Chapter 15
Physically Relevant Representations
From the examples and the discussion of the previous chapter, it appears that for
infinite systems the choice of the representation for the algebra of canonical variables (a basic preliminary step for even defining the dynamical problem) is a highly
non-trivial problem (unless the model is exactly soluble). Among the possible representations of the relevant algebra A, it is therefore convenient to isolate those which
are physically acceptable. For the moment, we restrict our discussion to the zero temperature case. The non-zero temperature case will be briefly discussed in Chap. 22.
On the basis of general physical considerations, we require the following conditions
for a physically relevant representation π.
I. (Existence of energy and momentum) The space and time translations are
described by strongly continuous groups of unitary operators U (a), U (t),
a ∈ R
s , t ∈ R.
By Stone’s theorem, this guarantees the existence of the generators P (the
momentum) and H (the energy), as well (densely) defined self-adjoint operators in the representation space H π . The existence of the energy is a necessary
condition for the representation to be physically realizable. The implementability
of the space translations is also necessary in relativistic quantum field theory, but
could be dispensed with in many body theory and, e.g. be replaced by the invariance under a discrete subgroup of the translations. In the sequel, for simplicity,
we shall not consider such more general cases.
II. (Stability or spectral condition) The spectrum σ(H ) of the Hamiltonian is
bounded from below. The relativistically invariant form of the spectral condition
is σ(H ) ≥ 0, H
2
− P
2
≥ 0.
Such a property guarantees that, under small (external) perturbations, the system
does not collapse to lower and lower energy states.
80
80 This condition is not required for non-zero temperature states, since in that case the reservoir can
feed the system and prevent it from collapsing.
© 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_15
101
Physically Relevant Representations
From the examples and the discussion of the previous chapter, it appears that for
infinite systems the choice of the representation for the algebra of canonical variables (a basic preliminary step for even defining the dynamical problem) is a highly
non-trivial problem (unless the model is exactly soluble). Among the possible representations of the relevant algebra A, it is therefore convenient to isolate those which
are physically acceptable. For the moment, we restrict our discussion to the zero temperature case. The non-zero temperature case will be briefly discussed in Chap. 22.
On the basis of general physical considerations, we require the following conditions
for a physically relevant representation π.
I. (Existence of energy and momentum) The space and time translations are
described by strongly continuous groups of unitary operators U (a), U (t),
a ∈ R
s , t ∈ R.
By Stone’s theorem, this guarantees the existence of the generators P (the
momentum) and H (the energy), as well (densely) defined self-adjoint operators in the representation space H π . The existence of the energy is a necessary
condition for the representation to be physically realizable. The implementability
of the space translations is also necessary in relativistic quantum field theory, but
could be dispensed with in many body theory and, e.g. be replaced by the invariance under a discrete subgroup of the translations. In the sequel, for simplicity,
we shall not consider such more general cases.
II. (Stability or spectral condition) The spectrum σ(H ) of the Hamiltonian is
bounded from below. The relativistically invariant form of the spectral condition
is σ(H ) ≥ 0, H
2
− P
2
≥ 0.
Such a property guarantees that, under small (external) perturbations, the system
does not collapse to lower and lower energy states.
80
80 This condition is not required for non-zero temperature states, since in that case the reservoir can
feed the system and prevent it from collapsing.
© 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_15
101