Action, Canonical Quantization, the Hamilton–Jacobi Equation, and Path Integrals: The Indistinguishable Bond Between Quantum and Classical Mechanics

I. A Neglected Fact

Textbooks typically describe classical mechanics and quantum mechanics as two entirely distinct worlds: classical mechanics is deterministic, continuous, and intuitive; quantum mechanics is probabilistic, discrete, and counter-intuitive. However, upon carefully examining the entire mathematical structure of quantum mechanics, one encounters an unsettling fact—every bone in the body of quantum mechanics is classical. Its "quantumness" manifests only in the replacement of a few specific joints, the physical meaning of which has never been truly understood, merely habituated.

II. Canonical Quantization: An Operator Garb on a Classical Skeleton

The Essence of the Procedure

The complete steps of Canonical Quantization (Dirac, 1925–1930) are as follows:

1. Write down the classical Lagrangian L(q,q˙)L(q,q˙) . This is purely classical mechanics.

2. Obtain the Hamiltonian H(q,p)H(q,p) via Legendre transformation. This remains purely classical mechanics. Define canonical momentum p=∂L/∂q˙p=∂L/∂q˙ and construct H=pq˙−LH=pq˙−L .

3. "Promote" qq and pp to operators and impose commutation relations:

[q^,p^]=iℏ[q^,p^]=iℏ

1. Replace q,pq,p in the classical Hamiltonian with q^,p^q^,p^ to obtain the quantum Hamiltonian operator H^H^ .

2. Write down the Schrödinger equation: iℏ∂∂t∣ψ⟩=H^∣ψ⟩iℏ∂t∂∣ψ⟩=H^∣ψ⟩ .

Scrutinizing this procedure: Of these five steps, the first two are purely classical; the fourth is merely substituting variables in a classical function with operators; and the structure of the fifth step is dictated by the commutation relations from the third. The entire physical content of the quantum theory—the form of interactions, the structure of potentials, the mode of coupling—derives entirely from the classical Lagrangian. Quantum mechanics generates no dynamical content independently.

What Did Quantum Mechanics Actually Add?

Strictly speaking, the quantization procedure adds only two things:

1. The Commutation Relation [q^,p^]=iℏ[q^,p^]=iℏ . This introduces the uncertainty principle and the possibility of discrete spectra. However, ℏℏ itself is an empirical constant, and the physical origin of the commutation relation is unexplained—it is a postulate, not a derivation.

2. The Hilbert Space Structure of the State Space. Classical phase space points (q,p)(q,p) are replaced by state vectors ∣ψ⟩∣ψ⟩ in a Hilbert space, and observables become Hermitian operators. Yet, the structure of the Hilbert space (inner product, superposition principle, probability interpretation) is likewise a postulate, not derived from deeper physical principles.

In other words, quantum mechanics adds only two postulates and one constant ( ℏℏ ) atop the complete dynamical framework of classical mechanics. Everything else—from the hydrogen atom to the Standard Model—is merely the mathematical expansion of the classical Lagrangian after operatorization.

If we consider spectral analysis and "Natural Quantization," or the dual ontology of particles possessing both a nuclear core and an extended field (as proposed by NQT), we may find that even these three additions are entirely traditional or classical in nature. This will be detailed in subsequent articles.

III. The Hamilton–Jacobi Equation: The "Wave Equation" Within Classical Mechanics

The Classical Hamilton–Jacobi Equation

In classical analytical mechanics, the Hamilton–Jacobi (HJ) equation holds a special status. Defining the principal function S(q,t)S(q,t) (Hamilton's principal function), we have:

∂S∂t+H(q,∂S∂q)=0∂t∂S+H(q,∂q∂S)=0

where p=∂S/∂qp=∂S/∂q . This is a partial differential equation for the scalar function SS . Note that its form is already that of a field equation, describing the propagation of the function SS in configuration space.

In classical mechanics, the level surfaces S(q,t)=constS(q,t)=const constitute "wavefronts." Classical particle trajectories are orthogonal to these wavefronts, just as light rays are orthogonal to equiphase surfaces in geometric optics. This is not an analogy—it is a mathematical isomorphism.

