Jacobi identity invariance

The absence of an E(3) field does not affect Lorentz symmetry, because in free space, the field equations of both 0(3) electrodynamics are Lorentz-invariant, so their solutions are also Lorentz-invariant. This conclusion follows from the Jacobi identity (30), which is an identity for all group symmetries. The right-hand side is zero, and so the left-hand side is zero and invariant under the general Lorentz transformation [6], consisting of boosts, rotations, and space-time translations. It follows that the B<3) field in free space Lorentz-invariant, and also that the definition (38) is invariant. The E(3) field is zero and is also invariant thus, B(3) is the same for all observers and E(3) is zero for all observers. 

The E(3) field is zero in frame K, and a Z boost means [from Eq. (403)] that it is zero in frame K. This is consistent with the fact that Ea> is a solution of an invariant equation, the Jacobi identity (30) of 0(3) electrodynamics. Finally, we can consider two further illustrative example boosts of El ]] in the X and Y directions, which both produce the following result ... 

In order to show that the Proca equation from gauge theory is gauge-invariant, it is convenient to consider the Jacobi identity... 

In the 0(3) invariant theory of the Aharonov-Bohm effect, the holonomy consists of parallel transport using 0(3) covariant derivatives and the internal gauge space is a physical space of three dimensions represented in the basis ((1),(2),(3)). Therefore, a rotation in the internal gauge space is a physical rotation, and causes a gauge transformation. The core of the 0(3) invariant explanation of the Aharonov-Bohm effect is that the Jacobi identity of covariant derivatives [46]... 

An attempt to solve the difficulties and inconsistencies arising from an approximated derivation of quantum-classical equations of motion was made some time ago [15] to restore the properties that are expected to hold within a consistent formulation of dynamics and statistical mechanics, and are instead missed by the existing approximate methods. We refer not only to the properties that the Lie brackets, which generate the dynamics, satisfy in a full quantum and full classical formulation, e.g., the bi-linearity and anti-symmetry properties, the Jacobi identity and the Leibniz rule12, but also to statistical mechanical properties, like the time translational invariance of equilibrium correlation functions [see eq.(8)]. 

We note that the Jacobi identity for the bracket A, B =(AX, LBX) is not needed for the manifold. tiA to be invariant and for (50) evaluated on it to become equivalent to (55). The skew symmetry of L suffices to guarantee both these properties. If however we begin with the time evolution generated by (50) and define the manifold M as the manifold on which (51) equals zero then the Jacobi identity is the integrability condition for (see Courant (1989)). 

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