Dynamics Reference

The Dynamics Reference chapter is the lookup layer for quantum time evolution and its equivalent formulations. Use it to recover a definition, formula, picture transformation, kernel, Green-function convention, phase-space normalization, diagnostic warning, or reading route without searching through an entire derivation.

These pages are compact by design. They state conventions and assumptions, but they do not replace the canonical concept and derivation pages linked from each entry. When a calculation depends on domains, boundary conditions, approximation status, or limiting procedures, follow the canonical link before using the compact formula.

What This Chapter Owns

This chapter is the canonical home for

• concise definitions of recurring dynamics terminology;
• a compact formula sheet spanning the major formulations;
• exact translations among Schrödinger, Heisenberg, interaction, and general unitary pictures;
• quick-reference propagator and Green-function tables;
• convention-fixed real-time and Euclidean path-integral formulas;
• convention-fixed Wigner, Weyl, star-product, and Moyal formulas;
• a cross-formulation catalog of common mistakes;
• a curated reading guide organized by use case and level.

The chapter does not own the derivation of the time-evolution operator, picture transformations, kernels, resolvents, path integrals, phase-space dynamics, or Floquet theory. Those remain in their subject chapters. It also does not own global symbol policy or formula cards across all volumes; those belong to the Reference.

Choose by Task

Need	Reference page	What it returns
Define a term quickly	Dynamics Glossary	definition, warning, canonical link
Recover a core equation	Formula Sheet	convention-fixed formula and route to derivation
Move states or operators between pictures	Picture Translation Table	dagger, sign, endpoint, and density-operator rules
Look up a standard time-domain kernel	Propagator Table	formula, domain, branch, and boundary reminder
Compare resolvent and response conventions	Green Function Table	$i0$, spectral, density-of-states, and free-kernel identities
Fix a path-integral normalization or source sign	Path Integral Conventions	real-time, Euclidean, measure, source, and Fourier choices
Fix a Wigner or star-product normalization	Phase-Space Conventions	transform, measure, marginal, and bracket conventions
Diagnose a suspicious result	Common Pitfalls	symptom, reason, and correction route
Choose a source or study sequence	Further Reading	annotated source categories and reading routes

Together these nine pages form the planned reference chapter.

Lookup Workflow

Use the narrowest page that answers the immediate question:

1. Name the object. Decide whether the problem concerns a state, observable, evolution operator, kernel, resolvent, Green function, path integral, phase-space symbol, or Floquet map.
2. Open the specific aid. Use the glossary for meaning, the formula sheet for structure, and a convention or table page for normalization and signs.
3. Read the assumptions beside the formula. Check time dependence, domain, boundary condition, basis, measure, and approximation status.
4. Follow the canonical link. Use the owning chapter for a derivation or interpretation that affects the calculation.
5. Run a structural check. Test units, limiting cases, unitarity, composition, normalization, marginals, or spectral identities as appropriate.
6. Record the convention. A result is not reproducible if the Fourier sign, $i0$ prescription, branch, or picture convention remains implicit.

The compact page is the beginning of a reliable lookup, not permission to omit hypotheses.

Convention Precedence

When two formulas look different, compare them in this order:

1. the physical model, Hilbert space, and operator domain;
2. the canonical concept or derivation page for that object;
3. the most specific convention page for the representation being used;
4. the reference table or formula sheet;
5. a glossary entry or broad overview.

This ordering does not rank pages by quality. It ranks them by specificity. A half-line propagator with Dirichlet boundary conditions cannot be corrected by quoting a full-line kernel from a table. A Wigner transform using another Fourier convention may be equivalent after rescaling even when its prefactor differs. A source that defines transformed states with $R$ rather than $R^\dagger$ must use the corresponding Hamiltonian sign.

Core Structural Checks

Evolution operators

Check

$$U(t_2,t_0) = U(t_2,t_1)U(t_1,t_0),$$ $$U(t,t_0)^\dagger U(t,t_0)=I,$$

and

$$U(t_0,t_0)=I.$$

For time-dependent Hamiltonians, determine whether time ordering is required.

Propagator kernels

Check the distributional initial condition,

$$K(x,t_0;x',t_0) = \delta(x-x'),$$

the Schrödinger equation at the final endpoint, composition over the correct measure, and the intended boundary conditions.

Green functions

Write the inverse equation and the prescription. Near the spectrum, retain the sign in

$$(E-H\pm i0)^{-1}.$$

Check whether the object is retarded, advanced, incoming, outgoing, time ordered, or Euclidean before comparing it with another source.

Path integrals

State the regulator, endpoint conditions, measure, real- or imaginary-time weight, source sign, and normalization. A formal $\mathcal Dq$ does not supply these choices by itself.

