Scattering generator

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"Eikonal matrix" redirects here. For the meaning in WKB approximation, see Eikonal approximation.

"N-matrix" redirects here. For a term in linear algebra, see Matrix (mathematics).

In scattering theory, the scattering generator or S-generator is an effective Hamiltonian that directly generates the interaction-picture time evolution from far past to far future.

The scattering generator is the log of the S-matrix in quantum mechanics and quantum field theory: $\hat{\chi }=i\hslash \log \hat{S}$.

Intuitively speaking, the scattering generator is an object that encapsulates and summarizes the entire history of scattering from $t=-\mathrm{\infty }$ to $t=+\mathrm{\infty }$, reproducing the final result within "one second."

The scattering generator is also defined in classical mechanics. In the classical scattering theory based on Hamiltonian formulation, the analog of the S-matrix is known as S-symplectomorphism, which is the canonical transformation mapping initial scattering states to final scattering states. The classical scattering generator is the generator of S-symplectomorphism. The quantum and classical scattering generators are related by the classical limit in a precise sense.

The scattering generator should be distinguished from the Magnusian, which is a term coined for incorporating the non-scattering cases.

History

Exponential representation of S-matrix

The idea of taking the log of S-matrix traces back to early days of quantum field theory and has been known as the exponential representation of S-matrix, which refers to the formula $$\begin{array}{r}\hat{S}=\exp \left(\frac{1}{i\hslash }\hat{\chi }\right).\end{array}$$ This approach also makes contact with the eikonal approximation, and hence $\hat{\chi }$ has been also called the eikonal matrix.

Modern take: unit-time flow

Modern literature has established the interpretation of $\hat{\chi }$ as the unit-time generator of scattering, although prototypical observations may trace back to the '80s. The idea is to reinterpret the above formula as an effective time evolution through dimensionless unit time: $$\begin{array}{r}\hat{S}=\exp \left(\frac{(\text{“}\mathrm{\Delta }t=1\text{”})}{i\hslash }\hat{\chi }\right).\end{array}$$ This means that the entire effect of classical scattering from $t=-\mathrm{\infty }$ to $t=+\mathrm{\infty }$, $\hat{S}$, is reproduced within "one second" ($\mathrm{\Delta }t=1$) by taking $\hat{\chi }$ as the Hamiltonian. In this sense, $\hat{\chi }$ could be viewed as an "effective Hamiltonian."

Manifest unitarity

The motivation behind the exponential representation is manifest unitarity in scattering, i.e., trivializing the conservation of probability. Provided hermiticity ${\hat{\chi }}^{†}=\hat{\chi }$, the S-matrix is automatically unitary as $$\begin{array}{r}{\hat{S}}^{†}\hat{S}=\exp \left(-\frac{1}{i\hslash }\hat{\chi }\right)\exp \left(\frac{1}{i\hslash }\hat{\chi }\right)=\hat{1}.\end{array}$$ A similar remark applies for the classical scattering generator as well, in which case one manifests the conservation of classical probability in the sense of Liouville theorem.

Definition

As is well-known, the S-matrix is concretely defined and computed by the Dyson series, which expands a time-ordered exponential. Similarly, the scattering generator is concretely defined and computed by the Magnus series. This is because the Magnus series computes the log of a time-ordered exponential by definition.

In quantum mechanics

For the quantum scattering generator $\hat{\chi }$, the Magnus series formula reads $$\begin{array}{rl}\hat{\chi }=& \int d{t}_1{\hat{V}}_{\text{I}}(t_1)+\frac{1}{2(i\hslash {)}^1}\underset{t_1>t_2}{\int }{d}^{2}t[{\hat{V}}_{\text{I}}(t_1),{\hat{V}}_{\text{I}}(t_2)]\\ & +\frac{1}{6(i\hslash {)}^2}\underset{t_1>t_2>t_3}{\int }{d}^{3}t\left([{\hat{V}}_{\text{I}}(t_1),[{\hat{V}}_{\text{I}}(t_2),{\hat{V}}_{\text{I}}(t_3)]]+[{\hat{V}}_{\text{I}}(t_3),[{\hat{V}}_{\text{I}}(t_2),{\hat{V}}_{\text{I}}(t_1)]]\right)+\cdots .\end{array}$$ This describes a sum of integrals whose integrands are nested commutators between the interaction-picture potential ${\hat{V}}_{\mathrm{I}}(t)$ at different times. Provided the free and interaction Hamiltonians are Hermitian, the scattering generator $\hat{\chi }$ is also a Hermitian operator.

