Is the ortho-complement of a proposition, in quantum logic/probability or hilbert space, the logical comple-ment of a proposition
such as 'spin up at direction x and 'spin down in direction y'
(mutually exhaustive, on the) same prepared spin system at the as same angle, ie on the same basis) and thus one can add them the probabilities to one, as in the usual probabilistic sense to .when the hilbert space inner product is zero (or rather one or zero, the kronecker delta).
Or is this a geometric notion that that allows one to add probabilities of events together that lie on distinct bases (bases/vectors on distinct angles of the same prepared spin system). Which is stronger (closer to a restricted form of cauchy additivity)
the on the very same vector in an orth-onormal basis; the very same basis.
, in the sense that they are disjoint A\cap B emptyset , and mutually exclusive. Or is its geometric, and relates to some kind of relation between events on distinct bases of a spin system.
When one speaks of an orthonormal bases or vectors that are mutually orthogonal , and are unit vectors that are mutually orthogonal, is this where ||u||=sqrt(x^2+y^2)=1 and ||v|=sqrt(x_i^2+y_i2)=1
u.v=x_2.v_2+ y_2.y_2=0;
does this denote, complementarity/non commutting, geometric orthonality, or logical orthogonality, that is events are disjoints in the logical sense (lie in the same basis or a commutting bases, and can be added) or are on distinct bases..
In other words
Is is there of way of using the inner product to determine whether the events are lie on non commutting and cannot be explicitely assume to add. I presume that the hilbert space inner product denoted logical orthogonality
x is amplitude moduli of spin up in direction of the vector u, with probability ||x||^2, and
y is the same but for spin down down in the same direction vector u with probability ||y||2
and x_i, y_i are the corresponding amplitude,moduli of spin up and spin down for the direction of the vector v,and probabilities,
||x_i||^2
||y_i||^2.
For example when Gleason or (someone before him deduced the orthogonal triple property x+y+m=1\toF(x)+F(y)+F(m)=W=
and odd-ness of the frame function,)F(-x))+F(x)=0,
or in the attached document,the stronger any given two property, x+y=1\to F(x)+F(y)=1, that these were deduced properties (which followed not just from their horizontal analogues normalization, and complement additivity alone but as a result of the equalities between distinct events , that may not be logically disjoint at all, due to the pecularities of quantum logic and non-contextuality) that would allow one to effectively resolve the frame function values for events
other than the generally nly (normally) esolvable events on probability spaces 1, 2 spaces which are closed under union and the unit and the frame funcrtion is ranked by the chances/mod squared, and is also a probability function, by allowing him to then insert the resolved F(1/3)=1/3 event on space 1, F(1/2)=1/2 on space 2 into space 3
presume that he did not merely say that we are presuming Given a normalization of the frame function, presumed to some such probability function, that on any specific bases,
(1)where any given amplitudes moduli squared sum to one, the frame function values do,three which lie on the very same basis sum to 1, as they are mutually exhaustive and exclusive; when the amplitudes mod squared do). A\cap,B=emptyset,\capC ,=emptyset A, B C, are mutually exclusive and exhaustive \Omega=A\vee B\vee C0, so P(A)+PR(B)+PR(C)=1
(2) and when he applied that the function is odd F(-x)+F(x)=0, or any given two, amplutdues mod squared which lies on the same basis, and are mutually exhaustive and exclusive sum to one , will have frame function values which sum to one, as instance of complement additivity PR(A)+PR(A_C)=1 A\cap A_c =\emptyset, A\vee A_c is mutually exhaustic, where A_c is teh set theoric complement . ie if Omega={A,B, C}, A_C=B \vee C
and some other property for instance complement additivty, any event and its mutually exclusive, exhaustive complement which sum to one, must have frame function values which do) but that he from this, and given the rank and symmetry of invoked by quantum non-contextuality and quantum logic, he meant this as a form of global addititity.
with probability
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