A Macroscopic Financial Field Model Based on Quantum Field Theory:

A Macroscopic Financial Field Model Based on Quantum Field Theory: Analysis of US Treasury, Japanese Government Bonds, Eurozone Bonds and Global Financial Market Dynamics

This paper presents a macroscopic financial field model based on quantum field theory (QFT) that analyzes the dynamic coupling relationships between US Treasury bonds (UST), Japanese Government Bonds (JGB), Eurozone bonds (EZB), and the global financial markets. By mapping financial markets as interacting fields, we capture the non-linear interactive effects of interest rates, monetary policy, and cross-border capital flows, while exploring systemic risk propagation pathways and potential phase transition phenomena. This model provides policymakers and investors with new risk assessment tools and intuitively presents field dynamics through visualization schemes.

Introduction: Interdisciplinary Correspondence Between Field Theory and Financial Systems

Quantum field theory (QFT), as a theoretical framework for describing particle interactions, has in recent years been applied to complex system modeling, including financial markets. This paper views interest rates, monetary policy, and global capital flows as fields, using Lagrangian formalism to characterize the coupling dynamics between US Treasury bonds (φ₁), Japanese Government Bonds (φ₂), Eurozone bonds (φ₄), and the global financial market (φ₃), analyzing their stability and potential crisis trigger points.

Research Motivation

• US Treasury (UST): As the anchor of the global reserve currency, its yield fluctuations dominate global capital flows.

• Japanese Government Bonds (JGB): Constrained by the Bank of Japan's (BOJ) Yield Curve Control (YCC), serving as a funding source for global carry trades.

• Eurozone Bonds (EZB): Benchmarked by German government bonds, influenced by European Central Bank (ECB) policies and fiscal differences among member states.

• Global Financial Markets: Representing the sum of risk assets, driven by US, Japanese, and European interest rate differentials and policy changes.

Core Question: When interest rate differentials between the US, Japan, and Europe widen, or when monetary policy shifts direction, how do field interactions trigger systemic risk?

Field Theory Mapping: Field Model of Financial Systems

We model financial markets as four interacting fields, defining their properties and coupling relationships:

US Treasury Field (φ₁)

• Physical Correspondence: Dominant field, similar to a scalar field in QFT, driving the ground state of the global financial system.

• Financial Properties: US 10-year Treasury yield (approximately 4.2% as of May 2025) serves as the global risk-free rate benchmark, with the Federal Open Market Committee (FOMC) and US Treasury issuance size acting as source terms for the field.

• Coupling Effects: Rising US bond yields (φ₁ field excitation) lead to USD appreciation, increasing depreciation pressure on JPY and EUR, affecting carry trades.

Japanese Government Bond Field (φ₂)

• Physical Correspondence: Low-volatility field, similar to a "supercooled vacuum state," constrained by BOJ's YCC policy (10-year JGB yield approximately 1.0% as of May 2025).

• Financial Properties: Japan's long-term low interest rate environment makes it a core funding source for global carry trades. BOJ's gradual relaxation of YCC in 2024 led to field dynamic symmetry breaking.

• Coupling Effects: Widening US-Japan interest differential (4.2% - 1.0% = 3.2%) triggers capital flow from Japan to the US, amplifying global market volatility.

Eurozone Bond Field (φ₄)

• Physical Correspondence: Medium-volatility field, between φ₁ and φ₂, driven by ECB policy and fiscal heterogeneity among member states.

• Financial Properties: Benchmarked by German 10-year government bonds (Bund, approximately 2.5% yield as of May 2025), the Eurozone bond market exhibits internal heterogeneity due to member states' credit risk differences (e.g., Italian bond yields around 4.5%).

• Coupling Effects: US-Eurozone interest differential (4.2% - 2.5% = 1.7%) triggers capital flows to the US; Eurozone-Japan differential (2.5% - 1.0% = 1.5%) makes the Eurozone a secondary source of carry trade funding.

Global Financial Market Field (φ₃)

• Physical Correspondence: Composite field, representing the sum of global risk assets (e.g., MSCI World Index, emerging market bonds).

• Financial Properties: Its volatility is driven by US, Japanese, and European interest differentials and cross-border capital flows. From 2023-2025, global markets experienced increased volatility due to Fed rate hikes, BOJ policy shifts, and ECB tightening.

• Coupling Effects: Interactions between US bond, Japanese bond, and Eurozone bond fields affect the stability of the φ₃ field through capital flows.

