Superluminal Transformations and Indeterminism

This paper presents a theory-independent no-go theorem demonstrating that any framework incorporating superluminal transformations must abandon at least one fundamental assumption—such as finite information, time-symmetric content, past memory, or preferred causal ordering—thereby implying that ontic indeterminacy arising from such transformations stems from unbounded informational content rather than finite information.

The Gist

The Big Picture: A Clash of Rules

Imagine the universe is a giant, complex storybook. Scientists have long debated how this story is written.

1. The Quantum View: The story is written with a "magic pen" that doesn't decide what happens next until you turn the page. The future is genuinely open and random.
2. The Classical View: The story was written all at once, from start to finish. If you knew the position of every single word perfectly, you could predict the entire book. The only reason we can't predict it is that we don't have enough information.

This paper asks a tricky question: What happens if we try to mix these two views with a third idea—traveling faster than light?

The authors, Amrapali Sen and Flavio Del Santo, argue that you cannot have a universe that:

• Has a limited amount of information (finite information).
• Has a clear past that is already "written" and a future that is still "open."
• Allows for faster-than-light travel (superluminal transformations).

They prove that if you try to combine all three, the storybook breaks. You have to give up one of the rules.

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Key Concepts Explained with Analogies

1. The "Infinite Precision" Trap (Classical Determinism)

In traditional physics, we assume numbers are like infinite rulers. You can measure a distance as 1.0, 1.01, 1.011, 1.0111... forever.

• The Problem: To write down a number with infinite digits, you need infinite information.
• The Paper's View: The authors suggest that in the real world, we don't have infinite rulers. We have finite rulers. We can only measure so precisely.
• The Result: Because our rulers are finite, the "future" isn't fully written yet. It's like a story where the author hasn't decided the next word. This creates indeterminism (randomness) even in classical physics, not just quantum physics.

2. The "Time-Traveling" Observer (Superluminal Transformations)

Imagine you are watching a movie.

• Normal Observer: You watch the movie from the beginning to the end. The past is fixed (you've seen it), and the future is unwritten.
• Superluminal Observer: Imagine a character who can run so fast they can jump to a different part of the movie reel instantly.
• The Twist: In physics, if you move fast enough (faster than light), your "now" changes. What looks like the "future" to you might look like the "past" to someone else.
• The Paper's Claim: Dragan and Ekert (previous researchers) suggested that this "time-jumping" ability is actually why the universe is random. They thought, "If you can jump around time, you can't know what happens next, so the universe is random."

3. The "No-Go" Theorem (The Impossible Triangle)

The authors set up a logical puzzle with four rules. They proved you can't have all four at once:

• Rule A (Finite Information): The universe has a limited storage capacity. You can't store infinite data (like infinite decimal numbers).
• Rule B (Time Symmetry): The universe is balanced. The amount of information needed to describe the past is roughly the same as the future.
• Rule C (Memory): The past is a solid record. Once something happens, it's "actualized" (it's a 1 or a 0). The future is a list of possibilities (a 0.5, or maybe a 0.3).
• Rule D (Faster-than-Light): You can have observers moving faster than light who swap space and time.

The Conflict:
Imagine a library (the universe) with a limited number of books (Finite Information).

• In a normal world, the Past Section is full of finished, bound books (Memory). The Future Section is full of blank pages waiting to be written.
• Now, imagine a "Time-Jumper" (Superluminal Observer) who runs into the library and swaps the Past and Future sections.
• Suddenly, the "Future" (blank pages) is in the Past section, and the "Past" (finished books) is in the Future section.
• The Problem: If the Past section now contains blank pages (possibilities), it requires more information to describe them than finished books do. But the library has a fixed limit on how many books it can hold.
• The Conclusion: The library runs out of space. The math breaks.

What Does This Mean?

The paper concludes that if faster-than-light travel is real, then our current understanding of "finite information" and "memory" must be wrong.

The Two Options Left:

1. Give up Finite Information: The universe must actually contain infinite information. This means the "rulers" are infinite, and the universe is actually deterministic (like a classical theory with real numbers). In this case, the randomness we see is just an illusion because we can't read the infinite details.
2. Give up the "Past is Fixed" idea: If the universe has finite information, then the "Past" cannot be a solid record. The past would have to be just as "fuzzy" and full of possibilities as the future. This would mean we have no true memory of what happened, which contradicts our everyday experience.

The Bottom Line

The paper argues that you cannot have a universe that is both finite (limited information) and allows for faster-than-light travel while keeping a clear distinction between a fixed past and an open future.

If you want faster-than-light travel, you likely have to accept that the universe is actually a giant, deterministic machine with infinite information hidden inside it, and the "randomness" we see is just a result of us not being able to see the whole picture.

In short: The paper doesn't prove faster-than-light travel is real. Instead, it says, "If you believe in faster-than-light travel, you probably have to believe the universe is actually deterministic and infinite, not random and finite."

