Interferometric discrepancy between the Schrödinger and Klein-Gordon wave equations in the non-relativistic limit due to their dissimilar phase velocities

This paper identifies a fundamental incompatibility between the non-relativistic limits of the Schrödinger and Klein-Gordon equations, demonstrating that the inclusion of rest energy in the latter leads to distinct phase velocities that cause unique wavefunction attenuation in Sagnac interferometers, a phenomenon absent in the former.

The Gist

Imagine you are watching a race between two runners, Runner S (representing the Schrödinger equation) and Runner K (representing the Klein-Gordon equation). Both are trying to describe how a tiny particle, like an electron or a neutron, moves through the world.

In the world of classical physics (like cars or baseballs), if you add a constant amount of energy to a system—say, you decide to measure height from the basement instead of the ground floor—the actual motion of the car doesn't change. It just shifts the "zero point" on your ruler.

But in the quantum world, things get weird. The paper argues that adding this "constant energy" (specifically the massive rest energy of the particle, $mc^2$) changes the speed of the wave crests (the peaks of the particle's wave) without changing the speed of the particle itself.

Here is the story of how the author, Frank Kowalski, uses a thought experiment to show that these two famous equations might actually disagree on what happens in a specific situation.

The Setup: The "Overtaking" Beam Splitter

Imagine a Sagnac interferometer. Think of this as a circular racetrack where a particle wave is split into two:

1. One wave goes Clockwise (CW).
2. One wave goes Counter-Clockwise (CCW).

They travel around the track and meet at the finish line to create an interference pattern (like ripples in a pond meeting).

Now, introduce a Moving Beam Splitter (BS). This is a gate or a mirror that the waves have to pass through.

• The CCW wave passes through the gate once while the gate is stationary.
• The CW wave encounters a gate that starts moving faster than the wave itself!

The Analogy: The Fast Train and the Slow Wave

Let's use a train analogy to understand the difference between the two theories.

The Wave: Imagine a long, slow-moving train of people walking in a line. The "wave crests" are the individual people in the line.
The Beam Splitter (BS): Imagine a fast-moving bus driving alongside the train.

Scenario A: The Schrödinger Equation (The "Slow Wave" Theory)

In this theory, the "wave crests" (the people walking) are relatively slow.

1. The bus (BS) starts moving and overtakes the walking people.
2. Because the bus is faster, it passes the people and leaves them behind.
3. The bus stops. The people, who were left behind, now catch up to the bus and pass through it again.
4. Later, the bus might move again, and the people pass through a third time.

The Result: The people (the wave) have to pass through the gate three times. Every time they pass through, the gate acts like a filter, making the group slightly smaller (attenuation). So, the final group of people is much smaller than it started.

Scenario B: The Klein-Gordon Equation (The "Fast Wave" Theory)

In this theory, the "wave crests" are moving incredibly fast because the theory includes the particle's "rest energy" (a huge chunk of energy just for existing).

1. The bus (BS) tries to overtake the walking people.
2. It can't. The people are moving so fast (near the speed of light, effectively) that the bus can never catch up to them.
3. The bus drives alongside them, but the people are always ahead.
4. The bus stops. The people have already passed through the gate once and are long gone.

The Result: The people only pass through the gate once. The final group is only slightly smaller, not tiny.

The Big Conflict

The paper points out a massive problem:

• Classical Physics says: Adding a constant energy shouldn't change the outcome of a race.
• Quantum Physics (Schrödinger) says: The wave gets attenuated (shrinks) because the gate overtook it three times.
• Quantum Physics (Klein-Gordon) says: The wave only shrinks a little because the gate never caught it.

Both theories are supposed to describe the same non-relativistic world (slow-moving particles). They should give the same answer. But because one theory includes the "rest energy" term and the other doesn't, they predict completely different results for this specific experiment.

Why Does This Matter?

