13 The Heisenberg group and the Schrödinger Representation

The Heisenberg group and the Schr¨odinger Representation

In our discussion of the free particle, we used just the actions of the groups ${\mathbf{R}}^3$ of spatial translations and the group $R$ of time translations, finding corresponding observables, the self-adjoint momentum and Hamiltonian operators $P$ and H. We’ve seen though that the Fourier transform allows a perfectly symmetrical treatment of position and momentum variables and the corresponding non-commuting position and momentum operators $Q_j$ and $P_j$

The $P_j$ and $Q_j$ operators satisfy relations known as the Heisenberg commutation relations, which first appeared in the earliest work of Heisenberg and collaborators on a full quantum-mechanical formalism in 1925. These were quickly recognized by Hermann Weyl as the operator relations of a Lie algebra representation, for a Lie algebra now known as the Heisenberg Lie algebra. The corresponding group is called the Heisenberg group by mathematicians, with physicists sometimes using the terminology “Weyl group” (which means something else to mathematicians). The state space of a quantum particle, either free or moving in a potential, will be a unitary representation of this group, with the group of spatial translations a subgroup.

Note that this particular use of a group and its representation theory in quantum mechanics is both at the core of the standard axioms and much more general than the usual characterization of the significance of groups as “symmetry groups”. The Heisenberg group does not in any sense correspond to a group of invariances of the physical situation (there are no states invariant under the group), and its action does not commute with any non-zero Hamiltonian operator. Instead it plays a much deeper role, with its unique unitary representation determining much of the structure of quantum mechanics.

中文目录

Chapter contents

• 13.1 The Heisenberg Lie algebra
• 13.2 The Heisenberg group
• 13.3 The Schr¨odinger representation
• 13.4 For further reading

Previous chapter: 12 Position and the Free Particle

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来源版本：2025-10-20

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