_{1}

^{*}

Barros discusses that [Jose Acacio de Barros, Int. J. Theor. Phys. 50, 1828 (2011)] Nagata derives inconsistencies from quantum mechanics [K. Nagata, Int. J. Theor. Phys. 48, 3532 (2009)]. Barros considers that the inconsistencies do not come from quantum mechanics, but from extra assumptions about the reality of observables. Here we discuss the fact that there is a contradiction within the quantum theory. We discuss the fact that only one expected value in a spin-1/2 pure state
〈σ
_{x}
〉
rules out the reality of the observable. We do not accept extra assumptions about the reality of observables. We use the actually measured results of quantum measurements (raw data). We use a single Pauli observable. We stress that we can use the quantum theory even if we give up the axiomatic system for the quantum theory.

Barros discusses that [

Here we discuss the fact that there is a contradiction within the quantum theory. We discuss the fact that only one expected value of a spin-1/2 pure state

On the other hand, the double-slit experiment [

If we modify this experiment so that one slit is closed, no interference pattern is observed. Thus, the state of both slits affects the final results. We can also arrange to have a minimally invasive detector at one of the slits to detect which slit the particle went through. When we do that, the interference pattern disappears [

We assume implementation of the double-slit experiment. There is a detector just after each slit. Thus interference figure does not appear, and we do not consider such a pattern. The possible values of the result of measurements are

Projective measurement theory does not meet Deutsch’s algorithm [

state lying in the

servables

measurements are

We consider whether projective measurement theory meets an easy detector model for Pauli observable. We try to implement double-slit experiment. There is a detector just after each slit. Thus interference figure does not appear, and we do not consider such a pattern. We assume that a source of spin-carrying particles emits them in a state, which can be described as an eigenvector of Pauli observable

At this stage we are in the following situation.

・ We cannot measure an expected value of a single spin observable in a state by using projective measurement theory.

First we discuss a contradiction within the quantum theory as follows [

Matrix theory is not compatible with probability theory. Matrix theory has axioms. Probability theory has axioms. These have axioms without a contradiction. Can we construct axioms for matrix theory and probability theory without a contradiction?

Let us consider joint probability.

We consider as follows: First we measure observable

On the other hand, the joint probability is depictured in terms of conditional probabilities:

From axioms of probability theory, we have

We cannot assign truth value “1” for the proposition (2) and for the proposition (4), simultaneously. We are in a contradiction. We cannot construct axioms for matrix theory and probability theory without the contradiction. There is a contradiction within the quantum theory.

The first point is actually that, conventional Quantum Mechanics discussions typically do not employ conditional probabilities correctly if at all. This is the central issue with Bell’s analysis leading to the idea that Quantum Mechanics requires non-locality or irreality and wave packet collapse and what not!

The two expected values of a spin-1/2 pure state

In this section, we discuss the fact that only one expected value of a spin-1/2 pure state

We assume implementation of the double-slit experiment [

We consider a quantum expected value

We introduce a hidden-variables theory for the quantum expected value of the Pauli observable

where

In what follows, we discuss the fact that we cannot assign the truth value “1” for the proposition (6). Assume the proposition (6) is true. We have the same proposition

An important note here is that the value of the right-hand-side of (6) is equal to the value of the right-hand- side of (7) because we only change a label.

We derive a necessary condition for the quantum expected value given in (6). We derive the possible value of the product

Here we use the fact

since the possible values of

We derive a necessary condition for the quantum expected value for the system in a pure spin-1/2 state

We do not assign the truth value “1” for two propositions (10) and (12), simultaneously. We are in a contradiction. We have to give up a hidden-variables theory for the expected value of the Pauli observable

Next we discuss the fact that there is a contradiction within the quantum theory by using a single Pauli observable [

We consider the relation between double-slit experiment and projective measurement. We try to implement double-slit experiment. There is a detector just after each slit. Thus interference figure does not appear, and we do not consider such a pattern. The actually measured results of quantum measurements are

Let

The above quantum expected value is zero if we consider only a wave function analysis.

We derive a necessary condition for the quantum expected value for the system in the pure spin-1/2 state

On the other hand, a mean value

where

Assume the quantum mean value with the system in an eigenvector

We can assume as follows by Strong Law of Large Numbers^{1},

In what follows, we show that we cannot assign the truth value “1” for the proposition (16) concerning projective measurement.

Assume the proposition (16) is true. By changing a label

An important note here is that the actually measured value of the right-hand-side of (16) is equal to the actually measured value of the right-hand-side of (18) because we only change the label. We have

Here

Thus we derive a proposition concerning the quantum mean value under an assumption that projective measurement is true (in a spin-1/2 system), that is

From Strong Law of Large Numbers, we have

Hence we derive the following proposition concerning projective measurement

We do not assign the truth value “1” for two propositions (14) (concerning a wave function analysis) and (23) (concerning projective measurement), simultaneously. We are in a contradiction.

We cannot accept the validity of the proposition (16) (concerning projective measurement) if we assign the truth value “1” for the proposition (14) (concerning a wave function analysis). In other words, such projective measurement does not meet the detector model for spin observable

We note here that there is much nonsense in the Physics literature regarding the theoretical formality for Spin. The formalism is correct so long as only one dimension is under consideration---a restriction that is fully acceptable in view of the fact that to engage spin empirically a Magnetic (B) field is required and it can have only one direction at the point of interacting with a charge. All formal talk of the spin of a particle in both the

In conclusions, Barros has discussed that Nagata has derived inconsistencies from quantum mechanics. Barros has considered that the inconsistencies do not come from quantum mechanics, but from extra assumptions about the reality of observables. Here we have discussed the fact that there is a contradiction within the quantum theory. We have discussed the fact that only one expected value of a spin-1/2 pure state

The author thanks Professor Tadao Nakamura.