Gauge Equivalence Classes in Yang-Mills Theory

Gauge equivalence classes are a cornerstone concept in the formulation and quantization of Yang-Mills fields. They represent the foundational idea that certain configurations of the gauge field, though distinct in form, are physically indistinguishable due to the underlying gauge symmetry. This principle is pivotal for the structure of the theory, impacting the quantization process, the definition of physical observables, and the Hamiltonian structure of the field theory.

Definition and Mathematical Formulation

The action of the gauge group $\mathcal{K}$ on the space of connections $\mathcal{D}$ is defined as follows:

$$ g \times A \mapsto g \circ A = -dgg^{-1} + gAg^{-1} $$

where $g \in \mathcal{K}$ represents an element of the gauge group, $A$ denotes a connection, $dgg^{-1}$ is the Maurer-Cartan form associated with $g$, and $gAg^{-1}$ signifies the adjoint action of $g$ on $A$. This action is free, which implies that the quotient space $\mathcal{D}/\mathcal{K}$, consisting of gauge equivalence classes, forms a smooth manifold.

Gauge Equivalence Classes

Each gauge equivalence class within the quotient space $\mathcal{D}/\mathcal{K}$ consists of all connections that can be transformed into each other by gauge transformations. These classes are essential for understanding the physical phase space of the Yang-Mills theory, which is effectively the space of these equivalence classes.

Role in Yang-Mills Hamiltonian

The Yang-Mills Hamiltonian, invariant under gauge transformations, is expressed in terms of the electric ($E$) and magnetic ($B$) fields:

$$ H = \int \frac{E^2 + B^2}{2} dx^3 $$

where the fields $E$ and $B$ satisfy the gauge-invariant Yang-Mills equations. The invariance of the Hamiltonian under gauge transformations underscores the significance of gauge equivalence classes in defining the theory's physical observables and dynamics.

Quantization and Gauge Fixing

For the non-perturbative quantization of the Yang-Mills field, gauge equivalence classes are crucial in defining the configuration space for the path integral. The measure in the path integral must uniquely integrate over each gauge equivalence class to account for the gauge symmetry. This requirement leads to the necessity of gauge fixing conditions, such as the Coulomb gauge or the Lorenz gauge, to select a unique representative from each class.

Geometric and Algebraic Structures

The interaction between gauge orbits and the gauge group's action on the space of connections unveils the deep geometric and algebraic structures within the Yang-Mills theory. For example, the moment map associated with the gauge action, relating to the divergence of the electric field, highlights the constraint equations that define physical states in the theory.

Conclusion

Gauge equivalence classes lay the groundwork for the physical interpretation, mathematical formulation, and quantization of Yang-Mills theories. They encapsulate the interplay between gauge symmetry and the theory's dynamical aspects, illustrating the intricate geometric structure that underlies gauge theories.

Related Concepts

• Yang-Mills Theory
• Path Integral Quantization
• Coulomb Gauge
• Lorenz Gauge
• Gauge Symmetry

For a deeper exploration of these topics and their mathematical formulations, consider delving into the detailed aspects of each concept through their respective pages.