Bargmann GroupEdit
The Bargmann group is a central extension of the Galilei group that captures how nonrelativistic quantum systems transform under the full set of Galilean symmetries. Named for Egon Bargmann, who clarified how mass enters the quantum-mechanical realization of these symmetries, the Bargmann group provides the correct mathematical setting in which nonrelativistic states carry both momentum and a mass-dependent phase. In quantum mechanics, projective representations of the Galilei group become true representations of the Bargmann group, with mass acting as a central charge. This framing helps explain why quantum states exhibit definite mass properties and how spin and internal structure fit into Galilean invariance. See how this connects to Galilei group, Egon Bargmann, and central extension of symmetry groups.
The Bargmann construction has become a standard reference point for understanding nonrelativistic invariance in quantum physics. By incorporating a U(1) phase factor associated with mass, the Bargmann group reconciles the mathematical requirements of unitary representations with the physical need to describe particles of definite mass. This approach underpins the way nonrelativistic particles of momentum, spin, and internal energy are modeled within a Galilean framework and links naturally to the formulation of the Schrödinger equation in a rotationally and Galilean-invariant setting. See also projective representations and representation theory in the nonrelativistic context.
Mathematical structure
Elements and extension: The Bargmann group is the central extension of the Galilei group by U(1) with mass M acting as a central generator. Concretely, a Galilean transformation combined with a U(1) phase forms an element of the Bargmann group, and the composition law carries a two-cocycle that encodes the mass-dependent phase. This construction is a standard example of a central extension in group theory.
Generators and relations: The relevant generators include angular momentum J, translations P, boosts K, and time translations H, with a central element M representing mass. The core commutation relations in the associated Lie algebra include
- [J_i, J_j] = i ε_ijk J_k
- [J_i, P_j] = i ε_ijk P_k
- [J_i, K_j] = i ε_ijk K_k
- [K_i, H] = i P_i
- [K_i, P_j] = i δ_ij M
- M commutes with all others (central) These relations formalize how rotations, boosts, and translations interact in a nonrelativistic quantum setting and illustrate how mass enters as a central charge.
Representations and physical interpretation: Irreducible unitary representations of the Bargmann group are labeled by mass m and spin s, and the corresponding quantum states carry both spatial information and spin degrees of freedom. For a free particle of mass m and spin s, the representation space is typically realized as a tensor product of a momentum-space or position-space sector with the spin space, and the action of a Galilean boost induces a mass-dependent phase that reflects the central extension. See spin and quantum mechanics for the broader context.
Relation to projective representations: The Galilei group has projective representations in quantum mechanics, and these projective representations are equivalently realized as true representations of the Bargmann group. This perspective clarifies why some quantum-mechanical transformations pick up phases when viewed strictly in terms of the Galilei group, and how the central extension resolves those ambiguities. See projective representations and representation theory for related material.
Representations and physical meaning
Mass and spin as fingerprints of symmetry: In the Bargmann framework, the mass m and the spin s are intrinsic labels of the representation. They determine how states transform under boosts and rotations and how the wavefunction acquires phases under Galilean motion. This makes mass a physically meaningful quantum number within Galilean invariance, not merely a parameter in the Hamiltonian.
Realizations in quantum mechanics: The canonical realization of the Bargmann group leads to the invariance of the Schrödinger equation under Galilean transformations, provided the wavefunction is equipped with the appropriate mass-dependent phase. This makes the Schrödinger dynamics compatible with the full nonrelativistic symmetry structure encoded by the Bargmann group. See Schrödinger equation and Galilei group for related concepts.
Practical consequences: The Bargmann picture helps organize how nonrelativistic particles with internal structure (e.g., spin, internal energy levels) transform under symmetry operations and how interference and coherence behave under boosts. It also informs the treatment of systems where mass appears effectively (through internal energy) and where symmetry considerations constrain possible states and transitions. See internal energy and spin for further context.
Mass, interference, and debates
The Bargmann mass superselection idea: A classic consequence of the Galilei-by-central-extension viewpoint is that, within a strict Galilean-invariant quantum description, superpositions of states with different mass are not compatible with a single irreducible representation. This leads to a statement sometimes summarized as a mass superselection rule: physical states effectively have definite mass with respect to Galilean symmetry. See mass superselection rule and Bargmann's theorem for core formulations.
Modern perspectives and caveats: In more general or less idealized settings, the status of a strict mass superselection rule is debated. When external fields, gravitational potentials, or relativistic corrections are important, the exact Galilean invariance is not strictly exact, and the neat separation into mass sectors can blur. Some approaches treat mass as a parameter that can be varied across a system or as an emergent quantity tied to internal energy, especially in effective theories or in contexts where Newton-Cartan or other nonrelativistic geometries are employed. See discussions surrounding group cohomology in physics, nonrelativistic quantum mechanics, and Newton-Cartan geometry for broader background.
Experimental and conceptual considerations: Experiments exploring interference with internal energy states or with composite systems have highlighted the nuanced role of mass-like quantities in interference phenomena. These lines of inquiry illuminate how symmetry and dynamics intersect, but they do not universally overturn the Bargmann-centered view; rather, they clarify the domain of its applicability and the boundaries where relativistic or gravitational effects become non-negligible. See quantum interference, internal energy, and relativistic limit for connected ideas.
Historical context
Origin and naming: The concept originates with Egon Bargmann, who clarified how continuous symmetry groups act on quantum states and showed how the central extension by mass yields the proper framework for nonrelativistic quantum mechanics. See Egon Bargmann and the development of representation theory in quantum physics.
Relation to broader symmetry groups: The Bargmann group sits alongside other fundamental symmetry groups in physics, notably the Poincaré group for relativistic spacetime and the various central extensions that appear in condensed matter and high-energy contexts. The idea of central charges and projective representations recurs across physics as a way to reconcile symmetry with quantum mechanics. See Poincaré group for comparison.
Influence on subsequent work: The Bargmann construction has informed approaches to spin, quantum kinematics, and the role of symmetry in nonrelativistic quantum theories, as well as discussions about how mass and internal structure are encoded in the geometry of spacetime symmetries. See spin and quantum mechanics for related threads.