## LATEX

### quantum perspectivism and quantum logic

The observer plays a fundamental role in quantum mechanics.  The first-person observers of the Copenhagen interpretations (which dominate the world) provide the local view of reality while the third-person observers of the many-world formalism provide the global view of reality.

Therefore, quantum mechanics is strongly perspectival in character which was formalized using the language of quantum logic in \cite{edwards}.  However, perspectivism in philosophy is a view due originally to Nietzsche  in which it is maintained that all reality is actually perspectival, i.e. there is no an objective reality out there independent from the knowing subject and free from interpretations and perspectives \cite{N0} (see also Leibniz and his theory of monads). Perspectives for Nietzsche provide an "optics" of knowledge and they constitute the fundamental condition of the conscious observer in his search for value and meaning in existence and life. Somewhat more precisely, perspectivism can be interpreted as a middle position between metaphysical realism and relativism akin of Putnam's internal realism \cite{AL}.

For our purposes, every physical theory is characterized by a certain logic ${\cal L}$. For example, the logic of classical mechanics is a Boolean algebra ${\cal L}$ which is an orthocomplemented distributive lattice based on the power set of the phase space ${\Sigma}$. In other words, Classical events (also called experimental propostions) are subsets of the phase space, pure states semantically decide the truth value of experimental propositions, and the corresponding Boolean algebra underlies Kolmogorovian probability theory.

On the other hand,  it was shown by Birkhoff and von Neumann in their seminal paper \cite{bv} that the logic ${\cal L}$ underlying quantum mechanics is given  by the Hilbert lattice of projection operators on the Hilbert space ${\bf H}$ which is a non-Boolean, non-distributive and orthocomplemented lattice. Therefore the quantum events (or experimental propositions) are given in this case by closed linear subspaces of the Hilbert space, i.e. by projection  operators while pure states decides the truth value only probabilistically, and the corresponding Hilbert lattice or logic of projectors underlies the standard Born’s rule.

The differences between the Boolean algebra of the phase space and the Hilbert lattice of projectors stems mostly from the logical disjunction operation ${\bf OR}$ which in the classical case is given by the union of subsets of the phase space whereas in the quantum case it is given by the direct sum of closed linear subspaces of the Hilbert space.

In the classical case there is therefore a single perspective (corresponding to the Boolean structure of the classical logic) relative to which we can observe every possible measurable property of the system but in the quantum case there is no single priveleged perspective but instead there is an intricate web of non-trivially interlocked classical perspectives each of which corresponds to a maximal Boolean subalgebra (also called a block) of the Hilbert lattice. A block corresponds naturally to a maximal number of compatible and therefore comeasurable observables which  can only be defined locally but not globally. In other words, every block is only locally measurable, i.e. it can not be defined globally independently of the simultaneous measurement of the other blocks.

The Hilbert lattice in two dimensions can be viewed as a non-trivial disjoint union (pasting) of blocks. However, by Gleason's theorem \cite{gleason} and its generalization the Kochen-Specker theorem \cite{KS67} the Hilbert lattice does not admit in general a Boolean reduction, i.e. there is no a homomorphism from the Hilbert lattice into a Boolean algebra or to a disjoint collection (pasting)  of Boolean algebras which is perhaps the best characterization of the measurement problem. It is worth pointing out that the insistence on the Boolean character is precisely the assumption of reality envisaged by the EPR argument \cite{EPR35}.

Thus, in quantum mechanics the Hilbert lattice admits a decomposition into maximal Boolean subalgebras or blocks which define an intricate web of non-trivially interlocked classical perspectives. These perspectives are complementarity to each other and their totality defines an omni-perspective (which is seeing everything  from everywhere as defined by Nietzsche originally) which is the maximal possible perspective allowed by quantum mechanics. These perspectives are associated naturally with the first-person observers of the Copenhagen interpretation.

The perspective of the third-person observer of the many-worlds formalism which sees coherent linear superpositions of classical states (such as a dead and alive cats) is  a non-perspective  (which is seeing everything from nowhere, i.e. a God’s eye view of a sort) which is logically impossible according to  Nietzsche. However, this impossibility is simply due as we have seen to the fact that the world is full of first-order observers who are not directly aware of coeherent  linear superpositions, i.e. consciousness is through and through classical.

Hence the complementarity between the many-worlds formalism and the Copenhagen interpetation proposed in this essay is nothing else but an extension of Bohr's complementarity principle which holds among first-person observers providing  classical perspectives on the world \cite{Bohr}.

### References

%\cite{Schrodinger}
\bibitem{Schrodinger}
J.D.~Trimmer,
"The Present Situation in Quantum Mechanics: A Translation of Schrodinger's 'Cat Paradox' Paper,''
Proceedings of the American Philosophical Society 124:5 (Oct. 10, 1980), 323-338.

%\cite{Wigner1}
\bibitem{Wigner1}
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%\cite{Bell:1964kc}
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%\cite{Bell:1964fg}
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J.~S.~Bell,
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%\cite{VN55}
\bibitem{VN55}
J.~von Neumann,
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First edition (1955), Princeton University Press, translated from the 1932 German original
by R. T. Beyer.

%\cite{Bohr}
\bibitem{Bohr}
N.~Bohr,
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%\cite{bv}
\bibitem{bv}
G.~Birkhoff, J.~ von Neumann,
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%\cite{mw2}
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%\cite{EPR35}
\bibitem{EPR35}
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%\cite{gleason}
\bibitem{gleason}
A.M.~Gleason,
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%\cite{KS67}
\bibitem{KS67}
S.~Kochen, E.P.~Specker,
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%\cite{zeh}
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H.~D.~Zeh,
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%\cite{N0}
\bibitem{N0}
Nietzsche, Friedrich Wilhelm, Walter Arnold Kaufmann, and R.J. Hollingdale. The Will to Power. New York: Vintage Books, 1964. §481, p. 267.

Friedrich Wilhelm Nietzsche and Walter Arnold Kaufmann. The Gay Science; With a Prelude in Rhymes and an Appendix of Songs. New York: Random House, 1974. Section 374, p. 336.)

Friedrich Nietzsche, On the Genealogy of Morality: A Polemic. trans. Maudemarie Clarke and Alan J. Swenswen. Indianapolis: Hackett Publishing, 1998.)

%\cite{edwards}
\bibitem{edwards}
D.~A.~Edwards,
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Synthese (1979) 42: 1. https://doi.org/10.1007/BF00413704.

%\cite{AL}
\bibitem{AL}
Anderson, R. Lanier,
“Truth and Objectivity in Perspectivism,” Synthese (1998) 115:1-32.