volume II: Synopsis
part III: Modern Physics
page 20: 'The wave function of the Universe'
Quantum mechanics is our best description of the physical Universe. We begin from elementary events and show how these events are assembled into larger and larger events, culminating in the total event which we call the Universe. The quantum mechanical description of the whole system is sometimes (for historical reasons) called the wave function of the Universe. This class of functions exists in transfinite function space.
One of the foundational equations of quantum mechanics is the Schrödinger equation, a differential equation with an infinite number of periodic solutions (hence the name 'wave' equation). These solutions ('eigenfunctions') may be added linearly to give further solutions which are called superpositions. Schrödinger equation - Wikipedia
Unlike the mathematical formalism of classical physics, quantum mechanics does not predict a definite outcome given a particular set of initial conditions. It gives instead a probability distribution for the different eigenfunctions (and corresponding 'eigenvalues') of a wave equation. An observation yields one of these eigenvalues. Many observations will yield a distribution of eigenvalues corresponding to the quantum mechanical prediction. The fact that only one eigenvalue is observed at a time is called the 'collapse of the wavefunction'. Wave function collapse - Wikipedia, Eigenfunction - Wikipedia
From a classical point of view, this collapse is the fundamental enigma of quantum mechanics. Although solutions to the wave equation evolve deterministically like a classical the solutions to classical differential equations, the outcomes of observation are only determined up to a probability distribution. This interpretation of quantum wave functions is known as Born's Rule. Born rule - Wikipedia
Many efforts have been made to overcome this situation. Some have proposed the existence of 'hidden variables' which control the outcome of observations. John Bell has shown that this solution is doubtful. Another approach is the 'many worlds' interpretation proposed by Hugh Everett and promoted by (among others) David Deutsch. Bell's theorem - Wikipedia, Everett III, Deutsch
The basic idea of the many worlds hypothesis is that all the solutions to the wave function are real and the wave function does not collapse. Instead, every quantum event is believed to split the Universe into many different worlds, each containing an outcome corresponding to one of the eigenfunctions of the wave equation. This idea sounds a little far fetched. First, it seems to imply the existence of a huge number of different Universes, which is inconsistent with the idea of Universe. Second, it would seem to violate the conservation of energy, since the realization of each eigenfunction requires energy. We believe that this hypothesis is based on a misunderstanding of the relationship of mathematical formalism to the real world. Mathematical formalism defines possibilities. Only observation yields realities.
Quantum mechanics enjoys a well established reputation for counter-intuitivity. Much of this reputation arises from the contrast between the probabilistic world of quantum mechanics and the deterministic 'billiard ball' world of classical physics. If we think of quantum mechanics as describing a communication process, however, things become clearer. We are natural communicators and and so have intuitive feeling for communication processes.
Since the whole world is a quantum world, quantum measurement involves the communication of two quantum systems, information passing between them. Error free communication requires that this communication be quantized. There are two reason for this. First, as Zurek points out, the quantum formalism demands that the communicating systems share a common sert of basis states if they are to share information. Zurek
Second, the mathematical theory of communication discovered by Shannon also requires quantization for the error free transmission of information. Error in communication occurs when different messages are confused. Shannon realized that to prevent confusion, messages should be placed as far apart in 'message space' as possible. In other words they should be as distinct or quantized. Claude E Shannon
From the communication point of view, each quantum system is a source. Each source has a source alphabet, the set of letters which the system can emit. In the case of quantum systems, the source alphabet is the set of eigenvalues corresponding to the shared eigenfunctions of the act of communication. Communication theory computes the 'source entropy' from the probabilities of emission of the individual letters. The sum of these probabilities, for both a quantum system and a communication source is normalized to one. In quantum mechanics, this normalization is enforced by the unitarity of quantum operators.
From the communication point of view, therefore, the superposition of eigenvalues does not exist simultaneously, in this or any other Universe, even though the mathematical formalism may seem to imply this. Instead, the set of eigenvalues is simply the set of possible letters to be emitted by the source. Since we are sources, we may imagine the set of eigenvalues as the set of all the things we could say. Even a lifetime of conversation is not enough to say everything that one could say.
Neither communication theory nor quantum mechanics is concerned with the meaning of the messages emitted by their sources, only with the probabilitioes of the letters from the source alphabet. Although the emission of letters appears to be random, this does not mean that the messages emitted by sources are meaningless. A further consequence of Shannon's theory is that an ideally coded message is indistinguishable from a random sequence, since such a sequence has the maximum entropy and the key to reducing errors in communication is to code messages so as to maximize the size of the message space, that is to maximize its entropy.
We can picture our world as sequences of events each of which can be decomposed into smaller events. Thus a party is made from the interactions of a roomful of people, talking, eating, drinking and so on. A particular conversation may be broken down into a series of utterances, each of which is an event. These events, though distinct, weld into a seamless whole which we call a party. A party itself is part of life, and each of our lives is part of the history of the human race, and so on.
