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| {{redirect|Scattering matrix|the meaning in linear electrical networks|Scattering parameters}}
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| {{For|the 1960s approach to particle physics|S-matrix theory}}
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| In [[physics]], the '''scattering matrix''' (or '''S-matrix''') relates the initial state and the final state of a physical system undergoing a [[scattering|scattering process]]. It is used in [[quantum mechanics]], [[scattering theory]] and [[quantum field theory]].
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| More formally, the S-matrix is defined as the [[unitary matrix]] connecting asymptotic particle states in the [[Hilbert space]] of physical states ([[scattering channel]]s). While the S-matrix may be defined for any background ([[spacetime]]) that is asymptotically solvable and has no horizons{{what?|date=January 2014}}, 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 representation]]s of the [[inhomogeneous space|inhomogeneous]] [[Lorentz group]]{{clarification needed|reason=Do they mean Poincaré group, or ?|date=January 2014}}; the S-matrix is the [[evolution operator]] between time equal to minus infinity (the distant past), and time equal to plus infinity (the distant future). 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 gap]], the [[quantum state|state]] in the asymptotic past and in the asymptotic future are both described by [[Fock space]]s.
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| ==History==
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| The S-matrix was first introduced by [[John Archibald Wheeler]] in the 1937 paper "'On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure'".<ref>John Archibald Wheeler, '[http://link.aps.org/abstract/PR/v52/p1107 On the Mathematical Description of Light Nuclei by the Method. of Resonating Group Structure]' Phys. Rev. 52, 1107–1122 (1937)</ref> 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".<ref name = "Mehra">[[Jagdish Mehra]], [[Helmut Rechenberg]], ''The Historical Development of Quantum Theory'' (Pages 990 and 1031) Springer, 2001 ISBN 0-387-95086-9, ISBN 978-0-387-95086-0</ref>
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| In the 1940s [[Werner Heisenberg]] developed, independently, the idea of the S-matrix. Due to the problematic divergences present in [[quantum field theory]] 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.<ref name = "Mehra"/>
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| After World War II, the clout of Heisenberg and his attachment to the S-matrix approach may have retarded development of alternative approaches and the closer study of sub-hadronic physics for a decade or more, at least in Europe: ''"Pretty much like medieval Scholastic Magisters were extremely inventive in defending the Church Dogmas and blocking the way to experimental science, some great minds in the sixties developed the S-Matrix dogma with great perfection and skill before it was buried down in the seventies after discovery of quarks and asymptotic freedom"'' <ref name="Sacha Migdal">[[Alexander Migdal]], [http://alexandermigdal.com/prose/paradise1.shtml Paradise Lost, Part 1]</ref>
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| ==Motivation==
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| In high-energy [[particle physics]] we are interested in computing the [[probability]] for different outcomes in [[scattering]] experiments. These experiments can be broken down into three stages:
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| 1. Collide together a collection of incoming [[subatomic particle|particle]]s (usually ''two'' particles with high energies).
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| 2. Allowing the incoming particles to interact. These interactions may change the types of particles present (e.g. if an [[electron]] and a [[positron]] [[annihilate]] they may produce two [[photon]]s).
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| 3. Measuring the resulting outgoing particles.
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| The process by which the incoming particles are transformed (through their [[interaction]]) into the outgoing particles is called [[scattering]]. 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 theory]] is used to do exactly this. It is assumed that the small-energy-density approximation is valid in these cases.
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| ===Use of S-matrices===
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| The S-matrix is closely related{{vague|date=January 2014}} to the transition [[probability amplitude]] in quantum mechanics and to [[cross section (physics)|cross sections]] of various interactions; the [[matrix element|elements]] (individual numerical entries) in the S-matrix are known as '''scattering amplitudes'''. [[pole (complex analysis)|Poles]] of the S-matrix in the complex-energy plane are identified with [[bound state]]s, virtual states or [[resonance (particle physics)|resonances]]. [[Branch point|Branch cuts]] of the S-matrix in the complex-energy plane are associated to the opening of a [[scattering channel]].
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| In the [[Hamiltonian (quantum mechanics)|Hamiltonian]] approach to [[quantum field theory]], the S-matrix may be calculated as a [[time-ordered]] [[matrix exponential|exponential]] of the integrated Hamiltonian in the [[interaction picture]]; it may also be expressed using [[Feynman's path integral]]s. In both cases, the [[perturbation theory (quantum mechanics)|perturbative]] calculation of the S-matrix leads to [[Feynman diagram]]s.
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| In [[scattering theory]], the '''S-matrix''' is an [[operator (physics)|operator]] mapping free particle ''in-states'' to free particle ''out-states'' ([[scattering channel]]s) in the [[Heisenberg picture]]. This is very useful because often we cannot describe the interaction (at least, not the most interesting ones) exactly.
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| ===Mathematical definition===
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| In [[Dirac notation]], we define <math>|0\rangle</math> as the [[vacuum state|vacuum quantum state]]<!-- do not produce seas of blue -->. If <math>a^{\dagger}(k)</math> is a creation operator, its [[hermitian conjugate]]<!-- wouldn’t [[adjoint operator]] be better? --> (destruction or annihilation operator) acts on the vacuum as follows:
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| :<math>a(k)\left |0\right\rangle = 0.</math>
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| Now, we define two kinds of [[creation and annihilation operators|creation/destruction operators]] acting on different [[Hilbert space]]s (initial space ''i'', final space ''f''), <math>a_i^\dagger (k)</math> and <math>a_f^\dagger (k)</math>.
