Copenhagen interpretation
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The Copenhagen interpretation is an interpretation of quantum mechanics, usually
understood to state that every particle is described by its wavefunction, which dictates the
1
probability for it to be found in any location following a measurement. Each measurement
causes a change in the state of the particle, known as wavefunction collapse.
Early twentieth century studies of the physics of very small-scale phenomena led to the
Copenhagen interpretation. The new experiments led to the discovery of phenomena that
could not be predicted on the basis of classical physics, and to new empirical generalizations
(theories) that described and predicted very accurately those micro-scale phenomena so
recently discovered. These generalizations, these models of the real world being observed at
this micro scale, could not be squared easily with the way objects are observed to behave on
the macro scale of everyday life. The predictions they offered often appeared counter-intuitive
to observers. Indeed, they touched off much consternation -- even in the minds of their
discoverers. "What can we make of these uncanny experimental results?" The Copenhagen
interpretation consists of attempts to explain the experiments and their mathematical
formulations in ways that do not go beyond the evidence to suggest more (or less) than is
actually there.
The work of relating the remarkable experiments and the abstract mathematical and
theoretical formulations that constitute quantum physics to the experience that all of us share
in the world of everyday life fell first to Niels Bohr and Werner Heisenberg in the course of
their collaboration in Copenhagen around 1927. Bohr and Heisenberg had stepped beyond the
world of empirical experiments, pragmatic predictions of such phenomena as the frequencies
of light emitted under various conditions and the observation that a discrete quantities of
energy must be postulated in order to avoid the paradoxes to which classical physics
inevitably led when it was pushed to extremes, and found a new world of quanta of energy,
entities that fit neither the classical ideas of particles nor the classical ideas of waves,
elementary particles that behaved in ways highly regular when many similar interactions were
analyzed yet highly unpredictable when one tried to predict things like individual trajectories
through a simple physical apparatus.
Not only did laboratory experiments disclose the fact, but the new theories predicted the
consequences that elementary particles are neither wave nor particle, that knowing the
position of a particle prevents us from knowing its direction and velocity (and vice-versa),
that the very fact of detecting whether a small object such as a photon or electron passes
through an apparatus by one path or another can change the end result of the experiment when
that small entity reaches a detection screen. The results of their own burgeoning
understanding disoriented Bohr and Heisenberg. And their results seemed to some people to
indicate, for instance, that the fact that a human being had observed some event changed the
reality of the event.
The Copenhagen interpretation was a composite statement about what could and could not be
legitimately stated in common language to complement the statements and predictions that
could be made in the language of instrument readings and mathematical operations. In other
words, it attempted to answer the question, "What do these amazing experimental results
really mean?"
Contents
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1 Overview
o 1.1 Principles
2 The meaning of the wave function
3 The nature of collapse
4 Acceptance among physicists
5 Consequences
6 Criticisms
7 Alternatives
8 See also
9 Notes and References
10 Further reading
11 Video Demonstration
12 External links
[edit] Overview
There is no definitive statement of the Copenhagen Interpretation[1] since it consists of the
views developed by a number of scientists and philosophers at the turn of the 20th Century.
Thus, there are a number of ideas that have been associated with the Copenhagen
interpretation. Asher Peres remarked that very different, sometimes opposite, views are
presented as the Copenhagen interpretation by different authors.[2]
[edit] Principles
1. A system is completely described by a wave function ψ, which represents an
observer's knowledge of the system. (Heisenberg)
2. The description of nature is essentially probabilistic. The probability of an event is
related to the square of the amplitude of the wave function related to it. (Max Born)
3. Heisenberg's uncertainty principle states the observed fact that it is not possible to
know the values of all of the properties of the system at the same time; those
properties that are not known with precision must be described by probabilities.
4. (Complementary Principle) Matter exhibits a wave-particle duality. An experiment
can show the particle-like properties of matter, or wave-like properties, but not both at
the same time.(Niels Bohr)
5. Measuring devices are essentially classical devices, and measure classical properties
such as position and momentum.
6. The Correspondence Principle of Bohr and Heisenberg: the quantum mechanical
description of large systems should closely approximate to the classical description.
[edit] The meaning of the wave function
The Copenhagen Interpretation denies that any wave function is anything more than an
abstraction, or is at least non-committal about its being a discrete entity or a discernible
component of some discrete entity.
There are some who say that there are objective variants of the Copenhagen Interpretation
that allow for a "real" wave function, but it is questionable whether that view is really
3
consistent with positivism and/or with some of Bohr's statements. Niels Bohr emphasized that
science is concerned with predictions of the outcomes of experiments, and that any additional
propositions offered are not scientific but rather meta-physical. Bohr was heavily influenced
by positivism. On the other hand, Bohr and Heisenberg were not in complete agreement, and
held different views at different times. Heisenberg in particular was prompted to move
towards realism.[3]
Even if the wave function is not regarded as real, there is still a divide between those who
treat it as definitely and entirely subjective, and those who are non-committal or agnostic
about the subject.
An example of the agnostic view is given by von Weizsäcker, who, while participating in a
colloquium at Cambridge, denied that the Copenhagen interpretation asserted: "What cannot
be observed does not exist". He suggested instead that the Copenhagen interpretation follows
the principle: "What is observed certainly exists; about what is not observed we are still free
to make suitable assumptions. We use that freedom to avoid paradoxes."[4]
The subjective view, that the wave function is merely a mathematical tool for calculating
probabilities of specific experiment, is a similar approach to the Ensemble interpretation.
[edit] The nature of collapse
All versions of the Copenhagen interpretation include at least a formal or methodological
version of wave function collapse,[5] in which unobserved eigenvalues are removed from
further consideration. (In other words, Copenhagenists have never rejected collapse, even in
the early days of quantum physics, in the way that adherents of the Many-worlds
interpretation do.) In more prosaic terms, those who hold to the Copenhagen understanding
are willing to say that a wave function involves the various probabilities that a given event
will proceed to certain different outcomes. But when one or another of those more- or lesslikely outcomes becomes manifest the other probabilities cease to have any function in the
real world. So if an electron passes through a double slit apparatus there are various
probabilities for where on the detection screen that individual electron will hit. But once it has
hit, there is no longer any probability whatsoever that it will hit somewhere else. Manyworlds interpretations say that an electron hits wherever there is a possibility that it might hit,
and that each of these hits occurs in a separate universe.
An adherent of the subjective view, that the wave function represents nothing but knowledge,
would take an equally subjective view of "collapse", understanding it as nothing more than an
observer becoming informed about something that was previously not known.
Some argue that the concept of collapse of a "real" wave function was introduced by John
Von Neumann in 1932 and was not part of the original formulation of the Copenhagen
Interpretation.[6]
[edit] Acceptance among physicists
According to a poll at a Quantum Mechanics workshop in 1997, the Copenhagen
interpretation is the most widely-accepted specific interpretation of quantum mechanics,
followed by the many-worlds interpretation.[7] Although current trends show substantial
4
competition from alternative interpretations, throughout much of the twentieth century the
Copenhagen interpretation had strong acceptance among physicists. Astrophysicist and
science writer John Gribbin describes it as having fallen from primacy after the 1980s.[8]
[edit] Consequences
The nature of the Copenhagen Interpretation is exposed by considering a number of
experiments and paradoxes.
1. Schrödinger's Cat - A cat is put in a box with a radioactive source and a radiation
detector. There is a 50-50 chance that a particle will be emitted and detected by the detector.
If a particle is detected, a poisonous gas will be released and the cat killed. Schrödinger set
this up as what he called a "ridiculous case" in which "The psi-function of the entire system
would express this by having in it the living and dead cat (pardon the expression) mixed or
smeared out in equal parts." He resisted an interpretation that would "so naively accepting as
valid a 'blurred model' for representing reality."[9] How can the cat be both alive and dead?
The Copenhagen Interpretation: The wave function reflects our knowledge of the system. The
wave function
cat is alive or dead.
simply means that there is a 50-50 chance that the
2. Wigner's Friend - Wigner puts his friend in with the cat. The external observer believes
the system is in the state
cat is alive. I.e. for him, the cat is in the state
different wave functions?
. His friend however is convinced that
. How can Wigner and his friend see
The Copenhagen Interpretation: Wigner's friend highlights the subjective nature of
probability. Each observer (Wigner and his friend) has different information and therefore
different wave functions. The distinction between the "objective" nature of reality and the
subjective nature of probability has led to a great deal of controversy. C.f. Bayesian versus
Frequentist interpretations of probability.
3. Double Slit Diffraction - Light passes through double slits and onto a screen resulting in a
diffraction pattern. Is light a particle or a wave?
The Copenhagen Interpretation: Light is neither. A particular experiment can demonstrate
particle (photon) or wave properties, but not both at the same time (Bohr's Complementary
Principle).
The same experiment can in theory be performed with any physical system: electrons,
protons, atoms, molecules, viruses, bacteria, cats, humans, elephants, planets, etc. In practice
it has been performed for light, electrons, buckminsterfullerene, and some atoms. Due to the
smallness of Planck's constant it is practically impossible to realize experiments that directly
reveal the wave nature of any system bigger than a few atoms but, in general, quantum
mechanics considers all matter as possessing both particle and wave behaviors. The greater
systems (like viruses, bacteria, cats, etc.) are considered as "classical" ones but only as an
approximation.
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4. EPR paradox. Entangled "particles" are emitted in a single event. Conservation laws
ensure that the measured spin of one particle must be the opposite of the measured spin of the
other, so that if the spin of one particle is measured, the spin of the other particle is now
instantaneously known. The most discomfiting aspect of this paradox is that the effect is
instantaneous so that something that happens in one galaxy could cause an instantaneous
change in another galaxy.
The Copenhagen Interpretation: Assuming wave functions are not real, wave function
collapse is interpreted subjectively. The moment one observer measures the spin of one
particle, he knows the spin of the other. However another observer cannot benefit until the
results of that measurement have been relayed to him, at less than or equal to the speed of
light.
Copenhagenists claim that interpretations of quantum mechanics where the wave function is
regarded as real have problems with EPR-type effects, since they imply that the laws of
physics allow for influences to propagate at speeds greater than the speed of light. However,
proponents of Many worlds [10] and the Transactional interpretation [11] [12] maintain that their
theories are fatally non-local.
The claim that EPR effects violate the principle that information cannot travel faster than the
speed of light can be avoided by noting that they cannot be used for signaling because neither
observer can control, or predetermine, what he observes, and therefore cannot manipulate
what the other observer measures. Relativistic difficulties about establishing which
measurement occurred first also undermine the idea that one observer is causing what the
other is measuring.[citation needed]
[edit] Criticisms
The completeness of quantum mechanics (thesis 1) was attacked by the Einstein-PodolskyRosen thought experiment which was intended to show that quantum physics could not be a
complete theory.
Experimental tests of Bell's inequality using particles have supported the quantum mechanical
prediction of entanglement.
The Copenhagen Interpretation gives special status to measurement processes without clearly
defining them or explaining their peculiar effects. In his article entitled "Criticism and
Counterproposals to the Copenhagen Interpretation of Quantum Theory," countering the view
of Alexandrov that (in Heisenberg's paraphrase) "the wave function in configuration space
characterizes the objective state of the electron." Heisenberg says,
Of course the introduction of the observer must not be misunderstood to imply that some kind
of subjective features are to be brought into the description of nature. The observer has,
rather, only the function of registering decisions, i.e., processes in space and time, and it does
not matter whether the observer is an apparatus or a human being; but the registration, i.e., the
transition from the "possible" to the "actual," is absolutely necessary here and cannot be
omitted from the interpretation of quantum theory.
-- Heisenberg, Physics and Philosophy, p. 137
6
Many physicists and philosophers have objected to the Copenhagen interpretation, both on the
grounds that it is non-deterministic and that it includes an undefined measurement process
that converts probability functions into non-probabilistic measurements. Einstein's comments
"I, at any rate, am convinced that He (God) does not throw dice."[13] and "Do you really think
the moon isn't there if you aren't looking at it?" exemplify this. Bohr, in response, said
"Einstein, don't tell God what to do".
Steven Weinberg in "Einstein's Mistakes", Physics Today, November 2005, page 31, said:
All this familiar story is true, but it leaves out an irony. Bohr's version of quantum
mechanics was deeply flawed, but not for the reason Einstein thought. The
Copenhagen interpretation describes what happens when an observer makes a
measurement, but the observer and the act of measurement are themselves treated
classically. This is surely wrong: Physicists and their apparatus must be governed by
the same quantum mechanical rules that govern everything else in the universe. But
these rules are expressed in terms of a wave function (or, more precisely, a state
vector) that evolves in a perfectly deterministic way. So where do the probabilistic
rules of the Copenhagen interpretation come from?
Considerable progress has been made in recent years toward the resolution of the
problem, which I cannot go into here. It is enough to say that neither Bohr nor Einstein
had focused on the real problem with quantum mechanics. The Copenhagen rules
clearly work, so they have to be accepted. But this leaves the task of explaining them
by applying the deterministic equation for the evolution of the wave function, the
Schrödinger equation, to observers and their apparatus.
The problem of thinking in terms of classical measurements of a quantum system becomes
particularly acute in the field of quantum cosmology, where the quantum system is the
universe.[14]
[edit] Alternatives
The Ensemble Interpretation is similar; it offers an interpretation of the wave function, but not
for single particles. The consistent histories interpretation advertises itself as "Copenhagen
done right". Consciousness causes collapse is often confused with the Copenhagen
interpretation.
If the wave function is regarded as ontologically real, and collapse is entirely rejected, a many
worlds theory results. If wave function collapse is regarded as ontologically real as well, an
objective collapse theory is obtained. Dropping the principle that the wave function is a
complete description results in a hidden variable theory.
Many physicists have subscribed to the null interpretation of quantum mechanics summarized
by the sentence "Shut up and calculate!". While it is sometimes attributed to Paul Dirac[15] or
Richard Feynman, it is probably a misquotation[16].
A list of alternatives can be found at Interpretation of quantum mechanics.
[edit] See also
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Afshar experiment
Bohr-Einstein debates
Consciousness causes collapse
Consistent Histories
Ensemble Interpretation
Interpretation of quantum mechanics
Philosophical interpretation of classical physics
Popper's experiment
[edit] Notes and References
1. ^ In fact Bohr and Heisenberg never totally agreed on how to understand the
mathematical formalism of quantum mechanics, and none of them ever used the term
“the Copenhagen interpretation” as a joint name for their ideas. Bohr once distanced
himself from what he considered to be Heisenberg's more subjective interpretation
Stanford Encyclopedia of Philosophy
2. ^ "There seems to be at least as many different Copenhagen interpretations as people
who use that term, probably there are more. For example, in two classic articles on the
foundations of quantum mechanics, Ballentine (1970) and Stapp(1972) give
diametrically opposite definitions of “Copenhagen.”", A. Peres, Popper's experiment
and the Copenhagen interpretation, Stud. History Philos. Modern Physics 33 (2002)
23, preprint
3. ^ "Historically, Heisenberg wanted to base quantum theory solely on observable
quantities such as the intensity of spectral lines, getting rid of all intuitive
(anschauliche) concepts such as particle trajectories in space-time [2]. This attitude
changed drastically with his paper [3] in which he introduced the uncertainty relations
– there he put forward the point of view that it is the theory which decides what can be
observed. His move from positivism to operationalism can be clearly understood as a
reaction on the advent of Schr¨odinger’s wave mechanics [1] which, in particular due
to its intuitiveness, became soon very popular among physicists. In fact, the word
anschaulich (intuitive) is contained in the title of Heisenberg’s paper [3]."Kiefer, C.
On the interpretation of quantum theory – from Copenhagen to the present day
4. ^ John Cramer on the Copenhagen Interpretation
5. ^ "To summarize, one can identify the following ingredients as being characteristic for
the Copenhagen interpretation(s)[...]Reduction of the wave packet as a formal rule
without dynamical significance"Kiefer, C. On the interpretation of quantum theory –
from Copenhagen to the present day
6. ^ "the “collapse” or “reduction” of the wave function. This was introduced by
Heisenberg in his uncertainty paper [3] and later postulated by von Neumann as a
dynamical process independent of the Schrodinger equation"Kiefer, C. On the
interpretation of quantum theory – from Copenhagen to the present day
7. ^ The Many Worlds Interpretation of Quantum Mechanics
8. ^ Gribbin, J. Q for Quantum
9. ^ Erwin Schrödinger, in an article in the Proceedings of the American Philosophical
Society, 124, 323-38.
10. ^ Michael price on nonlocality in Many Worlds
11. ^ Relativity and Causality in the Transactional Interpretation
12. ^ Collapse and Nonlocality in the Transactional Interpretation
13. ^ "God does not throw dice" quote
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14. ^ 'Since the Universe naturally contains all of its observers, the problem arises to
come up with an interpretation of quantum theory that contains no classical realms on
the fundamental level.'Kiefer, C. On the interpretation of quantum theory from
Copenhagen to the present day
15. ^ http://home.fnal.gov/~skands/slides/A-Quantum-Journey.ppt
16. ^ "Shut up and calculate" quote.
[edit] Further reading
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G. Weihs et al., Phys. Rev. Lett. 81 (1998) 5039
M. Rowe et al., Nature 409 (2001) 791.
J.A. Wheeler & W.H. Zurek (eds) , Quantum Theory and Measurement, Princeton
University Press 1983
A. Petersen, Quantum Physics and the Philosophical Tradition, MIT Press 1968
H. Margeneau, The Nature of Physical Reality, McGraw-Hill 1950
M. Chown, Forever Quantum, New Scientist No. 2595 (2007) 37.
T. Schürmann, A Single Particle Uncertainty Relation, Acta Physica Polonica B39
(2008) 587. [1]
[edit] Video Demonstration
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From Movie "Down the Rabbit Hole" (sequel to What the Bleep Do We Know!?)
[edit] External links
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Copenhagen Interpretation (Stanford Encyclopedia of Philosophy)
Physics FAQ section about Bell's inequality
The Copenhagen Interpretation of Quantum Mechanics
Preprint of Afshar Experiment
This Quantum World What is quantum mechanics trying to tell us about the nature of
Nature?
Retrieved from "http://en.wikipedia.org/wiki/Copenhagen_interpretation"
Categories: Fundamental physics concepts | Interpretations of Quantum Mechanics | Quantum
measurement | University of Copenhagen
Ensemble Interpretation
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Quantum mechanics
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Uncertainty principle
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The Ensemble Interpretation, or Statistical Interpretation of quantum mechanics, is an
interpretation that can be viewed as a minimalist interpretation; it is a quantum mechanical
interpretation that claims to make the fewest assumptions associated with the standard
mathematical formalization. At its heart, it takes the statistical interpretation of Max Born to
the fullest extent. The interpretation states that the wave function does not apply to an
individual system – or for example, a single particle – but is an abstract mathematical,
statistical quantity that only applies to an ensemble of similar prepared systems or particles.
Probably the most notable supporter of such an interpretation was Albert Einstein:
“
The attempt to conceive the quantum-theoretical description as the complete
description of the individual systems leads to unnatural theoretical
interpretations, which become immediately unnecessary if one accepts the
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„
interpretation that the description refers to ensembles of systems and not to
individual systems.
—Albert Einstein[1]
To date, probably the most prominent advocate of the Ensemble Interpretation is Leslie E.
Ballentine, Professor at Simon Fraser University, and writer of the graduate-level textbook
"Quantum Mechanics, A Modern Development".
The ensemble interpretation, unlike other interpretations to the Copenhagen
Interpretation[citation needed], does not attempt to justify, or otherwise derive, or explain quantum
mechanics from any deterministic process, or make any other statement about the real nature
of quantum phenomena; it is simply a statement as to the manner of wave function
interpretation. It is identical in all of its predictions as is the standard interpretations.
Contents
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1 Measurement and Collapse
o 1.1 Criticism
2 Single particles
o 2.1 Criticism
3 Schrödinger's cat
4 The frequentist probability variation
o 4.1 Criticism
5 The quantum Zeno effect
6 Earlier Classical Ensemble Ideas
7 References
8 External links
[edit] Measurement and Collapse
The attraction of the ensemble interpretation is that it immediately dispenses with the
metaphysical issues associated with reduction of the state vector, Schrödinger cat states, and
other issues related to the concepts of multiple simultaneous states. As the ensemble
interpretation postulates that the wave function only applies to an ensemble of systems, there
is no requirement for any single system to exist in more than one state at a time, hence, the
wave function is never physically required to be "reduced". This can be illustrated by an
example:
Consider a classical die. If this is expressed in Dirac notation, the "state" of the die can be
represented by a "wave" function describing the probability of an outcome given by:
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It is clear that on each throw, only one of the states will be observed, but it is also clear that
there is no requirement for any notion of collapse of the wave function/reduction of the state
vector, or for the die to physically exist in the summed state. In the ensemble interpretation,
wave function collapse would make as much sense as saying that the number of children a
couple produced, collapsed to 3 from its average value of 2.4.
The state function is not taken to be physically real, or be a literal summation of states. The
wave function, is taken to be an abstract statistical function, only applicable to the statistics of
repeated preparation procedures, similar to classical statistical mechanics. It does not directly
apply to a single experiment, only the statistical results of many.
[edit] Criticism
David Mermin sees the Ensemble interpretation as being motivated by an adherence ("not
always acknowledged") to classical principles.
"For the notion that probabilistic theories must be about ensembles implicitly assumes that
probability is about ignorance. (The “hidden variables” are whatever it is that we are ignorant
of.) But in a non-determinstic world probability has nothing to do with incomplete
knowledge, and ought not to require an ensemble of systems for its interpretation".
He also emphasises the importance of describing single systems, rather than ensembles.
"The second motivation for an ensemble interpretation is the intuition that because quantum
mechanics is inherently probabilistic, it only needs to make sense as a theory of ensembles.
Whether or not probabilities can be given a sensible meaning for individual systems, this
motivation is not compelling. For a theory ought to be able to describe as well as predict the
behavior of the world. The fact that physics cannot make deterministic predictions about
individual systems does not excuse us from pursuing the goal of being able to describe them
as they currently are."[2]
[edit] Single particles
According to proponents of this interpretation, no single system is ever required to be
postulated to exist in a physical mixed state so the state vector does not need to collapse.
It can also be argued that this notion is consistent with the standard interpretation in that, in
the CI, statements about the exact system state prior to measurement can not be made. That is,
if it were possible to absolutely, physically measure say, a particle in two positions at once,
then QM would be falsified as QM explicitly postulates that the result of any measurement
must be a single eigenvalue of a single eigenstate.
[edit] Criticism
Alfred Nuemaier find fault with the applicability of the Ensemble Interpretation to small
systems.
"Among the traditional interpretations, the statistical interpretation discussed by Ballentine in
Rev. Mod. Phys. 42, 358-381 (1970) is the least demanding (assumes less than the
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Copenhagen interpretation and the Many Worlds interpretation) and the most consistent one.
It explains almost everything, and only has the disadvantage that it explicitly excludes the
applicability of QM to single systems or very small ensembles (such as the few solar
neutrinos or top quarks actually detected so far), and does not bridge the gulf between the
classical domain (for the description of detectors) and the quantum domain (for the
description of the microscopic system)". (spelling emended) [3]
[edit] Schrödinger's cat
The Ensemble Interpretation states that superpositions are nothing but subensembles of a
larger statistical ensemble. That being the case, the state vector would not apply to individual
cat experiments, but only to the statistics of many similar prepared cat experiments.
Proponents of this interpretation state that this makes the Schrödinger's cat paradox a trivial
non issue. However, the application of state vectors to individual systems, rather than
ensembles, has explanatory benefits, in areas like single-particle twin-slit experiments and
quantum computing. (See here). As an avowedly minimalist approach, the Ensemble
Interpretation does not offer any specific alternative explanation for these phenomena.
[edit] The frequentist probability variation
The claim that the wave functional approach fails to apply to single particle experiments
cannot be taken as a claim that quantum fails in describing single-particle phenomena. In fact,
it gives correct results within the limits of a probablistic or stochastic theory.
Probability always require a set of multiple data, and thus single-particle experiments are
really part of an ensemble — an ensemble of individual experiments that are performed one
after the other over time. In particular, the interference fringes seen in the double slit
experiment require repeated trials to be observed.
[edit] Criticism
One possible criticism is that the procedure described is essentially frequentist. The
frequentist approach suffices for classical probability, but quantum mechanics is a theory of
quantum probability, which is more general.[4][5]
[edit] The quantum Zeno effect
Leslie Ballantine promoted the Ensemble Interpretation in his book "Quantum Mechanics, A
Modern Development". In it [6], he described what he called the "Watched Pot Experiment".
His argument was that, under certain circmstances, a repeatedly measured system, such as an
unstable nucleus, would be prevented from decaying by the act of measurement itself. He
initially presented this as a kind of reductio ad absurdum of wave function collapse. [7]
The effect has been shown to be real. (It is more widely known as the quantum Zeno effect).
He later wrote papers claiming that it could be explained without wave function collapse.[8]
[edit] Earlier Classical Ensemble Ideas
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Early proponents of statistical approaches regarded quantum mechanics as an approximation
to a classical theory. John Gribbin writes:
"The basic idea is that each quantum entity (such as an electron or a photon) has precise
quantum properties (such as position or momentum) and the quantum wavefunction is related
to the probability of getting a particular experimental result when one member (or many
members) of the ensemble is selected by an experiment"
However, hopes for turning quantum mechanics back into a classical theory were dashed.
Gribbin continues:
"There are many difficulties with the idea, but the killer blow was struck when individual
quantum entities such as photons were observed behaving in experiments in line with the
quantum wave function description. The Ensemble interpretation is now only of historical
interest."
[9]
Willem de Muynck describes an "objective-realist" version of the Ensemble Interpretation
featuring counterfactual definiteness and the "possessed values principle", in which values of
the quantum mechanical observables may be attributed to the object as objective properties
the object possesses independent of observation. He states that there are "strong indications, if
not proofs" that neither is a possible assumption.[10]
[edit] References
1.
2.
3.
4.
5.
6.
7.
^ Einstein: Philosopher-Scientist, ed. P.A. Schilpp (Harper & Row, New York)
^ Mermin, N.D. The Ithaca interpretation
^ Alfred Neumaier's FAQ
^ Stanford Encyclopedia of Philosophy
^ Baez, J. Bayesian probability Theory and Quantum Mechanics
^ Ballentine, L. Quantum Mechanics, A Modern Development(p 342)
^ "Like the old saying "A watched pot never boils", we have been led to the
conclusion that a contineously observed system never changes its state! This
conclusion is, of course false. The fallacy clearly results from the assertion that if an
observation indicates no decay, then the state vector must be |y_u>. Each successive
observation in the sequence would then "reduce" the state back to its initial value
|y_u>, and in the limit of continuous observation there could be no change at all. Here
we see that it is disproven by the simple empirical fact that [..] continuous observation
does not prevent motion. It is sometimes claimed that the rival interpretations of
quantum mechanics differ only in philosophy, and can not be experimentally
distinguished. That claim is not always true. as this example proves" ".Ballentine, L.
Quantum Mechanics, A Modern Development(p 342)
8. ^ "The quantum Zeno effect is not a general characteristic of continuous
measurements. In a recently reported experiment [Itano et al., Phys. Rev. A 41, 2295
(1990)], the inhibition of atomic excitation and deexcitation is not due to any
‘‘collapse of the wave function,’’ but instead is caused by a very strong perturbation
due to the optical pulses and the coupling to the radiation field. The experiment should
not be cited as providing empirical evidence in favor of the notion of ‘‘wave-function
collapse.’’" Physical Review
14
9. ^ John Gribbin, Q for Quantum
10. ^ Quantum Mechanics the Way I see it
[edit] External links