Quantum physics reveals that there exist 'smallest' or elementary events, whose size is measured by Max Planck's 'quantum of action'. Such an event is the emission or absorption of a photon from an atom, or the the change of an electron spin from 'up' to 'down'. Planck's constant is extremely small from our point of view, so that even the simplest event in human space comprises a huge number of elementary events. Quantum - Wikipedia
Quantum mechanics assigns to each such event a Hilbert space, in which the event is represented by the transformation of a vector by an operator. An example is the transformation of a vector representing an electron with spin 'up' to a vector representing an electron with spin 'down'. More complex events are modelled in more complex Hilbert spaces. Operator (physics) - Wikipedia
We imagine that the total transformation of the Universe embraces all elementary events through all time. The evidence we have to date suggests that the Universe, and hence the transformation of the Universe, has no end. The total transformation of the Universe is an infinite open set of events. The Hilbert space of the transformation of the Universe has a similar size to Cantors spasce pf transfinite numbers.
We may imagine the quantum transformations of the Universe as a hidden process (like our unconscious mind) which manipulates the probabilities of various observable events. We only see the results of this manipulation as they are communicated to us by events in the Universe, that is messages. We learn to see this spiritual background to physical events by decoding the messages we receive from the Universe, the totality of our experience. This process of exegesis we call learning and its resulting vision knowledge.
Since we are working on the hypothesis that the Universe is divine, every message we receive is a revelation from God, and all human experience is input to an evidence based scientific theology.
(revised 23 May 2013)
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|Bell, John S, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press 1987 Jacket: JB ... is particularly famous for his discovery of a crucial difference between the predictions of conventional quantum mechanics and the implications of local causality ... This work has played a major role in the development of our current understanding of the profound nature of quantum concepts and of the fundamental limitations they impose on the applicability of classical ideas of space, time and locality.
|de Witt, Bryce S and Neill Graham (eds) , and Hugh Everett III, J A Wheeler, B S DeWitt, L N Cooper, D van Vechten, N Graham (contributors), The Many-Worlds Interpretation of Quantum Mechanics, Princeton UP 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 is developed. The volume contains Dr Everett's short paper from 1957, "'Relative 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 De Witt, Graham and Cooper and van Vechtem provide further dicussion of the same theme. Together they constitute virtually the entire world output of scholarly commentary on the Everett interpretation.'
|Deutsch, David, The Fabric of Reality: The Science of Parallel Universes - and its Implications, Allen Lane Penguin Press 1997 Jacket: 'Quantum physics, evolution, computation and knowledge - these four strands of scientific theory and philosophy have, until now, remained incomplete explanations of the way the universe works. ... Oxford scholar DD shows how they are so closely intertwined that we cannot properly understand any one of them without reference to the other three. ...'
|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.'
|Hofstadter, Douglas R, Goedel Escher Bach: An Eternal Golden Braid, Basic/Harvester 1979 An illustrated essay on the philosophy of mathematics. Formal systems, recursion, self reference and meaning explored with a dazzling array of examples in music, dialogue, text and graphics.
|Holland, Peter R, The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics, Cambridge University Press 1993 Jacket: 'This book presents the first comprehensive exposition of the interpretation of quantum mechancs pioneered by Louis de Broglie and David Bohm. ... Developing the theme that a material system such as an electron is guided by a surrounding quantum wave, a detailed examination of the classic phenomena of quantum theory is presented ... . ... The theory provides a novel and satisfactory framework for analysing the classical limit of quantum mechanics and Heisenberg's relations, and implies a theory of measurement without wavefunction collapse. It also suggests a strikingly novel view of relativistic quantum theory, including the Dirac equation, quantum field theory and the wavefunction of the universe.'
|Smolin, Lee, The Life of the Cosmos, Oxford University Pres 1997 Jacket: 'Smolin posits that a process of self-organisation like that of biological evolution shapes the universe, as it develops and eventually reproduces through black holes, each of which may result in a big bang and a new universe. Natural selection may guide the appearance of the laws of physics, favouring those universes which best reproduce. . . . Smolin is one of the leading cosmologists at work today, and he writes with an expertise and a force of argument that will command attention throughout the world of physics.'
|Zurek, Wojciech Hubert, "Quantum origin of quantum jumps: Breaking of unitary symmetry induced by information transfer in the transition from quantum to classical", Physical Review A, 76, 5, 16 November 2007, page 052110-1--5. Abstract: 'Measurements transfer information about a system to the apparatus and then, further on, to observers and (often inadvertently) to the environment. I show that even imperfect copying essential in such situations restricts possible unperturbed outcomes to an orthogonal subset of all possible states of the system, thus breaking the unitary symmetry of its Hilbert space implied by the quantum superposition principle. Preferred outcome states emerge as a result. They provide a framework for 'wave-packet collapse', designating terminal points of quantum jumps and defining the measured observable by specifying its eigenstates. In quantum Darwinism, they are the progenitors of multiple copies spread throughout the environment — the fittest quantum states that not only survive decoherence, but subvert the environment into carrying information about them — into becoming a witness.'. back |
|Bell's theorem - Wikipedia Bell's theorem - Wikipedia 'In theoretical physics, Bell's theorem (a.k.a. Bell's inequality) is a no-go theorem, loosely stating that:
No physical theory of local hidden variables can reproduce all of the predictions of quantum mechanics.
