14
3 Symmetries in Classical Field Theory
∂
2 g k
∂z i ∂z j
(c) = 0, ∀c ∈ R
n
, i.e. g(z) = Az + a.
Equation (3.9) then becomes
∂
∂z l
U (Az + a) = (A
T A) li
∂
∂z i
U (z).
The invariance of the action integral up to a scale factor requires A
T A = λ1 and
U (Az + a) = λU (z)+ const; the normalization U (0) = 0 identifies the latter constant as U (a).
Having characterized the possible symmetries of (3.1), we may now ask whether
symmetry breaking can occur. For continuous groups this possibility seems to be in
conflict with Noether’s theorem.
Theorem 3.2
8 Let G be an N parameter Lie group of internal symmetries for the
classical system (3.1), then there exist N conserved currents
∂
μ J
a
μ (x, t) = 0, a = 1, . . . , N
(3.11)
and N conserved quantities
Q
a
(t) =
d
s x J
a
0 (x, t) = Q
a
(0),
(3.12)
which are the generators of the corresponding one-parameter subgroups {g
a
α , α ∈ R}
of symmetry transformations
δ
a u ≡ dg
a
α (u)/dα| α=0 = {u, Q
a
},
(3.13)
where the curly brackets denote the Poisson brackets.
For the proof, we refer to any standard textbook.
9,10
One should stress that for (3.12) some regularity properties of the solution are
needed, even if they are not spelt out in the standard accounts of the theorem. Actually,
the deep physical question of spontaneous breaking requires a more refined analysis
of the mathematical properties of the solutions and of their behaviour at infinity.
As we shall see, the problem of existence of “islands” or phases, stable under time
8 E. Noether, Nachr. d. Kgl. Ges. d. Wiss. Göttingen (1918), p. 235.
9 See, e.g. H. Goldstein, Classical Mechanics, 2nd. ed., Addison-Wesley 1980; E. L. Hill, Rev. Mod.
Phys. 23, 253 (1951); N.N. Bogoljubov and D.V. Shirkov, Introduction to the theory of quantized
fields, Interscience 1958, Sect. 2.5.
10 For the representations of Lie groups and their generators in classical systems, see D.G. Currie, T.F.
Jordan and E.C.G. Sudarshan, Rev. Mod. Phys. 35, 350 (1963); E.C.G. Sudarshan and N. Mukunda,
Classical Dynamics: A Modern Perspective, J. Wiley and Sons 1974.
3 Symmetries in Classical Field Theory
∂
2 g k
∂z i ∂z j
(c) = 0, ∀c ∈ R
n
, i.e. g(z) = Az + a.
Equation (3.9) then becomes
∂
∂z l
U (Az + a) = (A
T A) li
∂
∂z i
U (z).
The invariance of the action integral up to a scale factor requires A
T A = λ1 and
U (Az + a) = λU (z)+ const; the normalization U (0) = 0 identifies the latter constant as U (a).
Having characterized the possible symmetries of (3.1), we may now ask whether
symmetry breaking can occur. For continuous groups this possibility seems to be in
conflict with Noether’s theorem.
Theorem 3.2
8 Let G be an N parameter Lie group of internal symmetries for the
classical system (3.1), then there exist N conserved currents
∂
μ J
a
μ (x, t) = 0, a = 1, . . . , N
(3.11)
and N conserved quantities
Q
a
(t) =
d
s x J
a
0 (x, t) = Q
a
(0),
(3.12)
which are the generators of the corresponding one-parameter subgroups {g
a
α , α ∈ R}
of symmetry transformations
δ
a u ≡ dg
a
α (u)/dα| α=0 = {u, Q
a
},
(3.13)
where the curly brackets denote the Poisson brackets.
For the proof, we refer to any standard textbook.
9,10
One should stress that for (3.12) some regularity properties of the solution are
needed, even if they are not spelt out in the standard accounts of the theorem. Actually,
the deep physical question of spontaneous breaking requires a more refined analysis
of the mathematical properties of the solutions and of their behaviour at infinity.
As we shall see, the problem of existence of “islands” or phases, stable under time
8 E. Noether, Nachr. d. Kgl. Ges. d. Wiss. Göttingen (1918), p. 235.
9 See, e.g. H. Goldstein, Classical Mechanics, 2nd. ed., Addison-Wesley 1980; E. L. Hill, Rev. Mod.
Phys. 23, 253 (1951); N.N. Bogoljubov and D.V. Shirkov, Introduction to the theory of quantized
fields, Interscience 1958, Sect. 2.5.
10 For the representations of Lie groups and their generators in classical systems, see D.G. Currie, T.F.
Jordan and E.C.G. Sudarshan, Rev. Mod. Phys. 35, 350 (1963); E.C.G. Sudarshan and N. Mukunda,
Classical Dynamics: A Modern Perspective, J. Wiley and Sons 1974.