Quantum Mechanics as Wim Muynk sees it
Einstein's Reply to Criticisms
Kevin Aylwards's account of the Ensemble Interpretation
Detailed Ensemble Interpretation by Marcel Nooijen
Pechenkin, A.A. The early statistical interpretations of quantum mechanics
Krüger, T. An attempt to close the Einstein–Podolsky–Rosen debate
Retrieved from "http://en.wikipedia.org/wiki/Ensemble_Interpretation"
Categories: Quantum measurement | Interpretations of Quantum Mechanics | Quantum
mechanics
Hidden variable theory
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This article is about a class of quantum mechanics theories. For hidden variables in
economics, see latent variable. For other uses, see Hidden variables (disambiguation).
Historically, in Physics, Hidden Variable Theories were espoused by a minority of
physicists who argued that the statistical nature of quantum mechanics indicated that quantum
mechanics is "incomplete". Albert Einstein, the most famous proponent of hidden variables,
insisted that, "I am convinced God does not play dice"[1] — meaning that he believed that
physical theories must be deterministic to be complete.[2] It was thought that if hidden
variables exist, new physical phenomena beyond quantum mechanics are needed to explain
the universe as we know it.
The most famous such theory (because it gives the same answers as Quantum Mechanics, thus
invalidating the famous theorem by von Neumann that no hidden variable theory reproducing
the statistical predictions of QM is possible) is that of David Bohm, also known as the Causal
Interpretation of Quantum Mechanics. Bohm's (nonlocal) hidden variable is called the
Quantum Potential. Nowadays Bohm's theory is considered to be one of many Interpretations
of Quantum Mechanics which give a philosophical or realist meaning to the positivistic
calculations of the Quantum Mechanical procedure. It is in fact just a reformulation of
conventional Quantum Mechanics obtained by rearranging the equations and renaming the
variables. Nevertheless it is a hidden variable theory.
The major reference for Bohm's theory today is his posthumous book with Basil Hiley[3].
Contents
[hide]





