Draft:S-symplectomorphism

• Comment: Please read WP:NOTATEXTBOOK and rewrite Ldm1954 (talk) 12:21, 11 May 2026 (UTC)

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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. This parallel was first conceived as a qualitative analogy, based on the correspondence principle. In the modern formulation, the frameworks of phase space formulation or deformation quantization are employed to establish an exact relation.

The S-symplectomorphism is a purely classical concept, so it can be defined and computed within classical mechanics without invoking quantum mechanics or QFT.

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" or "canonical S-transformation." As an application, a geometric proof of the classical Levinson's theorem was provided.

A modern approach to the concept emerged during the development of a post-Minkowskian effective theory for spinning black holes, where "extracting the classical part of the S-matrix" is a key task.

The purely classical definition and formalization of S-symplectomorphism is established in the framework of symplectic or Poisson geometry. It is also obtained via the phase space formulation of S-matrix.

S-symplectomorphism finds various applications in scattering theory, classical particle mechanics, classical field theory, and gravitational physics. It leads to an "in-in" formalism for classical scattering (as an alternative to the Hamilton-Jacobi formalism), in connection with the Magnusian program.

Definition

Quantum time evolution as a unitary map

In quantum mechanics, the wavefunction evolves under the Schrödinger equation: $${\begin{aligned}{\frac {d}{dt}}\,{\big |}\Psi (t){\big \rangle }\,=\,{\frac {1}{i\hbar }}\,{\hat {H}}(t)\,{\big |}\Psi (t){\big \rangle }\,.\end{aligned}}$$ Here, ${\hat {H}}(t)$ is the time-dependent Hamiltonian operator. The solution to this equation is given in terms of the time-evolution operator ${\hat {U}}(t_{+},t_{-})$, which evolves the wavefunction from initial time $t_{-}$ to final time $t_{+}$ as $${\begin{aligned}{\big |}\Psi (t_{+}){\big \rangle }\,=\,{\hat {U}}(t_{+},t_{-})\,{\big |}\Psi (t_{-}){\big \rangle }\,.\end{aligned}}$$ Explicitly, the time-evolution operator is defined as the time-ordered exponential (Dyson series) formula $${\begin{aligned}{\hat {U}}(t_{+},t_{-})\,=\,{}&{}\mathrm {T} \exp {\bigg (}{\frac {1}{i\hbar }}\int _{t_{-}}^{t_{+}}dt\,\,{\hat {H}}(t){\bigg )}\,.\end{aligned}}$$ Provided ${\hat {H}}(t)$ is Hermitian, ${\hat {U}}(t_{+},t_{-})$ is a unitary operator. Physically, this implies the quantum-mechanical conservation of probability.

Classical time evolution as a symplectic map

In the Hamiltonian formulation of classical mechanics, the classical probability distribution (representing a statistical ensemble, for instance) evolves under the Liouville equation: $${\begin{aligned}{\frac {d}{dt}}\,\rho (t)\,=\,\{H(t),\rho (t)\}\,=\,X_{H(t)}\rho (t)\,.\end{aligned}}$$ Here, $X_{f}=\{f,\,\,\,\}$ denotes the Hamiltonian vector field of a phase space function $f$. By the mathematical equivalence between vector fields and first-order differential operators, one can view $X_{f}$ as a differential operator $g\mapsto \{f,g\}$ acting on phase space functions. With this understanding, the solution to the Liouville equation is given in the form $${\begin{aligned}\rho (t_{+})\,=\,U(t_{+},t_{-})^{*}[\,\rho (t_{-})\,]\,,\end{aligned}}$$ where $U(t_{+},t_{-})^{*}$ is a differential operator. This is the classical version of the time-evolution operator, concretely defined by the time-ordered exponential formula $${\begin{aligned}U(t_{+},t_{-})^{*}\,=\,{}&{}\mathrm {T} \exp {\bigg (}\int _{t_{-}}^{t_{+}}dt\,\,X_{H(t)}{\bigg )}\,.\end{aligned}}$$ Liouville's theorem implies that this differential operator computes the pullback of a symplectomorphism $U(t_{+},t_{-})$, namely a canonical transformation from the phase space at $t=t_{-}$ to the phase space at $t=t_{+}$. This $U(t_{+},t_{-})$ is the time-evolution symplectomorphism. Physically, the symplectic property of classical time evolution encodes the conservation of classical probability.