Lagrangian Model: Mathematical Description of Fields

We construct a Lagrangian density to describe the dynamics and coupling of the four fields:

L = (1/2)(∂μ φ₁)² + (1/2)(∂μ φ₂)² + (1/2)(∂μ φ₃)² + (1/2)(∂μ φ₄)² - V(φ₁, φ₂, φ₃, φ₄)

The potential function is:

V(φ₁, φ₂, φ₃, φ₄) = (m₁²/2)φ₁² + (m₂²/2)φ₂² + (m₃²/2)φ₃² + (m₄²/2)φ₄² + g₁ φ₁ φ₂ + g₂ φ₁ φ₃ + g₃ φ₂ φ₃ + g₄ φ₁ φ₄ + g₅ φ₂ φ₄ + g₆ φ₃ φ₄ + λ (φ₁ φ₂ φ₃ φ₄)²

Parameter Explanation:

• m₁², m₂², m₃², m₄²: "Mass" of each field, reflecting inherent volatility (high for US bonds, low for Japan, medium for Eurozone, high for global markets).

• g₁: Coupling strength between US and Japanese bond markets, related to interest differentials and JPY/USD exchange rate (approximately 150 as of May 2025).

• g₂, g₃: Influence of US bonds and Japanese bonds on the global financial market, related to cross-border capital flows.

• g₄: Coupling between US bonds and Eurozone bonds, related to US-Euro interest differential (1.7%) and EUR/USD exchange rate (approximately 1.08).

• g₅: Coupling between Japanese bonds and Eurozone bonds, related to Euro-Japan interest differential (1.5%) and JPY/EUR exchange rate (approximately 162).

• g₆: Influence of Eurozone bonds on the global market, related to Eurozone capital flows.

• λ: Nonlinear interaction term capturing amplification effects from multilateral monetary policy shifts (e.g., simultaneous adjustments by the Fed, BOJ, and ECB).

Equations of Motion

Using the Euler-Lagrange equations, the dynamics of each field are:

Equation 1 (φ₁):

◻φ₁ + m₁²φ₁ + g₁φ₂ + g₂φ₃ + g₄φ₄ + 2λφ₁φ₂²φ₃²φ₄² = 0

Equation 2 (φ₂):

◻φ₂ + m₂²φ₂ + g₁φ₁ + g₃φ₃ + g₅φ₄ + 2λφ₁²φ₂φ₃²φ₄² = 0

Equation 3 (φ₃):

◻φ₃ + m₃²φ₃ + g₂φ₁ + g₃φ₂ + g₆φ₄ + 2λφ₁²φ₂²φ₃φ₄² = 0

Equation 4 (φ₄):

◻φ₄ + m₄²φ₄ + g₄φ₁ + g₅φ₂ + g₆φ₃ + 2λφ₁²φ₂²φ₃²φ₄ = 0

These equations describe the nonlinear coupling and feedback among the fields.

Dynamics and Phase Transitions: Stability of Financial Markets

1. Stability Analysis of Fields

• US Treasury Field (φ₁): Fed rate hikes (2022-2023, benchmark rates rising to 5.25%-5.5%) excite the φ₁ field, pushing US bond yields to 4.2%, triggering capital flows from emerging markets and the Eurozone back to the US, compressing φ₃ and φ₄ fields.

• Japanese Bond Field (φ₂): BOJ’s YCC policy keeps φ₂ near static, but the 2024 relaxation of YCC (10-year JGB yield rising to 1.0%) breaks the symmetry of the field dynamics, triggering capital outflows and JPY depreciation (150 JPY/USD).

• Eurozone Bond Field (φ₄): ECB tightening (2023-2025, benchmark rates rising to 4%) increases φ₄ volatility; debt risk in Southern European countries (e.g., Italy, 4.5% yield) amplifies internal heterogeneity.

• Global Market Field (φ₃): Interest differentials—US-Japan (3.2%), US-Eurozone (1.7%), Eurozone-Japan (1.5%)—via coupling terms g₁, g₄, g₅, increase φ₃ volatility, potentially triggering risk asset repricing.

2. Phase Transitions and Systemic Risk

• Financial Phase Transitions: Rapid US bond yield increases (e.g., reaching 5% in October 2023) or BOJ’s full exit from YCC cause φ₁ or φ₂ fields to lose stability, triggering capital flow “vacuum bubble” collapses analogous to false vacuum decay in QFT.

• ECB Accelerated Rate Hikes or Eurozone Debt Crisis (e.g., Italy-Germany spread widening to 200 basis points) induce local phase transitions in φ₄, which propagate to φ₃ via g₆.

• Empirical Observations: From 2022-2023, rising US yields and JPY depreciation (from 130 to 150) increased global stock market volatility (VIX rising from 15 to 25). The 2022 Russia-Ukraine conflict caused Eurozone bond volatility, with German Bund yields rising from 0.2% to 2.5%, and VIX rising simultaneously, illustrating φ₄’s impact on φ₃.

Visualization Implementation

To intuitively display the interactive dynamics of the four fields, we designed the following charts using Chart.js:

1. Potential Energy Landscape of the Four Fields

Shows the potential energy surfaces of φ₁, φ₂, φ₃, and φ₄, reflecting each field’s stability.