1 Motivation
2 EPR Paradox & Bell's Theorem
3 Hidden-Variable Theory
4 References
5 See also
16
[edit] Motivation
Quantum mechanics is nondeterministic, meaning that it generally does not predict the
outcome of any measurement with certainty. Instead, it tells us what the probabilities of the
outcomes are. This leads to the situation where measurements of a certain property done on
two identical systems can give different answers. 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.
In other words, quantum mechanics as it stands might be an incomplete description of reality.
A minority[citation needed] of physicists maintain that underlying the probabilistic nature of the
universe is an objective foundation/property — the hidden variable. Most believe[citation needed],
however, that there is no deeper reality in quantum mechanics — experiments have shown a
vast class of hidden variable theories to be incompatible with observations.
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 quantum mechanics formalism looks like are also considered hidden variable
theories; for example Edward Nelson's stochastic mechanics.
[edit] EPR Paradox & Bell's Theorem
In 1935, Einstein, Podolsky and Rosen wrote a four-page paper titled "Can quantummechanical description of physical reality be considered complete?" that argued that such a
theory was in fact necessary, proposing the EPR Paradox as proof. In 1964, John Bell showed
through his famous theorem that if hidden variables exist, certain experiments could be
performed where the result would satisfy a Bell inequality. If, on the other hand, Quantum
entanglement is correct the Bell inequality would be violated. Another no-go theorem on
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[4](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.
[edit] Hidden-Variable Theory
A hidden-variable theory which is consistent with quantum mechanics would have to be nonlocal, maintaining the existence of instantaneous or faster than light noncausal relations
(correlations) between physically separated entities. The first hidden-variable theory was the
pilot wave theory by Louis de Broglie from the late 1920s. The currently best-known hiddenvariable theory, the Causal Interpretation, of the physicist and philosopher David Bohm,
created in 1952, is a non-local hidden variable theory. Those who believe the Bohm
Interpretation to be actually true (rather than just a model or interpretation) refer to it as
Bohmian Mechanics. We say they "reify" (make real) the Quantum Potential.
What Bohm did, based on an idea originally by Louis de Broglie, was to posit both the
quantum particle, e.g. an electron, and a hidden 'guiding wave' that governs its motion. Thus,
17
in this theory electrons are quite clearly particles. When you perform a double-slit experiment
(see wave-particle duality), they go through one slit rather than the other. However, their
choice of slit is not random but is governed by the guiding wave, resulting in the wave pattern
that is observed.
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 indeed Quantum Mechanics with which it is
exactly equivalent) are still locally causal but allow only nonlocal correlations (that is
information travel is still restricted to the speed of light). It points to a more holistic, mutually
interpenetrating and interacting view of the 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. The Bohm interpretation (as well as others) has also been the basis of some
books which attempt to connect physics with Eastern mysticism and consciousness.
Nevertheless this nonlocality is seen as a weakness of Bohm's theory by some physicists[5]
despite the fact that Quantum Mechanics itself is equally nonlocal.
Another possible weakness of Bohm's Theory is that some feel that it looks contrived. It was
deliberately designed to give predictions which are in all details identical to conventional
quantum mechanics. His aim was not to make a serious counterproposal but simply to
demonstrate that hidden-variables theories are indeed possible. His hope was that this could
lead to new insights and experiments that would lead beyond the current quantum theories.
Another type of deterministic theory[6] was recently introduced by Gerard 't Hooft. This
theory is motivated by the problems that are encountered when one tries to formulate a unified
theory of quantum gravity.
[edit] References
1. ^ private letter to Max Born, 4 December 1926, Albert Einstein Archives reel 8, item
180
2. ^ Einstein, A., Podolsky, B. and Rosen, N. (1935) Can Quantum-Mechanical
Description of Physical Reality Be Considered Complete?, Phys. Rev. 47, 777-780
3. ^ D.Bohm and B.J.Hiley, The Undivided Universe, Routledge, 1993, ISBN 0-41506588-7.
4. ^ Kwiat, P. G., et al. (1999) Ultrabright source of polarization-entangled photons,
Physical Review A 60, R773-R776
5. ^ "There is a certain irony here associated with the fact that most physicists (at least,
among those who have even heard of it) reject the de Broglie - Bohm theory because it
is explicitly non-local." Comment on Experimental realization of Wheeler’s delayedchoice GedankenExperiment - Travis Norsen
6. ^ 't Hooft, G. (1999) Quantum Gravity as a Dissipative Deterministic System, Class.
Quant. Grav. 16, 3263-3279
[edit] See also



Local hidden variable theory
Bell's theorem
Bell test experiments
18


Quantum mechanics
Bohm interpretation
Retrieved from "http://en.wikipedia.org/wiki/Hidden_variable_theory"
Categories: Quantum measurement
Transactional interpretation
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The transactional interpretation of quantum mechanics (TIQM) is an unusual interpretation
of quantum mechanics that describes quantum interactions in terms of a standing wave
formed by retarded (forward-in-time) and advanced (backward-in-time) waves. The
interpretation was first proposed by John G. Cramer in 1986. The author argues that it helps in
developing intuition for quantum processes, avoids the philosophical problems with the
Copenhagen interpretation and the role of the observer, and resolves various quantum
paradoxes.[1][2] Cramer uses TIQM in teaching quantum mechanics at the University of
Washington in Seattle.
The existence of both advanced and retarded waves as admissible solutions to Maxwell's
equations was proposed by Richard Feynman and John Archibald Wheeler in 1945 (cited in
original paper by J. Cramer). They used the idea to solve the problem of the self-energy of an
electron. Cramer revived their idea of two waves for his transactional interpretation of
quantum theory. While the ordinary Schrödinger equation does not admit advanced solutions,
its relativistic version does, and these advanced solutions are the ones used by TIQM.
Suppose a particle (such as a photon) emitted from a source could interact with one of two
detectors. According to TIQM, the source emits a usual (retarded) wave forward in time, the
"offer wave", and when this wave reaches the detectors, each one replies with an advanced
wave, the "confirmation wave", that travels backwards in time, back to the source. The phases
of offer and confirmation waves are correlated in such a way that these waves interfere
positively to form a wave of the full amplitude in the spacetime region between emitting and
detection events, and they interfere negatively and cancel out elsewhere in spacetime (i.e.,
before the emitting point and after the absorption point). The size of the interaction between
the offer wave and a detector's confirmation wave determines the probability with which the
particle will strike that detector rather than the other one. In this interpretation, the collapse of
the wavefunction does not happen at any specific point in time, but is "atemporal" and occurs
along the whole transaction, the region of spacetime where offer and confirmation waves
interact. The waves are seen as physically real, rather than a mere mathematical device to
record the observer's knowledge as in some other interpretations of quantum mechanics.
John Cramer has argued that the transactional interpretation is consistent with the outcome of
the Afshar experiment, while the Copenhagen interpretation and the many-worlds
interpretation are not.[3]
Contents
[hide]




1 See also
2 References
3 Further reading
4 External links
20
[edit] See also

Wheeler–Feynman absorber theory
[edit] References
1. ^ The Transactional Interpretation of Quantum Mechanics by John Cramer. Reviews
of Modern Physics 58, 647-688, July (1986)
2. ^ An Overview of the Transactional Interpretation by John Cramer. International
Journal of Theoretical Physics 27, 227 (1988)
3. ^ A Farewell to Copenhagen?, by John Cramer. Analog, December 2005.
[edit] Further reading

Tim Maudlin, Quantum Non-Locality and Relativity, Blackwell Publishers 2002,
ISBN 0-631-23220-6 (discusses a gedanken experiment designed to refute the TIQM)
[edit] External links


Pavel V. Kurakin, George G. Malinetskii, How bees can possibly explain quantum
paradoxes, Automates Intelligents (February 2, 2005). (This paper tells about a work
attempting to develop TIQM further)
Cramer's Transactional Interpretation and Causal Loop Problems (quant-ph/0408109)
by Ruth E Kastneran, an attempt to refute Maudlin's refutation
Retrieved from "http://en.wikipedia.org/wiki/Transactional_interpretation"
Categories: Interpretations of Quantum Mechanics | Quantum measurement
Many-worlds interpretation
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21
[show]Background
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The many-worlds interpretation or MWI (also known as relative state formulation, theory
of the universal wavefunction, parallel universes, many-universes interpretation or many
worlds), is an interpretation of quantum mechanics. Many-worlds denies the objective reality
of wavefunction collapse. Many-worlds then explains the subjective appearance of
wavefunction collapse with the mechanism of quantum decoherence. Consequently, manyworlds claims this resolves all the "paradoxes" of quantum theory since every possible
outcome to every event defines or exists in its own "history" or "world". In layman's terms,
this means that there are an infinite number of universes and that everything that could
possibly happen in our universe (but doesn't) does happen in another.
Proponents argue that MWI reconciles how we can perceive non-deterministic events (such as
the random decay of a radioactive atom) with the deterministic equations of quantum physics.
Prior to many worlds this had been viewed as a single "world-line". Many-worlds rather
views it as a many-branched tree where every possible branch of history is realised.
The relative state formulation is due to Hugh Everett[1] who formulated it in 1957. Later, this
formulation was popularised and renamed many worlds by Bryce Seligman DeWitt in the
1960s and '70s.[2][3][4][5] The decoherence approach to interpreting quantum theory has been
further explored and developed[6][7][8] becoming quite popular, taken as a class overall. MWI
is one of many Multiverse hypotheses in physics and philosophy. It is currently considered a
22
mainstream interpretation along with the other decoherence interpretations and the
Copenhagen interpretation.
The many worlds interpretation offers the possibility of deriving the probability interpretation
of quantum mechanics from other assumptions. In fact, this was first done by Everett and
DeWitt in the 1950's, but the old argument was criticized on philosophical grounds. In a
September 2007 conference[9] David Wallace reports on a proof by Deutsch and himself of
the Born Rule starting from Everettian assumptions[10] and this has been reported in the press
as support for parallel universes.[11][12]
Contents
[hide]


















