Information Lost
When we speak of the "weirdness" of quantum mechanics—Schrödinger’s cat, quantum entanglement, wavefunction collapse—few realize that these puzzles may stem from a deeper issue: quantum mechanics is fundamentally a spectral theory, not a spacetime theory.
Imagine listening to a symphony. The conventional way is to follow the unfolding melody in time, experiencing the articulation and transition of each note. But suppose I told you that quantum mechanics instead decomposes the entire piece into a superposition of sinusoidal waves of different frequencies and then declares, “This is the essence of music.” How would you react?
This analogy is not an exaggeration. Quantum mechanics indeed employs the Fourier transform to shift the physical world from the spacetime domain into the frequency domain. In doing so, it gains computational convenience at the cost of systematically discarding all local and instantaneous information. This loss is not a technical shortcoming but a structural blind spot inherent to the theoretical framework.
Chapter One: From Spacetime to Spectrum—A Silent Revolution
1.1 Fourier’s Magic and Its Price
In 1807, Joseph Fourier introduced a revolutionary mathematical tool: any periodic function can be decomposed into a superposition of sine waves. This seemingly pure mathematical insight became a cornerstone of quantum mechanics a century later.
The essence of the Fourier transform:
Time-domain signal f(t) ⟷ Frequency-domain signal F(ω)
Spatial wavefunction ψ(x) ⟷ Momentum-space wavefunction φ(p)
This transformation is akin to analyzing an oil painting through spectral decomposition of its pigments—you learn which colors were used and in what proportions, but the image, composition, and brushstrokes vanish entirely.
1.2 Quantum Mechanics’ Choice
Between 1925 and 1926, Heisenberg and Schrödinger independently formulated two versions of quantum mechanics. Intriguingly, both ultimately converged on spectral representations:
Heisenberg’s matrix mechanics operates directly among energy eigenstates (frequency modes).
Schrödinger’s wave mechanics describes wavefunction evolution, but solutions are invariably expanded in eigenstates.
Why the spectral choice? Because atomic spectra, blackbody radiation, and the photoelectric effect—the phenomena that triggered the quantum revolution—are all frequency-domain phenomena. Physicists naturally adopted the mathematical language best suited to describe them.
Yet they may not have fully appreciated the cost of this choice.
Chapter Two: The Disappearance of Magnetic Energy—The First Casualty
2.1 The Physical Reality of Magnetic Energy
In classical electromagnetism, magnetic energy is a tangible, spatially distributed quantity:
Magnetic energy density: u_B = B²/(2μ₀)
This quantity has clear physical meaning:
It can be measured pointwise.
It is continuously distributed in space.
Total energy is obtained by spatial integration.
It drives motors and induces currents.
Consider a simple bar magnet: every point in the surrounding space stores magnetic energy. This energy distribution determines the alignment of iron filings, the deflection of compass needles, and even the levitation force in maglev trains.
2.2 The Dilemma After Spectral Transformation
When we apply the Fourier transform to the magnetic field:
B(r,t) → B(k,ω)
Position r becomes wavevector k, and time t becomes frequency ω. Here arises the problem:
The magnetic energy density u_B(r) has no counterpart in spectral space!
Why? Because energy density is a local quantity in space, whereas spectral representation describes global modes. Just as you cannot locate “the loudness at beat two of measure three” in a frequency analysis of a musical score, you cannot recover spatial energy density from its spectral decomposition.
2.3 Quantum Mechanics’ “Creative” Resolution
Faced with the disappearance of magnetic energy, quantum mechanics introduced a series of conceptual constructs to “compensate”:
Electron spin:
Not actual rotation
No classical analogue
Purely intrinsic angular momentum
Generates magnetic moment μ = -gμ_B S
Exchange interaction:
H_ex = -J Σ S_i·S_jJ is an “exchange integral” (a purely mathematical construct)
Not genuine magnetic dipole interaction
Arises from symmetry requirements of the wavefunction
Pauli exclusion principle:
Two electrons cannot occupy the same quantum state
Generates “exchange energy”
Mimics magnetic repulsion
These concepts successfully explain experimental observations, but they function as mathematical patches rather than physical entities. The true magnetic energy—as a spatial energy density—is conceptually replaced in the quantum description.
Chapter Three: The Vanishing Particle Trajectory—From Path to Probability
3.1 The Puzzle of the Wilson Cloud Chamber
In 1911, Wilson invented the cloud chamber, allowing us for the first time to “see” particle trajectories—alpha particles and electrons left clear tracks in supersaturated vapor.
