| Summary: | It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>p</mi> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> group. According to Paul A. M. Dirac, from the uncertainty commutation relations for two variables, it possible to construct the de Sitter group <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, namely the Lorentz group applicable to three space-like variables and two time-like variables. By contracting one of the time-like variables in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, it is possible to construct the inhomogeneous Lorentz group <inline-formula> <math display="inline"> <semantics> <mrow> <mi>I</mi> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> which serves as the fundamental symmetry group for quantum mechanics and quantum field theory in the Lorentz-covariant world. This <inline-formula> <math display="inline"> <semantics> <mrow> <mi>I</mi> <mi>S</mi> <mi>O</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> group is commonly known as the Poincaré group.
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