[Paper Review] Fermions in Rotating Reference Frames

Adrian Manning|arXiv (Cornell University)|Dec 2, 2015

Atomic and Subatomic Physics Research20 references3 citations

TL;DR

This paper proposes a novel quantization scheme for fermions in rotating reference frames using cylindrical coordinates, addressing deficiencies in standard quantization that incorrectly predict vanishing radial current for single-particle states. By redefining field modes to represent free particles with definite radial momentum, the method enables consistent calculation of expectation values; explicit computation shows rotational non-inertial effects cancel, eliminating observable Zitterbewegung despite non-zero Bogoliubov mixing coefficients.

ABSTRACT

Current quantisations of fermions in cylindrical coordinates are shown to be inadequate in calculating some single-particle expectation values. This paper develops an alternate quantisation, applicable to one-particle states, which is generalised to rotating frames in cylindrical coordinates. Using this quantization, an explicit calculation of the velocity of a flat space free fermion as observed by a rotating observer is presented. This calculation demonstrates the validity of this quantisation and the cancellation of non-inertial rotational effects in the velocity, naively expected from the mixing of particle and anti-particle field operators.

Motivation & Objective

• To address the inadequacy of standard fermion quantization in cylindrical coordinates, which fails to describe free one-particle states with definite trajectories.
• To develop a generalized quantization method applicable to both flat and rotating spacetimes in cylindrical coordinates.
• To enable accurate calculation of single-particle expectation values—particularly velocity—under rotating reference frames.
• To resolve the paradox of apparent Zitterbewegung due to non-zero Bogoliubov coefficients in rotating frames.

Proposed method

• Introduce a new field quantization where quanta represent one-particle states with definite radial momentum, avoiding standing-wave solutions from symmetric boundary conditions.
• Use cylindrical tetrad formalism to derive the Dirac equation in rotating frames with constant angular velocity ω.
• Solve the Dirac equation using mode expansions in terms of Bessel functions, with quantum numbers tied to radial momentum and angular momentum.
• Compute Bogoliubov coefficients between flat and rotating frame quantizations to relate vacuum states and calculate expectation values.
• Apply the new quantization to compute the velocity expectation value of a free fermion as seen by a rotating observer.
• Simplify the resulting infinite sums using Bessel function identities and symmetry transformations to show cancellation of non-inertial effects.

Experimental results

Research questions

• RQ1 Why do standard quantization schemes in cylindrical coordinates fail to describe single-particle states with definite trajectories in rotating frames?
• RQ2 How can a consistent quantization of fermions be formulated in rotating reference frames using cylindrical coordinates?
• RQ3 Does the mixing of particle and antiparticle states in rotating frames—indicated by non-zero Bogoliubov β coefficients—lead to observable Zitterbewegung in the velocity of a free fermion?
• RQ4 Can the velocity of a free fermion as measured by a rotating observer be calculated consistently, and does it exhibit non-inertial effects?
• RQ5 What is the role of radial momentum quantization in ensuring the correct physical interpretation of one-particle states in non-inertial frames?

Key findings

• The standard quantization in cylindrical coordinates incorrectly predicts zero radial current for one-particle states due to unphysical boundary conditions at r=0.
• The proposed quantization, based on definite radial momentum modes, correctly describes free one-particle states and enables consistent calculation of expectation values.
• The velocity of a free fermion as observed by a rotating observer is found to be independent of rotation, with no observable Zitterbewegung despite non-zero Bogoliubov β coefficients.
• The infinite sums over angular momentum quantum numbers in the Bogoliubov coefficients simplify to a form identical to the flat-space case, confirming cancellation of non-inertial effects.
• The final expression for the velocity expectation value reduces to the inertial-frame result, demonstrating that rotational effects cancel in the physical observables.