Now we briefly introduce quantum language as follows.

Consider an operator algebra (i.e., an operator algebra composed of all bounded linear operators on a

Hilbert space H with the norm), and consider the pair, which is called a basic structure. Here, is a -algebra, and is a particular -algebra

(called a -algebra) such that is the weak closure of in.

The measurement theory (=quantum language = the linguistic interpretation) is classified as follows.

That is, when, the -algebra composed of all compact operators on a Hilbert space H, the (A₁) 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, the -algebra composed of all continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space ( cf. [6] [7] )), the (A₂) is called classical measurement theory.

Also, note that, when,

1) (=trace class), , (i.e., pre-dual space), thus, ().

Also, when,

2) = “the space of all signed measures on”, , , where is some measures on, thus, () (cf. [6] ).

Let be a -algebra, and let be the dual Banach space of. That is, , and the norm is defined by . Define the mixed state such that and for all such that. And define the mixed state space such that

A mixed state is called a pure state if it satisfies that “for some and” implies “”. Put

which is called a state space. It is well known (cf. [6] ) that , and, where . The latter implies that can be also identified with (called a spectrum space or simply spectrum) such as

For instance, in the above 2) we must clarify the meaning of the “value” of for and

. An element is said to be essentially continuous at, if there uniquely exists a complex number such that

(B) If converges to in the sense of weak^* topology of, that is,

then converges to.

And the value of is defined by the.

According to the noted idea (cf. [8] ), an observable in is defined as follows:

1) [-field] X is a set, (, the power set of X) is a -field of X, that is, “”, “”.

2) [Countable additivity] F is a mapping from to satisfying: a): for every, is a non-negative element in such that, b): and, where 0 and I is the

0-element and the identity in respectively, c): for any countable decomposition of (i.e., , ,), it holds that in the sense of weak^* topology in.

Measurement theory (A) is composed of two axioms (i.e., Axioms 1 and 2) as follows. With any system S, a basic structure 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 and an observable is represented by an observable in. Also, the measurement of the observable for the system S with the

state is denoted by (or more precisely,). An observer can obtain a measured value x () by the measurement.

The Axiom 1 presented 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 -algebraic formulation as follows.

Axiom 1 [Measurement]. The probability that a measured value x () obtained by the measurement belongs to a set is given by if is essentially continuous at.

Next, we explain Axiom 2. Let and be basic structures. A continuous linear operator (with weak^* topology) (with weak^* topology) is called a Markov operator, if it satisfies that 1): for any non-negative element in, 2):, where is the identity in,. In addition to the above 1) and 2), we assume that and .

It is clear that the dual operator satisfies that. If it holds that, the is said to be deterministic. If it is not deterministic, it is said to be non-

deterministic. Also note that, for any observable in, the is an observable in.

Now Axiom 2 is presented as follows (For details, see [4] ).

Axiom 2 [Causality]. Let. The causality is represented by a Markov operator.

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, see [4] .

Let be a quantum basic structure. Let be a countable set. Consider the projection valued observable in. Put

Axiom 1 says:

(D₁) The probability that a measured value is obtained by the measurement is given by

Also, the von Neumann-Lüders projection postulate (in the Copenhagen interpretation, cf. [9] [10] ) says:

(D₂) When a measured value is obtained by the measurement, the post-measurement state is given by

And therefore, when a next measurement is taken (where is arbitrary observable in), the probability that a measured value belongs to is given by

Problem 1. In the linguistic interpretation, the phrase: post-measurement state in the (D₂) is meaningless. Also, the above (=(D₁) + (D₂)) is equivalent to the simultaneous measurement, which does

not exist in the case that and do not commute. Hence the (D₂) is meaningless in general. Therefore, we have the following problem:

(E) Instead of the in, what observable should be chosen?

In the following section, I answer this problem within the framework of the linguistic interpretation.

Consider two basic structure and. Let be as in Section 2.1, and let be a complete orthonormal system in a Hilbert space K. Define the predual Markov operator by, for any,

or

Thus the Markov operator ( in Axiom 2) is defined by.

Define the observable in such that

Let be arbitrary observable in. Thus, we have the tensor observable in, where is the product -field.

Fix a pure state . Consider the measurement. Then, we see that

(F) the probability that a measured value obtained by the measurement belongs to is given by

(In a similar way, the same result is easily obtained in the case of (7)).

Thus, we see the following.

(G₁) if, then

(G₂) in case that a measured value belongs to, the conditional probability such that is given by

where it should be recalled that is arbitrary. Also note that the above (i.e., the projection postulate (G)) is a consequence of Axioms 1 and 2.

Considering the correspondence: (D) Û (G), that is,

namely,

there is a reason to assume that the true meaning of the (D) is just the (G). Also, note the taboo phrase “post- measurement state” is not used in (G₂) but in (D₂). Hence, we obtain the answer of Problem 1 (i.e.,).

Remark 1. So called Copenhagen interpretation may admit the post-measurement state (cf. [5] ). Thus, in this case, some may think that the post-measurement state is obtained by the formula (9). However,

this idea would not generally be approved. That is because, if the post-measurement state is admitted, a series of problems occur, that is, “When is a measurement taken?”, or “When does the wave function collapse happen?”, which is beyond Axioms 1 and 2. Readers should remember Wittgenstein’s famous word: “The limits of my language mean the limits of my world”, or “What we cannot speak about we must pass over in silence”.