In
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, the
scattering matrix (or
S-matrix) relates the initial state and the final state of a physical system undergoing a scattering process. It is used in
quantum mechanicsQuantum 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...
,
scattering theoryIn mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a...
and
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
.
More formally, the S-matrix is defined as the
unitary matrix connecting asymptotic particle states in the
Hilbert spaceThe 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...
of physical states (
scattering channelIn scattering theory, a scattering channel is a quantum state of the colliding system before or after the collision . The Hilbert space spanned by the states before collision is equal to the ones spanned by the states after collision which are both Fock spaces if there is a mass gap...
s). While the S-matrix may be defined for any background (
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
) that is asymptotically solvable and has no
horizonThe horizon is the apparent line that separates earth from sky, the line that divides all visible directions into two categories: those that intersect the Earth's surface, and those that do not. At many locations, the true horizon is obscured by trees, buildings, mountains, etc., and the resulting...
s, it has a simple form in the case of the
Minkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
. In this special case, the Hilbert space is a space of irreducible unitary representations of the inhomogeneous
Lorentz groupIn physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
; the S-matrix is the evolution operator between time equal to minus infinity, and time equal to plus infinity. It is defined only in the limit of zero energy density (or infinite particle separation distance). It can be shown that if a quantum field theory in Minkowski space has a
mass gapIn quantum field theory, the mass gap is the difference in energy between 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 as particles in plane-waves, the mass gap is the mass of the lightest...
, the state in the asymptotic past and in the asymptotic future are both described by
Fock spaceThe Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...
s.
History
The S-matrix was first introduced by
John Archibald WheelerJohn Archibald Wheeler was an American theoretical physicist who was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in explaining the basic principles behind nuclear fission...
in the 1937 paper "'On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure'". In this paper Wheeler introduced a
scattering matrix - a unitary matrix of coefficients connecting "the asymptotic behaviour of an arbitrary particular solution [of the integral equations] with that of solutions of a standard form".
In the 1940s
Werner HeisenbergWerner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
developed, independently, the idea of the S-matrix. Due to the problematic divergences present in
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
at that time Heisenberg was motivated to isolate the essential features of the theory that would not be affected by future changes as the theory developed. In doing so he was led to introduce a unitary "characteristic" S-matrix.
Motivation
In high-energy
particle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
we are interested in computing the
probabilityProbability 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...
for different outcomes in
scatteringScattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
experiments. These experiments can be broken down into three stages:
1. Collide together a collection of incoming
particleIn physics or chemistry, subatomic particles are the smaller particles composing nucleons and atoms. There are two types of subatomic particles: elementary particles, which are not made of other particles, and composite particles...
s (usually
two particles with high energies).
2. Allowing the incoming particles to interact. These interactions may change the types of particles present (e.g. if an
electronThe electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
and a
positronThe positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...
annihilate they may produce two
photonIn physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s).
3. Measuring the resulting outgoing particles.
The process by which the incoming particles are transformed (through their interaction) into the outgoing particles is called
scatteringScattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
. For particle physics, a physical theory of these processes must be able to compute the probability for different outgoing particles when we collide different incoming particles with different energies. The S-matrix in
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
is used to do exactly this. It is assumed that the small-energy-density approximation is valid in these cases.
Use of S-matrices
The S-matrix is closely related to the transition
probability amplitudeIn quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...
in quantum mechanics and to
cross sectionsA cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....
of various
interactionInteraction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the concept of interaction, as opposed to a one-way causal effect...
s; the elements (individual numerical entries) in the S-matrix are known as
scattering amplitudes.
Poles of the S-matrix in the
complex-energy planeA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
are identified with
bound stateIn physics, a bound state describes a system where a particle is subject to a potential such that the particle has a tendency to remain localised in one or more regions of space...
s, virtual states or
resonanceIn physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...
s.
Branch cutsIn the mathematical field of complex analysis, a branch point of a multi-valued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point...
of the S-matrix in the complex-energy plane are associated to the opening of a
scattering channelIn scattering theory, a scattering channel is a quantum state of the colliding system before or after the collision . The Hilbert space spanned by the states before collision is equal to the ones spanned by the states after collision which are both Fock spaces if there is a mass gap...
.
In the
HamiltonianIn quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
approach to
quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, the S-matrix may be calculated as a time-ordered
exponentialIn mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
of the integrated Hamiltonian in the
interaction pictureIn quantum mechanics, the Interaction picture is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of...
; it may be also expressed using Feynman's path integrals. In both cases, the perturbative calculation of the S-matrix leads to
Feynman diagramFeynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s.
In
scattering theoryIn mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a...
, the
S-matrix is an
operatorIn physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
mapping free particle
in-states to free particle
out-states (
scattering channelIn scattering theory, a scattering channel is a quantum state of the colliding system before or after the collision . The Hilbert space spanned by the states before collision is equal to the ones spanned by the states after collision which are both Fock spaces if there is a mass gap...
s) in the
Heisenberg pictureIn 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...
. This is very useful because often we cannot describe exactly the interaction (at least, not the most interesting ones).
Mathematical definition
In Dirac notation, we define
as the
vacuumIn quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
quantum state. If
is a creation operator, its hermitian conjugate (destruction or annihilation operator) acts on the vacuum as follows:
Now, we define two kinds of creation/destruction operators, acting on different
Hilbert spaceThe 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...
s (IN space
i, OUT space
f),
and
.
So now
It is possible to play the trick assuming that
and
are both invariant under translation and that the states
and
are eigenstates of the momentum operator
, by adiabatically turning on and off the interaction.
In the Heisenberg picture the states are time-independent, so we can expand initial states on a basis of final states (or vice versa) as follows:
Where
is the probability that the interaction transforms
into
According to
Wigner's theoremWigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT act on the Hilbert space of states....
,
must be a
unitary operatorIn functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
such that
. Moreover,
leaves the vacuum state invariant and transforms IN-space fields in OUT-space fields:
If
describes an interaction correctly, these properties must be also true:
If the system is made up with a single particle in momentum eigenstate
, then
The S-matrix element must be nonzero
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
momentum is conserved.
S-matrix and evolution operator U
Therefore
where
because
Substituting the explicit expression for
U we obtain:
By inspection it can be seen that this formula is not explicitly covariant.
Dyson series
The most widely used expression for the S-matrix is the
Dyson seriesIn scattering theory, the Dyson series, formulated by British-born American physicist Freeman Dyson, is a perturbative series, and each term is represented by Feynman diagrams. This series diverges asymptotically, but in quantum electrodynamics at the second order the difference from...
. This expresses the S-matrix operator as the
seriesA series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
:
-
where:
-
denotes time-ordering,
-
denotes the interaction Hamiltonian which describes the interactions in the theory.
See also
- Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
- LSZ reduction formula
In quantum field theory, the LSZ reduction formula is a method to calculate p-matrix elements from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the Lagrangian of some quantum field theory and leads to prediction of measurable quantities...
- Wick's theorem
Wick's theorem is a method of reducing high-order derivatives to a combinatorics problem . It is named after Gian-Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihilation operators to sums of products of pairs of these operators...