Extended framework for the hybrid Monte Carlo in lattice gauge theory

We develop an extended framework for the hybrid Monte Carlo (HMC) algorithm in lattice gauge theory by embedding the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>S</a:mi><a:mi>U</a:mi><a:mo stretchy="false">(</a:mo><a:mi>N</a:mi><a:mo stretchy="false">)</a:mo></a:math> group into the space of general complex matrices, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>M</e:mi><e:mi>N</e:mi></e:msub><e:mo stretchy="false">(</e:mo><e:mi mathvariant="double-struck">C</e:mi><e:mo stretchy="false">)</e:mo></e:math>. Auxiliary directions will be completely factorized in the path integral, and the embedding does not alter the expectation values of the original theory. We perform the molecular dynamics updates by using the matrix elements of <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>W</j:mi><j:mo>∈</j:mo><j:msub><j:mi>M</j:mi><j:mi>N</j:mi></j:msub><j:mo stretchy="false">(</j:mo><j:mi mathvariant="double-struck">C</j:mi><j:mo stretchy="false">)</j:mo></j:math> as the dynamical variables without group theoretic constraints. The framework enables us to introduce nonseparable Hamiltonians for the HMC in lattice gauge theory exactly, whose immediate application includes the Riemannian manifold HMC.