In physics, hidden-variable theories are held
by some physicists who argue that the state
of a physical system, as formulated by quantum
mechanics, does not give a complete description
for the system.
An example would be that quantum mechanics
is ultimately incomplete, and that a complete
theory would provide descriptive categories
to account for all observable behavior and
thus avoid any indeterminism.
The existence of indeterminacy for some measurements
is a characteristic of prevalent interpretations
of quantum mechanics; moreover, bounds for
indeterminacy can be expressed in a quantitative
form by the Heisenberg uncertainty principle.
Albert Einstein objected to the fundamentally
probabilistic nature of quantum mechanics,
and famously declared "I am convinced God
does not play dice".
Einstein, Podolsky, and Rosen argued that
"elements of reality" (hidden variables) must
be added to quantum mechanics to explain entanglement
without action at a distance.
Later, Bell's theorem suggested that local
hidden variables of certain types are impossible,
or that they evolve non-locally.
A famous non-local theory is the De Broglie–Bohm
theory.
== Motivation ==
Under the Copenhagen interpretation, quantum
mechanics is non-deterministic, meaning that
it generally does not predict the outcome
of any measurement with certainty.
Instead, it indicates what the probabilities
of the outcomes are, with the indeterminism
of observable quantities constrained by the
uncertainty principle.
The question arises whether there might be
some deeper reality hidden beneath quantum
mechanics, to be described by a more fundamental
theory that can always predict the outcome
of each measurement with certainty: if the
exact properties of every subatomic particle
were known the entire system could be modeled
exactly using deterministic physics similar
to classical physics.
In other words, it is conceivable that the
standard interpretation of quantum mechanics
is an incomplete description of nature.
The designation of variables as underlying
"hidden" variables depends on the level of
physical description (so, for example, "if
a gas is described in terms of temperature,
pressure, and volume, then the velocities
of the individual atoms in the gas would be
hidden variables").
Physicists supporting De Broglie–Bohm theory
maintain that underlying the observed probabilistic
nature of the universe is a deterministic
objective foundation/property—the hidden
variable.
Others, however, believe that there is no
deeper deterministic reality in quantum mechanics.A
lack of a kind of realism (understood here
as asserting independent existence and evolution
of physical quantities, such as position or
momentum, without the process of measurement)
is crucial in the Copenhagen interpretation.
Realistic interpretations (which were already
incorporated, to an extent, into the physics
of Feynman), on the other hand, assume that
particles have certain trajectories.
Under such view, these trajectories will almost
always be continuous, which follows both from
the finitude of the perceived speed of light
("leaps" should rather be precluded) and,
more importantly, from the principle of least
action, as deduced in quantum physics by Dirac.
But continuous movement, in accordance with
the mathematical definition, implies deterministic
movement for a range of time arguments; and
thus realism is, under modern physics, one
more reason for seeking (at least certain
limited) determinism and thus a hidden-variable
theory (especially that such theory exists:
see De Broglie–Bohm interpretation).
Although determinism was initially a major
motivation for physicists looking for hidden-variable
theories, non-deterministic theories trying
to explain what the supposed reality underlying
the quantum mechanics formalism looks like
are also considered hidden-variable theories;
for example Edward Nelson's stochastic mechanics.
== "God does not play dice" ==
In June 1926, Max Born published a paper,
"Zur Quantenmechanik der Stoßvorgänge" ("Quantum
Mechanics of Collision Phenomena") in the
scientific journal Zeitschrift für Physik,
in which he was the first to clearly enunciate
the probabilistic interpretation of the quantum
wave function, which had been introduced by
Erwin Schrödinger earlier in the year.
Born concluded the paper as follows:
Here the whole problem of determinism comes
up.
From the standpoint of our quantum mechanics
there is no quantity which in any individual
case causally fixes the consequence of the
collision; but also experimentally we have
so far no reason to believe that there are
some inner properties of the atom which conditions
a definite outcome for the collision.