Technical Summary

Technical Summary: Superluminal Transformations and Indeterminism

Problem Statement
The paper addresses a conceptual tension between two distinct sources of indeterminism in physics. On one hand, recent work (e.g., Gisin and Del Santo) suggests that classical physics can exhibit ontic indeterminacy if the assumption of infinite information (real-number representations) is abandoned in favor of finite information density. On the other hand, Dragan and Ekert have argued that extending Lorentz symmetry to include superluminal transformations (SpTs) provides a relativistic origin for quantum-like indeterminism. The central problem is to determine whether the indeterminism arising from finite information is compatible with the existence of non-order-preserving superluminal transformations. Specifically, the authors investigate whether a framework can simultaneously accommodate SpTs, finite information, and a consistent causal/temporal structure.

Methodology
The authors employ a theory-independent, structural approach using a "toy framework" rather than a specific physical model. They do not claim that superluminal theories describe physical reality but use them to explore the logical consistency of foundational principles.

The methodology involves:

1. Formalizing Indeterminism: Utilizing the concept of propensities (objective causal weights) to describe indeterministic causality. This framework distinguishes between "potential" causal connections (non-extremal propensities) and "actual" connections (extremal propensities of 0 or 1).
2. Information-Theoretic Modeling: Representing the state of a system as a causal multigraph $G(t)$ containing events (nodes) and causal links (edges) weighted by propensities. The information content $I(G(t))$ is quantified using Kolmogorov complexity, assuming a principle of finite information density (finite bits in finite regions).
3. Assumption Analysis: The authors formulate a set of four natural assumptions (A0–A4) and test their mutual consistency:
   • A0: Existence of non-order-preserving Superluminal Transformations (SpTs) that map timelike-separated events to spacelike-separated ones (reversing temporal order).
   • A1: Principle of finite information (total information $N$ is finite).
   • A2: Time symmetry of total information (the information capacity allocated to the past and future is approximately equal, $I_{past} \approx I_{future}$).
   • A3: The past holds memory (past events are actualized with extremal propensities $\{0, 1\}$, requiring fewer bits to encode than future potentialities).
   • A4: Time has a specific physical role (defining causal structure and the direction of actualization).

Key Contributions and Results
The primary contribution is a no-go theorem demonstrating that assumptions A0 through A4 cannot hold simultaneously.

• The Conflict: Under A1 (finite information) and A3 (past memory), the information required to encode the past is strictly less than that required for the future ($I_{past} < I_{future}$) because past propensities are extremal (1 bit) while future propensities are rational numbers requiring more bits. To satisfy A2 (time symmetry of total information), the information content of the events themselves ($I(E)$) would need to compensate, implying $I(E)_{past} > I(E)_{future}$.
• The Contradiction: Assumption A0 (non-order-preserving SpTs) implies that a superluminal boost can swap the roles of space and time coordinates. In such a transformed frame, the distinction between "past" and "future" based on temporal ordering is altered. If the transformation preserves the event structure (A4) but swaps the temporal ordering, the information distribution derived from A3 and A2 becomes inconsistent. Specifically, in the superluminal frame, the "past" and "future" information contents would appear symmetric ($I'_{past} \approx I'_{future}$), violating the asymmetry required by A3 (memory) or the finite bound of A1.
• The Conclusion: The conjunction of SpTs, finite information, time-symmetric information capacity, a determinate past, and a preferred causal time direction is logically impossible.

Interpretational Implications
The paper analyzes the consequences of relaxing each assumption to resolve the contradiction:

• Relaxing A0: If SpTs do not exist, the framework reduces to standard relativistic physics, where A1–A4 are consistent.
• Relaxing A1: If finite information is abandoned (allowing infinite information/real numbers), the universe could accommodate SpTs. This implies that any indeterminism associated with SpTs would not stem from finite information but would instead align with the ontology of deterministic classical theories based on real numbers.
• Relaxing A2 or A3: One could abandon the symmetry of total information or the notion that the past holds a determinate memory. This would imply a world where the past is as indeterminate as the future, or where the arrow of time lacks a fundamental informational asymmetry.
• Relaxing A4: One could deny that time has a specific physical role distinct from space, challenging the concept of a causal arrow of time.

Significance and Claims
The authors claim that their result clarifies the conceptual structure of indeterministic theories. They argue that the indeterminacy suggested by Dragan and Ekert to arise from SpTs cannot originate from a finite-information framework. Instead, if SpTs exist, they suggest an ontology with unbounded informational content (akin to real-number-based classical determinism).

The paper modestly positions itself as a conceptual tool rather than a proposal for new physics. It does not claim that superluminal phenomena are real but uses them to test the internal consistency of physical principles. The significance lies in highlighting the mutual tension between formal symmetries (like SpTs) and physical plausibility grounded in finite information and a determinate past. The authors suggest that this tension forces a choice: either SpTs are excluded from physical reality, or they are only permissible in a universe with infinite information, thereby decoupling SpT-induced indeterminism from the finite-information indeterminism found in modern classical interpretations.