Physical Coherence and Time's Emergence

This paper introduces the criterion of physical coherence to evaluate and challenge quantum gravity programs that attempt to derive time from a timeless fundamental level, specifically testing semiclassical and thermal time approaches against metaphysical incoherence arguments while setting a standard for their conceptual success.

The Gist

The Big Problem: Time Vanishes in the Deep Universe

Imagine you are looking at a map of the universe. At the level of everyday life, the map has roads, cities, and a timeline showing when things happen. But in the deepest theory of physics (Quantum Gravity), the map changes. When you zoom in all the way to the very bottom layer of reality, the concept of "time" disappears. There is no "before," no "after," and no "next." The fundamental equations of the universe look like a frozen photograph where nothing ever moves.

This creates a huge puzzle: If the universe is fundamentally frozen, how do we get the flowing, moving time we experience every day?

The "Middle Way" Solution

Scientists generally have three ways to handle this:

1. Fundamentalists: Say time is still there at the bottom, just hidden.
2. Eliminativists: Say time is an illusion and doesn't exist at all.
3. The Middle Way (The focus of this paper): Say time doesn't exist at the bottom, but it emerges (appears) at the top level, like steam rising from a pot of water. The water molecules aren't "wet" or "steamy" individually, but together they create wetness and steam.

The author, Eugene Chua, is interested in the Middle Way. He wants to know if the stories physicists tell to explain how time "emerges" from a timeless universe actually make sense.

The New Test: "Physical Coherence"

Chua introduces a new test called Physical Coherence.

Think of it like building a bridge.

• Metaphysical Coherence asks: "Does the concept of a bridge make sense in the universe of philosophy?" (This is a very abstract, "all-or-nothing" question).
• Physical Coherence asks: "Did the engineers use the right tools to build this specific bridge?"

Chua argues that even if a theory is mathematically perfect, it fails the Physical Coherence test if the tools used to build it secretly require time to work.

The Analogy: Imagine you are trying to bake a cake using only ingredients that don't include flour. You claim you can make a cake. But, to mix the batter, you use a blender. If the blender only works if you plug it into a power outlet that requires electricity (which you claimed didn't exist), your cake recipe is physically incoherent. You can't use a tool that needs electricity to prove electricity doesn't exist.

Chua's rule is simple: You cannot use a tool that secretly assumes time exists to prove that time emerges from a world where time doesn't exist. That would be circular reasoning.

Testing Two Popular Recipes

Chua applies this test to two famous "Middle Way" recipes used by physicists.

1. The Semiclassical Recipe (The "Clock" Approach)

The Idea: Physicists try to use gravity (the shape of space) as a clock to measure time for matter.
The Problem: To make this work, they use a tool called Decoherence.

• The Analogy: Imagine you have a messy room (a quantum superposition) and you want to tidy it up so you can see a clear path (time). You use a vacuum cleaner (Decoherence) to suck up the mess.
• The Catch: In standard physics, a vacuum cleaner only works if you push it back and forth over time. It's a dynamic process.
• The Verdict: Chua argues that if the universe is fundamentally frozen (no time), you can't push the vacuum cleaner. You can't use a time-dependent tool to create time. Therefore, this recipe currently fails the Physical Coherence test.

2. The Thermal Recipe (The "Heat" Approach)

The Idea: Physicists try to say time emerges from Thermodynamics (heat and equilibrium). If a system is in "thermal equilibrium," we can define a "thermal time."
The Problem: To use this, they need to define Equilibrium.

• The Analogy: Imagine a cup of coffee. It is in equilibrium when it stops changing temperature. But "stopping to change" implies that time was passing for it to change in the first place.
• The Catch: Standard definitions of equilibrium rely on things settling down over time. If the universe is frozen, nothing can "settle down."
• The Verdict: Chua argues that defining equilibrium without time is like trying to describe a "still photo" of a race without mentioning the runners moving. The current definitions of equilibrium secretly assume time exists, making this recipe physically incoherent as well.

Why This Matters

Chua isn't saying these theories are definitely wrong or that time can't emerge. He is saying: "Stop and check your tools."

• If a physicist uses a tool that secretly needs time, they haven't actually explained how time comes from nothing; they've just smuggled time back in through the back door.
• This test forces scientists to be honest. They must either:
  1. Find a new way to use these tools that doesn't secretly rely on time.
  2. Admit that their current explanation is incomplete.

The Conclusion

The paper is a challenge to the physics community. It says: "Don't just show us the math; show us the physical story."

If you want to explain how a river (time) flows from a dry desert (timelessness), you can't just say, "Look, here is water!" if the only way you got the water was by using a hose connected to a tap that you claimed didn't exist. You have to explain how the water appeared without the tap.

Chua's Physical Coherence is the litmus test to see if these "Middle Way" theories are truly building time from nothing, or if they are just pretending to do so while secretly using time as a crutch.

Technical Summary

Problem
The paper addresses the "problem of time" in quantum gravity, specifically the challenge of recovering the emergence of time from a fundamentally timeless ontology. In canonical quantum gravity (CQG), the imposition of constraints leads to the Wheeler-DeWitt equation ($\hat{H}|\Psi\rangle = 0$), which lacks a time parameter and dynamical evolution. While "eliminativists" argue time is illusory and "fundamentalists" argue it is hidden but fundamental, the "middle way" posits that time does not exist fundamentally but emerges at a non-fundamental level.

