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" in S-matrix.

Related Topics:
Quantum mechanics - Scattering theory - Quantum field theory - Matrix - Out-channels

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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.

Related Topics:
Unitary matrix - Hilbert space - Scattering channel - Spacetime - Horizon - Minkowski space - Inhomogeneous - Lorentz group - Evolution operator - Quantum field theory - Fock space

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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.

Related Topics:
Probability amplitude - Quantum mechanics - Cross section - Interaction - Poles - Complex-energy plane - Bound state - Resonance - Branch cuts - Scattering channel

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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.

Related Topics:
Hamiltonian - Quantum field theory - Time-ordered - Exponential - Dirac picture - Feynman's path integral - Perturbative - Feynman diagram

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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).

Related Topics:
Scattering theory - Operator - Scattering channel - Heisenberg picture

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In Dirac notation, we define left |0 ight angle as the void (or vacuum) quantum state. If a^{dagger}(k) is a creation operator, its hermitian conjugate (destruction or annihilation operator) acts on the void as follows:

Related Topics:
Dirac notation - Quantum state

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:a(k)left |0 ight angle = 0

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Now, we define two kinds of creation/destruction operators, acting on different Hilbert spaces (IN space i, OUT space f), a_i^dagger (k) and a_f^dagger (k).

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So now

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:mathcal H_mathrm{IN} = operatorname{span}{ left| I, k_1ldots k_n ight angle = a_i^dagger (k_1)cdots a_i^dagger (k_n)left| I, 0 ight angle},

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:mathcal H_mathrm{OUT} = operatorname{span}{ left| F, p_1ldots p_n ight angle = a_f^dagger (p_1)cdots a_f^dagger (p_n)left| F, 0 ight angle}.

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It is possible to prove that left| I, 0 ight angle and left| F, 0 ight angle are both invariant under translation and that the states left| I, k_1ldots k_n ight angle and left| F, p_1ldots p_n ight angle are eigenstates of the momentum operator mathcal P^mu.

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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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:left| I, k_1ldots k_n ight angle = C_0 + sum_{m=1}^infty int{d^4p_1ldots d^4p_mC_m(p_1ldots p_m)left| F, p_1ldots p_n ight angle}

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Where left|C_m ight|^2 is the probability that the interaction transforms left| I, k_1ldots k_n ight angle into left| F, p_1ldots p_n ight angle

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According to Wigner's theorem, S must be a unitary operator such that left langle I,eta ight |Sleft | I,lpha ight angle = S_{lphaeta} = left langle F,eta | I,lpha ight angle. Moreover, S leaves the void invariant and transforms IN-space fields in OUT-space fields:

Related Topics:
Wigner's theorem - Unitary operator

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:Sleft|0 ight angle = left|0 ight angle

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:phi_f=S^{-1}phi_f S

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If S 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 left| k ight angle, then Sleft| k ight angle=left| k ight angle

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The S-Matrix element must be non zero if and only if momentum is conserved.

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