(Chart configuration code omitted for brevity)

2. Coupling Field Volatility Heatmap

Displays the relationship between US-Japan coupling (g₁), US-Euro coupling (g₄), and global market volatility (φ₃).

(Chart configuration code omitted for brevity)

3. Financial Vacuum Bubble Animation

Using D3.js to simulate abrupt capital flow changes under the four-field interaction, highlighting local instability in φ₄ triggered by ECB policy or Eurozone debt crises.

Conclusion and Policy Implications

Theoretical Contributions

• Field Theory Perspective: Models US, Japanese, and Eurozone bond markets and the global financial market as coupled fields, capturing nonlinear dynamics and systemic risk propagation.

• Phase Transition Analysis: Reveals market instability triggered by monetary policy shifts (BOJ’s YCC exit, ECB rate hikes) or Eurozone internal debt crises.

• Risk Detection: Identifies potential formation and collapse of financial “vacuum bubbles” through field potential and coupling parameters.

Policy Recommendations

• Central Bank Coordination: The Fed, BOJ, and ECB should enhance policy communication to avoid rapid multilateral interest differential expansions that trigger capital flow crises.

• Risk Monitoring: Develop field theory-based risk indicators to track the impact of US-Japan (g₁), US-Euro (g₄), and Eurozone internal spreads on the φ₃ field.

• Investment Strategies: Investors can utilize the four-field coupling model to forecast carry trade risks and returns, designing multi-asset allocation strategies.

Future Research Directions

• Refine the internal structure of the φ₄ field by modeling interactions among subfields representing Germany, Italy, and other member states.

• Calibrate coupling parameters g₁ through g₆ and λ using historical data (bond yields and exchange rates from 2022-2025).

• Simulate field phase transition behavior under extreme scenarios such as global debt crises or geopolitical shocks.

References

• Baaquie, B. E. (2010). Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates. Cambridge University Press.

• Ilinski, K. (2001). Physics of Finance: Gauge Modelling in Non-Equilibrium Pricing. Wiley.

• Bank of Japan, Monetary Policy Reports (2023-2025).

• Federal Reserve, FOMC Statements (2023-2025).

• European Central Bank, Monetary Policy Decisions (2023-2025).

• Deutsche Bundesbank, German Bund Yield Data (2025).

• MSCI World Index Data, Bloomberg Terminal (2025).

U.S. Treasury Field (φ₁)

Linear relationship with 1:1 field strength to energy ratio. Represents the dominant influence of U.S. Treasury markets.

Japanese Government Bond Field (φ₂)

Lowest energy scaling (0.1x), indicating more stable, lower volatility characteristics of Japanese bond markets.

Global Financial Field (φ₃)

Highest energy scaling (1.1x base), representing amplified global market dynamics and interconnectedness.

Eurozone Bond Field (φ₄)

Moderate energy scaling (0.8x), reflecting the intermediate volatility of European bond markets.

Key Observations:

• • All fields show linear energy-field strength relationships
• • Global Financial Field exhibits the steepest energy gradient
• • Japanese bonds show the most conservative energy scaling
• • Clear hierarchy: Global > U.S. Treasury > Eurozone > Japanese bonds

US-Japan Coupling (g₁)

Shows higher coupling strengths (1-5) with corresponding volatility range of 15-35

US-Eurozone Coupling (g₄)

Shows lower coupling strengths (0.5-2.5) with volatility range of 10-30

      GitHub Link : https://github.com/fullyloaded/

φ₁ - U.S. Treasury Field

The most stable field providing systematic capital attraction. Blue particles represent capital inflows.

φ₂ - Japanese Government Bonds Field

High-stability safe-haven field. Orange particles show safe-haven capital flows to Japanese bonds.

φ₃ - Global Financial Field

The largest field representing overall global financial market dynamics and interconnectedness.

φ₄ - Eurozone Bonds Field (Crisis Focus)

The most unstable field. During ECB policy changes or debt crises, it generates capital flow disruptions and vacuum bubble effects.

Animation Effects Explanation:

• • Normal State: Particles flow smoothly between fields, representing normal capital allocation
• • φ₄ Crisis State: Eurozone field generates repulsion forces, causing capital outflows and vacuum bubble formation
• • Vacuum Bubble Effect: Red circles represent areas where market liquidity suddenly vanishes
• • Field Interactions: Attractive and repulsive forces between fields simulate real capital flow dynamics
• • Video Export: Click "Export Video" to record 10 seconds of animation and auto-download WebM format video

This paper integrates field-theoretic analysis of US, Japanese, and Eurozone bond markets with the global financial market, providing a comprehensive theoretical framework and visualization scheme.