S-matrix - WikiHQ

In physics, the S-matrix or scattering matrix is a matrix that relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).

More formally, in the context of QFT, the S-matrix is defined as the unitary matrix connecting sets of asymptotically free particle states (the in-states and the out-states) in the Hilbert space of physical states: a multi-particle state is said to be free (or non-interacting) if it transforms under Lorentz transformations as a tensor product, or direct product in physics parlance, of one-particle states as prescribed by equation below. Asymptotically free then means that the state has this appearance in either the distant past or the distant future.

While the S-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no event 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 Poincaré group); the S-matrix is the evolution operator between <math>t= - \infty </math> (the distant past), and <math>t= + \infty </math> (the distant future). It is defined only in the limit of zero energy density (or infinite particle separation distance).

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.

History

The initial elements of S-matrix theory are found in Paul Dirac's 1927 paper "Über die Quantenmechanik der Stoßvorgänge". The S-matrix was first properly introduced by John Archibald Wheeler in the 1937 paper "On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure". In this paper Wheeler introduced a scattering matrix – a unitary matrix of coefficients connecting "the asymptotic behaviour of an arbitrary particular solution [of the integral equations] with that of solutions of a standard form", but did not develop it fully.

In the 1940s, Werner Heisenberg independently developed and substantiated the idea of the S-matrix. Because of the problematic divergences present in quantum field theory at that time, Heisenberg was motivated to isolate the essential features of the theory that would not be affected by future changes as the theory developed. In doing so, he was led to introduce a unitary "characteristic" S-matrix.

<math display="block">\begin{pmatrix}C \\ D \end{pmatrix} = \begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix}\begin{pmatrix} A \\ B \end{pmatrix}</math> and its components can be derived from the components of the S-matrix via: <math>M_{11}=1/S_{12}^*= 1/S_{21} ^* {,}\ M_{22}= M_{11}^*</math> and <math>M_{12}=-S_{11}^*/S_{12}^* = S_{22}/S_{12} {,}\ M_{21} = M_{12}^*</math>, whereby time-reversal symmetry is assumed.

In the case of time-reversal symmetry, the transfer matrix <math>\mathbf{M}</math> can be expressed by three real parameters:

<math display="block">M = \frac{1}{\sqrt{1-r^2 \begin{pmatrix} e^{i\varphi} & -r\cdot e^{-i\delta} \\ -r\cdot e^{i\delta} & e^{-i\varphi} \end{pmatrix}</math>

with <math>\delta,\varphi \in [0;2\pi]</math> and <math>r\in [0;1]</math> (in case there would be no connection between the left and the right side)

Finite square well

The one-dimensional, non-relativistic problem with time-reversal symmetry of a particle with mass m that approaches a (static) finite square well, has the potential function with

<math display="block">V(x) = \begin{cases}

-V_0 & \text{for}~~ |x| \le a ~~ (V_0 > 0) \quad\text{and}\\[1ex]

0 & \text{for}~~ |x|>a

\end{cases}</math>

The scattering can be solved by decomposing the wave packet of the free particle into plane waves <math>A_k\exp(ikx)</math> with wave numbers <math>k>0</math> for a plane wave coming (faraway) from the left side or likewise <math>D_k\exp(-ikx)</math> (faraway) from the right side.

The S-matrix for the plane wave with wave number has the solution: <math>S_{12} = S_{21} = \frac{\exp(-2ika)}{\cosh(2\kappa a)-i\sinh(2\kappa a)\frac{k^2-{\kappa}^2}{2k\kappa</math>

and likewise: <math>S_{11}=-i\frac{k^2+\kappa^2}{2k\kappa}\sinh(2\kappa a)\cdot S_{12}</math> and also in this case <math>S_{22}=S_{11}</math>.

The transmission is <math>T_k=|S_{21}|^2=|S_{12}|^2=\frac{1}{1+(\sinh(2\kappa a))^2\frac{(k^2+\kappa^2)^2}{4k^2\kappa^2</math>.

Transmission coefficient and reflection coefficient

The transmission coefficient from the left of the potential barrier is, when ,

The reflection coefficient from the left of the potential barrier is, when ,

Similarly, the transmission coefficient from the right of the potential barrier is, when ,

The reflection coefficient from the right of the potential barrier is, when ,

The relations between the transmission and reflection coefficients are

and

This identity is a consequence of the unitarity property of the S-matrix.