1 Outline
2 Wavefunction collapse and the problem of interpretation
3 Advantages
4 Objections
5 Brief overview
6 Relative state
7 Comparative properties and experimental support
o 7.1 Copenhagen interpretation
o 7.2 Quantum suicide
o 7.3 The universe and false vacuum
o 7.4 Many-minds
8 Axiomatics
9 An example
10 Partial trace and relative state
o 10.1 Decohered states as relative states
o 10.2 Multiple branching
o 10.3 General quantum operations
o 10.4 Branching
11 Quantum probabilities explained by continuous branching
12 Acceptance among physicists
13 Speculative implications
o 13.1 Choice and travel
o 13.2 Quantum suicide
o 13.3 Time travel
14 Many worlds in literature and science fiction
15 See also
16 External links
17 Notes and references
18 Further reading
[edit] Outline
Although several versions of MWI have been proposed since Hugh Everett's original work,[1]
they all contain one key idea: the equations of physics that model the time evolution of
23
systems without embedded observers are sufficient for modelling systems which do contain
observers; in particular there is no observation-triggered wavefunction collapse which the
Copenhagen interpretation proposes. Provided the theory is linear with respect to the
wavefunction the exact form of the quantum dynamics modelled, be it the non-relativistic
Schrödinger equation, relativistic quantum field theory or some form of quantum gravity or
string theory, does not alter the validity of MWI since MWI is a metatheory applicable to all
linear quantum theories, and there is no experimental evidence for any non-linearity of the
wavefunction in physics.[13][14] MWI's main conclusion is that the universe (or multiverse in
this context) is composed of a quantum superposition of very many, possibly infinitely many,
increasingly divergent, non-communicating parallel universes or quantum worlds.[5]
The idea of MWI originated in Everett's Princeton Ph.D. thesis "The Theory of the Universal
Wavefunction",[5] developed under his thesis advisor John Archibald Wheeler, a shorter
summary of which was published in 1957 entitled "Relative State Formulation of Quantum
Mechanics" (Wheeler contributed the title "relative state";[15] Everett originally called his
approach the "Correlation Interpretation"). The phrase "many worlds" is due to Bryce
DeWitt,[5] who was responsible for the wider popularisation of Everett's theory, which had
been largely ignored for the first decade after publication. DeWitt's phrase "many-worlds" has
become so much more popular than Everett's "Universal Wavefunction" or Everett-Wheeler's
"Relative State Formulation" that many forget that this is only a difference of terminology;
the content of all three papers is the same.
The many-worlds interpretation shares many similarities with later, other "post-Everett"
interpretations of quantum mechanics which also use decoherence to explain the process of
measurement or wavefunction collapse. MWI treats the other histories or worlds as real since
it regards the universal wavefunction as the "basic physical entity"[16] or "the fundamental
entity, obeying at all times a deterministic wave equation".[17] The other decoherent
interpretations, such as many histories, consistent histories, the Existential Interpretation etc,
either regard the extra quantum worlds as metaphorical in some sense, or are agnostic about
their reality; it is sometimes hard to distinguish between the different varieties. MWI is
distinguished by two qualities: it assumes realism[16][17], which it assigns to the wavefunction,
and it has the minimal formal structure possible, rejecting any hidden variables, quantum
potential, any form of a collapse postulate (i.e. Copenhagenism) or mental postulates (such as
the many-minds interpretation makes).
Many worlds is often referred to as a theory, rather than just an interpretation, by those who
propose that many worlds can make testable predictions (such as David Deutsch) or is
falsifiable (such as Everett) or that all the other, non-MWI, are inconsistent, illogical or
unscientific in their handling of measurements; Hugh Everett argued that his formulation was
a metatheory, since it made statements about other interpretations of quantum theory; that it
was the "only completely coherent approach to explaining both the contents of quantum
mechanics and the appearance of the world"[2].
[edit] Wavefunction collapse and the problem of
interpretation
As with the other interpretations of quantum mechanics, the many-worlds interpretation is
motivated by behavior that can be illustrated by the double-slit experiment. When particles of
light (or anything else) are passed through the double slit, a calculation assuming wave-like
24
behavior of light is needed to identify where the particles are likely to be observed. Yet when
the particles are observed in this experiment, they appear as particles (i.e. at definite places)
and not as non-localized waves.
The Copenhagen interpretation of quantum mechanics proposed a process of "collapse" in
which an indeterminate quantum system would probabilistically collapse down onto, or select,
just one determinate outcome to "explain" this phenomenon of observation. Wavefunction
collapse was widely regarded as artificial and ad-hoc, so an alternative interpretation in which
the behavior of measurement could be understood from more fundamental physical principles
was considered desirable.
Everett's Ph.D. work provided such an alternative interpretation. Everett noted that for a
composite system (for example that formed by a particle interacting with a measuring
apparatus, or more generally by a subject (the "observer") observing an object (the "observed"
system) the statement that a subsystem (i.e. the observer or the observed) has a well-defined
state is meaningless -- in modern parlance the subsystem states have become entangled -- we
can only specify the state of one subsystem relative to the state of the other subsystem, i.e. the
state of the observer and the observed are correlated. This led Everett to derive from the
unitary, deterministic dynamics alone (i.e. without assuming wavefunction collapse) the
notion of a relativity of states of one subsystem relative to another.
Everett noticed that the unitary, deterministic dynamics alone decreed that after an
observation is made each element of the quantum superposition of the combined subjectobject wavefunction contains two relative states: a "collapsed" object state and an associated
observer who has observed the same collapsed outcome; what the observer sees and the state
of the object are correlated. The subsequent evolution of each pair of relative subject-object
states proceeds with complete indifference as to the presence or absence of the other elements,
as if wavefunction collapse has occurred, which has the consequence that later observations
are always consistent with the earlier observations. Thus the appearance of the object's
wavefunction's collapse has emerged from the unitary, deterministic theory itself. (This
answered Einstein's early criticism of quantum theory, that the theory should define what is
observed, not for the observables to define the theory[18] .)
Since Everett stopped doing research in theoretical physics shortly after obtaining his Ph.D.,
much of the elaboration of his ideas was carried out by other researchers and forms the basis
of much of the decoherent approach to quantum measurement.
[edit] Advantages


MWI removes the observer-dependent role in the quantum measurement process by
replacing wavefunction collapse with quantum decoherence. Since the role of the
observer lies at the heart of most, if not all, "quantum paradoxes" this automatically
resolves a number of problems; see for example Schrödinger's cat thought-experiment,
the EPR paradox, von Neumann's "boundary problem" and even wave-particle duality.
Quantum cosmology also becomes intelligible, since there is no need anymore for an
observer outside of the universe.
MWI allows quantum mechanics to become a realist, deterministic, local theory
making it more akin to classical physics (including the theory of relativity), at the
expense of losing counterfactual definiteness.
25
The simplest way to see that the many-worlds metatheory is a local theory is to note that it
requires that the wavefunction obey some relativistic wave equation, the exact form of which
is currently unknown, but which is presumed to be locally Lorentz invariant at all times and
everywhere. This is equivalent to imposing the requirement that locality is enforced at all
times and everywhere. Therefore many-worlds is a local theory. Another way of seeing this is
examine how macrostates evolve. Macrostates descriptions of objects evolve in a local
fashion. Worlds split as the macrostate description divides inside the light cone of the
triggering event. Thus the splitting is a local process, transmitted causally at light or sub-light
speeds.
—Michael Clive Price[3]


MWI (or other, broader multiverse considerations) provides a context for the anthropic
principle which may provide an explanation for the fine-tuned universe.
MWI, being a decoherent formulation, is axiomatically more streamlined than the
Copenhagen and other collapse interpretations; and thus favoured under certain
interpretations of Ockham's razor. Of course there are other decoherent interpretations
that also possess this advantage with respect to the collapse interpretations.
[edit] Objections

The many worlds interpretation is very vague about the ways to determine when
splitting happens, and nowadays usually the criterion is that the two branches have
decohered. However, present day understanding of decoherence does not allow a
completely precise, self contained way to say when the two branches have
decohered/"do not interact", and hence many worlds interpretation remains arbitrary.
This is the main objection opponents of this interpretation raise,[citation needed] saying that
it is not clear what is precisely meant by branching, and point to lack of self contained
criterion specifying branching to be described.
MWI response: the decoherence or "splitting" or "branching" is complete when the
measurement is complete. In Dirac notation a measurement is complete when:
where O[i] represents the observer having detected the object system in the i-th state.
Before the measurement has started the observer states are identical; after the
measurement is complete the observer states are orthonormal.[5][1] Thus a measurement
defines the branching process: the branching is as well- or ill- defined as the
measurement is. Thus branching is complete when the measurement is complete.
Since the role of the observer and measurement per se plays no special role in MWI
(measurements are handled as all other interactions are) there is no need for a precise
definition of what an observer or a measurement is -- just as in Newtonian physics no
precise definition of either an observer or a measurement was required or expected. In
all circumstances the universal wavefunction is still available to give a complete
description of reality.
Objections response: the MWI response states no special role nor need for precise
definition of measurement in MWI, yet uses the word "measurement" in part of its
main argument.
MWI response: "measurements" are treated a subclass of interactions, which induce
subject-object correlations in the combined wavefunction. There is nothing special
26
about measurements (they don't trigger any wave function collapse, for example); they
are just another unitary time development process.
Also, it is a common misconception to think that branches are completely separate. In
Everett's formulation, they may in principle quantum interfere with each other in the
future,[19] although this requires all "memory" of the earlier branching event to be lost,
so no observer ever sees another branch of reality.[citation needed]
Fundamentally, any arguments based on the definition of "branching" or
"decoherence" are simply based on a misunderstanding of MWI.

There is circularity in Everett's measurement theory. Under the assumptions made by
Everett, there are no 'good observations' as defined by him, and since his analysis of
the observational process depends on the latter, it is void of any meaning. The concept
of a 'good observation' is the projection postulate in disguise and Everett's analysis
simply derives this postulate by having assumed it, without any discussion.[20] Talk of
probability in Everett presumes the existence of a preferred basis to identify
measurement outcomes for the probabilities to range over. But the existence of a
preferred basis can only be established by the process of decoherence, which is itself
probabilistic.[21]
MWI response: Everett's treatment of observations / measurements covers both
idealised good measurements and the more general bad or approximate cases.[22] Thus
it is legitimate to analyse probability in terms of measurement; no circularity is
present.

