29.2 Higgs Mechanism in Local Gauges
207
j μ (x) = ∂
ν F μν (x) + L μ (x),
(29.4)
where L μ (x) is an “unphysical” field which must have vanishing matrix elements
< Ψ, L μ > between physical states, in order to avoid violation of Maxwell equations in physical expectations (see (29.5) below).
For example, in the Feynman–Gupta–Bleuler (FGB) gauge one has
− A μ (x) = j μ (x) = ∂
ν F μν (x) − ∂ μ ∂
ν A ν (x),
(29.5)
and the subspace of physical vectors Ψ is identified by the subsidiary (Gupta–Bleuler)
condition
(∂
ν A ν )
−
Ψ = 0,
where ∂ A
− denotes the negative energy part of the free field ∂ A. Such a condition
implies the vanishing of the expectations < Ψ, L μ Ψ >.
Such features are clearly displayed by the local (covariant) quantization of the
free vector potential
197 but can be argued to be present in general if locality holds.
198
After these premises we can state the following.
Theorem 29.1 (Higgs theorem in local gauges) In the local gauges of a U (1) gauge
theory, with a gauge fixing invariant under the global U (1) group, the U (1) symmetry
breaking with order parameter
< δ A >= i lim
R→∞
< [ j 0 ( f R α), A] > = 0, A ∈ F,
implies that the Fourier transform of the two-point function < j 0 (x)A > contains
a δ(k
2
) (Goldstone modes).
However, such a singularity cannot be ascribed to the energy–momentum
spectrum of a physical vector Ψ , as a consequence of the weak Gauss law
< Ψ, ( j μ − ∂
ν F μν )Ψ >= 0, i.e. the Goldstone modes are not physical.
Proof. The proof crucially exploits locality and weak Gauss law. The existence of
the δ(k
2
) singularity follows from a slight extension of the proof of the Goldstone
theorem in the absence of positivity (see Chap. 27).
Moreover, by the locality of A one has
lim
R→∞
< [∂
i F 0i ( f R α), A] >= 0
and therefore the symmetry breaking condition may also be written as
197 See, e.g. S.S. Schweber, An Introduction to Relativistic Quantum Field Theory, Harper and Row
1961, Chap. 19.
198 F. Strocchi, Selected Topics on the General Properties of Quantum Field theory, World Scientific
1993, Chaps. VI, VII; the interplay between locality and Gauss’ law is discussed, e.g. in F. Strocchi,
Elements of Quantum Mechanics of Infinite Systems, World Scientific 1985, Part C, Chap. II.
207
j μ (x) = ∂
ν F μν (x) + L μ (x),
(29.4)
where L μ (x) is an “unphysical” field which must have vanishing matrix elements
< Ψ, L μ > between physical states, in order to avoid violation of Maxwell equations in physical expectations (see (29.5) below).
For example, in the Feynman–Gupta–Bleuler (FGB) gauge one has
− A μ (x) = j μ (x) = ∂
ν F μν (x) − ∂ μ ∂
ν A ν (x),
(29.5)
and the subspace of physical vectors Ψ is identified by the subsidiary (Gupta–Bleuler)
condition
(∂
ν A ν )
−
Ψ = 0,
where ∂ A
− denotes the negative energy part of the free field ∂ A. Such a condition
implies the vanishing of the expectations < Ψ, L μ Ψ >.
Such features are clearly displayed by the local (covariant) quantization of the
free vector potential
197 but can be argued to be present in general if locality holds.
198
After these premises we can state the following.
Theorem 29.1 (Higgs theorem in local gauges) In the local gauges of a U (1) gauge
theory, with a gauge fixing invariant under the global U (1) group, the U (1) symmetry
breaking with order parameter
< δ A >= i lim
R→∞
< [ j 0 ( f R α), A] > = 0, A ∈ F,
implies that the Fourier transform of the two-point function < j 0 (x)A > contains
a δ(k
2
) (Goldstone modes).
However, such a singularity cannot be ascribed to the energy–momentum
spectrum of a physical vector Ψ , as a consequence of the weak Gauss law
< Ψ, ( j μ − ∂
ν F μν )Ψ >= 0, i.e. the Goldstone modes are not physical.
Proof. The proof crucially exploits locality and weak Gauss law. The existence of
the δ(k
2
) singularity follows from a slight extension of the proof of the Goldstone
theorem in the absence of positivity (see Chap. 27).
Moreover, by the locality of A one has
lim
R→∞
< [∂
i F 0i ( f R α), A] >= 0
and therefore the symmetry breaking condition may also be written as
197 See, e.g. S.S. Schweber, An Introduction to Relativistic Quantum Field Theory, Harper and Row
1961, Chap. 19.
198 F. Strocchi, Selected Topics on the General Properties of Quantum Field theory, World Scientific
1993, Chaps. VI, VII; the interplay between locality and Gauss’ law is discussed, e.g. in F. Strocchi,
Elements of Quantum Mechanics of Infinite Systems, World Scientific 1985, Part C, Chap. II.