated.'

[page 90]

Wigner page 52: 'This is not so with the general theory of relativity. The basic premise of this theory is that coordinates are only auxiliary quantities which can be given arbitrary values for every event.' [but they are strongly constrained by the need for continuity]

That is, the space-time coordinates for an event are not intrinsic to the event itself, which is a private communication between the participants in the event who, like lovers, are oblivious to their surroundings [ie can only couple pairwise at a given time]. It is only when a higher layer makes use of the communications between peers of the lower layer for it own purposes that the relationship between events acquires meaning and the coordinates of 'atomic' events (like the execution of a logical function) become important to some more complex event (the computation of the product of two numbers).

Arithmetic and logic meet in the binary domain, arithmetic giving higher meaning to binary logical operations.

'Evidently, the usual statements about future positions of particles as specified by their coordinates, are not meaningful statements in [general] relativity. This is a point which cannot be emphasized strongly enough and is the basis of a much deeper dilemma than the more technical questions of the Lorentz invariance of the quantum field equations.'

'Real coordinates' are relationships established in a higher layer (a 'user') between elements of a lower layer ('an alphabet of tools').

I have been able to think myself so far from orthodoxy because I am an isolated particle, scientifically speaking part of neither the theological or the physical establishment.

[page 91]

Wigner page 53: 'Relativistic Quantum Theory of Elementary Systems: . . . Two cases have been distinguished: the particle either can or cannot be transformed to rest. . . . if a particle cannot be transformed to rest, its velocity must always be equal to the velocity of light. Every other velocity can be transformed to rest. The rest-mass of these particles is zero because a non-zero rest-mass would entail infinite energy if moving with light velocity.'

On the other hand 'rest-mass' is meaningless for a particle that cannot be transformed to rest! So photons and gravitons are 'outside' the mechanism of special relativity. Space-time has no meaning for them and so on the layered model they are lower than the space-time layer. They do not have space-time coordinates, which is consistent, however, with the coordinate free approach of general relativity.

Higher layers are invisible to lower layers and so although a higher layer may manipulate lower layer, the lower layer does not see it and so is indifferent to it. This indifference (we suppose) explains the universality of the velocity of light. From this point of view direction of motion and spin (polarization) are attributes given to photons by the higher layers using them.

page 54: 'Instead of the question: "Why do particles with zero rest-mass have only two directions of polarization?" the slightly different question 'Why do particles with a finite rest mass have more than two directions of polarization:" is proposed.

= Why can't the angular momentum of a particle with finite rest mass be parallel to its velocity?

[page 92]

Wigner page 54: 'The statement that the spin is parallel to the velocity is a relativistically invariant statement [for particles with m = 0]

Time is reversible in the domain where Turing machines are reversible.

Wigner page 63: We want to see space-time as an emergent property of the Universe, somehow made by using massless particle photon and graviton which are never at rest to create the space-time Universe of experience. How? From this point of view, the ur-dimension is velocity, specifically c. In special relativity c is a local phenomenon, all observers in all inertial frames observing the same c. We try to explain this by the coding delay associated with error free communication. The simplest coding takes one bit from the source and maps it to one signal in the channel and vice versa. In a Universe with no choice of signals or messages, this process is essentially error free and so, we suspect, instantaneous. Salart et al

'The events of special relativity are coincidences, that is collisions between particles.'

The less violent among us might see this as mapping from one particle to another. On the Trinitarian analogy, we might say gravitation = father, photon = son or vice versa, and the system works by mapping [photon to graviton] and back. There yet being no space-time there is no velocity involved in the process but we may associate a quantum of action with each instance of mapping, and from the frequency of mapping derive an energy.

Assuming c = 1, we can equate distance and time,

[page 93]

given that they exist. [We might say the possibility of this equations derives from them being the 'same thing' underneath]

Wigner page 69:

'. . . states will be generated by looking at the same state -- the standard state -- [for us something near the initial singularity] from various coordinate systems. Hence each Lorentz frame of reference will define a state of the system, -- the state which the standard state appears to be from the point of view of this coordinate system. . . . two states of the system will be identical only if the Lorentz frames of reference which define them are identical.

