This site documents the Divine Mind:
the Universe that enables us to create ourselves
page 14: Measurement—the interface between the Minkowski and Hilbert spaces
Table of contents
14.1: The interface between the Minkowski and Hilbert spaces
14.2: Are we dealing with two particles and communications between them?
14.3: Or does every quantum event split the universe in two?
14.4: Zurek: observations involve interactions in Hilbert space
14.5: The role of observers in classical physics
14.6: The role of observers in quantum physics
14.7: Measurement and creation
14.1: The interface between the Minkowski and Hilbert spaces
We live in classical Minkowski spacetime whose properties were established by Einstein's special theory of relativity. This space is descended from the intuitively obvious space and time studied by Isaac Newton. Newton saw that we move around in a three dimensional Euclidean space and the durations of events are measured by an independent and unstoppable flow of time. While we can freely move in any of the three directions in space, our motion in time moves inevitably from past to future. Many people have imagined time machines to enable us to move back in time and recover our lost youth but such morion seems to be impossible in Minkowski space and we inevitably age. Special relativity - Wikipedia
Another serious problem with Newton's picture of the world is that although he saw that motion obeys three simple laws, there does not seem to be any explanation of these laws. His model of the solar system worked beautifully but he was not happy with the need to postulate action at a distance to connect the Sun, planets and moons together.
People had been aware of static electricity and the use of natural magnetism for navigation since antiquity but it was not until Volta made a practical battery that electric current became available for experiments. Ampere, Faraday, Maxwell and many others found the relationship between electric current and magnetism and in less than a century engineers exploited electrodynamics to produce a new energy economy. Faraday's notion of field made action at a distance more intelligible and Maxwell and Hertz showed that light is an electromagnetic phenomenon.
Newton found that white light is the superposition of a spectrum of colours and spectroscopes were developed to measure the precise frequencies of electromagnetic radiation. Kirchoff postulated a relationship between frequency and temperature which led Planck to make the first step toward quantum mechanics. Kirchoff's law of thermal radiation - Wikipedia, Planck's Law - Wikipedia
Niels Bohr took the first step toward explaining atomic spectra using the electronic structure of atoms and by the late 1920s quantum theory had advanced to a point where physicists could make relatively precise calculations of the relationships between electronic structure and radiation frequencies. By the 1960s the development of quantum field theory and computers meant that the measured energies (eigenvalues of many atomic events and their observed probabilities of occurrence could be computed with considerable accuracy. The statistical element in this model upset people like Einstein who thought that physical processes should be deterministic. He felt that the statistical element in quantum mechanics was hiding a failure to understand what was really happening. One expression of his dissatisfaction with quantum mechanics is his 1933 Herbert Spencer lecture. The aura of uncertainty around quantum mechanics arising from the relationship between the invisible processes in Hilbert space and the visible process in Minkowski space became a perennial issue in quantum theory often known as the measurement problem. Measurement problem - Wikipedia, Albert Einstein (1933): On the Method of Theoretical Physics: Herbert Spencer Lecture 1933
The Hilbert space representation of a quantum state is a vector which may be the sum of the orthonormal basis states corresponding to the dimension of a Hilbert space. The principal difference between Hilbert spaces is their number of dimensions. Quantum mechanics, like the theory of networks, is symmetrical with respect to complexity. Both are built from an atom: in quantum mechanics the atomic operator is the quantum of action, the operation involved where two quantum states interact such as the absorption or emission of a photon by an atom. In the theory of computation we have the not-and or Sheffer stroke operator. Multiple instances of these operators, properly connected to a suitable memory, can perform all the functions needed to execute both quantum and classical computations. Orthonormal basis - Wikipedia, Nielsen & Chuang (2016): Quantum Computation and Quantum Information, Odysseus Makridis : The Sheffer Stroke
Our conjectures about this hidden quantum mechanical structure are based on observing the interactions of observable particles. One of these particles may be prepared in a particular state, to some extent known, which serves as the measurement operator. We arrange for it to interact with a particle in an unknown state. The information we want is carried by the resulting particle(s) which embody the result of our measurement. What we see are eigenvalues, for instance frequencies or energies, which correspond to the eigenfunctions of the operator we use for our measurement.
The theory predicts that there are as many possible eigenvalues as the dimension of the measurement operator. The terms collapse or reduction of the wave function refer to the fact that individual observations only ever reveal just one of the possible states of an unknown system. In this respect, a quantum measurement is equivalent to the emission of one symbol from a communication source. The spectrum of the measurement operator corresponds to the alphabet of the source.
