250
Appendix D: Global and Local Gauge Symmetries
gauge symmetries, involves superselected quantum numbers given by the spectrum
of (polynomial functions of) the generators of the gauge group.
This is clearly displayed by Quantum Electrodynamics, where the generator of the
global U (1) gauge transformations describes the superselected (observable!) electric
charge of the states.
31
The operational meaning of a non-abelian global gauge symmetry is simply displayed by the theory of a free massive Dirac field ψ transforming as the fundamental representation of an internal U (2) = SU (2) × U (1) symmetry group, with the
observables declared to be pointwise invariant under U (2) (global gauge group).
Actually, the algebra of observables A may be characterized as the pointwise invariant subalgebra of the field algebra F (generated by ψ).
As discussed in Chap. 12, the existence of the free Hamiltonian selects the Fock
representation of F, where the gauge symmetry U (2) is unbroken, with generators
Q, Q
i
, i = 1, 2, 3.
The unitary operators U(λ) ≡ e
iλQ
, T
(2)
(μ) ≡ e
iμ
i (Q
i )
2 , λ, μ ∈ R, belong to
the algebra of observables A, actually to its centre Z and the points of their spectra, respectively, exp [iλ n], n ∈ Z, exp [iμ j ( j + 1)], j ∈
1
2 N, label inequivalent
representations of A, namely disjoint phases, with superselected quantum numbers
n, j ( j + 1). In the Fock representation of F the fermion fields act as intertwiners
between them.
Therefore, the existence of an unbroken global gauge symmetry corresponds to
the existence of disjoint phases and their physical labelling.
The next natural question is whether it is possible to break a global gauge symmetry and, in this case, what is its physical meaning. This issue has been debated in
the literature and one may even find the statement that a gauge symmetry is not a
symmetry and therefore cannot be broken.
Clearly, in order to define a gauge symmetry one must introduce a larger algebra,
typically a field algebra F, and the symmetry breaking operator cannot belong to the
observable algebra. Then, the question is the characterization of the representations of
the observable algebra defined by a ground/vacuum state on F, which, e.g. completely
breaks the global gauge group down to the identity.
Roughly, in such representations the spectra of (the invariant polynomial functions
of) the generators of the global gauge group get concentrated at infinity.
We have already seen an example of such a phenomenon in Chap. 17, Sect. 17.2,
and Chap. 13, Sect. 4, for the free Bose gas. Indeed, in this case the unitary
observable operator V(λ) R ≡ exp iλ Q R , where Q R ≡
d
3 x f R (x) [ψ
∗
(x) ψ(x) − ¯
ρ],
¯
ρ ≡< ψ
∗
(x) ψ(x) > is the local charge which generates the U (1) global gauge transformations, does not converge to a unitary operator, as R → ∞ (a meaningful physical property); indeed, ||Q R Ψ 0 || =
√ ¯
ρ||ψ
∗
F ( f R )Ψ 0 || → R→∞ ∞, and, therefore, for
any A ∈ F, ||Q R AΨ 0 || = || [Q R , A] Ψ 0 + AQ R Ψ 0 || → ∞ .
31 For the general proof of the charge superselection rule, see F. Strocchi and A.S. Wightman,
Jour. Math. Phys. 15, 2198 (1974); for an updated account, see F. Strocchi, An Introduction NonPerturbative Foundations of Quantum Field Theory, Oxford University Press 2013, improved reprint
2016, hereafter referred to as F. Strocchi [13, 16], Chap. 7, Sect. 5.
Appendix D: Global and Local Gauge Symmetries
gauge symmetries, involves superselected quantum numbers given by the spectrum
of (polynomial functions of) the generators of the gauge group.
This is clearly displayed by Quantum Electrodynamics, where the generator of the
global U (1) gauge transformations describes the superselected (observable!) electric
charge of the states.
31
The operational meaning of a non-abelian global gauge symmetry is simply displayed by the theory of a free massive Dirac field ψ transforming as the fundamental representation of an internal U (2) = SU (2) × U (1) symmetry group, with the
observables declared to be pointwise invariant under U (2) (global gauge group).
Actually, the algebra of observables A may be characterized as the pointwise invariant subalgebra of the field algebra F (generated by ψ).
As discussed in Chap. 12, the existence of the free Hamiltonian selects the Fock
representation of F, where the gauge symmetry U (2) is unbroken, with generators
Q, Q
i
, i = 1, 2, 3.
The unitary operators U(λ) ≡ e
iλQ
, T
(2)
(μ) ≡ e
iμ
i (Q
i )
2 , λ, μ ∈ R, belong to
the algebra of observables A, actually to its centre Z and the points of their spectra, respectively, exp [iλ n], n ∈ Z, exp [iμ j ( j + 1)], j ∈
1
2 N, label inequivalent
representations of A, namely disjoint phases, with superselected quantum numbers
n, j ( j + 1). In the Fock representation of F the fermion fields act as intertwiners
between them.
Therefore, the existence of an unbroken global gauge symmetry corresponds to
the existence of disjoint phases and their physical labelling.
The next natural question is whether it is possible to break a global gauge symmetry and, in this case, what is its physical meaning. This issue has been debated in
the literature and one may even find the statement that a gauge symmetry is not a
symmetry and therefore cannot be broken.
Clearly, in order to define a gauge symmetry one must introduce a larger algebra,
typically a field algebra F, and the symmetry breaking operator cannot belong to the
observable algebra. Then, the question is the characterization of the representations of
the observable algebra defined by a ground/vacuum state on F, which, e.g. completely
breaks the global gauge group down to the identity.
Roughly, in such representations the spectra of (the invariant polynomial functions
of) the generators of the global gauge group get concentrated at infinity.
We have already seen an example of such a phenomenon in Chap. 17, Sect. 17.2,
and Chap. 13, Sect. 4, for the free Bose gas. Indeed, in this case the unitary
observable operator V(λ) R ≡ exp iλ Q R , where Q R ≡
d
3 x f R (x) [ψ
∗
(x) ψ(x) − ¯
ρ],
¯
ρ ≡< ψ
∗
(x) ψ(x) > is the local charge which generates the U (1) global gauge transformations, does not converge to a unitary operator, as R → ∞ (a meaningful physical property); indeed, ||Q R Ψ 0 || =
√ ¯
ρ||ψ
∗
F ( f R )Ψ 0 || → R→∞ ∞, and, therefore, for
any A ∈ F, ||Q R AΨ 0 || = || [Q R , A] Ψ 0 + AQ R Ψ 0 || → ∞ .
31 For the general proof of the charge superselection rule, see F. Strocchi and A.S. Wightman,
Jour. Math. Phys. 15, 2198 (1974); for an updated account, see F. Strocchi, An Introduction NonPerturbative Foundations of Quantum Field Theory, Oxford University Press 2013, improved reprint
2016, hereafter referred to as F. Strocchi [13, 16], Chap. 7, Sect. 5.