Gauge transformation

gradient transform

A transformation in classical and quantum field theory which alters non-observable properties of fields (e.g. potentials) without changing the physically-meaningful measurable magnitudes (e.g. intensities). The name "gauge transformation" (or "gradient transformation" ) arose in the classical theory of electromagnetic fields. In this theory the four-dimensional electromagnetic vector potential $ A _ {n} ( x) $, $ n = 0, 1, 2, 3 $, is introduced in an ambiguous manner, since the so-called gauge transformations of the second kind:

$$ \tag{1 } A _ {n} ( x) \rightarrow A _ {n} ^ \prime ( x) = \ A _ {n} ( x) + \frac{\partial f }{\partial x ^ {n} } , $$

with an arbitrary function $ f $ having partial derivatives of the first and second orders, do not affect the values of the components of the anti-symmetric electromagnetic field tensor

$$ F _ {kl} = \ \frac{\partial A _ {l} }{\partial x ^ {k} } - \frac{\partial A _ {k} }{\partial x ^ {l} } ,\ \ k, l = 0, 1, 2, 3. $$

These components are equal to the physically-observable components of the intensity vectors of the electric field $ E _ \alpha = F _ {\alpha 0 } $, $ \alpha = 1, 2, 3 $, and of the magnetic field $ H _ {1} = F _ {23} $; $ H _ {2} = - F _ {31} $; $ H _ {3} = F _ {12} $. Under a gauge transformation of the second kind the field equations

$$ \sum _ { k } g ^ {kk} \left \{ \frac{\partial ^ {2} A _ {n} }{\partial x ^ {k} \partial x ^ {k} } - \frac{\partial ^ {2} A _ {k} }{\partial x ^ {k} \partial x ^ {n} } \right \} = 0, $$

$$ g ^ {11} = g ^ {22} = g ^ {33} = - g ^ {00} = - 1, $$

remain unchanged; this is the so-called gauge invariance of field theory. If the function $ f $ is suitably chosen, the $ A _ {n} $ can be made to satisfy some additional condition (the so-called gauge condition), whereby the form of the field equations can be simplified. Thus, the Lorentz condition, which is linear with respect to the vector potential $ A _ {n} $, $ n = 0, 1, 2, 3 $, stating that its four-dimensional divergence vanishes:

$$ \tag{2 } \sum _ { k } g ^ {kk} \frac{\partial A _ {k} ( x) }{\partial x ^ {k} } = 0, $$

leads to d'Alembert's equation for $ A _ {n} ( x) $( cf. d'Alembert operator):

$$ \tag{3 } - \sum _ { k } g ^ {kk} \frac{\partial ^ {2} A _ {n} }{\partial x ^ {k} \partial x ^ {k} } = \ \square A _ {n} ( x) = 0. $$

The Lorentz condition does not completely determine the $ A _ {n} $ since, in theory, the invariance with respect to a so-called specialized gauge transformation (1) of the second kind with a function $ f _ {0} $ that satisfies d'Alembert's equation $ \square f _ {0} ( x) = 0 $ remains. However, if $ f _ {0} $ is selected in a certain manner (for a Lorentz coordinate system), the condition $ A _ {0} = 0 $ will be met. The Lorentz condition (2) is then reduced to the condition $ \mathop{\rm div} \mathbf A = 0 $ for the three-dimensional vector potential, viz. the electromagnetic field is to be transverse. If the field is complex-valued, the theory must also be invariant with respect to gauge transformations of the first kind for the wave functions of the field and their derivatives:

$$ \tag{4 } \Phi ( x) \rightarrow \Phi ^ \prime ( x) = \ e ^ {i \alpha } \Phi ( x); $$

$$ \Phi ^ {*} ( x) \rightarrow \Phi ^ {*\prime} ( x) = e ^ {- i \alpha } \Phi ^ {*} ( x), $$