Here no cloning says no identical frames.

The scheme embodies the what we see depends on how we look paradigm, and given that relations of looking (= transformation, communication) are real as in Thomas' god and Einstein's physics) we might see how the same logical operation (eg mapping from photon to graviton) can be seen and be different from different points of view, just as each operation the the Central Processing Unit of a computer is differentiated from every other by its relationship to all the other operations in the execution of the program.

Let our ur-alphabet be C, P, T (Wigner page 74) Streater & Wightman Space inversion, time inversion, particle conjugation. For every annihilation there is a creation.

#bbinclude#="MacBook:Users:jeffrey:Sites:NT:NT_text:aBBIncludes:back_end.txt" -->

Related sites

Concordat Watch

Revealing Vatican attempts to propagate its religion by international treaty


Copyright:

You may copy this material freely provided only that you quote fairly and provide a link (or reference) to your source.


Further reading

Books

Click on the "Amazon" link below each book entry to see details of a book (and possibly buy it!)

Ashby, W Ross, An Introduction to Cybernetics, Methuen 1964 'This book is intended to provide [an introduction to cybernetics]. It starts from common-place and well understood concepts, and proceeds step by step to show how these concepts can be made exact, and how they can be developed until they lead into such subjects as feedback, stability, regulation, ultrastability, information, coding, noise and other cybernetic topics' 
Amazon
  back
Everett III, Hugh, and Bryce S Dewitt, Neill Graham (editors), The Many Worlds Interpretation of Quantum Mechanics, Princeton University Press 1973 Jacket: 'A novel interpretation of quantum mechanics, first proposed in brief form by Hugh Everett in 1957, forms the nucleus around which this book has developed. The volume contains Dr Everett's short paper from 1957, "'Relativge State' formulation of quantum mechanics" and a far longer exposition of his interpretation entitled "The Theory of the Universal Wave Function" never before published. In addition other papers by Wheeler, DeWitt, Graham, Cooper and van Vechten provide further discussion of the same theme. Together they constitute virtually the entire world output of scholarly commentary on the Everett interpretation.' 
Amazon
  back
Hughes, I S, Elementary Particles, Cambridge Univerity Press 1991 Jacket: 'This is an extensively revised and updated edition of a text that has established itself as one of the standard undergraduate books on the subject of elementary particle physics.' 
Amazon
  back
Khinchin, A I, Mathematical Foundations of Information Theory (translated by P A Silvermann and M D Friedman), Dover 1957 Jacket: 'The first comprehensive introduction to information theory, this book places the work begun by Shannon and continued by McMillan, Feinstein and Khinchin on a rigorous mathematical basis. For the first time, mathematicians, statisticians, physicists, cyberneticists and communications engineers are offered a lucid, comprehensive introduction to this rapidly growing field.' 
Amazon
  back
Nicolis, G , and Ilya Prigogine, Self Organisation in Nonequilibrium Systems: From Dissipative Structures to Order through Fluctuations, Wiley Interscience 1977 General Introduction: 'The aim of the present monograph can ... be expressed as the studiy of self-organisation in non-equilibrium systems, characterised by the appearance of dissipative structures through the amplification of appropriate fluctuations. ... The natural approach to the problem of the emergence of new patterns is bifurcation theory. The purpose of this theory is to study the possible branching of solutions that may arise under certain conditions. We have tried to present a readable introduction to this rapidly expanding field ... Our main emphasis is in physical examples and simple but representative models, and our aim is to give the reader an idea of the variety of space-time structures that may arise through bifurcation. ... ' 
Amazon
  back
Park, David Allen, Introduction to the Quantum Theory, McGraw-Hill Book Company 1992  
Amazon
  back
Streater, Raymond F, and Arthur S Wightman, PCT, Spin, Statistics and All That, Princeton University Press 2005 Amazon product description: ' PCT, Spin and Statistics, and All That is the classic summary of and introduction to the achievements of Axiomatic Quantum Field Theory. This theory gives precise mathematical responses to questions like: What is a quantized field? What are the physically indispensable attributes of a quantized field? Furthermore, Axiomatic Field Theory shows that a number of physically important predictions of quantum field theory are mathematical consequences of the axioms. Here Raymond Streater and Arthur Wightman treat only results that can be rigorously proved, and these are presented in an elegant style that makes them available to a broad range of physics and theoretical mathematics.' 
