When “Ω = 1” Shatters. The Challenge of Defining Probability "Without" an "All"

Abstract

It is clear is that the classical equation P(Ω)=1 cannot survive intact in a world where even Ω lacks a fixed meaning. The “ground truth” of probability must therefore be sought not in an external denominator, but in the internal consistency conditions that any rational, information-limited agent—or any self-consistent physical law—must obey when the very stage of events is in flux. As a toy example, we explore how in fluctuating space-time the ground truth of probability dissolves. A fully fledged stochastic general relativity—a measure on the space of metrics modulo diffeomorphisms—has scarcely been explored, but early efforts suggest it could either (i) illuminate how quantum gravity emerges, or (ii) be quantum gravity in disguise.

Introduction

In standard quantum mechanics, probability is well-defined via the Born rule, which assigns probabilities to measurement outcomes based on a system's wavefunction at a specific time. This framework assumes a fixed background space-time: time progresses linearly, and space-like hypersurfaces can be cleanly sliced to define system states. For example, in non-relativistic quantum mechanics or even special relativity, the directions of time-like and space-like separations are globally fixed, allowing us to specify configurations at a "time" and compute conditional probabilities for future states.

However, in general relativity, space-time is dynamical. The metric tensor, which encodes the geometry of space-time, determines which directions are time-like and which are space-like. If this metric fluctuates—due to quantum effects, stochastic processes, or other mechanisms—the distinction between time and space becomes ambiguous. Without a fixed temporal direction or a clear foliation into spatial slices, the foundational elements of probability crumble:

• State Definition: Probabilities require a system state at a given time, but if "time" is ill-defined, what does "state at a time" mean?
• Conditional Probabilities: Probabilistic theories often describe transitions from one state to another over time (e.g.,$P(\text{state at } t_2 | \text{state at } t_1)$), but without a consistent time axis, conditioning becomes problematic.
• Sample Space: Probability theory rests on a sample space of possible outcomes, yet in fluctuating space-time, the set of possible configurations or histories is unclear.
• Difficulty: "If [the metric tensor] is itself fluctuating, you don't know a priori which directions are the space directions and which directions are the time directions, so you can't even obviously phrase a probabilistic theory." This is the core epiphany—probability, as traditionally understood, may not be definable when the very structure enabling its definition (space-time) is uncertain.

Ground Truth for Probability: Foundations and Challenges

To address the "ground truth" of probability, we must first clarify its standard formulation in mathematics and physics, then examine how it falters in fluctuating space-time, and finally propose how it might be reimagined.

Standard Probability Theory

In mathematics, probability is formalized via Kolmogorov's axioms:

• Sample Space ($\Omega$): A set of all possible outcomes.
• Sigma-Algebra ($\mathcal{F}$): A collection of subsets of $\Omega$ (events) to which probabilities are assigned.
• Probability Measure (P): A function $P: \mathcal{F} \to [0,1]$ satisfying $P(\Omega) = 1$ and additivity for disjoint events.

In physics, this translates to:

• Statistical Mechanics: $\Omega$ is the set of microstates, and (P) is determined by an ensemble (e.g., microcanonical or canonical).
• Quantum Mechanics: $\Omega$ is the set of possible measurement outcomes, and (P) is given by the Born rule over a Hilbert space.

The "ground truth" here is the existence of a well-defined $\Omega$ —the "denominator," the total set of possibilities (the "1")—and a consistent way to assign probabilities to its subsets. This assumes a stable context, such as a fixed space-time background.

Breakdown in Fluctuating Space-Time

In a dynamical, fluctuating space-time, the ground truth unravels:

• No Clear Sample Space: Is $\Omega$ the set of all possible metric tensors? All space-time geometries? All histories? Without a fixed structure, $\Omega$ lacks a clear boundary. For instance, in general relativity, solutions to the Einstein field equations depend on initial conditions and matter distributions, but if the metric fluctuates stochastically, the set of "allowed" configurations becomes ambiguous.
• No Fixed Denominator: The "all," the total probability normalizing to 1, requires a complete enumeration of possibilities. In fluctuating space-time, this enumeration is undefined—do we include all topologies, all curvature states, all fluctuations? The absence of a stable "1" undermines normalization.
• Event Definition: Events (subsets of $\Omega$) rely on distinguishing "what happens when," but with no consistent time, events like "system state at (t)" lose meaning.