With time-reversal symmetry, the S-matrix is symmetric and hence <math>T_{\rm L}=|S_{21}|^2=|S_{12}|^2=T_{\rm R}</math> and <math>R_{\rm L} = R_{\rm R}</math>.

Optical theorem in one dimension

In the case of free particles , the S-matrix is

<math display="block"> S=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.</math>

Whenever is different from zero, however, there is a departure of the S-matrix from the above form, to

<math display="block"> S = \begin{pmatrix} 2ir & 1+2it \\ 1+2it &2ir^* \frac{1+2it}{1-2it^*} \end{pmatrix}.</math>

This departure is parameterized by two complex functions of energy, and .

From unitarity there also follows a relationship between these two functions,

<math display="block">|r|^2+|t|^2 = \operatorname{Im}(t).</math>

The analogue of this identity in three dimensions is known as the optical theorem.

Definition in quantum field theory

Interaction picture

A straightforward way to define the S-matrix begins with considering the interaction picture. Let the Hamiltonian be split into the free part and the interaction , . In this picture, the operators behave as free field operators and the state vectors have dynamics according to the interaction . Let

<math display="block">\left|\Psi(t)\right\rangle</math>

denote a state that has evolved from a free initial state

<math display="block">\left|\Phi_{\rm i}\right\rangle.</math>

The S-matrix element is then defined as the projection of this state on the final state

<math display="block">\left\langle\Phi_{\rm f}\right|.</math>

Thus

<math display="block">S_{\rm fi} \equiv \lim_{t \rightarrow +\infty} \left\langle\Phi_{\rm f}|\Psi(t)\right\rangle \equiv \left\langle\Phi_{\rm f}\right|S\left|\Phi_{\rm i}\right\rangle,</math>

where is the S-operator. The great advantage of this definition is that the time-evolution operator evolving a state in the interaction picture is formally known,

where denotes the time-ordered product. Expressed in this operator,

<math display="block">S_{\rm fi} = \lim_{t_2 \rightarrow +\infty}\lim_{t_1 \rightarrow -\infty}\left\langle\Phi_{\rm f}\right|U(t_2, t_1)\left|\Phi_{\rm i}\right\rangle,</math>

from which

<math display="block">S = U(\infty, -\infty).</math>

Expanding using the knowledge about gives a Dyson series,

<math display="block">S = \sum_{n=0}^\infty \frac{(-i)^n}{n!}\int_{-\infty}^\infty dt_1\cdots \int_{-\infty}^\infty dt_n T\left[V(t_1)\cdots V(t_n)\right],</math>

or, if comes as a Hamiltonian density <math>\mathcal{H}</math>,

<math display="block">S = \sum_{n=0}^\infty \frac{(-i)^n}{n!}\int_{-\infty}^\infty dx_1^4\cdots \int_{-\infty}^\infty dx_n^4 T\left[\mathcal{H}(x_1)\cdots \mathcal{H}(x_n)\right].</math>

Being a special type of time-evolution operator, is unitary. For any initial state and any final state one finds

<math display="block">S_{\rm fi} = \left\langle\Phi_{\rm f}|S|\Phi_{\rm i}\right\rangle = \left\langle\Phi_{\rm f} \left|\sum_{n=0}^\infty \frac{(-i)^n}{n!}\int_{-\infty}^\infty dx_1^4\cdots \int_{-\infty}^\infty dx_n^4 T\left[\mathcal{H}(x_1)\cdots \mathcal{H}(x_n)\right]\right| \Phi_{\rm i}\right\rangle .</math>

This approach is somewhat naïve in that potential problems are swept under the carpet. This is intentional. The approach works in practice and some of the technical issues are addressed in the other sections.

In and out states

Here a slightly more rigorous approach is taken in order to address potential problems that were disregarded in the interaction picture approach of above. The final outcome is, of course, the same as when taking the quicker route. For this, the notions of in and out states are needed. These will be developed in two ways, from vacuum, and from free particle states. Needless to say, the two approaches are equivalent, but they illuminate matters from different angles.

