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Mathematical foundations of quantum mechanics
240
Book
reviews
Chapters 6 and 11 are devoted to the conditions of the problems, the use of check computations for improving results, and a brief analysis of the influence of rounding-off errors in certain methods. The book is easy to read and contains numerical examples.
a large
number of illustrative
0 .K.
G.W. UACK-, Mathematical Amsterdam, W.A. Benjamin,
foundations
Inc.,
of
quantum
1963, X + 137 pp.,
mechanics,
Faddeeo
Vew York -
3.95 $.
THF book under review is based on lectures on the mathematical foundations of quantum mechanics delivered by Prof. Mackey at Harvard University in 1960. The author remarks in the foreword that his aim when devis. ing the course was to present quantum mechanics and certain branches of classical physics from a point of view closer to pure mathematics than is usual in physics faculties. The lectures are basically concerned with explaining the mathematical structure of the theory, and not with treating methods of solving specific problems. The classical books by Neumann and Weyl on quantum mechanics have undoubtedly had a considerable influence on the author. Nevertheless, his lectures are original in content and clearly reflect his personal ideas and tastes, Though only quantum mechanics is mentioned in the title, a substantia: part of the book is devoted to the foundations of classical mechanics. In addition, there is a short chapter on applications of group theory and of the quantum mechanics of the atom, The author achieves a number of aims by treating classical before quantum mechanics. Firstly, he is able to illustrate by means of a more customary example fundamental concepts characterizing a physical system states, or a dynamic group. Secondly, as such as observable quantities, the treatment progresses he gradually introduces the mathematical appathe theory of selfadjoint operators ratus required later. For instance, in Hilbert space appears naturally while describing a linear classical dynamic system with an infinite number of degrees of freedom, an example of which is provided by an electromagnetic wave in vacua. The final statements of the fundamental concepts of classical mechanics are given in terms of modern differential geometry. The phase space is a co-tangential fibre space, the base of which is the coordinate
Book
reviews
241
manifold; observable quantities are described by arbitrary functions on the phase space, and states by arbitrary measures on it. With every observable quantity there is naturally associated a one-parameter group of transformations of the phase space - contact transformations. The role of energy is explained thus: the group of contact transformations generated by this observable quantity describes the variation of the system in time. The most original and interesting point in the book is the statement of the fundamental axioms of which quantum mechanics in its modern form is a consequence. The raw materials of the axioms are the concepts of an observable quantity and of the state of a system. Every state associates with every observable quantity a probability distribution for its possible values. The author formulates six intuitively natural axioms, which reduce a system of observable quantities and states to the concept of a partially ordered set with orthogonal complements (logic of the system) with a strictly convex family of probability measures specified on it. The states and observable quantities of classical mechanics satisfy these axioms. The logic of classical mechanics is a Boolean algebra of subsets of the phase space. Quantum mechanics is distinguished by means of an extra, seventh axiom, according to which the logic of quantum mechanics is isomorphic to a partially ordered set of all closed subspaces of a separable complex Hilbert space. The author himself observes that this axiom differs considerably from the previous ones and appears to be extremely arbitrary. To justify it, he analyses the different possible choices of logic, different from the logic of classical mechanics, and at the same time reasonably easily performed. It is noted that the structure of closed subspaces of Hilbert space over the real number, complex number or quaternion field, as also the structure of selfadjoint idempotents in factors types II and III, are the simplest possibilities in this sense. Complex Hilbert space is distinguished from others by the fact that a natural correspondence between observable quantities and one-parameter symmetry groups exists in it - a property occurring in classical mechanics. The author emphasizes, however, the need for a detailed study of other spaces. The statement of the fundamental concepts of quantum mechanics, accepted at the present time, follows from the given axioms, including the seventh. Observable magnitudes have a one-to-one relation with selfadjoint operators in complex Hilbert space, states correspond to the positive nuclear operators with a trace equal to unity. The probability distribution of a quantity A in a state ,Mis given by the formula
242
f? -+ tr (P-&n,
Book
reviews
where E is a Bore1 set on the real axis, and Pi
is the
spectral family of the selfadjoint operator A. Dynamics is described by a one-parameter group of unitary transformations. The reproducing operator of this group is the energy operator of the system. The further treatment is concerned with constructing the energy operator for different concrete systems. The author derives in a unique way a method of “canonical quantization” suitable for constructing the quantum-mechanical snalogue of a classical mechanical system, the coordinate manifold of which is a Euclidearl space. An extremely interesting section contains an extension of the rules of canonical quantization to the case of classical systems with arbitrary coordinate manifolds. The author emphasizes, however, that the rules described are quite arbitrary and not unique. A special section is devoted to the quantization of linear systems with an infinite number of degrees of freedom, i.e. the theory of quantized free fields. Such, briefly, are the contents of Prof. Mackey’ s book. The style is highly informal, with the author making full use of the book’s special form, namely, a transcript of a course of lectures. Considerable profit and pleasure should be derived from the book, both by the mathematician commencing a study of quantummechanics, and by the theoretical physicist wishing to go deeper into familiar concepts. The WY Publishing House has done well to publish a Russian translation. 1 .D.
