Phase and Proper Time: Quantum-Geometric Bridge

“Phase can be viewed as a quantum counter of proper time.”

🎯 Core Proposition

Proposition (Phase-Proper Time Correspondence):

For a particle of mass $m$ propagating along worldline $\gamma$, its quantum phase can be expressed as:

$$\boxed{\varphi =\frac{m{c}^2}{\hslash }{\int }_{\gamma }d\tau }$$

Where:

• $\varphi$: quantum phase
• $m$: rest mass
• $c$: speed of light
• $\hslash$: reduced Planck constant
• $\tau$: proper time along worldline

Physical meaning:

• Left side ($\varphi$): quantum phase, pure quantum concept
• Right side ($\int d\tau$): proper time, pure geometric concept
• Relationship: This equation establishes a bridge between quantum and geometry.

💡 Intuitive Image: Wave Oscillation

Classical Analogy: Pendulum Clock

Imagine a pendulum clock:

    |
   \|/
    O  ← Pendulum
   / \

Oscillation period $T$ measures time.

Number of oscillations $N=t/T$ counts swings.

Analogy:

• Proper time $\tau$ ↔ time $t$
• Phase $\varphi$ ↔ number of oscillations $N$
• $\varphi =(m{c}^2/\hslash )\tau$ ↔ $N=t/T$

Physical meaning: Phase $\varphi$ can be understood as a counter of particle’s intrinsic “oscillation”.

Quantum Wave Packet

Consider wave function of free particle:

$$\psi (x,t)=A{e}^{i(kx-\omega t)}$$

Phase:

$$\varphi (x,t)=kx-\omega t$$

Plane wave frequency:

$$\omega =\frac{E}{\hslash }=\frac{m{c}^2}{\hslash }\gamma$$

Where $\gamma =1/\sqrt{1-v^2/c^2}$ is Lorentz factor.

Proper time: For particle at rest, $d\tau =dt/\gamma$.

Relationship:

$$\varphi =-\omega t=-\frac{m{c}^2}{\hslash }\gamma t=-\frac{m{c}^2}{\hslash }\tau$$

(Negative sign is convention)

📐 Path Integral Derivation

Classical Action

In relativity, classical action of particle along worldline:

$$S[\gamma ]=-m{c}^2{\int }_{\gamma }d\tau =-mc{\int }_{\gamma }\sqrt{-g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }}$$

Why this form?

• Action dimension: $[S]=\text{energy}\times \text{time}$
• $m{c}^2$ is rest energy
• $\int d\tau$ is proper time
• Negative sign from metric signature convention

Quantum Path Integral

In quantum mechanics, propagation amplitude from point $A$ to point $B$:

$$K(B,A)=\int \mathcal{D}\gamma \exp \left(\frac{i}{\hslash }S[\gamma ]\right)$$

Semiclassical limit: $\hslash \to 0$

Stationary phase condition: $\delta S=0$ → classical geodesic ${\gamma }_{\text{cl}}$

Principal phase:

$$\varphi =\frac{S[{\gamma }_{\text{cl}}]}{\hslash }=-\frac{1}{\hslash }{\int }_{{\gamma }_{\text{cl}}}m{c}^{2}d\tau$$

Key: Ignoring negative sign (phase convention), we get:

$$\boxed{\varphi =\frac{m{c}^2}{\hslash }\int d\tau }$$

graph TB
    A["Classical Action<br/>S = -mc² ∫dτ"] --> P["Path Integral<br/>K ~ exp(iS/ℏ)"]
    P --> SC["Semiclassical Limit<br/>ℏ → 0"]
    SC --> PH["Principal Phase<br/>φ = S/ℏ"]
    PH --> R["Phase-Time Correspondence<br/>φ = (mc²/ℏ)∫dτ"]

    style A fill:#e1f5ff
    style R fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

🧮 Calculation in Flat Spacetime

Minkowski Spacetime

In flat spacetime $d{s}^2=-c^{2}d{t}^2+d{x}^2+d{y}^2+d{z}^2$:

For particle moving along $x$ direction:

$$v=\frac{dx}{dt}$$

Proper time:

$$d\tau =dt\sqrt{1-\frac{v^2}{c^2}}=\frac{dt}{\gamma }$$

Phase change rate:

$$\frac{d\varphi }{dt}=\frac{m{c}^2}{\hslash }\frac{d\tau }{dt}=\frac{m{c}^2}{\hslash \gamma }$$

Energy relationship:

In relativity, energy $E=\gamma m{c}^2$, frequency $\omega =E/\hslash =\gamma m{c}^2/\hslash$.