Note: This construction assumes any classical Hamiltonian system that is defined on a space equipped with a Poisson bracket. Therefore it could be a particle, a system of particle, a field, a system of fields, or a system of particles and fields.

Interaction picture in classical mechanics

To switch to the scattering context, one needs to implement the interaction picture in classical mechanics. The standard setup of scattering problem presumes a time-dependent Hamiltonian that splits into "free" and "interaction" parts: $${\begin{aligned}H(t)\,=\,H^{\circ }+V(t)\,.\end{aligned}}$$ For simplicity, suppose the free Hamiltonian $H^{\circ }$ is time-independent. The interaction Hamiltonian $V(t)$ should satisfy good fall-off conditions at $t\to \pm \infty$.

Let $U(t_{+},t_{-})$ and $U^{\circ }(t_{+},t_{-})$ be the time-evolution symplectomorphisms for $H(t)$ and $H^{\circ }$, respectively. The classical interaction picture is implemented by the pullback from time $t$ to a "reference time slice" at $t_{0}$ of one's choice, which means to simply plug in free-theory trajectory: $${\begin{aligned}\rho _{\mathrm {I} }(t)\,&=\,U^{\circ }(t_{0},t)^{*}[\,\rho (t)\,]\,=\,\rho (t)\circ U^{\circ }(t,t_{0})\,,\\V_{\mathrm {I} }(t)\,&=\,U^{\circ }(t_{0},t)^{*}[\,V(t)\,]\,=\,V(t)\circ U^{\circ }(t,t_{0})\,.\end{aligned}}$$ Here, $\circ$ denotes the composition of maps. The Liouville equation in the classical interaction picture is $${\begin{aligned}{\frac {d}{dt}}\,\rho _{\mathrm {I} }(t)\,=\,\{V_{\mathrm {I} }(t),\rho _{\mathrm {I} }(t)\}\,=\,X_{V_{\mathrm {I} }(t)}\rho _{\mathrm {I} }(t)\,.\end{aligned}}$$

S-symplectomorphism

Finally, the S-symplectomorphism $S$ is the time-evolution symplectomorphism for $t=-\infty$ to $t=+\infty$ in the classical interaction picture. Concretely, its pullback is computed as a differential operator by the time-ordered exponential formula $${\begin{aligned}S^{*}\,=\,{}&{}\mathrm {T} \exp {\bigg (}\int _{-\infty }^{+\infty }dt\,\,X_{V_{\text{I}}(t)}{\bigg )}\,.\end{aligned}}$$ It follows that the classical probability distribution at far past is mapped to that at far future as $${\begin{aligned}\rho _{\mathrm {I} }(+\infty )\,=\,{}&{}S^{*}[\,\rho _{\mathrm {I} }(-\infty )\,]\,.\end{aligned}}$$

In a more careful approach, one defines the map $S$ by the demanding a well-behaved limit $${\begin{aligned}S\,=\,{}&{}\lim _{t_{\pm }\to \pm \infty }{\Big (}\,U^{\circ }(t_{0},t_{+})\circ U(t_{+},,t_{-})\circ U^{\circ }(t_{-},t_{0})\,{\Big )}\,,\end{aligned}}$$ which connects to the Møller operator construction. An instructive calculation is $${\begin{aligned}&S^{*}[\,\rho _{\mathrm {I} }(-\infty )\,]\\&\,=\,\rho _{\mathrm {I} }(-\infty )\circ S^{-1}\,,\\&\,=\,\lim _{t_{\pm }\to \pm \infty }{\Big (}\,\rho _{\mathrm {I} }(t_{-})\circ U^{\circ }(t_{0},t_{-})\circ U(t_{-},t_{+})\circ U^{\circ }(t_{+},t_{0})\,{\Big )}\,,\\&\,=\,\lim _{t_{\pm }\to \pm \infty }{\Big (}\,\rho (t_{-})\circ U(t_{-},t_{+})\circ U^{\circ }(t_{+},t_{0})\,{\Big )}\,,\\&\,=\,\lim _{t_{\pm }\to \pm \infty }{\Big (}\,\rho (t_{+})\circ U^{\circ }(t_{+},t_{0})\,{\Big )}\,=\,\lim _{t_{\pm }\to \pm \infty }\rho _{\mathrm {I} }(t_{+})\,=\,\rho _{\mathrm {I} }(+\infty )\,.\end{aligned}}$$

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. It encodes the conservation of classical probability.

