Reverse-Engineering Quantum Mechanics, VIII.

The Transition |M⟩ → |Q⟩

The transition from (classical) Mechanics to Quantics.

Update: Before you read this—or worse, feed it into an AI to see what this is about—read the Post-amble to this post (which should really be the Preamble to the Preamble).

Preamble

OK, this is the post where I report on the quite unexpected (at least, to me) thing that occurred when I investigated what happens if you relax the assumption that the variations at the endpoint of the Lagrangian action functional vanish. I did this in a couple of different ways, but the one that is compelling from a physics perspective is when that non-vanishing is due to a physical postulate, informed by experiment, as opposed to merely a mathematical exploration done just for the fun of it (which, truth be told, was indeed fun, and helped the further exploration of the physical postulate).¹

So, what is the unexpected thing? Well, it seems to be the case that the mathematics of quantum mechanics—what Lévy-Leblond calls ‘quantics’, see Part V—emerges from Lagrangian Mechanics if you relax the assumption of vanishing variations: specifically, if those variations have a finite extension—such as is implied by the experimental evidence of quantum physics—as well as by paying attention to the mathematics required by needing to conform to what empirical experiments tell us. In short, instead of assuming (classical) idealised point-particles, which imply infinitely-sharp endpoints on the action integral, we instead use the empirically-guided fundamental postulate that the quantum entities—‘quantons’, in Lévy-Leblond’s terminology (Part V)—have continuous extension (what he called pantopy). The extent of their ‘pantopic extension’ is found from experiments which show that non-relativistic quantum “particles” have an associated “wavelength”—the de Broglie wavelength \(\lambda_{dB}\)—which is related to Planck’s constant, \(h\). Indeed, Planck’s constant is the only ‘magic number’ that needs to be input into the purely classical framework used here, and it comes directly from experiment, just like \(c\) in relativity, or \(G\) in Newtonian (and Einsteinian) gravity.

That is, one two-fold physical postulate: a qualitative assumption of extension together with a quantitative value, Planck’s constant, implied by that assumption, both of which are provided by empirical observation. When these are applied to classical Lagrangian Mechanics, it reproduces the mathematics of Quantics, where certain necessary mathematical choices that need to be made are also guided by empirical results. The assumption is what I will call the Extended Boundary Condition (i.e., that the boundaries are not infinitely sharp but are extended), where the extension, by way of the de Broglie wavelength, is governed by the value of Planck’s constant, \(h\) (the quantitative value the universe provides).

As I noted in the previous post, we need to take seriously the fact that action is one of the fundamental quantities of the universe. In this view, the global action structure that Quantics needs to utilise is already built into the machinery of Mechanics (Lagrangian, Hamiltonian and Hamilton-Jacobi) all of which, however, treat it as merely a mathematical convenience rather than as the fundamental physical reality. In this case, therefore, the transition \(\vert M \rangle\) → \(\vert Q \rangle\) from Mechanics to Quantics is not so much the introduction of something weird and new and strange, but simply the recognition and promotion of the action from a mathematical variational device to the fundamental physical quantity it actually is, and that the extension which quantons display experimentally needs to be taken seriously within the mathematical formalism. With this quite simple starting point, the mathematics of Quantics emerges from Mechanics surprisingly straight-forwardly, provided one ensures the mathematics is properly guided by the actual empirical physics which prompts the postulated change in the mathematics.

But first a note. This post is written with a view to rigorously deriving Quantics from Lagrangian Mechanics, showing the full derivation of the Feynman short-time propagator kernel and then the full propagator itself and how it leads to the path integral, and to do so in a relatively self-contained way. Therefore, it has a ‘fair bit’ of mathematics going on (I warned you this was coming last time in Part VII), so if that is not your bag, then just skip over it and read the associated commentary. The mathematics is there to demonstrate that the commentary is well founded. The key finding, though, is that it is possible to derive the Feynman path integral approach to Quantics from four key assumptions, three physically-motivated and one mathematically, by way of two distinct approaches, one empirical (in the main text), the other variational (in the Appendix). The fact that it was possible to do this means—by the property of transitivity—that everything that flows from Feynman’s path integral formulation of Quantics also follows from this derivation, including the Schrödinger equation, the derivation of which from the path integral formalism is very well-known (e.g., Feynman and Hibbs 1965; Shankar 1994; Derbes 1996). Feynman’s formulation of Quantics is equivalent to the other two (Heisenberg, Schrödinger), so being able to derive it from empirically-guided first principles in the classical Lagrangian formalism is a somewhat unexpected (but intriguing) result.

And it all starts with a different boundary condition on the action functional in Mechanics.

The Extended Boundary Condition (EBC)

Recall from Part VII, that in standard Lagrangian Mechanics, the Principle of Least (i.e., Stationary) Action relies on the specific boundary condition \(\delta q(t_f) = \delta q(t_i) = 0\); that is, that variations \(\delta q\) are assumed to vanish at the endpoints \(t_i\) and \(t_f\). This is a consequence of the so-named Dirichlet Boundary Condition (fixed value); if the value is fixed, then variations in that value must \(=0\). This assumption is compatible with the fact that classical particles are treated as point-like (i.e., zero-dimensional) so that the paths they follow are 1-dimensional trajectories in 3D space. These trajectories are thereby carried over as such during the forward mapping of 3D coordinates into the generalised coordinates \(q_i\) in configuration space. That is, the ‘endpoints’ are assumed to be precise points in space and time, and so therefore also when the 3D coordinates are transformed into those in configuration space.

However, experimental evidence shows us that the interactions of quantons do not happen at idealised zero-dimensional points (Dirac \(\delta\)s), but rather over a finite interaction width, related to the de Broglie wavelength. To take this experimental fact seriously in the mathematics, the boundary conditions for the action functional cannot be based on the classical assumption of point-like particles and thus similarly sharp boundary conditions. A quanton would not arrive at a detector apparatus as a sharp point at a sharp time \(t_f\), but rather as an extended object whose arrival ‘event’ is ‘smeared’ over a duration \(\Delta t\) and a spatial region defined by its own extent. The boundary conditions therefore need to be similarly extended over that width.

A clarification is in order here, however. The ‘extension’ I am referring to is the interaction width or interaction length of the quanton. Scattering experiments show that, for example, the electron’s ‘core’, if it even has one at all, would be in the less than attometre (possibly zeptometre) range (i.e., \(<10^{-18}-10^{-21}\)m), while for a non-relativistic electron its de Broglie wavelength tends to be in the nanometre to picometre range (i.e., \(10^{-9}\) to \(10^{-12}\)m). So, this ‘extension’ is interactional, which is what has been well-established empirically. The question of whether this is a ‘real’ extension is secondary (and unnecessary)—and this analysis remains entirely agnostic with respect to that ontological claim. Our interest is in the empirical fact that it obviously interacts as if it has extension. And that is what we wish to properly model mathematically. It will require modelling the quantum entity with a non-zero-width distribution rather than treating it as an idealised Dirac-delta point-particle.

