In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the cluster decomposition property states that
experiment
An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome oc ...
s carried out far from each other cannot influence each other. Usually applied to
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, it requires that
vacuum expectation value
In quantum field theory the vacuum expectation value (also called condensate or simply VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. ...
s of
operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
localized in
bounded regions factorize whenever these regions becomes sufficiently distant from each other. First formulated by Eyvind H. Wichmann and James H. Crichton in 1963 in the context of the
S-matrix
In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
More forma ...
, it was conjectured by
Steven Weinberg
Steven Weinberg (; May 3, 1933 – July 23, 2021) was an American theoretical physicist and Nobel laureate in physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic interactio ...
that in the
low energy limit the cluster decomposition property, together with
Lorentz invariance
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
and
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, inevitably lead to quantum field theory.
String theory
In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
satisfies all three of the conditions and so provides a counter-example against this being true at all energy scales.
Formulation
The S-matrix
describes the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
for a process with an initial state
evolving into a final state
. If the initial and final states consist of two clusters, with
and
close to each other but far from the pair
and
, then the cluster decomposition property requires the S-matrix to factorize
:
as the distance between the two clusters increases. The physical interpretation of this is that any two spatially well separated experiments
and
cannot influence each other. This condition is fundamental to the ability to doing physics without having to know the
state
State may refer to:
Arts, entertainment, and media Literature
* ''State Magazine'', a monthly magazine published by the U.S. Department of State
* ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, United States
* ''Our S ...
of the entire
universe
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
. By expanding the S-matrix into a sum of a product of connected S-matrix elements
, which at the perturbative level are equivalent to
connected Feynman diagrams, the cluster decomposition property can be restated as demanding that connected S-matrix elements must vanish whenever some of its clusters of particles are far apart from each other.
This position space formulation can also be reformulated in terms of the
momentum space
In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension.
Position space (also real space or coordinate space) is the set of all ''position vectors'' r in space, and h ...
S-matrix
. Since its
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
ation gives the position space connected S-matrix, this only depends on position through the exponential terms. Therefore, performing a uniform
translation
Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
in a direction
on a subset of particles will effectively change the momentum space S-matrix as
:
By
translational invariance
In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by .
In physics and mathematics, continuous translational symmetry is the invariance of a system of equatio ...
, a translation of all particles cannot change the S-matrix, therefore
must be proportional to a momentum conserving
delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
to ensure that the translation exponential factor vanishes. If there is an additional delta function of only a subset of momenta corresponding to some cluster of particles, then this cluster can be moved arbitrarily far through a translation without changing the S-matrix, which would violate cluster decomposition. This means that in momentum space the property requires that the S-matrix only has a single delta function.
Cluster decomposition can also be formulated in terms of
correlation functions
The cross-correlation matrix of two random vectors is a matrix containing as elements the cross-correlations of all pairs of elements of the random vectors. The cross-correlation matrix is used in various digital signal processing algorithms.
D ...
, where for any two operators
and
localized to some region, the vacuum expectation values factorize as the two operators become distantly separated
:
This formulation allows for the property to be applied to theories that lack an S-matrix such as
conformal field theories
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...
. It is in terms of these
Wightman functions
In mathematical physics, the Wightman axioms (also called Gårding–Wightman axioms), named after Arthur Wightman, are an attempt at a mathematically rigorous formulation of quantum field theory. Arthur Wightman formulated the axioms in the e ...
that the property is usually formulated in
axiomatic quantum field theory Axiomatic quantum field theory is a mathematical discipline which aims to describe quantum field theory in terms of rigorous axioms. It is strongly associated with functional analysis and operator algebras, but has also been studied in recent years ...
. In some formulations, such as Euclidean
constructive field theory, it is explicitly introduced as an
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
.
Properties
If a theory is constructed from
creation and annihilation operators
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually ...
, then the cluster decomposition property automatically holds. This can be seen by expanding out the S-matrix as a sum of Feynman diagrams which allows for the identification of connected S-matrix elements with connected Feynman diagrams.
Vertices arise whenever creation and annihilation operators commute past each other leaving behind a single momentum delta function. In any
connected diagram with V vertices, I internal lines and L loops, I-L of the delta functions go into fixing internal momenta, leaving V-(I-L) delta functions unfixed. A form of
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...
states that any graph with C disjoint connected components satisfies C = V-I+L. Since the connected S-matrix elements correspond to C=1 diagrams, these only have a single delta function and thus the cluster decomposition property, as formulated above in momentum space in terms of delta functions, holds.
Microcausality, the
locality
Locality may refer to:
* Locality (association), an association of community regeneration organizations in England
* Locality (linguistics)
* Locality (settlement)
* Suburbs and localities (Australia), in which a locality is a geographic subdivis ...
condition requiring commutation relations of local operators to vanish for
spacelike separations, is a sufficient condition for the S-matrix to satisfy cluster decomposition. In this sense cluster decomposition serves a similar purpose for the S-matrix as microcausality does for
fields
Fields may refer to:
Music
*Fields (band), an indie rock band formed in 2006
*Fields (progressive rock band), a progressive rock band formed in 1971
* ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010)
* "Fields", a song by ...
, preventing
causal
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
influence from propagating between regions that are distantly separated. However, cluster decomposition is weaker than having no
superluminal causation since it can be formulated for classical theories as well.
One key requirement for cluster decomposition is that it requires a unique
vacuum state
In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. The word zero-point field is sometimes used as ...
, with it failing if the vacuum state is a
mixed state. The rate at which the correlation functions factorize depends on the spectrum of the theory, where if it has
mass gap
In quantum field theory, the mass gap is the difference in energy between the lowest energy state, the vacuum, and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of ...
of mass
then there is an exponential falloff
while if there are
massless particle
In particle physics, a massless particle is an elementary particle whose invariant mass is zero. There are two known gauge boson massless particles: the photon (carrier of electromagnetism) and the gluon (carrier of the strong force). However, glu ...
s present then it can be as slow as
.
References
{{DEFAULTSORT:Cluster decomposition
Quantum field theory
Axiomatic quantum field theory
Theorems in quantum mechanics