S-symplectomorphism

In physics, the S-symplectomorphism or scattering symplectomorphism is a symplectomorphism (i.e., canonical transformation) that relates the initial state and the final state of a classical physical system undergoing a scattering process. It is used in classical mechanics, scattering theory, and modern applications of quantum field theory (QFT) to classical physics.

The S-symplectomorphism is the classical counterpart of the S-matrix. Although this correspondence was first conceieved in the qualitative sense, the modern formulation shows that it is not a mere analogy. The relationship between the S-symplectomorphism and the S-matrix has been rigorously established in terms of the faithful $\hslash \to 0$ limit by utilizing the framework known as phase space formulation.

The S-symplectomorphism is a purely classical concept, so it can be defined and computed within classical mechanics without invoking quantum mechanics or QFT. Intuitively speaking, Hamiltonian mechanics geometrically interprets classical time evolution as the flow of an incompressible fluid filling the phase space. In a scattering problem, the question is how the initial state at far past ($t=-\mathrm{\infty }$) evolves into a final state at far future ($t=+\mathrm{\infty }$). In this analogy, the S-symplectomorphism is the incompressible flow that directly maps the initial fluid configuration to the final fluid configuration.

History

The idea of S-symplectomorphism was first proposed by Hunziker in 1968. Subsequent works in the '70s and the '80s developed the idea further, calling it "S-map," "S-transformation," or "canonical S-transformation." For example, a proof of the classical Levinson's theorem was provided an interesting and insightful application of S-symplectomorphism.

A modern rediscovery of the concept emerged during the development of a post-Minkowskian effective theory for spinning black holes, where "extracting the classical part of the quantum scattering" is a key task. Previously this problem was handled by the Kosower-Maybee-O'Connell (KMO'C) formalism, but the use of phase space sigma models naturally redirected researchers to envision an alternative approach.

Eventually, the purely classical definition and formalization of S-symplectomorphism was provided in the framework of symplectic and Poisson geometries, and its exact relationship with the S-matrix was also established via the frameworks of phase space formulation or deformation quantization. This redefined the status of S-symplectomorphism from a qualitative analogy to an object with quantitative correspondence with the S-matrix.

Today, the framework of S-symplectormophism is applied to classical mechanics of relativistic particles, post-Minkowskian effective theory, classical field theory, and quantum field theory, in connection with the Magnusian program.

Symplectic Property

As per the Liouville theorem, the S-symplectomorphism is an "incompressible flow" that preserves the area element of the phase space, i.e., the symplectic form. This is the symplectic property of classical scattering, paralleling the unitary property of quantum scattering.

To manifest this symplectic property, one can employ the Magnus expansion. This leads to the exponential representation of S-symplectomorphism, which reads $$S=\exp (\{\chi ,\}).$$ Here, $\chi$ is referred to as the (classical) scattering generator or classical eikonal. (In the quantum case, this has been known as the N-matrix. In the case of bulk-to-bulk evolution, the term Magnusian was coined.) The Magnus series formula (to be compared with the Dyson series formula) reads $$\begin{array}{rl}\chi =& \int d{t}_1\tilde{V}(t_1)+\frac{1}{2}\underset{t_1>t_2}{\int }{d}^{2}t\{\tilde{V}(t_1),\tilde{V}(t_2)\}\\ & +\frac{1}{6}\underset{t_1>t_2>t_3}{\int }{d}^{3}t\left(\{\tilde{V}(t_1),\{\tilde{V}(t_2),\tilde{V}(t_3)\}\}+\{\tilde{V}(t_3),\{\tilde{V}(t_2),\tilde{V}(t_1)\}\}\right),\end{array}$$ which describes a sum of integrals whose integrands are nested Poisson brackets between the interaction-picture potential $\tilde{V}(t)$ at different times.

Scattering generator as an effective Hamiltonian

The scattering generator $\chi$ is an "effective Hamiltonian" that generates the infinite time evolution from $t=-\mathrm{\infty }$ to $t=+\mathrm{\infty }$ in the interaction picture within "one second" (meaning dimensionless unit time); this insight was used to establish the relationship between scattering generator and the on-shell action in Hamilton-Jacobi formulation. See also Magnusian.

Impulse of Classical Observables

For a phase space function ${𝒪}$ representing a classical observable, its impulse due to the scattering process is computed as $$\mathrm{\Delta }{𝒪}=(S^{-1}{)}^{*}{𝒪}-{𝒪},$$ where $(S^{-1}{)}^{*}$ denotes the pullback of the inverse map $S^{-1}$. Since $S^{-1}$ is the map from the final phase space to the initial phase space, the pullback $(S^{-1}{)}^{*}$ maps functions living in the initial phase space to functions living in the final phase space. Hence $(S^{-1}{)}^{*}{𝒪}$ is the image of the initial-time observable ${𝒪}$ at the final-time. This explains the above formula.

Nested bracket formula

By using the exponential representation of S-symplectomorphism described above, the impulse formula can be written as $$\begin{array}{rl}\mathrm{\Delta }{𝒪}& =(e^{\{\chi ,\}}{𝒪})-{𝒪}\\ & =\{\chi ,{𝒪}\}+\frac{1}{2!}\{\chi ,\{\chi ,{𝒪}\}\}+\frac{1}{3!}\{\chi ,\{\chi ,\{\chi ,{𝒪}\}\}\}+\cdots .\end{array}$$ This is the nested bracket formula that computes the impulse of any classical observable ${𝒪}$ from the classical scattering generator $\chi$.

