Action and Observables

Natural Units of Time and Length

To describe the dynamics of ant-foraging systems such as 2 and 3, we focus on the fundamental unit of motion: the ant step. Each ant step is characterized by two parameters: its duration and its step length. Let $\tau$ denote the mean duration of a single ant step, and $\lambda$ the mean step length.

For a given system—defined as a particular species of ants foraging in a relatively stable thermal environment—both $\tau$ and $\lambda$ may be treated as constants. They are intrinsic properties of the species and the environment, and therefore serve as natural units of time and length for describing the system’s motion. The step frequency

\[f = \frac{1}{\tau}\]

represents the expected number of actions executed per unit time by a single ant (action agent). The step length $\lambda$ represents the unit dispacement along its path. Together, they define a characteristic speed,

\[c = \frac{\lambda}{\tau} = \lambda f.\]

Given a stable environment, the speed $c$ is the same along all ant paths, up to some stochastic fluctuations.

Action, Energy, and Momentum

Let $h$ denote the step motion of an ant crossing a fixed reference point relative to the thermal environment. The physical action associated with this motion can be precisely defined as

\[h \equiv \overline{E} \tau \equiv \overline{p} \lambda,\]

where $\overline{E}$ is the mean energy required to execute the action, $\tau$ is the natural unit of time, $\overline{p}$ is the mean momentum associated with each step motion, and $\lambda$ is the natural unit of length.

Under this definition, the unit energy and unit momentum can be expressed in terms of the fundamental units $h$, $\tau$, and $\lambda$:

\[\overline{E} = \frac{h}{\tau} = h f, \quad \overline{p} = \frac{h}{ \lambda}.\]

These relations highlight that energy and momentum are not independent primitives, but rather observables derived from the temporal and spatial realization of the same physical action.

To distinguish the direction of motion, we denote by $h^+$ the action of an ant crossing a reference point in one chosen direction, and by $h^-$ the action crossing the same point in the opposite direction.

A system consisting of two opposing actions doubles its energy expenditure:

\[\begin{aligned} \overline{E}_{\pm} &= \overline{E}_{+} + \overline{E}_{-} \\ &= \frac{h^+}{\tau^+} + \frac{h^-}{ \tau^-} \\ &= 2\frac{h}{\tau} \gt 0, \end{aligned}\]

but produces no net change in its center of mass,

\[\begin{aligned} \vec{p}_{\pm} &= \vec{p}_{+} + \vec{p}_{-} \\ &= (\hat{\mathbf{v}}_{+} + \hat{\mathbf{v}}_{-}) \frac{h}{\lambda} \\ &= (\hat{\mathbf{v}}_{+} - \hat{\mathbf{v}}_{+}) \frac{h}{\lambda} = 0, \end{aligned}\]

where \(\hat{\mathbf{v}}_{+} \; \text{and} \; \hat{\mathbf{v}}_{-} = - \hat{\mathbf{v}}_{+}\) denote the unit vectors pointing in opposite directions. These relations highlight a fundamental difference between scalar and vector quantities: observables such as time and energy are scalars: their quantities increase monotonically as the system evolves. The cumulative action carried by the system therefore serves as a natural clock for tracking its evolution.

By contrast, vector quantities such as momentum and displacement depend sensitively on the directional organization of individual actions. Coordinated actions reinforce one another, while uncorrelated or oppositely directed actions may partially or completely cancel. The directional nature of vector quantities enables the emergence of coherent, large-scale spatiotemporal patterns, even when the underlying microscopic actions remain largely stochastic.

Uncertainty Principle

To describe the motion of an ant-foraging system, one must observe at least a complete action—that is, a singe ant step. This establishes the limits of what can be meaningfully measured in such a system:

\[\begin{aligned} \Delta t \geq \tau, \quad \Delta x &\geq \lambda, \\ \Delta E \geq \overline{E}, \quad \Delta p &\geq \overline{p}. \end{aligned}\]

Attempts to probe or describe the system below these limits introduce uncertainty, as no single event can be fully resolved. The observable quantities such as time $\Delta t$ and energy $\Delta E$, as well as length $\Delta x$ and momentum $\Delta p$, are not independent, and their relation is often expressed as the uncertainty principle:

\[\Delta{E} \Delta{t} \geq \overline{E}\tau = h, \quad \\ \Delta p \Delta x \geq \overline{p} \lambda = h,\]

where $h$ denotes the action of a single ant step.

The uncertainty principle does not arise from imperfect instruments or experimental techniques, but from the transactional nature of the underlying system, which advances one action at a time. Such transactional systems are what physicists often refer to as quantum systems. In a quantum system, any observable derived from the underlying action—such as time, length, speed, energy, and momentum—cannot be meaningfully defined or measured at scales finer than a single action.

Action Density

We define $\rho(x,t)$ as the observable action density at the spacetime point $(x,t)$. The expected number of net actions observable within the unit spacetime region

\[\left[x-\frac{\lambda}{2},x+\frac{\lambda}{2} \right] \times \left[t-\frac{\tau}{2},t+\frac{\tau}{2} \right]\]

is given by

\[P(x, t) = \int_{t-\frac{\tau}{2}}^{t+\frac{\tau}{2}} \int_{x-\frac{\lambda}{2}}^{x+\frac{\lambda }{2}} \rho(x', t') \, \mathrm{d}x' \, \mathrm{d}t'.\]

$P(x,t) = +1$ if a forward motion or positive action ($h^+$) is observed, $-1$ if a backward motion or negative action ($h^-$) is observed, and $0$ if no net motion is observed. The sign of $P(x,t)$ encodes the direction of the action, while its magnitude represents the local probability of observing an action event within the unit spacetime. Because no two action agents (i.e., ants) can occupy the same location simultaneously, this exclusion principle ensures that the local probability never exceeds one.