Faddeeu
Book
reviews
Chapters 6 and 11 are devoted to the conditions of the problems, the use of check computations for improving results, and a brief analysis of the influence of rounding-off errors in certain methods. The book is easy to read and contains numerical examples.
a large
number of illustrative
0 .K.
G.W. UACK-, Mathematical Amsterdam, W.A. Benjamin,
foundations
Inc.,
of
quantum
1963, X + 137 pp.,
mechanics,
Faddeeo
Vew York -
3.95 $.
THF book under review is based on lectures on the mathematical foundations of quantum mechanics delivered by Prof. Mackey at Harvard University in 1960. The author remarks in the foreword that his aim when devis. ing the course was to present quantum mechanics and certain branches of classical physics from a point of view closer to pure mathematics than is usual in physics faculties. The lectures are basically concerned with explaining the mathematical structure of the theory, and not with treating methods of solving specific problems. The classical books by Neumann and Weyl on quantum mechanics have undoubtedly had a considerable influence on the author. Nevertheless, his lectures are original in content and clearly reflect his personal ideas and tastes, Though only quantum mechanics is mentioned in the title, a substantia: part of the book is devoted to the foundations of classical mechanics. In addition, there is a short chapter on applications of group theory and of the quantum mechanics of the atom, The author achieves a number of aims by treating classical before quantum mechanics. Firstly, he is able to illustrate by means of a more customary example fundamental concepts characterizing a physical system states, or a dynamic group. Secondly, as such as observable quantities, the treatment progresses he gradually introduces the mathematical appathe theory of selfadjoint operators ratus required later. For instance, in Hilbert space appears naturally while describing a linear classical dynamic system with an infinite number of degrees of freedom, an example of which is provided by an electromagnetic wave in vacua. The final statements of the fundamental concepts of classical mechanics are given in terms of modern differential geometry. The phase space is a co-tangential fibre space, the base of which is the coordinate
Book
reviews
241
manifold; observable quantities are described by arbitrary functions on the phase space, and states by arbitrary measures on it. With every observable quantity there is naturally associated a one-parameter group of transformations of the phase space - contact transformations. The role of energy is explained thus: the group of contact transformations generated by this observable quantity describes the variation of the system in time. The most original and interesting point in the book is the statement of the fundamental axioms of which quantum mechanics in its modern form is a consequence. The raw materials of the axioms are the concepts of an observable quantity and of the state of a system. Every state associates with every observable quantity a probability distribution for its possible values. The author formulates six intuitively natural axioms, which reduce a system of observable quantities and states to the concept of a partially ordered set with orthogonal complements (logic of the system) with a strictly convex family of probability measures specified on it. The states and observable quantities of classical mechanics satisfy these axioms. The logic of classical mechanics is a Boolean algebra of subsets of the phase space. Quantum mechanics is distinguished by means of an extra, seventh axiom, according to which the logic of quantum mechanics is isomorphic to a partially ordered set of all closed subspaces of a separable complex Hilbert space. The author himself observes that this axiom differs considerably from the previous ones and appears to be extremely arbitrary. To justify it, he analyses the different possible choices of logic, different from the logic of classical mechanics, and at the same time reasonably easily performed. It is noted that the structure of closed subspaces of Hilbert space over the real number, complex number or quaternion field, as also the structure of selfadjoint idempotents in factors types II and III, are the simplest possibilities in this sense. Complex Hilbert space is distinguished from others by the fact that a natural correspondence between observable quantities and one-parameter symmetry groups exists in it - a property occurring in classical mechanics. The author emphasizes, however, the need for a detailed study of other spaces. The statement of the fundamental concepts of quantum mechanics, accepted at the present time, follows from the given axioms, including the seventh. Observable magnitudes have a one-to-one relation with selfadjoint operators in complex Hilbert space, states correspond to the positive nuclear operators with a trace equal to unity. The probability distribution of a quantity A in a state ,Mis given by the formula
242
f? -+ tr (P-&n,
Book
reviews
where E is a Bore1 set on the real axis, and Pi
is the
spectral family of the selfadjoint operator A. Dynamics is described by a one-parameter group of unitary transformations. The reproducing operator of this group is the energy operator of the system. The further treatment is concerned with constructing the energy operator for different concrete systems. The author derives in a unique way a method of “canonical quantization” suitable for constructing the quantum-mechanical snalogue of a classical mechanical system, the coordinate manifold of which is a Euclidearl space. An extremely interesting section contains an extension of the rules of canonical quantization to the case of classical systems with arbitrary coordinate manifolds. The author emphasizes, however, that the rules described are quite arbitrary and not unique. A special section is devoted to the quantization of linear systems with an infinite number of degrees of freedom, i.e. the theory of quantized free fields. Such, briefly, are the contents of Prof. Mackey’ s book. The style is highly informal, with the author making full use of the book’s special form, namely, a transcript of a course of lectures. Considerable profit and pleasure should be derived from the book, both by the mathematician commencing a study of quantummechanics, and by the theoretical physicist wishing to go deeper into familiar concepts. The WY Publishing House has done well to publish a Russian translation. 1 .D.
Faddeeu