[Paper Review] Lectures on BRS invariance for massive boson fields

José M. Gracia-Bondı́a|arXiv (Cornell University)|Aug 21, 2008

Particle physics theoretical and experimental studies8 references3 citations

TL;DR

This paper provides a rigorous treatment of BRS invariance in massive spin-1 gauge theories, focusing on the Stückelberg formalism and the role of Krein operators in ghost field quantization. It resolves foundational issues in gauge theory quantization by introducing a non-local but causally consistent Krein operator, ensuring unitarity and gauge invariance in the presence of massive vector bosons.

ABSTRACT

These notes correspond to lectures given at the Villa de Leyva Summer School in Colombia (July 2007). Our main purpose in this short course on BRS invariance of gauge theories is to illuminate corners of the theory left in the shade by standard treatments. The plan is as follows. First a review of Utiyama's "general gauge theory". Promptly we find a counterexample to it in the shape of the massive spin-1 Stueckelberg gauge field. This is not fancy, as the massive case is the most natural one to introduce BRS invariance in the context of free quantum fields. Mathematically speaking, the first part of the course uses Utiyama's notation, and thus has the flavour and non-intrinsic notation of standard physics textbooks. Next we deal with boson fields on Fock space and BRS invariance in connection with the existence of Krein operators; the attending rigour points are then addressed.

Motivation & Objective

• To clarify the foundational issues in BRS invariance for massive vector fields, particularly in the context of the Stückelberg formalism.
• To address the lack of intrinsic geometric structure in standard treatments of gauge theories by emphasizing Utiyama’s first-principles derivation of gauge symmetry.
• To establish a mathematically consistent framework for BRS symmetry in massive spin-1 fields using Fock space and Krein operators.
• To resolve the conflict between locality, unitarity, and gauge invariance in the presence of ghost fields by constructing a proper Krein operator.
• To demonstrate that the standard ghost Lagrangian can be reformulated using a symmetric derivative structure to preserve gauge invariance and compatibility with the Krein metric.

Proposed method

• Uses Utiyama’s method to derive the minimal coupling of gauge fields to matter fields from global to local symmetry, emphasizing the role of the covariant derivative.
• Applies the Stückelberg formalism to introduce a massive vector field with a scalar auxiliary field, preserving gauge invariance and enabling a consistent quantization scheme.
• Introduces ghost fields $u, ilde{u}$ as free quantum fields on Fock space, with a Krein operator $\eta_{\rm gh}$ defined via a unitary transformation $S$ to ensure $\eta$-hermiticity.
• Constructs the ghost number operator $Q_{\rm gh}$ and defines the Krein operator $\eta_{\rm gh} = S I S^{-1} = T(\sigma_1)$, which ensures locality and correct ghost number transformation properties.
• Reformulates the interaction term $T_1$ using symmetric derivatives $\partial^\mu$ to ensure gauge invariance and compatibility with the $\eta_{\rm gh}$-hermitian structure.
• Establishes that the S-matrix is unitary on the physical subspace by combining $\eta$-unitarity with gauge invariance, using $\eta = \eta_A \otimes \eta_{\rm gh}$.

Experimental results

Research questions

• RQ1 How can BRS invariance be consistently implemented in massive vector field theories, particularly in the absence of massless gauge invariance?
• RQ2 What is the role of the Krein operator in ensuring unitarity and causality in the presence of ghost fields in massive gauge theories?
• RQ3 Why does the standard ghost Lagrangian fail to be $\eta$-hermitian, and how can it be corrected using symmetric derivatives?
• RQ4 How does the choice of Krein operator affect the locality and hermiticity of the ghost fields and their interactions?
• RQ5 What is the relationship between the Poincaré group representation and the $\eta$-unitarity structure in massive versus massless Yang-Mills theories?

Key findings

• The Stückelberg formalism provides a consistent framework for massive spin-1 fields that preserves gauge invariance and allows for a BRS quantization scheme.
• The ghost fields $u, \tilde{u}$ must be treated with a non-trivial Krein operator $\eta_{\rm gh}$ to ensure $\eta$-hermiticity and causality, avoiding the use of $u^\dagger, \tilde{u}^\dagger$.
• The operator $I = (-)^{N_{-1}}$ is not suitable as a Krein operator due to its non-locality in ghost number, but the transformed operator $\eta_{\rm gh} = S I S^{-1} = T(\sigma_1)$ restores locality and correct transformation properties.
• The interaction term $T_1$ can be rewritten using symmetric derivatives $\partial^\mu$ to ensure gauge invariance and compatibility with the $\eta_{\rm gh}$-hermitian structure, differing from the standard form by a $Q_{\rm gh}$-coboundary.
• The S-matrix is unitary on the physical subspace due to the combination of $\eta$-unitarity and gauge invariance, with $\eta = \eta_A \otimes \eta_{\rm gh}$, ensuring consistency in massive theories.
• In the massive case, the Poincaré representation is unitary and commutes with all charges, making $\eta$-unitarity automatic, unlike in the massless case where $\eta_A$ is introduced for covariance.