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(Redirected from S-matrix)
In quantum mechanics, scattering theory or quantum field theory, the 'S-matrix' relates the final state in the infinite future (out-channels) and the initial state in the infinite past (in-channels). The "S" stands for "scattering" or "Strahlung" (radiation).
More mathematically, the S-matrix is defined as the unitary matrix connecting asymptotic particle states in the Hilbert space of physical states (scattering channels). While the S-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no horizons, it has a simple form in the case of the Minkowski space. In this special case, the Hilbert space is a space of irreducible unitary representations of the inhomogeneous Lorentz group; the S-matrix is the evolution operator between time equal to minus infinity, and time equal to plus infinity. It can be shown that if a quantum field theory in Minkowski space has a mass gap, the state in the asymptotic past and in the asymptotic future are both described by Fock spaces.
The S-matrix is closely related to the transition probability amplitude in quantum mechanics and to cross sections of various interactions; the elements (individual numerical entries) in the S-matrix are known as 'scattering amplitudes'. Poles of the S-matrix in the complex-energy plane are identified with bound states, virtual states or resonances. Branch cuts of the S-matrix in the complex-energy plane are associated to the opening of a scattering channel.
In the Hamiltonian approach to quantum field theory, the S-matrix may be calculated as a time-ordered exponential of the integrated Hamiltonian in the Dirac picture; it may be also expressed using Feynman's path integrals. In both cases, the perturbative calculation of the S-matrix leads to Feynman diagrams.
In scattering theory, the 'S-matrix' is an operator mapping free particle ''in-states'' to free particle ''out-states'' (scattering channels) in the Heisenberg picture. This is very useful because we cannot describe exactly the interaction (at least, the most interesting ones).
In Dirac notation, we define as the void (or vacuum) quantum state. If is a creation operator, its hermitian conjugate (destruction or annihilation operator) acts on the void as follows:
:
Now, we define two kinds of creation/destruction operators, acting on different Hilbert spaces (IN space ''i'', OUT space ''f''), and .
So now
:
:
It is possible to prove that and are both invariant under translation and that the states and are eigenstates of the momentum operator .
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 theorem, must be a unitary operator such that . Moreover, leaves the void 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 non zero if and only if momentum is conserved.
:
:
So we have where
:
because
:
Substituting the explicit expression for ''U'' we obtain:
:
You can see that this formula is not explicitly covariant.
:
The task is to find an expression for the S-Matrix element using the reduction formula. Before starting to accomplish this, it is useful to show the following ''trick'':
:
:
We will use this in the following calculation:
:
This operation is called ''particle extraction''.
:
This is true because ''p'' is not equal to ''k''.
:
:
:
Remembering that ''f'' functions are solutions of Klein-Gordon equation:
:
where stands for the D'Alembertian. Substituting this in previous equation we get (integrating by parts two times):
:
Now we repeat these operations for all the particle in the system, and finally we get:
:
This is, of course, the simplest case with only four interacting particles.
Now we Fourier transform (it is not ''exactly'' a Fourier transformation) the reduction formula ''F'' and we get:
:
There is a theorem that states (proof omitted) that the S-matrix elements are the residuals of ''f'' calculated on mass-shell:
:
The matter is that we do not have an explicit expression for , so we have to make a perturbative expansion with .
In the end, we obtain:
:
Wick's theorem is named after Gian-Carlo Wick.
'Definition of' ''contraction'':
:
Which means that
In the end, we approach at Wick's theorem:
'T' ''Wick's theorem''
The T-product of a time-ordered free fields string can be expressed in the following manner:
:
:
Applying this theorem to S-matrix elements, we discover that normal-ordered terms acting on void state give a null contribute to the sum. We conclude that ''m'' is even and only completely contracted terms remain.
:
:
where ''p'' is the number of interaction fields (or, equivalently, the number of interacting particles) and ''n'' is the development order (or the number of vertices of interaction). For example, if
This is analogous to the corresponding theorem in statistics for the moments of a Gaussian distribution.
Feynman diagram.
''The Theory of the Scattering Matrix'' (Barut, 1967).
In quantum mechanics, scattering theory or quantum field theory, the 'S-matrix' relates the final state in the infinite future (out-channels) and the initial state in the infinite past (in-channels). The "S" stands for "scattering" or "Strahlung" (radiation).
