Consistent histories
Encyclopedia
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
. This interpretation of quantum mechanics is based on a consistency criterion that then allows probabilities to be assigned to various alternative histories of a system such that the probabilities for each history obey the rules of classical probability while being consistent with the Schrödinger equation
. In contrast to some interpretations of quantum mechanics, particularly the Copenhagen interpretation, the framework does not include "wavefunction collapse" as a relevant description of any physical process, and emphasizes that measurement theory is not a fundamental ingredient of quantum mechanics.
(here 
labels different histories) is a sequence of Proposition
s
specified at different moments of time 
(here 
labels the times). We write this as:
and read it as "the proposition
is true at time 
and then the proposition 
is true at time 
and then 
". The times 
are strictly ordered and called the temporal support of the history.
Inhomogeneous histories are multiple-time propositions which cannot be represented by a homogeneous history. An example is the logical OR
of two homogeneous histories:
.
These propositions can correspond to any set of questions that include all possibilities.
Examples might be the three propositions 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 large number of propositions each one specifying the location of the keys in some small region of space.
Each single-time proposition
can be represented by a projection operator 
acting on the system's Hilbert space (we use "hats" to denote operators). It is then useful to represent homogeneous histories by the time-ordered tensor product
of their single-time projection operators. This is the history projection operator (HPO) formalism developed by Christopher Isham
and
naturally encodes the logical structure of the history propositions. The homogeneous history
is represented by the projection operator
This definition can be extended to define projection operators that represent inhomogeneous histories too.
The symbol
indicates that the factors in the product are ordered chronologically according to their values of 
: the "past" operators with smaller values of 
appear on the right side, and the "future" operators with greater values of 
appear on the left side.
This definition can be extended to inhomogeneous histories as well.
Central to the consistent histories is the notion of consistency. A set of histories
is consistent (or strongly consistent) if
for all
. Here 
represents the initial density matrix
, and the operators are expressed in the Heisenberg picture
.
The set of histories is weakly consistent if
for all
.
of history
is simply
which obeys the axioms of probability if the histories
come from the same (strongly) consistent set.
As an example, this means the probability of "
OR 
" equals the probability of "
" plus the probability of "
" minus the probability of "
AND 
", and so forth.
.
Quantum decoherence implies that irreversible macroscopic phenomena (hence, all classical measurements) render histories automatically consistent, which allows one to recover classical reasoning and "common sense" when applied to the outcomes of these measurements. More precise analysis of decoherence allows (in principle) a quantitative calculation of the boundary between the classical domain and the quantum domain covariance. According to Roland Omnès
:
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 Consistent Histories, the Hilbert space
, and the Hamiltonian must be supplemented by a set selection rule. However, Griffiths holds the opinion that asking the question of which set of histories will "actually occur" is a misinterpretation of the theory; histories are a tool for description of reality, not separate alternate realities.
The proponents of this Consistent Histories interpretation, such as Murray Gell-Mann
, James Hartle
, Roland Omnès
and Robert B. Griffiths 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.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, the consistent histories approach is intended to give a modern interpretation of quantum mechanics
Interpretation of quantum mechanics
An interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing, many of these experiments are open to different interpretations...
, generalising the conventional Copenhagen interpretation
Copenhagen interpretation
The Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...
and providing a natural interpretation of quantum cosmology
Quantum cosmology
In theoretical physics, quantum cosmology is a field attempting to study the effect of quantum mechanics on the formation of the universe, or its early evolution, especially just after the Big Bang...
. This interpretation of quantum mechanics is based on a consistency criterion that then allows probabilities to be assigned to various alternative histories of a system such that the probabilities for each history obey the rules of classical probability while being consistent with the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
. In contrast to some interpretations of quantum mechanics, particularly the Copenhagen interpretation, the framework does not include "wavefunction collapse" as a relevant description of any physical process, and emphasizes that measurement theory is not a fundamental ingredient of quantum mechanics.
Histories
A homogeneous history (here labels different histories) is a sequence of PropositionProposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
s
specified at different moments of time (here labels the times). We write this as:and read it as "the proposition
is true at time and then the proposition is true at time and then ". The times are strictly ordered and called the temporal support of the history.Inhomogeneous histories are multiple-time propositions which cannot be represented by a homogeneous history. An example is the logical OR
Logical disjunction
In logic and mathematics, a two-place logical connective or, is a logical disjunction, also known as inclusive disjunction or alternation, that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are...
of two homogeneous histories:
.These propositions can correspond to any set of questions that include all possibilities.
