S matrix

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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 \left |0\right\rangle 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:

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 void 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 non zero if and only if momentum is conserved.

Contents

[edit] 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

So we have 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)}}

You can see that this formula is not explicitly covariant.

[edit] L.S.Z. (Lehman, Symanzik, Zimmermann) reduction formula

F_n(x_1\dots x_n)=\left\langle 0|\mathcal T\phi(x_1)\dots\phi(x_n)|0\right\rangle

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:

\left(\lim_{x_0\to \infty} - \lim_{x_0 \to \infty}\right)f^*\partial_0^{\leftrightarrow}\phi=\int_{-\infty}^{\infty} {dx_0\,\left( f^*\ddot\phi-\ddot f^*\phi\right)},
\lim_{t_1,t_2\to\infty}\int_{t_1}^{t_2}{d\tau\, \frac{\partial}{\partial t}\int{d^3x\,\psi(x,t)}}=\left(\lim_{x_0\to \infty} - \lim_{x_0 \to \infty}\right)\int{d^3x\,\psi(x,t)}.

We will use this in the following calculation:

S_{fi}=\left \langle F,k_1, k_2 | I,p_1,p_2\right\rangle=\left \langle F,k_1, k_2 | a_i^\dagger(p_2)|I,p_1\right\rangle

This operation is called particle extraction.

=\left \langle F,k_1, k_2 | a_i^\dagger(p_2)-a_f^\dagger(p_2)|I,p_1\right\rangle

This is true because p is not equal to k.

=-i\int{d^3x\, f^*(p_2,x)\partial_0^\leftrightarrow \left \langle F,k_1, k_2 | \phi_i(x)-\phi_f(x)|I,p_1\right\rangle}
=i\left(\lim_{t\to \infty} - \lim_{t \to \infty}\right)\int{d^3x\, f^*(p_2,t)\partial_0^\leftrightarrow \left \langle F,k_1, k_2 | \phi(x)|I,p_1\right\rangle}
=i\int{d^4x\, \left \langle F,k_1, k_2 | f^*\ddot \phi - \ddot f^*\phi|I,p_1\right\rangle}

Remembering that f functions are solutions of Klein-Gordon equation:

\left( \Box + m^2 \right ) f^*=0=\ddot f^* - \nabla^2 f^* + m^2 f^* \Rightarrow \ddot f^*=\left( \nabla^2-m^2\right)f^*

where \Box stands for the D'Alembertian. Substituting this in previous equation we get (integrating by parts two times):

S_{fi}=i\int{d^4x\, f^*(p_2,x)\left(\Box_x+m^2\right )\left \langle F,k_1, k_2 | \phi(x)|I,p_1\right\rangle}.

Now we repeat these operations for all the particle in the system, and finally we get:

S_{fi}=(i)^4\int{d^4x_1\, d^4x_2\, d^4y_1\, d^4y_2\, f^*(p_1,x_1)f^*(p_2,x_2)f(k_1,y_1)f(k_2,y_2)\left(\Box_{x_1}+m^2\right )\left(\Box_{x_2}+m^2\right )\left(\Box_{y_1}+m^2\right )\left(\Box_{y_2}+m^2\right )\left \langle 0|\mathcal T\phi(x_1)\phi(x_2)\phi(y_1)\phi(y_2)|0\right\rangle}.

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:

f_{mn}(q_1\dots 1_{m+n})=\int{d^4x_1\cdots d^4x_n\, d^4y_1\cdots d^4y_m\, \frac{e^{-iq_1x_1}}{\sqrt{(2\pi)^32\omega_k}} \cdots\frac{e^{-iq_{n+m}x_{n+m}}}{\sqrt{(2\pi)^32\omega_k}} F_{nm}(x_1\dots x_n,y_1\dots y_m)}.

There is a theorem that states (proof omitted) that the S-matrix elements are the residuals of f calculated on mass-shell:

S_{fi}=(i)^{n+m}\lim_{q_i\to m^2}(m^2-q_1)\cdots(m^2-q_{n+m})f_{nm}(q_1\dots 1_{n+m}).

The matter is that we do not have an explicit expression for φ(x), so we have to make a perturbative expansion with φi(x).

In the end, we obtain:

F_p(x)=\left \langle 0 |\mathcal T\phi(x_1)\dots\phi(x_p)| 0 \right \rangle=\frac{\left \langle 0 |\mathcal T e^{-i\int d\tau\, V_i(\tau)} \phi_i(x_1)\dots\phi_i(x_p)| 0 \right \rangle}{\left \langle 0 |e^{-i\int{d\tau\, V_i(\tau)}}| 0 \right \rangle}.

[edit] Wick's theorem

Wick's theorem 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 void state give a null contribute 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.

See also Feynman diagram.

[edit] Bibliography

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

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