Comparison:

$$\frac{d\varphi }{dt}=\frac{m{c}^2}{\hslash \gamma }\ne \omega$$

Why different?

Because $\varphi$ is phase along worldline (proper time), while $\omega t$ is phase along coordinate time.

Correct relationship:

$$\frac{d\varphi }{d\tau }=\frac{m{c}^2}{\hslash }=\text{constant}$$

This is Lorentz invariant.

Particle at Rest

For particle at rest ($v=0$, $\gamma =1$):

$$\tau =t$$

$$\varphi =\frac{m{c}^2}{\hslash }t$$

$$\omega =\frac{E}{\hslash }=\frac{m{c}^2}{\hslash }$$

This is Compton frequency!

$${\omega }_C=\frac{m{c}^2}{\hslash }\approx 10^{20}\text{ Hz (electron)}$$

Physical meaning: Even at rest, particle “oscillates” at Compton frequency.

🌀 Curved Spacetime

Schwarzschild Spacetime

In Schwarzschild metric:

$$d{s}^2=-\left(1-\frac{2M}{r}\right)c^{2}d{t}^2+{\left(1-\frac{2M}{r}\right)}^{-1}d{r}^2+r^{2}d{\Omega }^2$$

For radially free-falling particle:

$$d\tau =\sqrt{-g_{\mu \nu }\frac{d{x}^{\mu }}{ds}\frac{d{x}^{\nu }}{ds}}ds$$

Phase:

$$\varphi =\frac{m{c}^2}{\hslash }\int d\tau$$

Gravitational redshift:

For observer at rest at $r=r_0$, proper time $d\tau =\sqrt{1-2M/r_0}\cdot dt$.

Compton frequency (local):

$${\omega }_{\text{local}}=\frac{m{c}^2}{\hslash }$$

Coordinate frequency (distant observer):

$${\omega }_{\text{coord}}=\sqrt{1-\frac{2M}{r_0}}{\omega }_{\text{local}}$$

Redshift: Gravity makes phase evolution “slower”.

graph LR
    L["Local Observer<br/>ω_local = mc²/ℏ"] --> R["Distant Observer<br/>ω_coord < ω_local"]

    G["Gravitational Field<br/>g_tt < -1"] -.->|"Redshift"| R

    style G fill:#ffe1e1
    style L fill:#e1f5ff
    style R fill:#fff4e1

FRW Universe

In expanding universe $d{s}^2=-d{t}^2+a(t{)}^{2}d{\mathbf{x}}^2$:

For comoving observer ($d\mathbf{x}=0$):

$$d\tau =dt$$

$$\varphi =\frac{m{c}^2}{\hslash }t$$

Phase frequency:

$$\frac{d\varphi }{dt}=\frac{m{c}^2}{\hslash }$$

But: For photons propagating in universe, frequency redshifts!

$$\omega (t)=\frac{{\omega }_e\cdot a(t_e)}{a(t)}$$

Explanation: Photons have zero mass ($m=0$), formula $\varphi =(m{c}^2/\hslash )\int d\tau$ does not apply.

Need phase definition for massless particles (covered in next article).

🔬 Experimental Verification

1. Compton Scattering

$${\lambda }^{\prime }-\lambda =\frac{h}{mc}(1-\cos \theta )$$

Compton wavelength: ${\lambda }_C=h/(mc)=2\pi \hslash /(mc)$

Compton frequency: ${\omega }_C=2\pi c/{\lambda }_C=m{c}^2/\hslash$

Highly consistent with $\varphi =(m{c}^2/\hslash )\tau$.

2. Neutron Interference (COW Experiment)

Phase difference between upper and lower paths in gravitational field:

$$\Delta \varphi =\frac{m}{\hslash }\int (g\cdot h)dt=\frac{m{c}^2}{\hslash }\Delta \tau$$

Where $\Delta \tau$ is proper time difference caused by gravity.

Experimental result: Strongly supports phase-time relationship.

