^{1}

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As a trial, though thinking of general concepts, of our scientific challenge, we consider whether the Charge-Parity-Time (CPT) symmetry can be almighty even in a photon. This is the main aim of this paper. In what follows, we discuss our argumentations dividing the conjecture into two parts. Rotational invariance of physical laws is an accepted principle in Newton’s theory. We show that it leads to an additional constraint on local realistic theories with mixture of ten-particle Greenberger-Horne-Zeilinger state. This new constraint rules out such theories even in some situations in which standard Bell inequalities allow for explicit construction of such theories. This says new hypothesis to the number of ten. Next, it turns out Zermelo-Fraenkel set theory has contradictions. Further, the von Neumann’s theory has a contradiction by using ±1/. We solve the problem of von Neumann’s theory while escaping from all contradictions made by Zermelo-Fraenkel set theory, simultaneously. We assume that the results of measurements are

These days, an interesting experimental research was reported [

The results of first argumentation, dealing with consequences for the assumption of rotational invariance for tests of non-locality, have been published [

Non-locality in quantum physics means the possibility of distributing correlations that cannot be due to previously shared randomness, without signaling [

Leggett-type nonlocal realistic theory [

by using entangled states (two spins

Rotational invariance of physical laws is a generally accepted principle in Newton’s theory. It states that the value of a correlation function does not depend on the coordinate systems used by the observers. The measure- ment setup classifies realistic theories [

Many of the recent advances in quantum information theory suggest that the highly-non-local properties of quantum states that lead to violations of Bell inequalities can be used as a resource to achieve success in some tasks, which are locally impossible. Examples can serve quantum cryptography and quantum communication complexity [

We aim to show that the fundamental property of the known laws of Newton’s theory, their rotational invariance can be used to find new hypothesis by using disqualification of experimentally accessible local rea- listic theories with ten particles in a new state.

This argumentation is as follows. Assume that we have some correlation function. This correlation function has a form which is rotationally invariant (cf. Equation (2)). We want to build rotationally invariant local realistic theory for the rotationally invariant correlation function. We see that the demand that the resulting correlation function must be rotationally invariant leads to a generalized Bell inequality [

Next, we discuss von Neumann’s theory cannot meet the Deutsch-Jozsa algorithm. We discuss axiomatic theory has a contradiction. So we think that we give confusion to many scientists. Let confused scientists feel relieved by solving the problems mentioned above. We do not think that the solution is only one presented here. We think it is one of solutions. Let us start reviewing the axiomatic set theory.

Zermelo-Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. It has a single primitive ontological notion, that of a hereditary well-founded set, and a single ontological assumption, namely that all individuals in the universe of discourse are such sets. ZFC is a one-sorted theory in first-order logic. The signature has equality and a single primitive binary relation, a set membership, which is usually denoted Î. The formula

There is a solution escaping from all of contradictions made by ZFC axioms and von Neumann’s theory for the Deutsch-Jozsa algorithm. A solution of von Neumann’s theory for the Deutsch-Jozsa algorithm have already published. So, we summarize this part to be included in this big paper.

We solve the problem of von Neumann’s theory while escaping from both contradictions made by Zermelo- Fraenkel set theory, simultaneously. We assume that the results of measurements are

In what follows, we discuss our argumentations dividing the conjecture into two parts.

This paper is organized as follows.

In Section 2, we discuss generalized Bell inequality.

In Section 3, we discuss mixture of ten-qubit GHZ state.

In Section 4, we discuss violation of rotational invariance of local realistic models with mixture of GHZ state.

In Section 5, we discuss summary of the first argumentation.

In Section 6, we discuss whether there are solutions of the problems of ZFC axioms and von Neumann’s theory for the Deutsch-Jozsa algorithm.

In Section 7, we discuss the first type of contradiction made by the axiomatic theory. This type of contra- diction is escapable by using max or min of possible values of expected value under study.

In Section 8, we discuss the second type of contradiction made by the axiomatic theory. This type of con- tradiction is escapable by assuming that zero does not exist.

In Section 9, we discuss the solution escaping from both of contradictions made by the axiomatic theory and von Neumann’s theory for the Deutsch-Jozsa algorithm.

In Section 10, we discuss the summary of the second argumentation.

In Section 11, we discuss implications of our argumentation.

Section 12 summarizes this paper.