From Hamilton–Jacobi to Schrödinger

Schrödinger’s discovery (1926) of his equation proceeded directly from the Hamilton–Jacobi equation. By positing the wavefunction as:

ψ=AeiS/ℏψ=AeiS/ℏ

and substituting it into the Schrödinger equation iℏ∂ψ∂t=−ℏ22m∇2ψ+Vψiℏ∂t∂ψ=−2mℏ2∇2ψ+Vψ , one finds that in the limit ℏ→0ℏ→0 , SS satisfies the Hamilton–Jacobi equation exactly.

This implies that the Schrödinger equation is simply the wave-generalization of the Hamilton–Jacobi equation—endowing the classical equiphase "wavefronts" with real amplitude and the capacity for interference. Conversely, classical Hamilton–Jacobi theory already contained the germ of a wave structure; quantum mechanics merely unfolded this germ.

The Insight from WKB Approximation

The WKB (Wentzel–Kramers–Brillouin) approximation is the technical realization of this relationship. Expanding SS as a power series in ℏℏ :

S=S0+ℏS1+ℏ2S2+…S=S0+ℏS1+ℏ2S2+…

The zeroth-order term S0S0 satisfies the classical Hamilton–Jacobi equation, while higher-order terms provide quantum corrections. This is not merely an approximation trick; it reveals the analytic structure of quantum mechanics: quantum theory is a formal power series expansion of classical theory in ℏℏ , with classical theory being the leading term. They are not two different theories, but different orders of approximation of the same mathematical structure.

IV. Path Integrals: The Quantum Summation of Classical Action

The Structure of Feynman Path Integrals

Feynman (1948) reformulated quantum mechanics as a path integral:

⟨qf,tf∣qi,ti⟩=∫D[q(t)] eiS[q(t)]/ℏ⟨qf,tf∣qi,ti⟩=∫D[q(t)]eiS[q(t)]/ℏ

where S[q(t)]=∫titfL(q,q˙) dtS[q(t)]=∫titfL(q,q˙)dt is the classical action.

Examining every component of this formula:

• The Phase S[q(t)]/ℏS[q(t)]/ℏ in the Integrand: This is the classical action, divided by ℏℏ . The weight of every path is determined entirely by the classical Lagrangian.

• The Summation ∫D[q(t)]∫D[q(t)] : The sum over all possible paths—this is the only thing quantum mechanics adds. Classical mechanics selects only the path that extremizes the action ( δS=0δS=0 , i.e., the Euler–Lagrange equations); quantum mechanics sums over all paths, weighted by the classical action.

• The Classical Limit: When S≫ℏS≫ℏ , the phases of non-extremal paths oscillate rapidly and cancel out (stationary phase approximation). Only paths near δS=0δS=0 contribute—precisely recovering the Principle of Least Action of classical mechanics.

The Deep Relationship Revealed by Path Integrals

The path integral is not just an equivalent formulation of quantum mechanics; it exposes the true essence of the relationship between quantum and classical mechanics:

1. Quantum Mechanics Has No Independent Dynamics. The only dynamical quantity appearing in the path integral is the classical action SS . No independent "quantum potential" or "quantum force" is introduced. Quantum effects arise entirely from the democratic summation over classical dynamics—letting all paths participate rather than selecting only the extremal one.

2. The Role of ℏℏ is to Control the Degree of "Democracy." ℏ→0ℏ→0 means only the extremal path matters—returning to classical "dictatorship." A finite ℏℏ means paths deviating from the extremum also have a voice—quantum "democracy." Yet, the content of what each path says (i.e., its phase) is still determined by the classical action.

3. Quantum Mechanics is a "Weighted Sum" of Classical Mechanics. If classical mechanics is a deterministic theory that selects the optimal path, then quantum mechanics is a theory that coherently superposes all paths according to classical weights. The difference lies only in "selection" vs. "superposition," not in the dynamics themselves.

V. The Situation in Field Theory: Even More Thoroughly Classical

In Quantum Field Theory (QFT), this deep entanglement between classical and quantum is even more pronounced.

