On characterizing the standard quantum logics

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a complete projective logic. Then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a natural representation as the lattice of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle dot comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ⟨ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ⋅ </mml:mo> <mml:mo>,</mml:mo> <mml:mo> ⋅ </mml:mo> </mml:mrow> <mml:mo fence="false" stretchy="false"> ⟩ </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle { \cdot , \cdot } \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -closed subspaces of a left vector space <italic>V</italic> over a division ring <italic>D</italic> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle dot comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ⟨ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ⋅ </mml:mo> <mml:mo>,</mml:mo> <mml:mo> ⋅ </mml:mo> </mml:mrow> <mml:mo fence="false" stretchy="false"> ⟩ </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle {\cdot ,\cdot } \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a definite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta"> <mml:semantics> <mml:mi> θ </mml:mi> <mml:annotation encoding="application/x-tex">\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bilinear symmetric form on <italic>V</italic> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta"> <mml:semantics> <mml:mi> θ </mml:mi> <mml:annotation encoding="application/x-tex">\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> being some involutive antiautomorphism of <italic>D</italic> . Now a well-known theorem of Piron states that if <italic>D</italic> is isomorphic to the real field, the complex field or the sfield of quaternions, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta"> <mml:semantics> <mml:mi> θ </mml:mi> <mml:annotation encoding="application/x-tex">\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is continuous, and if the dimension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is properly restricted, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is just one of the standard Hilbert space logics. Here we also assume <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a complete projective logic. Then if every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta"> <mml:semantics> <mml:mi> θ </mml:mi> <mml:annotation encoding="application/x-tex">\theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -fixed element of <italic>D</italic> is in the center of <italic>D</italic> and can be written as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="plus-or-minus d theta left-parenthesis d right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo> ± </mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi>d</mml:mi> <mml:mi> θ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mm