Fred Y. Ye, IJNSR, 2017; 1:4

Research Article IJNSR (2017) 1:4

International Journal of Natural Science and Reviews (ISSN:2576-5086)

A Vortex Formulation of Quantum Setting Discrete Quantum States into Continuous Space-time

Fred Y. Ye 1, 2*

1 School of Information Management, Nanjing University, Nanjing 210023, CHINA 2International Joint Informatics Laboratory (IJIL), UI-NU, Champaign-Nanjing

ABSTRACT

Any quantum state can be described by a vortex, which is math- *Correspondence to Author: ematically a multi-vector and physically a united-measure. Fred Y. Ye When the vortex formulation of quantum physics is introduced, School of Information Management, Hamil-ton principle keeps its core position in physical analysis. Nanjing University, Nanjing 210023, While the global characteristics are described by Lagranrian CHINA; International Joint Informat- function for dynamics and double complex core function for ics Laboratory (IJIL), UI-NU, Cham- stable states, Schrödinger equation and gauge symmetries paign-Nanjing reveal local char-acteristics. The vortex-based physics provides a new unified understanding of wave-particle duality and uncertainty, quantum entanglement and teleportation, as How to cite this article: well as quantum information and computation, with setting Fred Y. Ye. A Vortex Formulation of discrete quantum states into con-tinuous space-time for Quantum Physics Setting Discrete keeping concordance of methodology in processing micro- Quantum States into Continuous particle and macro-galaxy. Two fundamental experiments are Space-time. International Journal suggested to correct and verify the physical for-mulation. of Natural Science and Reviews, 2017; 1:4.

Keywords: Vortex; vortex formulation; quantum mechanism; quantum state; quantum physics; space-time eSciPub LLC, Houston, TX USA. Website: http://escipub.com/

http://escipub.com/international-journal-of-natural-science-and-reviews/ 0001 Fred Y. Ye, IJNSR, 2017; 1:4 INTRODUCTION The path integral formulation of quantum There are a few famous formulations of mechanics is a description of quantum theory quantum physics, in which we mention three that generalizes the action principle of classical [4] main representative formulations: Schrödinger mechanics . It replaces the classical notion of wave formulation, Heisenberg matrix a single, unique classical trajectory for a system formulation and Feynman path integral with a sum, or functional integral, over an infinity formulation [1]. of quantum-mechanically possible trajectories to compute a quantum amplitude. This In wave formulation, the state of a system at a formulation has proven crucial to the given time is described by a complex wave subsequent development of theoretical physics, function, also referred to as state vector in a because manifest Lorentz covariance (time and complex vector space. The Schrödinger space components of quantities enter equations equation describes how wave functions change in the same way) is easier to achieve than in the in time. In the formulation, the operators stay operator formalism of canonical quantization. fixed while the Schrödinger equation changes Unlike previous methods, the path integral the basis with time. However, the wave packet allows a physicist to easily change coordinates will also spread out as time progresses, which between very different canonical descriptions of means that the position becomes more the same quantum system. Another advantage uncertain with time. This also has the effect of is that it is in practice easier to guess the correct turning a position eigenstate, which can be form of the Lagrangian of a theory, which thought of as an infinitely sharp wave packet, naturally enters the path integrals, than the into a broadened wave packet that no longer Hamiltonian. represents a definite position eigenstate [2]. In this paper, a vortex formulation is suggested, In matrix formulation, it is the operators which and the bracket notations are including three change in time while the basis of the space main types as follows. remains fixed. A fixed basis is, in some ways, more mathematically pleasing. This formulation (1) Dirac bra–ket notation < | and | >: the left also generalizes more easily to relativity. Matrix part < |, called “bra”, represents a row vector, mechanics was the first conceptually while the right part | >, called “ket”, denotes a autonomous and logically consistent formulation column vector; and < | > represents an inner of quantum physics. product, producing a scalar, while the | > < | denotes an outer product, producing a tensor. Dirac’s interaction gave a link of matrix formulation and wave formulation, via manifest (2) Commutated notation: It is defined that [X, Y] in Dirac's bra–ket notation [3], where matrix = XY – YX for measures X and Y. Mostly, [X, Y] formulation and wave formulation are ≠ [Y, X]. Particularly, following non-commutation equivalence of both the basis and the operators, relation contributes quantum mechanism carrying time-dependence. The interaction [X ,Y]  XY YX  i (1) picture allows for operators to act on the state vector at different times and forms the basis for in which i is the imaginary unit and ℏ is the quantum field theory and other methods. reduced Planck's constant (ℏ =h/2π).

