Why Time Cannot Flow Backward: Path Entropy and the Cost of Memory

The Deepest Secret of Time

Why does time only flow forward?

This is one of physics’ deepest mysteries.

You can go left or right, jump up or crouch down, but you cannot return to yesterday.

Space is symmetric (left-right, up-down, forward-backward all possible), but time is asymmetric—only one direction.

Even more bizarre: Physical laws themselves are time-symmetric.

• Newton’s equation: $F=ma$, still holds when video is reversed
• Maxwell’s equations: Light propagation, still solutions when reversed
• Schrödinger’s equation: $i\hslash {\partial }_t|\psi ⟩=H|\psi ⟩$, still holds when $t\to -t$

If physical laws don’t care about time direction, why does reality have an arrow?

Today we reveal the answer:

The arrow of time comes from observation—each measurement leaves an indelible footprint in information space.

Act I: Reversible Physics, Irreversible Reality

Can Movies Be Played Backward?

Imagine filming a video, then playing it backward:

• Billiard collisions: Completely reasonable (Newton’s laws time-symmetric)
• Planetary motion: Completely reasonable (Kepler’s laws time-symmetric)
• Light reflection: Completely reasonable (Maxwell’s equations time-symmetric)

But:

• Egg breaking: Looks weird when reversed (fragments assembling into egg?)
• Smoke diffusion: Looks weird (smoke converging into a stream?)
• Human aging: Looks weird (old people becoming young?)

Microscopic reversible, macroscopic irreversible.

Why?

Second Law of Thermodynamics

Traditional answer: Entropy always increases.

Entropy is “disorder,” and nature tends from order to disorder:

• Egg breaking (order → disorder)
• Smoke diffusion (concentrated → dispersed)
• Human aging (structure → degradation)

But this only pushes the question back: Why does entropy increase?

Act II: Loschmidt’s Paradox—How Do Reversible Collisions Produce Irreversibility?

In 1876, physicist Loschmidt raised a sharp question:

If each particle collision is reversible (Newton’s laws time-symmetric), how can collective behavior of many particles be irreversible?

Paradox: Reversible + Reversible + … + Reversible = Irreversible?

This is like: Symmetry + Symmetry = Asymmetry?

Boltzmann’s Answer

Boltzmann said: Not necessarily irreversible, but overwhelmingly probabilistically irreversible.

Imagine a box of gas:

• All molecules in left half (low entropy)
• Wait a bit, molecules diffuse throughout box (high entropy)

Theoretically molecules can spontaneously return to left half (time reversal), but probability is:

$$P\sim e^{-N}$$

where $N\sim 10^{23}$ (number of molecules).

This probability is so small the universe’s age isn’t enough to wait for it to happen once.

So, entropy increase isn’t a law, but statistical law of overwhelming probability.

But Problem Remains

Boltzmann’s explanation has a loophole: Why is initial state low entropy?

If universe was high entropy (uniform distribution) at some past moment, how did it spontaneously become low entropy (structured)?

This is the past hypothesis—need to artificially assume universe’s initial state was low entropy.

Why? Nobody knows.

Act III: Observation Creates Arrow

GLS framework gives a revolutionary answer:

Arrow of time is not a premise of second law of thermodynamics, but product of observation.

π-Semantic Collapse

Recall earlier: Quantum measurement is anchor switching.

Each observation, observer chooses a new reference frame (anchor point), wave function collapses to definite state.

This process produces path entropy:

$${\Sigma }_k=D({\rho }_k\Vert {\rho }^{\text{ss}})-D({\rho }_{k-1}\Vert {\rho }^{\text{ss}})-\ln \frac{{\mathbb{P}}^{†}(x_k|{\rho }_{k-1})}{\mathbb{P}(x_k|{\rho }_{k-1})}$$

In plain language: Each observation leaves a “footprint” in information space, size of this footprint is path entropy.

Path Entropy Non-Negative

Key theorem (Belavkin, Spohn):

$$⟨{\Sigma }_{0:t}⟩\ge 0$$

On average, path entropy is always non-negative—footprints don’t disappear by themselves.

Like walking on sand:

• Each step leaves footprint (path entropy increases)
• Footprints don’t smooth themselves (unless external force)

Arrow of time is cumulative direction of these footprints.

Act IV: No Observers, No Time

Now, astonishing inference:

Without observation (measurement), there’s no path entropy, no arrow of time.

Reversibility of Quantum World

Unmeasured quantum systems evolve completely reversibly:

$$|\psi (t)⟩=e^{-iHt/\hslash }|\psi (0)⟩$$

You can let $t\to -t$, system will “flow backward” to initial state.

