Primary field
Encyclopedia
In theoretical physics
, a primary field is a field operator in quantum field theory
- especially conformal field theory
or a theory with supersymmetry
- that is invariant under the positive frequency modes of the Virasoro algebra
- or under one half of the supersymmetries and superconformal generators.
In conformal field theory, a primary field is equivalent to a tensor field with well-defined weights and simple operator product expansion
with the stress-energy tensor
. In the state-operator correspondence, a primary field is associated with a highest-weight vector. One can also discuss quasi-primary fields, which transform as tensor representations of the global conformal group (as opposed to primary fields, which vary under local conformal transformations).
A descendant field or a secondary field is, on the contrary, a field that is not primary but a field that can be written as the action of negative frequency modes of the Virasoro algebra on a primary field. In terms of representation theory
, each irreducible representation contains one primary field and many descendant fields.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, a primary field is a field operator in quantum field theory
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...
- especially conformal field theory
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
or a theory with supersymmetry
Supersymmetry
In particle physics, supersymmetry is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartners...
- that is invariant under the positive frequency modes of the Virasoro algebra
Virasoro algebra
In mathematics, the Virasoro algebra is a complex Lie algebra, given as a central extension of the complex polynomial vector fields on the circle, and is widely used in conformal field theory and string theory....
- or under one half of the supersymmetries and superconformal generators.
In conformal field theory, a primary field is equivalent to a tensor field with well-defined weights and simple operator product expansion
Operator product expansion
- 2D Euclidean quantum field theory :In quantum field theory, the operator product expansion is a Laurent series expansion of two operators...
with the stress-energy tensor
Stress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
. In the state-operator correspondence, a primary field is associated with a highest-weight vector. One can also discuss quasi-primary fields, which transform as tensor representations of the global conformal group (as opposed to primary fields, which vary under local conformal transformations).
A descendant field or a secondary field is, on the contrary, a field that is not primary but a field that can be written as the action of negative frequency modes of the Virasoro algebra on a primary field. In terms of representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, each irreducible representation contains one primary field and many descendant fields.
See also
- Vertex algebra
- Knizhnik–Zamolodchikov equations
- Fusion rule