Recent metaphysical critiques (e.g., Baron, Miller, and Tallant 2022) argue that the middle way is metaphysically incoherent because standard accounts of emergence (composition, realization) rely on spatiotemporal relations that do not exist in a fundamentally non-spatiotemporal universe. The author argues that these metaphysical objections are external to the physical project and often rule out all emergence claims "from on high" without engaging with the specific technical details of quantum gravity programs. The paper seeks to determine if these programs are physically coherent on their own terms, independent of external metaphysical definitions.

Methodology
The author develops a new criterion termed physical coherence. This criterion is distinct from metaphysical coherence and is designed to be sensitive to the specific formal tools and physical interpretations used within individual quantum gravity programs.

The methodology involves a three-part "recipe" analysis for any program attempting to derive time from no-time:

1. Formalism: (a) A fundamental timeless mathematical theory; (b) The introduction of formal tools (approximations, ansätze, idealizations); (c) A resulting non-fundamental temporal mathematical theory.
2. Interpretation: (a') The fundamental theory is interpreted as timeless; (b') The formal tools are given a physical interpretation; (c') The resulting theory is interpreted as temporal.

Physical Coherence is satisfied only if:

• The formal derivation (a) $\to$ (b) $\to$ (c) is mathematically consistent.
• The interpretive chain (a') $\to$ (b') $\to$ (c') is consistent. Crucially, the physical interpretation of the formal tools (b') must be justifiable within the context of the fundamentally timeless ontology (a') without implicitly or explicitly assuming the existence of time or dynamics that the theory aims to derive. If the tools require time to be applied or interpreted, the program suffers from physical incoherence (circularity).

The paper applies this criterion to two specific programs: Semiclassical Time and Thermal Time, and briefly uses Group Field Theory (GFT) as a warm-up example.

Key Contributions and Results

1. Group Field Theory (GFT) as a Case Study:
   The paper analyzes GFT, where spacetime is argued to emerge from non-spatiotemporal group elements via phase transitions (condensate states). Citing Mozota Frauca (2023a), the author notes that standard interpretations of phase transitions rely on thermodynamic variables (temperature, pressure) changing over time. In a fundamentally timeless ontology, there is no time for these transitions to occur. Thus, interpreting the GFT phase transition as a genuine physical process risks circularity, suggesting GFT may currently fail the test of physical coherence.

2. Semiclassical Time:
   This program (e.g., Kiefer 2007) attempts to derive a Schrödinger-like equation for matter fields by treating gravity as a clock, using the Born-Oppenheimer and WKB approximations and decoherence.

   • The Decoherence Problem: The author argues that the standard physical interpretation of decoherence (either via environmental interaction or decoherent histories) is inherently dynamical. It requires a temporal evolution (e.g., the decay of interference terms $\langle E_i(t)|E_j(t)\rangle$) to suppress interference and select a branch.
   • The Tension: Kiefer's own account suggests time arises from decoherence, yet decoherence is required to derive the time-dependent equation. The author contends that without a new, non-dynamical interpretation of decoherence, the program is physically incoherent because it assumes the very time it seeks to explain to justify the application of its formal tools.
   • Response to Huggett and Thébault (2025): The paper addresses recent defenses of semiclassical time which claim the Born-Oppenheimer approximation is a purely formal technique not requiring time. The author counters that while the math may be atemporal, the physical interpretation of the approximation (e.g., treating a "heavy" subsystem as fixed relative to a "light" one) implicitly relies on dynamical notions of relative motion or energy separation. Without a physical story that avoids time, the formal validity does not guarantee physical coherence.

3. Thermal Time Hypothesis (TTH):
   This program (Connes and Rovelli 1994) derives time from thermodynamic equilibrium states (KMS states) in a timeless universe.

   • The Equilibrium Problem: The author argues that defining thermal equilibrium (and thus KMS states) standardly requires time. Equilibrium is defined as a state where properties are time-independent and the system has evolved to a stationary state.
   • Critique of Timeless Definitions: The paper examines Rovelli's attempt to define equilibrium via the factorizability of statistical states (subsystems in relative equilibrium). The author (citing Chua 2025) demonstrates that this definition is physically inadequate: it is too stringent (failing for systems with "Maxwell's demon" partitions) and insufficient (failing to distinguish between non-interacting systems at different temperatures).
   • Result: To make the definition work, one must appeal to background dynamics (interaction over time) to justify the factorization. Consequently, TTH appears physically incoherent because it requires time to define the equilibrium states necessary to define time.

Significance and Claims
The paper claims that physical coherence serves as a necessary condition for the successful emergence of time, distinct from and complementary to metaphysical coherence.

• Operability: Unlike metaphysical critiques that rule out all emergence "from on high," physical coherence acts as a "litmus test" for specific programs. It allows for the possibility that some programs might succeed if they can provide non-circular physical interpretations of their formal tools.
• Diagnostic Power: The analysis pinpoints specific, implicit temporal assumptions in current leading programs (decoherence in semiclassical time; equilibrium in thermal time) that render them physically incoherent under standard interpretations.
• Future Direction: The paper does not claim these programs are definitively dead. Rather, it challenges proponents to clarify the conceptual foundations of their programs. If proponents can provide a new, non-temporal physical interpretation for the formal tools (e.g., a "primitive" decoherence or a non-dynamical definition of equilibrium), the programs may pass the test. If not, the programs have not yet earned the conclusion that time emerges from no-time.

The author emphasizes that establishing physical coherence is non-trivial work for philosophers of physics and physicists, requiring a deep engagement with the technical details of quantum gravity rather than broad metaphysical stipulations.