Let \(\sigma\) denote the characteristic spatial extension of the interaction length/width of a quanton. In the non-relativistic regime, experiment (e.g., the electron diffraction experiment of Davisson and Germer) shows this extension is governed by the de Broglie wavelength: \(\sigma \sim \lambda_{\text{dB}} = {h}/{p}\), where \(p\) is the quanton’s momentum.

This is what provides the physics-based rationale to apply the mathematical Extended Boundary Condition (EBC): the variations at the boundaries of the action functional are not zero, but are instead conditioned by the finite (interaction) width of the quanton itself.

Under the EBC, the variation \(\delta q\) is no longer zero displacement of an idealised zero-dimensional point. Rather, it is now a fuzzily smeared out variation over the quanton’s characteristic width \(\sigma\). The action functional \(S[q]\) no longer deals solely with a single infinitely-thin 1-dimensional classical trajectory, but with an, as it were, ‘fuzzy cylinder’, centred on and surrounding the classical trajectory which becomes its central ‘spine’, and which represents the quanton’s interaction extent as it traces a path. In the terminology introduced in Part V, this fuzzy cylinder was termed the “perihodon” (the “around path”).

Mathematically, it also means that we can’t ignore the higher-order terms in the Taylor series expansion of the action functional beyond the conventional first variation \(\delta S\). For a point particle, these higher-order variations are negligible. But for a quanton with extension, the second, \(\delta^2S\), third, \(\delta^3S\), and so on to even higher, variations cannot be ignored a priori. It is the finite interaction width that ‘feels’ the higher-order variations that a point-particle would not.

This simple relaxation of a mathematical assumption about the nature of the boundary condition, from sharp points to extended distributions, is the step that unlocks the transition from Mechanics \(\vert M \rangle\) to Quantics \(\vert Q\rangle\), because it forces the variational calculus to account for the quanton’s effective interaction width. For example, we will note later below that the higher order term at \(\delta^2S\) seems to account for the “quantum potential” \(V_Q\) found in the Schrödinger equation (Part III), by way of the Madelung/Bohm decomposition of \(\psi\), which forms an important part of the de Broglie-Bohm theory. Before then, however, the Heisenberg Inequalities arise even more directly than that.

The Heisenberg Inequalities

In Part V, in his discussions of reforming the nomenclature of quantum mechanics, Lévy-Leblond came down hard on the notion that the inherent extension of the quanton giving rise to non-sharp values for its physical attributes constituted “uncertainty” in those values, rather than simply a ‘spread’ or ‘dispersion’ or ‘spectrum width’. The Heisenberg “Uncertainty” relations he therefore recast as the Heisenberg Inequalities. Here, we show how these arise naturally from the quanton’s extension. There are a couple of ways of doing this, one of which—mostly heuristic—is based on a geometric argument arising from the form the boundary term takes. A more rigorous derivation follows this.

The initial clue that relaxing the Dirichlet boundary condition might be worth trying out as a purely mathematical postulate came from noticing the boundary term in the classical case. Recall (Part VII) that for a classical path, the boundary term has the form \[
\left[ \frac{\partial L}{\partial\dot{q}_i}\delta q \right]_{t_i}^{t_f} .
\] Also recall that \(\partial L/\partial\dot{q}_i\) is the definition of the conjugate momentum, \(p_i\). The variation in \(S\) from the boundary term therefore takes the form: \[
\delta S_B = p(t_f)\delta q(t_f) – p(t_i)\delta q(t_i) .
\] In symplectic geometry, this represents the canonical one-form (or Liouville one-form, or symplectic potential), and the above expression represents the change in symplectic potential between the endpoints. But, since the momentum \(p_i\) is determined from the partial derivative with respect to the time derivative of the position \(q_i\), if the position \(q_i\) is not sharply-defined but has a spatial extension of \(\sigma_q\), then the conjugate momentum \(p_i\) must also end up not being sharply-defined and having its own associated spread in value, \(\sigma_p\), related to and dependent upon \(\sigma_q\).

That is, due to the EBC, the endpoints are not sharp but are regions in phase space of finite extent. The ‘area’ of this region in the \((q, p)\) phase ‘plane’ is given by the product of the spreads in \(q\) and \(p\): \[
\text{Area} \sim \sigma_q\,\sigma_p = \Delta q\, \Delta p \;.
\] Dimensionally, this area \(\Delta S\) in phase space corresponds to the action, since momentum \(\times\) position has dimensions of action. But action is known from experiment (black-body radiation, photo-electric effect) to have a ‘graininess’: namely, Planck’s constant, \(h\). So, the minimum action difference \(\Delta S\) between two distinct states in phase space (that is, the Area \(\Delta S\)) must be at least \(h\). That is, \[
\Delta S \geq h \implies \Delta q\,\Delta p \gtrsim h
\] which should be very familiar from standard quantum mechanics. But notice that this inequality was derived solely from the non-vanishing of the boundary term due to the EBC and the implications that has via the geometry of phase space, rather than anything to do with the disruption of accuracy due to measurement of the quantum system, or from the non-commuting of quantum operators. This is admittedly a somewhat heuristic ‘derivation’ of the Heisenberg position-momentum inequality, founded upon the empirical fact of the granularity of action and the symplectic geometry of phase space. It is simply a geometrical inequality due to the finite extent of the quanton, arising from relaxing the boundary condition on the Lagrangian formulation.

A less heuristic and much more rigorous direct derivation comes solely from the empirical fact of the extension of a quanton itself together with classical Fourier analysis, specifically the Fourier duality of a distribution. If a function \(F(x)\) is localised to within a width of \(\Delta x\), then its Fourier transform \(\hat{F}(k)\) must have a width \(\Delta k\), such that \(\Delta x\,\Delta k \gtrsim 1\), where the precise proportionality factor (of order \(\sim1\)) depends on how the ‘width’ is defined, whether, e.g., root-mean-square (RMS), or full-width-half-maximum (FWHM), etc. Here \(k\) (where \(k=2\pi/\lambda\)) is the wave number or spatial frequency and \(\lambda\) is the wavelength. We know from experiment (electron diffraction) that the de Broglie relation for momentum is \(p=h/\lambda = \hbar k\), where \(\hbar=h/2\pi\), which implies \(k=p/\hbar\). Putting this back into the relation yields (since \(\hbar\) is a constant), \[
\Delta x\, \Delta (p/\hbar)\gtrsim 1 \implies \Delta x \Delta p \gtrsim \hbar
\] The energy-time inequality also arises from the extension of the quanton. The time taken for an entity with spatial extension \(\sigma\) travelling at a velocity \(v\) to pass a certain point will be a duration \(\Delta t \sim \sigma / v\). We know from experiment (photo-electric effect) that the energy \(E\) of a quanton is given by the Einstein relation \[
E=hf=\hbar\omega
\] where \(f\) is the frequency and \(\omega=2\pi f\) is the angular frequency. Classical Fourier analysis again tells us that a signal of finite duration \(\Delta t\) will have a corresponding spread in its frequency spectrum \(\Delta f \gtrsim 1/\Delta t\), where, again, the precise proportionality factor (\(\sim1\)) depends on how the ‘width’ is defined. Putting this back into the relation gives \[
\Delta E = h \Delta f \gtrsim h / \Delta t
\] which yields the familiar expression
\[
\Delta E \, \Delta t \gtrsim h \;.
\]
Again, this is a classical result from wave mechanics applied to the observed spatial extension of the quanton which, like the one above for position-momentum, only ‘goes quantum’ when we make use of the empirical relation between energy and frequency found from experiment. Lévy-Leblond has noted (1977, 182–83) that the use of the term “uncertainty” is particularly inapt (and even inept) in this case, since the non-sharp spectrum width of a short-duration pulse is recognised as inherent rather than due to any putative “uncertainty” in its value.

