7 Noether Theorem and Symmetry Breaking
39
As shown by the above discussion, in general the continuity equation for J
i
μ may
fail to give rise to a conserved charge by a mechanism which is closely related to
that of symmetry breaking in quantum systems.
32,33
For the vanishing of the flux at infinity, a crucial role is played by the condition
∇ϕ 0 ∈ L
2
(R
s
), which is related to the invariance of the Hilbert sector under space
translations and characterizes the physical sectors. This is no longer the case in gauge
field theories, since the energy–momentum density involves the covariant derivative
(∇ + A)ϕ (where A denotes the gauge potential), rather than ∇ϕ. This opens the
way to the Higgs mechanism of symmetry breaking
34 for which the boundary effects
give rise to a charge leaking at infinity.
35
32 J. Goldstone, Nuovo Cim. 19, 154 (1961); J. Goldstone, A. Salam and S. Weinberg, Phys. Rev.
127, 965 (1962); Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); 124, 246 (1961).
33 For a simple account see F. Strocchi, Elements of Quantum Mechanics of Infinite Systems, World
Scientific 1985.
34 P.W. Higgs, Phys. Lett. 12, 132 (1964); T.W. Kibble, Proc. Oxford Int. Conf. Elementary Particles,
Oxford, Oxford Univ. Press 1965; G.S. Guralnik, C.R. Hagen and T.W. Kibble, in Advances in
Particle Physics Vol. 2, R.L. Cool and R.E. Marshak eds., Interscience, New York, 1968 and refs.
therein. See also the references in the footnote below.
35 G. Morchio and F. Strocchi, in Fundamental Problems of Gauge Field Theory, G. Velo and A.S.
Wightman eds., Plenum 1986; F. Strocchi, in Fundamental Aspects of Quantum Theory, V. Gorini
and A. Frigerio eds., Plenum 1986.
39
As shown by the above discussion, in general the continuity equation for J
i
μ may
fail to give rise to a conserved charge by a mechanism which is closely related to
that of symmetry breaking in quantum systems.
32,33
For the vanishing of the flux at infinity, a crucial role is played by the condition
∇ϕ 0 ∈ L
2
(R
s
), which is related to the invariance of the Hilbert sector under space
translations and characterizes the physical sectors. This is no longer the case in gauge
field theories, since the energy–momentum density involves the covariant derivative
(∇ + A)ϕ (where A denotes the gauge potential), rather than ∇ϕ. This opens the
way to the Higgs mechanism of symmetry breaking
34 for which the boundary effects
give rise to a charge leaking at infinity.
35
32 J. Goldstone, Nuovo Cim. 19, 154 (1961); J. Goldstone, A. Salam and S. Weinberg, Phys. Rev.
127, 965 (1962); Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); 124, 246 (1961).
33 For a simple account see F. Strocchi, Elements of Quantum Mechanics of Infinite Systems, World
Scientific 1985.
34 P.W. Higgs, Phys. Lett. 12, 132 (1964); T.W. Kibble, Proc. Oxford Int. Conf. Elementary Particles,
Oxford, Oxford Univ. Press 1965; G.S. Guralnik, C.R. Hagen and T.W. Kibble, in Advances in
Particle Physics Vol. 2, R.L. Cool and R.E. Marshak eds., Interscience, New York, 1968 and refs.
therein. See also the references in the footnote below.
35 G. Morchio and F. Strocchi, in Fundamental Problems of Gauge Field Theory, G. Velo and A.S.
Wightman eds., Plenum 1986; F. Strocchi, in Fundamental Aspects of Quantum Theory, V. Gorini
and A. Frigerio eds., Plenum 1986.