More mathematically, the S-matrix is defined as the unitary matrix connecting asymptotic particle states in the Hilbert space of physical states (scattering channels). While the S-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no horizons, it has a simple form in the case of the Minkowski space. In this special case, the Hilbert space is a space of irreducible unitary representations of the inhomogeneous Lorentz group; the S-matrix is the evolution operator between time equal to minus infinity, and time equal to plus infinity. It can be shown that if a quantum field theory in Minkowski space has a mass gap, the state in the asymptotic past and in the asymptotic future are both described by Fock spaces.
The S-matrix is closely related to the transition probability amplitude in quantum mechanics and to cross sections of various interactions; the elements (individual numerical entries) in the S-matrix are known as 'scattering amplitudes'. Poles of the S-matrix in the complex-energy plane are identified with bound states, virtual states or resonances. Branch cuts of the S-matrix in the complex-energy plane are associated to the opening of a scattering channel.
In the Hamiltonian approach to quantum field theory, the S-matrix may be calculated as a time-ordered exponential of the integrated Hamiltonian in the Dirac picture; it may be also expressed using Feynman's path integrals. In both cases, the perturbative calculation of the S-matrix leads to Feynman diagrams.
In scattering theory, the 'S-matrix' is an operator mapping free particle ''in-states'' to free particle ''out-states'' (scattering channels) in the Heisenberg picture. This is very useful because we cannot describe exactly the interaction (at least, the most interesting ones).
In Dirac notation, we define as the void (or vacuum) quantum state. If is a creation operator, its hermitian conjugate (destruction or annihilation operator) acts on the void as follows:
:
Now, we define two kinds of creation/destruction operators, acting on different Hilbert spaces (IN space ''i'', OUT space ''f''), and .
So now
:
:
It is possible to prove that and are both invariant under translation and that the states and are eigenstates of the momentum operator .
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 theorem, must be a unitary operator such that . Moreover, leaves the void 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 non zero if and only if momentum is conserved.
| Contents |
| S-matrix and evolution operator ''U'' |
| L.S.Z. (Lehman, Symanzik, Zimmermann) reduction formula |
| Wick's theorem |
| See also |
| Bibliography |
S-matrix and evolution operator ''U''
:
:
So we have where
:
because
:
Substituting the explicit expression for ''U'' we obtain:
:
You can see that this formula is not explicitly covariant.
L.S.Z. (Lehman, Symanzik, Zimmermann) reduction formula
:
The task is to find an expression for the S-Matrix element using the reduction formula. Before starting to accomplish this, it is useful to show the following ''trick'':
:
:
We will use this in the following calculation:
:
This operation is called ''particle extraction''.
:
This is true because ''p'' is not equal to ''k''.
:
:
:
Remembering that ''f'' functions are solutions of Klein-Gordon equation:
:
where stands for the D'Alembertian. Substituting this in previous equation we get (integrating by parts two times):
:
Now we repeat these operations for all the particle in the system, and finally we get:
:
This is, of course, the simplest case with only four interacting particles.
Now we Fourier transform (it is not ''exactly'' a Fourier transformation) the reduction formula ''F'' and we get:
:
There is a theorem that states (proof omitted) that the S-matrix elements are the residuals of ''f'' calculated on mass-shell:
:
The matter is that we do not have an explicit expression for , so we have to make a perturbative expansion with .
In the end, we obtain:
:
Wick's theorem
Wick's theorem is named after Gian-Carlo Wick.
'Definition of' ''contraction'':
:
Which means that
In the end, we approach at Wick's theorem:
'T' ''Wick's theorem''
The T-product of a time-ordered free fields string can be expressed in the following manner:
:
:
Applying this theorem to S-matrix elements, we discover that normal-ordered terms acting on void state give a null contribute to the sum. We conclude that ''m'' is even and only completely contracted terms remain.
:
:
where ''p'' is the number of interaction fields (or, equivalently, the number of interacting particles) and ''n'' is the development order (or the number of vertices of interaction). For example, if
This is analogous to the corresponding theorem in statistics for the moments of a Gaussian distribution.
See also
Feynman diagram.
Bibliography
''The Theory of the Scattering Matrix'' (Barut, 1967).
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