Appendix D: Global and Local Gauge Symmetries
251
Similarly, in the case of a broken U (2) gauge group the observable unitary operators V R (λ) ≡ e
iλQ R , T
(2)
R (μ) ≡ e
iμ
i (Q
i
R )
2 do not converge to unitary operators as
R → ∞, distinctively from the unbroken case.
Much more problematic is the possible physical/empirical meaning of a local
gauge symmetry, as we shall discuss below.
Even if the standard way of introducing a local gauge group is as an extension of a
global gauge group G, by allowing the group parameters to be C
∞ functions of space
time, the local gauge group G should be better (and distinctively) characterized by
localized group parameters, i.e. by arbitrary C
∞ (bounded) functions of space time
of compact support in space.
From a strategical point of view a local gauge symmetry has a twofold role. The
introduction of local fields, which transform under a local gauge group G, leads to
a more tractable dynamical problem, in terms of Lagrangian or Hamiltonian field
equations, not otherwise available in a direct approach to the time evolution of the
observables. The time evolution of the local field algebra F is then instrumental for
determining that of its observable (gauge invariant) subalgebra A.
Another point for the introduction of auxiliary local variables with non-trivial
transformation under a local gauge group G is the identification of the local algebra
of observables A as the subalgebra of the local field algebra F which is pointwise
invariant under G; the invariance under the related global gauge group G is not
enough, as clearly displayed in quantum electrodynamics (QED), where, e.g. the
field operator ¯
ψ( f ) ψ(g) is invariant under the global U (1) gauge group, but it is not
a local observable field.
This use of a local gauge group is particularly relevant in relativistic quantum
field theories, where the relativistic locality of the observables is a crucial property,
and the local observable algebra A is not a priori given, without referring to the G
invariant functions of the local fields.
The invariance of the Lagrangian or of the Hamiltonian under G guarantees the
stability of A under time evolution.
Such a strategy goes under the name of Gauge Principle, believed to be the deep
physical basis of the present theory of elementary particles; from this point of view,
local gauge symmetries are raised to the rank of fundamental algebraic symmetries
of the description of the physical world.
However, at a closer inspection one finds that a local gauge symmetry G reduces to
the identity not only on the observables, but also on the states (apart from its possible
topological invariants, see Appendix F); as we shall argue below, G appears only at
the intermediate steps for the construction of the dynamical physical theory, being
doomed to lose any operational (and philosophical) meaning at the end.
The first obstruction for a physical/empirical meaning of a local gauge symmetry
is that a dynamics described by a G invariant Lagrangian is incompatible with a
deterministic time evolution of the field algebra, i.e. a time evolution for which the
initial value Cauchy problem has one and only one solution.
251
Similarly, in the case of a broken U (2) gauge group the observable unitary operators V R (λ) ≡ e
iλQ R , T
(2)
R (μ) ≡ e
iμ
i (Q
i
R )
2 do not converge to unitary operators as
R → ∞, distinctively from the unbroken case.
Much more problematic is the possible physical/empirical meaning of a local
gauge symmetry, as we shall discuss below.
Even if the standard way of introducing a local gauge group is as an extension of a
global gauge group G, by allowing the group parameters to be C
∞ functions of space
time, the local gauge group G should be better (and distinctively) characterized by
localized group parameters, i.e. by arbitrary C
∞ (bounded) functions of space time
of compact support in space.
From a strategical point of view a local gauge symmetry has a twofold role. The
introduction of local fields, which transform under a local gauge group G, leads to
a more tractable dynamical problem, in terms of Lagrangian or Hamiltonian field
equations, not otherwise available in a direct approach to the time evolution of the
observables. The time evolution of the local field algebra F is then instrumental for
determining that of its observable (gauge invariant) subalgebra A.
Another point for the introduction of auxiliary local variables with non-trivial
transformation under a local gauge group G is the identification of the local algebra
of observables A as the subalgebra of the local field algebra F which is pointwise
invariant under G; the invariance under the related global gauge group G is not
enough, as clearly displayed in quantum electrodynamics (QED), where, e.g. the
field operator ¯
ψ( f ) ψ(g) is invariant under the global U (1) gauge group, but it is not
a local observable field.
This use of a local gauge group is particularly relevant in relativistic quantum
field theories, where the relativistic locality of the observables is a crucial property,
and the local observable algebra A is not a priori given, without referring to the G
invariant functions of the local fields.
The invariance of the Lagrangian or of the Hamiltonian under G guarantees the
stability of A under time evolution.
Such a strategy goes under the name of Gauge Principle, believed to be the deep
physical basis of the present theory of elementary particles; from this point of view,
local gauge symmetries are raised to the rank of fundamental algebraic symmetries
of the description of the physical world.
However, at a closer inspection one finds that a local gauge symmetry G reduces to
the identity not only on the observables, but also on the states (apart from its possible
topological invariants, see Appendix F); as we shall argue below, G appears only at
the intermediate steps for the construction of the dynamical physical theory, being
doomed to lose any operational (and philosophical) meaning at the end.
The first obstruction for a physical/empirical meaning of a local gauge symmetry
is that a dynamics described by a G invariant Lagrangian is incompatible with a
deterministic time evolution of the field algebra, i.e. a time evolution for which the
initial value Cauchy problem has one and only one solution.