Amazon
  back
Veltman, Martinus, Diagrammatica: The Path to the Feynman Rules, Cambridge University Press 1994 Jacket: 'This book provides an easily accessible introduction to quantum field theory via Feynman rules and calculations in particle physics. The aim is to make clear what the physical foundations of present-day field theory are, to clarify the physical content of Feynman rules, and to outline their domain of applicability. ... The book includes valuable appendices that review some essential mathematics, including complex spaces, matrices, the CBH equation, traces and dimensional regularization. ...' 
Amazon
  back
von Neumann, John, and Robert T Beyer (translator), Mathematical Foundations of Quantum Mechanics, Princeton University Press 1983 Jacket: '. . . a revolutionary book that caused a sea change in theoretical physics. . . . JvN begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which JvN regards as the definitive form of quantum mechanics. . . . Regarded as a tour de force at the time of its publication, this book is still indispensible for those interested in the fundamental issues of quantum mechanics.' 
Amazon
  back
Wheeler, John Archibald, and Wojciech Hubert Zurek, Quantum Theory and Measurement (Princeton Series on Physics), Princeton University Press 1983 Amazon customer review: 'This is a must-own collection for anyone studying or working in quantum physics. These are the original papers concerning the so-called problem of measurement. Minority views are included; for instance, both parts of Bohm's 1952 paper are here. Not only physicists, but also historians and philosophers of science, will want to read these papers.' Paul E Oppenheimer 
Amazon
  back
Wigner, Eugene, Symmetries and Reflections: Scientific Essays , MIT Press 1970 Jacket: 'This volume contains some of Professor Wigner's more popular papers which, in their diversity of subject and clarity of style, reflect the author's deep analytical powers and the remarkable scope of his interests. Included are articles on the nature of physical symmetry, invariance and conservation principles, the structure of solid bodies and of the compound nucleus, the theory of nuclear fission, the effects of radiation on solids, and the epistemological problems of quantum mechanics. Other articles deal with the story of the first man-made nuclear chain reaction, the long term prospects of nuclear energy, the problems of Big Science, and the role of mathematics in the natural sciences. In addition, the book contains statements of Wigner's convictions and beliefs. as we as memoirs of his friends Enrico Fermi and John von Neumann. Eugene P. Wigner is one of the architects of the atomic age. He worked with Enrco Fermi at the Metallurgical Laboratory of the University of Chicago at the beginning of the Manhattan Project, and he has gone on to receive the highest honours that science and his country can bestow, including the Nobel Prize for physics, the Max Planck Medal, the Enrico Fermi Award and the Atoms for Peace Award. '. 
Amazon
  back
Papers
Pennycuick, C J, "The soaring flight of vultures", Scientific American, 229, 6, December 1973, page 102-109. 'The six common vultures of East Africa can make a round trip of as much as 200 kilometres by skilfully riding updrafts. How they do so is examined with the aid of a powered glider.' . back
Salart, Daniel, et al, "Testing the speed of 'spooky action at a distance'", Nature, 454, , 14 August 2008, page 861-864. 'Correlations are generally described by one of two mechanisms: either a first event influences a second one by sending information encoded in bosons or other physical carriers, or the correlated events have some common causes in their shared history. Quantum physics predicts an entirely different kind of cause for some correlations, named entanglement. This reveals itself in correlations that violate Bell inequalities (implying that they cannot be described by common causes) between space-like separated events (implying that they cannot be described by classical communication). Many Bell tests have been performed, and loopholes related to locality and detection have been closed in several independent experiments. It is still possible that a first event could influence a second, but the speed of this hypothetical influence (Einstein's 'spooky action at a distance') would need to be defined in some universal privileged reference frame and be greater than the speed of light. Here we put stringent experimental bounds on the speed of all such hypothetical influences. We performed a Bell test over more than 24 hours between two villages separated by 18 km and approximately east–west oriented, with the source located precisely in the middle. We continuously observed two-photon interferences well above the Bell inequality threshold. Taking advantage of the Earth's rotation, the configuration of our experiment allowed us to determine, for any hypothetically privileged frame, a lower bound for the speed of the influence. For example, if such a privileged reference frame exists and is such that the Earth's speed in this frame is less than 10-3 times that of the speed of light, then the speed of the influence would have to exceed that of light by at least four orders of magnitude.. back
