25.3 The Goldstone Theorem with Mathematical Flavour
185
In conclusion, in the limit k → 0 the imaginary part of the matrix elements of the
energy spectral projection between the states j 0 ( f )Ψ 0 and A Ψ 0 is given by ˜
J (ω).
As in Sect. 25.1, the stability under time evolution of the algebra A 0 on which β
λ
is generated by Q R implies that J (t) is independent of time so that ˜
J (ω) ∼ δ(ω);
moreover, by the above argument, ω = 0 is the limit of the energy spectral support
when k → 0.
The ground state cannot contribute to ˜
J (ω) since, for real symmetric test functions
˜
g(ω), ˜
h(k), < j 0 ( f )d E( ˜
g) d E( ˜
h) > 0 < A > 0 is real. The infinite lifetime is implied
by the continuity in k of the energy spectrum, which shrinks to zero when k → 0.
The above version of the Goldstone theorem improves the standard treatment (for
non-relativistic systems) in i) identifying the relevant hypotheses, in a way which
looks applicable to the physically interesting cases, ii) emphasizing the role of the
localization properties of the dynamics, and iii) predicting the existence of quasiparticle Goldstone bosons.
As we shall see, the existence of stable Goldstone particles is related to relativistic
locality and spectrum.
Remark. In the case of a continuous symmetry group G broken only by the effect
of a (small) symmetry breaking term g H 1 in the otherwise symmetric Hamiltonian
(briefly softly broken Wigner symmetry), one may start by classifying the states
according to representations of G and set up a perturbative expansion in g for computing energy shifts, mixings, etc. This strategy, pioneered by Wigner, has been
widely used in atomic and in nuclear physics, with the powerful technical help of
the Wigner–Eckart theorem, which allows one to control the matrix elements of an
operator O between states belonging to definite representations of G.
In striking contrast, a spontaneously broken symmetry group G, while exactly
realized through its representations by the algebra A of observables (or more generally of canonical variables), is completely lost at the level of the states, which can
not be even approximately classified in terms of representations of G. This a priori
precludes the use of the Wigner–Eckart relations for computing symmetry breaking
effects.
Under general assumptions, one may nevertheless prove the existence of generalized Wigner–Eckart relations for the matrix elements of operators O between states
obtained from the ground state 0 by operators B, C ∈ A belonging to definite representations of G (C. Heissenberg and F. Strocchi, Corrections to Wigner–Eckart
relations by spontaneous symmetry breaking, arXiv: 2007.03539 [quant-ph]). Actually, for a spontaneously broken continuous symmetry group G, these generalized
Wigner–Eckart relations are identical to the standard ones up to a correction encoded
in a simple “tadpole” term involving Goldstone bosons. This allows for a perturbative calculation of symmetry breaking effects, like mass splitting, even in the case of
spontaneously broken symmetries.
More specifically, considering the Wigner–Eckart relations derived from the
matrix elements (B 0 , δ
a O C 0 ), with B, O, C, ∈ A and δ
a O the infinitesimal
variation of O under a one-dimensional subgroup of G, locally generated by j
a
0 , the
tadpole corrections are terms of the form
185
In conclusion, in the limit k → 0 the imaginary part of the matrix elements of the
energy spectral projection between the states j 0 ( f )Ψ 0 and A Ψ 0 is given by ˜
J (ω).
As in Sect. 25.1, the stability under time evolution of the algebra A 0 on which β
λ
is generated by Q R implies that J (t) is independent of time so that ˜
J (ω) ∼ δ(ω);
moreover, by the above argument, ω = 0 is the limit of the energy spectral support
when k → 0.
The ground state cannot contribute to ˜
J (ω) since, for real symmetric test functions
˜
g(ω), ˜
h(k), < j 0 ( f )d E( ˜
g) d E( ˜
h) > 0 < A > 0 is real. The infinite lifetime is implied
by the continuity in k of the energy spectrum, which shrinks to zero when k → 0.
The above version of the Goldstone theorem improves the standard treatment (for
non-relativistic systems) in i) identifying the relevant hypotheses, in a way which
looks applicable to the physically interesting cases, ii) emphasizing the role of the
localization properties of the dynamics, and iii) predicting the existence of quasiparticle Goldstone bosons.
As we shall see, the existence of stable Goldstone particles is related to relativistic
locality and spectrum.
Remark. In the case of a continuous symmetry group G broken only by the effect
of a (small) symmetry breaking term g H 1 in the otherwise symmetric Hamiltonian
(briefly softly broken Wigner symmetry), one may start by classifying the states
according to representations of G and set up a perturbative expansion in g for computing energy shifts, mixings, etc. This strategy, pioneered by Wigner, has been
widely used in atomic and in nuclear physics, with the powerful technical help of
the Wigner–Eckart theorem, which allows one to control the matrix elements of an
operator O between states belonging to definite representations of G.
In striking contrast, a spontaneously broken symmetry group G, while exactly
realized through its representations by the algebra A of observables (or more generally of canonical variables), is completely lost at the level of the states, which can
not be even approximately classified in terms of representations of G. This a priori
precludes the use of the Wigner–Eckart relations for computing symmetry breaking
effects.
Under general assumptions, one may nevertheless prove the existence of generalized Wigner–Eckart relations for the matrix elements of operators O between states
obtained from the ground state 0 by operators B, C ∈ A belonging to definite representations of G (C. Heissenberg and F. Strocchi, Corrections to Wigner–Eckart
relations by spontaneous symmetry breaking, arXiv: 2007.03539 [quant-ph]). Actually, for a spontaneously broken continuous symmetry group G, these generalized
Wigner–Eckart relations are identical to the standard ones up to a correction encoded
in a simple “tadpole” term involving Goldstone bosons. This allows for a perturbative calculation of symmetry breaking effects, like mass splitting, even in the case of
spontaneously broken symmetries.
More specifically, considering the Wigner–Eckart relations derived from the
matrix elements (B 0 , δ
a O C 0 ), with B, O, C, ∈ A and δ
a O the infinitesimal
variation of O under a one-dimensional subgroup of G, locally generated by j
a
0 , the
tadpole corrections are terms of the form