The radical problem facing our understanding of quantum mechanics and the development of quantum computation is illustrated by the difference between a classical bit (binary digit) and its quantum analogue, the qubit. A classical bit has just two states, usually represented by 0 and 1. These states are orthogonal, one is not the other. A qubit on the other hand is a vector formed in a two dimensional Hilbert space by adding the orthogonal basis states | 0⟩ and |1⟩ and normalizing the result to preserve the probability structure of a quantum source.
This vector ia understood to have a continuous spectrum of states represented by the equation |qubit⟩ = a| 0⟩ + b|1⟩, where a and b are complex numbers subject to the constraint that |a|2 + |b|2 = 1. When we observe a qubit, however, all we ever see is | 0⟩ or |1⟩ with relative frequency P( | 0⟩ ) = |a|2, P( |1⟩ ) = |b|2. The infinite amount of information which we suppose to have been represented by the qubit turns out to be at best just 1 classical bit. It has collapsed. Designers of quantum computations must try to devise some way to take advantage of this (allegedly) hidden information. Nielsen and Chuang write:
Understanding the hidden quantum information is a question that we grapple with for much of this book and which lies at the heart of what makes quantum mechanics a powerful tool for information processing. Qubit - Wikipedia, Nielsen & Chuang (2000): Quantum Computation and Quantum Information, page 16
The essence of the historical measurement problem, sometimes called the reduction or the collapse of the wave function, is why actual measurements only yield one definite result and ignore the wide spectrum of states presumed to be represented by the standard quantum mechanical formalism.
14.2: Are we dealing with two elementary particles and communications between them?
We are quite familiar with the Newtonian space in which we live. We know that it is filled with tiny elementary particles that bind to one another to produce everything that we see from ants to the Earth, stars supernovas and galaxies. Newtonian physics takes care of most of our day to day dynamical problems at the human scale. Special and general relativity extend our understanding to Minkowski space where speeds approach the velocity of light and gravitation determines the large scale structure of the Universe. Minkowski space - Wikipedia, Elementary particle - Wikipedia, Theory of relativity - Wikipedia
Behind all this, explaining its behaviour, are natural quantum computations in Hilbert space, invisible but quite well known after more than a century of intense study. Engineering applications of quantum theory are now ubiquitous in modern technology.
Measurement in a general sense is everywhere and is closely related to the encoding and decoding of information discussed on page 11: Quantization: the mathematical theory of communication. It forms the theoretical foundation of all the arts and sciences. It may be simple, like using a ruler to discover that I am 1580 mm tall, or very complex, like the transformation of something we have seen into a spoken description. The inverse transcription, from natural language to kinematic imagery is now the subject of intense research in artificial intelligence. Artificial intelligence - Wikipedia, Open AI (2024_02_15)
Some of it is clear and definitive, like the encoding of the measurements and specifications of an industrial artefact. Some of it involves personal and aesthetic judgements that are not so well defined, like books intended to represent the rise and fall of empires or novels, operas, movies and songs intended to capture the actions and feelings of networks of interacting people.
Every measurement involves a transformation between two spaces. Shakespeare wrote his play Hamlet by creatively transforming the space of historical information available to him. Since it was written, his play has been transformed by thousands of different producers, theatre companies, composers, and cinematographers. It has fed into the imaginations of millions of people. Royal Shakespeare Company: A 400 Year Stage History
Some have thought there is human psychologial involvement in quantum measurement, but this seems unnecessary. From the point of view of this site special operations carried out under laboratory conditions to study quantum behaviour are no exception to the general process of quantum interaction. Quantum measurement is ubiquitous. It is the foundation of the world occurring continually at the simplest ontological level, a step beyond the initial singularity. At that level it is the process by which the Universe communicates with itself. Every communication between discrete particles is a measurement in which they exchange information. In contrast to some of the implications of field theory, this site takes the view that quantum communications are one-on-one events quite similar to the conversations between individual living entities like people and animals.
14.3: Or does every quantum event split the universe in two?