Shannon, Claude E, "Communication in the Presence of Noise", Proceedings of the IEEE, 86, 2, February 1998, page 447-457. Reprint of Shannon, Claude E. "Communication in the Presence of Noise." Proceedings of the IEEE, 37 (January 1949) : 10-21. 'A method is developed for representing any communication system geometrically. Messages and the corresponding signals are points in two function spaces, and the modulation process is a mapping of one space into the other. Using this representation, a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect. Formulas are found for the maximum rate of transmission of binary digits over a system when the signal is perturbed by various types of noise. Some of the properties of "ideal" systems which transmit this maximum rate are discussed. The equivalent number of binary digits per second of certain information sources is calculated.' . back
Turing, Alan, "On Computable Numbers, with an application to the Entscheidungsproblem", Proceedings of the London Mathematical Society, 2, 42, 12 November 1937, page 230-265. 'The "computable" numbers maybe described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost as easy to define and investigate computable functions of an integrable variable or a real or computable variable, computable predicates and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique. I hope shortly to give an account of the rewlations of the computable numbers, functions and so forth to one another. This will include a development of the theory of functions of a real variable expressed in terms of computable numbers. According to my definition, a number is computable if its decimal can be written down by a machine'. back
Zurek, Wojciech Hubert, "Decoherence, einselection, and the quantum origins of the classical", Review of Modern Physics, 75, , 2003, page 715-775. The manner in which states of some quantum systems become effectively classical is of great significance for the foundations of quantum physics, as well as for problems of practical interest such as quantum engineering. In the past two decades it has become increasingly clear that many (perhaps all) of the symptoms of classicality can be induced in quantum systems by their environments. Thus decoherence is caused by the interaction in which the environment in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the Universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal “Schrödinger-cat states.” The classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit. Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information that has predictive power. When the measured quantum system is microscopic and isolated, this restriction on the predictive utility of its correlations with the macroscopic apparatus results in the effective “collapse of the wave packet.” The existential interpretation implied by einselection regards observers as open quantum systems, distinguished only by their ability to acquire, store, and process information. Spreading of the correlations with the effectively classical pointer states throughout the environment allows one to understand “classical reality” as a property based on the relatively objective existence of the einselected states. Effectively classical pointer states can be “found out” without being re-prepared, e.g, by intercepting the information already present in the environment. The redundancy of the records of pointer states in the environment (which can be thought of as their “fitness” in the Darwinian sense) is a measure of their classicality. A new symmetry appears in this setting. Environment-assisted invariance or envariance sheds new light on the nature of ignorance of the state of the system due to quantum correlations with the environment and leads to Born’s rules and to reduced density matrices, ultimately justifying basic principles of the program of decoherence and einselection.. back
Links
Emmy Noether English translation of 'Invariant variation problems' "Invariante Variationsprobleme," Nachr. v. d. Ges. d. Wiss. zu Göttingen 1918, pp 235-257 English translation by M. A. Tavel. Reprinted from "Transport Theory and Statistical Mechanics" 1(3), 183-207 (1971). Provided to this site by M.A. Tavel and Henry M. Paynter. back
Emmy Noether - Wikipedia Emmy Noether - Wikipedia, the free encyclopedia 'Amalie Emmy Noether, . . . (23 March 1882 – 14 April 1935) was a German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of rings, fields, and algebras. In physics, Noether's theorem explains the fundamental connection between symmetry and conservation laws.' back