In classical mechanics

For the classical scattering generator $\chi$, the Magnus series formula reads $$\begin{array}{rl}\chi =& \int d{t}_1{V}_{\text{I}}(t_1)+\frac{1}{2}\underset{t_1>t_2}{\int }{d}^{2}t\{V_{\text{I}}(t_1),V_{\text{I}}(t_2)\}\\ & +\frac{1}{6}\underset{t_1>t_2>t_3}{\int }{d}^{3}t\left(\{V_{\text{I}}(t_1),\{V_{\text{I}}(t_2),V_{\text{I}}(t_3)\}\}+\{V_{\text{I}}(t_3),\{V_{\text{I}}(t_2),V_{\text{I}}(t_1)\}\}\right)+\cdots ,\end{array}$$ which can be deduced by taking the classical limit to the above quantum formula in spirit of correspondence principle and canonical quantization. This assumes a Hamiltonian system defined on a phase space equipped with a Poisson bracket. $\chi$ is a function on the phase space. $V_{\text{I}}(t)$ is a time-dependent function on the phase space, encoding the classical interaction Hamiltonian in the interaction picture.

More precisely, the frameworks of phase space formulation and deformation quantization have been employed to establish the relationship between $\hat{\chi }$ and $\chi$ in a rigorous fashion. Most generally, the classical scattering generator is well-defined on Poisson manifolds.

The S-symplectomorphism $S$, i.e., the canonical transformation from the initial phase space to the final phase space, arises by exponentiating the Hamiltonian vector field of $\chi$.

Use

In quantum mechanics

In the interaction picture, the quantum Liouville equation reads $$\begin{array}{r}\frac{d}{dt}{\hat{\rho }}_{\mathrm{I}}(t)=\frac{1}{i\hslash }[{\hat{V}}_{\mathrm{I}}(t),{\hat{\rho }}_{\mathrm{I}}(t)],\end{array}$$ where ${\hat{\rho }}_{\mathrm{I}}(t)$ is the density matrix in the interaction picture. Solving this equation gives rise to the S-matrix $\hat{S}$ as $$\begin{array}{r}{\hat{\rho }}_{\mathrm{I}}(+\mathrm{\infty })=\hat{S}{\hat{\rho }}_{\mathrm{I}}(-\mathrm{\infty })\hat{S}{}^{-1}={\mathrm{A}\mathrm{d}}_{\hat{S}}{\hat{\rho }}_{\mathrm{I}}(-\mathrm{\infty }),\end{array}$$ in terms of the adjoint action ${\mathrm{A}\mathrm{d}}_{\hat{S}}$. The formula $\hat{S}=e^{\hat{\chi }/i\hslash }$ then implies $$\begin{array}{rl}{\hat{\rho }}_{\mathrm{I}}(+\mathrm{\infty })& =\exp \left({\mathrm{a}\mathrm{d}}_{\hat{\chi }/i\hslash }\right){\hat{\rho }}_{\mathrm{I}}(-\mathrm{\infty }),\\ & =\exp \left(\frac{1}{i\hslash }[\hat{\chi },]\right){\hat{\rho }}_{\mathrm{I}}(-\mathrm{\infty }),\\ & ={\hat{\rho }}_{\mathrm{I}}(-\mathrm{\infty })+\frac{1}{i\hslash }[\hat{\chi },{\hat{\rho }}_{\mathrm{I}}(-\mathrm{\infty })]+\frac{1}{2!(i\hslash {)}^2}[\hat{\chi },[\hat{\chi },{\hat{\rho }}_{\mathrm{I}}(-\mathrm{\infty })]]+\cdots ,\end{array}$$ which describes a sum of nested commutators.

This describes that the entire time evolution of the quantum (ensemble) state from ${\hat{\rho }}_{\mathrm{I}}(-\mathrm{\infty })$ to ${\hat{\rho }}_{\mathrm{I}}(+\mathrm{\infty })$ is reproduced by a unit-time adjoint action of the scattering generator $\hat{\chi }$.

In classical mechanics

In classical Hamiltonian mechanics, an analogous formula holds for the classical probability distribution $\rho (t)$, representing a statistical ensemble and evolving under the classical Liouville equation: $$\begin{array}{rl}{\rho }_{\mathrm{I}}(+\mathrm{\infty })& =\exp \left(\{\chi ,\}\right){\rho }_{\mathrm{I}}(-\mathrm{\infty }),\\ & ={\rho }_{\mathrm{I}}(-\mathrm{\infty })+\{\chi ,{\rho }_{\mathrm{I}}(-\mathrm{\infty })\}+\frac{1}{2!}\{\chi ,\{\chi ,{\rho }_{\mathrm{I}}(-\mathrm{\infty })\}\}+\cdots .\end{array}$$ This is the nested Poisson bracket formula in the S-symplectomorphism framework. Certainly, the effect of the time evolution on the classical state, from ${\rho }_{\mathrm{I}}(-\mathrm{\infty })$ to ${\rho }_{\mathrm{I}}(+\mathrm{\infty })$, is generated by the unit-time Hamiltonian flow of $\chi$.

Conventions

There exist different conventions for the scattering generator. Some authors prefer using the definition $\hat{N}=-i\hslash \log \hat{S}$ and the name N-matrix.

References

This article "Scattering generator" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Scattering generator. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

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