A most intriguing (and probably false) response to the quantum measurement problem was devised by Hugh Everett III in his 1957 PhD thesis. It attracted authors like David Deutsch who used Everett's idea to build a theory of everything. Bruce de Witt (1973) et. al: The Many-Worlds Interpretation of Quantum Mechanics, David Deutsch (1997): The Fabric of Reality: The Science of Parallel Universes - and its Implications
Everett's was seeking a quantum theory of gravitation. This raises a problem which faced by Einstein. The geometry of gravitation is Gaussian since, like Gauss's Theorema egregium, it describes the shape of spacetime by its intrinsic properties with no external reference system. Newton, on the other hand, took a God's eye view of the solar system, looking at it from the outside. Everett's idea is that there is an unresolved paradox in the measurement process. The conventional or external observation formulation of quantum mechanics assumes that the observer and the system observed are independent parts of the Universe and the observer's activity is confined to their particular experiment or experience. On the other hand, if we imagine that there is just one wave function for the whole Universe, this cannot be externally observed, since the observers are part of the system that they are observing. Theorema Egregium - Wikipedia, Hugh Everett III (1957): "Relative State" Formulation of Quantum Mechanics
If one takes the mathematical formulation of quantum mechanics literally, the external observation approach may appear to make no sense. In the introduction to his 1957 PhD thesis, Everett outlines his problem:
The task of quantizing general relativity raises serious questions about the meaning of the present formulation and interpretation of quantum mechanics when applied to so fundamental a structure as the space-time geometry itself. This paper seeks to clarify the foundations of quantum mechanics. It presents a reformulation of quantum theory in a form believed suitable for application to general relativity.The aim is not to deny or contradict the conventional formulation of quantum theory, which has demonstrated its usefulness in an overwhelming variety of problems, but rather to supply a new, more general and complete formulation, from which the conventional interpretation can be deduced.
The relationship of this new formulation to the older formulation is therefore that of a metatheory to a theory, that is, it is an underlying theory in which the nature and consistency, as well as the realm of applicability, of the older theory can be investigated and clarified.
He continues: The special postulates in the old theory which deal with observation are omitted in the new theory. He begins with a succinct statement of the problematic duality of quantum mechanics:
There are two fundamentally different ways in which the state function can change:Process 1 : The discontinuous change brought about by the observation of a quantity with eigenstates Φ1, Φ2· · · , in which the state Ψ will be changed to the state Φj with probability |(Ψ, Φj)|2.
Process 2 : The continuous, deterministic change of state of an isolated system with time according to a wave equation ∂Ψ/∂t = AΨ, where A is a linear operator.
This formulation describes a wealth of experience. No experimental evidence is known which contradicts it. . . .
How is one to apply the conventional formulation of quantum mechanics to the space-time geometry itself? The issue becomes especially acute in the case of a closed universe. There is no place to stand outside the system to observe it.
Everett's answer, the many worlds hypothesis, is to devise a relative state interpretation for quantum mechanics that does not require the conventional observation process and the rejection of all but one of the eigenstates Φj. To achieve this he postulates that every observation creates multiple branches of the world. One branch is our world in which a particular observer sees what they see, and there is a infinity of other unique worlds where other instances of the same observer and the same event realize the other outcomes of that observation implicit in the uncollapsed wave function. The wave function does not collapse; all the predicted outcomes are realized, but in different universes. This process, it would seem, is recursive, so every observation in each of the infinity of other worlds also creates an infinity of other worlds. We might ask what happens to the pinciple of conservation of energy in this case where every event involving a single quantum of action creates a whole new Universe!
Since the infinity of universes created by each of the enormous number of quantum observations occurring at every moment is invisible, we will return to this subject on page 15: Quantum amplitudes and logical processes are invisible
The hypothesis of invisible universes takes this interpretation out of the realm of science. Here I prefer a more conventional approach which treats quantum events as instances of communication. Shannon's mathematical definition of a communication source has statistical properties that demand unitarity in the evolution of quantum systems, but it does not demand that a source emit all possible signals in its repertoire at any given moment. Instead, symbols are emitted one at a time in sequence as we do in normal speech. This points to a grain of truth in Everett's idea. When we (or fundamental particles) are talking to one another we may think of each of our minds as a high dimensional Hilbert space from which each of us is choosing a response to what the other says.
This seems to be where Einstein missed the point of quantum theory. The principle of general covariance that helped him reach his field equation is based on the idea that observation has no influence on the system observed. This is not the case in quantum theory. The theory describes conversation, not dictation. The observed system does not dictate to the observer, it also listens. (See page 11: Quantization: the mathematical theory of communication). General covariance - Wikipedia
14.4: Zurek: observations involve interactions in Hilbert space
Zurek's suggests that the alleged collapse of the wave function is a necessary consequence of the transmission of information between two quantum systems. The distinction between observer and observed is fictitious, in the sense that a quantum process is simply the communication channel in Hilbert space between two sources in Minkowski space. The mathematical theory of communication treats the space of all possible communications between two sources but its results apply to each particular communication. The mathematical expression of quantum mechanics may work in the same way. Wojciech Hubert Zurek (2008): Quantum origin of quantum jumps: breaking of unitary symmetry induced by information transfer and the transition from quantum to classical
We often think of a measurement as an interaction between a classical and a quantum system, but in reality it is the interaction of two quantum systems. One classical system is the source of a state which we call the measurement operator. The operator interacts with an unknown state attached to another classical system, yielding a classically observable result, the outcome of this interaction. A measurement interrupts an isolated system by injecting another process, represented by a measurement operator, into the isolated system. This is analogous to one person interrupting another by starting a conversation.
