**7.1 HUP environment**

Under the variant construction, variant measurements can be organized into multiple sets of simultaneous measurements. Each element in a *N* bit vector provides only a small portion of information, collected measurements are independent of special positions. Under this condition, there is no essential HUP environment for the variant construction. 0-1 groups and their measurements are naturally parallel . They can be processed in simultaneous conditions. Considering these properties, such group measurements do not correspond with the requirements of Heisenberg single particle environments. Viewed as a whole, the system of the variant construction has discrete and separate properties that serve to facilitate complex local interactions for any selected group.

From a measurement viewpoint, the parallel parameters of the variant measurements enable them to exist in different interactive models simultaneously. This set of simultaneous properties exhibits significant differences between the original wave functions and the variant construction.

#### **7.2 Weakness of BCP**

The main weakness of the BCP lies deep in the very logic on which it is founded. In his approach to QM, Bohr applied then extant classical principles of logic using static YES/NO approaches to dynamic particle and wave measurements. However, the complex nature of QM phenomena means that such a classical logic framework cannot fully support this quaternion organization or fully model the dynamic systems involved. This is the main reason why the BCP requires the application of exclusive properties to pairs of opposites.

The variant construction provides quaternion measurement groups. This property naturally supports QM-like structures. Useful configurations can be chosen for further development.

The main experimental evidence following Bohr in rejecting particle models are sets of wave interference distributions generated in long duration and very low intensity single photon experiments. These experiments show intrinsic wave interference patterns under many environments. Understandably, such data have long been held to be strongly indicitive of wave properties within even single quanta. Consequently, it has been deemed natural and necessary to apply wave descriptions and analysis tools in the search for QM solutions.

However, evidence residing within the main visual distributions of this chapter, serves to show that statistical distributions under a conditional probability environment naturally link to intrinsic wave properties in the majority of situations. Nearly all interesting distributions show obvious wave properties. Notably, such intrinsic wave distributions may be sufficient to allow a satisfactory alternative explanation of experimental results generated in long duration and very low intensity single photon experiments.

#### **7.3 The BCP for a special subset of QM**

20 Measurement Systems

Four anti-diagonal positions are linked to symmetry and anti-symmetry pairs, twelve other pairs of functions belong to non-symmetry and non-anti-symmetry conditions. Their meta

Under the variant construction, variant measurements can be organized into multiple sets of simultaneous measurements. Each element in a *N* bit vector provides only a small portion of information, collected measurements are independent of special positions. Under this condition, there is no essential HUP environment for the variant construction. 0-1 groups and their measurements are naturally parallel . They can be processed in simultaneous conditions. Considering these properties, such group measurements do not correspond with the requirements of Heisenberg single particle environments. Viewed as a whole, the system of the variant construction has discrete and separate properties that serve to facilitate complex

From a measurement viewpoint, the parallel parameters of the variant measurements enable them to exist in different interactive models simultaneously. This set of simultaneous properties exhibits significant differences between the original wave functions and the variant

The main weakness of the BCP lies deep in the very logic on which it is founded. In his approach to QM, Bohr applied then extant classical principles of logic using static YES/NO approaches to dynamic particle and wave measurements. However, the complex nature of QM phenomena means that such a classical logic framework cannot fully support this quaternion organization or fully model the dynamic systems involved. This is the main reason

The variant construction provides quaternion measurement groups. This property naturally supports QM-like structures. Useful configurations can be chosen for further development. The main experimental evidence following Bohr in rejecting particle models are sets of wave interference distributions generated in long duration and very low intensity single photon experiments. These experiments show intrinsic wave interference patterns under many environments. Understandably, such data have long been held to be strongly indicitive of wave properties within even single quanta. Consequently, it has been deemed natural and necessary to apply wave descriptions and analysis tools in the search for QM solutions.

However, evidence residing within the main visual distributions of this chapter, serves to show that statistical distributions under a conditional probability environment naturally link to intrinsic wave properties in the majority of situations. Nearly all interesting distributions show obvious wave properties. Notably, such intrinsic wave distributions may be sufficient to allow a satisfactory alternative explanation of experimental results generated in long duration

why the BCP requires the application of exclusive properties to pairs of opposites.

elements can be identified by the relevant variant expressions.

**7. Core debated issues under variant construction**

local interactions for any selected group.

and very low intensity single photon experiments.

**7.1 HUP environment**

construction.

**7.2 Weakness of BCP**

We may deduce that there is (only) a special subset of QM for which the BCP is satisfied. Under the variant construction there are six distinct logical configurations that can be used to support 0-1 vectors. Of these six, Bohr's approach is suitable for only the two schemes of pure static YES or NO. Meanwhile, the other four variant, invariant and mixed configurations lie outside the BCP framework. From this viewpoint, Bohr offers insight into important special circumstances of QM rather than provides an all embracing general solution.