Ought we to hope later to discover such properties
... and determine them in individual cases?
Or ought we to believe that the agreement
of theory and experiment—as to the impossibility
of prescribing conditions for a causal evolution—is
a pre-established harmony founded on the nonexistence
of such conditions?
I myself am inclined to give up determinism
in the world of atoms.
But that is a philosophical question for which
physical arguments alone are not decisive.
Born's interpretation of the wave function
was criticized by Schrödinger, who had previously
attempted to interpret it in real physical
terms, but Albert Einstein's response became
one of the earliest and most famous assertions
that quantum mechanics is incomplete:
Quantum mechanics is very worthy of regard.
But an inner voice tells me that this is not
yet the right track.
The theory yields much, but it hardly brings
us closer to the Old One's secrets.
I, in any case, am convinced that He does
not play dice.
Niels Bohr reportedly replied to Einstein's
later expression of this sentiment by advising
him to "stop telling God what to do."
== Early attempts at hidden-variable theories
==
Shortly after making his famous "God does
not play dice" comment, Einstein attempted
to formulate a deterministic counter proposal
to quantum mechanics, presenting a paper at
a meeting of the Academy of Sciences in Berlin,
on 5 May 1927, titled "Bestimmt Schrödinger's
Wellenmechanik die Bewegung eines Systems
vollständig oder nur im Sinne der Statistik?"
("Does Schrödinger's wave mechanics determine
the motion of a system completely or only
in the statistical sense?").
However, as the paper was being prepared for
publication in the academy's journal, Einstein
decided to withdraw it, possibly because he
discovered that, contrary to his intention,
it implied non-separability of entangled systems,
which he regarded as absurd.At the Fifth Solvay
Congress, held in Belgium in October 1927
and attended by all the major theoretical
physicists of the era, Louis de Broglie presented
his own version of a deterministic hidden-variable
theory, apparently unaware of Einstein's aborted
attempt earlier in the year.
In his theory, every particle had an associated,
hidden "pilot wave" which served to guide
its trajectory through space.
The theory was subject to criticism at the
Congress, particularly by Wolfgang Pauli,
which de Broglie did not adequately answer.
De Broglie abandoned the theory shortly thereafter.
== Declaration of completeness of quantum
mechanics, and the Bohr–Einstein debates
==
Also at the Fifth Solvay Congress, Max Born
and Werner Heisenberg made a presentation
summarizing the recent tremendous theoretical
development of quantum mechanics.
At the conclusion of the presentation, they
declared:
[W]hile we consider ... a quantum mechanical
treatment of the electromagnetic field ... as
not yet finished, we consider quantum mechanics
to be a closed theory, whose fundamental physical
and mathematical assumptions are no longer
susceptible of any modification....
On the question of the 'validity of the law
of causality' we have this opinion: as long
as one takes into account only experiments
that lie in the domain of our currently acquired
physical and quantum mechanical experience,
the assumption of indeterminism in principle,
here taken as fundamental, agrees with experience.
Although there is no record of Einstein responding
to Born and Heisenberg during the technical
sessions of the Fifth Solvay Congress, he
did challenge the completeness of quantum
mechanics during informal discussions over
meals, presenting a thought experiment intended
to demonstrate that quantum mechanics could
not be entirely correct.
He did likewise during the Sixth Solvay Congress
held in 1930.
Both times, Niels Bohr is generally considered
to have successfully defended quantum mechanics
by discovering errors in Einstein's arguments.
== EPR paradox ==
The debates between Bohr and Einstein essentially
concluded in 1935, when Einstein finally expressed
what is widely considered his best argument
against the completeness of quantum mechanics.
Einstein, Podolsky, and Rosen had proposed
their definition of a "complete" description
as one that uniquely determines the values
of all its measurable properties.
Einstein later summarized their argument as
follows:
Consider a mechanical system consisting of
two partial systems A and B which interact
with each other only during a limited time.
