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 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 transformation or 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.

(Preface to The Principles of Quantum Mechanics, First Edition, p.vi, in the Fourth Edition, p.viii)

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.

Chapter 1 of The Principles of Quantum Mechanics, First Edition, 1930, pp.14-15.

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

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.

Chapter 1 of The Principles of Quantum Mechanics, Fourth Edition, 1956, pp.14-15.

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:

• 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.

Feynman's thesis was titled "The Principle of Least Action in Quantum Mechanics." Following a section II called "Least Action in Classical Mechanics," Feynman's section III was called "Least Action in Quantum Mechanics," in which Section III.1 was called "The Lagrangian in Quantum Mechanics," the same title as Dirac's paper.

Feynman's thesis did not refer to Dirac's paper but to the new sections added to Dirac's classic text, "The Principles of Quantum Mechanics, in the 1935 and all later editions as "The Action Principle."

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