Zurek begins with a concise definition of standard quantum mechanics in six propositions. The first three describe its mathematical mechanism:
(1) the quantum state of a system is represented by a vector in its Hilbert space;
(2) a complex system is represented by a vector in the tensor product of the Hilbert spaces of the constituent systems;
(3) the evolution of isolated quantum systems is unitary, governed by the Schrödinger equation:
i ℏ ∂|ψ⟩ / ∂t = H |ψ⟩ where H is the energy (or Hamiltonian) operator.
The other three show how the mathematical formalism in Hilbert space couples to the observed world:
(4) immediate repetition of a measurement yields the same outcome;
(5) measurement outcomes are restricted to an orthonormal set { | sk ⟩ } of eigenstates of the measured observable [ie the measurement operator associated with the classical measuring system];
(6) the probability of finding a given outcome is pk = |⟨sk||ψ⟩|2, where |ψ⟩; is the preexisting state of the [measured] system.
Schrödinger equation - Wikipedia, Born rule - Wikipedia
Zurek writes:
The aim of this paper is to point out that already the (symmetric and uncontroversial) postulates (1) - (3) necessarily imply selection of some preferred set of orthogonal states – that they impose the broken symmetry that is at the heart of the collapse postulate (4).
Zurek examines a system in 2D Hilbert space, noting that the complexity invariance of quantum mechanics enables an extension of the argument to a space of any dimension.
In the Hilbert space HS the unknown state vector |ψS⟩ is the superposition of two states: |ψS⟩ = α|v⟩ + β|w⟩.
The measurement operator A0, embodied in some apparatus, measures state S:
|ψS⟩|A0⟩ = (α|v⟩ + β|w⟩)A0 = α|v⟩|Av⟩ + β|w⟩|Aw⟩ = |ΦSA⟩.
|ΦSA⟩ is now a vector in the tensor product of the observing and observed Hilbert spaces and holds the information transferred from S to A: the state of A now contains a record of S.
The composite system is normed and linear, so ⟨A0||A0⟩ = ⟨Av||Av⟩ = ⟨Aw||Aw⟩ = 1.
So
⟨ψS||ψS⟩ - ⟨ΦSA||ΦSA⟩ = 2Rα*β<v||w⟩ (1 - ⟨Av||Aw⟩ = 0.
so ⟨v||w⟩ (1 - ⟨Av||Aw⟩ = 0.
then if ⟨v||w⟩ ≠ 0, information transfer must have failed since
⟨Av||Aw⟩ = 1, so 1 - ⟨Av||Aw⟩ = 0.
or else
⟨v||w⟩ = 0 so ⟨Av||Aw⟩ may have any value. So, to give a result, the basis vectors |v⟩, |w⟩ of the unknown state must be orthogonal.
Conclusion: 'The overlap ⟨v||w⟩ must be exactly 0 for ⟨Av||Aw⟩ to differ from unity.
He concludes:
Selection of an orthonormal basis induced by information transfer – the need for spontaneous symmetry breaking that arises from the unitary axioms of quantum mechanics (i, iii) is a general and intriguing result.
From this point of view, the so called collapse of the waves function is a form of quantum natural selection which picks out a visible state <v||w> = 0 with an orthogonal basis from a set of unknown states. It establishes that a classical communication source, the output of a measurement, only emits one symbol at a time.
14.5: The role of observers in classical physics . . .
Einstein radically revised classical physics with his theories of special and general relativity. His work struck deeper however, into the methodology of physics, summed up in principles of covariance. The core idea is that the Universe is the same, regardless of the state of motion of any observer. The transformation of the observed reality to what the observer actually sees must therefore be a function of the relationship between the observer and the observed system. When everything is moving inertially the Lorentz transformation enables each observer to transform what they see in another frame to what it would look like in their own frame and vice versa. Covariance and contravariance of vectors - Wikipedia.
