What is general, is meant by the orthogonality relation x⊥y
in the functional equation: orthogonal additivity below (1)
(1)∀(x,y)∈(dom(F)∩[x⊥y]):F(x+y)=F(x)+F(y)
Particularly as used in the following two references listed below (by De Zelah and Ratz, in the context of quantum spin half born rule derivations
What is general, is meant by the orthogonality relation x⊥y
denotes x is orthogonal to y.
.
(1)∀(x,y)∈(dom(F)∩[x⊥y]):F(x+y)=F(x)+F(y)
See Rätz, Jürg, On orthogonally additive mappings, Aequationes Math. 28, 35-49 (1985). ZBL0569.39006.
2.See 'Comment on `Gleason-type theorem including Qubits and pro-jective measurements: the Born rule beyond quantum physics' ", by Michael J. W. Hall ">https://www.researchgate.net/profile/Francisco_De_Zela
where x⊥y
denote: -'logical/set theoretical Disjointedness' of events, on the same basis.That is mutual exclusivity? Or - or geometrical orthogonality events . That is, on perpendicular/possibly non-commuting vectors/bases in spin 12 system? (for example, spin up x direction versus spin up ydirection)? - OR something else?
That is, within QM, in the Hilbert inner product metric, what does it mean for the frame function two events to be explicitly allowed to add as per x⊥y
, in functional equation, Orthogonal additivity.
.
Does z⊥m
denote events mean events whose amplitude modul-i squared (P(A),PR(B))=(z,m)respectively and which lie on the same basis/vector in a spin system, for example
:
A spin up on basis in x direction,P(A)=||amplitude(spin up_x)||2=z
and¬A spin down on basis in x direction;P(¬A)=||amplitude(spin down_x)||2=m
2.Versus Non commuting bases:
A spin up in x direction
andA spin down in y direction
Corresponding to the logical and geometric notions roughly, respectively
How does one distinguish this, from events that are non-commuting from events& orthogonal events in the geometric sense from events, that are orthogonal in the logical sense and disjoint events on the same basis?
Is there a distinct operator, inner product in quantum mechanics (which equals zero) which determines when the frame function probabilities of events can explicitly be assumed to add?
That is disjoint, in the sense of **Kolmogorov that is disjoint or mutually exclusive, or its analog in quantum mechanics as in (1)?
(1)F(X∪Y)=F(X)+F(Y)where in (1);X∩Y=∅,∧X,Y∈Ω,,X,Y are mutually exclusive and lie on the same basis
(1a)P(A∪AC)=1,A∪AC=Ω=⊤∧A∪AC=∅
(1.b)∀(Ai)∈Ω;where,Ai∈F,the singletons, atoms, of which are in the algebra of events F;P(∪n=|Ω|i=1Ai)=[∑i=1n=|Ω|P(Ai)]=1
where, in 1(b)[∪n=|Ω|i=1Ai]=⊤∧∀(j);j≠i⟺Ai∩Aj=∅
This being opposed to: when the events on distinct vectors, that are geometrically orthogonal or non commuting/complementary bases/distinct bases, which are not explicitly specified to add, they may happen to, as was derived for n≥3
, where this is really a form of the much stronger global ,Cauchy additivity equation (2) which relate to unique-ness , when one has (1), in addition ? (2)(2)∀(x,y)∈dom(unitsphere);F(x+y)=F(x)+F(y)
where x+y,,x,y, can be translated as ||x+y||2,||x|2,||y||2 Of these two notions, to which does orthogonal additivity relate to, is probabilism (or quantum probabilism) with some further richness properties that make it closer to (2)in a restricted form
That is the sense of being on bases that are at right angles or being at 90
degrees from one another, not simultaneously measurable such as (A),(B):
as
(A)spin-up at angle y with||spin−up−y||2=P(spin up, measured@ angle,y)12
(B)spin-up at angle x with||spin−upxy||2=P(spin up, measured@ angle,x)=12
Both seem to use the Hilbert space inner product? I presume one is for bases, and another is for events.
Does this relate to when the events commute, are on vectors at right angles from each other, or rather do commute and lie on the same basis. If the former is just standard probabilism.
Article On the conditional Cauchy functional equation of orthogonal additivity
An arbitrary wave functionin Hilbert space may be written as
psi) = sum over i of i)(i psi) = sum over i of P(i) psi)
where P(i)=i)(i is a projector to the state i)
Each P(i) squared is equal to itself, The product of any two is zero for different índices.
As a ressult of a measurment, one of the P(i) has eigenvalue 1, all the rest zero, and it is
determined that the particle is in state i, mutually exclusive to all other meauement posible results.
There is then a kind of additivity, and mutually exclusive posible results
I =sum over all i of P(i)
is the unit operator.
Im using ) and ( for the ket an bra rspectively in Dirac language.
Read Diracs Q. M.
as for orthogonality it is in (i,j) = 1 if i=j and 0 if i not equal to j
ah yes, is it 1, when its one and the same basis ie spin up x versus spin down x direction,for the same prepared state, m and otherwise zero, when the events, spin up, spin down, on distinct bases at angles, x,y (ie at right angles from each other),spin up x versus spin up y ,say on a spin 1/2 system on same prepared state m,. Ie so 1, denote disjoint in the traditional sense, and zero denotes completely complementary, impossibility to measure simultanously, although both are considered orthogonal in the sense that they denote simultaneously impossible events
for isntance is 0 , for orthogonality when the events are on geometrically orthogonal basis vectors,(90 degree difference roughly) maximally non commutting bases, and 1 when they are on the same vector; are these both considered a form of orthogonality
- Badges
- Science topic
- Similar topics
- Philosophy
- Philosophy Of Science