These derivations show that, due to the extension of the quanton, the Heisenberg relations are not ‘uncertainties’, but are inherent inequalities on the simultaneous sharpness of the values of conjugate variables arising from the geometric constraints of the quanton’s finite extension, scaled by the fundamental quantum of action, \(h\).

The Dynamics of the Quanton

Now we need to move into the actual dynamics: how does a quantum entity evolve in time?

To describe the evolution of a quanton we need a mathematical tool that maps its initial state (i.e., its interaction extent, modelled by a distribution \(\phi\)) to another state later in time. In the language of differential equations, this tool is known (loosely) as a Green’s function or kernel (more strictly, a fundamental solution).

In Mechanics, the state of the system at some time \(t\) is found by applying the evolution operator, represented by the Green’s function, to the initial distribution, usually a Dirac delta ‘function’ (which is really a ‘distribution’) \(\delta(x – x_{\text{cl}})\), where \(x_{\text{cl}}\) is the classical path (this is a distribution that is zero everywhere except for a sharp spike which is at \(x_{\text{cl}}\)). Thus, evolution of a zero-dimensional point is described by a distribution which mathematically reflects this assumption of a zero-dimensional point, and produces a path which is a 1-dimensional trajectory.

However, for an extended quanton, the input cannot be a Dirac delta but rather a spatially extended distribution \(\phi(x,t)\) that has a finite width.

The entity which is to be described by the distribution \(\phi(x,t)\) is itself constrained by three conditions which arise from empirical observation:

1. scale/width: the Einstein-de Broglie relations show that a (non-relativistic) quanton has a finite interaction extension of spatial width \(\sigma\sim \lambda = h/p\) (de Broglie relation) which implies a momentum proportional to wave number \(p=\hbar k\), and an energy related to its frequency \(E=hf=\hbar\omega\) (Einstein relation). Planck’s constant sets the scale for conversion between ‘particle’ properties (energy and momentum) and ‘wave’ properties (frequency and wave number).
2. coherence: the entity maintains coherent phase relationships across its extension, which is evidenced by diffraction and interference patterns in single-entity experiments, such as the double-slit experiment. This would not be possible for an incoherent ‘smear’. This condition is actually a logical consequence of the first condition (width) and the empirical observation of interference and diffraction. And
3. quadratic dispersion: kinetic energy measurements and electron diffraction experiments establish that for non-relativistic matter, energy scales quadratically with momentum (i.e., \(E = p^2/2m\)). This is a dynamical condition.

A further non-empirical condition needs to be stated, namely a generalisation principle (largely understood implicitly but not often articulated explicitly):

4. correspondence: if a mathematical structure works in the classical case, then its generalisation to the quantum regime must be such that the formalism must reduce to the classical case in the appropriate classical limit. This is the mathematical condition.

These four conditions, three empirical and one mathematical, are collectively sufficient to allow selection of a unique evolution operator from the many possibilities afforded by mathematics. The term “propagator” eventually came to be used as standard, along with Green’s function, in quantum theory (Schulman 1981; Bayin 2018). Feynman tended to use the term ‘kernel’ (Feynman and Hibbs 1965). Here I will use the term ‘kernel’ for the short-time, or infinitesimal propagator, \(K_\epsilon\) and ‘propagator’ for the full (long-time) propagator \(K\) found from \(K_\epsilon\) by extension to finite time.

There are actually two pathways that lead to the propagator \(K\), one empirical, the other variational, each of which makes use of the above four conditions, albeit with a different emphasis and a different order of derivation. Here in the main text I will derive \(K\) by way of the empirical approach, because it is the more rigorous of the two and shows how all four of the above conditions uniquely select the form of the kernel. The variational approach is more heuristic and more in keeping with the basic idea of deriving Quantics from (classical) Lagrangian Mechanics, so I have included it here as an Appendix, to provide a slightly more intuitive derivation. They both, of course, end up producing the same object: the Feynman path integral.

Empirical Derivation of the Propagator \(K\)

If the quantum entity has extension, is coherent, and obeys quadratic dispersion, the evolution operator must be the unique linear map that preserves these properties for any initial distribution satisfying them. That is, the evolution rule (the ‘propagator’) in this case must be capable of transporting this extended entity while preserving its internal coherence as an entity. We therefore seek an infinitesimal kernel (or short-time propagator), \(K_\epsilon\), which governs how to advance the quanton’s state from a position \(y\) at one instant at time \(t\) to a new position \(x\) at the next instant at \(t+\epsilon\), where \(\epsilon\) is infinitesimally small, that respects the requirement that the entity remains stable and coherent during this evolution.

Mathematically, the requirement for coherence (condition 2) implies that the time evolution must be reversible and norm-preserving. These together imply that it is ‘unitary’. This comes from Liouville’s Theorem from Mechanics: it basically says that the number of states in phase space is conserved during time evolution; here this corresponds to the quanton’s distribution \(\phi\) being conserved from one instant to the next. In conventional quantum mechanics, this is interpreted as the conservation of probability. A real-valued kernel (like the heat kernel) would describe diffusion; the distribution would spread out and flatten, losing phase coherence information and becoming merely a ‘smear’, but this would contradict the empirical fact of quanton interference (and thus coherence). Therefore, the kernel cannot be real-valued but must be complex-valued, which implies that it will have a form \(\propto e^{i\varphi(x, y, \epsilon)}\), where \(\varphi\) is a real-valued phase function.