Assume that we have a set of N spins

measurements (observables) for such spins are parameterized by a unit vector

where

If an experimental correlation function admits rotationally invariant tensor structure familiar from Newton’s theory, we can introduce the following form:

where

where

Assume that one knows the values of all

where

Those assumptions are now demystify by Hess and Phillip [

Let us parametrize the arbitrary unit vector in a spherical coordinate system defined by

where

We shall show that the scalar product of rotationally invariant local realistic correlation function

with the rotationally invariant experimental correlation function, that is

is bounded by a specific number dependent on

where

We use decomposition (5). We introduce the usual measure

that is for

In what follows, we derive the upper bound (8). Since the rotationally invariant local realistic theory is an average over

where

and

Let us analyze the structure of this integral (11). It is easy to notice that (11) is a sum, with coefficients given by

and

Notice that we deal here with integrals, or rather scalar products of

The normalized functions

three-dimensional real functional space, which we shall call

where

with a normalized vector

Note that the sum in (18) over the components of this vector is just

It remains to show the upper bound on the norm

possible value of the scalar product between

where

where the dot between three-dimensional vectors denotes the usual scalar product in

Finally, we have

The relation (24) is a generalized N-qubit Bell inequality with the entire range of measurement settings. Rotationally invariant local hidden variable theories,

where we use the orthogonality relation

(25) suggests that the value of (25) does not have to be smaller than (24). That is there may be such correlation functions

We shall present an important quantum state. We assume

where

tions of the Cartesian coordinate system. The unitary realizing this rotation is given by [

with

global phase which does not contribute to correlations, of the same form as

where

preted as the reduction factor of the interferometric contrast observed in the ten-particle correlation experiment.

We present here a simple, but important example of violation (24). Imagine 10 observers who can choose between three orthogonal directions of spin measurement,

Let us assume that the source of 10 entangled spin-carrying particles emits them in a state, which can be described as a mixture of Greenberger-Horne-Zeilinger correlations, given in (28). We can show that if the observers limit their settings to

components of

It is easy to see that

Then, we have

spins the rotational invariance puts an additional, non trivial, constraint on a local realistic theory. For

despite the fact that there exists a local realistic theory for the actually measured values of the correlation function, the rotational invariance principle disqualifies this theory. As it is shown in [

then there always exists an explicit local realistic theory for the set of correlation function values

Nevertheless the rotational invariance principle excludes local realistic theories for

situation is such for

theory for the values of the correlation function for the settings chosen in the experiment. But these theories must be consistent with each other, if we want to extend their validity beyond the 2^{10} settings to which each of

them pertains. Our result clearly indicates that this is impossible for

to reconstruct the 2^{10} data points, when compared with each other, must be inconsistent-therefore they are invalidated. The theories must contradict each other. In other words the explicit theories, given in [

Please note that all information needed to get this conclusion can be obtained in a three-orthogonal-settings- per-observer experiments with ten particles. Simply to get both the value of (25)

In summary of the first argumentation, we have shown that rotational invariance of physical laws leads to an additional constraint on local realistic theories with ten particles new state. This new constraint has ruled out such theories even in some situations in which standard Bell inequalities allow for explicit construction of such theories. The whole analysis has been performed without any additional mathematical assumptions on the form of local realistic theories.

The interesting feature is that Bell’s theorem rules out realistic interpretation of some quantum mechanical predictions, and therefore of quantum mechanics in general, provided one assumes locality. Locality is a con- sequence of the general symmetries of the Poincaré group of the Special Relativity Theory. However it is a direct consequence of the Lorentz transformations (boosts), as they define the light-cone. As our discussion shows a subgroup of the Poincaré group, rotations of the Cartesian coordinates, introduces an additional con- straint on the local realistic models.

The results presented are the result of the definition of

There is a solution escaping from both contradictions made by ZFC axioms and von Neumann’s theory for the Deutsch-Jozsa algorithm. A solution of von Neumann’s theory for the Deutsch-Jozsa algorithm have already published. So, we summarize this part to be included in this big paper.

