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# S matrix

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.

## 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: $S\left|0\right\rangle = \left|0\right\rangle$
φf = S − 1φfS

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.