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S matrix

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Scattering matrix redirects here. For the meaning in linear electrical networks, see scattering parameters.

In physics, the Scattering matrix (S-matrix) relates the initial state and the final state for an interaction of particles. It is used in quantum mechanics, scattering theory and quantum field theory.

More formally, 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.

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Explanation

Use of S-matrices

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).

Mathematical Definition

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

$a(k)\left |0\right\rangle = 0$

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 $a_f^\dagger (k)$ .

So now

$\mathcal H_\mathrm{IN} = \operatorname{span}\{ \left| I, k_1\ldots k_n \right\rangle = a_i^\dagger (k_1)\cdots a_i^\dagger (k_n)\left| I, 0\right\rangle\},$

$\mathcal H_\mathrm{OUT} = \operatorname{span}\{ \left| F, p_1\ldots p_n \right\rangle = a_f^\dagger (p_1)\cdots a_f^\dagger (p_n)\left| F, 0\right\rangle\}.$

It is possible to prove that $\left| I, 0\right\rangle$ and $\left| F, 0\right\rangle$ are both invariant under translation and that the states $\left| I, k_1\ldots k_n \right\rangle$ and $\left| F, p_1\ldots p_n \right\rangle$ are eigenstates of the momentum operator $\mathcal P^\mu$ .

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:

$\left| I, k_1\ldots k_n \right\rangle = C_0 + \sum_{m=1}^\infty \int{d^4p_1\ldots d^4p_mC_m(p_1\ldots p_m)\left| F, p_1\ldots p_n \right\rangle}$

Where $\left|C_m\right|^2$ is the probability that the interaction transforms $\left| I, k_1\ldots k_n \right\rangle$ into $\left| F, p_1\ldots p_n \right\rangle$

According to Wigner's theorem, $S$ must be a unitary operator such that $\left \langle I,\beta \right |S\left | I,\alpha\right\rangle = S_{\alpha\beta} = \left \langle F,\beta | I,\alpha\right\rangle$ . Moreover, $S$ leaves the vacuum state invariant and transforms IN-space fields in OUT-space fields:

$S\left|0\right\rangle = \left|0\right\rangle$

φ f = S - 1 φ f S

If $S$ describes an interaction correctly, these properties must be also true:

If the system is made up with a single particle in momentum eigenstate $\left| k\right\rangle$ , then $S\left| k\right\rangle=\left| k\right\rangle$

The S-matrix element must be nonzero if and only if momentum is conserved.

S-matrix and evolution operator U

$a\left(k,t\right)=U^{-1}(t)a_i\left(k\right)U\left( t \right)$

$\phi_f=U^{-1}(\infty)\phi_i U(\infty)=S^{-1}\phi_i S$

Therefore $S=e^{i\alpha}U(\infty)$ where

$e^{i\alpha}=\left\langle 0|U(\infty)|0\right\rangle^{-1}$

because

$S\left|0\right\rangle = \left|0\right\rangle.$

Substituting the explicit expression for U we obtain:

$S=\frac{1}{\left\langle 0|U(\infty)|0\right\rangle}\mathcal T e^{-i\int{d\tau V_i(\tau)}}$

By inspection it can be seen that this formula is not explicitly covariant.

LSZ reduction formula

The LSZ reduction formula is used to calculate predictions of S-matrix elements based on the field being analyzed.

Wick's theorem

Also at Wick's theorem

Wick's theorem is a method of reducing high-order derivatives to a combinatorics problem.(Philips, 2001) It is named after Gian-Carlo Wick.

Definition of contraction:

$\mathcal C(x_1, x_2)=\left \langle 0 |\mathcal T\phi_i(x_1)\phi_i(x_2)|0\right \rangle=\overline{\phi_i(x_1)\phi_i(x_2)}=i\Delta_F(x_1-x_2) =i\int{\frac{d^4k}{(2\pi)^4}\frac{e^{-ik(x_1-x_2)}}{(k^2-m^2)+i\epsilon}}.$

Which means that $\overline{AB}=\mathcal TAB-:AB:$

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:

$\mathcal T\Pi_{k=1}^m\phi(x_k)=:\Pi\phi_i(x_k):+\sum_{\alpha,\beta}\overline{\phi(x_\alpha)\phi(x_\beta)}:\Pi_{k\not=\alpha,\beta}\phi_i(x_k):+$

$\mathcal +\sum_{(\alpha,\beta),(\gamma,\delta)}\overline{\phi(x_\alpha)\phi(x_\beta)}\;\overline{\phi(x_\gamma)\phi(x_\delta)}:\Pi_{k\not=\alpha,\beta,\gamma,\delta}\phi_i(x_k):+\cdots.$

Applying this theorem to S-matrix elements, we discover that normal-ordered terms acting on vacuum state give a null contribution to the sum. We conclude that m is even and only completely contracted terms remain.

$F_m^i(x)=\left \langle 0 |\mathcal T\phi_i(x_1)\phi_i(x_2)|0\right \rangle=\sum_\mathrm{pairs}\overline{\phi(x_1)\phi(x_2)}\cdots \overline{\phi(x_{m-1})\phi(x_m})$

$G_p^{(n)}=\left \langle 0 |\mathcal T:v_i(y_1):\dots:v_i(y_n):\phi_i(x_1)\cdots \phi_i(x_p)|0\right \rangle$

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 $v=gy^4 \Rightarrow :v_i(y_1):=:\phi_i(y_1)\phi_i(y_1)\phi_i(y_1)\phi_i(y_1):$

This is analogous to the corresponding theorem in statistics for the moments of a Gaussian distribution.

Bibliography

Barut (1967). The Theory of the Scattering Matrix.

Tony Philips (11 2001). Finite-dimensional Feynman Diagrams. What's New In Math. American Mathematical Society. Retrieved on 2007-10-23.

Category: Scattering theory

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "S_matrix". A list of authors is available in Wikipedia.