Appendix D: Global and Local Gauge Symmetries
255
Only depends on the behaviour of the solutions at space infinity and therefore is
invariant under local deformations, displaying a sort of topological content. The
charge Q
a , being given by a flux at infinity, has a non-local content.
Thus, the elaborate intricacy of choosing a field algebra F transforming as a
representation of a local gauge group G, a Lagrangian density L invariant under G,
a gauge fixing which breaks G invariance to the extent of yielding a deterministic
time evolution of F, a vacuum representation of F and a subsidiary condition which
selects the physical states are merely instrumental for eventually finding the physical
representations and the time evolution of the local algebra of observables, characterized as the local subalgebra of F pointwise invariant under G. In this process, the
starting local gauge group G is doomed to disappear, reducing to the identity at the
end; the surviving relevant effect of such a procedure appears to be the validity of
local Gauss laws on the physical states, for the currents which generate the related
global gauge group G.
This is the physically relevant property which characterizes the so-called (local)
gauge theories with respect to the standard field theories.
In fact, it is directly responsible for the main physical properties of such theories
with respect to standard quantum field theories
37 :
1) states carrying a (global) charge defined by a current which satisfies a local
Gauss, briefly a Gauss charge, cannot be localized;
2) particles carrying a Gauss charge are not Wigner particles, since they cannot
have a definite mass (called infraparticles); this applies in particular to electrically charged particles;
3) an (unbroken) Gauss charge defines a superselection rule, namely one cannot
observe a coherent superposition of states with different charges; this applies in
particular to the electric charge;
4) the local Gauss law, implying a Coulomb delocalization of the charged fields
provides a direct mechanism for evading the Goldstone theorem.
Such information are not directly provided in the Wilson gauge, defined by a
Lagrangian invariant under the local gauge group, without the addition of a gauge
fixing, which yields only the vacuum sector of the observables; non-trivial charged
fields are not represented in the vacuum sector and states carrying a Gauss charge
cannot be constructed in terms of a charged field algebra.
In conclusion, the pretentious “Gauge Principle”, advocated as “the most fundamental cornerstone of modern theoretical physics”, appears to be downgraded to
a merely technical tool which guarantees the validity of the much more significant
local Gauss laws on the physical states, appearing as the physically relevant and
characteristic property of gauge theories.
37 This point is discussed and stressed in F. Strocchi [13, 16], Chap. 7.
255
Only depends on the behaviour of the solutions at space infinity and therefore is
invariant under local deformations, displaying a sort of topological content. The
charge Q
a , being given by a flux at infinity, has a non-local content.
Thus, the elaborate intricacy of choosing a field algebra F transforming as a
representation of a local gauge group G, a Lagrangian density L invariant under G,
a gauge fixing which breaks G invariance to the extent of yielding a deterministic
time evolution of F, a vacuum representation of F and a subsidiary condition which
selects the physical states are merely instrumental for eventually finding the physical
representations and the time evolution of the local algebra of observables, characterized as the local subalgebra of F pointwise invariant under G. In this process, the
starting local gauge group G is doomed to disappear, reducing to the identity at the
end; the surviving relevant effect of such a procedure appears to be the validity of
local Gauss laws on the physical states, for the currents which generate the related
global gauge group G.
This is the physically relevant property which characterizes the so-called (local)
gauge theories with respect to the standard field theories.
In fact, it is directly responsible for the main physical properties of such theories
with respect to standard quantum field theories
37 :
1) states carrying a (global) charge defined by a current which satisfies a local
Gauss, briefly a Gauss charge, cannot be localized;
2) particles carrying a Gauss charge are not Wigner particles, since they cannot
have a definite mass (called infraparticles); this applies in particular to electrically charged particles;
3) an (unbroken) Gauss charge defines a superselection rule, namely one cannot
observe a coherent superposition of states with different charges; this applies in
particular to the electric charge;
4) the local Gauss law, implying a Coulomb delocalization of the charged fields
provides a direct mechanism for evading the Goldstone theorem.
Such information are not directly provided in the Wilson gauge, defined by a
Lagrangian invariant under the local gauge group, without the addition of a gauge
fixing, which yields only the vacuum sector of the observables; non-trivial charged
fields are not represented in the vacuum sector and states carrying a Gauss charge
cannot be constructed in terms of a charged field algebra.
In conclusion, the pretentious “Gauge Principle”, advocated as “the most fundamental cornerstone of modern theoretical physics”, appears to be downgraded to
a merely technical tool which guarantees the validity of the much more significant
local Gauss laws on the physical states, appearing as the physically relevant and
characteristic property of gauge theories.
37 This point is discussed and stressed in F. Strocchi [13, 16], Chap. 7.