The requirement for quadratic dispersion (condition 3) applies a further constraint by fixing the form of \(\varphi\). From standard classical wave theory, the dispersion relation \(\omega(k)\) can be used to compute the Green’s function, using standard classical Fourier decomposition. For an impulse that has moved a distance \((x-y)\) over a time step of \(\epsilon\) from some arbitrary instant taken as \(t=0\), this takes the form: \[
K_{\epsilon}(x, \epsilon;y,0) = \int e^{ik(x-y)}e^{-i\omega(k)\epsilon}\, dk \,.\quad(1)
\] This essentially decomposes a disturbance into plane wave components, propagates each component according to the dispersion relation, and reassembles it again. Now, we know from the Einstein-de Broglie relations (condition 1) that \(p=h/\lambda=\hbar k\), and \(E=hf=\hbar\omega\). So, the quadratic dispersion condition \(E=p^2/2m\) gives \(\omega= \hbar k^2/2m\). Substituting for \(\omega\) back into the integral (1) yields (for a general time-step \(t\to t+\epsilon\)) \[
K_{\epsilon}(x,t+\epsilon;y,t) = \int e^{ik(x-y)}e^{-i\frac{\hbar k^2}{2m}\epsilon}\, dk \,.\quad(2)
\] This is a standard Gaussian (Fresnel) integral from classical optics, the solution of which has the form \[
K_\epsilon(x, t+\epsilon; y, t) = A(\epsilon) \exp\left(\frac{i}{\hbar}\frac{m(x-y)^2}{2\epsilon}\right)\quad(3)
\] where \(A(\epsilon)\) is a normalisation factor, \(x\) is the new position at the infinitesimal later time \(t+\epsilon\), and \(y\) was the old position at the earlier time \(t\). The form of (3) has arisen from the mathematical necessity to describe any entity that is extended, coherent, and obeys the quadratic dispersion relation of our universe. The Gaussian form also confers stability (i.e., the boundedness of the distribution) as well as the consistency of time evolution (i.e., evolution over \(t_1\to t_2\to t_3\) should equal the time evolution from \(t_1\to t_3\)). Indeed, any other functional form would either fail the time consistency property or give rise to instabilities that are not compatible with a stable, coherent entity.

The term \(m(x-y)^2/2\epsilon\) in (3) is immediately recognisable as the classical action for a free mass \(m\) moving a displacement \((x-y)\) over a time \(\epsilon\). The action arising as the phase of the kernel is a consequence of the quadratic dispersion relation, which is an empirical energy condition (i.e., from \(E=\hbar\omega=p^2/2m\)). The action is generally of the form \(S=\int L\,dt\), where the Lagrangian \(L\) is the difference between the kinetic \(T\) and potential \(V\) energies, \(L=T-V\). In the above kernel, only the kinetic energy term \(T\) is present in the phase, so the phase for the free mass is \(\varphi_{\text{free}}=\frac{i}{\hbar}\int T\,dt\).

Now, a further requirement from empirical observation needs to be considered. If a potential \(V\) were also present, this would modify the energy condition of the system from \(E=T\) to \(E=T+V\), so the presence of a potential \(V\) would modify the phase accumulation relation. In order to preserve consistency with the classical action functional (to which it must reduce in the appropriate limit, condition 4), the sign of the potential in the phase relation must follow that of the classical Lagrangian. Thus the phase relation in the kernel needs to be \(\varphi_{\text{full}}=\frac{i}{\hbar}\int L(T-V)\,dt\equiv\frac{i}{\hbar}S\). When the method of stationary phase \(\delta\varphi=0\) is applied, this implies \(\delta S=0\) which is the required classical result. In an infinitesimal time \(\epsilon\), a potential term \(V\) would modify the phase relation as \(-(\frac{i}{\hbar})\epsilon V\) (i.e., \(\frac{i}{\hbar}\int -V\, dt\), where \(dt=\epsilon\)).

So, to recap, conditions 2 and 3 above set the form of the kernel, while condition 1 gives the constant needed to set both the scale and what would be an otherwise undetermined constant in the phase factor. All three are necessary. Note that no ‘quantum’ postulates were utilised, just classical wave theory with empirical conditions. The requirement that the phase must track the local energy at a spatial point means summing the local contributions to the energy at that point. And, since in the classical limit the action functional must reduce to the classical principle of stationary action (condition 4), this requires that the full Lagrangian be present in the kernel’s phase relation.

Thus the short-time propagator kernel \(K_\epsilon\) (with the potential \(V\) averaged over its values at \(x\) and \(y\)) is \[
K_\epsilon(x, t+\epsilon; y, t) = A(\epsilon) \exp\left[\frac{i}{\hbar}\left(\frac{m(x-y)^2}{2\epsilon} -\epsilon V\frac{(x+y)}{2}\right)\right]\,.\quad(4)
\] This kernel (4) can now be used to derive the Schrödinger equation for \(\psi(x,t)\) from: \[
\psi(x, t+\epsilon) = \int_{-\infty}^{\infty} K_\epsilon(x, t+\epsilon;y,t)\,\psi(y,t)\, dy \;.\quad(5)
\] This was lots of fun to do, by the way, and I debated whether to include it here as another Appendix, but it is something that you can already find done elsewhere in detail, if it is of interest (e.g., Feynman and Hibbs 1965; Shankar 1994; Derbes 1996).

We now need to determine the full (long-time) propagator \(K(x_f, t_f; x_i,t_i)\). For the evolution to be consistent, this must be obtained by integrating over \(N-1\) intermediate states, composing the infinitesimal kernel \(K_\epsilon\) over \(N\) time slices \(\epsilon=T/N\) for a finite time \(T = t_f-t_i\), then taking the continuum limit \(N\to\infty\) (equivalently \(\epsilon\to0\)) (cf. Feynman 1948, 2005).

