254
Appendix D: Global and Local Gauge Symmetries
There is a vast literature on the second Noether theorem, as well as on its implications and relevance for gauge theories.
34 However, in the textbook accounts of
the theorem, often it is not sufficiently emphasized that a crucial assumption of the
theorem is the exact invariance of the Lagrangian density under the (full) local group
G; even invariance up to a total divergence would not suffice for the derivation of the
local Gauss law, not to speak of the invariance of L broken by the gauge fixing.
Therefore, the second Noether theorem does not apply to a Lagrangian which
describes a well defined field dynamics, since the necessary gauge fixing does not
allow arbitrary C
∞ gauge functions of space and time of compact support in space;
then, in this case the conserved currents J
a
μ do not satisfy the local Gauss law.
Indeed, in the Feynman gauge of quantum electrodynamics the electric current
satisfies j μ = ∂
ν F μν − ∂ μ ∂
ν A ν ; in fact, the parameters ε(x) which define the residual
local gauge group are restricted to satisfy ε = 0, so that the vanishing of the term
with ∂ μ ∂ ν ε in the variation of L requires the antisymmetry of δL/δ ∂ μ A ν only up to
a term proportional to g μν .
Even though the local Gauss law fails for the field algebra F, a careful analysis
of the relation between the choice of the gauge fixing and the identification of the
physical states allow to rescue the validity of the local Gauss law on the physical
states, as it will be shown below. In fact, one may prove that, quite generally the
operator L
a
μ ≡ J
a
μ − ∂
ν G
a
μν , which describes the violation of the local Gauss law
arising as a consequence of the gauge fixing, has vanishing matrix elements on the
physical states.
This is not surprising in quantum electrodynamics, where the local Gauss law
corresponds to the Maxwell equations of observable operators and the selection of
the physical states must therefore guarantee its validity in their matrix elements (as it
may explicitly be checked in the Feynman–Gupta–Bleuler gauge and in the Coulomb
gauge).
Less obvious is the proof that such a property also holds in the non-abelian case.
35
In the non-abelian case the local Gauss law is not a relation between observable
operators, but one may prove that the operator L
a
μ maps physical states into physical
states and therefore it is well defined on the physical Hilbert space; then, the validity
of the local Gauss law may be read as the vanishing of the gauge invariant (and
therefore observable) operator
a L
a
μ (x)
† L
aμ
(x).
From a differential geometrical point of view, the local Gauss law states that
the current J
a
μ defines a differential form which is a δ-boundary or co-exact.
36 This
implies that the charge
Q
a
=
d
3 x J
a
0 (x, 0)
34 See the detailed account in K. Brading and H.R. Brown, Noether’s Theorems and Gauge Symmetries, arXiv:hep-th/0009058; Symmetries and Noether’s Theorems, in K. Brading and E. Castellani
eds., Symmetries in Physics: Philosophical Reflections, Cambridge University Press 2003.
35 See discussion on the local Gauss law in F. Strocchi [13, 16]; in particular, Chap. 7, Sect. 4 for
the discussion in the Becchi–Rouet–Stora–Tyutin (BRST) quantization and Chap. 8, Sect. 2 for the
quantization in the temporal gauge.
36 G. de Rham, Differential Manifolds, Springer 1984.
Appendix D: Global and Local Gauge Symmetries
There is a vast literature on the second Noether theorem, as well as on its implications and relevance for gauge theories.
34 However, in the textbook accounts of
the theorem, often it is not sufficiently emphasized that a crucial assumption of the
theorem is the exact invariance of the Lagrangian density under the (full) local group
G; even invariance up to a total divergence would not suffice for the derivation of the
local Gauss law, not to speak of the invariance of L broken by the gauge fixing.
Therefore, the second Noether theorem does not apply to a Lagrangian which
describes a well defined field dynamics, since the necessary gauge fixing does not
allow arbitrary C
∞ gauge functions of space and time of compact support in space;
then, in this case the conserved currents J
a
μ do not satisfy the local Gauss law.
Indeed, in the Feynman gauge of quantum electrodynamics the electric current
satisfies j μ = ∂
ν F μν − ∂ μ ∂
ν A ν ; in fact, the parameters ε(x) which define the residual
local gauge group are restricted to satisfy ε = 0, so that the vanishing of the term
with ∂ μ ∂ ν ε in the variation of L requires the antisymmetry of δL/δ ∂ μ A ν only up to
a term proportional to g μν .
Even though the local Gauss law fails for the field algebra F, a careful analysis
of the relation between the choice of the gauge fixing and the identification of the
physical states allow to rescue the validity of the local Gauss law on the physical
states, as it will be shown below. In fact, one may prove that, quite generally the
operator L
a
μ ≡ J
a
μ − ∂
ν G
a
μν , which describes the violation of the local Gauss law
arising as a consequence of the gauge fixing, has vanishing matrix elements on the
physical states.
This is not surprising in quantum electrodynamics, where the local Gauss law
corresponds to the Maxwell equations of observable operators and the selection of
the physical states must therefore guarantee its validity in their matrix elements (as it
may explicitly be checked in the Feynman–Gupta–Bleuler gauge and in the Coulomb
gauge).
Less obvious is the proof that such a property also holds in the non-abelian case.
35
In the non-abelian case the local Gauss law is not a relation between observable
operators, but one may prove that the operator L
a
μ maps physical states into physical
states and therefore it is well defined on the physical Hilbert space; then, the validity
of the local Gauss law may be read as the vanishing of the gauge invariant (and
therefore observable) operator
a L
a
μ (x)
† L
aμ
(x).
From a differential geometrical point of view, the local Gauss law states that
the current J
a
μ defines a differential form which is a δ-boundary or co-exact.
36 This
implies that the charge
Q
a
=
d
3 x J
a
0 (x, 0)
34 See the detailed account in K. Brading and H.R. Brown, Noether’s Theorems and Gauge Symmetries, arXiv:hep-th/0009058; Symmetries and Noether’s Theorems, in K. Brading and E. Castellani
eds., Symmetries in Physics: Philosophical Reflections, Cambridge University Press 2003.
35 See discussion on the local Gauss law in F. Strocchi [13, 16]; in particular, Chap. 7, Sect. 4 for
the discussion in the Becchi–Rouet–Stora–Tyutin (BRST) quantization and Chap. 8, Sect. 2 for the
quantization in the temporal gauge.
36 G. de Rham, Differential Manifolds, Springer 1984.