An intermediate step: a stochastic, non-quantum version of general relativity. But even here, defining $\Omega$ is non-trivial. A classical probabilistic ensemble of space-times might treat each geometry as an outcome, yet assigning a measure (P)—say, a uniform distribution—raises questions: Uniform over what? Metrics? Initial data? And how do global constraints (e.g., global hyperbolicity) restrict $\Omega$ ?

Reimagining Probability

To define probability without a fixed probability space, we must rethink its foundations:

1. Histories-Based Approach:
   • Instead of states at a time, consider probabilities over entire space-time histories, akin to the consistent histories interpretation of quantum mechanics. Here, $\Omega$ is the set of all possible four-dimensional geometries satisfying certain conditions (e.g., Einstein’s equations with stochastic terms).
   • Probability might be assigned via a measure over this space, perhaps inspired by path integrals (non-quantum approach). The challenge is specifying this measure without a fixed background.
2. Relational Probability:
   • Define probabilities relative to an observer or reference frame within the space-time. For example, a local foliation could temporarily fix space-like slices, allowing conditional probabilities relative to that perspective. However, this is observer-dependent and conflicts with general relativity’s covariance.
3. Non-Markovian Stochasticity:
   • Non-Markovian processes, where the future depends on the entire past, not just the current state. In fluctuating space-time, the metric’s evolution might depend on its full history, suggesting $\Omega$ as a space of trajectories with memory. Probability could emerge from statistical rules governing these trajectories, though defining them remains complex.
4. Coarse-Graining:
   • At large scales, where fluctuations average out, a quasi-fixed space-time might emerge, allowing traditional probabilities. The ground truth then lies in how these effective probabilities arise from a fluctuating substrate, though this sidesteps the fundamental issue.

None of these fully resolves the absence of a denominator. Conjecture—A "fully probabilistic generalization of general relativity" might resemble quantum gravity—suggests that probability’s ground truth may require a paradigm shift, possibly mimicking quantum interference via classical stochasticity (as in the indivisible stochastic approach).

Connection to Non-Ergodicity

Non-ergodicity arises when a system’s long-time behavior does not sample its entire phase space, so time averages differ from ensemble averages. In standard physics, ergodicity assumes a fixed time evolution over a well-defined $\Omega$ , enabling statistical predictions. In fluctuating space-time:

• No Global Time: Without a consistent time parameter, "long-time behavior" loses meaning, breaking the ergodic hypothesis.
• Restricted Exploration: If space-time fluctuations create inaccessible regions (e.g., due to topological constraints or singularities), the system may not explore all of $\Omega$, even if $\Omega$ were defined.
• Non-Markovian Dynamics: Link non-ergodicity to non-Markovian processes. In general relativity, the global nature of the theory (e.g., constraints across the entire manifold) suggests that local, step-by-step evolution (Markovian) may not apply, reinforcing probabilistic challenges.

For example, a stochastic general relativity might exhibit non-ergodic behavior if certain geometries dominate due to historical dependencies, not uniform sampling. This ties to the observation that even classical general relativity resists Markovian formulation, hinting that probability must account for global, non-local effects.

Implications and Speculation

This is problematic in semi-classical quantum gravity, where expectation values of quantum fields are treated as classical sources in Einstein’s equations. This conflates measurement-based probabilities with "things happening," a fallacy unless justified by a broader interpretation (e.g., many worlds). In fluctuating space-time, this practice is even less tenable, as the probabilistic framework itself is undefined.