The general problem involving accelerated motion is more complex. Unlike Newton, Einstein was working inside the Universe, one of an immense number of different points of view. In order to get an arithmetic grip on the geometry of nature Einstein used Gaussian coordinates which map real numbers to geometric points. Unlike Cartesian coordinates, however, Gaussian coordinates describe topological spaces which may be bent and stretched as long as they are not torn. The Gauss co-ordinate system has to take the place of a body of reference. The following statement corresponds to the fundamental idea of the general principle of relativity: "All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature". Gaussian curvature - Wikipedia, Albert Einstein (1916, 2005): Relativity: The Special and General Theory, page 123
The key to getting a deterministic mathematical theory out of this somewhat arbitrary coordinate system is that the only observable points in nature are events and the space-time intervals between them. Whatever coordinate systems we choose must be constrained to give a one to one correspondence between identical spacetime intervals and identical differences in Gaussian coordinates. An observation is an event, so that the foundation of classical science is equivalent to the foundation of general relativity: all observers, no matter what their perspective must, to make sense, agree on what they actually see transformed to their own rest frames. Einstein exploited the freedom of the Gaussian coordinate system to establish the relationship between energy and spacetime distance to determine the large scale structure of the Universe. He expressed this as a field equation which connects every point in the Universe to its neighbours by contact, without an action at a distance. Hawking & Ellis (1975): The Large Scale Structure of Space-Time, John Baez & Emory Bunn: The Meaning of Einstein's Equation
14.6: The role of observers in quantum physics
The difference between classical and quantum physics is that all classical phenomena are considered to be completely independent of the fact that they are being observed. This is the foundation of the idea of general covariance which Einstein used to derive his field equation. Gravitation is not linear, however, because the gravitational field carries energy, and therefore acts upon itself. We might say that it observes itself. It is thus in a sense conscious as explained on page 17: Gravitation and quantum theory—in the beginning where we will look at both gravitation and quantum theory as instances of the world communicating with itself, something akin to the mental phenomenon of consciousness.
If we think of Einstein's general covariance in human terms, it is very like dictation. I dictate and you write, and you are not permitted to talk back to me. The quantum world is much more natural. It involves conversation. Every communication is a meeting. There are always two actors and they are changed by the meeting. Because there are two basically uncorrelated actors the outcome of every conversation has a random element as quantum mechanics demonstrates.
A successful meeting occurs when people understand one another. In quantum mechanical terms this means sharing an eigenvector as Zurek shows above. Although both the state vectors in a quantum meeting may be the superposition of a large number of basis vectors, information is only transferred when the same state is shared by both observer and observed.
The metric in Hilbert space measures the distance between states. The Born rule shows that the probability of observing states that are close together is higher than the probability for states that are far apart. In quantum mechanics the distance between states is not a matter of space but of angle or phase.
It is often thought that the evolution of unobserved quantum systems is deterministic but this is not the case in the same way as it is in the theory of communication. There the encoding and decoding of a message is required to yield an output identical to the original input.
In the unobserved quantum world, however, the implicit unitarity imposed by the wave equation ensures that a vector as it evolves remains a complete set of events in the probabilistic sense, but does not demand that the sequence of events retains its exact identity as the theory of communication requires. All that is required is that only one of the possible events happens at a time and the sum of their probabilities remains 1. Because the quantum system is in perpetual motion, we can at best estimate a probability that the observer and the observed system will share a particular state at the moment of observation. This is what the Born rule predicts.
14.7: Measurement and creation
Von Neumann shows that quantum mechanical measurement creates entropy. This may seem counterintuitive: the alleged annihilation of quantum states implicit in measurement process would seem to decrease the entropy of the system. Nevertheless observation leads to the selection of a real state, the outcome of the measurement. John von Neumann (2014): Mathematical Foundations of Quantum Mechanics, Chapter V §3 Reversibility and Equilibrium Problems
Everywhere, the Universe is measuring itself, and at the most basic level this is happening at the interface between quantum mechanics and spacetime, a conversation between the invisible world of Hilbert space and the visible world of Minkowski space.
The spacetime in which we live acts as our interface with the quantum world. Every move we make sends signals to this invisible world for processing and the answer comes back to us as the results of our actions. Our physical actions follow a similar cycle. The potential to move my finger arises in my mind which is a complex information processing system whose every move is coupled to moves in my observable body. We will discuss this in terms of visibility and invisibility on the next page.
(Revised Wednesday 7 August 2024)
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Notes and references
Further reading
Books
de Witt (1973), Bryce S., and Neill Graham, Hugh Everett III, J. A. Wheeler, L. N. Cooper, D. van Vechten, N. Graham (contributors and editors), 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 discussion of the same theme. Together they constitute virtually the entire world output of scholarly commentary on the Everett interpretation.'
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Deutsch (1997), 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. . . .'
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Einstein (1916, 2005), 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 1916, 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 simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated.' page 3
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Einstein (1916, 2005), 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 1916, 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 simplest and most intelligible form, and on the whole, in the sequence and connection in which they actually originated.' page 3
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Hawking (1975), Steven W, and G F R Ellis, The Large Scale Structure of Space-Time, Cambridge UP 1975 Preface: Einstein's General Theory of Relativity . . . leads to two remarkable predictions about the universe: first that the final fate of massive stars is to collapse behind an event horizon to form a 'black hole' which will contain a singularity; and secondly that there is a singularity in our past which constitutes, in some sense, a beginning to our universe. Our discussion is principally aimed at developing these two results.'
