't Hooft-Polyakov monopole
Encyclopedia
In theoretical physics
, the 't Hooft–Polyakov monopole is a topological soliton similar to the Dirac monopole but without any singularities. It arises in the case of a Yang–Mills theory with a gauge group G, coupled to a Higgs field which spontaneously breaks
it down to a smaller group H via the Higgs mechanism
. It was first found independently by Gerard 't Hooft
and Alexander Polyakov.
Unlike the Dirac monopole, the 't Hooft–Polyakov monopole is a smooth solution with a finite total energy
. The solution is localized around
. Very far from the origin, the gauge group G is broken to H, and the 't Hooft–Polyakov monopole reduces to the Dirac monopole.
However, at the origin itself, the G gauge symmetry is unbroken and the solution is non-singular also near the origin. The Higgs field
is proportional to
where the adjoint indices are identified with the three-dimensional spatial indices. The gauge field at infinity is such that the Higgs field's dependence on the angular directions is pure gauge. The precise configuration for the Higgs field and the gauge field near the origin is such that it satisfies the full Yang–Mills–Higgs equations
of motion.
Σ. Then, for finite energies, as we move along each direction towards spatial infinity, the state along the path approaches a point on the vacuum manifold Σ. Otherwise, we would not have a finite energy. In topologically trivial 3 + 1 dimensions, this means spatial infinity is homotopically equivalent to the topological sphere S2. So, the superselection sector
s are classified by the second homotopy group of Σ, π2(Σ).
In the special case of a Yang–Mills–Higgs theory, the vacuum manifold is isomorphic to the quotient space G/H and the relevant homotopy group is π2(G/H). Note that this doesn't actually require the existence of a scalar Higgs field. Most symmetry breaking mechanisms (e.g. technicolor) would also give rise to a 't Hooft–Polyakov monopole.
It's easy to generalize to the case of d + 1 dimensions. We have πd−1(Σ).
theory. Cosmic inflation
remedies the situation by diluting any primordial abundance of magnetic monopoles.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, the 't Hooft–Polyakov monopole is a topological soliton similar to the Dirac monopole but without any singularities. It arises in the case of a Yang–Mills theory with a gauge group G, coupled to a Higgs field which spontaneously breaks
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
it down to a smaller group H via the Higgs mechanism
Higgs mechanism
In particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....
. It was first found independently by Gerard 't Hooft
Gerardus 't Hooft
Gerardus 't Hooft is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G...
and Alexander Polyakov.
Unlike the Dirac monopole, the 't Hooft–Polyakov monopole is a smooth solution with a finite total energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
. The solution is localized around
. Very far from the origin, the gauge group G is broken to H, and the 't Hooft–Polyakov monopole reduces to the Dirac monopole.However, at the origin itself, the G gauge symmetry is unbroken and the solution is non-singular also near the origin. The Higgs field
is proportional to
where the adjoint indices are identified with the three-dimensional spatial indices. The gauge field at infinity is such that the Higgs field's dependence on the angular directions is pure gauge. The precise configuration for the Higgs field and the gauge field near the origin is such that it satisfies the full Yang–Mills–Higgs equations
Yang–Mills–Higgs equations
In mathematics, the Yang–Mills–Higgs equations are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a Higgs field, given by a section of a vector bundle...
of motion.
Mathematical details
Suppose the vacuum is the vacuum manifoldVacuum manifold
In quantum field theory, the vacuum state may be degenerate. Each pure vacuum state generates its own superselection sector. The space of all pure vacuum states often has a manifold structure and is called the vacuum manifold....
Σ. Then, for finite energies, as we move along each direction towards spatial infinity, the state along the path approaches a point on the vacuum manifold Σ. Otherwise, we would not have a finite energy. In topologically trivial 3 + 1 dimensions, this means spatial infinity is homotopically equivalent to the topological sphere S2. So, the superselection sector
Superselection sector
In Quantum mechanics, superselection extends the concept of selection rules.Superselection rules are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables....
s are classified by the second homotopy group of Σ, π2(Σ).
In the special case of a Yang–Mills–Higgs theory, the vacuum manifold is isomorphic to the quotient space G/H and the relevant homotopy group is π2(G/H). Note that this doesn't actually require the existence of a scalar Higgs field. Most symmetry breaking mechanisms (e.g. technicolor) would also give rise to a 't Hooft–Polyakov monopole.
It's easy to generalize to the case of d + 1 dimensions. We have πd−1(Σ).
Monopole problem
The "monopole problem" refers to the cosmological implications of Grand unification theories (GUT). Since monopoles are generically produced in GUT during the cooling of the universe, and since they are expected to be quite massive, their existence threatens to overclose it. This is considered a "problem" within the standard Big BangBig Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
theory. Cosmic inflation
Cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...
remedies the situation by diluting any primordial abundance of magnetic monopoles.