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| So now
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| :<math>\mathcal H_\mathrm{IN} = \operatorname{span}\{ \left| I, k_1\ldots k_n \right\rangle = a_i^\dagger (k_1)\cdots a_i^\dagger (k_n)\left| I, 0\right\rangle\},</math>
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| :<math>\mathcal H_\mathrm{OUT} = \operatorname{span}\{ \left| F, p_1\ldots p_n \right\rangle = a_f^\dagger (p_1)\cdots a_f^\dagger (p_n)\left| F, 0\right\rangle\}.</math>
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| It is possible to play the trick assuming that <math>\left| I, 0\right\rangle</math> and <math>\left| F, 0\right\rangle</math> are both invariant under translation and that the states <math>\left| I, k_1\ldots k_n \right\rangle</math> and <math>\left| F, p_1\ldots p_n \right\rangle</math> are [[eigenstate]]s of the momentum operator <math>\mathcal P^\mu</math>, by adiabatically turning on and off the interaction.
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| 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:
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| :<math>\left| I, k_1\ldots k_n \right\rangle = C_0 \left| F, 0\right\rangle\ + \sum_{m=1}^\infty \int{d^4p_1\ldots d^4p_mC_m(p_1\ldots p_m)\left| F, p_1\ldots p_m \right\rangle}</math>
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| Where <math>\left|C_m\right|^2</math> is the probability that the interaction transforms <math>\left| I, k_1\ldots k_n \right\rangle</math> into <math>\left| F, p_1\ldots p_m \right\rangle</math>
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| According to [[Wigner's theorem]], <math>S</math> must be a [[unitary operator]] such that <math>\left \langle I,\beta \right |S\left | I,\alpha\right\rangle = S_{\alpha\beta} = \left \langle F,\beta | I,\alpha\right\rangle</math>. Moreover, <math>S</math> leaves the vacuum state invariant and transforms IN-space fields in OUT-space fields:
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| :<math>S\left|0\right\rangle = \left|0\right\rangle</math>
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| :<math>\phi_f=S\phi_i S^{-1}</math>
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| If <math>S</math> describes an interaction correctly, these properties must be also true:
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| * If the system is made up with a single particle in momentum eigenstate <math>\left| k\right\rangle</math>, then <math>S\left| k\right\rangle=\left| k\right\rangle</math>
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| * The S-matrix element may be nonzero only where the output state has the same total [[momentum]] as the input state.
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| ===S-matrix and evolution operator ''U''===
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| Define a time-dependent creation and annihilation operator as follow
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| :<math>a^{\dagger}\left(k,t\right)=U^{-1}(t)a^{\dagger}_i\left(k\right)U\left( t \right)</math>
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| :<math>a\left(k,t\right)=U^{-1}(t)a_i\left(k\right)U\left( t \right)</math>
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| Hence
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| :<math>\phi_f=U^{-1}(\infty)\phi_i U(\infty)=S^{-1}\phi_i S.</math>
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| where we have
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| :<math>S= e^{i\alpha}\, U(\infty)</math> .
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| We allow a phase difference given by
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| :<math>e^{i\alpha}=\left\langle 0|U(\infty)|0\right\rangle^{-1}</math>
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| because for <math>S</math>:
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| :<math>S\left|0\right\rangle = \left|0\right\rangle \Longrightarrow \left\langle 0|S|0\right\rangle = \left\langle 0|0\right\rangle =1</math>
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| Substituting the explicit expression for ''U'' we obtain:
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| :<math>S=\frac{1}{\left\langle 0|U(\infty)|0\right\rangle}\mathcal T e^{-i\int{d\tau H_{\rm{int}}(\tau)}}.</math>
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| where <math> H_{\rm{int}}</math> is the interaction part of the hamiltonian and <math> \mathcal T </math> is the time ordering.
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| By inspection it can be seen that this formula is not explicitly covariant.
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| ==Dyson series==
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| {{Main|Dyson series}}
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| The most widely used expression for the S-matrix is the [[Dyson series]]. This expresses the S-matrix operator as the [[Series (mathematics)|series]]:
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| : <math>S = \sum_{n=0}^\infty \frac{(-i)^n}{n!} \int \cdots \int d^4x_1 d^4x_2 \ldots d^4x_n T [ H_{\rm{int}}(x_1) H_{\rm{int}}(x_2) \cdots H_{\rm{int}}(x_n)] </math>
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| where:
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| * <math>T[\cdots]</math> denotes [[time-ordering]],
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| * <math>\; H_{\rm{int}}(x)</math> denotes the [[interaction Hamiltonian]] density which describes the interactions in the theory.
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| ==See also==
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| {{Portal|Mathematics}}
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| *[[Feynman diagram]]
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| *[[LSZ reduction formula]]
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| *[[Wick's theorem]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{cite book |first=A.O.|last=Barut |year=1967 |title=The Theory of the Scattering Matrix}}
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| *{{cite web |url=http://www.math.sunysb.edu/~tony/whatsnew/column/feynman-1101/feynman1.html |title=Finite-dimensional Feynman Diagrams |accessdate=2007-10-23 |author=Tony Philips |date=November 2001 |work=What's New In Math |publisher=[[American Mathematical Society]]}}
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| {{DEFAULTSORT:S-Matrix}}
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| [[Category:Quantum field theory]]
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| [[Category:Scattering theory]]
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| [[Category:Matrices]]
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Gallstones are known as the main games primary forex. The Jewels are previously used to purchase resources along who has speeding up numerous vital tasks. The Gemstones can also be would buy bonus items. Apart from that, it can possibly let the leader observable any undesired debris if you would like to obtain a much more gems. Players can quickly obtain Gems through answering numerous tasks or certainly using the clash of clans identify available online.
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