Representational Time: Theories & Applications

• Representational time is the encoding, emergence, and manipulation of temporal information across diverse systems, from quantum mechanics to neural networks.
• Operator-based and relational approaches define time through mathematical constructs like commutation relations, time-POVMs, and emergent clock models in background-independent theories.
• In computational and neuroscientific contexts, techniques such as time-embeddings, dynamic alignment, and decoding of neural signals enable precise temporal representations for prediction and decision-making.

Representational time refers to the explicit or implicit encoding, emergence, manipulation, or inference of temporal relations in physical theories, neural systems, mathematical models, or artificial networks. Across foundational physics, cognition, neuroscience, and machine learning, “representational time” denotes the concrete or formal mechanism by which temporal information becomes accessible, is acted upon, or serves as an internal construct for prediction, decision-making, or computation.

1. Operator-Based and Quantum Conceptions of Representational Time

Quantum theory classically treats time as an external parameter, with observables represented as self-adjoint operators acting on a Hilbert space. Attempts to build a time operator $T$ conjugate to the Hamiltonian $H$ rely on the canonical commutation relation (CCR)

$[T, H] = i.$

For free particles, constructions such as the Aharonov–Bohm operator

$T = - \frac{1}{2m} (Q P^{-1} + P^{-1} Q)$

(where $Q$ and $P$ are the canonical position and momentum operators) yield operators that formally satisfy the CCR with $H = P^2/(2m)$. However, because the physical Hamiltonian $H$ is typically semi-bounded, Pauli's theorem prohibits the existence of self-adjoint $T$ satisfying the CCR globally. This motivates a taxonomy:

• Type (1) Time Operators: Self-adjoint, both $T$ and $H$ have spectrum $\mathbb{R}$; not viable for semi-bounded $H$.
• Type (2) "Weak Weyl" Operators: Merely symmetric, satisfy a restricted CCR; these serve as generalized “pre-observables”.
• Type (3) "Normal" Operators: Self-adjoint on dense subdomains (e.g., harmonic oscillator’s Galapon operator), potentially with bounded spectrum.

The exponentiated counterpart, $$\exp(itH) \exp(isT) = \exp(-ist) \exp(isT) \exp(itH)$$, underpins the standard notion of conjugate variables and, when satisfied in any form, signals representational time as an observable in quantum theory.

POVMs generalize the notion of observables, allowing symmetric non-self-adjoint operators to be interpreted operationally. For a time operator $T$, the first moment of a time-POVM $F(S)$,

$T = \int x\,dF(x),$

connects the mathematical object to measured outcomes, even if $T$'s spectrum is not entirely real—a crucial distinction when reconciling theoretical constructions with empirical evidence. From the standpoint of the Copenhagen interpretation, time splits into “external” (classical), “observation-based,” and “internal” (quantum) forms, with only type (B) and type (C) time being strictly representational within quantum mechanics. The time–energy uncertainty relation,

$\Delta T \; \Delta E \gtrsim \hbar,$

finds its rigorous home in this perspective, indicating the operational limitations inherent to time measurement (Fujimoto, 2014).

2. Temporal Relationalism and Emergence from Change

In background-independent theories such as General Relativity, time is not a fundamental parameter. Temporal Relationalism (TR) asserts that at the fundamental level, there is no time for the universe as a whole; time emerges as a secondary, derived label from the relational change of configurations: $$S = \sqrt{2}\int ds\,\sqrt{W}, \quad ds = \left( M(Q)dQ \cdot dQ \right)^{1/2}$$ with $W = E - V(Q)$. Time arises as “ephemeris” or “Jacobi” time,

$t^{\text{(em)}} = \int \frac{ds}{\sqrt{2W}}$

which is not inserted by hand but distilled from the totality of change. The primary quadratic constraint

$\mathcal{C} = \frac{||P||^2}{2} - W(Q) = 0$

is central, with its quantum version yielding the Wheeler–DeWitt equation, $\widehat{\mathcal{H}} \Psi = 0$, manifesting the “Frozen Formalism” problem: the apparent absence of time at the quantum level.

By replacing velocities with “changes” and reworking the standard Legendre-square to include a differential anti-Routhian,

$$dA(Q, P, dc) = dJ(Q, dQ, dc) - P \cdot dQ,\quad P := \frac{dJ}{dQ},$$

the formalism achieves strict reparametrization invariance and background independence. The Hamiltonian and Hamilton-Jacobi formalisms are adapted: $dH := P \cdot dQ - dJ,\qquad \frac{dH}{dQ} = -dP$ with emergent time read off as an internal clock abstracted from the constraint (sometimes referred to as Machian time) (Anderson, 2015, Anderson, 2015). This approach naturally incorporates and solves the interconnected facets of the "Problem of Time" in both classical and quantum settings.

