We discover a class of spacetime symmetries unique to time-periodic systems, which we term ``mixing symmetry'' due to its combination of space and time coordinates in the symmetry transformation. We systematically enumerate the symmetry groups and classify the corresponding Floquet-Bloch band theories by utilizing the winding number of quasienergy. Moreover, we provide a comprehensive scheme for the experimental realization of these symmetries. The particle propagator exhibits an intriguing pattern that remains invariant even under transformations mixing space and time coordinates. We anticipate that this distinct feature can be observed in current cold atom experiments.
Advances in the study of relativistic quantum phase space have established the set of Linear Canonical Transformations (LCTs) as a candidate for the fundamental symmetry group associated with relativistic quantum physics. In this framework, for a spacetime of signature
(N+,N−), the symmetry of the relativistic quantum phase space is described by the LCT group, isomorphic to the symplectic Lie group
Sp(2N+,2N−), which preserves the canonical commutation relations (CCRs) and treats spacetime coordinates and momenta operators on an equal footing. In this work, we investigate the contraction structure of the Lie algebra associated with the LCT group for signature
(1,4), clarifying how familiar spacetime symmetry groups emerge from this more fundamental quantum phase space symmetry. Using the Inönü-Wigner group contraction formalism, we examine each limit case corresponding to the possible combinations of asymptotic values of two fundamental length scale parameters associated with the theory, namely a minimum length
ℓ and a maximum length
L, which may be identified respectively with the Planck length and the de Sitter radius. We explicitly analyze how contractions of the LCT Lie algebra lead to the physically relevant de Sitter algebra
so(1,4) and, in the flat-curvature limit, to the Poincaré algebra
iso(1,3) of four-dimensional spacetime. This provides an explicit mechanism through which relativistic spacetime symmetry can emerge from a deeper symplectic structure of quantum phase space.
We investigate a family of invertible phases of matter with higher-dimensional exotic excitations in even spacetime dimensions, which includes and generalizes the Kitaev’s chain in 1+1d. The excitation has \mathbb{Z}_2 ℤ 2 higher-form symmetry that mixes with the spacetime Lorentz symmetry to form a higher group spacetime symmetry. We focus on the invertible exotic loop topological phase in 3+1d. This invertible phase is protected by the \mathbb{Z}_2 ℤ 2 one-form symmetry and the time-reversal symmetry, and has surface thermal Hall conductance not realized in conventional time-reversal symmetric ordinary bosonic systems without local fermion particles and the exotic loops. We describe a UV realization of the invertible exotic loop topological order using the SO(3)_- S O ( 3 ) − gauge theory with unit discrete theta parameter, which enjoys the same spacetime two-group symmetry. We discuss several applications including the analogue of ``fermionization’’ for ordinary bosonic theories with \mathbb{Z}_2 ℤ 2 non-anomalous internal higher-form symmetry and time-reversal symmetry.