Yet quantum mechanics asserts: particles have no definite trajectories.
How is this contradiction resolved? The standard answer is that droplet condensation in the cloud chamber “measures” the particle’s position, repeatedly collapsing the wavefunction and producing an apparently continuous path.
But this explanation raises problems:
Why does collapse always yield a continuous trajectory?
Why do trajectories conform to classical expectations?
How can frequent “measurements” avoid destroying quantum coherence?
3.2 The Enigma of the Double-Slit Experiment
The disappearance of trajectories is most striking in the double-slit experiment:
Classical expectation:
Particle passes through left slit or right slit
Screen shows two bright bands
Quantum reality:
Interference fringes appear
Particle “passes through both slits”?
Observing the path destroys interference
Feynman famously said, “Nobody understands quantum mechanics,” largely referring to this fundamental loss of path information. In the spectral representation, a particle is a superposition of momentum modes—there is no concept of “path” at all.
3.3 The Paradox of Quantum Tunneling
Quantum tunneling further defies intuition: particles can “penetrate” potential barriers higher than their energy.
Classical analogies fail:
Not “digging a tunnel”
Not “borrowing energy”
Not “jumping over”
Spectral explanation: the wavefunction decays exponentially inside the barrier but retains non-zero amplitude on the other side. While mathematically correct, this offers no physical picture. Where is the particle inside the barrier? How does it get from A to B?
The spectral representation cannot answer these questions because it contains no spacetime trajectory information.
Chapter Four: The Abstraction of Many-Body Interactions
4.1 From Newtonian Gravity to Quantum Entanglement
In classical physics, interactions possess clear spacetime characteristics:
Newtonian gravity: F = GMm/r²
Force decays with inverse square of distance
Directionally well-defined
Field strength computable pointwise
Coulomb force: F = kq₁q₂/r²
Same inverse-square law
Explicit locality
Simple superposition principle
But in the spectral representation of quantum mechanics, what becomes of these local interactions?
4.2 The Magic of Momentum Space
Consider the Coulomb interaction between two electrons. In quantum mechanics:
Coordinate space: V(r) = e²/|r₁ - r₂|
Momentum space: V(k) = 4πe²/k²
This transformation appears equivalent, but its physical meaning is radically altered:
The concept of “distance” is lost
Interaction becomes a function of momentum transfer
Local action becomes global coupling
4.3 Quantum Entanglement: An Inevitable Spectral Correlation
In the EPR paradox, quantum entanglement appears “natural” in the spectral picture:
Two particles share a single wavefunction: |Ψ⟩ = (|↑↓⟩ - |↓↑⟩)/√2
This does not imply mysterious spatial connections, but rather that both particles belong to the same spectral mode. Measuring one collapses the entire mode, instantly determining the state of the other.
The apparent “faster-than-light correlation” arises largely from our attempt to describe spectral phenomena using spacetime language.
Chapter Five: Deeper Consequences
5.1 The Root of the Measurement Problem
“The measurement causes wavefunction collapse”—this central mystery of quantum mechanics stems fundamentally from the incompatibility between spectral representation and spacetime observation.
Nature of observation:
We always observe at specific spacetime points
Instruments record local events
Data carry spacetime labels
Theoretical dilemma:
Wavefunction is a spectral superposition
Lacks spacetime localization
Must “collapse” to match observation
This is not a flaw of measurement devices, but a fundamental conflict between two modes of description.
5.2 Quantum Field Theory’s Illusion of Resolution
Quantum field theory (QFT) attempts to reconcile this conflict by introducing field operators φ(x,t), seemingly restoring spacetime description. But closer inspection reveals:
Field operator expansion:
φ(x,t) = Σ_k [a_k e^{i(kx - ωt)} + a_k† e^{-i(kx - ωt)}]
This remains a superposition of frequency modes (k, ω)! The so-called “local field” is merely a mathematical recombination of spectral modes.
The trick of virtual particles:
Electromagnetic interaction → exchange of virtual photons
Strong interaction → exchange of virtual gluons
Weak interaction → exchange of virtual W/Z bosons
These “virtual particles” lack definite spacetime trajectories; they are mathematical tools in spectral space, artificially endowed with particle-like names.
