[Paper Review] Localization of particles in quantum field theory

W. Westra|arXiv (Cornell University)|Dec 15, 2010

Quantum Mechanics and Applications63 references3 citations

TL;DR

This paper proposes a covariant, local probabilistic interpretation of scalar quantum field theory by replacing the non-local Newton-Wigner position operator with a bilinear, covariant position operator that enables consistent particle localization. By incorporating negative-frequency modes and using the Wheeler propagator and Hadamard two-point function, the framework restores Einstein causality and derives Heisenberg’s uncertainty relations from relativistic wave functions, enabling a first-quantized, Copenhagen-like interpretation of QFT.

ABSTRACT

We put forward an interpretation of scalar quantum field theory as relativistic quantum mechanics by curing well known problems related to locality. A probabilistic interpretation of quantum field theory similar to quantum mechanics is difficult if particle localization is defined using the Newton-Wigner position operator as it is non-local and non-covariant. An alternative bilinear covariant position operator is discussed which incorporates a time operator that can be exponentiated to a unitary operator. Moreover, it satisfies an algebra that unifies special relativity and quantum mechanics and has the same form for particles with spin. Higher power position operators are derived which yield Heisenberg's uncertainty relations. Our ideas are illustrated with a relativistic wave function whose probability density can be perfectly localized.

Motivation & Objective

• To resolve the non-locality and causality problems in standard relativistic quantum mechanics and quantum field theory.
• To provide a consistent first-quantized formulation of scalar QFT with a probabilistic interpretation akin to non-relativistic quantum mechanics.
• To replace the non-covariant, non-local Newton-Wigner position operator with a bilinear, covariant alternative that supports higher-power moment operators.
• To restore the Heisenberg uncertainty principle within a relativistic quantum field theory framework.
• To establish a Lorentz-invariant probability density that includes both positive and negative frequency modes.

Proposed method

• Introduce a new probability density using the field and its Hilbert transform, equivalent to the standard Klein-Gordon form but expressed in a basis that treats positive and negative frequency components symmetrically.
• Use the Wheeler propagator (real part of the Feynman propagator) as the time-ordered evolution kernel, ensuring causal time evolution and replacing the acausal Feynman propagator.
• Define a bilinear, covariant position operator that generalizes the Schrödinger picture position operator and avoids non-local terms in both position and momentum space.
• Derive higher-power position operators that are Hermitian with respect to relativistically normalized wave functions, enabling derivation of uncertainty relations.
• Construct a probability density from the Hadamard two-point function (statistical propagator), ensuring positivity and Lorentz invariance without restricting to positive frequencies.
• Show that the time-ordered evolution of real scalar fields is governed by the commutator two-point function, distinct from the probability inner product defined by the Hadamard function.

Experimental results

Research questions

• RQ1 Can a local, covariant, and probabilistically consistent particle localization be achieved in scalar quantum field theory?
• RQ2 How can Einstein causality be restored in relativistic wave function evolution without violating locality?
• RQ3 Can the Heisenberg uncertainty principle be derived from a relativistic wave function formalism using a new position operator?
• RQ4 What is the role of negative-frequency modes in maintaining causality and probability conservation in relativistic quantum mechanics?
• RQ5 How can the inner product and probability density be re-expressed in a manifestly Lorentz-invariant and causal form?

Key findings

• The proposed bilinear position operator is covariant, local, and generalizes the Schrödinger picture position operator without non-local terms in position or momentum space.
• The use of negative-frequency modes with the Wheeler propagator restores Einstein causality, ensuring wave function evolution respects light-cone structure.
• A new probability density is derived using the Hilbert transform and the Hadamard two-point function, which is positive definite and Lorentz-invariant, including both positive and negative frequency components.
• The framework allows derivation of Heisenberg’s uncertainty relations from relativistic wave functions via higher-power moment operators, restoring a core principle of quantum mechanics in QFT.
• The time evolution of wave functions is governed by the commutator two-point function (Wheeler propagator), while the probability density is defined by the Hadamard function, decoupling dynamics from the inner product.
• The standard normalization of the Klein-Gordon field is generalized to include both frequency components, yielding a Lorentz-invariant Born rule: $ P(k) = \frac{1}{2} \int dk_0 \delta(k^2 + m^2) |f(k)|^2 $.