. . .
What is powerful about Bell's theorem is that it doesn't refer to any particular physical theory. What makes Bell's theorem unique and powerful is that it shows that nature violates the most general assumptions behind classical pictures, not just details of some particular models. No combination of local deterministic and local random variables can reproduce the phenomena predicted by quantum mechanics and repeatedly observed in experiments.' back |
|Born rule - Wikipedia Born rule - Wikipedia, the free encyclopedia 'The Born rule (also called the Born law, Born's rule, or Born's law) is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of the Copenhagen interpretation of quantum mechanics. There have been many attempts to derive the Born rule from the other assumptions of quantum mechanics, with inconclusive results. . . . The Born rule states that if an observable corresponding to a Hermitian operator A with discrete spectrum is measured in a system with normalized wave function (see Bra-ket notation), then
the measured result will be one of the eigenvalues λ of A, and
the probability of measuring a given eigenvalue λi will equal <psi,|Pi|psi> where Pi is the projection onto the eigenspace of A corresponding to λi'. back |
|Claude E Shannon A Mathematical Theory of Communication 'The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages.' back |
|Eigenfunction - Wikipedia Eigenfunction - Wikipedia, the free encyclopedia 'In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has Af =
λf for some scalar, λ, the corresponding eigenvalue.' back |
|Operator (physics) - Wikipedia Operator (physics) - Wikipedia, the free encyclopedia ' . . . The mathematical description of quantum mechanics is built upon the concept of an operator.
Physical pure states in quantum mechanics are unit-norm vectors in a certain vector space (a Hilbert space). Time evolution in this vector space is given by the application of a certain operator, called the evolution operator. Since the norm of the physical state should stay fixed, the evolution operator should be unitary. Any other symmetry, mapping a physical state into another, should keep this restriction.
Any observable, i.e., any quantity which can be measured in a physical experiment, should be associated with a self-adjoint linear operator. The operators must yield real eigenvalues, since they are values which may come up as the result of the experiment. Mathematically this means the operators must be Hermitian . . . ' back |
|Quantum - Wikipedia Quantum - Wikipedia, the free encyclopedia 'In physics, a quantum (plural: quanta) is an indivisible entity of a quantity that has the same units as the Planck constant and is related to both energy and momentum of elementary particles of matter (called fermions) and of photons and other bosons. The word comes from the Latin "quantus," for "how much." Behind this, one finds the fundamental notion that a physical property may be "quantized", referred to as "quantization". This means that the magnitude can take on only certain discrete numerical values, rather than any value, at least within a range.' back |
|Schrödinger equation - Wikipedia Schrödinger equation - Wikipedia, the free encyclopedia 'In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1926, describes the space- and time-dependence of quantum mechanical systems. It is of central importance in non-relativistic quantum mechanics, playing a role for microscopic particles analogous to Newton's second law in classical mechanics for macroscopic particles. Microscopic particles include elementary particles, such as electrons, as well as systems of particles, such as atomic nuclei.' back |
|Wave function collapse - Wikipedia Wave function collapse - Wikipedia, the free encyclopedia 'In quantum mechanics, wave function collapse (also called collapse of the state vector or reduction of the wave packet) is the phenomenon in which a wave function—initially in a superposition of several different possible eigenstates—appears to reduce to a single one of those states after interaction with an observer. In simplified terms, it is the reduction of the physical possibilities into a single possibility as seen by an observer.' back |
|Wojciech Hubert Zurek Quantum origin of quantum jumps: breaking of unitary symmetry induced by information transfer and the transition from quantum to classical 'Submitted on 17 Mar 2007 (v1), last revised 18 Mar 2008 (this version, v3))
"Measurements transfer information about a system to the apparatus, and then further on -- to observers and (often inadvertently) to the environment. I show that even imperfect copying essential in such situations restricts possible unperturbed outcomes to an orthogonal subset of all possible states of the system, thus breaking the unitary symmetry of its Hilbert space implied by the quantum superposition principle. Preferred outcome states emerge as a result. They provide framework for the ``wavepacket collapse'', designating terminal points of quantum jumps, and defining the measured observable by specifying its eigenstates. In quantum Darwinism, they are the progenitors of multiple copies spread throughout the environment -- the fittest quantum states that not only survive decoherence, but subvert it into carrying information about them -- into becoming a witness.' back |