5.3 The Origin of Information Paradoxes
The black hole information paradox and the non-locality of quantum information both stem from the same issue:
Classical information:
Requires spacetime carrier
Has definite location
Obeys causality
Quantum information:
Resides in spectral space
Distributed non-locally
Appears to transcend causality?
Where is a “qubit” located? What path does “information” take in quantum entanglement?
The spectral representation cannot answer these questions.
Chapter Six: Lessons from the History of Science
6.1 From Geocentrism to Heliocentrism
Ptolemy’s geocentric model accurately predicted planetary positions but required complex epicycles and deferents. Copernicus’ heliocentric model was initially less precise but revealed a truer physical picture.
Quantum mechanics may be in a similar position:
Computationally precise (like Ptolemy)
Conceptually complex (epicycles ≈ virtual particles?)
Lacking a coherent physical image
6.2 The Rise and Fall of Caloric Theory
In the 18th century, caloric theory explained many thermal phenomena:
Heat conduction (flow of caloric)
Heat capacity (storage of caloric)
Latent heat (phase change of caloric)
It was ultimately replaced by kinetic theory. The key insight: heat is not a substance but a manifestation of motion.
Could quantum mechanics’ “wavefunction,” “virtual particles,” and “quantum fields” be modern-day “caloric”?
Chapter Seven: Possible Paths Forward
7.1 Wavelet Transform: A Time-Frequency Compromise
Wavelet transforms offer partial time-frequency localization:
Advantages:
Retain partial time information
Retain partial frequency information
Enable multi-scale analysis
Attempts at application:
Wavelet mechanics (not mainstream)
Signal processing (engineering success)
Potential physical insights
7.2 Insights from the Path Integral
Feynman’s path integral formulation seemingly restores the notion of trajectories:
Z = ∫ D[path] e^{iS[path]/ℏ}
But this integrates over all possible paths, not a specific trajectory. Nevertheless, it hints that a deeper spacetime description may exist.
7.3 Hints from the Holographic Principle
The holographic principle suggests that information within a volume can be encoded on its boundary. This implies that neither spacetime nor spectrum is fundamental—they may be dual projections of a deeper reality.
Chapter Eight: Philosophical Reflections
8.1 Realism vs. Instrumentalism
The spectral nature of quantum mechanics intensifies this philosophical debate:
Realist stance:
Physical theories should describe objective reality
Spectral representation omits essential aspects of reality
A more complete theory is needed
Instrumentalist stance:
Theories are merely predictive tools
Spectral representation is sufficient if it works
No need for a “true picture”
8.2 Epistemological Limits
Our senses and instruments operate in spacetime. But if reality is fundamentally spectral (or both), how can we comprehend it?
This is not merely a physical question, but a profound epistemological challenge.
Conclusion: Re-examining Physical Reality
Quantum mechanics chose the spectral representation, achieving remarkable success—but at a cost:
What was lost:
Spatially localized magnetic energy
Spacetime trajectories of particles
Locality of interactions
Spatiotemporal carriers of information
Clarity of causal relations
What was gained:
Precise energy-level calculations
Mathematical description of wave-particle duality
Symmetries and conservation laws
Quantum information processing
Was this trade-off necessary? Was it worth it? Can it be transcended?
Future Directions
A genuine breakthrough may require:
New mathematical frameworks: beyond the limitations of Fourier transforms
Conceptual revolution: redefining “particle,” “field,” and “interaction”
Experimental tests: probing the boundaries and failures of spectral descriptions
Philosophical reflection: What is physical reality? How do we know it?
Final Thought
Einstein once said, “God does not play dice.” Today, we may need to ask a deeper question:
Does quantum mechanics describe nature itself—or merely its spectral projection?
If the latter, then the “quantum weirdness” that has puzzled physicists for a century may simply be the inevitable consequence of choosing an inadequate descriptive language. It would be like using spectral analysis of a musical score to understand Beethoven’s emotional expression—technically feasible, but missing the soul of the music.
The next revolution in physics may begin with the realization that what we need is not a more precise spectral theory, but a new language that unifies spacetime and spectrum. Such a language would restore magnetic energy as a physical reality, make particle trajectories visible again, re-establish the locality of interactions—and yet preserve the computational power of quantum mechanics.
This would not be a step backward, but a return to completeness.
As Einstein said, “I want to know God’s thoughts; the rest are details.” Perhaps God’s thoughts are neither purely spacetime nor purely spectral, but a perfect unity of both—a descriptive language we have yet to discover, in which all quantum “weirdness” becomes perfectly natural.