1. Phase Velocity is Real (in a way): Usually, physicists say the "phase velocity" (the speed of the wave peaks) doesn't matter because it doesn't carry information. But this paper argues that if a physical object (the gate) moves faster than those peaks, it physically interacts with them differently, changing the outcome.
2. The "Rest Energy" Dilemma: The Schrödinger equation (which we use for almost all chemistry and standard quantum mechanics) ignores the rest energy. The Klein-Gordon equation (a more "relativistic" version) includes it. This experiment suggests that including that rest energy term might actually break the rules for slow-moving particles in a way we didn't expect.
3. The Measurement: If we could build a machine where a beam splitter moves faster than a slow neutron (which is possible with modern laser cooling), we could see which theory is right.
   • If the signal drops significantly (3x attenuation), Schrödinger is right.
   • If the signal stays strong (1x attenuation), Klein-Gordon is right.

The Takeaway

The author is essentially saying: "We have two rulebooks for how particles move. They usually agree, but if you set up a race where the finish line moves faster than the runners, the rulebooks disagree on how many times the runners cross the line. This suggests that one of our fundamental assumptions about energy in quantum mechanics might need a rewrite."

It's a reminder that even in the "slow" world of non-relativistic quantum mechanics, the way we define energy can have surprising, measurable consequences.

Technical Summary

Here is a detailed technical summary of the paper "Interferometric discrepancy between the Schrödinger and Klein-Gordon wave equations in the non-relativistic limit due to their dissimilar phase velocities" by Frank V. Kowalski.

1. Problem Statement

The paper addresses a fundamental theoretical inconsistency between the two primary formulations of non-relativistic quantum mechanics: the Schrödinger equation and the non-relativistic limit of the Klein-Gordon equation.

• The Core Conflict: In classical mechanics, adding a constant energy offset (such as the rest energy $mc^2$) to the Hamiltonian does not alter the equations of motion. In quantum mechanics, however, such an addition alters the phase velocity ($v_p = \omega/k$) of the wavefunction, even though the probability density function (PDF, $|\Psi|^2$) remains unchanged because the global phase factor cancels out.
• The Discrepancy:
  • Schrödinger Equation: Phase velocity $v_p = v_g/2$ (where $v_g$ is the group/particle velocity).
  • Non-relativistic Klein-Gordon Equation: Includes the rest energy term, resulting in $v_p \approx c^2/v_g$.
• The Question: Since both equations are supposed to describe the same non-relativistic physics, they should yield identical experimental predictions. Kowalski proposes a scenario where these differing phase velocities lead to divergent, measurable outcomes regarding wave attenuation in an interferometer.

2. Methodology

The author employs a thought experiment involving a Sagnac interferometer with a moving beamsplitter (BS) to test the predictions of both equations.

• The Setup: A particle beam (momentum eigenstate or Gaussian wave group) is split into clockwise (CW) and counter-clockwise (CCW) paths. A specific beamsplitter within the interferometer undergoes a trajectory where it accelerates, moves at a constant velocity $V$, and decelerates.
• The Critical Condition: The BS is designed to move at a speed $V$ that exceeds the phase velocity ($v_p$) of the particle in the Schrödinger formulation ($V > v_g/2$).
• Analytical Approaches:
  1. Retarded Phase Method: Calculates the phase at the output based on the time a wave crest was at the input, emphasizing the role of phase velocity in time-dependent path lengths.
  2. Frame Transformation: Analyzes the interaction by transforming between the laboratory frame and the rest frame of the moving BS.
  3. Comparison: The behavior of the wavefunction is compared for:
     • Electromagnetic waves (where $v_p = c$).
     • The Klein-Gordon equation (where $v_p \approx c^2/v_g \gg c$).
     • The Schrödinger equation (where $v_p = v_g/2$).

3. Key Contributions

• Identification of a Regime of Incompatibility: The paper identifies a specific kinematic regime (where a boundary moves faster than the wave's phase velocity) where the Schrödinger and Klein-Gordon equations predict fundamentally different physical phenomena, despite both being considered valid non-relativistic limits.
• Amplitude Attenuation vs. Phase Shift: The author demonstrates that the interaction with a super-sonic (relative to phase velocity) beamsplitter results in amplitude attenuation without a phase shift for the Schrödinger equation, a phenomenon absent in the Klein-Gordon and EM cases.
• Re-evaluation of the Born Rule Context: The work suggests that the interpretation of the probability density (Born rule) may require re-evaluation in regimes where the phase velocity is lower than the velocity of the interacting apparatus.

