Linguistic Interpretation of Quantum Mechanics ; Projection Postulate

As the fundamental theory of quantum information science, recently I proposed the linguistic interpretation of quantum mechanics, which was characterized as the linguistic turn of the Copenhagen interpretation of quantum mechanics. This turn from physics to language does not only extend quantum theory to classical theory but also yield the quantum mechanical world view. Although the wave function collapse (or more generally, the post-measurement state) is prohibited in the linguistic interpretation, in this paper I show that the phenomenon like wave function collapse can be realized. That is, the projection postulate is completely clarified in the linguistic interpretation.


The Linguistic Interpretation of Quantum Mechanics
Recently in [1]- [4], I proposed quantum language (i.e., the linguistic interpretation of quantum mechanics, or measurement theory), which was characterized as the linguistic turn of the Copenhagen interpretation of quantum mechanics.This turn from physics to language does not only extend quantum theory to classical theory but also yield the quantum mechanical world view.Also, I believe that the linguistic interpretation is the true colors of the Copenhagen interpretation, though there are a lot of opinions about the Copenhagen interpretation (cf.[5]).
As mentioned in a later section (Section 1.3 (C)), the wave function collapse (or more generally, the postmeasurement state) is prohibited in the linguistic interpretation.Thus, some asked me "How is the projection postulate?".This question urges me to write this paper.The reader who would like to know only my answer may skip this section and read from Section 2.

Preparations
Now we briefly introduce quantum language as follows.
Consider an operator algebra ( ) B H .The measurement theory (=quantum language = the linguistic interpretation) is classified as follows.
( ) ( A : quantum system theory when A measurement theory A : classical system theory when , the * C -algebra composed of all compact operators on a Hilbert space H, the (A 1 ) is called quantum measurement theory (or, quantum system theory), which can be regarded as the linguistic aspect of quantum mechanics.Also, when  is commutative (that is, when  is characterized by ( ) C Ω , the * C -algebra composed of all continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space Ω ( cf. [6] [7])), the (A 2 ) is called classical measurement theory.
Also, note that, when ( ) ) , Tr Also, when ( ) *  = "the space of all signed measures on Ω ", where ν is some measures on Ω , thus, ( [6]).Let can be also identified with Ω (called a spectrum space or simply spectrum) such as For instance, in the above 2) we must clarify the meaning of the "value" of ( ) then ( ) And the value of ( ) 0 F ρ is defined by the α .According to the noted idea (cf.[8]), an observable 2) [Countable additivity] F is a mapping from  to  satisfying: a): for every Ξ ∈  , ( ) where 0 and I is the 0-element and the identity in  respectively, c): for any countable decomposition { } , , , , ) sense of weak * topology in  .

Axiom 1 [Measurement] and Axiom 2 [Causality]
Measurement theory (A) is composed of two axioms (i.e., Axioms 1 and 2) as follows.With any system S, a ba-   can be associated in which the measurement theory (A) of that system can be formulated.A state of the system S is represented by an element  ( ) ( ) . In addition to the above 1) and 2), we assume that ( ) Φ is said to be deterministic.If it is not deterministic, it is said to be nondeterministic.Also note that, for any observable ( ) is an observable in 1  .Now Axiom 2 is presented as follows (For details, see [4]).Axiom 2 [Causality].Let 1 2 t t ≤ .The causality is represented by a Markov operator , :

The Linguistic Interpretation
In the above, Axioms 1 and 2 are kinds of spells, (i.e., incantation, magic words, metaphysical statements), and thus, it is nonsense to verify them experimentally.Therefore, what we should do is not "to understand" but "to use".After learning Axioms 1 and 2 by rote, we have to improve how to use them through trial and error.We can do well even if we do not know the linguistic interpretation (=the manual to use Axioms 1 and 2).However, it is better to know the linguistic interpretation, if we would like to make progress quantum language early.
The essence of the manual is as follows: (C) Only one measurement is permitted.And thus, the state after a measurement is meaningless since it cannot be measured any longer.Hence, the wave function collapse is prohibited.We are not concerned with the problem: "When is a measurement taken?".Also, the causality should be assumed only in the side of system, however, a state never moves.Thus, the Heisenberg picture should be adopted, and thus, the Schrödinger picture should be prohibited.and so on.For details, [4].

The Wave Function Collapse (i.e., the Projection Postulate)
From here, I devote myself to quantum system (A 1 ) (and not classical system (A 2 )).And therefore, when a next measurement

Problem: The von
), the probability that a measured value belongs to ( ) Problem 1.In the linguistic interpretation, the phrase: post-measurement state in the (D 2 ) is meaningless.Also, the above (=(D 1 ) + (D 2 )) is equivalent to the simultaneous measurement where it should be recalled that O F is arbitrary.Also note that the above (i.e., the projection postulate (G)) is a consequence of Axioms 1 and 2.
C -algebra, and let *  be the dual Banach space of  .That is, all F ∈  such that 0 F ≥ .And define the mixed state space ( ) Also, the measurement of the observable O for the system S with the state ρ is denoted by below is a kind of mathematical generalization of Born's probabilistic interpretation of quantum mechanics.And thus, it is a statement without reality.Now we can present Axiom 1 in the * W -algebraic formulation as follows.Axiom 1[Measurement].The probability that a measured value x ( X ∈ ) obtained by the measurement ( von Neumann-Lüders projection postulate (in the Copenhagen interpretation, cf.[9] [10]) says: (D 2 ) When a measured value 0

(
In a similar way, the same result is easily obtained in the case of (7)).Thus, we see the following.
(i.e., an operator algebra composed of all bounded linear operators on a Hilbert space H with the norm B H