Topology, symmetry, and phase transitions in lattice gauge ... - tuprints

G A U G E T H E O R I E S
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Before diving into the meat of this thesis, we will briefly review some formal aspects
of gauge theories. The intention is to establish conventions, notation, and a
context for the subsequent chapters.
2.1 gauge invariance and geometry
Remarkably, both of the pillars of modern physics are based on a local symmetry
principle. For Einstein’s General Relativity the symmetry is diffeomorphism
invariance of spacetime itself. For the field theories of the Standard Model, it is
an internal gauge symmetry of the field content that lives upon a spacetime background.
In each case, the theory is most beautifully formulated using a geometric
perspective.
Start with a complex scalar field φ(x) ∈ C for some spin-0 particle with mass m.
The free Lagrangian,
is invariant under global gauge transformations,
L = 1 2 (∂ µφ ∗ )(∂ µ φ) − 1 2 m2 φ ∗ φ, (2.1)
φ(x) → e −ieθ φ(x), (2.2)
that map the vector space C of field values on to itself. The electric charge e tells
us how rapidly the phase of φ rotates with θ. Demanding invariance under local
gauge transformations,
φ(x) → e −ieθ(x) φ(x), (2.3)
we may imagine that φ(x) maps to a unique copy of C at each spacetime point x,
and a local gauge transformation is equivalent to a change of basis for the field
space. Fiber bundles provide the appropriate framework for a geometric formulation.
1 In the language of fiber bundles, the vector space C is called the standard fiber,
while spacetime M is the base space. In this case, the fiber bundle is a manifold that
locally has the form of a product M × C but may have additional non-trivial global
structure. The gauge group is comprised of the set of gauge transformations of a
fiber on to itself, which is the abelian group U(1) in this case.
The derivative operator compares field values living at different spacetime points,
so a consistent way of transporting vectors from one fiber to another is required.
Such a connection replaces the ordinary derivative with a covariant one,
D µ φ = (∂ µ + ieA µ )φ, D µ φ ∗ = (∂ µ − ieA µ )φ ∗ . (2.4)
The introduction of a vector potential A µ (x) accounts for the infinitesimal change of
coordinates (phase) as we move between fibers belonging to neighboring spacetime
points.
1 See Ref. [21] for a readable introduction.
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