On the Limits of the Thermofield-Double Interpretation of the Minkowski Vacuum

TL;DR

This paper critically examines the common interpretation of the Minkowski vacuum as a thermofield double state, showing that this analogy has limitations and is not exact, especially for higher-derivative correlators.

Contribution

It demonstrates that the Minkowski vacuum's TFD interpretation is an approximation and provides an alternative representation that challenges the standard view.

Findings

Mixed-derivative correlators match between methods

Higher-derivative correlators show persistent mismatches

A new non-thermal entangled-state representation is constructed

Abstract

The Minkowski vacuum is often presented in textbooks and reviews as a thermofield double (TFD) state, an entangled state of field modes in the left and right Rindler wedges. This picture is widely used to explain the Unruh effect, motivate entanglement entropy calculations, and connect quantum field theory to black hole thermodynamics and AdS/CFT. However, we show that this interpretation, while elegant, is not exact. We explicitly compute two-point functions and their derivatives for a massless scalar field in two-dimensional Minkowski space, comparing results obtained from canonical quantization with those obtained by assuming a TFD form of the vacuum. Mixed-derivative correlators agree perfectly, but higher-derivative correlators show systematic mismatches that persist even for points well away from horizons and are not removed by infrared regularization. To further test this picture, we construct an alternate coordinate system that divides Minkowski spacetime into two disconnected regions, apply the same derivation that leads to the standard TFD expression, and obtain a new "entangled-state" representation of the vacuum that is not thermal. This demonstrates that the appearance of a TFD structure is a feature of the derivation method rather than a fundamental property of the vacuum. Our results clarify the limits of interpreting the Minkowski vacuum as a literal TFD state, emphasizing that while it captures key thermal features, it should be viewed as a powerful calculational tool rather than a precise statement about Hilbert space structure.