3. Atomic Clocks

Atomic clocks on GPS satellites, relative to ground have:

• Gravitational redshift (different $g_{tt}$)
• Motion time dilation (different velocity)

Combined effect:

$$\Delta \varphi =\frac{m{c}^2}{\hslash }\Delta \tau$$

GPS system corrects about 38 microseconds per day, completely consistent with relativity prediction.

📊 Phase as Time Scale

Phase is “Absolute”

Key insight:

In quantum mechanics, phase $\varphi$ has gauge freedom (additive constant), but phase difference $\Delta \varphi$ is physically observable.

Phase-time relationship:

$$\Delta \varphi =\frac{m{c}^2}{\hslash }\Delta \tau$$

Meaning:

• Left side: quantum observable (interference fringes)
• Right side: geometric proper time
• Phase can serve as operational definition of time.

Time Standards

Traditional time standards:

• Astronomical time (Earth rotation)
• Atomic clocks (cesium atom transitions)

Quantum-geometric time standard:

• Define time using phase $\varphi$
• $\tau =(\hslash /m{c}^2)\varphi$

Advantages:

• Universal (applies to all particles)
• Quantum-geometric unified
• Lorentz invariant

graph TB
    P["Quantum Phase φ"] --> T["Time Definition<br/>τ = (ℏ/mc²)φ"]

    T --> O1["Operational Definition<br/>Interference Experiment"]
    T --> O2["Theoretical Definition<br/>Proper Time"]
    T --> O3["Standard Definition<br/>Atomic Clock"]

    O1 -.->|"Consistent"| O2
    O2 -.->|"Consistent"| O3
    O3 -.->|"Consistent"| O1

    style P fill:#e1f5ff
    style T fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

💡 Profound Meaning

Unification of Quantum and Geometry

Traditional view:

• Quantum: wave function $\psi$, phase $\varphi$
• Geometry: metric $g_{\mu \nu }$, proper time $\tau$
• The two are independent

GLS view:

• $\varphi =(m{c}^2/\hslash )\int d\tau$
• Phase and geometry are equivalent in mathematical structure
• Quantum-geometric unification

Nature of Time

Question: What is time?

Traditional answers:

• Newton: absolute time
• Einstein: relative time (depends on observer)
• Quantum: external parameter

GLS answer:

• Time can be understood as geometric projection of phase
• $\tau =(\hslash /m{c}^2)\varphi$
• Time is essentially related to phase

Meaning of Compton Frequency

Why is Compton frequency ${\omega }_C=m{c}^2/\hslash$ so fundamental?

Answer: It is particle’s intrinsic “clock frequency”.

$$\frac{d\varphi }{d\tau }=\frac{m{c}^2}{\hslash }={\omega }_C$$

Physical meaning:

• Each particle carries an “intrinsic clock”
• Frequency determined by mass
• Mass determines intrinsic clock frequency

$$m=\frac{\hslash {\omega }_C}{c^2}$$

Profound: $E=m{c}^2$ and $E=\hslash \omega$ are unified.

📝 Key Formulas Summary

🎓 Further Reading

• Path integrals: R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals (1965)
• Relativistic action: L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields (1975)
• COW experiment: R. Colella, A.W. Overhauser, S.A. Werner, PRL 34, 1472 (1975)
• GLS theory: unified-time-scale-geometry.md
• Previous: 00-time-overview_en.md - Unified Time Overview
• Next: 02-scattering-phase_en.md - Scattering Phase and Group Delay

🤔 Exercises

1. Conceptual understanding:

   • Why is phase proportional to proper time, not coordinate time?
   • What is the physical meaning of Compton frequency?
   • Can phase serve as operational definition of time?

2. Calculation exercises:

   • Calculate Compton frequency ${\omega }_C$ of electron
   • For particle with velocity $v=0.6c$, calculate $d\varphi /dt$
   • In Schwarzschild spacetime at $r=3M$, calculate gravitational redshift

3. Physical applications:

   • How to explain GPS satellite time correction using phase-time relationship?
   • How does neutron interference experiment verify $\varphi \propto \tau$?
   • Can cosmological redshift be explained using phase? (Hint: see article 7)

4. Advanced thinking:

   • How to define phase for massless particles (photons)?
   • How to generalize phase-time relationship in quantum gravity?
   • Can we define a Lorentz-invariant “absolute time” using phase?

Next step: After understanding equivalence of phase and proper time, we will explore time in scattering theory—Wigner-Smith group delay!