Chapter 28
An Extension of Goldstone Theorem
to Non-symmetric Hamiltonians
The Goldstone theorem and its rigorous predictions on the energy spectrum at zero
momentum can be extended
184 to the case in which the Hamiltonian H is not symmetric, but it has simple transformation properties, in the sense that the multiple
commutators of H and the charge Q generate a finite dimensional Lie algebra,
briefly
[Q
i
, H ] = c
i
k Q
k
.
The invariance of the dynamics is then replaced by
I. (Covariance group of the dynamics)
There exists a Lie group G of
∗ -automorphisms α
g
, g ∈ G, of a subalgebra
A 0 ⊆ A, which contains the dynamics α t as a one-parameter subgroup; for simplicity, in the following, α
g is assumed to commute with the space translations α x .
II. (Local generation of the covariance group)
The covariance group α
g
, g ∈ G is locally generated by charge densities
δ
i A ≡ ∂α
g
(A)/∂g i | g=0 = i lim
R→∞
[Q
i
R , A], A ∈ A 0 ,
(28.1)
Q
i
R (t) = α
t
(Q
i
R ) =
dx f R (x) j
i
0 (x, t)),
and the charge density commutators are absolutely integrable (for large |x|) as tempered distributions in t (the local charge generating α
t is the infrared regularized
Hamiltonian H L ).
185
184 G. Morchio and F. Strocchi, Ann. Phys. 185, 241 (1988).
185 As remarked in the standard case, the above commutators as well as the following ones are
understood as bilinear forms on a dense set of states in each relevant representation; actually, all
what is needed is their expectations on the ground state.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_28
199
An Extension of Goldstone Theorem
to Non-symmetric Hamiltonians
The Goldstone theorem and its rigorous predictions on the energy spectrum at zero
momentum can be extended
184 to the case in which the Hamiltonian H is not symmetric, but it has simple transformation properties, in the sense that the multiple
commutators of H and the charge Q generate a finite dimensional Lie algebra,
briefly
[Q
i
, H ] = c
i
k Q
k
.
The invariance of the dynamics is then replaced by
I. (Covariance group of the dynamics)
There exists a Lie group G of
∗ -automorphisms α
g
, g ∈ G, of a subalgebra
A 0 ⊆ A, which contains the dynamics α t as a one-parameter subgroup; for simplicity, in the following, α
g is assumed to commute with the space translations α x .
II. (Local generation of the covariance group)
The covariance group α
g
, g ∈ G is locally generated by charge densities
δ
i A ≡ ∂α
g
(A)/∂g i | g=0 = i lim
R→∞
[Q
i
R , A], A ∈ A 0 ,
(28.1)
Q
i
R (t) = α
t
(Q
i
R ) =
dx f R (x) j
i
0 (x, t)),
and the charge density commutators are absolutely integrable (for large |x|) as tempered distributions in t (the local charge generating α
t is the infrared regularized
Hamiltonian H L ).
185
184 G. Morchio and F. Strocchi, Ann. Phys. 185, 241 (1988).
185 As remarked in the standard case, the above commutators as well as the following ones are
understood as bilinear forms on a dense set of states in each relevant representation; actually, all
what is needed is their expectations on the ground state.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0_28
199