In
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experi ...
, background field method is a useful procedure to calculate the
effective action
In quantum field theory, the quantum effective action is a modified expression for the classical action taking into account quantum corrections while ensuring that the principle of least action applies, meaning that extremizing the effective act ...
of a
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 a ...
by expanding a quantum field around a classical "background" value ''B'':
:
.
After this is done, the Green's functions are evaluated as a function of the background. This approach has the advantage that the
gauge invariance
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
is manifestly preserved if the approach is applied to
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
.
Method
We typically want to calculate expressions like
:
where ''J''(''x'') is a source,
is the
Lagrangian density
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 ...
of the system, ''d'' is the number of dimensions and
is a field.
In the background field method, one starts by splitting this field into a classical background field ''B''(''x'') and a field η(''x'') containing additional quantum fluctuations:
:
Typically, ''B''(''x'') will be a solution of the classical equations of motion
:
where ''S'' is the action, i.e. the space integral of the Lagrangian density. Switching on a source ''J''(''x'') will change the equations into
:
.
Then the action is expanded around the background ''B''(''x''):
:
The second term in this expansion is zero by the equations of motion. The first term does not depend on any fluctuating fields, so that it can be brought out of the path integral. The result is
:
The path integral which now remains is (neglecting the corrections in the dots) of Gaussian form and can be integrated exactly:
:
where "det" signifies a
functional determinant In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order (representing a linear transformation from a finite-dimensional vector space to itself) to the ...
and ''C'' is a constant. The power of minus one half will naturally be plus one for
Grassmann fields.
The above derivation gives the Gaussian approximation to the functional integral. Corrections to this can be computed, producing a diagrammatic expansion.
See also
*
BF theory The BF model or BF theory is a topological field, which when quantized, becomes a topological quantum field theory. BF stands for background field B and F, as can be seen below, are also the variables appearing in the Lagrangian of the theory, whic ...
*
Effective action
In quantum field theory, the quantum effective action is a modified expression for the classical action taking into account quantum corrections while ensuring that the principle of least action applies, meaning that extremizing the effective act ...
References
*
*
*
* {{ cite journal
, last1=Abbott
, first1=L. F.
, title=Introduction to the Background Field Method
, journal=Acta Phys. Pol. B
, volume=13
, page=33
, year=1982
, url=http://isites.harvard.edu/fs/docs/icb.topic721083.files/Background_field_abbott.pdf
Quantum field theory