The action density definition of $\rho(x,t)$ extends naturally to two- and three-dimensional space, where $x$ represents a position vector. For notational simplicity, we continue to use a single symbol $x$ throughout, with the understanding that it may denote $(x,y)$ in 2D or $(x,y,z)$ in 3D.

For a given time $\mathbf{t}$, the spatial integral

\[P(x, \mathbf{t}) = \int_{x-\frac{\lambda}{2}}^{x+\frac{\lambda }{2}} \rho(x', \mathbf{t}) \, \mathrm{d}x'\]

provides a snapshot of the system at that instant. In this snapshot, each “pixel” represents a spatial region of size $\lambda$ and has a value $P(x,\mathbf{t}) \in [-1,+1]$. The magnitude indicates the progression of the action within the unit space centered at $x$, while its sign encodes the direction of the action occurring at that location.

For a fixed reference point $\mathbf{x}$, the temporal integral

\[P(\mathbf{x}, t) = \int_{t-\frac{\tau}{2}}^{t+\frac{\tau}{2}} \rho(\mathbf{x}, t') \, \mathrm{d}t'\]

yields a discrete action signal passing through $\mathbf{x}$. Each unit pulse corresponds to a passing action event.

Integrating the action density $\rho(x,t)$ over a large spacetime region ($\Delta t \gg \tau$ and $\Delta x \gg \lambda$) computes the net effect of all actions within that region:

\[P(x, t) = \int_{t-\frac{\Delta t}{2}}^{t+\frac{\Delta t}{2}} \int_{x-\frac{\Delta x}{2}}^{x+\frac{\Delta x}{2}} \rho(x', t') \, \mathrm{d}x' \, \mathrm{d}t'.\]

To illustrate it, consider two observable actions at different resolutions:

• $h_i$: an individual ant moves across a fixed reference point;

• $h_c$: a group of ants transport a large load across the same point.

Each action is characterized by a pair of spatiotemporal parameters. For the individual action $h_i$, these are the mean step duration $\tau_i$ and the step length $\lambda_i$. For the collective action $h_c$, they are the mean transport duration $\tau_c$ and the spatial extent of the load $\lambda_c$. These parameters satisfy the relation: $\tau_c \gg \tau_i$ and $\lambda_c \gg \lambda_i$

At fine resolution ($\Delta t \sim \tau_i$ and $\Delta x \sim \lambda_i$), the system is resolved into discrete step motions of individual ants. At coarser resolution ($\Delta t \sim \tau_c$ and $\Delta x \sim \lambda_c$), the system is resolved into the collective action, where uncorrelated individual actions average out and coordinated actions give rise to the smooth transport of the load.

The microscopic dynamics of the system are captured by the density of the individual action $h_i$, denoted $\rho_i(x,t)$, while the macroscopic dynamics are described by the density of the collective action $h_c$, denoted $\rho_c(x,t)$. These two descriptions are related through spatiotemporal coarse-graining:

\[\rho_c(x,t) = \frac{1}{\lambda_c \tau_c } \int_{t-\frac{\tau_c}{2}}^{t+\frac{\tau_c}{2}} \int_{x-\frac{\lambda_c}{2}}^{x+\frac{\lambda_c}{2}} \rho_i(x',t') \, \mathrm{d}x' \, \mathrm{d}t'.\]

The normalization factor $(\lambda_c \tau_c)^{-1}$ ensures that $\rho(x,t)$ retains the probabilistic interpretation of action density. The coarse-graining procedure averages out stochastic motions at the individual level while preserving coherent, large-scale patterns that emerge at the system level.

Observable Signals

The propagation of an action event along a physical trajectory produces distinct energy and momentum signatures. The observable signals are related to the action density $\rho(x,t)$:

\[E(x,t) = \frac{h}{\tau} \rho(x,t),\] \[\vec{p}(x,t) = \frac{h}{\lambda} \hat{\mathbf{v}}_x \rho(x,t),\]

where $h$ denotes the unit action, $\tau$ its duration, and $\lambda$ its step length. The unit vector $\hat{\mathbf{v}}_x$ specifies the direction of unit displacement at position $x$.

If the moving entity—whether an ant or a transported load—carries an electric charge $n^{\pm}$, the action signal manifests as an electric current:

\[I(x,t) = n^{\pm} \, \rho(x,t).\]

More generally, any physical signal observable along an action path can be derived from the corresponding action density:

\[\mathcal{O}(x,t) = \hat{o} \, \rho(x,t),\]

where $\hat{o}$ denotes the unit observable associated with the action $h$. Examples include:

• the amount of heat generated per step,
• the number of oxygen molecules consumed,
• the number of carbon dioxide molecules produced,
• or any other quantifiable effect resulting from the execution of a single action.

Note that $\rho(x,t)$ is defined as the expected number of observable (net) actions per unit spacetime, rather than as the density of individual action agents. An action path can thus be interpreted either as the trajectory of a single ant or as the collective path of many ants, with individuals stochastically joining or leaving the path at any time. The action density links microscopic events to macroscopic observables, providing a consistent description of the system’s dynamics across spatial and temporal scales.

Quantum Mechanics Reimagined (ant = kinesin/step motor = action agent)