Indefinite probabilities in quantum spacetime: A deepening of unpredictability

This paper demonstrates that employing the $SU_q(2)$ quantum group to model rotational symmetry in spin-$\frac{1}{2}$ systems leads to non-commuting probability operators and an associated uncertainty principle, thereby establishing a framework of "indefinite probabilities" that fundamentally prevents observers from sharply measuring their relative orientation.

The Gist

Imagine you are trying to figure out how two people are standing relative to each other in a room. In our everyday world, if Alice sends Bob a bunch of arrows (representing spinning particles) pointing in specific directions, and Bob measures them with his own set of arrows, he can calculate a precise "rotation map" to tell exactly how his room is turned compared to Alice's. In this classical world, the "probability" of an arrow pointing up or down is just a number, like 50% or 75%. It's a fixed, predictable statistic.

This paper suggests that if we look at the universe through the lens of quantum gravity (the theory of how space and time work at the tiniest possible scales), this simple picture breaks down. The authors, Vittorio D'Esposito, Giuseppe Fabiano, and Domenico Frattulillo, propose a radical new idea: Probabilities themselves can be fuzzy and undefined.

Here is the breakdown of their discovery using simple analogies:

1. The "Fuzzy" Compass

In standard quantum mechanics, a particle (like an electron) doesn't have a definite position until you measure it. But once you measure it, you get a result, and the chance of getting that result is a solid number.

The authors argue that in a "quantum spacetime," even the chance (the probability) isn't a solid number. Instead, the probability is like a fuzzy compass needle.

• Normal World: If you ask, "What is the chance of heads?" the answer is a fixed number, like 0.5.
• Quantum Spacetime: The answer isn't a number; it's a "quantum object" that can be in a superposition of different chances. It's as if the coin flip itself hasn't decided how likely it is to land on heads until you measure the likelihood.

2. The "Braided" Messengers

To make this math work, the authors use a concept called braiding. Imagine you have two people, Alice and Bob, trying to talk to each other.

• In a normal room, if Alice speaks and Bob listens, their voices don't interfere with each other's ability to speak.
• In this quantum world, the "messengers" (the particles) and the "receivers" (the measuring devices) are so deeply intertwined that they are braided together, like two strands of hair twisted into a braid.

Because of this braiding, the rules of the game change. If Alice tries to measure the probability of a particle spinning "up," and Bob tries to measure the probability of it spinning "right," these two measurements cannot be done at the same time with perfect precision.

3. The "Unmeasurable" Angle

The paper's biggest punchline is about relative orientation.

• The Goal: Alice and Bob want to know exactly how much Bob's room is rotated compared to Alice's.
• The Problem: To find this out, they need to measure the probabilities of spin outcomes.
• The Result: Because the probabilities are "indefinite" (they don't have fixed values simultaneously), Alice and Bob cannot ever determine their relative angle with perfect precision.

It's like trying to measure the angle between two rulers, but the rulers themselves are made of mist. No matter how good your eyes are or how many times you look, you can never get a sharp, exact number for the angle. The "angle" itself becomes a fuzzy, quantum thing.

4. Why This Matters (According to the Paper)

The authors call this "Doubly Quantum Mechanics."

• First Level of Uncertainty: In normal quantum mechanics, you can't predict what the result of a measurement will be (e.g., will the coin be heads or tails?).
• Second Level of Uncertainty: In this new framework, you can't even predict what the odds are of getting heads or tails.

They argue that this isn't just a limitation of our technology; it is a fundamental feature of the universe. Even if you had infinite time, infinite money, and perfect instruments, you still couldn't pin down the probabilities or the relative orientation of two observers. The "fuzziness" is baked into the fabric of space and time itself.

Summary

Think of the universe as a giant, complex dance.

• Classical Physics: The dancers move on a solid floor. You can predict exactly where they will be.
• Standard Quantum Physics: The dancers are on a foggy floor. You can't see exactly where they are, but you know the rules of their movement (the probabilities).
• This Paper's "Quantum Spacetime": The floor itself is made of fog. Not only can't you see the dancers, but the rules of the dance (the probabilities) are also shifting and undefined. You can't even agree on the "odds" of the next step, making the relative position of the dancers fundamentally unknowable.

The paper concludes that this "indefinite probability" is a natural consequence of treating space and time as quantum objects, fundamentally deepening the unpredictability of reality.

Technical Summary

Technical Summary: Indefinite Probabilities in Quantum Spacetime

Problem Statement
The paper addresses a foundational gap in the interface between quantum theory and gravity. While general relativity dictates that physical content resides solely in relational quantities (e.g., the relative orientation between observers), standard quantum mechanics treats these relational geometrical parameters as classical, commuting numbers. The authors posit that in a regime where quantum gravity effects are significant, geometrical quantities (such as spatial directions) acquire non-classical features, such as non-commutativity. Consequently, the probabilities of measurement outcomes, which are derived from these geometrical relations, should themselves exhibit non-classical behavior. The central problem is to determine how the transition from classical to quantum spacetime affects the nature of probability itself, specifically whether probabilities can be assigned definite values simultaneously or if they become "indefinite."