The propagator \(K\) is thus found from (where \(x_N = x_f\) is the final position and \(x_0 = x_i\) is the initial position) \[
K(x_f, t_f; x_i, t_i) = \int dx_{N-1} \, K_\epsilon(x_N, t_N; x_{N-1},t_{N-1}) \, \dots \int dx_1 \, K_\epsilon(x_1,t_1; x_0, t_0)\,,\quad(6)
\] where the short-time kernel \(K_\epsilon\) for the step from \(x_j\) to \(x_{j+1}\) is now, \[
K_\epsilon(x_{j+1},t_{j+1};x_j,t_j) = A(\epsilon)\exp\left[\frac{i}{\hbar}\left(\frac{m(x_{j+1}-x_j)^2}{2\epsilon}-\epsilon V \frac{(x_{j+1} + x_j)}{2}\right)\right].\quad(7)
\] As noted, the term inside the exponential function multiplied by \(i/\hbar\) in (7) is exactly the action for that infinitesimal time step \(\epsilon\) from \(j\to j+1\). Let this segment be denoted by \(S_j\). Then \[
S_j \approx \frac{m}{2} \left(\frac{(x_{j+1}-x_j)^2}{\epsilon}\right) – \epsilon V \frac{(x_{j+1} + x_i)}{2} = \int_{t_j}^{t_{j+1}}\tfrac{1}{2} m \dot{x}^2 – V\,dt.\quad(8)
\]
When exponential functions are multiplied, their exponents add, so the product of all the exponentials in the \(K_\epsilon\) in the above multiple integral (6) takes the form \[
\prod_{j=0}^{N-1} \exp\left(\frac{i}{\hbar} S_j\right) = \exp\left(\frac{i}{\hbar}\sum_{j=0}^{N-1} S_j\right)\quad(9)
\] so that the integral (6) itself takes the form \[
K(x_f, t_f; x_i, t_i) = \int dx_{N-1} \cdots \int dx_1 \,A(\epsilon)^N \exp\left(\frac{i}{\hbar}\sum_{j=0}^{N-1} S_j\right).\quad(10)
\] In the continuum limit as \(N\to\infty\), the summation becomes an integral: \[
\exp\left(\frac{i}{\hbar}\sum_{j=0}^{N-1} S_j\right) \xrightarrow[N\to\infty]{} \exp\left(\frac{i}{\hbar}\int_{t_i}^{t_f} L(x,\dot{x})\,dt\right) = e^{(i/\hbar) S[x(t)]}\,.\quad(11)
\] The multiple integrals \(\int\dots\int\) over the intermediate positions \(x_1\dots x_{N-1}\) in (10) combined with the normalisation factors \(A^N\) are the functional measure. In the limit \(N\to\infty\), this is conventionally denoted in a compact way: \[
\int dx_{N-1} \dots dx_1 \left[A(\epsilon)\right]^N \xrightarrow[N\to\infty]{} \int \mathcal{D}[x(t)]\,.\quad(12)
\] Combining (10), (11) and (12) yields the expression for the full (long-time) propagator, \(K\): \[
K(x_f, t_f; x_i, t_i) = \int e^{i S[x(t)]/\hbar} \;\mathcal{D}[x(t)]\quad(13)
\] where the script-D functional measure \(\mathcal{D}\) indicates that this is a path integral (Feynman’s notation in 2010, 35), and this is precisely Feynman’s path integral expression for the propagator. Also see the Appendix for the alternative variational derivation.

The upshot of this is that we have arrived at Feynman’s path integral via two different routes. The empirical route used the three empirical constraints (1-3) on the quanton and derived the short-time kernel as the unique Green’s function consistent with these constraints. The phase of the kernel was found to coincide (by condition 4) with the classical action. Composing this with itself over finite time led to the full propagator. The variational route (Appendix) began with the classical action functional and asked what happens if the sharp boundary conditions appropriate to a point particle were relaxed, by dint of condition 1 (scale/width). This meant having to integrate over all paths between the extended boundary conditions, with the weighting of each path forced to be \(e^{(i/\hbar)S}\), by correspondence (condition 4, recovering \(\delta S=0\) in the classical limit) and coherence (constraint 2, preserving phase relationships). The specific dynamical condition 3 does not enter until the path integral needs to be evaluated.

Comparison with the Feynman Path Integral

We have reproduced the mathematical form of the Feynman path integral both from first principles (empirical derivation) and by generalisation of the variational functional (extended boundary condition). But the underlying assumed ontology is very different.

Feynman’s approach was founded upon a comment in a paper by Dirac that the action functional (time integral of the Lagrangian) was analogous to a contact transformation function in quantum mechanics, and that “the phase of the wave function corresponds to Hamilton’s principle [sic] function in classical theory” (Dirac 1933). (This was of course Schrödinger’s inspired guess when he inferred his equation from the Hamilton-Jacobi equation via the optico-mechanical analogy; see Part I).

He then used this as the basis of a way to derive an expression for what he called the kernel or amplitude \(K\) and did so operating under the prevailing view of assuming quasi-point-like particles, and therefore also 1-dimensional infinitely-thin paths. His intuition told him, however, that it is not just the classical trajectory \(x_{\text{cl}}\) that is important, where the phase \(S_{\text{cl}}/\hbar\) is stationary, but that those ‘nearby’ trajectories that have a phase ‘close’ to the stationary phase would also contribute constructively, while those that are further away will have phases that oscillate so much that their effective contribution is cancelled out by destructive interference. His derivation went as follows: Each individual trajectory \(\phi[x(t)]\) would have a phase proportional to the action “for the corresponding classical system” (Feynman and Hibbs 2010, 29): \[
\phi[x(t)] = \text{const}\; e^{(i/\hbar) S[x(t)]}
\] and the ‘amplitude’ \(K(b,a)\) is then the sum of all contributions for every path from \(a\) to \(b\): \[
K(b,a) = \sum_{\text{paths from $a$ to $b$}} \phi[x(t)]
\] He ultimately rendered this as (Feynman and Hibbs 2010, 35): \[
K(b,a) = \int_a^b e^{(i/\hbar)\,S[b,a]}\,\mathcal{D}[x(t)],
\] where \(S[b,a]\) is the classical action for each path from \(a\) to \(b\). His method therefore consisted of including everything, summing over every imaginable path, including impossible trajectories, those with infinite velocity, and backwards time-jumps. The impossible paths don’t disappear, their phases just oscillate so wildly that they cancel out destructively, leaving only those paths near the classical ‘spine’ where \(\delta S=0\) whose phases align constructively. Thus, in this view, Nature tries everything, but only the ‘reasonable’ paths remain. This is exclusion by cancellation.

I’ve had the original (1965) version of Feynman’s book for over 35 years, ever since my grad student days. It was while again recently trying to understand how the path integral looks that I realised that exactly the same mathematical object would be derived if the quantum entity itself has an interaction extent which is the coherence width that influences phase contributions in Feynman’s approach. The sum in this case is not of infinitely-many particle trajectories (most of them impossible), but rather the mathematical decomposition of a quanton’s coherence extent into its constituent modes. The integration measure \(\mathcal{D}[x]\) is effectively restricted to only those continuous coherent mode histories that match up with the quanton’s coherence width. There are never any impossible or infinite-velocity paths in this derivation because the starting premise of extension rules them out as unphysical descriptions of the entity. Only valid paths are ever summed, and the sum naturally yields the evolution of the distributed entity. This is inclusion by coherence.

The key thing to notice is what the limits on the integrals mean. For Feynman, the integrals over the \(x_i\) are taken from \(-\infty\) to \(+\infty\), which is ‘sum everything, everywhere’, and so includes all the trajectories, including the impossible ones. For the integral (10) above, while it is also the case that the limits formally go from \(-\infty\) to \(+\infty\), the coherence width of the quanton drops off very rapidly close to the classical trajectory, so that the main contributions come from the region immediately surrounding it, and the rest of the contributions away from it are as effectively zero as all the cancellations in the Feynman integral. Thus, cancellation vs (coherent) inclusion.

These different approaches (incredibly!) produce the same mathematical object. Feynman’s insight was to show that if you start by summing everything, Nature selects only the correct form by cancellation. Our derivations show that this cancellation is unnecessary if you start with extended entities since you only ever need to sum over the coherent modes of the entity. The unphysical paths are automatically excluded by the definition of the state space. Feynman’s mathematical virtuosity actually got to where it needed to go, even despite its infinitely-many impossible sideways detours along the way.