Oracle machine - Wikipedia Oracle machine - Wikipedia, the free encyclopedia 'In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as a Turing machine with a black box, called an oracle, which is able to decide certain decision problems in a single operation. The problem can be of any complexity class. Even undecidable problems, like the halting problem, can be used.' back
Quantum electrodynamics - Wikipedia Quantum electrodynamics - Wikipedia, the free encyclopedia 'Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. QED was developed by a number of physicists, beginning in the late 1920s. It basically describes how light and matter interact. More specifically it deals with the interactions between electrons, positrons and photons. QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons. It has been called "the jewel of physics" for its extremely accurate predictions of quantities like the anomalous magnetic moment of the electron, and the Lamb shift of the energy levels of hydrogen. back
Richard P. Feynman Nobel Lecture Nobel Lecture, December 11, 1965: 'We have a habit in writing articles published in scientific journals to make the work as finished as possible, to cover all the tracks, to not worry about the blind alleys or to describe how you had the wrong idea first, and so on. So there isn't any place to publish, in a dignified manner, what you actually did in order to get to do the work, although, there has been in these days, some interest in this kind of thing. Since winning the prize is a personal thing, I thought I could be excused in this particular situation, if I were to talk personally about my relationship to quantum electrodynamics, rather than to discuss the subject itself in a refined and finished fashion. Furthermore, since there are three people who have won the prize in physics, if they are all going to be talking about quantum electrodynamics itself, one might become bored with the subject. So, what I would like to tell you about today are the sequence of events, really the sequence of ideas, which occurred, and by which I finally came out the other end with an unsolved problem for which I ultimately received a prize.' back
Wojciech Hubert Zurek Decoherence, einselection and the quantum origins of the classical 'Decoherence is caused by the interaction with the environment which in effect monitors certain observables of the system, destroying coherence between the pointer states corresponding to their eigenvalues. This leads to environment-induced superselection or einselection, a quantum process associated with selective loss of information. Einselected pointer states are stable. They can retain correlations with the rest of the Universe in spite of the environment. Einselection enforces classicality by imposing an effective ban on the vast majority of the Hilbert space, eliminating especially the flagrantly nonlocal “Schr¨odinger cat” states. Classical structure of phase space emerges from the quantum Hilbert space in the appropriate macroscopic limit: Combination of einselection with dynamics leads to the idealizations of a point and of a classical trajectory. In measurements, einselection replaces quantum entanglement between the apparatus and the measured system with the classical correlation. Only the preferred pointer observable of the apparatus can store information that has predictive power. When the measured quantum system is microscopic and isolated, this restriction on the predictive utility of its correlations with the macroscopic apparatus results in the effective “collapse of the wavepacket”. Existential interpretation implied by einselection regards observers as open quantum systems, distinguished only by their ability to acquire, store, and process information. Spreading of the correlations with the effectively classical pointer states throughout the environment allows one to understand ‘classical reality’ as a property based on the relatively objective existence of the einselected states: They can be “found out” without being re-prepared, e.g, by intercepting the information already present in the environment. The redundancy of the records of pointer states in the environment (which can be thought of as their ‘fitness’ in the Darwinian sense) is a measure of their classicality. A new symmetry appears in this setting: Environment - assisted invariance or envariance sheds a new light on the nature of ignorance of the state of the system due to quantum correlations with the environment, and leads to Born’s rules and to the reduced density matrices, ultimately justifying basic principles of the program of decoherence and einselection. back

www.naturaltheology.net is maintained by The Theology Company Proprietary Limited ACN 097 887 075 ABN 74 097 887 075 Copyright 2000-2020 © Jeffrey Nicholls