Exponential representation

The symplectic property of the S-symplectomorphism is manifested by its exponential representation: $$S\,=\,\exp(\{\chi ,\,\,\,\})\,=\,\exp(X_{\chi })\,.$$ Here, $\chi$ is referred to as the (classical) scattering generator or classical eikonal. Note that the exponentiation of a Hamiltonian vector field $X_{\chi }$ is a symplectomorphism by construction.

Scattering generator from Magnus expansion

The Dyson expansion computes $S$ as a giant sum of differential operators of all degrees, which does not manifest the symplectic property Instead, $\chi$ can be computed by the Magnus expansion.

The definition of S-symplectomorphism and scattering generator implies that $$X_{\chi }\,=\,\log \mathrm {T} \exp {\bigg (}\int _{t_{-}}^{t_{+}}dt\,\,X_{V_{\text{I}}(t)}{\bigg )}\,.$$ The Magnus expansion computes the log of the time-ordered exponential. By using the Jacobi identity of the Poisson bracket in the form $[X_{f},X_{g}]=X_{\{f,g\}}$, it follows that $${\begin{aligned}\chi \,=\,{}&{}\int dt_{1}\,\,V_{\text{I}}(t_{1})+{\frac {1}{2}}\int _{t_{1}>t_{2}}d^{2}t\,\,\{V_{\text{I}}(t_{1}),V_{\text{I}}(t_{2})\}\\{}&{}+{\frac {1}{6}}\int _{t_{1}>t_{2}>t_{3}}d^{3}t\,\,{\Big (}\,{\{V_{\text{I}}(t_{1}),\{V_{\text{I}}(t_{2}),V_{\text{I}}(t_{3})\}\}+\{V_{\text{I}}(t_{3}),\{V_{\text{I}}(t_{2}),V_{\text{I}}(t_{1})\}\}}\,{\Big )}+\cdots \,.\end{aligned}}$$ This is the explicit formula for the scattering generator (classical eikonal) $\chi$ from Magnus expansion, where the integrands are nested Poisson brackets between the interaction-picture potential $V_{\mathrm {I} }(t)$ at different times.

Impulse of Classical Observables

Suppose a phase space function ${\mathcal {O}}$ representing a classical observable. The inverse of S-symplectomorphism, $S^{-1}$, is the map from the final phase space to the initial phase space. Thus, the pullback $(S^{-1})^{*}$ maps functions living in the initial phase space to functions living in the final phase space. Hence $(S^{-1})^{*}[\,{\mathcal {O}}\,]$ is the image of the initial-time observable ${\mathcal {O}}$ at the final time. As a result, the impulse is computed as $${\mathit {\Delta }}{\mathcal {O}}=(S^{-1})^{*}[\,{\mathcal {O}}\,]-{\mathcal {O}}\,.$$

Nested bracket formula

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

Note: the nested bracket formula should be distinguished from the impulse formulae that one encounters in the Hamilton-Jacobi formalism, which employ a single Poisson bracket. The scattering generator is an object defined on the in phase space, whereas the Hamilton-Jacobi principal functions are inherently functions of both in and out variables due to their identity as on-shell actions.

