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Philosophers

Mortimer Adler
Rogers Albritton
Alexander of Aphrodisias
Samuel Alexander
William Alston
Anaximander
G.E.M.Anscombe
Anselm
Louise Antony
Thomas Aquinas
Aristotle
David Armstrong
Harald Atmanspacher
Robert Audi
Augustine
J.L.Austin
A.J.Ayer
Alexander Bain
Mark Balaguer
Jeffrey Barrett
William Barrett
William Belsham
Henri Bergson
George Berkeley
Isaiah Berlin
Richard J. Bernstein
Bernard Berofsky
Robert Bishop
Max Black
Susanne Bobzien
Emil du Bois-Reymond
Hilary Bok
Laurence BonJour
George Boole
Émile Boutroux
Daniel Boyd
F.H.Bradley
C.D.Broad
Michael Burke
Lawrence Cahoone
C.A.Campbell
Joseph Keim Campbell
Rudolf Carnap
Carneades
Nancy Cartwright
Gregg Caruso
Ernst Cassirer
David Chalmers
Roderick Chisholm
Chrysippus
Cicero
Randolph Clarke
Samuel Clarke
Anthony Collins
Antonella Corradini
Diodorus Cronus
Jonathan Dancy
Donald Davidson
Mario De Caro
Democritus
Daniel Dennett
Jacques Derrida
René Descartes
Richard Double
Fred Dretske
John Dupré
John Earman
Laura Waddell Ekstrom
Epictetus
Epicurus
Austin Farrer
Herbert Feigl
Arthur Fine
John Martin Fischer
Frederic Fitch
Owen Flanagan
Luciano Floridi
Philippa Foot
Alfred Fouilleé
Harry Frankfurt
Richard L. Franklin
Bas van Fraassen
Michael Frede
Gottlob Frege
Peter Geach
Edmund Gettier
Carl Ginet
Alvin Goldman
Gorgias
Nicholas St. John Green
H.Paul Grice
Ian Hacking
Ishtiyaque Haji
Stuart Hampshire
W.F.R.Hardie
Sam Harris
William Hasker
R.M.Hare
Georg W.F. Hegel
Martin Heidegger
Heraclitus
R.E.Hobart
Thomas Hobbes
David Hodgson
Shadsworth Hodgson
Baron d'Holbach
Ted Honderich
Pamela Huby
David Hume
Ferenc Huoranszki
Frank Jackson
William James
Lord Kames
Robert Kane
Immanuel Kant
Tomis Kapitan
Walter Kaufmann
Jaegwon Kim
William King
Hilary Kornblith
Christine Korsgaard
Saul Kripke
Thomas Kuhn
Andrea Lavazza
Christoph Lehner
Keith Lehrer
Gottfried Leibniz
Jules Lequyer
Leucippus
Michael Levin
Joseph Levine
George Henry Lewes
C.I.Lewis
David Lewis
Peter Lipton
C. Lloyd Morgan
John Locke
Michael Lockwood
Arthur O. Lovejoy
E. Jonathan Lowe
John R. Lucas
Lucretius
Alasdair MacIntyre
Ruth Barcan Marcus
Tim Maudlin
James Martineau
Nicholas Maxwell
Storrs McCall
Hugh McCann
Colin McGinn
Michael McKenna
Brian McLaughlin
John McTaggart
Paul E. Meehl
Uwe Meixner
Alfred Mele
Trenton Merricks
John Stuart Mill
Dickinson Miller
G.E.Moore
Thomas Nagel
Otto Neurath
Friedrich Nietzsche
John Norton
P.H.Nowell-Smith
Robert Nozick
William of Ockham
Timothy O'Connor
Parmenides
David F. Pears
Charles Sanders Peirce
Derk Pereboom
Steven Pinker
Plato
Karl Popper
Porphyry
Huw Price
H.A.Prichard
Protagoras
Hilary Putnam
Willard van Orman Quine
Frank Ramsey
Ayn Rand
Michael Rea
Thomas Reid
Charles Renouvier
Nicholas Rescher
C.W.Rietdijk
Richard Rorty
Josiah Royce
Bertrand Russell
Paul Russell
Gilbert Ryle
Jean-Paul Sartre
Kenneth Sayre
T.M.Scanlon
Moritz Schlick
Arthur Schopenhauer
John Searle
Wilfrid Sellars
Alan Sidelle
Ted Sider
Henry Sidgwick
Walter Sinnott-Armstrong
J.J.C.Smart
Saul Smilansky
Michael Smith
Baruch Spinoza
L. Susan Stebbing
Isabelle Stengers
George F. Stout
Galen Strawson
Peter Strawson
Eleonore Stump
Francisco Suárez
Richard Taylor
Kevin Timpe
Mark Twain
Peter Unger
Peter van Inwagen
Manuel Vargas
John Venn
Kadri Vihvelin
Voltaire
G.H. von Wright
David Foster Wallace
R. Jay Wallace
W.G.Ward
Ted Warfield
Roy Weatherford
C.F. von Weizsäcker
William Whewell
Alfred North Whitehead
David Widerker
David Wiggins
Bernard Williams
Timothy Williamson
Ludwig Wittgenstein
Susan Wolf