3. Thermodynamic and Informational Approaches

An alternative to geometric or relational emergence is the thermodynamic/statistical view. In relaxation processes governed by a probability distribution,

$$P(x;t,\beta) = \sqrt{ \frac{\beta k}{2\pi(1-e^{-2\gamma t})} } \exp\left( - \frac{\beta k (x - x_0 e^{-\gamma t})^2}{2(1-e^{-2\gamma t})} \right),$$

with $\gamma = \beta k D$, both $t$ and inverse temperature $\beta$ are inferred parameters. Fisher information geometry is used to quantify their mutual regulation: $$g_{\mu\nu} = \int dx\; P(x;t, \beta) \big[ \partial_\mu \ln P \big] \big[ \partial_\nu \ln P \big]$$ with $(\mu, \nu) \in \{t, \beta\}$. The scalar curvature is universally $R = -1$, indicating a hyperbolic geometry on the $(t, \beta)$ parameter manifold. Here, representational time is not a primitive in the model but is instead a statistically inferred property that is informationally dual to temperature; its physical significance is emergent from the probabilistic evolution of the system (Tanaka, 2020).

The thermal time hypothesis (TTH) generalizes these ideas to quantum gravity: in a fundamentally timeless setting (e.g., the Wheeler–DeWitt equation $\hat{H}\Psi = 0$), time is derived from the modular automorphism group $\alpha_t^\omega$ associated with a faithful state $\omega$,

$\alpha_t^\omega(A) = \Delta^{it} A\,\Delta^{-it},$

where $\Delta$ is the modular operator. Time is thus defined as the modular parameter of the unique flow arising in an equilibrium (KMS) state. However, a significant controversy is the circularity in TTH: the identification of equilibrium and the physical relevance of modular flow appear to depend on an antecedent notion of dynamics, potentially undermining TTH’s standing as a foundational account of emergent time (Chua, 11 Jul 2024).

4. Neuroscientific and Cognitive Instantiations

Representational time in neural systems and human cognition manifests as the strategy by which time-varying information is encoded, transformed, or internally modeled:

• In human cognition, the “time compaction” hypothesis posits that temporal structure in dynamic scenes is internally represented as static spatial maps. Empirical studies show that exposure to predictive static scenes improves (or, for incongruent exposure, impairs) subsequent discrimination of dynamic situations, a process more prominent in men than women. Mathematical modeling of learning and memory in these tasks relies on exponential decay of recall probability:

  $P(\text{recall}) = e^{-a d}$

where $a$ is the recall decay rate and $d$ the delay. This reflects a strategy in which dynamic, temporal information is mapped to static spatial memory, enabling rapid, anticipatory decision-making (Villacorta-Atienza et al., 2018).

• At the population neural level, representational time is evident in cross-modality contrastive learning analyses. Neural embeddings are constructed that optimally decode instantaneous time labels (e.g., 33 ms frame windows in visual cortex), with representational drift quantified as a drop in decoding accuracy across sessions. Such drift is disproportionately detrimental for fast-varying features (e.g., optic flow) compared to slower features (e.g., scene texture), suggesting that distinct compensation mechanisms operate for different timescales. These studies reveal a neural substrate that is both temporally precise and flexibly robust under ongoing dynamical change (Wang et al., 2023).
• In the olfactory cortex, representational drift over days is modeled as a competition between spontaneous geometric mean–reverting synaptic fluctuations and rapid learning via STDP. This yields a balance in which memory remains robust to slow drift, with learning “anchoring” population codes to behaviorally relevant manifolds and defining a representational time over which the structure is stable or plastic (Morales et al., 18 Dec 2024).

5. Machine Learning Formulations: Time-Embeddings and Explicit Temporal Representation

Modern machine learning extends representational time to artificial systems through explicit time-embedding, temporal alignment, and sequence modeling:

• In self-supervised learning for time series (T-Rep), explicit time-embeddings $\tau_t$ are learned alongside spatial features, capturing trend, periodicity, and distribution shifts. These embeddings are constrained to a probabilistic simplex, i.e.,

  $$(\tau_t)_k = \frac{\sigma(h_\psi(t))_k}{\sum_{j=1}^K \sigma(h_\psi(t))_j},$$

with $\sigma$ the sigmoid nonlinearity, $h_\psi$ a learnable MLP, and $K$ the embedding dimension. Two pretext tasks—time-embedding divergence prediction (with Jensen–Shannon divergence) and time-conditioned forecasting—enforce that the latent space reflects temporal proximity and can regress to future (or past) representations given only the time index. This framework outperforms alternatives on classification, forecasting, and anomaly detection, particularly under missing data regimes. The learned representations remain smooth and interpretable over time, directly reflecting multi-scale temporal information (Fraikin et al., 2023).

