174
25 Breaking of Continuous Symmetries: Goldstone’s Theorem
II. Generation of the Symmetry by a Local Charge
Much more problematic and subtle is condition II, the precise formulation of which
involves properties with important physical consequences.
i) Local charge as an integral of a density
First, for technical reasons (see below), it is convenient to smooth out the sharp
boundary in (25.2), by introducing a C
∞ function of compact support
150 (for simplicity, we omit the boldface notation for the variable x ∈ R
s )
f R (x) = f (|x|/R), f ∈ D(R),
f (x) = 1, for |x| ≤ 1, f (x) = 0, for |x| ≥ 1 + ε,
(25.12)
and replace the definition of Q R (t) in (25.2), (25.6), by
Q R (t) =
dx f R (x) j 0 (x, t) ≡ j 0 ( f R , t).
(25.13)
Even with such a proviso, the limit R → ∞, i.e. the formal integral Q(t) =
dx j 0 (x, t), does not exist and therefore it does not define an operator, since by
(25.4) the current density does not “decrease” for |x| → ∞.
Much better are the convergence properties of the integral of the commutator
J (x, t) = i [ j 0 (x, t), A],
with a local operator A, since J (x, t) at least vanishes for |x| → ∞, by asymptotic
abelianess and actually has compact support if j 0 and A satisfy the relativistic locality
property (Chap. 14, (14.2)).
It is implicit in (25.1) that J (x, t) must be at least integrable in x. For a mathematical control of the proof, one actually needs that J (x, t) is absolutely integrable
in x for large |x|.
151 Thus, one must supplement the condition of local generation
by a charge with the integrability condition of the charge density commutators.
also briefly called charge integrability condition.
It means that the ground state expectation values of the charge density commutators are absolutely integrable in x for large |x|, as tempered distributions in t, i.e.
∀g ∈ S(R)
<
dt g(t) [ j 0 (x, t), A] > 0
(25.14)
is absolutely integrable in x for large |x|.
150 As we shall discuss below, for relativistic systems also a smearing in time is necessary to cope
with the ultraviolet singularities.
151 G. Morchio and F. Strocchi, Comm. Math. Phys. 99, 153 (1985); J. Math. Phys. 28 622 (1987).
The crucial role of such a condition for the non-relativistic version of the Goldstone theorem and
the need for a careful handling of the distributional and measure theoretical problems do not seem
to have been noted in the vast previous literature.
25 Breaking of Continuous Symmetries: Goldstone’s Theorem
II. Generation of the Symmetry by a Local Charge
Much more problematic and subtle is condition II, the precise formulation of which
involves properties with important physical consequences.
i) Local charge as an integral of a density
First, for technical reasons (see below), it is convenient to smooth out the sharp
boundary in (25.2), by introducing a C
∞ function of compact support
150 (for simplicity, we omit the boldface notation for the variable x ∈ R
s )
f R (x) = f (|x|/R), f ∈ D(R),
f (x) = 1, for |x| ≤ 1, f (x) = 0, for |x| ≥ 1 + ε,
(25.12)
and replace the definition of Q R (t) in (25.2), (25.6), by
Q R (t) =
dx f R (x) j 0 (x, t) ≡ j 0 ( f R , t).
(25.13)
Even with such a proviso, the limit R → ∞, i.e. the formal integral Q(t) =
dx j 0 (x, t), does not exist and therefore it does not define an operator, since by
(25.4) the current density does not “decrease” for |x| → ∞.
Much better are the convergence properties of the integral of the commutator
J (x, t) = i [ j 0 (x, t), A],
with a local operator A, since J (x, t) at least vanishes for |x| → ∞, by asymptotic
abelianess and actually has compact support if j 0 and A satisfy the relativistic locality
property (Chap. 14, (14.2)).
It is implicit in (25.1) that J (x, t) must be at least integrable in x. For a mathematical control of the proof, one actually needs that J (x, t) is absolutely integrable
in x for large |x|.
151 Thus, one must supplement the condition of local generation
by a charge with the integrability condition of the charge density commutators.
also briefly called charge integrability condition.
It means that the ground state expectation values of the charge density commutators are absolutely integrable in x for large |x|, as tempered distributions in t, i.e.
∀g ∈ S(R)
<
dt g(t) [ j 0 (x, t), A] > 0
(25.14)
is absolutely integrable in x for large |x|.
150 As we shall discuss below, for relativistic systems also a smearing in time is necessary to cope
with the ultraviolet singularities.
151 G. Morchio and F. Strocchi, Comm. Math. Phys. 99, 153 (1985); J. Math. Phys. 28 622 (1987).
The crucial role of such a condition for the non-relativistic version of the Goldstone theorem and
the need for a careful handling of the distributional and measure theoretical problems do not seem
to have been noted in the vast previous literature.