An automorphism preserving the physically specified background structure of spacetime. With only a Minkowski metric and fixed orientations, the appropriate external symmetries are the corresponding Poincaré group elements. An added preferred vector field reduces this group. This is distinct from merely changing coordinates.
A transformation taking solutions of a specified set of dynamical equations into solutions, with the theory's fixed background data held fixed. For a Lagrangian, invariance of the action up to a boundary term is a useful sufficient criterion. Symmetry of the equations does not imply that every individual solution has that symmetry: initial data and boundary conditions can select a direction or state of motion.
Invariance under a spatial reflection. A one-dimensional Quantum Hamiltonian invariant under admits energy eigenfunctions of definite even or odd midpoint parity. Even and odd functions are orthogonal on a reflection-symmetric interval.
A plane through which reflection leaves an object unchanged. In a molecular skeleton, sites exchanged by a symmetry preserving all substituent identities can be chemically equivalent in an achiral NMR spectroscopy experiment.
A point such that replacing every position relative to it by leaves the structure unchanged. Molecular orbitals can then be classified as or according to their parity.
Invariance of an action, equation or configuration under a translation. For a continuous spatial translation symmetry, Noether's theorem supplies the conserved momentum; spacetime translation gives the canonical stress-energy tensor.
A scalar field unchanged by every translation parallel to a specified plane depends only on the coordinate normal to that plane. For electrostatics between ideal infinite parallel plates, this reduces Laplace's equation to . Finite plate edges break the exact symmetry and produce fringing field.
A discrete symmetry has transformations separated from the identity rather than connected to it by continuously varying symmetry parameters.
A continuous symmetry belongs to a group whose parameters can vary continuously. Spontaneous breaking of such a symmetry can produce Goldstone bosons.
A helical symmetry combines an axial rotation and a proportional axial translation. A helix with is unchanged as a set by this transformation. If a mechanical Lagrangian is invariant, the Noether conserved quantity for a mechanical point symmetry is , where the two momenta include all moving components. Pure axial rotation and pure axial translation need not separately be symmetries. The helical symmetry of a potential is a related special case in which the constraint is replaced by an invariant potential.[ref]
If zero conserved helical momentum reduces to , a closed transverse projection has axial displacement . Here and is the signed area, with winding multiplicity for self-intersections. This is a path-dependent displacement even when the transverse coordinates return to their initial values. It follows from the momentum constraint and Green's theorem; it is not a prediction that every chosen closed curve is dynamically realizable.
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In physics, symmetry refers to a property or characteristic of a system that remains invariant under certain transformations. This can involve spatial transformations, such as translations, rotations, and reflections, as well as other types such as time reversibility or particle interchange. Symmetry plays a critical role in understanding physical laws and phenomena, often leading to conservation laws and simplifying complex problems.