Modelling financial time series with $φ^{“4”}$ quantum field theory
ArXiv ID: 2512.17225 “View on arXiv”
Authors: Dimitrios Bachtis, David S. Berman, Arabella Schelpe
Abstract
We use a $φ^{“4”}$ quantum field theory with inhomogeneous couplings and explicit symmetry-breaking to model an ensemble of financial time series from the S$&$P 500 index. The continuum nature of the $φ^4$ theory avoids the inaccuracies that occur in Ising-based models which require a discretization of the time series. We demonstrate this using the example of the 2008 global financial crisis. The $φ^{“4”}$ quantum field theory is expressive enough to reproduce the higher-order statistics such as the market kurtosis, which can serve as an indicator of possible market shocks. Accurate reproduction of high kurtosis is absent in binarized models. Therefore Ising models, despite being widely employed in econophysics, are incapable of fully representing empirical financial data, a limitation not present in the generalization of the $φ^{“4”}$ scalar field theory. We then investigate the scaling properties of the $φ^{“4”}$ machine learning algorithm and extract exponents which govern the behavior of the learned couplings (or weights and biases in ML language) in relation to the number of stocks in the model. Finally, we use our model to forecast the price changes of the AAPL, MSFT, and NVDA stocks. We conclude by discussing how the $φ^{“4”}$ scalar field theory could be used to build investment strategies and the possible intuitions that the QFT operations of dimensional compactification and renormalization can provide for financial modelling.
Keywords: Quantum Field Theory (QFT), φ⁴ Theory, Econophysics, Machine Learning Weights, Dimensional Compactification, Equities (S&P 500, Individual Stocks)
Complexity vs Empirical Score
- Math Complexity: 9.0/10
- Empirical Rigor: 3.0/10
- Quadrant: Lab Rats
- Why: The paper employs advanced quantum field theory concepts (ϕ⁴ theory, symmetry breaking, renormalization) and dense mathematical formulations, placing it in the high-complexity range, while its empirical validation is limited to a modest subset of stocks (20) and theoretical forecasting without live trading or extensive backtesting, resulting in low rigor.
flowchart TD
A["Research Goal:<br/>Model S&P 500 financial time series<br/>& forecast stock prices"] --> B["Methodology:<br/>φ^4 Quantum Field Theory<br/>(Inhomogeneous couplings, symmetry breaking)"]
B --> C["Data Input:<br/>S&P 500 Index & AAPL, MSFT, NVDA stocks"]
C --> D["Computational Process:<br/>Train ülf;^4 ML Algorithm<br/>(Extract couplings & scaling exponents)"]
D --> E["Outcome 1:<br/>Reproduces high kurtosis<br/>(Captures 2008 crisis shocks better than Ising)"]
D --> F["Outcome 2:<br/>Forecast price changes<br/>(Investment strategy foundation)"]
E & F --> G["Conclusion:<br/>φ^4 theory generalizes better for financial data<br/>Insights via dimensional compactification & renormalization"]