Scientists

David Albert
Michael Arbib
Walter Baade
Bernard Baars
Jeffrey Bada
Leslie Ballentine
Marcello Barbieri
Gregory Bateson
John S. Bell
Mara Beller
Charles Bennett
Ludwig von Bertalanffy
Susan Blackmore
Margaret Boden
David Bohm
Niels Bohr
Ludwig Boltzmann
Emile Borel
Max Born
Satyendra Nath Bose
Walther Bothe
Jean Bricmont
Hans Briegel
Leon Brillouin
Stephen Brush
Henry Thomas Buckle
S. H. Burbury
Melvin Calvin
Donald Campbell
Sadi Carnot
Anthony Cashmore
Eric Chaisson
Gregory Chaitin
Jean-Pierre Changeux
Rudolf Clausius
Arthur Holly Compton
John Conway
Jerry Coyne
John Cramer
Francis Crick
E. P. Culverwell
Antonio Damasio
Olivier Darrigol
Charles Darwin
Richard Dawkins
Terrence Deacon
Lüder Deecke
Richard Dedekind
Louis de Broglie
Stanislas Dehaene
Max Delbrück
Abraham de Moivre
Bernard d'Espagnat
Paul Dirac
Hans Driesch
John Eccles
Arthur Stanley Eddington
Gerald Edelman
Paul Ehrenfest
Manfred Eigen
Albert Einstein
George F. R. Ellis
Hugh Everett, III
Franz Exner
Richard Feynman
R. A. Fisher
David Foster
Joseph Fourier
Philipp Frank
Steven Frautschi
Edward Fredkin
Benjamin Gal-Or
Lila Gatlin
Michael Gazzaniga
Nicholas Georgescu-Roegen
GianCarlo Ghirardi
J. Willard Gibbs
Nicolas Gisin
Paul Glimcher
Thomas Gold
A. O. Gomes
Brian Goodwin
Joshua Greene
Dirk ter Haar
Jacques Hadamard
Mark Hadley
Patrick Haggard
J. B. S. Haldane
Stuart Hameroff
Augustin Hamon
Sam Harris
Ralph Hartley
Hyman Hartman
John-Dylan Haynes
Donald Hebb
Martin Heisenberg
Werner Heisenberg
John Herschel
Basil Hiley
Art Hobson
Jesper Hoffmeyer
Don Howard
William Stanley Jevons
Roman Jakobson
E. T. Jaynes
Pascual Jordan
Ruth E. Kastner
Stuart Kauffman
Martin J. Klein
William R. Klemm
Christof Koch
Simon Kochen
Hans Kornhuber
Stephen Kosslyn
Daniel Koshland
Ladislav Kovàč
Leopold Kronecker
Rolf Landauer
Alfred Landé
Pierre-Simon Laplace
David Layzer
Joseph LeDoux
Gilbert Lewis
Benjamin Libet
David Lindley
Seth Lloyd
Hendrik Lorentz
Josef Loschmidt
Ernst Mach
Donald MacKay
Henry Margenau
Owen Maroney
Humberto Maturana
James Clerk Maxwell
Ernst Mayr
John McCarthy
Warren McCulloch
N. David Mermin
George Miller
Stanley Miller
Ulrich Mohrhoff
Jacques Monod
Emmy Noether
Alexander Oparin
Abraham Pais
Howard Pattee
Wolfgang Pauli
Massimo Pauri
Roger Penrose
Steven Pinker
Colin Pittendrigh
Max Planck
Susan Pockett
Henri Poincaré
Daniel Pollen
Ilya Prigogine
Hans Primas
Henry Quastler
Adolphe Quételet
Lord Rayleigh
Jürgen Renn
Emil Roduner
Juan Roederer
Jerome Rothstein
David Ruelle
Tilman Sauer
Jürgen Schmidhuber
Erwin Schrödinger
Aaron Schurger
Sebastian Seung
Thomas Sebeok
Franco Selleri
Claude Shannon
Charles Sherrington
David Shiang
Abner Shimony
Herbert Simon
Dean Keith Simonton
Edmund Sinnott
B. F. Skinner
Lee Smolin
Ray Solomonoff
Roger Sperry
John Stachel
Henry Stapp
Tom Stonier
Antoine Suarez
Leo Szilard
Max Tegmark
Teilhard de Chardin
Libb Thims
William Thomson (Kelvin)
Richard Tolman
Giulio Tononi
Peter Tse
Francisco Varela
Vlatko Vedral
Mikhail Volkenstein
Heinz von Foerster
Richard von Mises
John von Neumann
Jakob von Uexküll
C. S. Unnikrishnan
C. H. Waddington
John B. Watson
Daniel Wegner
Steven Weinberg
Paul A. Weiss
Herman Weyl
John Wheeler
Wilhelm Wien
Norbert Wiener
Eugene Wigner
E. O. Wilson
Günther Witzany
Stephen Wolfram
H. Dieter Zeh
Ernst Zermelo
Wojciech Zurek
Konrad Zuse
Fritz Zwicky

