232
Appendix A: Long Range Dynamics and Vacuum Seizing
hence, π(σ ∞ ) is aligned to h and β
λ is not broken. Such a representation may be
obtained by choosing free boundary conditions, so that σ ∂V tends to get aligned to h.
An alternative less natural strategy is to impose the boundary condition σ ∂V = θ,
with θ a fixed (free) parameter, so that one obtains a dynamics described by the finite
volume Hamiltonian
H
θ
V =
i∈V
σ i · (h − J θ).
In a factorial representation defined by a ground/equilibrium state invariant under
translations, π(σ) is aligned to h − J θ and β
λ is broken, unless a fine tuning is made
forcing the alignment of θ to h.
11
A.2 Boundary Counter Terms
The algebraic structure described above and displayed by physically interesting models (see the following Appendices B, C) could be somewhat trivialized by declaring
that the algebraic dynamics is that defined by a priori fixing the variables at infinity
to their c-number values in a factorial representation π of the algebra A.
On one side, such a choice recovers an essential localization of the dynamics,
close to that of the standard algebraic approach for non-relativistic systems, but on
the other side it looses much of the algebraic description in terms of α
t and its
symmetries.
In fact, i) the dynamics α
t
π is in general implementable only in the (factorial) representation π, ii) different phases of the same physical system get described by different algebraic structures, loosing the unifying picture of a common one-parameter
group of time translations α
t , with different realizations ascribed to the values taken
by variables at infinity, iii) the possibility of relating different phases in terms of
spontaneous symmetry breaking is in general precluded.
It is clear that the above choice provides only a partial account (that given by one
representation) of a much richer algebraic structure. In particular, one misses the
general mechanism by which symmetries of the infrared cutoff Hamiltonian H V are
no longer symmetries of the effective dynamics in factorial representations.
Moreover, one fails to recognize an important physical effect: in contrast with the
essentially local case (i.e. interactions decreasing for large space separations much
faster than |x|
−s+1 in s space dimensions), in the presence of long range interactions
(e.g. decreasing as |x|
−2 ), the coupling of a localized A with operators B ∂V , localized
around the boundary ∂V , does not vanish in the limit V → ∞, so that the time
evolution of A involves variables at infinity.
11 This mimics ’t Hooft position about the appearance of the θ angle in QCD, as discussed in G.
Morchio and F. Strocchi, Boundary Terms, Long Range Effects and Chiral Symmetry Breaking,
in Fields and Particles, Schladming 1990, H. Mitter and W. Schweiger eds., Springer 1991, pp.
171–214.
Appendix A: Long Range Dynamics and Vacuum Seizing
hence, π(σ ∞ ) is aligned to h and β
λ is not broken. Such a representation may be
obtained by choosing free boundary conditions, so that σ ∂V tends to get aligned to h.
An alternative less natural strategy is to impose the boundary condition σ ∂V = θ,
with θ a fixed (free) parameter, so that one obtains a dynamics described by the finite
volume Hamiltonian
H
θ
V =
i∈V
σ i · (h − J θ).
In a factorial representation defined by a ground/equilibrium state invariant under
translations, π(σ) is aligned to h − J θ and β
λ is broken, unless a fine tuning is made
forcing the alignment of θ to h.
11
A.2 Boundary Counter Terms
The algebraic structure described above and displayed by physically interesting models (see the following Appendices B, C) could be somewhat trivialized by declaring
that the algebraic dynamics is that defined by a priori fixing the variables at infinity
to their c-number values in a factorial representation π of the algebra A.
On one side, such a choice recovers an essential localization of the dynamics,
close to that of the standard algebraic approach for non-relativistic systems, but on
the other side it looses much of the algebraic description in terms of α
t and its
symmetries.
In fact, i) the dynamics α
t
π is in general implementable only in the (factorial) representation π, ii) different phases of the same physical system get described by different algebraic structures, loosing the unifying picture of a common one-parameter
group of time translations α
t , with different realizations ascribed to the values taken
by variables at infinity, iii) the possibility of relating different phases in terms of
spontaneous symmetry breaking is in general precluded.
It is clear that the above choice provides only a partial account (that given by one
representation) of a much richer algebraic structure. In particular, one misses the
general mechanism by which symmetries of the infrared cutoff Hamiltonian H V are
no longer symmetries of the effective dynamics in factorial representations.
Moreover, one fails to recognize an important physical effect: in contrast with the
essentially local case (i.e. interactions decreasing for large space separations much
faster than |x|
−s+1 in s space dimensions), in the presence of long range interactions
(e.g. decreasing as |x|
−2 ), the coupling of a localized A with operators B ∂V , localized
around the boundary ∂V , does not vanish in the limit V → ∞, so that the time
evolution of A involves variables at infinity.
11 This mimics ’t Hooft position about the appearance of the θ angle in QCD, as discussed in G.
Morchio and F. Strocchi, Boundary Terms, Long Range Effects and Chiral Symmetry Breaking,
in Fields and Particles, Schladming 1990, H. Mitter and W. Schweiger eds., Springer 1991, pp.
171–214.