A scheme for Galilei general relativistic quantum mechanics

Abstract Although it is usual to read that non-relativistic quantum mechanics is covariant, and even invariant, under the Galilei transformations, this topic has scarcely been addressed in the standard literature on the theory. This fact has, at least, two undesirable consequences. On the one hand, the meanings of the concepts of covariance and invariance are not precisely elucidated. On the other hand, one is left without a clear idea about in what sense the covariance and/or invariance of quantum mechanics can be asserted. This situation has its counterpart in the field of the interpretation of quantum mechanics: the relevance of the Galilei group is rarely discussed in the impressive amount of literature on the subject. The general premise underlying the present chapter is that the relationship between interpretation and Galilei transformations deserves to be seriously analyzed: the fact that the theory is covariant under the Galilei group does not guarantee the same property for the interpretation because, in general, interpretations add interpretative postulates to the formal structure of the theory. On this basis, and after clarifying the concepts of covariance and invariance, this chapter considers in what sense quantum mechanics is covariant and under what conditions it is invariant under the transformations of the Galilei group. This conceptual framework will provide the tools to study the result of the application of the Galilei transformations to the Modal-Hamiltonian interpretative postulates.Abstract Although it is usual to read that non-relativistic quantum mechanics is covariant, and even invariant, under the Galilei transformations, this topic has scarcely been addressed in the standard literature on the theory. This fact has, at least, two undesirable consequences. On the one hand, the meanings of the concepts of covariance and invariance are not precisely elucidated. On the other hand, one is left without a clear idea about in what sense the covariance and/or invariance of quantum mechanics can be asserted. This situation has its counterpart in the field of the interpretation of quantum mechanics: the relevance of the Galilei group is rarely discussed in the impressive amount of literature on the subject. The general premise underlying the present chapter is that the relationship between interpretation and Galilei transformations deserves to be seriously analyzed: the fact that the theory is covariant under the Galilei group does not guarantee the same property for the interpretation because, in general, interpretations add interpretative postulates to the formal structure of the theory. On this basis, and after clarifying the concepts of covariance and invariance, this chapter considers in what sense quantum mechanics is covariant and under what conditions it is invariant under the transformations of the Galilei group. This conceptual framework will provide the tools to study the result of the application of the Galilei transformations to the Modal-Hamiltonian interpretative postulates.

Feb 19, 20260 citations

Foundations of Relativistic Quantum Mechanics

Mark Goodall

NatureJul 1, 19624 citations

Nonrelativistic Quantum Mechanics for Particles with Arbitrary Spin

William J. Hurley

Motivated by the difficulties which persist in the construction of a consistent description of interacting relativistic particles with spin greater than 1, we examine the analogous problem within the framework of nonrelativistic quantum mechanics and find that all of the difficulties vanish in the (Galilei-invariant) limit. It is found that a unique, first-order, Galilei-covariant wave equation describing massive, spin- $s$ particles follows from general invariance assumptions and a minimality condition on the number of components of the wave function. The minimal theory has $6s+1$ components ( $2s+1$ of which are independent) and admits a consistent quantum-mechanical interpretation. An external electromagnetic field interaction is introduced via the minimal-coupling replacement, and, in contrast with the relativistic case, a consistent theory emerges for arbitrary spin. The Galilean spin- $s$ particles so described are found to possess only an electric charge and a magnetic dipole with a $g$ factor of $\frac{1}{s}$ . The Galilei-invariant addition of arbitrary moment terms is also described. The extension of the formalism to second-quantized spin- $s$ fields is discussed, and it is found that in that case, too, the difficulties are peculiar to the relativistic case. Reasons for the simplicity of the Galilei case are presented. Finally, for the sake of completeness, the first-order form of the Schr\"odinger theory is presented, as well as examples of theories which violate the minimality condition.Motivated by the difficulties which persist in the construction of a consistent description of interacting relativistic particles with spin greater than 1, we examine the analogous problem within the framework of nonrelativistic quantum mechanics and find that all of the difficulties vanish in the (Galilei-invariant) limit. It is found that a unique, first-order, Galilei-covariant wave equation describing massive, spin-$s$ particles follows from general invariance assumptions and a minimality condition on the number of components of the wave function. The minimal theory has $6s+1$ components ($2s+1$ of which are independent) and admits a consistent quantum-mechanical interpretation. An external electromagnetic field interaction is introduced via the minimal-coupling replacement, and, in contrast with the relativistic case, a consistent theory emerges for arbitrary spin. The Galilean spin-$s$ particles so described are found to possess only an electric charge and a magnetic dipole with a $g$ factor of $\frac{1}{s}$. The Galilei-invariant addition of arbitrary moment terms is also described. The extension of the formalism to second-quantized spin-$s$ fields is discussed, and it is found that in that case, too, the difficulties are peculiar to the relativistic case. Reasons for the simplicity of the Galilei case are presented. Finally, for the sake of completeness, the first-order form of the Schr\"odinger theory is presented, as well as examples of theories which violate the minimality condition.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsMay 15, 197137 citations

Relativistic Quantum Mechanics

Sol Wieder

Elsevier eBooksJan 1, 19730 citations

Relativistic Quantum Mechanics

Walter Greiner

Springer eBooksJan 1, 1990232 citations

s

particles follows from general invariance assumptions and a minimality condition on the number of components of the wave function. The minimal theory has

6s+1

components (

2s+1

of which are independent) and admits a consistent quantum-mechanical interpretation. An external electromagnetic field interaction is introduced via the minimal-coupling replacement, and, in contrast with the relativistic case, a consistent theory emerges for arbitrary spin. The Galilean spin-

s

particles so described are found to possess only an electric charge and a magnetic dipole with a

g

factor of

\frac{1}{s}

. The Galilei-invariant addition of arbitrary moment terms is also described. The extension of the formalism to second-quantized spin-

s

fields is discussed, and it is found that in that case, too, the difficulties are peculiar to the relativistic case. Reasons for the simplicity of the Galilei case are presented. Finally, for the sake of completeness, the first-order form of the Schr\"odinger theory is presented, as well as examples of theories which violate the minimality condition.