# Alexian reference frame

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The Alexian reference frame is the transformation of a real physical system whereby the dynamics become pure imaginary via the transformations:

${\displaystyle x^{\mu }\rightarrow ig^{\mu \nu }\sigma _{\nu \alpha }\,x^{\alpha },}$

where ${\displaystyle \sigma }$ are the Pauli matrices.

Constraints on physical systems that are invariant under Alexian shifts of reference frame are said to be Alexian conditions.

It has been claimed that real systems that are Alexian are dual to the imaginary analogue, but this then leads to the Alexian paradox, whereby a real system and the imaginary system are not covariant under the Alexian transformation with respect to the unitary time-evolution of the observables of the system.

Most scientists consider the Alexian frame of reference to be inconsistent.