Vacuum manifold
Encyclopedia
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.
Vacuum manifolds arise during the process of spontaneous symmetry breaking
from a group G to a subgroup H and the corresponding vacuum manifold has to be a realization
of G and contain the quotient space
G/H. In many cases, it would simply be G/H, although it could be larger.
Not all vacuum manifolds arise due to spontaneous symmetry breaking. Supersymmetric models often contain moduli space
s which is another name for the vacuum manifold.
In many cases, the vacuum manifold is parameterized by the values of permissible vacuum expectation value
s. This is not the case for spontaneous symmetry breaking due to fermion condensation, though.
If the vacuum manifold is homotopically
nontrivial, it's possible for there to be topological sectors.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, the vacuum state
Vacuum state
In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
may be degenerate. Each pure vacuum state generates its own 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....
. The space of all pure vacuum states often has a manifold structure and is called the vacuum manifold.
Vacuum manifolds arise during the process of spontaneous symmetry breaking
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....
from a group G to a subgroup H and the corresponding vacuum manifold has to be a realization
Realization
realization and realisation may be:* Realization , a sport climbing route in Ceüse, France* Realization , the pricing of security at market value* Realization , an actually observed value of random variable...
of G and contain the quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
G/H. In many cases, it would simply be G/H, although it could be larger.
Not all vacuum manifolds arise due to spontaneous symmetry breaking. Supersymmetric models often contain moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
s which is another name for the vacuum manifold.
In many cases, the vacuum manifold is parameterized by the values of permissible vacuum expectation value
Vacuum expectation value
In quantum field theory the vacuum expectation value of an operator is its average, expected value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle...
s. This is not the case for spontaneous symmetry breaking due to fermion condensation, though.
If the vacuum manifold is homotopically
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
nontrivial, it's possible for there to be topological sectors.