A stochastic general relativity might serve as a stepping stone, testing whether classical probabilities over geometries can mimic quantum effects (e.g., via non-Markovian interference). The ground truth for probability may then shift from a static $\Omega$ to a dynamically evolving, relational, or history-based structure, though constructing this rigorously remains an open challenge.

Conclusion

In fluctuating space-time, the ground truth of probability—traditionally a fixed sample space and measure—dissolves due to the absence of a stable time and space framework. Without an "all" to normalize against, probability becomes elusive. Approaches like histories, relational definitions, or non-Markovian stochasticity offer paths forward, but each struggles to define a denominator without imposing artificial structure. Non-ergodicity exacerbates this, highlighting the need for a new probabilistic paradigm tailored to dynamical space-time—one that may bridge classical stochasticity and quantum gravity. This remains a profound, unresolved question at the frontier of theoretical physics.

Inspiration

Harvard Scientist: "There is No Quantum Multiverse" | Jacob Barandes

AI Reasoning. Grok

The query asks us to deeply explore the conceptual challenges of defining probabilities in a fluctuating space-time, as highlighted in the provided prompt, and to articulate the "ground truth" for the concept of probability in such a context. It also requests a connection to non-ergodicity. The core epiphany from the prompt is that when space-time itself is dynamical and fluctuating—as it is in general relativity or potential stochastic generalizations—traditional probabilistic frameworks falter because they rely on a fixed notion of time and space, which is absent in this scenario. Below, I present a detailed analysis addressing these points at a PhD level.

So I'm trying to understand the task deeply, especially the part about probabilities in the context of fluctuating space-time. The author seems to be saying that in quantum mechanics, particularly when dealing with space-time that isn't fixed, like in general relativity, it's hard to even define what probability means. Let me see if I can unpack this.

First, in standard quantum mechanics, we have a clear notion of time. We can talk about the state of a system at a particular time, and how it evolves from one time to another. Probabilities come into play when we make measurements; the Born rule tells us the probability of getting a certain outcome. But this all relies on having a fixed background space-time, where we know what "time" means and can slice the universe into space-like hypersurfaces.

Now, when space-time itself is dynamical, as in general relativity, this becomes problematic. In general relativity, the metric tensor determines the geometry of space-time, telling us which directions are space-like and which are time-like. But if the metric is fluctuating—say, due to quantum effects—then we don't have a fixed notion of time or space. So, how do we even define what a "state at a time" is? Without that, it's unclear how to set up a probabilistic framework, because probabilities typically involve conditioning on past events or states at earlier times.

The author mentions that in special relativity, even though space-time is relativistic, the directions of space and time are still fixed globally. But in general relativity, especially if the metric is fluctuating, this isn't the case. So, the challenge is to define probabilities in a context where the very structure of space-time is uncertain.

This ties into the idea of non-ergodicity, I think. Ergodicity is about how a system explores its possible states over time, and in physics, it's often assumed that time averages equal ensemble averages. But if time itself is fluctuating, or if the system doesn't have a well-defined time evolution, then ergodicity might not hold, and our usual probabilistic tools might not apply.

We can also point out that there's been little work on a stochastic version of general relativity, which is interesting. Before jumping to quantum gravity, maybe we should try to understand a probabilistic version of gravity first. That could be a stepping stone.

Let me try to think about what a stochastic version of general relativity might look like. In standard general relativity, the Einstein field equations deterministically relate the curvature of space-time to the distribution of matter and energy. A stochastic version might involve some randomness in how space-time curves in response to matter, or perhaps in the initial conditions.

But the author suggests that a fully probabilistic generalization might actually be related to quantum gravity. That's a bold idea. Maybe by incorporating probability in a deep way into general relativity, we could capture some quantum aspects without needing the full machinery of quantum mechanics.

Wait, but quantum mechanics is already probabilistic, so maybe there's a connection there. In the indivisible stochastic approach mentioned, it seems like they're trying to build quantum mechanics from a purely probabilistic foundation, without starting from Hilbert spaces. So perhaps a similar approach could be applied to gravity.