We solve the problem of von Neumann’s theory while escaping from both contradictions made by Zermelo- Fraenkel set theory, simultaneously. We assume that the results of measurements are

First of all we discuss a contradiction made by Zermelo-Fraenkel set theory with the axiom of choice. Assume all axioms of Zermelo-Fraenkel set theory with the axiom of choice is true. We consider statistics. We assume that the results of measurements are

We derive the possible value of the product (the square of mean)

Thus we have the following proposition when

The expected value which is the average of the results of measurements is given by

We assume that the possible values of the actually measured result

Same expected value is given by

We only change the labels as

We derive a proposition concerning the expected value given in (36) under the assumption that the possible values of the actually measured result are

We do not assign the truth value “1” for two propositions (35) and (40) simultaneously if

Assume all axioms of Zermelo-Fraenkel set theory with the axiom of choice is true. We assume zero exists. We consider statistics. We assume that the results of measurements are

We derive the possible value of the product (the square of mean)

Thus we have the following proposition when

The expected value which is the average of the results of measurements is given by

We assume that the possible values of the actually measured result

We only change the labels as

By using these facts we derive a necessary condition for the expected value given in (44). We derive the possible values of the product

We derive a proposition concerning the expected value given in (44) under the assumption that the possible values of the actually measured result are

We do not assign the truth value “1” for two propositions (43) and (48) simultaneously. Thus we are in a contradiction due to assuming that zero exists. This type of contradiction is escapable if we skip zero.

We assume zero does not exist. Thus we escape from the second type of contradiction made by Zermelo- Fraenkel set theory with the axiom of choice. We assume that the results of measurements are

We assume the possible value of the product

We assume the following proposition:

This expected value which can be the average of the results of measurements is given by

We assume that the possible values of the actually measured result

Same expected value can be given by

We change the labels as

We can derive a proposition concerning the expected value given in (52) under the assumption that the possible values of the actually measured result are

We can assign the truth value “1” for two propositions (51) and (56) simultaneously. We are not in a con- tradiction. Thus we escape from the first type of contradiction made by Zermelo-Fraenkel set theory with the axiom of choice.

One of wrong point of the axiomatic set theory is assuming zero. So we do not assume the existence of zero. Also the wrong point of von Neumann’s theory for quantum computation is assuming the results of measure- ments are

In summary of the second argumentation, one of wrong point of the axiomatic set theory has been assuming zero. So we do not have assumed the existence of zero. Also the wrong point of von Neumann’s theory for quantum computation has been assuming the results of measurements are

We have concluded the following with discussions: Our discussion implies that there will be some mass for a photon. Since there is not zero, the mass will not be zero. Now the speed of the photon is finite. We suspect that some fields give resistance to the photon at this stage. In the future the mass will be increased so that we can see it experimentally.

We find there is not system of units in physics because there is not

Also we think it is dangerous to define subtraction. It may be introducing a contradiction to axiomatic set theory.

We would expect that the big-bang hypothesis would be true. We would then infer that the particle antiparticle pair would be constructed. The number of particles would be two. This would disqualify of a local realistic model with two particles due to quantum effect (Violation Charge). Then the voluntary violation of time symmetry would be true, so that the number of particles would be six. This would disqualify of a local realistic model with six particles due to quantum effect (Violation Time). Then the voluntary violation of the constant value of gravity would be true. Finally we would infer that the number of particles here would be ten. This would disqualify of a local realistic model with ten particles due to quantum effect as well (Violation Parity).

The Charge, Parity, and Time (CPT) symmetry would be violated. We know such results of experiments [

An important conjecture is that a photon and an anti-photon pair collapse. Since the photon would be massive, it is theoretically possible. It is believed that the spin of a photon has

It is interesting to create by using optical fiber device, which is something like that an electric current flows. If so, we would create new computer device by using optical fiber. We do not know what the device likes.

Finally, we point out spin

In conclusions, we have presented a theoretical conjecture of violation of the CPT symmetry even in a photon. This has been the main aim of this paper. We have discussed our argumentations dividing the conjecture into two parts. Rotational invariance of physical laws has been an accepted principle in Newton’s theory. We have shown that it leads to an additional constraint on local realistic theories with mixture of ten-particle Greenberger- Horne-Zeilinger state. This new constraint has ruled out such theories even in some situations in which standard Bell inequalities allow for explicit construction of such theories. Recently, it has turned out Zermelo-Fraenkel set theory has a contradiction. Further, the von Neumann’s theory has had a contradiction by using

We thank Professor K. Niizeki and Dr. C.-L. Ren for valuable discussions.

Koji Nagata,Tadao Nakamura, (2015) Whether the CPT Symmetry Can Be Almighty Even in a Photon. Open Access Library Journal,02,1-14. doi: 10.4236/oalib.1101806