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^(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 = \operatorname\,]

[\mathcal H_\mathrm = \operatorname\.]

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_^\infty \int]

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_ = \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]

[\phi_f=S^\phi_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 non zero if and only if momentum is conserved.

S-matrix and evolution operator U

[a(k,t)=U^(t)a_i(k)U(t)]

[\phi_f=U^(\infty)\phi_i U(\infty)=S^\phi_i S]

So we have [S=e^U(\infty)] where

[e^=\left\langle 0|U(\infty)|0\right\rangle^]

because

[S\left|0\right\rangle = \left|0\right\rangle.]

Substituting the explicit expression for U we obtain:

[S=\frac\mathcal T e^}]

You can see that this formula is not explicitly covariant.

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_ - \lim_\right)f^*\partial_0^\phi=\int_^,]

[\lim_\int_^\int}=\left(\lim_ - \lim_\right)\int.]

We will use this in the following calculation:

[S_=\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]

[=i\left(\lim_ - \lim_\right)\int]

[=i\int]

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_=i\int.]

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

[S_=(i)^4\int+m^2\right )\left(\Box_+m^2\right )\left(\Box_+m^2\right )\left(\Box_+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_(q_1\dots 1_)=\int}}\cdots\fracx_}}}F_(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_=(i)^\lim_(m^2-q_1)\cdots(m^2-q_)f_(q_1\dots 1_).]

The matter is that we do not have an explicit expression for [\phi(x)], so we have to make a perturbative expansion with [\phi_i(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}| 0 \right \rangle}}| 0 \right \rangle}.]

Wick's theorem

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=i\Delta_F(x_1-x_2)=i\int\frac}}.]

Which means that [\overline=\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_^m\phi(x_k)=:\Pi\phi_i(x_k):+\sum_\overline:\Pi_\phi_i(x_k):+\sum_\overline\;\overline:\Pi_\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\overline\cdots\overline)\phi(x_m})]

[G_p^=\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.

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).

 

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