From Classical to Quantum

In classical mechanics, “action” $S$ picks out a single path by extremizing
$$ S[q] = \int_{t_i}^{t_f} L\bigl(q,\dot q,t\bigr)\mathrm{d}t. $$
Richard Feynman’s genius was to say: don’t pick just one path, sum over all paths, weighting each by $e^{iS/\hbar}$. In this path integral formulation, the quantum amplitude to go from $(q_i,t_i)$ to $(q_f,t_f)$ is
$$ \langle q_f,t_f | q_i,t_i \rangle = \int \mathcal{D}[q(t)]\exp\Bigl(\tfrac{i}{\hbar}S[q]\Bigr). $$
This approach not only reproduces Schrödinger’s equation but shines in quantum field theory, statistical mechanics, and even quantum gravity.

Key 2019 Literature

Below are a few representative reviews and breakthroughs up to 2019:

  • Coherent-State Path Integrals in Many-Body Physics
    Zhang & Duan (2019) review the use of action-based path integrals for Bose–Einstein condensates and spin systems, emphasizing non-equilibrium dynamics and topology.

  • Resurgence and Trans-Series in Quantum Field Theory
    Dunne & Ünsal (2019) develop the mathematical framework of resurgent trans-series to understand factorial divergences in perturbation theory and their connection to nonperturbative saddle points.

  • Action and Topological Phases
    Kitaev & Laumann (2019) explore how topological field theories arise from path integrals with modified action terms, laying groundwork for fault-tolerant quantum computation.

Action in Modern Field Theory

The variational principle extends naturally to fields $\phi_a(x^\mu)$. One defines the action functional
$$ S[\phi] = \int \mathcal{L}\bigl(\phi_a,\partial_\mu\phi_a\bigr)\mathrm{d}^4x, $$
where $\mathcal{L}$ is the Lagrangian density. Extremizing $S$ yields the Euler–Lagrange equations for fields, e.g. the Klein–Gordon or Yang–Mills equations.

  • Gauge Theories (QED, QCD):
    $\mathcal{L} = -\tfrac14 F_{\mu\nu}F^{\mu\nu} + \bar\psi(i\gamma^\mu D_\mu - m)\psi$.
  • General Relativity:
    The Einstein–Hilbert action
    $\displaystyle S = \frac{1}{16\pi G}\int R\sqrt{-g}\mathrm{d}^4x$ reproduces Einstein’s field equations.

Contemporary research often studies modifications of these actions, higher-derivative terms, topological $\theta$-terms, and couplings to new fields, to probe dark matter, inflation, and beyond-Standard-Model physics.

Open Questions & Frontiers

  1. Nonperturbative Phenomena & Resurgence
    How do complex saddles (“Lefschetz thimbles”) control real-time dynamics? Can resurgence unify perturbative and nonperturbative sectors in QFT?

  2. Quantum Gravity Path Integrals
    Does a well-defined gravitational path integral exist? How do we sum geometries in a background-independent way?

  3. Discrete Variational Integrators
    Can numerical simulations preserve symplectic structure and gauge invariance exactly, using action-based discretization?

  4. Higher-Form and Generalized Symmetries
    How do extended objects (strings, branes) and their conserved charges emerge from variational principles?

In our next blog series, we’ll explore one of these frontiers in detail, stay tuned!

References

  • Feynman, R.P., & Hibbs, A.R. (1965). Quantum Mechanics and Path Integrals.
  • Zhang, L., & Duan, H. (2019). “Coherent-State Path Integrals in Many-Body Physics.” Rev. Mod. Phys.
  • Dunne, G.V., & Ünsal, M. (2019). “Resurgence and Trans-Series in Quantum Field Theory.” J. Math. Phys.
  • Kitaev, A., & Laumann, C. (2019). “Topological Phases and Quantum Computation.” Annu. Rev. Condens. Matter Phys.
  • Peskin, M.E., & Schroeder, D.V. (1995). An Introduction to Quantum Field Theory.
  • Weinberg, S. (1996). The Quantum Theory of Fields, Vol. I: Foundations.