From vacuum

If is a creation operator, its hermitian adjoint is an annihilation operator and destroys the vacuum,

<math display="block">a(k)\left |*, 0\right\rangle = 0.</math>

In Dirac notation, define

<math display="block">|*, 0\rangle</math>

as a vacuum quantum state, i.e. a state without real particles. The asterisk signifies that not all vacua are necessarily equal, and certainly not equal to the Hilbert space zero state . All vacuum states are assumed Poincaré invariant, invariance under translations, rotations and boosts, on two distinct complete sets (Fock spaces; initial space , final space ). These operators satisfy the usual commutation rules,

<math display="block">\begin{align}

{[a_{\rm i,o}(\mathbf{p}), a^\dagger_{\rm i,o}(\mathbf{p}')]} &= i\delta(\mathbf{p} - \mathbf{p'}),\\

{[a_{\rm i,o}(\mathbf{p}), a_{\rm i,o}(\mathbf{p'})]} &= {[a^\dagger_{\rm i,o}(\mathbf{p}), a^\dagger_{\rm i,o}(\mathbf{p'})]} = 0.

\end{align}</math>

The action of the creation operators on their respective vacua and states with a finite number of particles in the in and out states is given by

<math display="block">\begin{align}

\left| \mathrm{i}, k_1\ldots k_n \right\rangle &= a_i^\dagger (k_1)\cdots a_{\rm i}^\dagger (k_n)\left| i, 0\right\rangle,\\

\left| \mathrm{f}, p_1\ldots p_n \right\rangle &= a_{\rm f}^\dagger (p_1)\cdots a_f^\dagger (p_n)\left| f, 0\right\rangle,

\end{align}</math>

where issues of normalization have been ignored. See the next section for a detailed account on how a general state is normalized. The initial and final spaces are defined by

<math display="block">\mathcal H_{\rm i} = \operatorname{span}\{ \left| \mathrm{i}, k_1\ldots k_n \right\rangle = a_{\rm i}^\dagger (k_1)\cdots a_{\rm i}^\dagger (k_n)\left| \mathrm{i}, 0\right\rangle\},</math>

<math display="block">\mathcal H_{\rm f} = \operatorname{span}\{ \left| \mathrm{f}, p_1\ldots p_n \right\rangle = a_{\rm f}^\dagger (p_1)\cdots a_{\rm f}^\dagger (p_n)\left| \mathrm{f}, 0\right\rangle\}.</math>

The asymptotic states are assumed to have well defined Poincaré transformation properties, i.e. they are assumed to transform as a direct product of one-particle states. This is a characteristic of a non-interacting field. From this follows that the asymptotic states are all eigenstates of the momentum operator ,

<math display="block">|\mathrm{i}, 0\rangle = |\mathrm{f}, 0\rangle = |*,0\rangle \equiv |0\rangle.</math>

The interaction is assumed adiabatically turned on and off.

Heisenberg picture

The Heisenberg picture is employed henceforth. In this picture, the states are time-independent. A Heisenberg state vector thus represents the complete spacetime history of a system of particles. This definition conforms with the direct approach used in the interaction picture. Also, due to unitary equivalence,

<math display="block">\langle\Psi_\beta^+|S|\Psi_\alpha^+\rangle = S_{\beta\alpha} = \langle\Psi_\beta^-|S|\Psi_\alpha^-\rangle.</math>

As a physical requirement, must be a unitary operator. This is a statement of conservation of probability in quantum field theory. But

<math display="block">\langle\Psi_\beta^-|S|\Psi_\alpha^-\rangle = S_{\beta\alpha} = \langle\Psi_\beta^-|\Psi_\alpha^+\rangle.</math>

By completeness then,

<math display="block">S|\Psi_\alpha^-\rangle = |\Psi_\alpha^+\rangle,</math>

so S is the unitary transformation from in-states to out states.