Another point is about expectation values in quantum mechanics. The author criticizes the common practice of treating expectation values as if they represent classical averages of things happening, when in fact, according to the standard interpretation, they only make sense in the context of measurements. This is a valid point; in the Copenhagen interpretation, quantum mechanics is fundamentally about measurement outcomes, not about what's "really happening" in between measurements.

But in quantum gravity, we often do things like take expectation values of quantum fields and plug them into the Einstein equations, as if they were classical sources. This is problematic because it mixes quantum and classical concepts in a way that might not be consistent.

So, to summarize, the core issue is that in a theory where space-time is dynamical and possibly fluctuating, it's hard to define a consistent probabilistic framework because the usual notions of time and space are not fixed. This makes it challenging to even formulate what we mean by probability in such a context.

For a PhD-level analysis, I think we need to delve into the foundations of probability theory itself. Typically, probability is defined on a sample space, with events being subsets of that space, and a probability measure assigning numbers to those events. In physics, the sample space often corresponds to possible configurations or histories of the system.

But in a fluctuating space-time, what is the sample space? Is it the set of all possible space-time geometries? Or is it something else? Moreover, how do we define conditional probabilities when the notion of "earlier" and "later" might not be well-defined globally?

Perhaps one way to approach this is through the concept of histories or paths in space-time. In the path integral formulation of quantum mechanics, we sum over all possible paths, weighted by their actions. Maybe in quantum gravity, we need to consider sums over all possible space-time geometries, but that's already what people do in approaches like causal dynamical triangulations or quantum Regge calculus.

However, the author seems to be suggesting something different: a purely probabilistic, non-quantum approach to fluctuating space-time. That is, not invoking quantum superpositions, but just classical probabilities over different space-time configurations.

But then, how would that differ from just having a statistical ensemble of classical space-times? And would that capture any quantum features? Probably not, unless the probabilities are set up in a very specific way, perhaps mimicking the interference patterns of quantum mechanics.

Wait, actually, in the indivisible stochastic approach, they do manage to reproduce quantum phenomena using classical probabilities, but with non-Markovian dynamics. So maybe something similar could be done for gravity.

Let me recall what non-Markovian means. In a Markov process, the future state depends only on the current state, not on the history. Non-Markovian processes have memory; the future depends on the past in a more complex way. In the context of space-time, perhaps the evolution of the geometry depends on its entire history, not just its current state.

But in general relativity, the Einstein equations are local in time; given the metric and its first derivative at a time, you can evolve it forward. However, in a fully covariant formulation, general relativity is about the entire space-time manifold satisfying certain conditions globally.

So maybe a stochastic version would involve probabilities over entire space-time histories, rather than step-by-step evolution in time. That could be a way to handle the fact that time itself is dynamical.

In fact, this reminds me of the consistent histories interpretation of quantum mechanics, where one considers probabilities for entire histories, not just for outcomes at a single time.

Perhaps in quantum gravity, we need a similar approach: probabilities for entire space-time geometries, with some consistency conditions to ensure that they make sense.

But the author mentions that even in classical general relativity, there are issues with formulating it as a Markov process, due to the global nature of the theory.

For example, in general relativity, the topology of space-time can be non-trivial, and there are global constraints like the Gauss-Bonnet theorem or the requirement that the space-time be globally hyperbolic. These global properties might not fit easily into a local, step-by-step probabilistic framework.

So, to define probabilities in such a context, we might need to consider the space of all possible space-time geometries that satisfy certain conditions, and assign probabilities to them. But what those conditions are, and how to assign the probabilities, is far from clear.

Moreover, in quantum gravity, we expect that space-time might not even be a smooth manifold at the smallest scales; there might be quantum fluctuations that make the concept of a classical space-time geometry inadequate.

But sticking to the classical case for now, let's think about what "ground truth" for probability means in this context.