We cannot be sure that the universe is a quantum multiverse until we have a theory of
everything and, in particular, a successful theory of quantum gravity.[23] If the final
theory of everything is non-linear with respect to wavefunctions then many-worlds
would be invalid.[1][2][3][4][5]
MWI response: all accepted quantum theories of fundamental physics are linear with
respect to the wavefunction. Whilst quantum gravity or string theory may be nonlinear in this respect there is no evidence to indicate this at the moment.[13][14]
[edit] Brief overview
In Everett's formulation, a measuring apparatus M and an object system S form a composite
system, each of which prior to measurement exists in well-defined (but time-dependent)
states. Measurement is regarded as causing M and S to interact. After S interacts with M, it is
no longer possible to describe either system by an independent state. According to Everett, the
only meaningful descriptions of each system are relative states: for example the relative state
of S given the state of M or the relative state of M given the state of S.
27
Schematic representation of pair of "smallest possible" quantum mechanical systems prior to
interaction : Measured system S and measurement apparatus M. Systems such as S are
referred to as 1-qubit systems.
In DeWitt's formulation, the state of S after a sequence of measurements is given by a
quantum superposition of states, each one corresponding to an alternative measurement
history of S.
For example, consider the smallest possible truly quantum system S, as shown in the
illustration. This describes for instance, the spin-state of an electron. Considering a specific
axis (say the z-axis) the north pole represents spin "up" and the south pole, spin "down". The
superposition states of the system are described by (the surface of) a sphere called the Bloch
sphere. To perform a measurement on S, it is made to interact with another similar system M.
After the interaction, the combined system is described by a state that ranges over a sixdimensional space (the reason for the number six is explained in the article on the Bloch
sphere). This six-dimensional object can also be regarded as a quantum superposition of two
"alternative histories" of the original system S, one in which "up" was observed and the other
in which "down" was observed. Each subsequent binary measurement (that is interaction with
a system M) causes a similar split in the history tree. Thus after three measurements, the
system can be regarded as a quantum superposition of 8= 2 × 2 × 2 copies of the original
system S.
The accepted terminology is somewhat misleading because it is incorrect to regard the
universe as splitting at certain times; at any given instant there is one state in one universe.
28
Schematic illustration of splitting as a result of a repeated measurement.
[edit] Relative state
The goal of the relative-state formalism, as originally proposed by Everett in his 1957
doctoral dissertation, was to interpret the effect of external observation entirely within the
mathematical framework developed by Paul Dirac, von Neumann and others, discarding
altogether the ad-hoc mechanism of wave function collapse. Since Everett's original work,
there have appeared a number of similar formalisms in the literature. One such idea is
discussed in the next section.
The relative-state interpretation makes two assumptions. The first is that the wavefunction is
not simply a description of the object's state, but that it actually is entirely equivalent to the
object, a claim it has in common with some other interpretations. The second is that
observation or measurement has no special role, unlike in the Copenhagen interpretation
which considers the wavefunction collapse as a special kind of event which occurs as a result
of observation.
The many-worlds interpretation is DeWitt's popularisation of Everett's work, who had referred
to the combined observer-object system as being split by an observation, each split
corresponding to the different or multiple possible outcomes of an observation. These splits
generate a possible tree as shown in the graphic below. Subsequently DeWitt introduced the
term "world" to describe a complete measurement history of an observer, which corresponds
roughly to a single branch of that tree. Note that "splitting" in this sense, is hardly new or
even quantum mechanical. The idea of a space of complete alternative histories had already
been used in the theory of probability since the mid 1930s for instance to model Brownian
motion.
Partial trace as relative state. Light blue rectangle on upper left denotes system in pure state.
Trellis shaded rectangle in upper right denotes a (possibly) mixed state. Mixed state from
observation is partial trace of a linear superposition of states as shown in lower left-hand
corner.
Under the many-worlds interpretation, the Schrödinger equation, or relativistic analog, holds
all the time everywhere. An observation or measurement of an object by an observer is
modeled by applying the wave equation to the entire system comprising the observer and the
object. One consequence is that every observation can be thought of as causing the combined
29
observer-object's wavefunction to change into a quantum superposition of two or more noninteracting branches, or split into many "worlds". Since many observation-like events have
happened, and are constantly happening, there are an enormous and growing number of
simultaneously existing states.
If a system is composed of two or more subsystems, the system's state will be a superposition
of products of the subsystems' states. Once the subsystems interact, their states are no longer
independent. Each product of subsystem states in the overall superposition evolves over time
independently of other products. The subsystems states have become correlated or entangled
and it is no longer possible to consider them independent of one another. In Everett's
terminology each subsystem state was now correlated with its relative state, since each
subsystem must now be considered relative to the other subsystems with which it has
interacted.
Successive measurements with successive splittings
[edit] Comparative properties and experimental support
One of the salient properties of the many-worlds interpretation is that observation does not
require an exceptional construct (such as wave function collapse) to explain it. Many
physicists, however, dislike the implication that there are infinitely many non-observable
alternate universes.
As of 2006, there are no practical experiments that distinguish between Many-Worlds and
Copenhagen. There may be cosmological, observational evidence.
[edit] Copenhagen interpretation
In the Copenhagen interpretation, the mathematics of quantum mechanics allows one to
predict probabilities for the occurrence of various events. In the many-worlds interpretation,
all these events occur simultaneously. What meaning should be given to these probability
calculations? And why do we observe, in our history, that the events with a higher computed
probability seem to have occurred more often? One answer to these questions is to say that
there is a probability measure on the space of all possible universes, where a possible universe
is a complete path in the tree of branching universes. This is indeed what the calculations
30
give. Then we should expect to find ourselves in a universe with a relatively high probability
rather than a relatively low probability: even though all outcomes of an experiment occur,
they do not occur in an equal way. As an interpretation which (like other interpretations) is
consistent with the equations, it is hard to find testable predictions of MWI.
[edit] Quantum suicide
There is a rather more dramatic test than the one outlined above for people prepared to put
their lives on the line: use a machine which kills them if a random quantum decay happens. If
MWI is true, they will still be alive in the world where the decay didn't happen and would feel
no interruption in their stream of consciousness. By repeating this process a number of times,
their continued consciousness would be arbitrarily unlikely unless MWI was true, when they
would be alive in all the worlds where the random decay was on their side. From their
viewpoint they would be immune to this death process. Clearly, if MWI does not hold, they
would be dead in the one world. Other people would generally just see them die and would
not be able to benefit from the result of this experiment. See Quantum suicide.
[edit] The universe and false vacuum
Some Cosmologists argue that the universe is in a False vacuum state. There is also the claim
that the universe should have already experienced quantum tunnelling to a true vacuum state.
This has not happened. That may increase the probability that many-worlds is true.
See Possibility that the universe is a false vacuum.
[edit] Many-minds
The many-worlds interpretation should not be confused with the many-minds interpretation
which postulates that it is only the observers' minds that split instead of the whole world.
[edit] Axiomatics
The existence of many worlds in superposition is not accomplished by introducing some new
axiom to quantum mechanics, but on the contrary by removing the axiom of the probabilistic
collapse of the wave packet: All the possible consistent states of the measured system and the
measuring apparatus (including the observer) are present in a physically real quantum
superposition, not just formally mathematical superposition, as in other interpretations. (Such
a superposition of consistent state combinations of different systems is called an entangled
state.)
Hartle[24] showed that in Everett's relative-state theory, Born's probability law
The probability of an observable A to have the value a in a normalized state
absolute square of the eigenvalue component of the state corresponding to the
is the
eigenvalue a:
no longer has to be considered an axiom or postulate. It can rather be derived from the other
axioms of quantum mechanics. All that has to be assumed is that if the state
is an
eigenstate
of the observable A, then the result a of the measurement is certain. This means
31
that a second axiom of quantum mechanics can be removed. Hartle's derivation only works in
a theory (like Everett's) that does not cut away ("collapse") any superposition components of
the wave function. In other interpretations it is not comprehensible why the absolute square is
used and not some other arbitrary, more complicated expression of the eigenvalue component
say, the square root or some polynomial of its norm.
As a consequence Everett's interpretation or metatheory is an alternative formulation of
quantum theory requiring fewer axioms than previously required and thus favoured by
interpretations of the "Occam's razor" heuristic that emphasize simplicity of the mathematical
or logical structure of a theory (as opposed to interpretations that emphasize a minimal
number of hypothesized entities or some other aspect).
One might argue that postulating the existence of many worlds is some kind of axiomatic
assumption, but each world is merely an element in the quantum superposition of the
universal wavefunction; quantum superpositions are a common and indispensable part of all
interpretations of quantum theory, as is most clearly illustrated in the path integral
formulation of quantum mechanics. Even the simple reflection of a photon from a mirror
becomes amazingly convoluted when looked at from this perspective, as the photon follows
all paths instead of just following the incident and reflected rays, and destructively interferes
with itself on all paths save the classical. Everett's theory just considers it a real phenomenon
in nature and applies it to macroscopic systems in the same way as it is conventionally applied
to microscopic systems.
[edit] An example
MWI describes measurements as a formation of an entangled state which is a perfectly linear
process (in terms of quantum superpositions) without any collapse of the wave function. For
illustration, consider a Stern-Gerlach experiment and an electron or a silver atom passing this
apparatus with a spin polarization in the x direction and thus a superposition of a spin up and
a spin down state in z-direction. As a measuring apparatus, take a tracking chamber or another
nonabsorbing particle detector; let the electron pass the apparatus and reach the same site in
the end on either way so that except for the z-spin polarization the state of the electron is
finally the same regardless of the path taken (see The Feynman Lectures on Physics for a
detailed discussion of such a setup). Before the measurement, the state of the electron and the
measuring apparatus is:
The state is factorizable into a tensor factor for the electron and another factor for the
measurement apparatus. After the measurement, the state is:
32
The state is no longer factorizable -- regardless of the vector basis chosen. As an illustration,
understand that the following state is factorizable:
since it can be written as
(which might be not so obvious if another vector basis is chosen for the states).
The state of the above experiment is decomposed into a sum of two so-called entangled states
("worlds") both of which will have their individual history without any interaction between
the two due to the physical linearity of quantum mechanics (the superposition principle): All
processes in nature are linear and correspond to linear operators acting on each
superposition component individually without any notice of the other components being
present.
This would also be true for two non-entangled superposed states, but the latter can be detected
by interference which is not possible for different entangled states (without reversing the
entanglement first): Different entangled states cannot interfere; interactions with other
systems will only result in a further entanglement of them as well. In the example above, the
state of a Schrödinger cat watching the scene will be factorizable in the beginning (before
watching)
but not in the end:
This example also shows that it's not the whole world that is split up into "many worlds", but
only the part of the world that is entangled with the considered quantum event. This splitting
tends to extend by interactions and can be visualised by a zipper or a DNA molecule which
are in a similar way not completely opened instantaneously but gradually, element by
element.
Imaginative readers will even see the zipper structure and the extending splitting in the
formula:
33
If a system state is entangled with many other degrees of freedom (such as those in amplifiers,
photographs, heat, sound, computer memory circuits, neurons, paper documents) in an
experiment, this amounts to a thermodynamically irreversible process which is constituted
of many small individually reversible processes at the atomic or subatomic level as is
generally the case for thermodynamic irreversibility in classical or quantum statistical
mechanics. Thus there is -- for thermodynamic reasons -- no way for an observer to
completely reverse the entanglement and thus observe the other worlds by doing interference
experiments on them. On the other hand, for small systems with few degrees of freedom this
is feasible, as long as the investigated aspect of the system remains unentangled with the rest
of the world.
The MWI thus solves the measurement problem of quantum mechanics by reducing
measurements to cascades of entanglements.
The formation of an entangled state is a linear operation in terms of quantum superpositions.
Consider for example the vector basis
and the non-entangled initial state
The linear (and unitary and thus reversible) operation (in terms of quantum superpositions)
corresponding to the matrix
(in the above vector basis) will result in the entangled state
[edit] Partial trace and relative state
The state transformation of a quantum system resulting from measurement, such as the double
slit experiment discussed above, can be easily described mathematically in a way that is
consistent with most mathematical formalisms. We will present one such description, also
called reduced state, based on the partial trace concept, which by a process of iteration, leads
to a kind of branching many worlds formalism. It is then a short step from this many worlds
formalism to a many worlds interpretation.
For definiteness, let us assume that system is actually a particle such as an electron. The
discussion of reduced state and many worlds is no different in this case than if we considered
any other physical system, including an "observer system". In what follows, we need to
consider not only pure states for the system, but more generally mixed states; these are
described by certain linear operators on the Hilbert space H. Indeed, as the various
measurement scenarios point out, the set of pure states is not closed under measurement.
34
Mathematically, density matrices are statistical mixtures of pure states. Operationally a mixed
state can be identified to a statistical ensemble resulting from a specific lab preparation
process.
[edit] Decohered states as relative states
Suppose we have an ensemble of particles, prepared in such a way that its state S is pure. This
means that there is a unit vector
in H (unique up to phase) such that S is the projection
operator given in bra-ket notation by
Now consider an experimental setup to determine whether the particle has a particular
property: For example the property could be that the location of the particle is in some region
A of space. The experimental setup can be regarded either as a measurement of an observable
or as a filter. As a measurement, it measures the observable Q which takes the value 1 if the
particle is found in A and 0 otherwise. As a filter, it filters in those particles in the ensemble
which have the stated property of being in A and filtering out the others.
Mathematically, a property is given by a self-adjoint projection E on the Hilbert space H:
Applying the filter to an ensemble of particles, some of the particles of the ensemble are
filtered in, and others are filtered out. Now it can be shown that the operation of the filter
"collapses" the pure state in the following sense: it prepares a new mixed state given by the
density operator
where F = 1 - E.
To see this, note that as a result of the measurement, the state of the particle immediately after
the measurement is in an eigenvector of Q, that is one of the two pure states
with respective probabilities
The mathematical way of presenting this mixed state is by taking the following convex
combination of pure states:
which is the operator S1 above.
35
Remark. The use of the word collapse in this context is somewhat different that its use in
explanations of the Copenhagen interpretation. In this discussion we are not referring to
collapse or transformation of a wave into something else, but rather the transformation of a
pure state into a mixed one.
The considerations so far, are completely standard in most formalisms of quantum mechanics.
Now consider a "branched" system whose underlying Hilbert space is
where H2 is a two-dimensional Hilbert space with basis vectors
and
. The branched
space can be regarded as a composite system consisting of the original system (which is now
a subsystem) together with a non-interacting ancillary single qubit system. In the branched
system, consider the entangled state
We can express this state in density matrix format as
. This multiplies out to:
The partial trace of this mixed state is obtained by summing the operator coefficients of
and
in the above expression. This results in a mixed state on H. In fact, this
mixed state is identical to the "post filtering" mixed state S1 above.
To summarize, we have mathematically described the effect of the filter for a particle in a
pure state ψ in the following way:

The original state is augmented with the ancillary qubit system.

