In
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, a non-topological soliton (NTS) is a
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
field configuration possessing, contrary to a
topological one, a conserved
Noether charge
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in ...
and stable against transformation into usual particles of this field for the following reason. For fixed charge ''Q'', the mass sum of ''Q'' free particles exceeds the energy (mass) of the NTS so that the latter is energetically favorable to exist.
The interior region of an NTS is occupied by
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
different from the ambient vacuum. The vacuums are separated by the surface of the NTS representing a
domain wall
A domain wall is a type of topological soliton that occurs whenever a discrete symmetry is spontaneously broken. Domain walls are also sometimes called kinks in analogy with closely related kink solution of the sine-Gordon model or models with pol ...
configuration (
topological defect
A topological soliton occurs when two adjoining structures or spaces are in some way "out of phase" with each other in ways that make a seamless transition between them impossible. One of the simplest and most commonplace examples of a topological ...
), which also appears in field theories with broken
discrete symmetry
In mathematics and geometry, a discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square' ...
. Infinite domain walls contradict
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount (lexicographer), Thomas Blount's ''Glossographia'', and in 1731 taken up in ...
, but the surface of an NTS is closed and finite, so its existence would not be contradictory. If the topological domain wall is closed, it shrinks because of wall tension; however, due to the structure of the NTS surface, it does not shrink since the decrease of the NTS volume would increase its energy.
Introduction
Quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
has been developed to describe the elementary particles. However, in the mid 1970s it was found out that this theory predicts one more class of stable compact objects: non-topological solitons (NTS). The NTS represents an unusual coherent state of matter, called also bulk matter. Models were suggested for the NTS to exist in forms of stars, quasars, the dark matter and nuclear matter.
A NTS configuration is the lowest energy solution of classical equations of motion possessing a spherical symmetry. Such a solution has been found for a rich variety of field
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
s. One can associate the
conserved charge
Conservation is the preservation or efficient use of resources, or the conservation of various quantities under physical laws.
Conservation may also refer to:
Environment and natural resources
* Nature conservation, the protection and manageme ...
with global, local,
Abelian and
non-Abelian symmetry. It appears to be possible that the NTS configuration exists with both
bosons
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer spi ...
as well as with
fermions. In different models either one and the same field carries the charge and binds the NTS, or there are two different fields: charge carrier and binding field.
The spatial size of the NTS configuration may be elementary small or astronomically large, depending on the model fields and constants. The NTS size could increase with its energy until the gravitation complicates its behavior and finally causes the collapse. In some models, the NTS charge is bounded by the stability (or metastability) condition.
Simple examples
One field
For a complex scalar field with the U(1) invariant Lagrange density
:
the NTS is a ball with radius R filled with the field
. Here
is a constant inside the ball except for a thin surface coat where it sharply drops to the global U(1) symmetrical minimum of
. The value
is adjusted so that it minimises the energy of the configuration
:
Since the ''U''(1) symmetry gives the conserved current
the ball possesses the conserved charge
:
The minimization of the energy (1) with R gives
:
The charge conservation allows the decay of the ball into Q particles exactly. This decay is energetically unprofitable if the sum mass Qm exceed the energy (2). Therefore, for the NTS existence it is necessary to have
:
The thin wall approximation, which was used above, allows to omit the gradient term
in the expression for energy (1), since
. This approximation is valid for
and is justified by the exact solution of the motion equation.
Two fields
The NTS configuration for a couple of interacting scalar fields
is sketched here.
The Lagrange density
:
is invariant under U(1) transformation of the complex scalar field
Let this field depends on time and coordinate simply as
. It carries the conserved charge
. In order to check that the energy of the configuration is smaller than Qm, one should either to calculate this energy numerically or to use the variational method. For trial functions
and
for ''r'' < ''R'',
:
the energy in the large Q limit is approximately equal to
.
The minimization with R gives the upper estimation
for the energy of the exact solution of motion equations
and
.
It is indeed smaller than
for Q exceeding the crucial charge
:
Fermion plus scalar
If instead of boson, fermions carry the conserved charge, an NTS also exists. At this time one could take
:
''N'' is the number of fermion species in the theory. Q can't exceed N due to the
Pauli exclusive principle
In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
if the fermions are in the coherent state. This time the NTS energy E is bound by
:
See Friedberg/Lee.
Stability
Classical stability
The condition