• In spiking neural networks (SNNs), representational time is intrinsic to the temporal dynamics of discrete-time leaky integrate-and-fire (LIF) architectures. The time-to-spike (latency $T$) controls partitioning of the input space: a single neuron with $T$ time steps generates up to $O(T^2)$ parallel hyperplanes, leading to piecewise constant mappings where the number of input regions, and thus representational granularity, scales with latency. Theoretical bounds quantify network size needed for universal approximation:

  $$n_1 = \left\lceil \frac{\text{diam}_\infty(\Omega)\ \Gamma}{\varepsilon} \right\rceil + 1$$

where $\Gamma$ is the Lipschitz constant, $\varepsilon$ the accuracy, and $n_1$ the first hidden layer width. Experiments confirm that increased latency, rather than depth, is the principal axis for enhancing representational power in these networks (Nguyen et al., 23 May 2025).

• In representation learning for sequential data, e.g., global temporal alignment via differentiable dynamic time warping (DTW), temporal structure is enforced and exploited via alignment losses and cycle consistency, ensuring the latent space respects the global order of the process (e.g., actions in a video) (Hadji et al., 2021).

6. Topological and Geometric Invariants of Representational Time

Time-varying representational structure in high-dimensional biological or artificial systems can be robustly characterized through both geometric and topological invariants:

• Topological RSA (tRSA) applies nonlinear monotonic transforms to similarity matrices, constructing geo-topological matrices that compress noise-sensitive metric extremes while preserving local neighborhood topology:

  $$GT_{l,u}(d_{ij}) = \begin{cases} 0 & d_{ij} \leq l \ (d_{ij} - l) / (u - l) & l < d_{ij} < u \ 1 & d_{ij} \geq u \end{cases}$$

This approach achieves robust identification of neural regions and DNN layers, and outperforms purely geometric or topological metrics in practical settings.

• Temporal Topological Data Analysis (tTDA) extends these ideas to time, computing “RDM movies” via sliding windows and aligning them with Procrustes analysis (pMDS), allowing visualization and quantification of trajectory, divergence, and convergence of category-specific representations over time in neural data. Adaptive geo–topological dependence measures (AGTDM) further enable discovery of multivariate dependencies in noisy, high-dimensional temporal data.

These advances suggest that the “meaningful invariants” of representational time are often topological, rather than strictly geometric, across both biological and computational systems (Lin, 21 Aug 2024, Lin et al., 2019).

7. Philosophical, Epistemic, and Observer-Dependent Aspects

Representational time also encompasses the philosophical and observer-dependent dimension:

• In frameworks grounded in the observer, representational time splits into discrete, universal sequential time $n$ (which indexes all update events in potential knowledge) and relational time $t$ (which quantifies measurable temporal distances between events and is observer-relative). The formal structure includes evolution operators (e.g., $S(n+1) \subset u_1 S(n)$) and object identity conditions (e.g., $S_{O_O}(n) \cap S_{O_O}(n+1) \neq \emptyset$), with a continuous evolution parameter $\sigma$ bridging discrete subjective updates and continuous dynamics (Östborn, 3 Nov 2024).
• In special relativity and philosophy of time, the existence of observer-dependent “now” is emphasized: in eternalism, all times are equally real, but the temporal ordering of events is perspective-dependent, being dictated by the Lorentz transformation:

  $t = \gamma (t' + v x'/c^2)$

where simultaneity—and thus present "truth"—depends on the observer's inertial frame. Perspectival realism thus holds that what is “now” is determined by the choice of perspective, not by a cosmic absolute (Slavov, 2020).

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Representational time, as codified by these diverse approaches, is a multifaceted construct: operator-based and emergent in quantum foundations, relational and background-independent in gravitation, statistically inferred in non-equilibrium and informational models, topologically robust in neural and artificial systems, and observer-relative in epistemic and philosophical doctrines. Across contexts, representational time operationalizes the manner in which systems, theories, or agents encode, recover, or experience temporal order and dynamics, with implications for foundational physics, brain computation, adaptive artificial intelligence, and the structure of physical law.