Chapter 4
Examples of Classical Mechanical
Projections
4.1 The Heisenberg Group (CCR)
4.1.1 A physical system consisting of the finite number N of nonrelativistic (apriori
mutually distinguishable) point particles is described in the conventional QM by an
infinite dimensional unitary irreducible representation of the 2n + 1 - dimensional
Heisenberg group G (n := N ν, ν is the dimension of the one-particle configuration
space); cf. also [37, Sect. 3.3-b]. The Heisenberg group G is a central extension by
R of the commutative group R
2n (which can be identified with the classical flat
phase space R
2n
= T
∗
R
n ), compare [321] and [346]. The (scalar multiples of the)
selfadjoint generators X j , j = 1, 2, . . . 2n, of the representation correspond to basic
‘kinematical’ observables of the system. The choice of X j ’s is conveniently made in
such a way, that on corresponding domains (e.g. on D G ) the commutation relations
(CCR) are fulfilled:
[X j , X k ] = i S jk X 0 for j, k = 1, 2, . . . 2n;
(4.1.1a)
[X j , X 0 ] = 0, j = 1, 2, . . . 2n.
(4.1.1b)
Here the elements S jk of the 2n × 2n real matrix S are defined:
S j j+n = −S j+n j = 1, j = 1, 2, . . . n, S jk = 0 otherwise.
(4.1.2)
Hence S
−1
= S
T
= −S where S
T is the transposed matrix to S. From (4.1.1b) we
see, that
X 0 = I, (I is the identity of L(H)).
(4.1.3)
The parameter ∈ R, = 0, ( := the ‘Planck constant’, if its value is chosen
properly) classifies all infinite-dimensional unitary irreducible representations of G;
representations corresponding to various values of are mutually inequivalent, [346].
© Springer Nature Switzerland AG 2020, corrected publication 2020
P. Bóna, Classical Systems in Quantum Mechanics,
https://doi.org/10.1007/978-3-030-45070-0_4
49
Examples of Classical Mechanical
Projections
4.1 The Heisenberg Group (CCR)
4.1.1 A physical system consisting of the finite number N of nonrelativistic (apriori
mutually distinguishable) point particles is described in the conventional QM by an
infinite dimensional unitary irreducible representation of the 2n + 1 - dimensional
Heisenberg group G (n := N ν, ν is the dimension of the one-particle configuration
space); cf. also [37, Sect. 3.3-b]. The Heisenberg group G is a central extension by
R of the commutative group R
2n (which can be identified with the classical flat
phase space R
2n
= T
∗
R
n ), compare [321] and [346]. The (scalar multiples of the)
selfadjoint generators X j , j = 1, 2, . . . 2n, of the representation correspond to basic
‘kinematical’ observables of the system. The choice of X j ’s is conveniently made in
such a way, that on corresponding domains (e.g. on D G ) the commutation relations
(CCR) are fulfilled:
[X j , X k ] = i S jk X 0 for j, k = 1, 2, . . . 2n;
(4.1.1a)
[X j , X 0 ] = 0, j = 1, 2, . . . 2n.
(4.1.1b)
Here the elements S jk of the 2n × 2n real matrix S are defined:
S j j+n = −S j+n j = 1, j = 1, 2, . . . n, S jk = 0 otherwise.
(4.1.2)
Hence S
−1
= S
T
= −S where S
T is the transposed matrix to S. From (4.1.1b) we
see, that
X 0 = I, (I is the identity of L(H)).
(4.1.3)
The parameter ∈ R, = 0, ( := the ‘Planck constant’, if its value is chosen
properly) classifies all infinite-dimensional unitary irreducible representations of G;
representations corresponding to various values of are mutually inequivalent, [346].
© Springer Nature Switzerland AG 2020, corrected publication 2020
P. Bóna, Classical Systems in Quantum Mechanics,
https://doi.org/10.1007/978-3-030-45070-0_4
49