Chapter 11
Quantum Mechanics. Algebraic
Structure and States
We briefly review the basic structure of quantum mechanics (QM) with the aim
of covering both the case of systems with a finite number of degrees of freedom
(ordinary QM) as well as the case of systems with an infinite number of degrees of
freedom (briefly infinite systems).
46
For this purpose, it is useful to recall that in the original formulation of QM the
emphasis has been on the canonical structure, in terms of the canonical variables q, p,
in analogy with the classical case. The quantization conditions, which mark the basic
difference between classical and quantum mechanics, amounts to replace the Poisson brackets structure, for N degrees of freedom, with the canonical commutation
relations (CCR)
[q i , p j ] = iδ i j , [q i , q j ] = 0 = [p i , p j ], i, j = 1, 2, . . . , N ,
(11.1)
where δ i j denotes the Kronecker symbol and for simplicity units have been chosen
such that = 1. In this way, the abelian algebra of the classical canonical variables
is turned into the non-abelian Heisenberg algebra A H .
47
The CCR imply that the canonical variables q, p cannot both be represented by
self-adjoint bounded operators in a Hilbert space.
48 This is the source of technical
mathematical problems (domain questions, etc.), so that it is more convenient to use
as basic variables the so-called Weyl operators
46 For an elementary introduction to the quantum mechanics of infinite systems, see, e.g. F. Strocchi,
Elements of Quantum Mechanics of Infinite Systems, World Scientific 1985, hereafter referred to as
[S 85].
47 W. Heisenberg, The Physical Principles of the Quantum Theory, Dover 1930; P.A.M. Dirac, The
Principles of Quantum Mechanics, Oxford University Press 1986.
48 In fact, the CCR imply inq n−1 = q n p − pq n and by taking the norms one has n||q n−1 || ≤
2||q n−1 || ||q|| || p||, i.e. ||q|| || p|| ≥ n/2.
© 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_11
73
Quantum Mechanics. Algebraic
Structure and States
We briefly review the basic structure of quantum mechanics (QM) with the aim
of covering both the case of systems with a finite number of degrees of freedom
(ordinary QM) as well as the case of systems with an infinite number of degrees of
freedom (briefly infinite systems).
46
For this purpose, it is useful to recall that in the original formulation of QM the
emphasis has been on the canonical structure, in terms of the canonical variables q, p,
in analogy with the classical case. The quantization conditions, which mark the basic
difference between classical and quantum mechanics, amounts to replace the Poisson brackets structure, for N degrees of freedom, with the canonical commutation
relations (CCR)
[q i , p j ] = iδ i j , [q i , q j ] = 0 = [p i , p j ], i, j = 1, 2, . . . , N ,
(11.1)
where δ i j denotes the Kronecker symbol and for simplicity units have been chosen
such that = 1. In this way, the abelian algebra of the classical canonical variables
is turned into the non-abelian Heisenberg algebra A H .
47
The CCR imply that the canonical variables q, p cannot both be represented by
self-adjoint bounded operators in a Hilbert space.
48 This is the source of technical
mathematical problems (domain questions, etc.), so that it is more convenient to use
as basic variables the so-called Weyl operators
46 For an elementary introduction to the quantum mechanics of infinite systems, see, e.g. F. Strocchi,
Elements of Quantum Mechanics of Infinite Systems, World Scientific 1985, hereafter referred to as
[S 85].
47 W. Heisenberg, The Physical Principles of the Quantum Theory, Dover 1930; P.A.M. Dirac, The
Principles of Quantum Mechanics, Oxford University Press 1986.
48 In fact, the CCR imply inq n−1 = q n p − pq n and by taking the norms one has n||q n−1 || ≤
2||q n−1 || ||q|| || p||, i.e. ||q|| || p|| ≥ n/2.
© 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_11
73