| Summary: | After a brief introduction of Born’s reciprocal relativity theory is presented, we review the construction of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mi>e</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>e</mi><mi>d</mi></mrow></semantics></math></inline-formula> quaplectic group that is given by the semi-direct product of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula> with the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mi>e</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>e</mi><mi>d</mi></mrow></semantics></math></inline-formula> (noncommutative) Weyl–Heisenberg group corresponding to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mi>o</mi><mi>n</mi><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi></mrow></semantics></math></inline-formula> fiber coordinates and momenta <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><msub><mi>X</mi><mi>a</mi></msub><mo>,</mo><msub><mi>X</mi><mi>b</mi></msub><mo>]</mo><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>; <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><msub><mi>P</mi><mi>a</mi></msub><mo>,</mo><msub><mi>P</mi><mi>b</mi></msub><mo>]</mo><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>. This construction leads to more general algebras given by a two-parameter family of deformations of the quaplectic algebra, and to further algebraic extensions involving antisymmetric tensor coordinates and momenta of higher ranks <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><msub><mi>X</mi><mrow><msub><mi>a</mi><mn>1</mn></msub><msub><mi>a</mi><mn>2</mn></msub><mo>⋯</mo><msub><mi>a</mi><mi>n</mi></msub></mrow></msub><mo>,</mo><msub><mi>X</mi><mrow><msub><mi>b</mi><mn>1</mn></msub><msub><mi>b</mi><mn>2</mn></msub><mo>⋯</mo><msub><mi>b</mi><mi>n</mi></msub></mrow></msub><mo>]</mo><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>; <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><msub><mi>P</mi><mrow><msub><mi>a</mi><mn>1</mn></msub><msub><mi>a</mi><mn>2</mn></msub><mo>⋯</mo><msub><mi>a</mi><mi>n</mi></msub></mrow></msub><mo>,</mo><msub><mi>P</mi><mrow><msub><mi>b</mi><mn>1</mn></msub><msub><mi>b</mi><mn>2</mn></msub><mo>⋯</mo><msub><mi>b</mi><mi>n</mi></msub></mrow></msub><mo>]</mo><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>. We continue by examining algebraic extensions of the Yang algebra in extended noncommutative phase spaces and compare them with the above extensions of the deformed quaplectic algebra. A solution is found for the exact analytical mapping of the noncommuting <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi><mi>μ</mi></msup><mo>,</mo><msup><mi>p</mi><mi>μ</mi></msup></mrow></semantics></math></inline-formula> operator variables (associated to an 8D curved phase space) to the canonical <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Y</mi><mi>A</mi></msup><mo>,</mo><msup><mi mathvariant="sans-serif">Π</mi><mi>A</mi></msup></mrow></semantics></math></inline-formula> operator variables of a flat 12D phase space. We explore the geometrical implications of this mapping which provides, in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mi>l</mi><mi>a</mi><mi>s</mi><mi>s</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>l</mi></mrow></semantics></math></inline-formula> limit, the embedding functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>Y</mi><mi>A</mi></msup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow><mo>,</mo><msup><mi mathvariant="sans-serif">Π</mi><mi>A</mi></msup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of an 8D curved phase space into a flat 12D phase space background. The latter embedding functions determine the functional forms of the base spacetime metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>g</mi><mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the fiber metric of the vertical space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>h</mi><mrow><mi>a</mi><mi>b</mi></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and the nonlinear connection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mrow><mi>a</mi><mi>μ</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> associated with the 8D cotangent space of the 4D spacetime. Consequently, we find a direct link between noncommutative curved phase spaces in lower dimensions and commutative flat phase spaces in higher dimensions.
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