Lorentz invariance is another crucial requirement on the S-matrix. The S-operator represents the quantum canonical transformation of the initial in states to the final out states. Moreover, leaves the vacuum state invariant and transforms in-space fields to out-space fields,

<math display="block">S\left|0\right\rangle = \left|0\right\rangle</math>

<math display="block">\phi_\mathrm{f}=S\phi_\mathrm{i} S^{-1} ~.</math>

In terms of creation and annihilation operators, this becomes

<math display="block">a_{\rm f}(p)=Sa_{\rm i}(p)S^{-1}, a_{\rm f}^\dagger(p)=Sa_{\rm i}^\dagger(p)S^{-1},</math>

hence

<math display="block">\begin{align}

S|\mathrm{i}, k_1, k_2, \ldots, k_n\rangle

&= Sa_{\rm i}^\dagger(k_1)a_{\rm i}^\dagger(k_2) \cdots a_{\rm i}^\dagger(k_n)|0\rangle =

Sa_{\rm i}^\dagger(k_1)S^{-1}Sa_{\rm i}^\dagger(k_2)S^{-1} \cdots Sa_{\rm i}^\dagger(k_n)S^{-1}S|0\rangle \\[1ex]

&=a_{\rm o}^\dagger(k_1)a_{\rm o}^\dagger(k_2) \cdots a_{\rm o}^\dagger(k_n)S|0\rangle

=a_{\rm o}^\dagger(k_1)a_{\rm o}^\dagger(k_2) \cdots a_{\rm o}^\dagger(k_n)|0\rangle

=|\mathrm{o}, k_1, k_2, \ldots, k_n\rangle.

\end{align}</math>

A similar expression holds when operates to the left on an out state. This means that the S-matrix can be expressed as

<math display="block">S_{\beta\alpha} = \langle \mathrm{o}, \beta|\mathrm{i}, \alpha \rangle = \langle \mathrm{i}, \beta|S|\mathrm{i}, \alpha \rangle = \langle \mathrm{o}, \beta|S|\mathrm{o}, \alpha \rangle.</math>

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

If the system is made up with a single particle in momentum eigenstate , then . This follows from the calculation above as a special case.
The S-matrix element may be nonzero only where the output state has the same total momentum as the input state. This follows from the required Lorentz invariance of the S-matrix.

Evolution operator U

Define a time-dependent creation and annihilation operator as follows,

<math display="block">\begin{align}

a^{\dagger}{\left(k,t\right)} &= U^{-1}(t) \, a^{\dagger}_{\rm i}{\left(k\right)} \, U{\left( t \right)} \\[1ex]

a{\left(k,t\right)} &= U^{-1}(t) \, a_{\rm i}{\left(k\right)} \, U{\left( t \right)} \, ,

\end{align}</math>

so, for the fields,

<math display="block">\phi_{\rm f}=U^{-1}(\infty)\phi_{\rm i} U(\infty)=S^{-1}\phi_{\rm i} S~,</math>

where

<math display="block">S= e^{i\alpha}\, U(\infty).</math>

We allow for a phase difference, given by

<math display="block">e^{i\alpha}=\left\langle 0|U(\infty)|0\right\rangle^{-1} ~,</math>

because for ,

Substituting the explicit expression for , one has

<math display="block">S=\frac{1}{\left\langle 0|U(\infty)|0\right\rangle}\mathcal T e^{-i\int{d\tau H_{\rm{int(\tau)~,</math>

where <math> H_{\rm{int</math> is the interaction part of the Hamiltonian and <math> \mathcal T </math> is the time ordering.

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

Dyson series

The most widely used expression for the S-matrix is the Dyson series. This expresses the S-matrix operator as the series:

<math display="block">S = \sum_{n=0}^\infty \frac{(-i)^n}{n!} \int \cdots \int d^4x_1 d^4x_2 \ldots d^4x_n T [ \mathcal{H}_{\rm{int(x_1) \mathcal{H}_{\rm{int(x_2) \cdots \mathcal{H}_{\rm{int(x_n)] </math>

where:

<math>T[\cdots]</math> denotes time-ordering,
<math>\; \mathcal{H}_{\rm{int(x)</math> denotes the interaction Hamiltonian density which describes the interactions in the theory.

The not-S-matrix

Since the transformation of particles from black hole to Hawking radiation could not be described with an S-matrix, Stephen Hawking proposed a "not-S-matrix", for which he used the dollar sign ($), and which therefore was also called "dollar matrix".

Remarks

Notes

References

§125

History

Finite square well

Transmission coefficient and reflection coefficient

Optical theorem in one dimension

Definition in quantum field theory

Interaction picture

In and out states

From vacuum

Heisenberg picture

Evolution operator U

Dyson series

The not-S-matrix

See also

Remarks

Notes

References