Presentations

Biosemiotics
Free Will
Mental Causation
James Symposium
 
Paul Dirac

Paul (P. A. M.) Dirac formulated the most elegant version of the mathematical principles of quantum mechanics after reading the proof copy of Werner Heisenberg's paper on the new "matrix mechanics."

A few months after the completion of matrix mechanics by Heisenberg’s mentor Max Born and Born’s assistant Pascual Jordan, Erwin Schrödinger developed his "wave mechanics." Dirac and Schrödinger independently showed the new wave mechanics was mathematically and physically equivalent to the Heisenberg picture, despite the extraordinary differences between the two quantum theories.

Almost two decades after Albert Einstein had said

It is therefore my opinion that the next stage in the development of theoretical physics will bring us a theory of light that can be understood as a kind of fusion of the wave and emission theories of light,
Dirac's transformation theory gave us that "fusion" between waves and particles.

Dirac combined the matrix and wave formulations using abstract symbolic methods from classical mechanics called Poisson brackets and canonical transformations.

In his textbook The Principles of Quantum Mechanics, Paul Dirac introduced the new concepts of superposition of quantum states, the projection postulate, the axiom of measurement, and indeterminacy using simple examples with polarized photons.

Dirac's examples suggest a very simple and inexpensive experiment that we call the Dirac 3-polarizers experiment to demonstrate the notions of quantum states, the preparation of quantum systems in states with known properties, the superposition of states, the measurement of various properties, the representation of a state vector in another basis set of vectors, and the infamous "collapse" or "reduction" of the wave function and the resulting indeterministic projection into one of the proper basis states.

In their Copenhagen interpretation of quantum mechanics, Niels Bohr and Heisenberg said that the results of quantum measurements must be expressible in classical concepts because it is the language that humans can understand. By contrast, Dirac argued that the non-intuitive concepts of quantum mechanics, though impossible to understand in terms of classical concepts, could be mastered through long familiarity with them.

The new theories, if one looks apart from their mathematical setting, are built up from physical concepts which cannot be explained in terms of things previously known to the student, which cannot even be explained adequately in words at all. Like the fundamental concepts (e.g. proximity, identity) which every one must learn on his arrival into the world, the newer concepts of physics can be mastered only by long familiarity with their properties and uses.

A Photon Interferes Only With Itself

In 1930, Dirac famously described a photon as interfering only with itself.

Consider a beam of light to be split into two components of equal intensity, which are made to interfere. According to the old corpuscular theory we would say that each of the two components contains an equal number of photons and we should then require that a photon in one component could interfere with one in the other. Under certain conditions they would have to annihilate one another, and under others to produce four photons. This contradicts the idea of photons being discrete particles and is, besides, in disagreement with the conservation of energy, which should hold for each process in detail and not be merely statistically true.

The answer that quantum mechanics gives to the difficulty is that one should consider each photon to go partly into each of the two components, in the way allowed by the idea of the superposition of states. Each photon then interferes only with itself. Interference between two different photons can never occur. The solution of Maxwell’s equations that forms the wave picture of the phenomenon represents one of the photons and not the whole assembly of photons.

In his later editions Dirac made the explanation more clear...

The scientist who "realized that the connection between light waves and photons must be of a statistical character" was of course Einstein.