Methodology
The authors employ the framework of Doubly Quantum Mechanics (DQM), a deformation of quantum theory where both the phase space of physical systems and their geometrical configurations are described by quantum degrees of freedom. The specific methodology involves:

1. Quantum Group Deformation: The standard rotational symmetry group $SU(2)$ is promoted to its quantum deformation, $SU_q(2)$. In this framework, the group parameters are non-commutative operators rather than complex numbers. The deformation parameter $q$ is treated as a function of the Planck length and cosmological constant, with $q \to 1$ recovering the classical limit.
2. Operator-Valued Probabilities: Spin states and Stern-Gerlach apparatus orientations are described by $q$-spinors (operator-valued vectors). Probabilities of spin measurement outcomes are constructed as expectation values of projectors acting on a Hilbert space of "geometry states" ($H_A$). This promotes probability from a real number to a self-adjoint operator ($P$) acting on $H_A$.
3. Braiding Formalism: To ensure the full covariance of the theory under $SU_q(2)$ transformations, the authors introduce braiding relations. This is a crucial step where cross-commutation relations between distinct copies of the quantum group (representing different spin beams or measurement apparatuses) are fixed. Without braiding, the commutation relations would not be preserved under rotation. The braiding relations (involving the $R$-matrix) ensure that operators associated with different measurement setups do not commute.
4. Perturbative Analysis: The commutator between two distinct probability operators is calculated. The analysis is expanded to the leading order in the deformation parameter $(1-q)$, assuming $q$ is close to 1, to derive a tractable uncertainty principle.

Key Contributions and Results

• Indefinite Probabilities: The primary result is the demonstration that probability operators associated with different measurement setups (e.g., measuring spin along different axes or using different apparatuses) do not commute. Specifically, the commutator $[P_i(\uparrow), P_j(\uparrow)]$ is non-zero.
• Uncertainty Principle for Probabilities: The non-commutativity leads to a fundamental uncertainty principle between probability measurements. For two probability operators $P_i$ and $P_j$, the uncertainty relation is derived as:
  $$\Delta P_i \Delta P_j \geq \frac{(1-q)}{4} \left| \langle (\mathbf{b}_{m_i} + \mathbf{b}_{m_j}) \cdot (\mathbf{b}_{n_i} \times \mathbf{b}_{n_j}) - (\mathbf{b}_{n_i} + \mathbf{b}_{n_j}) \cdot (\mathbf{b}_{m_i} \times \mathbf{b}_{m_j}) \rangle \right|$$
  where $\mathbf{b}_n$ and $\mathbf{b}_m$ are quantum direction operators corresponding to the spin beam and Stern-Gerlach apparatus, respectively. This implies that if one probability is sharply measured, the outcome of a subsequent measurement of another probability cannot be predicted with certainty.
• Non-Commutative Rotation Matrices: The paper applies this result to the communication protocol between two observers (Alice and Bob) attempting to determine their relative orientation. The rotation matrix elements $R_{ij}$, which relate the two frames, are expressed in terms of probability operators ($R_{ij} \approx 2P_{ij} - 1$). Due to the non-commutativity of the underlying probabilities, the elements of the rotation matrix do not commute.
• Fundamental Limit on Orientation Measurement: Consequently, two observers cannot sharply measure their relative orientation, even with infinite resources and precision. The relative orientation becomes a "genuinely quantum object" subject to intrinsic quantum fluctuations.

Significance and Claims
The paper claims to introduce the concept of indefinite probabilities for the first time in the literature. The significance lies in deepening the unpredictability of quantum theory:

• Standard quantum mechanics introduces indeterminacy in measurement outcomes (the result of a single trial).
• This work suggests that in quantum spacetime, indeterminacy extends to the probabilities of those outcomes themselves. Probabilities cannot be assigned definite values simultaneously for different measurement contexts.

The authors emphasize that this is a fundamental feature of the spacetime fabric, not a limitation of measurement devices or resources. It arises from the non-commutative structure of spacetime symmetries (via $SU_q(2)$) and the resulting non-commutativity of the geometry states. This result strengthens previous findings on quantum reference frames by showing that the very relational quantities used to define frames (relative orientation) are subject to quantum uncertainty. The paper concludes that this framework necessitates a revision of the understanding of quantum mechanics, where the "background" geometry is itself a quantum degree of freedom that is disturbed by the act of probability measurement.