Strong CP 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.