Some time before the discovery of quantum mechanics people realized that the connexion between light waves and photons must be of a statistical character. What they did not clearly realize, however, was that the wave function gives information about the probability of one photon being in a particular place and not the probable number of photons in that place. The importance of the distinction can be made clear in the following way. Suppose we have a beam of light consisting of a large number of photons split up into two components of equal intensity. On the assumption that the intensity of a beam is connected with the probable number of photons in it, we should have half the total number of photons going into each component. If the two components are now made to interfere, we should require a photon in one component to be able to interfere with one in the other. Sometimes these two photons would have to annihilate one another and other times they would have to produce four photons. This would contradict the conservation of energy. The new theory, which connects the wave function with probabilities for one photon, gets over the difficulty by making each photon go partly into each of the two components. Each photon then interferes only with itself. Interference between two different photons never occurs.

Regarding Dirac's claim that the wave function gives us "information about the probability of one photon being in a particular place and not the probable number of photons in that place," we should note that Einstein, and Born years later, strongly held both to be true. And we can give the reason.

Dirac's quantum mechanics associates the quantum wave function with possibilities and a quantum particle with actualization of a possibility. Evaluating the Schrödinger equation lets us calculate the probabilities for each possibility, to an extraordinary degree of accuracy. Although the calculation involves abstract complex quantities and the motion through space of immaterial information about those possibilities, the result is both understandable (if non-intuitive because never experienced in our macroscopic world) and it is visualizable.

We solve the Schrödinger equation given the boundary conditions to get the wave function. The boundary conditions are different when either one or two slits are open. So the probabilities of finding particles at the back screen are different, producing different interference fringes. These probabilities tell us where particles will be found, whichever slit the particles come through.

Conservation laws (for energy, mass, charge, etc.) suggest that a particle comes through a single slit. It cannot divide into two photons or two electrons, or two buckyballs, despite Dirac's "manner of speaking."

The resulting interference is described on the two-slit experiment page...

Remembering that the double-slit interference appears even if only one particle at a time is incident on the two slits, we see why many say that the particle interferes with itself. But it is the wave function alone that is interfering with itself. Whichever slit the particle goes through, it is the probability amplitude ψ, whose squared modulus |ψ|2 gives us the probability of finding a particle somewhere, the interference pattern. It is what it is because the two slits are open.

The reason that the wave function for a single photon also describes the probabilities for large numbers of particles being found (despite Dirac above saying it does not describe "the probable number of photons in that place.") is because the wave function is determined by the physical environment, which is the same for all photons coming through the two slits. One wave function rules them all.

The standard interpretation of quantum mechanics is based on three simple premises:

When you hear or read that electrons are both waves and particles, think "either-or" -
first a wave of possibilities, then an actual particle.
  • Quantum systems evolve in two ways:
    • the first is the wave function deterministically exploring all the possibilities for interaction, interfering with itself as it travels,
    • the second is the particle randomly choosing one of those possibilities to become actual.

  • No knowledge can be gained by a "conscious observer" unless new information has already been irreversibly recorded in the universe. That information can be created and recorded in three places:
    • in the target quantum system,
    • in the combined target system and the measuring apparatus (which may just be the environment),
    • it may then become knowledge in an observer's mind.

  • In our two-stage model of free will, an agent first freely generates alternative possibilities, then evaluates them and chooses one, adequately determined by its motives, reasons, desires, etc. First come "free alternatives," then "willed actions." Just as with quantum processes - first possibilities, then actuality.
    The measuring apparatus is quantal, not deterministic nor "classical." It need only be statistically determined and capable of recording the irreversible information about an interaction. The human mind is similarly only statistically determined.

We try to visualize some of these concepts, including Dirac's three polarizers, the two-slit experiment, and the Einstein-Podolsky-Rosen thought experiment.

References
The Fundamental Equations of Quantum Mechanics, 1925

On the Theory of Quantum Mechanics, 1926

Relativity Quantum Mechanics with an Application to Compton Scattering, 1926

The Physical Interpretation of the Quantum Dynamics, 1927

The Quantum Theory of the Emission and Absorption of Radiation, 1927

From the Preface to The Principles of Quantum Mechanics, First Edition, 1930

Chapter 1 of The Principles of Quantum Mechanics, First Edition, 1930

The Lagrangian in Quantum Mechanics, 1933

On the Analogy Between Quantum and Classical Mechanics, 1945

Chapter 1 of The Principles of Quantum Mechanics, Fourth Edition, 1956

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