Quantum Field Theory Is Not Merely Quantum Mechanics Applied to Low Energy Effective Degrees of Freedom

TL;DR

This paper argues that quantum field theory possesses holistic features that cannot be fully captured by simply applying quantum mechanics to low-energy degrees of freedom, especially in curved spacetime contexts.

Contribution

It highlights the limitations of viewing quantum field theory as just quantum mechanics on effective degrees of freedom, emphasizing the holistic aspects and naturalness issues in renormalization.

Findings

Holistic aspects of quantum field theory are essential and cannot be reduced to quantum mechanics.

Renormalization procedures in curved spacetime are natural from QFT but ad hoc from a quantum mechanics perspective.

Implications for the cosmological constant problem are discussed.

Abstract

It is commonly assumed that quantum field theory arises by applying ordinary quantum mechanics to the low energy effective degrees of freedom of a more fundamental theory defined at ultra-high-energy/short-wavelength scales. We shall argue here that, even for free quantum fields, there are holistic aspects of quantum field theory that cannot be properly understood in this manner. Specifically, the ``subtractions'' needed to define nonlinear polynomial functions of a free quantum field in curved spacetime are quite simple and natural from the quantum field theoretic point of view, but are at best extremely ad hoc and unnatural if viewed as independent renormalizations of individual modes of the field. We illustrate this point by contrasting the analysis of the Casimir effect, the renormalization of the stress-energy tensor in time-dependent spacetimes, and anomalies from the point of quantum field theory and from the point of view of quantum mechanics applied to the independent low energy modes of the field. Some implications for the cosmological constant problem are discussed.