The Action Manifold and the Dynamical Response Operator

This brings us to the action functional itself.

In the previous Part, the operator \(e^\delta S\) was introduced. It has the expanded form \[
e^\delta S[q] = \sum_{n=0}^\infty \frac{1}{n!}\delta^n = \mathbb{1}\cdot S[q] + \delta S[q] + \frac{1}{2!}\delta^2S[q] + \frac{1}{3!}\delta^3 S[q]+ \dots\, .
\] This expands the action functional \(S[q]\) into an infinite (Taylor) series of progressively higher-order variations centred around the classical path. It is possible to conceive of this as an Action Manifold, \(\mathscr{A}\), which is the infinite-dimensional space of all possible paths allowed by the action around the classical path.

The Dynamical Response Operator \(\mathscr{D}\) was also introduced which acts on the classical action functional \(S[q]\) (the ‘seed’) in such a way as to ‘extract’ the dynamics by way of the Dynamical Response Functional \(\mathscr{D}S\): \[
\mathscr{D}S = \left(e^\delta – \mathbb{1}\right)S = \delta S + \frac{1}{2!}\delta^2S + \frac{1}{3!}\delta^3S + \dots .
\] In this way, the Action Manifold \(\mathscr{A}\) can be considered as simultaneously containing both the ‘seed’ \(S\) as well as the ‘dynamics’ \(\mathscr{D}S\), since it can be viewed as the following decomposition: \[
\mathscr{A} = e^\delta S[q] = S[q] + (e^\delta – 1)S[q] = S[q] + \mathscr{D}S[q]\;.
\] The ‘zeroth’ variation term is then the classical action \(S\) itself (the ‘seed’, excluded from \(\mathscr{D}\)). In the standard point-particle classical limit, \(\sigma\to0\), the boundaries are fixed (Dirichlet), the first variation \(\delta S\) is infinitesimal, higher-order terms essentially vanish, the Dynamical Response Functional effectively reduces to the first variation, \(\mathscr{D}S\to\delta S\), and we recover (classical) Mechanics from \(\mathscr{D}S\to\delta S=0\). That is, the first variation \(\delta S\) is the one that yields the classical Euler-Lagrange equations (which represent force equations in 3D) when the endpoints are fixed (Dirichlet).

But when we adopt the Extended Boundary Condition, the endpoints of the first variation \(\delta\) have a finite width, which yields the Heisenberg Inequalities. The quanton no longer propagates only on a single sharp Dirac-delta 1-dimensional trajectory, but rather as a fuzzy cylinder centred on the classical trajectory ‘spine’ and representing the interaction extent of the quanton. As a result of its extension, the higher-order variations of the action manifold beyond the first can no longer be considered negligible but may become essential parts of the dynamics. The empirical question then becomes, how many variations are sufficient to replicate the experimental results found so far, and what new directions (or possible results) might be revealed as we explore the higher variations (see below)?

Thus, the transition \(\vert M \rangle \to \vert Q\rangle\), from Mechanics to Quantics is not so much a change in the laws of physics, as it is simply a change in the order of the dynamical response we must consider, as we move from the 1st variation (classical) to the higher variations (quantum) in the infinite series of \(\mathscr{D}S\), since the apparent ontology of the universe, shown from experiment, seems to require us to respect the finite interaction extent of the quanton.² The results of this exploration suggest that Quantics could be regarded as the (classical) Mechanics of extended entities. And that suggests that the Level 1 ontology in the ladder of abstraction from Parts I-III could reasonably be postulated as the quanton, which possesses both discreteness in number and extension, which might solve the Fenchurch Problem of Part VI. The way this might translate into, for example, Hamilton-Jacobi theory would be a very interesting thing to explore. I suspect it would give rise to the Schrödinger equation double-quickly by the need to include the second variation owing to the finite extent of the quantum entity.

The following section now sketches out some of the other preliminary results of this investigation, none of which are at the stage of being ready to report in full just yet, as I am still exploring these for their implications. But what has surfaced so far is highly suggestive, so it is laid out here in outline form for those who might want to investigate this for themselves.

Where to Next?

The idea of relaxing the Dirichlet boundary condition and to adopt a ‘fuzzy’ Extended one initially came from noticing the form of the conventional boundary term in the action functional for \(\delta S=0\), and was simply a mathematical curiosity explored ‘just for the fun of it’. However, assuming a primitive ontology of a quanton with a spatial interaction extent required a change in the mathematical apparatus used to extract the dynamics from the action, which is manifested precisely as the Extended Boundary Condition. That is what I meant in the Preamble above about a physics-motivated change to the mathematics—if a quanton has a finite interaction extent, the usual Dirichlet condition is no longer appropriate nor well-motivated physically.

Similarly, the idea of looking at the complete infinite series of variations defined by the Dynamical Response Operator \(\mathscr{D}S\) came from noticing that the second variation \(\delta^2S\) already makes itself felt in the Schrödinger equation. That is, it turns out that the second variation of the action is already in Quantics, albeit mostly out of sight, but it has been found by researchers on several occasions. It is also where entanglement (implexity in Lévy-Leblond’s terminology) first shows up. In Part III, we saw how the de Broglie-Bohm (dBB) theory has an ‘extra’ term \(V_Q = -\frac{\hbar^2}{2m} \frac{\nabla^2 R}{R}\) in the Hamilton-Jacobi equation which is derived from substituting the form \(\psi=Re^{iS/\hbar}\) into the Schrödinger equation. This term was explicitly non-local (pantopic) and is called the ‘quantum potential’ in dBB theory. In Part IV, I mentioned that Madelung (1927) also found the same term in his formulation of the Schrödinger equation as a hydrodynamic equation; there it was interpreted as a ‘quantum pressure’. In the previous Part I mentioned that in functional analysis applied to Mechanics, the second variation \(\delta^2S\) of the action functional gives rise to two mathematical objects of some interest: the (functional) Hessian which is the matrix of second-order variational derivatives of the action; and the van Vleck Determinant which is the determinant of a particular form of the Hessian.

It is precisely in these objects that both ‘non-locality’ and ‘entanglement’ seem to show up for the first time, non-locality in the quantum potential \(V_Q\), which depends on the Van Vleck Determinant, and something that looks a lot like entanglement as off-diagonal elements in the Hessian and van Vleck determinant in the multi-particle case. In essence, it seems that \(V_Q\) is rather more like the quantum pressure of Madelung, in that it represents a force resisting the ‘compression’ of the quanton field by the ‘curvature’ of the action manifold acting upon the extended quanton. It essentially prevents the extended quanton’s distribution from collapsing to a point, or the fuzzy-tubular bundle of paths from collapsing to a classical line trajectory.