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Nielsen (2000), Michael A, and Isaac L Chuang, Quantum Computation and Quantum Information, Cambridge University Press 2000 Review: A rigorous, comprehensive text on quantum information is timely. The study of quantum information and computation represents a particularly direct route to understanding quantum mechanics. Unlike the traditional route to quantum mechanics via Schroedinger's equation and the hydrogen atom, the study of quantum information requires no calculus, merely a knowledge of complex numbers and matrix multiplication. In addition, quantum information processing gives direct access to the traditionally advanced topics of measurement of quantum systems and decoherence.' Seth Lloyd, Department of Quantum Mechanical Engineering, MIT, Nature 6876: vol 416 page 19, 7 March 2002.
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Nielsen (2016), Michael A., and Isaac L Chuang, Quantum Computation and Quantum Information, Cambridge University Press 2016 Review: A rigorous, comprehensive text on quantum information is timely. The study of quantum information and computation represents a particularly direct route to understanding quantum mechanics. Unlike the traditional route to quantum mechanics via Schroedinger's equation and the hydrogen atom, the study of quantum information requires no calculus, merely a knowledge of complex numbers and matrix multiplication. In addition, quantum information processing gives direct access to the traditionally advanced topics of measurement of quantum systems and decoherence.' Seth Lloyd, Department of Quantum Mechanical Engineering, MIT, Nature 6876: vol 416 page 19, 7 March 2002.
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Links
Albert Einstein (1933), On the Method of Theoretical Physics: Herbert Spencer Lecture 1933, ' It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience. back |
Artificial intelligence - Wikipedia, Artificial intelligence - Wikipedia, the free encyclopedia, ' Artificial intelligence (AI) is the intelligence of machines or software, as opposed to the intelligence of living beings, primarily of humans. It is a field of study in computer science that develops and studies intelligent machines. Such machines may be called AIs. AI technology is widely used throughout industry, government, and science. Some high-profile applications are: advanced web search engines (e.g., Google Search), recommendation systems (used by YouTube, Amazon, and Netflix), interacting via human speech (such as Google Assistant, Siri, and Alexa), self-driving cars (e.g., Waymo), generative and creative tools (ChatGPT and AI art), and superhuman play and analysis in strategy games (such as chess and Go).' 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 <ψ|Pi|ψ> where Pi is the projection onto the eigenspace of A corresponding to λi'.' back |
Claude Shannon (1949), Communication in the Presence of Noise, '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 at this maximum rate are discussed. The equivalent number of binary digits per second for certain information sources is calculated.' [C. E. Shannon , “Communication in the presence of noise,” Proc. IRE, vol. 37, pp. 10–21, Jan. 1949.] back |
Codec - Wikipedia, Codec - Wikipedia, the free encyclopedia, 'A codec is a device or computer program that encodes or decodes a data stream or signal. Codec is a portmanteau of coder/decoder. . . . IA coder or encoder encodes a data stream or a signal for transmission or storage, possibly in encrypted form, and the decoder function reverses the encoding for playback or editing. Codecs are used in videoconferencing, streaming media, and video editing applications. In the mid-20th century, a codec was a device that coded analog signals into digital form using pulse-code modulation (PCM). Later, the name was also applied to software for converting between digital signal formats, including companding functions. ' back |
Covariance and contravariance of vectors - Wikipedia, Covariance and contravariance of vectors - Wikipedia, the free encyclopedia, 'In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. . . . Vectors exhibit this behavior of changing scale inversely to changes in scale to the reference axes: they are contravariant.. . . The components of dual vectors change in the same way as changes to scale of the reference axes: they are covariant.' back |
Electric current - Wikipedia, Electric current - Wikipedia, the free encyclopedia, ' An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. The moving particles are called charge carriers, which may be one of several types of particles, depending on the conductor. In electric circuits the charge carriers are often electrons moving through a wire. In semiconductors they can be electrons or holes. In an electrolyte the charge carriers are ions, while in plasma, an ionized gas, they are ions and electrons. back |
Elementary particle - Wikipedia, Elementary particle - Wikipedia, the free encyclopedia, ' In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include the fundamental fermions (quarks, leptons, antiquarks, and antileptons), which generally are "matter particles" and "antimatter particles", as well as the fundamental bosons (gauge bosons and the Higgs boson), which generally are "force particles" that mediate interactions among fermions. A particle containing two or more elementary particles is a composite particle.' back |
Gaussian curvature - Wikipedia, Gaussian curvature - Wikipedia, the free encyclopedia, ' In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way it is isometrically embedded in space. This result is the content of Gauss's Theorema egregium.' back |