4. Results

A. Plane-Wave Interference

• Schrödinger Prediction:
  • When the BS moves faster than the phase velocity ($V > v_p$), the BS overtakes the wave crests of the CW path.
  • The wave crests are "left behind" the BS.
  • Upon the BS stopping, these lagging crests traverse the BS a second and third time.
  • Result: The CW wave experiences three traversals of the BS, leading to significant amplitude attenuation (proportional to $T^3$, where $T$ is transmission). The CCW path (moving opposite to BS) is not overtaken and traverses only once.
  • Interference Pattern: The output intensity $I_{Sch}$ changes from the stationary case ($I_0$) to a new value ($I_{Sch}$) due to the differential attenuation between CW and CCW paths.
• Klein-Gordon & EM Prediction:
  • The phase velocity is extremely high ($v_p \approx c^2/v_g$ or $c$). The BS can never exceed this speed.
  • The BS never overtakes the wave crests.
  • Result: Both CW and CCW waves traverse the BS exactly once.
  • Interference Pattern: The output intensity $I_{KG}$ remains identical to the stationary case ($I_0$). No attenuation difference occurs.

B. Wave Group Interference

• Schrödinger: A Gaussian wave group moving through the system experiences distortion. Components with $v_p < V$ are overtaken and attenuated three times; components with $v_p > V$ are attenuated once. This leads to a distorted wave group at the output with reduced amplitude.
• Klein-Gordon: All momentum eigenstates within the wave group have $v_p \gg V$. The entire wave group traverses the BS three times physically, but because the BS never overtakes the wave, the attenuation corresponds to only one pass. The wave group structure remains intact, and the interference pattern is unchanged from the stationary case.

C. Physical Mechanism

The attenuation is not due to energy loss or inelastic scattering (which would require a rest-energy term). Instead, it is a kinematic effect: the moving boundary condition interacts with the wave multiple times because the boundary moves faster than the wave crests can propagate away from it. The Klein-Gordon equation, by including $mc^2$, ensures the phase velocity is always superluminal relative to the particle, preventing this "overtaking" scenario.

5. Significance

• Fundamental Incompatibility: The paper highlights a regime where the non-relativistic Klein-Gordon equation and the Schrödinger equation yield mutually exclusive predictions. If the Schrödinger prediction (attenuation due to overtaking) is correct, the inclusion of the rest energy term in the non-relativistic limit (Klein-Gordon) is invalid for describing such interactions. Conversely, if the Klein-Gordon prediction holds, the Schrödinger equation fails in this regime.
• Experimental Feasibility: The author notes that while difficult, this could be tested using ultracold neutrons or laser-cooled atoms.
  • Neutron speeds: 5–1000 m/s.
  • Atom speeds: < 0.3 m/s (with laser cooling).
  • A moving beamsplitter (e.g., a grating or mirror) moving at speeds exceeding the phase velocity of these slow particles is physically achievable.
• Implications for Quantum Theory: The results challenge the standard assumption that adding a constant rest energy term is merely a global phase factor with no physical consequence. The paper argues that in dynamic, time-dependent boundary conditions, this "global" phase factor manifests as a measurable difference in wave propagation and interference.

Conclusion

Kowalski's work proposes a critical test for the foundations of non-relativistic quantum mechanics. By exploiting the difference in phase velocities caused by the inclusion (or exclusion) of rest energy, the paper suggests that the Schrödinger equation predicts a unique "overtaking" attenuation effect that the Klein-Gordon equation forbids. Experimental verification of this effect would necessitate a revision of how the non-relativistic limit of relativistic equations is understood and how the probability density is interpreted in dynamic systems.