The somewhat more tenuous entanglement argument rests upon noticing the same situation in Mechanics—the two masses on a spring example discussed by Norsen (2022). If this is examined in purely classical terms, the Hessian and Van Vleck Determinant have off-diagonal elements which encode the (Norsen’s term) “intertwinement” of the two masses, which is irreducible: that is, it cannot be separated into distinct separate actions for each mass. However, given that we know the forward map from 3D coordinates to configuration space coordinates, this is not in the least bit mysterious, since we also know the underlying ontology (a spring) giving rise to the intertwinement. In Quantics, however (as argued in Part VI), we know neither the forward map, nor the ontology: we only have the configuration space representation, which is not bijective with 3D space. Thus, finding off-diagonal terms in the Hessian or Van Vleck Determinant in Quantics would show that the action is similarly additively non-separable, which would be very suggestive that something at least related to entanglement might be going on, however it actually arises. But that is something that needs much closer scrutiny to see if it actually maps across to the quantum realm.

So, the second variation \(\delta^2S\) represents the curvature of the action functional (in function space), while the third variation \(\delta^3S\) represents the skewness or torsion, and this higher-order term will account for diffraction and tunneling, by way of catastrophe theory, such as through Airy functions (and Pearcey integrals at fourth order) when the Hessian has zero eigenvalues or the van Vleck Determinant vanishes at a caustic. This has already been extensively studied in the seminal work of Berry and Mount (1972) and Berry (1976), and will be examined more closely in Part IX.

As well, the measurement ‘problem’ very likely becomes simply an ordinary interaction between two quantons, one of which is part of a (macroscopic) detector apparatus (and so strongly constrained). I suspect the most likely mechanism will therefore be related to decoherence, given the decades of work that has gone into investigating that approach, and the promising results of that investigation (Schlosshauer 2007). If that does turn out to be a viable pathway, it would imply that no observer is needed, nor any ‘shifty split’ Heisenberg ‘cut’ to separate out the classical observer from the quantum world. The well-documented structural problems of the Copenhagen Interpretation (e.g., John Bell’s well-known critique discussed in Part IV) would thereby be bypassed.

The question of intrinsic angular moment (‘spin’) is probably best deferred to the later exploration on the generalisation to relativity (in Part X) of the present purely non-relativistic formulation, although we will in Part IX likely examine Lévy-Leblond’s (1967) paper which found that spin is not necessarily a strictly relativistic phenomenon. Hestenes, too, beginning in (1966), has extensively examined the question of ‘spin’ as a non-relativistic phenomenon in 3D.

So, to sum up. Relaxing the Dirichlet Boundary Condition on the classical Lagrangian due to the inherent extension of the quanton yields the Extended Boundary Condition, which generates the first aspect of ‘quantum’ effects found so far: the Heisenberg Inequalities. Pursuing this further, based on four conditions—three empirical, one mathematical—it was possible to derive the Feynman path integral from both variational (strongly heuristic) as well as empirical (considerably more rigorous) starting points. Preliminary explorations suggest that the extended nature of the quanton implies that higher-order variations than just the first are required to capture the full dynamics at play. This suggests the possibility that the mathematics of Quantics is the Mechanics of extended entities on an Action Manifold \(\mathscr{A}=S+\mathscr{D}S\) using the Extended Boundary Condition. Classical Mechanics is then simply the truncation of this to the first variation, carried out with the Dirichlet boundary condition. The further implications that arise from this remarkably simple starting ansatz will be examined in the next Part.

Next: Part VIII.5 – ‘Post-amble’ to the Previous Post

Appendix – Variational Derivation of the Propagator \(K\)

The variational derivation is marginally less exhausting than the empirical one carried out in the main text above, and somewhat closer in spirit to what Feynman did, given that he started with the assumption of action=phase in \(e^{iS/\hbar}\), whereas it was derived from first principles in the empirical derivation. Unlike the empirical derivation we will not need all four conditions explicitly, only three (but the fourth does enter implicitly), and they enter at different places and with different emphases than in the empirical derivation. However, the fourth condition does end up entering explicitly when one seeks to calculate the propagator from the Lagrangian.

As noted in Part VII, the principle of stationary action in Mechanics is used to find the classical trajectory when \(\delta S=0\), which is based on the Dirichlet condition of exact endpoints, and vanishing variations at the endpoints. The critical assumption is that the endpoints are sharp. This makes perfect sense for a particle assumed to be a mathematical point, and representable by a Dirac delta.

However, experimental evidence shows that quantum entities have finite interaction extension, and do not interact like idealised mathematical points. Rather, they behave like continuous distributions. A point particle will follow a single path, but an extended entity entering a slit does not follow a single path—its different parts traverse different paths simultaneously. So the question becomes how to mathematically describe the evolution of an extended entity that explores multiple paths simultaneously?

Now, let \(\phi(x,t)\) describe the state of a quantum entity with extension, namely, a quanton (that is, \(\phi\) describes its extent). Since it is a continuous distribution, its evolution must obey superposition; that is, the total effect of a distributed source is the sum of the effects of its individual parts. In classical field theory, the evolution of a system from an initial distribution \(\phi(x_i, t_i)\) to a final distribution \(\phi(x_f,t_f)\) is governed by a Green’s function or propagator \(K(x_f,t_f;x_i,t_i)\). Specifically, \[
\phi(x_f,t_f) = \int_{-\infty}^\infty K(x_f,t_f;x_i,t_i)\,\phi(x_i,t_i)\,dx_i \quad (A1)
\] where \(x_f\) is any point in the final distribution, \(x_i\) is any point in the initial distribution, and the propagator \(K\) is the weight representing the influence of \(x_i\) at the initial time \(t_i\) on \(x_f\) at the final time \(t_f\).

Since the entity has extension, the connection between the initial and final distributions cannot simply be a single line trajectory; it must be a superposition of multiple connections. Specifically, because the different parts of the entity will move along all physically coherent connections between \(x_i\) and \(x_f\), the kernel \(K\) must be constructed as a sum of all such coherent connections connecting them. Thus, \(K=\sum_{\text{all paths }x(t)}W[x]\), where \(W[x]\) is a weight assigned to each general path \(x(t)\). So, the question now becomes: what sort of form must the weight \(W[x]\) take?

Experiment shows (condition 2) that the entity displays interference and diffraction, which produces regions both of non-zero density (i.e., \(\rho=\vert\phi\vert^2\neq0\)), as well as zero density (from cancellations, i.e., \(\vert\phi\vert^2=0\)). Cancellation is impossible if the weights associated with the paths are simple positive real numbers. Rather, cancellation requires opposition of values. To allow for constructive addition and destructive cancellation in a linear sum of superpositions, the weights \(W[x]\) need to be complex numbers, \(W[x]=e^{i\varphi[x]}\) where \(\varphi[x]\) is a real-valued phase associated with the path \(x(t)\). Now the question becomes: what is the value of the phase \(\varphi[x]\)?