The QFT path integral is:

Z=∫D[ϕ] eiS[ϕ]/ℏZ=∫D[ϕ]eiS[ϕ]/ℏ

where S[ϕ]=∫d4x L(ϕ,∂μϕ)S[ϕ]=∫d4xL(ϕ,∂μϕ) is the action of the classical field theory. All quantum predictions of the Standard Model—from the anomalous magnetic moment in QED to asymptotic freedom in QCD—are derived from the classical Lagrangian LSMLSM via path integrals and perturbative expansions.

• Feynman Rules are Determined Entirely by the Classical Lagrangian. Propagators come from quadratic terms, vertices from interaction terms, and coupling constants are simply the parameters in the classical Lagrangian. The entire perturbative structure of QFT—Feynman diagrams, loop integrals, renormalization—is a mathematical expansion of the classical Lagrangian within the path integral framework.

• Classical Field Equations are the Saddle-Point Approximation of QFT. Setting δS/δϕ=0δS/δϕ=0 returns the classical field equations (e.g., Maxwell’s equations, Dirac equation, Yang-Mills equations). Quantum corrections are fluctuations around this saddle point, represented by loop diagrams.

VI. What Does This All Mean?

Quantum Mechanics is Not an Independent Theory

From the above analysis, a stark conclusion emerges: Quantum mechanics is not a self-standing theoretical system, but a specific mathematical generalization of classical mechanics. All its dynamical content comes from classical mechanics; it contributes only a mathematical framework (Hilbert space, commutation relations) and a constant ( ℏℏ ).

This stands in sharp contrast to the impression given by textbooks—that quantum mechanics is a brand-new theory fundamentally different from classical mechanics. Differences certainly exist: discrete spectra, tunneling, interference, entanglement. But these "quantum phenomena" can all be traced back to the path summation and coherent superposition of the classical action. They are natural consequences of classical dynamics under the condition ℏ≠0ℏ=0 , not results of some independent "quantum principle."

The "Quantum Revolution" May Have Been Exaggerated

If the mathematical structure of quantum mechanics is so deeply rooted in classical mechanics, then describing it as a "revolution that overturned the classical worldview" warrants re-examination. The amount of truly new physical content may be far less than commonly believed. The core novelties might be only:

1. Action has a minimum unit ℏℏ (quantization condition).

2. Systems do not select a single path but coherently "experience" all paths.

And even these two points may have deeper classical origins. If particles are not points but possess finite spatial scale field structures (as NQT proposes), then the intrinsic vibrational modes of the field naturally introduce discreteness, and the spatial extension of the field naturally allows for "propagation along multiple paths simultaneously." In this scenario, ℏℏ is no longer an inexplicable fundamental constant but perhaps a characterization of the particle's finite size and field structure.

Clues to Deeper Understanding

The fact that classical and quantum mechanics are so indistinguishable should not be viewed as a coincidence or mere formal convenience, but as a clue pointing to deeper physics. It suggests that quantum mechanics may not be the ultimate language of nature, but rather a manifestation of some more fundamental, unified dynamical theory under specific conditions—just as geometric optics is the short-wavelength limit of wave optics, which in turn is an approximate layer of electromagnetic field theory.

VII. Conclusion

The Principle of Least Action runs through the entire road from classical mechanics to quantum field theory. The Hamilton–Jacobi equation is a wave structure already present within classical mechanics; the Schrödinger equation is its expansion; and the Path Integral places the classical action at the absolute core of quantum theory. Quantum mechanics has never detached itself from the dynamical content of classical mechanics; it has merely changed how we use this content—from "selecting the extremum" to "superposing all."

This fact itself is the most powerful dissolution of quantum "weirdness." If the entire flesh and bone of quantum theory are classical, then describing it as "incomprehensible" or "fundamentally broken from the classical world" is not profound—it is a misreading of its own theoretical structure. The relationship between quantum and classical mechanics is not one of revolution and overthrow, but of the trunk and branches of the same great tree. Understanding this is a crucial step toward a truly unified physical theory.