The pure state of the original system is replaced with a pure entangled state of the
augmented system and

The post-filter state of the system is the partial trace of the entangled state of the
augmented system.
[edit] Multiple branching
In the course of a system's lifetime we expect many such filtering events to occur. At each
such event, a branching occurs. In order for this to be consistent with the branching structure
as depicted in the illustration above, we must show that if a filtering event occurs in one path
from the root node of the tree, then we may assume it occurs in all branches. This shows that
the tree is highly symmetric, that is for each node n of the tree, the shape of the tree does not
change by interchanging the subtrees immediately below that node n.
36
In order to show this branching uniformity property, note that the same calculation carries
through even if original state S is mixed. Indeed, the post filtered state will be the density
operator:
The state S1 is the partial trace of
This means that to each subsequent measurement (or branching) along one of the paths from
the root of the tree to a leaf node corresponds to a homologous branching along every path.
This guarantees the symmetry of the many-worlds tree relative to flipping child nodes of each
node.
Superposition over paths through observation tree
[edit] General quantum operations
In the previous two sections, we have represented measurement operations on quantum
systems in terms of relative states. In fact there is a wider class of operations which should be
considered: these are called quantum operations. Considered as operations on density
operators on the system Hilbert space H, these have the following form:
where I is a finite or countably infinite index set. The operators Fi are called Kraus
operators.
37
Theorem. Let
Then
Moreover, the mapping V defined by
is such that
If γ is a trace-preserving quantum operation, then V is an isometric linear map
where the Hilbert direct sum is taken over copies of H indexed by elements of I. We can
consider such maps Φ as imbeddings. In particular:
Corollary. Any trace-preserving quantum operation is the composition of an isometric
imbedding and a partial trace.
This suggests that the many worlds formalism can account for this very general class of
transformations in exactly the same way that it does for simple measurements.
[edit] Branching
In general we can show the uniform branching property of the tree as follows: If
and
where
and
38
then a calculation shows
This also shows that in between the measurements given by proper (that is, non-unitary)
quantum operations, one can interpolate arbitrary unitary evolution.
[edit] Quantum probabilities explained by continuous
branching
Dr. David Deutsch along with Oxford colleagues have demonstrated mathematically that the
bush-like branching structure created by the universe splitting into parallel versions of itself
can explain the probabilistic nature of quantum outcomes. In the New Scientist article on the
discovery, Andy Albrecht, a physicist at the University of California at Davis, is quoted as
saying "This work will go down as one of the most important developments in the history of
science." Deutsch and his Oxford colleaques are thus seen to apparently bolster March - May
'07 internet postings of Dr. David Anacker (to physics cognoscenti including Lisa Randall,
Lee Smolin, David Deutsch, G. T'Hooft, S. Glashow, S. Weinberg, M. Kaku, L. Susskind,
et.al.) via internet archive earlier establishing agreement between predictive statistics of the
Everett and Copenhagen interpretations.[12]
[edit] Acceptance among physicists
There is a wide range of claims that are considered "many worlds" interpretations. It is often
noted by those who do not believe in MWI[25] that Everett himself was not entirely clear as to
what he meant; however MWI adherents believe they fully understand Everett's meaning,
pointing to his stated belief in quantum immortality (which requires absolute belief in the
reality of all the many worlds) and the reality of all components the uncollapsed universal
wavefunction[26].
"Many worlds"-like interpretations are now considered fairly mainstream within the quantum
physics community. For example, a poll of 72 leading physicists conducted by the American
researcher David Raub in 1995 and published in the French periodical Sciences et Avenir in
January 1998 recorded that nearly 60% thought many worlds interpretation was "true". Max
Tegmark (see reference to his web page below) also reports the result of a poll taken at a 1997
quantum mechanics workshop. According to Tegmark, "The many worlds interpretation
(MWI) scored second, comfortably ahead of the consistent histories and
Bohm interpretations." Other such unscientific polls have been taken at other conferences: see
for instance Michael Nielsen's blog [4] report on one such poll. Nielsen remarks that it
appeared most of the conference attendees "thought the poll was a waste of time". MWI
sceptics (for instance Asher Peres) argue that polls regarding the acceptance of a particular
interpretation within the scientific community, such as those mentioned above, cannot be used
as evidence supporting a specific interpretation's validity. However, others note that science is
39
a group activity (for instance, peer review) and that polls are a systematic way of revealing the
thinking of the scientific community.
A 2005 minor poll on the Interpretation of Quantum Mechanics workshop at the Institute for
Quantum Computing University of Waterloo produced contrary results, with the MWI as the
least favored.[5]
One of MWI's strongest advocates is David Deutsch.[27] According to Deutsch the single
photon interference pattern observed in the double slit experiment, can be explained by
interference of photons in multiple universes. Viewed in this way, the single photon
interference experiment is indistinguishable from the multiple photon interference
experiment. In a more practical vein, in one of the earliest papers on quantum computing,[28]
he suggested that parallelism that results from the validity of MWI could lead to "a method by
which certain probabilistic tasks can be performed faster by a universal quantum computer
than by any classical restriction of it". Deutsch has also proposed that when reversible
computers become conscious that MWI will be testable (at least against "naive"
Copenhagenism) via the reversible observation of spin.[29]
Asher Peres was an outspoken critic of MWI, for example in a section in his 1993 textbook
with the title Everett's interpretation and other bizarre theories. In fact, Peres questioned
whether MWI is really an "interpretation" or even if interpretations of quantum mechanics are
needed at all. Indeed, the many-worlds interpretation can be regarded as a purely formal
transformation, which adds nothing to the instrumentalist (i.e. statistical) rules of the quantum
mechanics. Perhaps more significantly, Peres seems to suggest that positing the existence of
an infinite number of non-communicating parallel universes is highly suspect as it violates
those interpretations of Occam's Razor that seek to minimize the number of hypothesized
entities. Proponents of MWI argue precisely the opposite, by applying Occam's Razor to the
set of assumptions rather than multiplicity of universes. In Max Tegmark's formulation, the
alternative to many worlds is the undesirable "many words", an allusion to the complexity of
von Neumann's collapse postulate).
MWI is considered by some to be unfalsifiable and hence unscientific because the multiple
parallel universes are non-communicating, in the sense that no information can be passed
between them. Others[29] claim MWI is directly testable. Everett regarded MWI as falsifiable
since any test that falsifies conventional quantum theory would also falsify MWI.[6]
According to Martin Gardner MWI has two different interpretations: real or unreal, and
claims that Stephen Hawking and Steve Weinberg favour the unreal interpretation.[30] Gardner
also claims that the interpretation favoured by the majority of physicists is that the other
worlds are not real in the same way as our world is real, whereas the "realist" view is
supported by MWI experts David Deutsch and Bryce DeWitt. However Stephen Hawking is
on record as a saying that the other worlds are as real as ours[31] and Tipler reports Hawking
saying that MWI is "trivially true" (scientific jargon for "obviously true", which Gardner
seems not to realise) if quantum theory applies to all reality[32]. Roger Penrose agrees with
Hawking that QM applied to the universe implies MW, although he considers the current lack
of a successful theory of quantum gravity negates the claimed universality of conventional
QM.[23]
[edit] Speculative implications
40
Speculative physics deals with questions also discussed in science fiction.
[edit] Choice and travel
Under the Many-Worlds interpretation, it is theoretically possible that every choice a person
makes results in the creation of two or more 'new' universes: one for each 'option' in a given
choice. Price gives evidence for both sides to the speculation. On the one hand he says that
quantum effects rarely or never affect human decisions. On the other hand he says that all
possible decisions are realized in some worlds.
In quantum terms each neuron is an essentially classical object. Consequently quantum noise
in the brain is at such a low level that it probably doesn't often alter, except very rarely, the
critical mechanistic behaviour of sufficient neurons to cause a decision to be different than we
might otherwise expect. (...) If both sides of a choice are selected in different worlds why
bother to spend time weighing the evidence before selecting? The answer is that whilst all
decisions are realised, some are realised more often than others - or to put to more precisely
each branch of a decision has its own weighting or measure which enforces the usual laws of
quantum statistics.
—Michael Clive Price[7]
It is further speculated that it might be possible to move 'between' these universes, of which
there would be an infinite number or a very large finite number. Price believes that travel
between worlds is impossible.
According to our present knowledge of physics whilst it is possible to detect the presence of
other nearby worlds, through the existence of interference effects, it is impossible travel to or
communicate with them. (...) the interfering worlds can't influence each other in the sense that
an experimenter in one of the worlds can arrange to communicate with their own, already
split-off, quantum copies in other worlds. (...) Since each component of a linear solution
evolves with complete indifference as to the presence or absence of the other terms/solutions
then we can conclude that no experiment in one world can have any effect on another
experiment in another world. Hence no communication is possible between quantum worlds.
—Michael Clive Price[8]
[edit] Quantum suicide
It has been claimed that there is an experiment that would clearly differentiate between the
many-worlds interpretation and other interpretations of quantum mechanics. It involves a
quantum suicide machine and an experimenter willing to risk death. However, at best, this
would only decide the issue for the experimenter; bystanders would learn nothing. The flip
side of quantum suicide is quantum immortality.
Another speculation is that the separate worlds remain weakly coupled (e.g. by gravity)
permitting "communication between parallel universes". This requires that gravity be a
classical force and not quantized.
The many-worlds interpretation has some similarity to modal realism in philosophy, which is
the view that the possible worlds used to interpret modal claims actually exist. Unlike
41
philosophy, however, in quantum mechanics counterfactual alternatives can influence the
results of experiments, as in the Elitzur-Vaidman bomb-testing problem or the Quantum Zeno
effect.
[edit] Time travel
The many-worlds interpretation could be one possible way to resolve the paradoxes that one
would expect to arise if time travel turns out to be permitted by physics (permitting closed
timelike curves and thus violating causality). Entering the past would itself be a quantum
event causing branching, and therefore the timeline accessed by the time traveller simply
would be another timeline of many. In that sense, it would make the Novikov self-consistency
principle unnecessary.
[edit] Many worlds in literature and science fiction
Main article: Parallel universe (fiction)
The many-worlds interpretation (and the somewhat related concept of possible worlds) have
been associated to numerous themes in literature, art and science fiction.
Some of these stories or films violate fundamental principles of causality and relativity, and
are extremely misleading since the information-theoretic structure of the path space of
multiple universes (that is information flow between different paths) is very likely
extraordinarily complex. Also see Michael Clive Price's FAQ referenced in the external links
section below where these issues (and other similar ones) are dealt with more decisively.
Another kind of popular illustration of many worlds splittings, which does not involve
information flow between paths, or information flow backwards in time considers alternate
outcomes of historical events. According to many worlds, most of the historical speculations
entertained within the alternate history genre are realised in parallel universes.
[edit] See also







Fabric of Reality
Interpretation of quantum mechanics
Many-minds interpretation
Multiverse
Multiple histories
Quantum decoherence
Quantum immortality - a thought experiment.
[edit] External links
Find more about Many-worlds interpretation on
Wikipedia's sister projects:
Dictionary definitions
42
Textbooks
Quotations
Source texts
Images and media
News stories
Learning resources











