Reciprocal relativity of noninertial frames: quantum mechanics

Abstract

Noninertial transformations on time-position-momentum-energy space {t,q,p,e} with invariant Born-Green metric ds^2=-dt^2+dq^2/c^2+(1/b^2)(dp^2-de^2/c^2) and the symplectic metric -de/\dt+dp/\dq are studied. This U(1,3) group of transformations contains the Lorentz group as the inertial special case. In the limit of small forces and velocities, it reduces to the expected Hamilton transformations leaving invariant the symplectic metric and the nonrelativistic line element ds^2=dt^2. The U(1,3) transformations bound relative velocities by c and relative forces by b. Spacetime is no longer an invariant subspace but is relative to noninertial observer frames. Born was lead to the metric by a concept of reciprocity between position and momentum degrees of freedom and for this reason we call this reciprocal relativity. For large b, such effects will almost certainly only manifest in a quantum regime. Wigner showed that special relativistic quantum mechanics follows from the projective representations of the inhomogeneous Lorentz group. Projective representations of a Lie group are equivalent to the unitary reprentations of its central extension. The same method of projective representations of the inhomogeneous U(1,3) group is used to define the quantum theory in the noninertial case. The central extension of the inhomogeneous U(1,3) group is the cover of the quaplectic group Q(1,3)=U(1,3)*s H(4). H(4) is the Weyl-Heisenberg group. A set of second order wave equations results from the representations of the Casimir operators.

References in corpus (7)

• Strong Dynamics and Electroweak Symmetry Breaking
• Generalized Lorentz invariance with an invariant energy scale
• Lorentz violation at high energy: concepts, phenomena and astrophysical constraints
• Born-Infeld Kinematics
• Reciprocal relativity of noninertial frames and the quaplectic group
• Born reciprocity and the granularity of space-time
• Poincare and Heisenberg quantum dynamical symmetry: Casimir invariant field equations of the quaplectic group

Cited by in corpus (13)

• World-line Quantisation of a Reciprocally Invariant System
• Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators
• Resolving Curvature Singularities in Holomorphic Gravity
• Hermitian Gravity and Cosmology
• Relativity group for noninertial frames in Hamilton's mechanics
• Hamilton relativity group for noninertial states in quantum mechanics
• Beyond the Relativistic Point Particle: A Reciprocally Invariant System and its Generalisation
• Maximal quantum mechanical symmetry: Projective representations of the inhomogenous symplectic group
• Noninertial Symmetry Group of Hamilton's Mechanics
• Projective Representations of the Inhomogeneous Hamilton Group: Noninertial Symmetry in Quantum Mechanics
• Constraint quantisation of a worldline system invariant under reciprocal relativity. II
• Noninertial symmetry group with invariant Minkowski line element consistent with Heisenberg quantum commutation relations
• Relativity implications of the quantum phase