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Complexified Minkowski space is defined as {{math|1=''M<sub>c</sub>'' = ''M'' ⊕ ''iM'' }}.<ref>Y. Friedman, A Physically Meaningful Relativistic Description of the Spin State of an Electron, Symmetry 2021, 13(10), 1853; https://doi.org/10.3390/sym13101853 {{Webarchive|url=https://web.archive.org/web/20230813114023/https://www.mdpi.com/2073-8994/13/10/1853 |date=2023-08-13 }}</ref> Its real part is the Minkowski space of [[four-vectors]], such as the [[four-velocity]] and the [[four-momentum]], which are independent of the choice of [[orientation (geometry)|orientation]] of the space. The imaginary part, on the other hand, may consist of four pseudovectors, such as [[angular velocity]] and [[magnetic moment]], which change their direction with a change of orientation. A [[pseudoscalar]] {{math|''i''}} is introduced, which also changes sign with a change of orientation. Thus, elements of {{math|''M<sub>c</sub>''}} are independent of the choice of the orientation.
The [[inner product]]-like structure on {{math|''M<sub>c</sub>''}} is defined as {{math|1=''u'' ⋅ ''v'' = ''η''(''u'',''v'')}} for any {{math|''u'',''v'' ∈ ''M<sub>c</sub>''}}. A relativistic pure [[Spin (physics)|spin]] of an [[electron]] or any half spin particle is described by {{math|''ρ'' ∈ '' M<sub>c</sub>''}} as {{math|1=''ρ'' = ''u+is''}}, where {{math|''u''}} is the four-velocity of the particle, satisfying {{math|1=''u''<sup>2</sup> = 1}} and {{mvar|s}} is the 4D spin vector,<ref>Jackson, J.D., Classical Electrodynamics, 3rd ed.; John Wiley \& Sons: Hoboken, NJ,
===Generalized Minkowski space===
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