Pure quantum evolution has no time direction.

Measurement Breaks Reversibility

But once measured:

• Wave function collapses to definite state
• Produces path entropy $\Sigma >0$
• Irreversible!

You cannot “cancel” measurement—information already acquired, footprint already left.

Subjectivity of Time

So:

• Objective physics (Schrödinger equation): No time direction
• Subjective experience (path entropy from observation): Has time direction

Time passing is observer’s experience, not objective property of universe.

This doesn’t mean “time is illusion,” but: Time is emergent product of interaction between observer and world.

Act V: The Cost of Memory

Why do you remember the past, not the future?

Memory Is Information Storage

Memory requires physical carriers:

• Brain’s neural synapses
• Computer hard drives
• Words on paper

Storing information requires changing physical state—and changing physical state produces entropy.

Landauer Principle

In 1961, physicist Landauer proved:

Erasing 1 bit of information produces at least $k_{B}T\ln 2$ entropy.

In other words: Information deletion has minimum energy cost.

$$\Delta Q\ge k_{B}T\ln 2$$

This isn’t engineering limitation, but law of thermodynamics.

Memory = Irreversible

So:

• Remembering (storing information): Requires changing physical state, produces entropy
• Forgetting (deleting information): Requires erasure, produces entropy

Whether remembering or forgetting, both are irreversible thermodynamic processes.

You remember the past because past events left physical traces in your brain (changes in neural network weights).

You don’t remember the future because future events haven’t happened, no traces left.

Memory defines time direction.

Act VI: Quantum Chaos and Arrow of Time

Now we introduce a profound concept: quantum chaos.

OTOC—Propagation of Disorder

Define Out-of-Time-Order Correlator (OTOC):

$$C(t)=⟨[W(t),V(0){]}^{†}[W(t),V(0)]⟩$$

It measures: How a local perturbation $V(0)$ “disrupts” global observable $W(t)$ over time.

In chaotic systems, $C(t)$ grows exponentially:

$$C(t)\sim e^{{\lambda }_{L}t}$$

where ${\lambda }_L$ is Lyapunov exponent—strength of chaos.

Loschmidt Echo

Another measure: Loschmidt echo

$$\mathcal{L}(t)=|⟨\psi (0)|e^{iHt/\hslash }{e}^{-i(H+\delta H)t/\hslash }|\psi (0)⟩{|}^2$$

It measures: Under tiny perturbation $\delta H$, can system “return” to initial state.

In chaotic systems, echo decays rapidly:

$$\mathcal{L}(t)\sim e^{-{\lambda }_{L}t}$$

Chaotic systems are extremely sensitive to initial conditions—tiny perturbations lead to completely different results.

MSS Chaos Upper Bound

In 2017, Maldacena, Shenker, Stanford proved a profound theorem:

In any quantum system, Lyapunov exponent has upper bound:

$${\lambda }_L\le \frac{2\pi k_{B}T}{\hslash }$$

Temperature limits growth speed of chaos!

Arrow of Time and Chaos

Chaos causes irreversibility:

• Initial state $\psi (0)$ evolves to $\psi (t)$
• Even if theoretically reversible, actually any tiny perturbation (measurement, environmental coupling…) makes system “forget” initial state

Chaos amplifies microscopic uncertainty, producing macroscopic irreversibility.

Arrow of time = path entropy from observation + information loss amplified by chaos

Act VII: The Universe’s Clock

If arrow of time comes from observation, then where does universe’s arrow of time come from?

Universe Has No External Observers

We observe Earth, Sun, galaxies, producing path entropy.

But who observes the entire universe?

Universe has no external observers—it observes itself.

Universe’s Self-Reference

Universe is the ultimate self-referential system:

• Universe contains observers (us)
• Observers observe universe
• Observation produces path entropy
• Path entropy defines time direction

Universe’s arrow of time is product of its self-observation.

Thermal History

Universe’s thermal history:

1. Big Bang: Extremely low entropy (quantum fluctuations)
2. Inflation: Exponential expansion, producing structure seeds
3. Matter dominance: Gravitational collapse forming galaxies, stars
4. Now: Stars burning, producing entropy (radiation)
5. Future: Black hole evaporation, thermal equilibrium (maximum entropy)

Universe goes from low entropy to high entropy—this is the largest arrow of time.

But why was initial state low entropy?

Hints from Quantum Gravity

Possible answer: Initial “low entropy” is natural state of quantum gravitational fluctuations.

