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 <math>\left |0\right\rangle</math> as the void (or vacuum) quantum state. If <math>a^{\dagger}(k)</math> is a creation operator, its hermitian conjugate (destruction or annihilation operator) acts on the void as follows:
- <math>a(k)\left |0\right\rangle = 0</math>
Now, we define two kinds of creation/destruction operators, acting on different Hilbert spaces (IN space i, OUT space f), <math>a_i^\dagger (k)</math> and <math>a_f^\dagger (k)</math>.
So now
- <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>
- <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>
It is possible to prove 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 eigenstates of the momentum operator <math>\mathcal P^\mu</math>. 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:
- <math>\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}</math>
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_n \right\rangle</math>
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 void invariant and transforms IN-space fields in OUT-space fields:
- <math>S\left|0\right\rangle = \left|0\right\rangle</math>
- <math>\phi_f=S^{-1}\phi_f S</math>
If <math>S</math> describes an interaction correctly, these properties must be also true:
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>
The S-Matrix element must be non zero if and only if momentum is conserved.
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S-matrix and evolution operator U
- <math>a(k,t)=U^{-1}(t)a_i(k)U(t)</math>
- <math>\phi_f=U^{-1}(\infty)\phi_i U(\infty)=S^{-1}\phi_i S</math>
So we have <math>S=e^{i\alpha}U(\infty)</math> where
- <math>e^{i\alpha}=\left\langle 0|U(\infty)|0\right\rangle</math>
because
- <math>S\left|0\right\rangle = \left|0\right\rangle.</math>
Substituting the explicit expression for U we obtain:
- <math>S=\frac{1}{\left\langle 0|U(\infty)|0\right\rangle}\mathcal T e^{-i\int{d\tau V_i(\tau)}}</math>
You can see that this formula is not explicitly covariant.
L.S.Z. (Lehman, Symanzik, Zimmermann) reduction formula
- <math>F_n(x_1\dots x_n)=\left\langle 0|\mathcal T\phi(x_1)\dots\phi(x_n)|0\right\rangle</math>
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:
- <math>\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)},</math>
- <math>\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)}.
</math>
We will use this in the following calculation:
- <math>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</math>
This operation is called particle extraction.
- <math>=\left \langle F,k_1, k_2 | a_i^\dagger(p_2)-a_f^\dagger(p_2)|I,p_1\right\rangle</math>
This is true because p is not equal to k.
- <math>=-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}</math>
- <math>=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}</math>
- <math>=i\int{d^4x\, \left \langle F,k_1, k_2 | f*\ddot \phi - \ddot f^*\phi|I,p_1\right\rangle}
</math>
Remembering that f functions are solutions of Klein-Gordon equation:
- <math>\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^*</math>
where <math> \Box </math> stands for the D'Alembertian. Substituting this in previous equation we get (integrating by parts two times):
- <math>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}.</math>
Now we repeat these operations for all the particle in the system, and finally we get:
- <math>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}.</math>
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:
- <math>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)}.</math>
There is a theorem that states (proof omitted) that the S-matrix elements are the residuals of f calculated on mass-shell:
- <math>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}).</math>
The matter is that we do not have an explicit expression for <math>\phi(x)</math>, so we have to make a perturbative expansion with <math>\phi_i(x)</math>.
In the end, we obtain:
- <math>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}.
</math>
Wick's theorem
Definition of contraction:
- <math>\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}}.</math>
Which means that <math>\overline{AB}=\mathcal TAB-:AB:</math>
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:
- <math>\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):+\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.
</math>
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.
- <math>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})</math>
- <math>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</math>
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 <math>v=gy^4 \Rightarrow :v_i(y_1):=:\phi_i(y_1)\phi_i(y_1)\phi_i(y_1)\phi_i(y_1):</math>
This is analogous to the corresponding theorem in statistics for the moments of a Gaussian distribution.
See also Feynman diagram.
See also
The article on Rayleigh scattering for an example of the application of the S-matrix.
Bibliography
The Theory of the Scattering Matrix (Barut, 1967).