4.1 The Heisenberg Group (CCR)
53
classical limit with the ‘unsmeared’ potential energy V (q) (up to the unessential
additive constant term in (4.1.20) ).
4.1.7 Notes. (i) The quantal correlation functions are constant on the orbits O ϕ ;
e.g.
T r(P
ϕ
x (X j − x j )(X k − x k )) = T r(P ϕ X j X k ),
for all j, k, and for all x ∈ R
2n
,
(4.1.22)
(ii) If the Hamiltonian operator A is quadratic in all the generators X j :
A :=
1
2
h
jk X j X k ⇒ f A (x) =
1
2
h
jk x j x k + const,
(4.1.23)
i.e. in this case the usual classical limit coincides with the classical projections. This
situation is analyzed in Sect. 4.2.
4.1.8 On the limit → 0.
All the previous results and considerations are equally valid for any nonvanishing value of the parameter . Any change of the value of the parameter might
be interpreted from the point of view of mathematics, either as a change of the
representation U (G) of the Heisenberg group G to an inequivalent one leaving the
correspondence of the generators
1
X j ∈ U (g) to fixed elements ξ j ∈ g of the Lie
algebra unchanged, or as a change of the basis {ξ j } in g into {λξ j } (corresponding
to a ‘reinterpretation’ (i.e. change of units) of parameters x occurring in (4.1.8) ),
leaving the choice of the representation fixed.
Let a physical interpretation of the generators X j be fixed (compare Sect. 1.2),
leaving the value of unspecified. If some empirical system is adequately described
by QM with the given interpretation of X j ’s, for some value of , then this value is
for the system unique (independently on any choices of generators of the evolution in
time— consider, e.g. the occurrence of in uncertainty relations). If two such systems
could form one composite system the mutually noninteracting parts of which they
are, then the value of for both systems is the same (interpretation of X j ’s fixed!),
since each of the subsystems taken separately determines for the whole system (we
have now a 2(n 1 + n 2 ) + 1—dimensional Heisenberg group, if the subsystems have
n 1 , resp. n 2 degrees of freedom).
These considerations show, that any change of the value of —if physically
interpreted—has to be connected with a change of interpretation of the generators
X j ∈ U (g). We obtain an example of such a reinterpretation, if we describe a system
consisting of a large number of particles: in a description of the center of mass motion
we can deal instead of with center of mass coordinates and total linear momenta
(which satisfy CCR with the experimental value of Planck constant) rather with
center of mass coordinates and averaged momenta per a particle.
If we keep the interpretation of X j ’s fixed, then for various values of we obtain
different theories. We shall describe a transition of → 0 in the context of the
53
classical limit with the ‘unsmeared’ potential energy V (q) (up to the unessential
additive constant term in (4.1.20) ).
4.1.7 Notes. (i) The quantal correlation functions are constant on the orbits O ϕ ;
e.g.
T r(P
ϕ
x (X j − x j )(X k − x k )) = T r(P ϕ X j X k ),
for all j, k, and for all x ∈ R
2n
,
(4.1.22)
(ii) If the Hamiltonian operator A is quadratic in all the generators X j :
A :=
1
2
h
jk X j X k ⇒ f A (x) =
1
2
h
jk x j x k + const,
(4.1.23)
i.e. in this case the usual classical limit coincides with the classical projections. This
situation is analyzed in Sect. 4.2.
4.1.8 On the limit → 0.
All the previous results and considerations are equally valid for any nonvanishing value of the parameter . Any change of the value of the parameter might
be interpreted from the point of view of mathematics, either as a change of the
representation U (G) of the Heisenberg group G to an inequivalent one leaving the
correspondence of the generators
1
X j ∈ U (g) to fixed elements ξ j ∈ g of the Lie
algebra unchanged, or as a change of the basis {ξ j } in g into {λξ j } (corresponding
to a ‘reinterpretation’ (i.e. change of units) of parameters x occurring in (4.1.8) ),
leaving the choice of the representation fixed.
Let a physical interpretation of the generators X j be fixed (compare Sect. 1.2),
leaving the value of unspecified. If some empirical system is adequately described
by QM with the given interpretation of X j ’s, for some value of , then this value is
for the system unique (independently on any choices of generators of the evolution in
time— consider, e.g. the occurrence of in uncertainty relations). If two such systems
could form one composite system the mutually noninteracting parts of which they
are, then the value of for both systems is the same (interpretation of X j ’s fixed!),
since each of the subsystems taken separately determines for the whole system (we
have now a 2(n 1 + n 2 ) + 1—dimensional Heisenberg group, if the subsystems have
n 1 , resp. n 2 degrees of freedom).
These considerations show, that any change of the value of —if physically
interpreted—has to be connected with a change of interpretation of the generators
X j ∈ U (g). We obtain an example of such a reinterpretation, if we describe a system
consisting of a large number of particles: in a description of the center of mass motion
we can deal instead of with center of mass coordinates and total linear momenta
(which satisfy CCR with the experimental value of Planck constant) rather with
center of mass coordinates and averaged momenta per a particle.
If we keep the interpretation of X j ’s fixed, then for various values of we obtain
different theories. We shall describe a transition of → 0 in the context of the