Relation to KMO'C formalism

The S-symplectomorphism or scattering generator approach to the impulse could be compared to the KMO'C approach. In the KMO'C formalism, the impulse of a quantum observable ${\hat {\mathcal {O}}}$ is obtained as $${\begin{aligned}{\mathit {\Delta }}{\hat {\mathcal {O}}}&\,=\,{\hat {S}}{}^{-1}{\hat {\mathcal {O}}}{\hat {S}}-{\hat {\mathcal {O}}}\,,\\&\,=\,{\frac {1}{i\hbar }}\,[{\hat {T}},{\hat {\mathcal {O}}}]+{\frac {1}{(i\hbar )^{2}}}\,{\hat {T}}^{\dagger }[{\hat {T}},{\hat {\mathcal {O}}}]\,,\end{aligned}}$$ where the second line arises by the usual split ${\hat {S}}={\hat {1}}+i{\hat {T}}/\hbar$ of the S-matrix. The first term, $[{\hat {T}},{\hat {\mathcal {O}}}]/i\hbar$, seems to admit a smooth classical limit to "$\{T,{\mathcal {O}}\}$". However, the dequantization of the T-matrix ${\hat {T}}$ diverges in the classical limit. At the same time, the second term ${\hat {T}}^{\dagger }[{\hat {T}},{\hat {\mathcal {O}}}]/(i\hbar )^{2}$ also diverges in the classical limit because it involves a raw operator product that is not packaged in terms of a commutator. An important check is to verify that these "superclassical terms" cancel.

To have a "standalone" classical framework, the adjoint action structure can be highlighted: $${\begin{aligned}{\mathit {\Delta }}{\hat {\mathcal {O}}}&\,=\,{\hat {S}}^{-1}{\hat {\mathcal {O}}}{\hat {S}}-{\hat {\mathcal {O}}}\,=\,(\mathrm {Ad} _{{\hat {S}}{}^{-1}}{\hat {\mathcal {O}}})-{\hat {\mathcal {O}}}\,.\end{aligned}}$$ As explained below, this translates to the phase space statement $${\begin{aligned}{\mathit {\Delta }}{\mathcal {O}}&\,=\,(S^{-1})^{*}[\,{\mathcal {O}}\,]-{\mathcal {O}}\,,\end{aligned}}$$ via employing the phase-space formulation and taking the limit $\hbar \to 0$. This limiting procedure is smooth and reproduces the nested bracket formula.

To see this directly, one employs the exponential representation of the S-matrix, $${\hat {S}}\,=\,e^{{\hat {\chi }}/i\hbar }\,.$$ Then the quantum impulse is $${\begin{aligned}{\mathit {\Delta }}{\hat {\mathcal {O}}}\,=\,(e^{-\mathrm {ad} _{{\hat {\chi }}/i\hbar }}{\hat {\mathcal {O}}})-{\hat {\mathcal {O}}}\,.\end{aligned}}$$ The dequantization of this formula is $${\begin{aligned}{\mathit {\Delta }}{\mathcal {O}}\,=\,(e^{-X_{\chi }}{\mathcal {O}})-{\mathcal {O}}\,=\,(e^{\{\,\,\,,\chi \}}{\mathcal {O}})-{\mathcal {O}}\,,\end{aligned}}$$ reproducing the nested bracket formula.

Precise Relation to S-matrix

In a precise sense, the S-symplectomorphism describes the classical limit of the adjoint action of the S-matrix in the phase space formulation, 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 $\hbar$-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 ${\mathcal {Q}}$ 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 $$\mathrm {Ad} _{\hat {S}}\circ {\mathcal {Q}}={\mathcal {Q}}\circ S^{\star }\,.$$ In the classical limit, the fuzzy S-diffeomorphism smoothly approaches to the S-symplectomorphism: $$\lim _{\hbar \to 0}S^{\star }=S\,.$$ Therefore, one establishes the precise equation $$S\,=\,\lim _{\hbar \to 0}{\Big (}\,{\mathcal {Q}}^{-1}\circ \mathrm {Ad} _{\hat {S}}\circ {\mathcal {Q}}\,{\Big )}\,,$$ where ${\mathcal {Q}}^{-1}$ is the dequantization map.

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

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.

Note that one may regard the fuzzy S-diffeomorphism $S^{\star }$ as the deformation quantization of the S-symplectomorphism $S$.

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 $\{J_{i},J_{j}\}=J_{k}\,\varepsilon ^{k}{}_{ij}$. 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 (Casimir of the Poisson algebra): $\{C,f\}=0$ for all $f$. For example, a modern formulation of relativistic massive spin is based on a 12-dimensional space with rank 10 Poisson bracket, with one of the Casimirs being the spin magnitude.

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 (Poisson 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
• Deformation quantization
• Scattering generator
• Magnusian