Eight types of symmetrically distinct vectorlike physical quantities.

Area of Science:

• Theoretical Physics
• Condensed Matter Physics
• Symmetry in Physics

Background:

• Physical quantities are often described using vector concepts.
• Spatiotemporal symmetry plays a crucial role in understanding physical laws.
• The nonrelativistic space-time rotation group O'(3) governs these symmetries.

Purpose of the Study:

• To investigate the full spectrum of spatiotemporal symmetries for vectorlike physical quantities.
• To identify and classify distinct classes of stationary physical quantities beyond the canonical polar vector.
• To demonstrate the existence of an octet of symmetrically distinct directional quantities.

Main Methods:

• Analysis of symmetry operations under the nonrelativistic space-time rotation group O'(3).
• Classification of vectorlike quantities based on their transformation properties.
• Examination of examples including electric dipole, magnetic toroidal, magnetization, and electric toroidal moments.

Main Results:

• Identified eight symmetrically distinct classes of stationary physical quantities.
• These include two polar vectors, two axial vectors, two chiral bidirectors, and two achiral bidirectors.
• Examples provided demonstrate these classifications, such as electric dipole and magnetic toroidal moments.

Conclusions:

• The canonical polar vector is only one of eight types of directional quantities.
• Recognizing these distinct classes provides a more complete understanding of spatiotemporal symmetry.
• This framework is applicable to various physical phenomena, including liquid crystals and antiferromagnetism.