since all the observable dynamic magnitudes must (in view of the condition of realness (Hermiticity)) be expressed solely in forms that are bilinear with respect to $ \Phi ^ {*} $ and $ \Phi $. Owing to the general principles of mechanics, the condition of invariance of a field theory with respect to the gauge transformations of the first kind means that certain observable physical magnitudes must be conserved — such as (electric) charges, which are expressed in terms of field functions, or, in other words, that conservation laws of these charges must exist. The corresponding Lagrangians and field equations should be invariant with respect to gauge transformations. For instance, in the case of complex-valued fields $ \psi ( x) $ interacting with the electromagnetic field $ A _ {n} ( x) $ that correspond to electric charges, the Lagrangians of the free fields and the Lagrangian of interaction and the field equation should be invariant with respect to gauge transformations of the type

$$ \tag{5 } \left . \begin{array}{c} \psi ( x) \rightarrow e ^ {i f ( x) } \psi ^ {*} ( x); \ \ \psi ^ {*} ( x) \rightarrow e ^ {- i f ( x) } \psi ^ {*} ( x); \\ A _ {n} ( x) \rightarrow A _ {n} ( x) + \frac{\partial f ( x) }{\partial x ^ {n} } , \\ \end{array} \right \} $$

i.e. with respect to the gauge transformations (1) and (4), where the phase multiplier in (4) may depend on the four-dimensional space-time coordinates and should coincide with the arbitrary scalar functions forming part of (1). In such a case the law of conservation of electric charge is fulfilled in the system of fields $ A _ {n} ( x) $ and $ \psi ( x) $. The gauge transformations (5) form an Abelian group of transformations in the sense that the gauge function $ f ( x) = g( x) + h( x) $ describes a gauge transformation representing two gauge transformations with functions $ g $ and $ h $ performed in any order. In constructing theories of interacting fields which are more involved than the example given above, a necessary condition for meeting the respective conditions of the conservation of any charges is the invariance of the Lagrangians and the field equations with respect to gauge transformations in which the gauge functions $ f $ are operators.

References

[1]	N.N. Bogolyubov, D.V. Shirkov, "Introduction to the theory of quantized fields" , Interscience (1959) (Translated from Russian)

Comments

Gauge field theories are at present a major area of research in mathematics and theoretical physics. Abstractly (mathematically) speaking, the starting object is a principal fibre bundle $ P $ over a manifold $ M $( $ = $ space-time) with structure group $ G $. A gauge potential is then a connection on this bundle, cf. also Yang–Mills field; the associated curvature of the connection represents the field strength tensor. The term gauge field is used for both the connection (form) and its associated curvature field. A gauge transformation is a bundle automorphism of $ P $( leaving $ M $ pointwise fixed). Such automorphisms form the group of gauge transformations. The structure group $ G $ of $ P $ is called the gauge group, e.g. $ U ( 1) $ in the case of electromagnetism. Cf. [a3] for a very useful glossary and dictionary of mathematical and physical terminology in this area.

There are deep connections between topology and gauge field theory. A major result of the interaction between these areas was the discovery of the exotic structures on $ \mathbf R ^ {4} $ by S. Donaldson and M. Freedman (cf., e.g., [a4]).

The rarely used phrase "gradient transformation" comes from the fact that in electromagnetism the electromagnetic potential $ A _ {i} $ is determined by the electromagnetic field tensor $ F _ {ij} $ up to a gradient vector (field).

References

[a1]	K. Moriyasu, "An elementary primer for gauge theory" , World Sci. (1983)
[a2]	L. O'Raifertaigh, "Group structure of gauge theories" , Cambridge Univ. Press (1986)
[a3]	M.E. Mayer, "Geometric aspects of gauge theory" L.P. Horowitz (ed.) Y. Ne'eman (ed.) , Proc. VII-Colloq. Group Theoretical Methods in Physics , Israel Physical Society & A. Hilger (1980) pp. 80–99
[a4]	M. Blaine Lawson jr., "The theory of gauge fields in four dimensions" , Amer. Math. Soc. (1985)
[a5]	M.F. Atiyah, "Geometry of Yang–Mills fields" , Accad. Naz. dei Lincei (1979)