de Broglie–Bohm theory

The de Broglie–Bohm theory, also known as the pilot wave theory, Bohmian mechanics, Bohm's interpretation, and the causal interpretation, is an interpretation of quantum mechanics. It postulates that in addition to the wavefunction, an actual configuration of particles exists, even when unobserved. The evolution over time of the configuration of all particles is defined by a guiding equation. The evolution of the wave function over time is given by the Schrödinger equation. The theory is named after Louis de Broglie (1892–1987) and David Bohm (1917–1992).

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${\displaystyle i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle ={\hat {H}}|\psi (t)\rangle }$ Schrödinger equation
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The theory is deterministic and explicitly nonlocal: the velocity of any one particle depends on the value of the guiding equation, which depends on the configuration of all the particles under consideration.

Measurements are a particular case of quantum processes described by the theory—for which it yields the same quantum predictions generally associated with the Copenhagen interpretation. The theory does not have a "measurement problem", due to the fact that the particles have a definite configuration at all times. The Born rule in de Broglie–Bohm theory is not a basic law. Rather, in this theory, the link between the probability density and the wave function has the status of a hypothesis, called the "quantum equilibrium hypothesis", which is additional to the basic principles governing the wave function.

The theory was historically developed in the 1920s by de Broglie, who, in 1927, was persuaded to abandon it in favour of the then-mainstream Copenhagen interpretation. David Bohm, dissatisfied with the prevailing orthodoxy, rediscovered de Broglie's pilot-wave theory in 1952. Bohm's suggestions were not then widely received, partly due to reasons unrelated to their content, such as Bohm's youthful communist affiliations. The de Broglie–Bohm theory was widely deemed unacceptable by mainstream theorists, mostly because of its explicit non-locality. On the theory, John Stewart Bell, author of the 1964 Bell's theorem wrote in 1982:

Bohm showed explicitly how parameters could indeed be introduced, into nonrelativistic wave mechanics, with the help of which the indeterministic description could be transformed into a deterministic one. More importantly, in my opinion, the subjectivity of the orthodox version, the necessary reference to the “observer”, could be eliminated. ...

But why then had Born not told me of this “pilot wave”? If only to point out what was wrong with it? Why did von Neumann not consider it? More extraordinarily, why did people go on producing “impossibility” proofs, after 1952, and as recently as 1978?... Why is the pilot wave picture ignored in text books? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show us that vagueness, subjectivity, and indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice?

Since the 1990s, there has been renewed interest in formulating extensions to de Broglie–Bohm theory, attempting to reconcile it with special relativity and quantum field theory, besides other features such as spin or curved spatial geometries.

The Stanford Encyclopedia of Philosophy article on quantum decoherence groups "approaches to quantum mechanics" into five groups, of which "pilot-wave theories" are one (the others are the Copenhagen interpretation, objective collapse theories, many-worlds interpretations and modal interpretations).

There are several equivalent mathematical formulations of the theory, and it is known by a number of names. The de Broglie wave has a macroscopic analogy termed the Faraday wave.