Bohr's QM construction is complete and useful in many theoretical and practical environments for static and static-like systems. However, the variant construction provides a more powerful and general mechanism to handle different dynamic systems with variant and invariant properties.

#### **7.4 The EPR contribution on variant construction**

From EPR proposed experiments and other theoretical considerations, Einstein demonstrated a depth of understanding of weakness inherent in the foundations of the QM approach. He clearly identified two operators with non-communication properties that failed to support simultaneous operations and recognized that this type of mechanism was still not explained in the Copenhagen interpretation.

Using the variant construction, EPR devices have the following correspondence:

$$\begin{cases} \text{S}\_1 &\rightarrow \{ \mu\_{\beta}, \upsilon\_{\beta}, \tilde{u}\_{\beta}, \tilde{v}\_{\beta} \dots \}; \\ \text{S}\_2 &\rightarrow \{ u\_{\beta}, \upsilon\_{\beta}, \tilde{u}\_{\beta}, \tilde{v}\_{\beta} \dots \}; \\ IM(\text{S}\_1, \text{S}\_2) &\rightarrow \{ M(u\_1), M(v\_1), M(\tilde{u}\_1), M(\tilde{v}\_1) \dots \}; \\ SM(\text{S}\_1, \text{S}\_2) &\rightarrow \{ M(u\_0), M(v\_0), M(\tilde{u}\_0), M(\tilde{v}\_0) \dots \}. \end{cases} \tag{18}$$

$$\begin{cases} \langle \mathcal{S}\_1, \mathcal{S}\_2, IM(\mathcal{S}\_1, \mathcal{S}\_2), SM(\mathcal{S}\_1, \mathcal{S}\_2) \rangle \rightarrow \\ \quad \quad \quad \quad \quad \quad \langle \{u\_{\beta}, v\_{\beta}, \tilde{u}\_{\beta}, \tilde{v}\_{\beta} \dots \}, \{u\_{\beta}, v\_{\beta}, \tilde{u}\_{\beta}, \tilde{v}\_{\beta} \dots \} \rangle \\ \quad \quad \quad \quad \quad \langle M(u\_1), M(v\_1), M(\tilde{u}\_1), M(\tilde{v}\_1) \dots \rangle, \\ \quad \quad \quad \quad \langle M(u\_0), M(v\_0), M(\tilde{u}\_0), M(\tilde{v}\_0) \dots \rangle \rangle \end{cases} \tag{19}$$

From this correspondence, many possible configurations of combinations and their subsets are available for future theoretical and experimental exploration.

Using the variant construction, rich configurations can be expressed. From such mapping, it can be seen to be nothing less than astounding that such meta constructions were identified by Einstein as far back as 1935.

#### **7.5 Afshar's experiments on variant construction**

Afshar's experiments apply anti-symmetry signals making the following correspondence:

$$\begin{cases} \psi\_1 \rightarrow \{v\_+\}; \\ \psi\_2 \rightarrow \{v\_1\}; \\ \sigma\_1 \rightarrow \{P\_H(v\_1|J)\}; \\ \sigma\_2 \rightarrow \{P\_H(v\_0|J)\}. \end{cases} \tag{20}$$

(c) D-P

<sup>361</sup> From Conditional Probability Measurements to Global Matrix Representations on Variant

Construction – A Particle Model of Intrinsic Quantum Waves for Double Path Experiments

(d) D-W

22 Measurement Systems

(a) Left

(b) Right

(g) D-P

<sup>363</sup> From Conditional Probability Measurements to Global Matrix Representations on Variant

Construction – A Particle Model of Intrinsic Quantum Waves for Double Path Experiments

(h) D-W

Representations. (a) Left; (b) Right; (c) D-P; (d)D-W in symmetry conditions; (e) Left; (f)

<sup>2</sup> Eight Matrices of Global Matrix

Fig. 4. (a-h) Even number groups: *<sup>N</sup>* <sup>=</sup> {12}, *<sup>f</sup>* <sup>∈</sup> *<sup>B</sup>*<sup>4</sup>

Right; (g) D-P; (h)D-W in anti-symmetry conditions.

24 Measurement Systems

(e) Left

(f) Right

Fig. 4. (a-h) Even number groups: *<sup>N</sup>* <sup>=</sup> {12}, *<sup>f</sup>* <sup>∈</sup> *<sup>B</sup>*<sup>4</sup> <sup>2</sup> Eight Matrices of Global Matrix Representations. (a) Left; (b) Right; (c) D-P; (d)D-W in symmetry conditions; (e) Left; (f) Right; (g) D-P; (h)D-W in anti-symmetry conditions.