General covariance - Wikipedia, General covariance - Wikipedia, the free encyclopedia, In theoretical physics, general covariance (also known as diffeomorphism covariance or general invariance) is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations. The essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws.' back |
Hugh Everett III (1957), "Relative State" Formulation of Quantum Mechanics, ' 1. Introduction The task of quantizing general relativity raises serious questions about the meaning of the present formulation and interpretation of quantum mechanics when applied to so fundamental a structure as the space-time geometry itself. This paper seeks to clarify the foundations of quantum mechanics. It presents a reformulation of quantum theory in a form believed suitable for application to general relativity. . . . The relationship of this new formulation to the older formulation is therefore that of a metatheory to a theory, that is, it is an underlying theory in which the nature and consistency, as well as the realm of applicability, of the older theory can be investigated and clarified.' back |
John Baez & Emory Bunn, The Meaning of Einstein's Equation , ' Abstract: This is a brief introduction to general relativity, designed for both students and teachers of the subject. While there are many excellent expositions of general relativity, few adequately explain the geometrical meaning of the basic equation of the theory: Einstein's equation. Here we give a simple formulation of this equation in terms of the motion of freely falling test particles. We also sketch some of the consequences of this formulation and explain how it is equivalent to the usual one in terms of tensors. Finally, we include an annotated bibliography of books, articles and websites suitable for the student of relativity.' back |
John Baez & Emory Bunn (2006), The Meaning of Einstein's Equation , ' Abstract: This is a brief introduction to general relativity, designed for both students and teachers of the subject. While there are many excellent expositions of general relativity, few adequately explain the geometrical meaning of the basic equation of the theory: Einstein's equation. Here we give a simple formulation of this equation in terms of the motion of freely falling test particles. We also sketch some of the consequences of this formulation and explain how it is equivalent to the usual one in terms of tensors. Finally, we include an annotated bibliography of books, articles and websites suitable for the student of relativity.' back |
John von Neumann (2014), Mathematical Foundations of Quantum Mechanics, ' Mathematical Foundations of Quantum Mechanics by John von Neumann translated from the German by Robert T. Beyer (New Edition) edited by Nicholas A. Wheeler. Princeton UP Princeton & Oxford. Preface: ' This book is the realization of my long-held intention to someday use the resources of TEX to produce a more easily read version of Robert T. Beyer’s authorized English translation (Princeton University Press, 1955) of John von Neumann’s classic Mathematische Grundlagen der Quantenmechanik (Springer, 1932).'This content downloaded from 129.127.145.240 on Sat, 30 May 2020 22:38:31 UTC back |
Kirchoff's law of thermal radiation - Wikipedia, Kirchoff's law of thermal radiation - Wikipedia, the free encyclopedia, 'Kirchhoff's law states that: For a body of any arbitrary material, emitting and absorbing thermal electromagnetic radiation at every wavelength in thermodynamic equilibrium, the ratio of its emissive power to its dimensionless coefficient of absorption is equal to a universal function only of radiative wavelength and temperature, the perfect black-body emissive power. back |
Measurement problem - Wikipedia, Measurement problem - Wikipedia, the free encyclopedia, ' In quantum mechanics, the measurement problem is the problem of definite outcomes: quantum systems have superpositions but quantum measurements only give one definite result. |
Minkowski space - Wikipedia, Minkowski space - Wikipedia, the free encyclopedia, ' By 1908 Minkowski realized that the special theory of relativity, introduced by his former student Albert Einstein in 1905 and based on the previous work of Lorentz and Poincaré, could best be understood in a four-dimensional space, since known as the "Minkowski spacetime", in which time and space are not separated entities but intermingled in a four-dimensional space–time, and in which the Lorentz geometry of special relativity can be effectively represented using the invariant interval x2 + y2 + z2 − c2 t2.' back |
Odysseus Makridis (Internet Encyclopedia of Philosophy), The Sheffer Stroke, ' The Sheffer Stroke is one of the sixteen definable binary connectives of standard propositional logic. The stroke symbol is “|” as in (p | q) ↔ (¬ p ∨ ¬ q) The linguistic expression whose logical behavior is presumed modeled by this logical connective is the truth-functional phrase “not both,” from which the name NAND originates. . . . The discovery of the Sheffer Stroke was achieved independently by Henry M. Sheffer in 1913 after it had been realized previously by Charles Sanders Peirce, as attested by a fragment written in 1880 (and, again, in 1902). This landmark discovery was hailed by such seminal figures in the history of logic as Ludwig Wittgenstein and Bertrand Russell.' back |