Let the ψ function [i.e., wavefunction ] before
their interaction be given.
Then the Schrödinger equation will furnish
the ψ function after the interaction has
taken place.
Let us now determine the physical state of
the partial system A as completely as possible
by measurements.
Then quantum mechanics allows us to determine
the ψ function of the partial system B from
the measurements made, and from the ψ function
of the total system.
This determination, however, gives a result
which depends upon which of the physical quantities
(observables) of A have been measured (for
instance, coordinates or momenta).
Since there can be only one physical state
of B after the interaction which cannot reasonably
be considered to depend on the particular
measurement we perform on the system A separated
from B it may be concluded that the ψ function
is not unambiguously coordinated to the physical
state.
This coordination of several ψ functions
to the same physical state of system B shows
again that the ψ function cannot be interpreted
as a (complete) description of a physical
state of a single system.
Bohr answered Einstein's challenge as follows:
[The argument of] Einstein, Podolsky and Rosen
contains an ambiguity as regards the meaning
of the expression "without in any way disturbing
a system."
... [E]ven at this stage [i.e., the measurement
of, for example, a particle that is part of
an entangled pair], there is essentially the
question of an influence on the very conditions
which define the possible types of predictions
regarding the future behavior of the system.
Since these conditions constitute an inherent
element of the description of any phenomenon
to which the term "physical reality" can be
properly attached, we see that the argumentation
of the mentioned authors does not justify
their conclusion that quantum-mechanical description
is essentially incomplete."
Bohr is here choosing to define a "physical
reality" as limited to a phenomenon that is
immediately observable by an arbitrarily chosen
and explicitly specified technique, using
his own special definition of the term 'phenomenon'.
He wrote in 1948:
As a more appropriate way of expression, one
may strongly advocate limitation of the use
of the word phenomenon to refer exclusively
to observations obtained under specified circumstances,
including an account of the whole experiment."
This was, of course, in conflict with the
definition used by the EPR paper, as follows:
If, without in any way disturbing a system,
we can predict with certainty (i.e., with
probability equal to unity) the value of a
physical quantity, then there exists an element
of physical reality corresponding to this
physical quantity.
[Italics in original]
== Bell's theorem ==
In 1964, John Bell showed through his famous
theorem that if local hidden variables exist,
certain experiments could be performed involving
quantum entanglement where the result would
satisfy a Bell inequality.
If, on the other hand, statistical correlations
resulting from quantum entanglement could
not be explained by local hidden variables,
the Bell inequality would be violated.
Another no-go theorem concerning hidden-variable
theories is the Kochen–Specker theorem.
Physicists such as Alain Aspect and Paul Kwiat
have performed experiments that have found
violations of these inequalities up to 242
standard deviations (excellent scientific
certainty).
This rules out local hidden-variable theories,
but does not rule out non-local ones.
Theoretically, there could be experimental
problems that affect the validity of the experimental
findings.
Gerard 't Hooft has disputed the validity
of Bell's theorem on the basis of the superdeterminism
loophole and proposed some ideas to construct
local deterministic models.
== Bohm's hidden-variable theory ==
Assuming the validity of Bell's theorem, any
deterministic hidden-variable theory that
is consistent with quantum mechanics would
have to be non-local, maintaining the existence
of instantaneous or faster-than-light relations
(correlations) between physically separated
entities.
The currently best-known hidden-variable theory,
the "causal" interpretation of the physicist
and philosopher David Bohm, originally published
in 1952, is a non-local hidden-variable theory.
Bohm unknowingly rediscovered (and extended)
the idea that Louis de Broglie had proposed
in 1927 (and abandoned) – hence this theory
is commonly called "de Broglie-Bohm theory".
Bohm posited both the quantum particle, e.g.
an electron, and a hidden 'guiding wave' that
governs its motion.
Thus, in this theory electrons are quite clearly
particles—when a double-slit experiment
is performed, its trajectory goes through
one slit rather than the other.
