Strong <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> phase and parity in the Hamiltonian formalism

We show using the Hamiltonian formalism that if parity is a good symmetry of QCD, then the strong <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>C</a:mi> <a:mi>P</a:mi> </a:math> phase <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mover accent="true"> <c:mi>θ</c:mi> <c:mo stretchy="false">¯</c:mo> </c:mover> </c:math> must be 0 or <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>π</g:mi> </g:math> . We find that for <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>P</i:mi> </i:math> to be a physical symmetry, it must leave the Hilbert space <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mi mathvariant="script">H</k:mi> <k:mi>θ</k:mi> </k:msub> </k:math> associated with the <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi>θ</n:mi> </n:math> -vacuum invariant ( <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mrow> <p:mi>P</p:mi> <p:mo>:</p:mo> <p:mtext> </p:mtext> <p:mtext> </p:mtext> <p:msub> <p:mrow> <p:mi mathvariant="script">H</p:mi> </p:mrow> <p:mrow> <p:mi>θ</p:mi> </p:mrow> </p:msub> <p:mo stretchy="false">→</p:mo> <p:msub> <p:mrow> <p:mi mathvariant="script">H</p:mi> </p:mrow> <p:mrow> <p:mi>θ</p:mi> </p:mrow> </p:msub> </p:mrow> </p:math> ), which is possible only for <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>θ</u:mi> <u:mo>=</u:mo> <u:mn>0</u:mn> </u:math> or <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mi>π</w:mi> </w:math> . We also show that forming linear combinations of states from different <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi>θ</y:mi> </y:math> sectors produces only classical statistical mixtures, consistent with superselection rules, confirming that <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msub> <ab:mi mathvariant="script">H</ab:mi> <ab:mi>θ</ab:mi> </ab:msub> </ab:math> is the most general Hilbert space for the quantum theory. Furthermore, we demonstrate that requiring <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"> <db:mo stretchy="false">[</db:mo> <db:mi>P</db:mi> <db:mo>,</db:mo> <db:mi mathvariant="normal">Ω</db:mi> <db:mo stretchy="false">]</db:mo> <db:mo>=</db:mo> <db:mn>0</db:mn> </db:math> —where <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:mi mathvariant="normal">Ω</ib:mi> </ib:math> is the generator of large gauge transformations—independently enforces <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"> <lb:mover accent="true"> <lb:mi>θ</lb:mi> <lb:mo stretchy="false">¯</lb:mo> </lb:mover> <lb:mo>=</lb:mo> <lb:mn>0</lb:mn> </lb:math> (mod <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"> <pb:mi>π</pb:mi> </pb:math> ), and that for complex quark mass matrix <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"> <rb:mi>M</rb:mi> </rb:math> , if a generalized parity operator <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"> <tb:mi mathvariant="script">P</tb:mi> </tb:math> is a symmetry, then the value of <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"> <wb:mi>θ</wb:mi> </wb:math> gets determined so that it exactly cancels <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"> <yb:mi>A</yb:mi> <yb:mi>r</yb:mi> <yb:mi>g</yb:mi> <yb:mi>D</yb:mi> <yb:mi>e</yb:mi> <yb:mi>t</yb:mi> <yb:mi>M</yb:mi> </yb:math> , again giving <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"> <ac:mrow> <ac:mover accent="true"> <ac:mrow> <ac:mi>θ</ac:mi> </ac:mrow> <ac:mrow> <ac:mo stretchy="false">¯</ac:mo> </ac:mrow> </ac:mover> <ac:mo>=</ac:mo> <ac:mn>0</ac:mn> <ac:mtext> </ac:mtext> <ac:mo stretchy="false">(</ac:mo> <ac:mi>mod</ac:mi> <ac:mtext> </ac:mtext> <ac:mtext> </ac:mtext> <ac:mi>π</ac:mi> <ac:mo stretchy="false">)</ac:mo> </ac:mrow> </ac:math> . These results establish the equivalence of the Hamiltonian and Lagrangian approaches to the strong <gc:math xmlns:gc="http://www.w3.org/1998/Math/MathML" display="inline"> <gc:mi>C</gc:mi> <gc:mi>P</gc:mi> </gc:math> problem.