Examples might be the three propositions 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 large number of propositions each one specifying the location of the keys in some small region of space.
Each single-time proposition
can be represented by a projection operator acting on the system's Hilbert space (we use "hats" to denote operators). It is then useful to represent homogeneous histories by the time-ordered tensor productTensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
of their single-time projection operators. This is the history projection operator (HPO) formalism developed by Christopher Isham
Christopher Isham
Christopher Isham is a theoretical physicist at Imperial College London. His main research interests are quantum gravity and foundational studies in quantum theory. He was the inventor of an approach to temporal quantum logic called the HPO formalism, and has worked on loop quantum gravity and...
and
naturally encodes the logical structure of the history propositions. The homogeneous history
is represented by the projection operatorThis definition can be extended to define projection operators that represent inhomogeneous histories too.
Consistency
An important construction in the consistent histories approach is the class operator for a homogeneous history:The symbol
indicates that the factors in the product are ordered chronologically according to their values of : the "past" operators with smaller values of appear on the right side, and the "future" operators with greater values of appear on the left side.This definition can be extended to inhomogeneous histories as well.
Central to the consistent histories is the notion of consistency. A set of histories
is consistent (or strongly consistent) iffor all
. Here represents the initial density matrixDensity matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
, and the operators are expressed in the Heisenberg picture
Heisenberg picture
In physics, the Heisenberg picture is a formulation of quantum mechanics in which the operators incorporate a dependency on time, but the state vectors are time-independent. It stands in contrast to the Schrödinger picture in which the operators are constant and the states evolve in time...
.
The set of histories is weakly consistent if
for all
.Probabilities
If a set of histories is consistent then probabilities can be assigned to them in a consistent way. We postulate that the probabilityProbability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
of history
is simplywhich obeys the axioms of probability if the histories
come from the same (strongly) consistent set.As an example, this means the probability of "
OR " equals the probability of "" plus the probability of "" minus the probability of " AND ", and so forth.Interpretation
The interpretation based on consistent histories is used in combination with the insights about quantum decoherenceQuantum decoherence
In quantum mechanics, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or probabilistically additive behavior...
.
Quantum decoherence implies that irreversible macroscopic phenomena (hence, all classical measurements) render histories automatically consistent, which allows one to recover classical reasoning and "common sense" when applied to the outcomes of these measurements. More precise analysis of decoherence allows (in principle) a quantitative calculation of the boundary between the classical domain and the quantum domain covariance. According to Roland Omnès
Roland Omnès
Roland Omnès is the author of several books which aim to close the gap between our common sense experience of the classical world and the complex, formal mathematics which is now required to accurately describe reality at its most fundamental level.- Biography :Omnès is currently Professor...
:
In order to obtain a complete theory, the formal rules above must be supplemented with a particular Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
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 Consistent Histories, the Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, and the Hamiltonian must be supplemented by a set selection rule. However, Griffiths holds the opinion that asking the question of which set of histories will "actually occur" is a misinterpretation of the theory; histories are a tool for description of reality, not separate alternate realities.
The proponents of this Consistent Histories interpretation, such as Murray Gell-Mann
Murray Gell-Mann
Murray Gell-Mann is an American physicist and linguist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles...
, James Hartle
James Hartle
James Burkett Hartle is an American physicist. He has been a professor of physics at the University of California, Santa Barbara since 1966, and he is currently a member of the external faculty of the Santa Fe Institute...
, Roland Omnès
Roland Omnès
Roland Omnès is the author of several books which aim to close the gap between our common sense experience of the classical world and the complex, formal mathematics which is now required to accurately describe reality at its most fundamental level.- Biography :Omnès is currently Professor...
and Robert B. Griffiths 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
Quantum Philosophy (book)
Quantum Philosophy is a book by the physicist Roland Omnès, in which he aims to show the non-specialist reader how modern developments in quantum mechanics allow the recovery of our common sense view of the world.- Book contents :...
, 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
EPR paradox
The EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...
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.