Unitarity and Lorentz in variance in QCD for a variety of gauges (2501.10315v1)

Abstract: We generalise the Kugo and Ojima formalism henceforth called KO) for the structure of state space in gauge theories, in four respects:- (i) We allow for a more general class of gauge-fixing including non-covariant cases. (ii) We display the two possible Hamiltonians allowed by the KO action. (iii) We give manifestly Lorentz invariant forms of KO, showing that they are not unique. (iv) We analyse the structure of Fock space, proving that the allowed states span exactly half of it (leaving aside pure transverse states).

• The paper expands the Kugo-Ojima formalism to incorporate a broader class of gauge-fixing conditions including non-covariant cases.
• It employs field equations, Fourier analysis, and canonical transformations to reveal conditions under which the free Hamiltonian diagonalizes.
• It demonstrates that allowed Fock states span only half the space, emphasizing the exclusive role of transverse states in preserving unitarity and Lorentz invariance.

Unitary and Lorentz Invariance in QCD for Various Gauges

This paper by A. Andraši and J. C. Taylor scrutinizes the applicability and generalization of the Kugo-Ojima (KO) formalism within Quantum Chromodynamics (QCD), particularly focusing on gauges beyond the traditional covariant gauge. Their paper embellishes the understanding of gauge theories by extending the KO framework in four distinctive ways: introducing a broader class of gauge-fixing conditions (including non-covariant cases), highlighting two possible Hamiltonians predicated on the KO action, presenting Lorentz covariant manifestations of the KO theory, and evaluating the structural composition of the Fock space to establish that permissible states exclusively span half of it.

The Kugo-Ojima formalism, drawing its foundational aspects from the construction of a nilpotent BRST operator $Q$ and the state-space structure within gauge theories, operates principally on the criteria that physical states are annihilated by $Q$. However, the paper exposes the intricacies and complexities that arise in the KO formalism when applied to gauges permitting a more comprehensive class than previously explored, notably those responding to a non-zero parameter, $s$.

A noteworthy contribution of the paper is its detailed explication of the generalized KO Lagrangian, which accommodates non-covariant gauges by incorporating a parameter $0$, reaching nonlinear effective descriptions when $s \neq 0$. Through formulating field equations and deducing Hamiltonians vis-à-vis variances of the canonical transformation, the paper argues for circumstances where the free Hamiltonian is only diagonal when $s = 0$, thus impacting the treatment of lightlike four-momentum.

In reviewing the free field asymptotic expansions and propagators, the paper meticulously details the Fourier transform of fields and analyses the interaction of creation and annihilation operators. This enables an interpretation that emphasizes the potential breaking of Lorentz invariance for non-zero $s$, despite overall covariant consistency when deriving propagators from KO expressions.

The profundity of the investigation culminates in demonstrating that allowed Fock states, besides the purely transverse states, span half of the Fock space. Through duality transformations and the examination of BRST operators $Q$ and its dual, this analysis affirms the criticality and specific utility of transverse states within the broader theoretical context.

The implications of these findings are profound, not only for advancing the KO formalism but also for their potential applications in refining our grasp of gauge theories under generalized conditions. Future advancements in AI and computational simulations in particle physics could leverage these theoretical insights to conceptualize more accurate models incorporating a wider spectrum of gauge conditions. Furthermore, the paper suggests that the ongoing exploration of general gauge theories could expand the boundaries of known physics while maintaining key invariance principles.

The paper provides a robust analytical scaffold for future theoretical endeavors aiming to reconcile unitary and Lorentz invariance within the frameworks of QCD and beyond.