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The Many-Worlds Interpretation of Quantum Mechanics.
Michael Price's Everett FAQ -- a very clear presentation of the theory with some
additional insights
Against Many-Worlds Interpretations by Adrian Kent
Everett's Relative-State Formulation of Quantum Mechanics
Many-Worlds Interpretation of Quantum Mechanics
Max Tegmark's web page
Many Worlds is a "lost cause" according to R. F. Streater
The many worlds of quantum mechanics
Henry Stapp's critique of MWI, focusing on the basis problem
Translation of Schrödinger's Cat paper.
Everett interpretation on arxiv.org
Scientific American report on the Many Worlds 50th anniversary conference at
Oxford
Highfield, Roger (September 21, 2007), Parallel universe proof boosts time travel
hopes, The Daily Telegraph,
<http://www.telegraph.co.uk/earth/main.jhtml?xml=/earth/2007/09/21/sciuni121.xml
>. Retrieved on 26 October 2007 .
[edit] Notes and references
1. ^ a b c d Hugh Everett, Relative State Formulation of Quantum Mechanics, Reviews of
Modern Physics vol 29, (1957) pp 454-462.
2. ^ a b Cecile M. DeWitt, John A. Wheeler eds, The Everett-Wheeler Interpretation of
Quantum Mechanics, Battelle Rencontres: 1967 Lectures in Mathematics and Physics
(1968)
3. ^ a b Bryce Seligman DeWitt, Quantum Mechanics and Reality, Physics Today,23(9)
pp 30-40 (1970) also April 1971 letters followup
4. ^ a b Bryce Seligman DeWitt, The Many-Universes Interpretation of Quantum
Mechanics, Proceedings of the International School of Physics "Enrico Fermi"
Course IL: Foundations of Quantum Mechanics, Academic Press (1972)
5. ^ a b c d e f Bryce Seligman DeWitt, R. Neill Graham, eds, The Many-Worlds
Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton
University Press (1973), ISBN 0-691-08131-X Contains Everett's thesis: The Theory
of the Universal Wavefunction, pp 3-140.
43
6. ^ H. Dieter Zeh, On the Interpretation of Measurement in Quantum Theory,
Foundation of Physics, vol. 1, pp. 69-76, (1970).
7. ^ Wojciech Hubert Zurek, Decoherence and the transition from quantum to classical,
Physics Today, vol. 44, issue 10, pp. 36-44, (1991).
8. ^ Wojciech Hubert Zurek, Decoherence, einselection, and the quantum origins of the
classical, Reviews of Modern Physics, 75, pp 715-775, (2003)
9. ^ Perimeter Institute, Many worlds at 50 conference, September 21-24, 2007
10. ^ Perimeter Institute, Seminar overview, Probability in the Everett interpretation: state
of play, David Wallace - Oxford University, 21 Sept 2007
11. ^ Breitbart.com, Parallel universes exist - study, Sept 23 2007
12. ^ a b Merali, Zeeya (2007-09-21), "Parallel universes make quantum sense", New
Scientist (no. 2622), <http://space.newscientist.com/article/mg19526223.700-paralleluniverses-make-quantum-sense.html>. Retrieved on 20 October 2007 (Summary
only).
13. ^ a b Steven Weinberg, Dreams of a Final Theory: The Search for the Fundamental
Laws of Nature (1993), ISBN 0-09-922391-0, pg 68-69
14. ^ a b Steven Weinberg Testing Quantum Mechanics, Annals of Physics Vol 194 #2
(1989), pg 336-386
15. ^ John Archibald Wheeler, Geons, Black Holes & Quantum Foam, ISBN 0-39331991-1. pp 268-270
16. ^ a b Everett 1957, section 3, 2nd paragraph, 1st sentence
17. ^ a b Everett [1956]1973, "Theory of the Universal Wavefunction", chapter 6 (e)
18. ^ "Whether you can observe a thing or not depends on the theory which you use. It is
the theory which decides what can be observed." Albert Einstein to Werner
Heisenberg, objecting to placing observables at the heart of the new quantum
mechanics, during Heisenberg's 1926 lecture at Berlin; related by Heisenberg in 1968,
quoted by Abdus Salam, Unification of Fundamental Forces, Cambridge University
Press (1990) ISBN 0-521-37140-6, pp 98-101
19. ^ Tegmark, Max The Interpretation of Quantum Mechanics: Many Worlds or Many
Words?, 1998. To quote: "What Everett does NOT postulate: “At certain magic
instances, the world undergoes some sort of metaphysical 'split' into two branches that
subsequently never interact.” This is not only a misrepresentation of the MWI, but
also inconsistent with the Everett postulate, since the subsequent time evolution could
in principle make the two terms...interfere. According to the MWI, there is, was and
always will be only one wavefunction, and only decoherence calculations, not
postulates, can tell us when it is a good approximation to treat two terms as noninteracting."
20. ^ Comments on the Everett FAQ, added comment May 13, 2003
21. ^ Many worlds interpretation shown to be circular, David J Baker, Princeton
University, 11 April 2006
22. ^ Everett [1956]1973, "Theory of the Universal Wavefunction", chapter V, section 4
"Approximate Measurements", pp. 100-103 (e)
23. ^ a b Penrose, Roger (August 1991). Roger Penrose Looks Beyond the ClassicQuantum Dichotomy. Sciencewatch. Retrieved on 2007-10-21.
24. ^ James Hartle, Quantum Mechanics of Individual Systems, American Journal of
Physics, vol 36 (1968), # 8
25. ^ Jeffrey A. Barrett, The Quantum Mechanics of Minds and Worlds, Oxford
University Press, 1999. According to Barret (loc. cit. Chapter 6) "There are many
many-worlds interpretations."
26. ^ Eugene Shikhovtsev's Biography of Everett
44
27. ^ David Deutsch, The Fabric of Reality: The Science of Parallel Universes And Its
Implications, Penguin Books (1998), ISBN 0-14-027541-X
28. ^ David Deutsch, Quantum theory, the Church-Turing principle and the universal
quantum computer, Proceedings of the Royal Society of London A 400, (1985) , pp.
97–117
29. ^ a b Paul C.W. Davies, J.R. Brown, The Ghost in the Atom (1986) ISBN 0-521-313163, pp. 34-38: "The Many-Universes Interpretation", pp83-105 for David Deutsch's test
of MWI
30. ^ A response to Bryce DeWitt, Martin Gardner, May 2002
31. ^ Award winning 1995 Channel 4 documentary "Reality on the rocks" [1] where
Hawking states that the other worlds are as real as ours
32. ^ Tipler, Frank J. (2006-11-26). What About Quantum Theory? Bayes and the Born
Interpretation. arXiv, Cornell University. Retrieved on 2007-10-20. Page 1: "It is wellknown that if the quantum formalism applies to all reality, both to atoms, to humans,
to planets and to the universe itself then the Many Worlds Interpretation is trivially
true (to use an expression of Stephen Hawking, expressed to me in a private
conversation)."
[edit] Further reading









Jeffrey A. Barrett, The Quantum Mechanics of Minds and Worlds, Oxford University
Press, Oxford, 1999.
Julian Brown, Minds, Machines, and the Multiverse, Simon & Schuster, 2000, ISBN
0-684-81481-1
Asher Peres, Quantum Theory: Concepts and Methods, Kluwer, Dordrecht, 1993.
Mark A. Rubin, Locality in the Everett Interpretation of Heisenberg-Picture Quantum
Mechanics, Foundations of Physics Letters, 14, (2001) , pp. 301-322, arXiv:quantph/0103079
David Wallace, Harvey R. Brown, Solving the measurement problem: de BroglieBohm loses out to Everett, Foundations of Physics, arXiv:quant-ph/0403094
David Wallace, Worlds in the Everett Interpretation, Studies in the History and
Philosophy of Modern Physics, 33, (2002), pp. 637-661, arXiv:quant-ph/0103092
Paul C.W. Davies, Other Worlds, (1980) ISBN 0-460-04400-1
John A. Wheeler and Wojciech Hubert Zurek (eds), Quantum Theory and
Measurement, Princeton University Press, (1983), ISBN 0-691-08316-9
James P. Hogan, The Proteus Operation, Science Fiction involving the Many-Worlds
Interpretation, time travel and World War 2 history., baen; Reissue edition (August 1,
1996) ISBN-10: 0671877577
Retrieved from "http://en.wikipedia.org/wiki/Many-worlds_interpretation"
Categories: Interpretations of Quantum Mechanics | Quantum measurement
Consistent histories
From Wikipedia, the free encyclopedia
Jump to: navigation, search
45
Quantum mechanics
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In quantum mechanics, the consistent histories approach is intended to give a modern
interpretation of quantum mechanics, generalising the conventional Copenhagen
interpretation and providing a natural interpretation of quantum cosmology. The theory is
based on a consistency criterion that then allows the history of a system to be described so
that the probabilities for each history obey the rules of classical probability while being
consistent with the Schrödinger equation.
46
According to this interpretation of quantum mechanics, the purpose of a quantum-mechanical
theory is to predict probabilities of various alternative histories. A history is defined as a
sequence (product) of projection operators at different moments of time:
The symbol T indicates that the factors in the product are ordered chronologically according
to their values of ti,j: the "past" operators with smaller values of t appear on the right side, and
the "future" operators with greater values of t appear on the left side.
These projection operators can correspond to any set of questions that include all possibilities.
Examples might be the three projections meaning "the electron went through the left slit", "the
electron went through the right slit" and "the electron didn't go through either slit". One of the
aims of the theory is to show that classical questions such as, "where are my keys?" are
consistent. In this case one might use a very large set of projections each one specifying the
location of the keys in some small region of space.
A history is a sequence of such questions, or—mathematically—the product of the
corresponding projection operators. The role of quantum mechanics is to predict the
probabilities of individual histories, given the known initial conditions.
Finally, the histories are required to be consistent, i.e.
for i,j different. Here ρ represents the initial density matrix, and the operators are expressed in
the Heisenberg picture. The consistency requirement allows us to postulate that the
probability of the history Hi is simply
which guarantees that the probability of "A or B" equals the probability of "A" plus the
probability of "B" minus the probability of "A and B", and so forth. The interpretation based
on consistent histories is used in combination with the insights about quantum decoherence.
Quantum decoherence implies that only special choices of histories are consistent, and it
allows a quantitative calculation of the boundary between the classical domain and the
quantum domain.
In some views the interpretation based on consistent histories does not change anything about
the paradigm of the Copenhagen interpretation that only the probabilities calculated from
quantum mechanics and the wave function have a physical meaning. In order to obtain a
complete theory, the formal rules above must be supplemented with a particular Hilbert space
and rules that govern dynamics, for example a Hamiltonian.
In the opinion of others this still does not make a complete theory as no predictions are
possible about which set of consistent histories will actually occur. That is the rules of CH,
the Hilbert space, and the Hamiltonian must be supplemented by a set selection rule.
47
The proponents of this modern interpretation, such as Murray Gell-Mann, James Hartle,
Roland Omnès, Robert B. Griffiths, and Wojciech Zurek argue that their interpretation
clarifies the fundamental disadvantages of the old Copenhagen interpretation, and can be used
as a complete interpretational framework for quantum mechanics.
In Quantum Philosophy, Roland Omnès provides a less mathematical way of understanding
this same formalism. The consistent histories approach can be interpreted as a way of
understanding which sets of classical questions can be consistently asked of a single quantum
system, and which sets of questions are fundamentally inconsistent, and thus meaningless
when asked together. It thus becomes possible to demonstrate formally why it is that the
questions which Einstein, Podolsky and Rosen assumed could be asked together, of a single
quantum system, simply cannot be asked together. On the other hand, it also becomes possible
to demonstrate that classical, logical reasoning often does apply, even to quantum
experiments – but we can now be mathematically exact about the limits of classical logic.
[edit] References



R. Omnès, Understanding Quantum Mechanics, Princeton University Press, 1999.
Chapter 13 describes consistent histories.
R. Omnès, Quantum Philosophy, Princeton University Press, 1999. See part III,
especially Chapter IX.
R. B. Griffiths, Consistent Quantum Theory, Cambridge University Press, 2003.
Retrieved from "http://en.wikipedia.org/wiki/Consistent_histories"
Categories: Interpretations of Quantum Mechanics | Quantum measurement
Quantum logic
From Wikipedia, the free encyclopedia
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48
[show]Fundamental concepts
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Quantum logic
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In mathematical physics and quantum mechanics, quantum logic is a set of rules for
reasoning about propositions which takes the principles of quantum theory into account. This
research area and its name originated in the 1936 paper by Garrett Birkhoff and John von
Neumann, who were attempting to reconcile the apparent inconsistency of classical boolean
logic with the facts concerning the measurement of complementary variables in quantum
mechanics, such as position and momentum.
Quantum logic can be formulated as a modified version of propositional logic. It has some
properties which clearly distinguish it from classical logic, most notably, the failure of the
distributive law of propositional logic:
p and (q or r) = (p and q) or (p and r),
where the symbols p, q and r are propositional variables. To illustrate why the distributive law
fails, consider a particle moving on a line and let
p = "the particle is moving to the right"
q = "the particle is to the left of the origin"
r = "the particle is to the right of the origin"
then the proposition "q or r" is true, so
p and (q or r) = true
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On the other hand, the propositions "p and q" and "p and r" are both false, since they assert
tighter restrictions on simultaneous values of position and momentum than is allowed by the
uncertainty principle. So,
(p and q) or (p and r) = false
Thus the distributive law fails.
Quantum logic has been proposed as the correct logic for propositional inference generally,
most notably by the philosopher Hilary Putnam, at least at one point in his career. This thesis
was an important ingredient in Putnam's paper Is Logic Empirical? in which he analysed the
epistemological status of the rules of propositional logic. Putnam attributes the idea that
anomalies associated to quantum measurements originate with anomalies in the logic of
physics itself to the physicist David Finkelstein. It should be noted, however, that this idea
had been around for some time and had been revived several years earlier by George
Mackey's work on group representations and symmetry.
The more common view regarding quantum logic, however, is that it provides a formalism for
relating observables, system preparation filters and states. In this view, the quantum logic
approach resembles more closely the C*-algebraic approach to quantum mechanics; in fact
with some minor technical assumptions it can be subsumed by it. The similarities of the
quantum logic formalism to a system of deductive logic may then be regarded more as a
curiosity than as a fact of fundamental philosophical importance.
Contents
[hide]
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1 Introduction
2 Projections as propositions
3 The propositional lattice of a quantum mechanical system
4 Statistical structure
5 Automorphisms
6 Non-relativistic dynamics
7 Pure states
8 The measurement process
9 Limitations of quantum logic
10 See also
11 References
12 External links
[edit] Introduction
In his classic treatise Mathematical Foundations of Quantum Mechanics, John von Neumann
noted that projections on a Hilbert space can be viewed as propositions about physical
observables. The set of principles for manipulating these quantum propositions was called
quantum logic by von Neumann and Birkhoff. In his book (also called Mathematical
Foundations of Quantum Mechanics) G. Mackey attempted to provide a set of axioms for this
50
propositional system as an orthocomplemented lattice. Mackey viewed elements of this set as
potential yes or no questions an observer might ask about the state of a physical system,
questions that would be settled by some measurement. Moreover Mackey defined a physical
observable in terms of these basic questions. Mackey's axiom system is somewhat
unsatisfactory though, since it assumes that the partially ordered set is actually given as the
orthocomplemented closed subspace lattice of a separable Hilbert space. Piron, Ludwig and
others have attempted to give axiomatizations which do not require such explicit relations to
the lattice of subspaces.
The remainder of the following article assumes the reader is familiar with the spectral theory
of self-adjoint operators on a Hilbert space. However, the main ideas can be understood using
the finite-dimensional spectral theorem.
[edit] Projections as propositions
The so-called Hamiltonian formulations of classical mechanics have three ingredients: states,
observables and dynamics. In the simplest case of a single particle moving in R3, the state
space is the position-momentum space R6. We will merely note here that an observable is
some real-valued function f on the state space. Examples of observables are position,
momentum or energy of a particle. For classical systems, the value f(x), that is the value of f
for some particular system state x, is obtained by a process of measurement of f. The
propositions concerning a classical system are generated from basic statements of the form

