252
Appendix D: Global and Local Gauge Symmetries
This is clearly displayed by free electrodynamics, where a G invariant Lagrangian
leads to the following evolution equations for the vector potential A μ :
A μ − ∂ μ ∂
ν A ν = 0;
(D.1)
then, if A μ (x, t) is a solution corresponding to the initial values A μ (x, 0), ˙
A μ (x, 0),
so is also A
μ (x, t) ≡ A μ (x, t) + ∂ μ χ(x, t), with χ(x, t) a local gauge function, vanishing in a neighbourhood of t = 0.
32
As a possible way out, it has been argued that such a loss of deterministic evolution
only affects the “redundant” gauge dependent variables and not the observables, so
that, from a physical point of view, its conceptual relevance may be considered as
marginal.
However, this position overlooks the role of a well defined dynamics of the auxiliary variables for solving the time evolution of the observables.
The problem is that such a loss of determinism is a serious obstruction for quantization, precluding the representation of the fields by quantum mechanical operators.
In fact, by the principles of quantum mechanics, the time evolution of the dynamical
variables must be described by the one-parameter group generated by the Hamiltonian, so that one automatically has a unique (and therefore deterministic) time
evolution.
Thus, for quantization one cannot use a Lagrangian which is invariant under the
full group G. A similar conclusion may be argued within the (non-rigorous) functional
integral approach to field quantization.
As it is well known, the solution of this difficulty is provided by the addition of a
so-called gauge fixing term, which destroys the invariance of the Lagrangian under
the full local gauge group G and therefore the local gauge symmetry G disappears
already at the level of the correlation functions of the fields, in terms of which the
theory is defined.
It is worthwhile to note that, contrary to what the name and the functional integral
argument might suggest, the gauge fixing does not have to completely fix the gauge;
the role of the gauge fixing is to reduce G invariance only to the extent of making
the dynamical problem of the field algebra well defined, namely a deterministic time
evolution.
In quantum electrodynamics, this is the case of the Feynman–Gupta–Bleuler
gauge, where the gauge fixing
1
2 (∂ A)
2 leads to a well defined hyperbolic evolution,
but does not eliminate the subgroup of G with gauge functions Λ(x, t), satisfying
Λ(x) = 0, with initial data of compact support in space (i.e. ∈ D(R
3
)).
Similarly, in the temporal gauge defined by the gauge condition A 0 = 0, one
has the residual subgroup of time-independent gauge transformations with gauge
function of compact support in space, and no conflict with deterministic evolution.
32 The loss of unique deterministic evolution, already at the classical level, has been the source of
puzzling discussions on local gauge theories by philosophers of science; see, e.g. the contributions
in K. Brading and E. Castellani eds., Symmetries in Physics: Philosophical Reflections, Cambridge
Univ. Press 2003.
Appendix D: Global and Local Gauge Symmetries
This is clearly displayed by free electrodynamics, where a G invariant Lagrangian
leads to the following evolution equations for the vector potential A μ :
A μ − ∂ μ ∂
ν A ν = 0;
(D.1)
then, if A μ (x, t) is a solution corresponding to the initial values A μ (x, 0), ˙
A μ (x, 0),
so is also A
μ (x, t) ≡ A μ (x, t) + ∂ μ χ(x, t), with χ(x, t) a local gauge function, vanishing in a neighbourhood of t = 0.
32
As a possible way out, it has been argued that such a loss of deterministic evolution
only affects the “redundant” gauge dependent variables and not the observables, so
that, from a physical point of view, its conceptual relevance may be considered as
marginal.
However, this position overlooks the role of a well defined dynamics of the auxiliary variables for solving the time evolution of the observables.
The problem is that such a loss of determinism is a serious obstruction for quantization, precluding the representation of the fields by quantum mechanical operators.
In fact, by the principles of quantum mechanics, the time evolution of the dynamical
variables must be described by the one-parameter group generated by the Hamiltonian, so that one automatically has a unique (and therefore deterministic) time
evolution.
Thus, for quantization one cannot use a Lagrangian which is invariant under the
full group G. A similar conclusion may be argued within the (non-rigorous) functional
integral approach to field quantization.
As it is well known, the solution of this difficulty is provided by the addition of a
so-called gauge fixing term, which destroys the invariance of the Lagrangian under
the full local gauge group G and therefore the local gauge symmetry G disappears
already at the level of the correlation functions of the fields, in terms of which the
theory is defined.
It is worthwhile to note that, contrary to what the name and the functional integral
argument might suggest, the gauge fixing does not have to completely fix the gauge;
the role of the gauge fixing is to reduce G invariance only to the extent of making
the dynamical problem of the field algebra well defined, namely a deterministic time
evolution.
In quantum electrodynamics, this is the case of the Feynman–Gupta–Bleuler
gauge, where the gauge fixing
1
2 (∂ A)
2 leads to a well defined hyperbolic evolution,
but does not eliminate the subgroup of G with gauge functions Λ(x, t), satisfying
Λ(x) = 0, with initial data of compact support in space (i.e. ∈ D(R
3
)).
Similarly, in the temporal gauge defined by the gauge condition A 0 = 0, one
has the residual subgroup of time-independent gauge transformations with gauge
function of compact support in space, and no conflict with deterministic evolution.
32 The loss of unique deterministic evolution, already at the classical level, has been the source of
puzzling discussions on local gauge theories by philosophers of science; see, e.g. the contributions
in K. Brading and E. Castellani eds., Symmetries in Physics: Philosophical Reflections, Cambridge
Univ. Press 2003.