Also, the slit passed through is not random
but is governed by the (hidden) guiding wave,
resulting in the wave pattern that is observed.
Since the location of where the particles
starts in the double-slit experiment is unknown
the initial position of the particle is the
hidden variable.
Such a view does not contradict the idea of
local events that is used in both classical
atomism and relativity theory as Bohm's theory
(and quantum mechanics) are still locally
causal (that is, information travel is still
restricted to the speed of light) but allow
non-local correlations.
It points to a view of a more holistic, mutually
interpenetrating and interacting world.
Indeed, Bohm himself stressed the holistic
aspect of quantum theory in his later years,
when he became interested in the ideas of
Jiddu Krishnamurti.
In Bohm's interpretation, the (non-local)
quantum potential constitutes an implicate
(hidden) order which organizes a particle,
and which may itself be the result of yet
a further implicate order: a superimplicate
order which organizes a field.
Nowadays Bohm's theory is considered to be
one of many interpretations of quantum mechanics
which give a realist interpretation, and not
merely a positivistic one, to quantum-mechanical
calculations.
Some consider it the simplest theory to explain
quantum phenomena.
Nevertheless, it is a hidden-variable theory,
and necessarily so.
The major reference for Bohm's theory today
is his book with Basil Hiley, published posthumously.A
possible weakness of Bohm's theory is that
some (including Einstein, Pauli, and Heisenberg)
feel that it looks contrived.
(Indeed, Bohm thought this of his original
formulation of the theory.)
It was deliberately designed to give predictions
that are in all details identical to conventional
quantum mechanics.
Bohm's original aim was not to make a serious
counter proposal but simply to demonstrate
that hidden-variable theories are indeed possible.
(It thus provided a supposed counterexample
to the famous proof by John von Neumann that
was generally believed to demonstrate that
no deterministic theory reproducing the statistical
predictions of quantum mechanics is possible.)
Bohm said he considered his theory to be unacceptable
as a physical theory due to the guiding wave's
existence in an abstract multi-dimensional
configuration space, rather than three-dimensional
space.
His hope was that the theory would lead to
new insights and experiments that would lead
ultimately to an acceptable one; his aim was
not to set out a deterministic, mechanical
viewpoint, but rather to show that it was
possible to attribute properties to an underlying
reality, in contrast to the conventional approach
to quantum mechanics.
== Recent developments ==
In August 2011, Roger Colbeck and Renato Renner
published a proof that any extension of quantum
mechanical theory, whether using hidden variables
or otherwise, cannot provide a more accurate
prediction of outcomes, assuming that observers
can freely choose the measurement settings.
Colbeck and Renner write: "In the present
work, we have ... excluded the possibility
that any extension of quantum theory (not
necessarily in the form of local hidden variables)
can help predict the outcomes of any measurement
on any quantum state.
In this sense, we show the following: under
the assumption that measurement settings can
be chosen freely, quantum theory really is
complete".
In January 2013, GianCarlo Ghirardi and Raffaele
Romano described a model which, "under a different
free choice assumption [...] violates [the
statement by Colbeck and Renner] for almost
all states of a bipartite two-level system,
in a possibly experimentally testable way".Also,
in the deterministic collapse theory of 2016
the non-local absolute phase constants of
the wave packets were taken as hidden variables
(cf.
). Collapse occurs when two wave packets spatially
overlap and satisfy a mathematical criterion,
which demands that their phase constants very
nearly coincide.
The wave packets then collapse to the overlap
volume.
In a measurement this mimics the action of
a point particle.
The phase constants are pseudorandom numbers,
in the sense of the deterministic chaos theory,
and the Born rules are derived under the assumption
that their distribution is uniform.
== See also ==
Local hidden-variable theory
Bell's theorem
Bell test experiments
Einstein's thought experiments
Quantum mechanics
Bohm interpretation
Spekkens toy model