In standard probability theory, the "ground truth" is the underlying sample space and the probability measure defined on it. But in physics, especially in theories like general relativity, the sample space isn't always clear.

For instance, in statistical mechanics, the sample space might be the set of all possible microstates of a system, and the probability measure comes from some ensemble, like the microcanonical or canonical ensemble.

In general relativity, if we want to think probabilistically, perhaps the sample space is the set of all possible solutions to the Einstein equations with certain boundary conditions or asymptotics.

But even then, defining a probability measure on this space is tricky. What would be a natural way to assign probabilities to different space-time geometries?

One might think of using a measure that's uniform in some sense, but what does "uniform" mean for space-time geometries? Alternatively, perhaps we can use a measure that's derived from a path integral, but that already brings in quantum ideas.

Wait, actually, in Euclidean quantum gravity, people do consider path integrals over metrics, but that's a quantum approach, not purely probabilistic.

So, for a purely classical probabilistic version, maybe we need to define a probability distribution over the space of metrics or over the space of initial data for the Einstein equations.

But even in the initial value formulation, there are constraints: the initial data must satisfy the constraint equations, which are part of the Einstein field equations.

So, the space of allowed initial data is not straightforward; it's a constrained submanifold of the space of all possible metrics and their derivatives.

Assigning a probability measure to such a space is non-trivial.

Furthermore, once we have initial data, the evolution is deterministic, so the probability would just be concentrated on the deterministic trajectory. But that's not what we want; we want a stochastic version where there's randomness in the evolution itself.

Alternatively, perhaps we can introduce noise into the Einstein equations, making them stochastic differential equations. For example, adding random terms to the stress-energy tensor or to the metric evolution.

But the author seems to be thinking of something more fundamental, not just adding small noise terms, but a fully probabilistic generalization.

I'm not sure what that would entail. Maybe it's about considering the metric as a random field, with some probability distribution over possible metric tensors.

In any case, I think the key point is that in standard probability theory, we need a well-defined sample space and a sigma-algebra of events, but in the context of fluctuating space-time, defining such a structure is challenging because the very notion of space and time is dynamic.

Perhaps one way to think about it is to use a relational approach, where we define probabilities relative to certain observers or reference frames within the space-time. For example, in relational quantum mechanics, probabilities are defined relative to an observer's perspective.

Similarly, maybe in a stochastic version of general relativity, probabilities could be defined relative to a particular slicing or foliation of space-time chosen by an observer.

But that seems observer-dependent, and we might want a more objective, observer-independent formulation.

Another idea is to use the concept of coarse-graining or effective theories. Maybe at large scales, where space-time is approximately fixed, we can define probabilities in the usual way, and then try to understand how these probabilities emerge from a more fundamental, fluctuating description.

But that might be too vague.

Let me try to connect this back to non-ergodicity. In non-ergodic systems, the long-time behavior doesn't explore the entire phase space, so time averages don't equal ensemble averages. In the context of space-time, if the dynamics are such that certain regions or configurations are not accessible, that could lead to non-ergodic behavior.

For example, in general relativity, there are space-times with closed time-like curves or other pathologies where the notion of a global time doesn't make sense. In such cases, defining probabilities might be particularly difficult.

Alternatively, perhaps in quantum gravity, the concept of probability needs to be generalized or replaced by something else, like in quantum measure theory or other exotic probability theories.

But I think the author's point is that even before getting to quantum gravity, there's a gap in our understanding: we don't have a good stochastic formulation of classical general relativity, and that might be a useful intermediate step.

So, to wrap up, the ground truth for probability in this context is elusive because the usual foundations—sample spaces, events, measures—rely on a fixed background structure, which is absent when space-time itself is fluctuating. We need new ways to think about probability that can accommodate dynamical space-time, perhaps by considering probabilities over entire histories or by finding invariant ways to define probabilistic concepts in a generally covariant theory.

This is a deep and challenging problem, and it's clear that much more work is needed to make progress on these foundational questions.