At Planck scale, spacetime itself is quantum fluctuations—no “before,” time itself emerges.

Time is born with universe, not pre-existing background.

Act VIII: Reversing Time?

Can arrow of time be reversed?

Time Reversal in Quantum Computers

In 2019, scientists successfully “reversed” time on quantum computers:

• Prepare simple state $|{\psi }_0⟩$
• Let it naturally evolve for a while $|\psi (t)⟩$ (entropy increases, information diffuses)
• Apply carefully designed reversal operation
• System returns to initial state $|{\psi }_0⟩$

Time “flowed backward”!

But note:

• Requires perfect control (no perturbations)
• Only effective for small systems (few qubits)
• Requires external information (knowing how to reverse)

This doesn’t violate second law of thermodynamics—because it’s an open system, has external information input.

Maxwell’s Demon

In 1867, Maxwell proposed a thought experiment:

A “demon” guards a door, lets fast molecules through, slow ones stay—system cools (entropy decreases)!

This seems to violate second law of thermodynamics.

Solution (Landauer, Bennett):

Demon needs memory (recording which molecules are fast/slow). Deleting memory produces entropy, offsetting system’s entropy decrease.

Total entropy (system + demon) still increases.

Reversing Memory?

Can memory be erased and past restored?

Theoretically, if you have complete information (precise microscopic state of universe), you can “calculate” the past.

But practically:

• Information gets lost (black holes, quantum entanglement, decoherence)
• Measurement itself changes system (uncertainty principle)
• Chaos amplifies tiny errors

Past is informationally unrecoverable.

Act IX: The Nature of Time

Let’s summarize understanding of time:

What Time Is Not

❌ Not absolutely flowing river (Newton) ❌ Not fourth dimension of space (Minkowski, partially correct) ❌ Not premise of second law of thermodynamics ❌ Not objective existence independent of observers

What Time Is

✅ Is accumulation of path entropy from observation ✅ Is direction of irreversible evolution of information ✅ Is thermodynamic cost of memory and forgetting ✅ Is emergent structure of conscious experience

Time is product of interaction between observer and universe—both objective (physical process) and subjective (experiential direction).

Three Types of Time

Physicists distinguish three types of time:

1. Coordinate time $t$: Parameter in equations (can be positive or negative)
2. Thermodynamic time: Direction of entropy increase (statistical arrow)
3. Psychological time: Flow of conscious experience (subjective arrow)

GLS framework unifies them:

• Coordinate time: Reversible
• Thermodynamic time: Path entropy $\Sigma$
• Psychological time: Flow of fixed points of self-referential loops

All three are essentially the same—all are irreversible evolution of information.

Act X: Eternity and Instant

Finally, philosophical reflection:

Einstein’s Consolation

After Einstein’s friend died, he wrote to console:

“For us believing physicists, the distinction between past, present, and future is only a stubborn illusion.”

His meaning: In relativity, spacetime is four-dimensional whole (block universe), all moments “simultaneously” exist.

But this ignores observer’s perspective.

Block Universe vs. Growing Universe

• Block universe: Time is one dimension of space, past present future all “already exist”
• Growing universe: Time is emergent, future not yet determined (quantum superposition)

GLS framework: Both are correct, depending on perspective.

• From “God’s perspective” (no observers): Block universe, no time direction
• From “observer’s perspective” (self-referential loop): Growing universe, time flows

Flow of time is real—for observers.

Living in the Present

If time is path entropy from observation, then:

“Present” is the only reality—past is memory (information traces), future is prediction (model inference).

What you can experience is only the self-referential fixed point of this current moment—the “you” of the previous moment has already disappeared (though leaving memory traces), the “you” of the next moment hasn’t yet emerged.

You are an instant existence, continuously emerging.

Take-Home Thoughts

Right now, you’re reading this text.

A few seconds ago, you read the previous paragraph—that moment has become the past, remaining in memory.

A few seconds later, you’ll read the end—that moment is still future, undetermined.

This flowing feeling—from past to present to future—is time.

It’s not the background of the universe, but footprints your observation leaves in information space.

Every thought, every heartbeat, every breath is an increase in path entropy, a step of the arrow of time.

You don’t flow in time—time emerges because of your observation.

When you stop observing (death), your time stops. But universe’s observation (through other consciousness) continues, time continues.

Time is collective dream of observers—wherever there’s consciousness, time flows.

Next: “The Cost of Free Will: Directed Information and the Physics of Choice”

We will see that free will is not metaphysics, but a measurable physical quantity—how much your choices can change the future, that’s how much freedom you have.