In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, and especially
quantum field theory, an auxiliary
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
is one whose equations of motion admit a single solution. Therefore, the
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 ...
describing such a field
contains an algebraic quadratic term and an arbitrary linear term, while it contains no kinetic terms (derivatives of the field):
:
The equation of motion for
is
:
and the Lagrangian becomes
:
Auxiliary fields generally do not propagate, and hence the content of any theory can remain unchanged in many circumstances by adding such fields by hand.
If we have an initial Lagrangian
describing a field
, then the Lagrangian describing both fields is
:
Therefore, auxiliary fields can be employed to cancel quadratic terms in
in
and linearize the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
.
Examples of auxiliary fields are the complex scalar field
F in a
chiral superfield
In theoretical physics, a supermultiplet is a representation of a supersymmetry algebra.
Then a superfield is a field on superspace which is valued in such a representation. Naïvely, or when considering flat superspace, a superfield can simply ...
,
the real scalar field
D in a
vector superfield
In theoretical physics, there are many theories with supersymmetry (SUSY) which also have internal gauge symmetries. Supersymmetric gauge theory generalizes this notion.
Gauge theory
A gauge theory is a mathematical framework for analysing gau ...
, the scalar field B in
BRST and the field in the
Hubbard–Stratonovich transformation The Hubbard–Stratonovich (HS) transformation is an exact mathematical transformation invented by Russian physicist Ruslan L. Stratonovich and popularized by British physicist John Hubbard. It is used to convert a particle theory into its respect ...
.
The
quantum mechanical
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
effect of adding an auxiliary field is the same as the
classical, since the
path integral over such a field is
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. To wit:
:
References
Quantum field theory