Open AI (2024_02_15), Video generation models as world simulators, ' We explore large-scale training of generative models on video data. Specifically, we train text-conditional diffusion models jointly on videos and images of variable durations, resolutions and aspect ratios. We leverage a transformer architecture that operates on spacetime patches of video and image latent codes. Our largest model, Sora, is capable of generating a minute of high fidelity video. Our results suggest that scaling video generation models is a promising path towards building general purpose simulators of the physical world.' back |
Orthonormal basis - Wikipedia, Orthonormal basis - Wikipedia, the free encyclopedia, 'In mathematics, an orthonormal basis of an inner product space V (i.e., a vector space with an inner product), or in particular of a Hilbert space H, is a set of elements whose span is dense in the space, in which the elements are mutually orthogonal and of magnitude 1. Elements in an orthogonal basis do not have to have a magnitude of 1 but must be mutually perpendicular. It is easy to change the vectors in an orthogonal basis by scalar multiples to get an orthonormal basis, and indeed this is a typical way that an orthonormal basis is constructed.' back |
Planck's Law - Wikipedia, Planck's Law - Wikipedia, the free encyclopedia, 'In physics, Planck's law describes the spectral radiance of electromagnetic radiation at all wavelengths from a black body at temperature T. As a function of frequency ν. back |
Qubit - Wikipedia, Qubit - Wikipedia, the free encyclopedia, 'A quantum bit, or qubit . . . is a unit of quantum information. That information is described by a state vector in a two-level quantum mechanical system which is formally equivalent to a two-dimensional vector space over the complex numbers. Benjamin Schumacher discovered a way of interpreting quantum states as information. He came up with a way of compressing the information in a state, and storing the information on a smaller number of states. This is now known as Schumacher compression. In the acknowledgments of his paper (Phys. Rev. A 51, 2738), Schumacher states that the term qubit was invented in jest, during his conversations with Bill Wootters.' back |
Royal Shakespeare Company, The stage history of Hamlet from the time Shakespeare wrote it to the present day., ' A 400 Year Stage History: Hamlet is the most complex and coveted role in classical theatre, attracting the leading actor of every age, and a few actresses as well, including a comically inventive Sarah Bernhardt, in the late nineteenth century, and Sarah Siddons, the great tragedienne of the late eighteenth century, who must be one of the few players to have tackled not only Hamlet but Ophelia, too. The play has rarely been off the stage throughout the 400 years since it was written and its stage history is accordingly extensive.' back |
Schrödinger equation - Wikipedia, Schrödinger equation - Wikipedia, the free encyclopedia, ' In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a quantum system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. . . . In classical mechanics Newton's second law, (F = ma), is used to mathematically predict what a given system will do at any time after a known initial condition. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system's wave function (also called a "state function").' back |
Special relativity - Wikipedia, Special relativity - Wikipedia, the free encyclopedia, ' Special relativity . . . is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein (after the considerable and independent contributions of Hendrik Lorentz, Henri Poincaré and others) in the paper "On the Electrodynamics of Moving Bodies". It generalizes Galileo's principle of relativity—that all uniform motion is relative, and that there is no absolute and well-defined state of rest (no privileged reference frames)—from mechanics to all the laws of physics, including both the laws of mechanics and of electrodynamics, whatever they may be. Special relativity incorporates the principle that the speed of light is the same for all inertial observers regardless of the state of motion of the source.' back |
Theorema Egregium - Wikpedia, Theorema Egregium - Wikipedia, the free encyclopedia, 'Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces. The theorem says that the Gaussian curvature of a surface can be determined entirely by measuring angles, distances and their rates on the surface itself, without further reference to the particular way in which the surface is embedded in the ambient 3-dimensional Euclidean space. Thus the Gaussian curvature is an intrinsic invariant of a surface.' back |
Theory of relativity - Wikipedia, Theory of relativity - Wikipedia, the free encyclopedia, ' The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in the absence of gravity. General relativity explains the law of gravitation and its relation to other forces of nature. It applies to the cosmological and astrophysical realm, including astronomy. The theory transformed theoretical physics and astronomy during the 20th century, superseding a 200-year-old theory of mechanics created primarily by Isaac Newton.' back |
Turing machine - Wikipedia, Turing machine - Wikipedia, the free encyclopedia, ' A Turing machine is a hypothetical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a computer. The "machine" was invented in 1936 by Alan Turingwho called it an "a-machine" (automatic machine). The Turing machine is not intended as practical computing technology, but rather as a hypothetical device representing a computing machine. Turing machines help computer scientists understand the limits of mechanical computation.' back |
Wojciech Hubert Zurek (2008), 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.' back |