Phase space

Test normalization and both marginals of a Wigner function. Record the Fourier convention, phase-space measure, and star-product sign before using Moyal dynamics.

Reference Page Map

Dynamics Glossary

Use the glossary when a word carries several neighboring meanings: propagator versus Green function, stationary state versus stationary vector, interaction picture versus approximation, or quasienergy versus energy. Each entry gives one canonical equation, a misuse warning, and the owning page.

Formula Sheet

Use the formula sheet to recall the structure of Schrödinger evolution, picture changes, Heisenberg equations, density-operator dynamics, product formulas, Dyson expansion, kernels, Green functions, path integrals, Wigner–Moyal dynamics, Floquet theory, and classical-limit bridges. Follow its links when a sign or domain matters.

Picture Translation Table

Use the translation table when moving a concrete calculation among pictures. It fixes

$$\lvert\psi_P\rangle = R^\dagger\lvert\psi_S\rangle$$

and derives every state, observable, density-operator, Hamiltonian, and endpoint-propagator rule from that convention. It also distinguishes a zero Heisenberg state generator from the transformed Hamiltonian observable.

Propagator Table

Use the propagator table for standard nonrelativistic kernels: free particles, constant force, infinite square well, harmonic oscillator, ring, and the semiclassical Van Vleck form. Confirm the geometry and boundary condition before inserting a formula into an integral.

Green Function Table

Use the Green-function table for resolvents, retarded and advanced boundary values, spectral functions, density of states, coordinate inverse kernels, and free-particle examples. The $i0$ sign and Fourier convention are part of every entry.

Path Integral Conventions

Use this page to translate among real-time, Euclidean, configuration-space, phase-space, source-dependent, and thermal path-integral expressions. It centralizes signs and normalizations so derivation pages do not silently drift apart.

Phase-Space Conventions

Use this page for Weyl symbols, Wigner transforms, marginals, many-degree-of-freedom measures, star products, Poisson brackets, Moyal brackets, and dynamics. It is the local authority when another source uses a different $2\pi\hbar$ placement.

Common Pitfalls

Use the pitfalls page when a calculation preserves norm but disagrees with a benchmark, mixes pictures, loses a boundary condition, treats a complex path-integral weight as a probability, confuses a kernel with a state, overreads Ehrenfest’s theorem, or forgets quasienergy modularity.

Further Reading

Use the reading guide to choose among undergraduate texts, graduate texts, mathematical treatments, path-integral references, Green-function sources, phase-space reviews, control literature, and QFT bridges. Its routes are organized by objective rather than prestige.

Canonical-Home Boundaries

• Foundations of Time Evolution owns $U(t,t_0)$, unitarity, time ordering, and Dyson evolution.
• Pictures of Quantum Mechanics owns picture definitions and transformation derivations.
• Operator Dynamics owns commutator evolution, density-operator equations, product formulas, and quantum maps.
• Propagators and Kernels owns transition amplitudes and time-domain kernels.
• Green Functions and Resolvents owns inverse operators, prescriptions, spectra, and density of states.
• Path Integral Formulation owns regulated sums over histories and their derivations.
• Phase-Space Formulation owns Weyl–Wigner–Moyal concepts and dynamics.
• Periodic and Driven Dynamics owns Floquet theory, rotating frames, quasienergies, and micromotion.
• Notebooks and Worked Examples owns reproducible numerical validations.

Reference aids cross-link these homes; they do not create parallel derivation homes.

Before Reusing a Formula

Confirm that the entry records or links to

• the object being calculated and its units;
• the Hilbert space, basis, and measure;
• the Hamiltonian’s time dependence and domain;
• boundary and initial conditions;
• Fourier, phase, and branch conventions;
• the $i0$ or contour prescription when relevant;
• exact, asymptotic, perturbative, or formal status;
• the range of parameters in which an approximation is controlled;
• at least one normalization, limiting-case, or structural check;
• a canonical derivation and an authoritative source.

If any item changes, the formula may require translation rather than direct substitution.

Cross-Links

• Quantum Dynamics
• Map of Quantum Dynamics
• Which Formulation Should I Use?
• Translation Table of Formulations
• Reference
• Formula Compendium
• Derivation Index
• Software, Notebooks, and Benchmarks

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• L. S. Schulman, Techniques and Applications of Path Integration, Dover, 2005.
• E. N. Economou, Green’s Functions in Quantum Physics, 3rd ed., Springer, 2006.
• M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution Functions in Physics: Fundamentals,” Physics Reports 106, 121–167 (1984), doi:10.1016/0370-1573(84)90160-1.