Relation to KMO'C formalism

The S-symplectomorphism or scattering generator approach to impulse formula can be compared to the KMO'C approach. In the KMO'C formalism, the impulse of a quantum observable $\widehat{{𝒪}}$ is obtained as $$\begin{array}{rl}\mathrm{\Delta }\widehat{{𝒪}}& ={\hat{S}}^{-1}\widehat{{𝒪}}\hat{S}-\widehat{{𝒪}},\\ & =\frac{1}{i\hslash }[\hat{T},\widehat{{𝒪}}]+\frac{1}{(i\hslash {)}^2}{\hat{T}}^{†}[\hat{T},\widehat{{𝒪}}],\end{array}$$ where the second line arises by the usual split $\hat{S}=\hat{1}+i\hat{T}/\hslash$ of the S-matrix. The first term, $[\hat{T},\widehat{{𝒪}}]/i\hslash$, admits a smooth classical limit to $\{T,{𝒪}\}$. However, the second term ${\hat{T}}^{†}[\hat{T},\widehat{{𝒪}}]/(i\hslash {)}^2$ does not and serves as the source of the superclassical term subtlety. This simply points out some potential inconveniences in having a "standalone" classical framework; the KMO'C formalism is still a useful and consistent method.

The nested bracket formula arises by highlighting the adjoint action structure: $$\begin{array}{rl}\mathrm{\Delta }\widehat{{𝒪}}& ={\hat{S}}^{-1}\widehat{{𝒪}}\hat{S}-\widehat{{𝒪}}=({\mathrm{A}\mathrm{d}}_{{\hat{S}}^{-1}}\widehat{{𝒪}})-\widehat{{𝒪}}.\end{array}$$ As explained below, this translates to the phase space statement $$\begin{array}{rl}\mathrm{\Delta }{𝒪}& =(S^{-1}{)}^{*}{𝒪}-{𝒪}.\end{array}$$ via employing the phase-space formulation and taking classical limit.

Precise Relation to S-matrix

The S-symplectomorphism essentially describes the classical limit of the adjoint action of the S-matrix, which can be shown as below.

A key fact is that the phase space formulation can be applied to scattering theory. In this case, the S-matrix $\hat{S}$ is formulated as a fuzzy diffeomorphism $S^{\star }$ on the $\hslash$-deformed phase space. According to ideas in non-commutative geometry, this is an automorphism of the C^∗-algebra. This means that $S^{\star }$ is a map that preserves the algebraic structure of quantum observables in the phase space formulation, such as pointwise addition and the star product (as the quantum operator algebra) $$S^{\star }[f+g]=S^{\star }[f]+S^{\star }[g],$$ $$S^{\star }[f\star g]=S^{\star }[f]\star S^{\star }[g].$$ Assuming a well-behaved and invertible quantization map ${𝒬}$ that maps phase space functions $f$ to operators $\hat{f}$ (such as the Weyl transform), the fuzzy diffeomorphism $S^{\star }$ is defined from the S-matrix by the intertwining relation $${\mathrm{A}\mathrm{d}}_{\hat{S}}\circ {𝒬}={𝒬}\circ S^{\star }.$$ In the classical limit, the fuzzy S-diffeomorphism approaches to the S-symplectomorphism: $$\underset{\hslash \to 0}{\lim }{S}^{\star }=S.$$ Therefore, one establishes the precise equation $$S=\underset{\hslash \to 0}{\lim }({{𝒬}}^{-1}\circ {\mathrm{A}\mathrm{d}}_{\hat{S}}\circ {𝒬}),$$ where ${{𝒬}}^{-1}$ is the dequantization map.

In summary, the adjoint action of the S-matrix, ${\mathrm{A}\mathrm{d}}_{\hat{S}}$, translates to the fuzzy S-diffeomorphism $S^{\star }$ via the intertwining by quantization map ${𝒬}$, whose $\hslash \to 0$ limit is the S-symplectomorphism.

To elaborate, the fuzzy S-diffeomorphism $S^{\star }$ preserves the algebraic structure of quantum observables: pointwise addition and the star product. Taking the classical limit yields a map $S$ that preserves the algebraic structure of classical observables. This means the Poisson algebra on phase space functions formed by pointwise addition, pointwise product, and the Poisson bracket. Namely, the pointwise product and Poisson bracket are the (semi-)classical vestiges of the star product.

Poisson S-diffeomorphism

Some classical systems admit their Hamiltonian formulation on odd-dimensional phase spaces, with well-defined Poisson bracket but with no symplectic form. A concrete example is the phase space of angular momentum or spin, which is the space ${\mathbb{ℝ}}^3$ of $(J_1,J_2,J_3)$ equipped with the Poisson bracket relation $\{J_i,J_j\}=J_k{\epsilon }^k{}_{ij}$. This phase space can be used for describing Rabi oscillation or rigid body motion, for instance.

In the meantime, some classical systems are formulated on phase spaces with a degenerate Poisson bracket, meaning that its "rank" is not full. Namely, there exists at least one function $C$ that has vanishing Poisson bracket between any other classical observable: $\{C,f\}=0$ for all $f$. For example, a modern formulation of relativistic massive spin is based on a 12-dimensional space with rank 10 Poisson bracket.

Mathematically speaking, these describe the cases where the phase space is a Poisson manifold but not a symplectic manifold. It is known that the scattering map from the initial phase space to the final phase space is well-defined even if the phase space is a Poisson manifold that is not symplectic. Moreover, the relation to the quantum S-matrix can also be established in a precise fashion in the context of deformation quantization.

In this Poisson case, the scattering map is called the Poisson S-diffeomorphism or scattering Poisson diffeomorphism since it preserves the Poisson structure (Poison bracket relation) of the phase space.

See also

• S-matrix
• Hamiltonian mechanics
• Symplectomorphism
• Canonical transformation
• Interaction picture
• Levinson's theorem
• Liouville's theorem (Hamiltonian)
• Phase-space formulation
• Magnusian

References

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

---