By condition 4, in the limit when the width of the entity \(\sigma\to0\), we must recover the classical result: namely, a single trajectory between \(y\) and \(x_f\), which is itself found from \(\delta S=0\). Now, consider a ‘general’ path \(x(t)\) connecting \(x_i\) to \(x_f\). If we sum the contributions from all possible paths \(x(t)\), the only way to ensure that the classical path \(x_{\text{cl}}(t)\) dominates when \(\sigma\to0\) is if the contributions interfere constructively near the classical stationary path and destructively elsewhere. Constructive interference occurs when phase is stationary \(\delta\varphi=0\), while destructive interference cancels out the non-stationary paths. The stationary point of the phase, \(\delta\varphi=0\), must coincide with the classical trajectory, which is the stationary point of the classical action, \(\delta S=0\). That is, \(\delta\varphi=0\iff\delta S=0\), so it must be the case that \(\varphi[x] = \alpha S[x]\) for some constant \(\alpha\), and so the contribution from each path must be \(W[x]=e^{i\alpha S[x]}\).

The constant \(\alpha\) is fixed by the scale condition (condition 1). Dimensional analysis shows that the constant must have dimensions of reciprocal action (i.e., \(\frac{1}{\text{action}}\)), so that the phase of the exponential is a dimensionless real number. Since wave mechanics deals in wave number and angular frequency, and phase angle is conventionally expressed in radians, not cycles, the appropriate terms from the scaling relations in condition 1 are those that use Planck’s constant \(h\) scaled by \(2\pi\) radians, namely \(\hbar\). Thus, the constant in the weight is \(1/\hbar\), so that \(\varphi=S/\hbar\) is the total accumulated phase in radians, and the full expression for the weight is \(W[x]=e^{(i/\hbar)S[x]}\).

To find the propagator $K$, first we need to find the infinitesimal propagator ‘kernel’ \(K_\epsilon\) that will evolve the distribution from position \(x\) at time \(t_0\) to a position \(x'\) at time \(t_0+\epsilon\), where \(\epsilon\) is infinitesimal. Over a time \(\epsilon\), the phase of the weight \(W[x]\) will change by an amount which is related to the change in the action \(S\) over the time step. This can be represented (taking \(t_0=0\) for simplicity) by \[
K_\epsilon(x',\epsilon,x,0) \propto\exp\left({\frac{i}{\hbar}S(x',x,\epsilon)}\right)\quad(A2)
\] where \(S(x',x,\epsilon)\) represents the action evaluated over the distance \((x'-x)\) in the infinitesimal time \(\epsilon\).

Any valid evolution law must allow for composition over intermediate times (i.e., evolution over \(t_1\to t_2\to t_3\) should equal the time evolution from \(t_1\to t_3\)) . So, if the interval is split into two halves, then evolving via the intermediate step should give the same result as evolving directly between the two end times. This implies that if the interval is split over \(N\) infinitesimal intervals of duration \(\epsilon\), then the (full) propagator \(K\) needs to be constructed from a product (composition) of the infinitesimal kernels \(K_\epsilon\) over the full time \(T=N\epsilon\). This ends up being a product of \(N\) steps, where each step integrates over the value of \(x_j\) at that time step. Thus, this is a multiple integral (where \(x_j\) represents the position at time step \(j\), and \(x_N = x_f\) is the final position, and the initial position \(x_i\) would then take the index value \(x_0\) in the sum): \[
K(x_f,t_f;x_i,t_i)=\lim_{N\to\infty} \int dx_1\,\dots \int dx_{N-1}\prod_{j=0}^{N-1}K_\epsilon(x_{j+1};x_j)\;.\quad(A3)
\] As found above in (A2), the weight for the infinitesimal kernel \(K_\epsilon\) for the single infinitesimal segment between \(x_j\) and \(x_{j+1}\) is determined from the classical action \(S_j[x_{j+1},x_j]\) accumulated over that interval. This can be rendered in the form \[
K_\epsilon(x_{j+1}, x_j) = A(\epsilon)\exp{(\tfrac{i}{\hbar}S_j)} \quad(A4)
\] where \(A(\epsilon)\) is a normalisation factor for the weight. Putting (A4) back into the expression (A3) for the full propagator \(K\) gives \[
K(x_f,t_f;x_i,t_i)=\lim_{N\to\infty} \int dx_1\, \dots \int dx_{N-1}\prod_{j=0}^{N-1}\left[A(\epsilon)\exp{(\tfrac{i}{\hbar}S_j)}\right].\quad(A5)
\] When exponential functions are multiplied, their exponents add, so the product of all the exponentials in (A5) takes the form \[
\prod_{j=0}^{N-1} \exp\left(\frac{i}{\hbar} S_j\right) = \exp\left(\frac{i}{\hbar}\sum_{j=0}^{N-1} S_j\right).\quad(A6)
\] The \(N\) products of the normalisation factor \(A(\epsilon)\) becomes a power \([A(\epsilon)]^N\) so that the integral (A5) itself takes the form \[
K(x_f, t_f; x_i, t_i) = \int dx_{N-1} \cdots \int dx_1 \,[A(\epsilon)]^N \exp\left(\frac{i}{\hbar}\sum_{j=0}^{N-1} S_j\right).\quad(A7)
\] Since the action is additive, the total action \(S[x]\) for the entire path \(x(t)\) is the continuum limit of the sum of the actions \(S_j\) for each individual segment: \[
\sum_{j=0}^{N-1} S_j \xrightarrow[N\to\infty]{} S[x],\quad(A8)
\] so the exponential term in (A7) converges to a global phase factor \(e^{(i/\hbar)S[x]}\).

The multiple integrals \(\int\dots\int\) over the intermediate positions \(x_1\dots x_{N-1}\) combined with the normalisation factors \(A^N\) in (A7) are the functional measure. In the limit \(N\to\infty\), this is conventionally denoted in a compact way: \[
\int dx_{N-1} \dots dx_1 \left[A(\epsilon)\right]^N \xrightarrow[N\to\infty]{} \int \mathcal{D}[x(t)].\quad(A9)
\] Combining (A7), (A8) and (A9) yields the expression for the full (long-time) propagator, \(K\): \[
K(x_f,t_f;x_i,t_i)=\int e^{(i/\hbar)S[x]}\;\mathcal{D}[x(t)], \quad(A10)
\] where the script-D functional measure \(\mathcal{D}\) indicates that this is a path integral (Feynman’s notation in 2010, 35). This is Feynman’s path integral expression for the propagator.

Note that in this derivation we did not make use of the explicit form of the Lagrangian in the action functional, so that it is independent of any specific form until such time as it becomes necessary to evaluate the integral, where the quadratic dispersion condition (condition 3) would become explicit in the kinetic energy term, \(T\), in \(S=\int L(T-V)\,dt\).