(c) D-P

<sup>365</sup> From Conditional Probability Measurements to Global Matrix Representations on Variant

Construction – A Particle Model of Intrinsic Quantum Waves for Double Path Experiments

(d) D-W

26 Measurement Systems

(a) Left

(b) Right

(g) D-P

<sup>367</sup> From Conditional Probability Measurements to Global Matrix Representations on Variant

Construction – A Particle Model of Intrinsic Quantum Waves for Double Path Experiments

(h) D-W

Representations. (a) Left; (b) Right; (c) D-P; (d)D-W in symmetry conditions; (e) Left; (f)

<sup>2</sup> Eight Matrices of Global Matrix

Fig. 5. (a-h) Odd number groups: *<sup>N</sup>* <sup>=</sup> {13}, *<sup>f</sup>* <sup>∈</sup> *<sup>B</sup>*<sup>4</sup>

Right; (g) D-P; (h)D-W in anti-symmetry conditions.

28 Measurement Systems

(e) Left

(f) Right

Fig. 5. (a-h) Odd number groups: *<sup>N</sup>* <sup>=</sup> {13}, *<sup>f</sup>* <sup>∈</sup> *<sup>B</sup>*<sup>4</sup> <sup>2</sup> Eight Matrices of Global Matrix Representations. (a) Left; (b) Right; (c) D-P; (d)D-W in symmetry conditions; (e) Left; (f) Right; (g) D-P; (h)D-W in anti-symmetry conditions.

Further insight may be found working from conditional probability measurements to global

<sup>369</sup> From Conditional Probability Measurements to Global Matrix Representations on Variant

Construction – A Particle Model of Intrinsic Quantum Waves for Double Path Experiments

Applying conditional probability models on interactive measurements and relevant statistical processes, two groups of parameters {*u*˜*β*, *v*˜*β*} describe left path, right path, D-P and D-W conditions with distinguishing symmetry and anti-symmetry properties. {*PH*(*u*˜*β*|*J*), *PH*(*v*˜*β*|*J*)} provide eight groups of distributions under symmetry and anti-symmetry forms. In addition, {*M*(*u*˜*β*), *M*(*v*˜*β*)} provide eight matrices to illustrate global

productions. How to overcome the limitations imposed by such complexity and how best to compare and contrast such simulations with real world experimentation will be key issues

Six predictions and four conjectures are offered for testing by further theoretical and

Thanks to Colin W. Campbell for help with the English edition, to The School of Software Engineering, Yunnan University and The Key Laboratory of Yunnan Software Engineering for financial supports to the Information Security research projects (2010EI02, 2010KS06) and

Afshar, S. (2005). Violation of the principle of complememtarity, and its implications, *Proc.*

Afshar, S. (2006). Violation of bohr's complememtarity: One slit or both?, *AIP Conf. Proc.* 810.

Afshar, S., Flores, E., McDonald, K. & Knoesel, E. (2007). Paradox in wave particle duality,

Aspect, A., Grangier, P. & Roger, G. (1982). Experimental realization of einstein -podolsky

Barrow, J. D., Davies, P. C. W. & Charles L. Harper, J. E. (2004). *SCIENCE AND ULTIMATE REALITY: Quantum Theory, Cosmology and Complexity*, Cambridge University Press.

Bohr, N. (1935). Can quantum-mechanical description of physical reality be considered

Bohr, N. (1949). *Discussion with Einstein on Epistemological Problems in Atomic Physics*, Evanston.


Ash, R. B. & Doléans-Dade, C. A. (2000). *Probability & Measure Theory*, Elsevier.

Bell, J. S. (1964). On the einstein-podolsky-rosen paradox, *Physics* 1. 195-200.

Aspect, A. (2007). To be or not to be local, *Nature* 446. 866-867.

complete?, *Physical Review* 48. 696-702.

Bohr, N. (1958). *Atomic Physics and Human Knowledge*, Wiley. Bolles, E. (2004). *Einstein Defiant*, Joseph Henry Press.

Born, M. (1971). *The Born Einstein Letters*, Walker and Company.

and exhaustive vector space

<sup>×</sup> <sup>2</sup>*N*) as ultra exponent

matrix representation on the variant construction.

behaviors under conditional environments.

in future work.

experimental work.

sub-CDIO project.

**11. References**

**10. Acknowledgements**

294-299.

*Lett.* 49. 91-94.

200-241.

*SPIE* 5866. 229-244.

*Found. Phys.* 37. 295-305.

The complexity of *n*-variable function space has a size of 22*<sup>n</sup>*

has 2*N*. Overall simulation complexity is determined by *O*(22*<sup>n</sup>*

Using quaternion structures,

$$\{\{\psi\_1, \psi\_2, \sigma\_1, \sigma\_2\} \rightarrow \langle \{v\_+\}, \{v\_1\}, \{P\_H(v\_1|f)\}, \{P\_H(v\_0|f)\}\}.\tag{21}$$

All Afshar's experiments are a special case of the EPR model.
