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American Journal of Physics

Why −i∇ is the momentum

Thomas F. Jordan

Published Dec 1, 1975• 15 citations
Publisher

Abstract

The operators −i∇, the translation generators, are related to mass and velocity by a simple argument that determines the form of the Hamiltonian, in terms of the translation generators, along with the Galilei generators, for a nonrelativistic particle. For a relativistic particle, a similar discussion is given for the analogous problem of determining the position operators for an irreducible unitary representation of the Poincaré group, at least well enough to identify the velocity operators and thus identify the momentum.

DOI
10.1119/1.9932

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The operators −i∇, the translation generators, are related to mass and velocity by a simple argument that determines the form of the Hamiltonian, in terms of the translation generators, along with the Galilei generators, for a nonrelativistic particle. For a relativistic particle, a similar discussion is given for the analogous problem of determining the position operators for an irreducible unitary representation of the Poincaré group, at least well enough to identify the velocity operators and thus identify the momentum.
The kind of operator algebra familiar in ordinary quantum mechanics is used to show formally that in an irreducible unitary representation of the Poincaré group for positive mass, the Newton–Wigner position operator is the only Hermitian operator with commuting components that transforms as a position operator should for translations, rotations, and time reversal and does not behave in a singular way that contradicts what can be learned from Lorentz transformations in the nonrelativistic limit.
Nonrelativistic Galilean quantum mechanics and the standard transition to relativistic Poincar\'e quantum mechanics is analyzed in terms of group theory. Special emphasis is given to the discussion of the relation between dynamics and geometry. Certain unsatisfactory features are pointed out and a new relativistic group G5{\mathcal{G}}_{5}G5​ is suggested as the symmetry group of dynamics. G5{\mathcal{G}}_{5}G5​ contains both the nonrelativistic Galilei group and the Poincar\'e group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of G5{\mathcal{G}}_{5}G5​ by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ correspond to infinite towers of states with increasing spin.
In the framework of limitable dynamical groups, a group Dst=Dt⊗SO(N)I, Dt=GE0(N) [multiplication in left half circle]Sp(2N, R),is discussed. It is shown that the physical representation U(Dst) describes interacting nonrelativistic particles of arbitrary spin in N dimensions, with linear combinations of second-order polynomials in momentum and position operators Pi, Qi with i = 1, ⋯, N, and spin-flip operators as Hamiltonians. GE0(N) is the central extension of the inhomogeneous, pure Galilei group, and SO(N)I a rotation group in N dimensions. Sp(2N, R) is the noncompact form of the symplectic group. It is proven that there exists no irreducible unitary representation of Dt describing particles with spin s > 0. Hence it was necessary to enlarge Dt to Dst. The physical free-particle group D0⊂Dst possesses projection parts in Sp(2N, R) as well as in SO(N)I. To calculate them, we decompose the physical representation U(D0) by group-theoretical methods such that the angular momentum splits into a spin and an orbital part. D0 is isomorphic to the central extension of the inhomogeneous Galilei group. The physical representation U(Dst) is calculated by the postulate that U(Dst) and U(D0) can be limited into each other, and that mass and spin conservation hold. The limitation is realized as a group contraction leading to a nonfaithful representation of the contracted group. We prove that U(Dst) is uniquely determined by mass and spin of the free particle which is obtained if the interaction in Dst is turned off. Furthermore, the generators are identified as functions of Pi, Qi, and spin-flip operators. Hence the limitation postulate leads to the physical representation of the abstract group Dst and to the identification of the described physical system.