Measurement of f yields a value in the interval [a, b] for some real numbers a, b.
It follows easily from this characterization of propositions in classical systems that the
corresponding logic is identical to that of some Boolean algebra of subsets of the state space.
By logic in this context we mean the rules that relate set operations and ordering relations,
such as de Morgan's laws. These are analogous to the rules relating boolean conjunctives and
material implication in classical propositional logic. For technical reasons, we will also
assume that the algebra of subsets of the state space is that of all Borel sets. The set of
propositions is ordered by the natural ordering of sets and has a complementation operation.
In terms of observables, the complement of the proposition {f ≥ a} is {f < a}.
We summarize these remarks as follows:

The proposition system of a classical system is a lattice with a distinguished
orthocomplementation operation: The lattice operations of meet and join are
respectively set intersection and set union. The orthocomplementation operation is set
complement. Moreover this lattice is sequentially complete, in the sense that any
sequence {Ei}i of elements of the lattice has a least upper bound, specifically the settheoretic union:
In the Hilbert space formulation of quantum mechanics as presented by von Neumann, a
physical observable is represented by some (possibly unbounded) densely-defined self-adjoint
operator A on a Hilbert space H. A has a spectral decomposition, which is a projection-valued
51
measure E defined on the Borel subsets of R. In particular, for any bounded Borel function f,
the following equation holds:
In case f is the indicator function of an interval [a, b], the operator f(A) is a self-adjoint
projection, and can be interpreted as the quantum analogue of the classical proposition

Measurement of A yields a value in the interval [a, b].
[edit] The propositional lattice of a quantum mechanical
system
This suggests the following quantum mechanical replacement for the orthocomplemented
lattice of propositions in classical mechanics. This is essentially Mackey's Axiom VII:

The orthocomplemented lattice Q of propositions of a quantum mechanical system is
the lattice of closed subspaces of a complex Hilbert space H where
orthocomplementation of V is the orthogonal complement V⊥.
Q is also sequentially complete: any pairwise disjoint sequence{Vi}i of elements of Q has a
least upper bound. Here disjointness of W1 and W2 means W2 is a subspace of W1⊥. The least
upper bound of {Vi}i is the closed internal direct sum.
Henceforth we identify elements of Q with self-adjoint projections on the Hilbert space H.
The structure of Q immediately points to a difference with the partial order structure of a
classical proposition system. In the classical case, given a proposition p, the equations
have exactly one solution, namely the set-theoretic complement of p. In these equations I
refers to the atomic proposition which is identically true and 0 the atomic proposition which is
identically false. In the case of the lattice of projections there are infinitely many solutions to
the above equations.
Having made these preliminary remarks, we turn everything around and attempt to define
observables within the projection lattice framework and using this definition establish the
correspondence between self-adjoint operators and observables : A Mackey observable is a
countably additive homomorphism from the orthocomplemented lattice of the Borel subsets
of R to Q. To say the mapping φ is a countably additive homomorphism means that for any
sequence {Si}i of pairwise disjoint Borel subsets of R, {φ(Si)}i are pairwise orthogonal
projections and
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Theorem. There is a bijective correspondence between Mackey observables and denselydefined self-adjoint operators on H.
This is the content of the spectral theorem as stated in terms of spectral measures.
[edit] Statistical structure
Imagine a forensics lab which has some apparatus to measure the speed of a bullet fired from
a gun. Under carefully controlled conditions of temperature, humidity, pressure and so on the
same gun is fired repeatedly and speed measurements taken. This produces some distribution
of speeds. Though we will not get exactly the same value for each individual measurement,
for each cluster of measurements, we would expect the experiment to lead to the same
distribution of speeds. In particular, we can expect to assign probability distributions to
propositions such as {a ≤ speed ≤ b}. This leads naturally to propose that under controlled
conditions of preparation, the measurement of a classical system can be described by a
probability measure on the state space. This same statistical structure is also present in
quantum mechanics.
A quantum probability measure is a function P defined on Q with values in [0,1] such that
P(0)=0, P(I)=1 and if {Ei}i is a sequence of pairwise orthogonal elements of Q then
The following highly non-trivial theorem is due to Andrew Gleason:
Theorem. Suppose H is a separable Hilbert space of complex dimension at least 3. Then for
any quantum probability measure on Q there exists a unique trace class operator S such that
for any self-adjoint projection E.
The operator S is necessarily non-negative (that is all eigenvalues are non-negative) and of
trace 1. Such an operator is often called a density operator.
Physicists commonly regard a density operator as being represented by a (possibly infinite)
density matrix relative to some orthonormal basis.
For more information on statistics of quantum systems, see quantum statistical mechanics.
[edit] Automorphisms
An automorphism of Q is a bijective mapping α:Q → Q which preserves the
orthocomplemented structure of Q, that is
53
for any sequence {Ei}i of pairwise orthogonal self-adjoint projections. Note that this property
implies monotonicity of α. If P is a quantum probability measure on Q, then E → α(E) is also
a quantum probability measure on Q. By the Gleason theorem characterizing quantum
probability measures quoted above, any automorphism α induces a mapping α* on the density
operators by the following formula:
The mapping α* is bijective and preserves convex combinations of density operators. This
means
whenever 1 = r1 + r2 and r1, r2 are non-negative real numbers. Now we use a theorem of
Richard Kadison:
Theorem. Suppose β is a bijective map from density operators to density operators which is
convexity preserving. Then there is an operator U on the Hilbert space which is either linear
or conjugate-linear, preserves the inner product and is such that
for every density operator S. In the first case we say U is unitary, in the second case U is antiunitary.
Remark. This note is included for technical accuracy only, and should not concern most
readers. The result quoted above is not directly stated in Kadison's paper, but can be reduced
to it by noting first that β extends to a positive trace preserving map on the trace class
operators, then applying duality and finally applying a result of Kadison's paper.
The operator U is not quite unique; if r is a complex scalar of modulus 1, then r U will be
unitary or anti-unitary if U is and will implement the same automorphism. In fact, this is the
only ambiguity possible.
It follows that automorphisms of Q are in bijective correspondence to unitary or anti-unitary
operators modulo multiplication by scalars of modulus 1. Moreover, we can regard
automorphisms in two equivalent ways: as operating on states (represented as density
operators) or as operating on Q.
[edit] Non-relativistic dynamics
In non-relativistic physical systems, there is no ambiguity in referring to time evolution since
there is a global time parameter. Moreover an isolated quantum system evolves in a
deterministic way: if the system is in a state S at time t then at time s > t, the system is in a
state Fs,t(S). Moreover, we assume
54

The dependence is reversible: The operators Fs,t are bijective.

The dependence is homogeneous: Fs,t = Fs − t,0.

The dependence is convexity preserving: That is, each Fs,t(S) is convexity preserving.

The dependence is weakly continuous: The mapping R→ R given by t → Tr(Fs,t(S) E)
is continuous for every E in Q.
By Kadison's theorem, there is a 1-parameter family of unitary or anti-unitary operators {Ut}t
such that
In fact,
Theorem. Under the above assumptions, there is a strongly continuous 1-parameter group of
unitary operators {Ut}t such that the above equation holds.
Note that it easily from uniqueness from Kadison's theorem that
Ut + s = σ(t,s)UtUs
where σ(t,s) has modulus 1. Now the square of an anti-unitary is a unitary, so that all the Ut
are unitary. The remainder of the argument shows that σ(t,s) can be chosen to be 1 (by
modifying each Ut by a scalar of modulus 1.)
[edit] Pure states
A convex combinations of statistical states S1 and S2 is a state of the form S = p1 S1 +p2 S2
where p1, p2 are non-negative and p1 + p2 =1. Considering the statistical state of system as
specified by lab conditions used for its preparation, the convex combination S can be regarded
as the state formed in the following way: toss a biased coin with outcome probabilities p1, p2
and depending on outcome choose system prepared to S1 or S2
Density operators form a convex set. The convex set of density operators has extreme points;
these are the density operators given by a projection onto a one-dimensional space. To see
that any extreme point is such a projection, note that by the spectral theorem S can be
represented by a diagonal matrix; since S is non-negative all the entries are non-negative and
since S has trace 1, the diagonal entries must add up to 1. Now if it happens that the diagonal
matrix has more than one non-zero entry it is clear that we can express it as a convex
combination of other density operators.
The extreme points of the set of density operators are called pure states. If S is the projection
on the 1-dimensional space generated by a vector ψ of norm 1 then
for any E in Q. In physics jargon, if
55
where ψ has norm 1, then
Thus pure states can be identified with rays in the Hilbert space H.
[edit] The measurement process
Consider a quantum mechanical system with lattice Q which is in some statistical state given
by a density operator S. This essentially means an ensemble of systems specified by a
repeatable lab preparation process. The result of a cluster of measurements intended to
determine the truth value of proposition E, is just as in the classical case, a probability
distribution of truth values T and F. Say the probabilities are p for T and q = 1 - p for F. By
the previous section p = Tr(S E) and q = Tr(S (I-E)).
Perhaps the most fundamental difference between classical and quantum systems is the
following: regardless of what process is used to determine E immediately after the
measurement the system will be in one of two statistical states:

If the result of the measurement is T

If the result of the measurement is F
(We leave to the reader the handling of the degenerate cases in which the denominators may
be 0.) We now form the convex combination of these two ensembles using the relative
frequencies p and q. We thus obtain the result that the measurement process applied to a
statistical ensemble in state S yields another ensemble in statistical state:
We see that a pure ensemble becomes a mixed ensemble after measurement. Measurement, as
described above, is a special case of quantum operations.
[edit] Limitations of quantum logic
Quantum logic provides a satisfactory foundation for a theory of reversible quantum
processes. Examples of such processes are the covariance transformations relating two frames
of reference, such as change of time parameter or the transformations of special relativity.
Quantum logic also provides a satisfactory understanding of density matrices. Quantum logic
can be stretched to account for some kinds of measurement processes corresponding to
56
answering yes-no questions about the state of a quantum system. However, for more general
kinds of measurement operations (that is quantum operations), a more complete theory of
filtering processes is necessary. Such an approach is provided by the consistent histories
formalism.
In any case, these quantum logic formalisms must be generalized in order to deal with supergeometry (which is needed to handle Fermi-fields) and non-commutative geometry (which is
needed in string theory and quantum gravity theory). Both of these theories use a partial
algebra with an "integral" or "trace". The elements of the partial algebra are not observables;
instead the "trace" yields "greens functions" which generate scattering amplitudes. One thus
obtains a local S-matrix theory (see D. Edwards).
Since around 1978 the Flato school ( see F. Bayen ) has been developing an alternative to the
quantum logics approach called deformation quantization (see Weyl quantization ).
[edit] See also


Quasi-set theory
HPO formalism (An approach to temporal quantum logic)
[edit] References
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
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S. Auyang, How is Quantum Field Theory Possible?, Oxford University Press, 1995.
F.Bayen,M.Flato,C.Fronsdal,A.Lichnerowicz and D.Sternheimer, Deformation theory
and quantization I,II,Ann. Phys. (N.Y.),111 (1978) pp. 61-110,111-151.
G. Birkhoff and J. von Neumann, The Logic of Quantum Mechanics, vol 37, 1936.
D. Cohen, An Introduction to Hilbert Space and Quantum Logic, Springer-Verlag,
1989. This is a thorough but elementary and well-illustrated introduction, suitable for
advanced undergraduates.
D. Edwards, The Mathematical Foundations of Quantum Field Theory: Fermions,
Gauge Fields, and Super-symmetry, Part I: Lattice Field Theories, International J. of
Theor. Phys., Vol. 20, No. 7 (1981).
D. Finkelstein, Matter, Space and Logic, Boston Studies in the Philosophy of Science
vol V, 1969
A. Gleason, Measures on the Closed Subspaces of a Hilbert Space, Journal of
Mathematics and Mechanics, 1957.
R. Kadison, Isometries of Operator Algebras, Annals of Mathematics, vol 54 pp 325338, 1951
G. Ludwig, Foundations of Quantum Mechanics, Springer-Verlag, 1983.
G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin,
1963 (paperback reprint by Dover 2004).
J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton
University Press, 1955. Reprinted in paperback form.
R. Omnès, Understanding Quantum Mechanics, Princeton University Press, 1999. An
extraordinarily lucid discussion of some logical and philosophical issues of quantum
mechanics, with careful attention to the history of the subject. Also discusses
consistent histories.
N. Papanikolaou, Reasoning Formally About Quantum Systems: An Overview, ACM
SIGACT News, 36(3), pp. 51-66, 2005.
57



C. Piron, Foundations of Quantum Physics, W. A. Benjamin, 1976.
H. Putnam, Is Logic Empirical?, Boston Studies in the Philosophy of Science vol. V,
1969
H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications, 1950.
[edit] External links

Stanford Encyclopedia of Philosophy entry on Quantum Logic and Probability Theory
Retrieved from "http://en.wikipedia.org/wiki/Quantum_logic"
Categories: Logic | Quantum measurement
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