http://escipub.com/international-journal-of-natural-science-and-reviews/ 0002 Fred Y. Ye, IJNSR, 2017; 1:4 (3) Poisson bracket notation { }: It is defined that The covariant derivative and the differential operators of two order derivatives are defined {f, g} =        = -{g, f} for functions f and g. respectively as The differential operators of one order 2     derivatives are defined as D  (   ) ;  g    ; g    (7)

   where we see that space-time metric g is    ;     (2) x naturally generated. In the paper, we use Greek in which four Dirac matrices are viewed as four subscripts μ, ν to denote 1, 2, 3, 4 and Latin orthonormal basis vectors for real space-time subscripts i, j to do 1, 2, 3. This is a foundational mathematical framework of physics.  I 0   0  k  Now we introduce some new ideas to  0   , k    (3)    k  0  I   0  understand quantum states via vortex, which can visually combine point particle, wave vector where  0 is time-like vector and and spin together, for setting discrete quantum states into continuous space-time.  k (k 1,2,3) space-like vectors. Similarly, the three Pauli matrices  k  ( 1, 2 , 3 ) are THEORETICAL FOUNDATIONS: Vortex- based physics viewed as three orthonormal basis vectors for three dimensional Euclidean space A vortex is mathematically represented by a multi-vector, and physically describes

linked-measure [7-9], which contributes a new 1 0 1 2 0  i 3 1 0  math-physical methodology to understand    ,   ,    (4) 1 0  i 0  0 1 physical reality [10].

Both { k } and { k } are to be interpreted Vortex formulation geometrically as meaningful space-time Mathematically, a vortex can be simply bivectors and not as operators. expressed by using a multi-vector Mk (k = 0, 1, 2, When we use space-time multi-vectors [5-6], for 3, 4) as follows each space-time point x on Dirac frame

M  M 0  M 1  M 2  M 3  M 4  {  ,  0,1,2,3}, there exists (9)  V  B  iU  i   A  B  ( , A, B)

  x  x , x   x (5) where ψ=φ+iθ is a complex scalar function (, but it describes mass), while The coordinates’ transformation will A=V+iU means a complex vector function be (potential function, but it describes field ) and B         x  x'    x ;   '    with          (1/ 2)B   as a unique bivector (spin (6) function, it can form spinor). One scalar ψ,one

http://escipub.com/international-journal-of-natural-science-and-reviews/ 0003 Fred Y. Ye, IJNSR, 2017; 1:4 vector A and one bivector B just describe a | M |,A, B | (,A, B) | | M | (16) vortex and construct a physical united- measure. M | ,A, B | | (,A, B) | | M | (17) Its revision (revised conjugation) is denoted as

~ The addition, subtraction, and multiplication of M ~ ~ ~ ~ ~ ~ multi-vectors produce also multi-vectors. With M  M  M  M  M  M  0 1 2 3 4 (10) definitions  V  B  iU  i  ( , A,B) ~ ~ M |  | M  M |  | M  M |  | M   0 and And its space-time conjugation is ~ ~ | M  M || M  M || M   M | 0 , the M  iMi  M  M  M  M  M 0 1 2 3 4 (11)   A  B  ( ,A, B) addition and multiplication will lead to various composite and superposition states, such as Physically, a linked-measure means that ψ, A, and B can be meantime measured, which just M |  | M  2 |  2A|  2B| (18) constructs the mass-energy structure, including ~ ~ mass, momentum and M |  | M  2 |  2A|  2B| (19) (spin), of a vortex. At micro-particle level, a vortex M and its space-time conjugation M M |  | M  2 | - 2A|  2B| (20) describe particle and anti-particle, while the M ~ | M  M | 2 |  2A|  2B| (21) and its revised conjugation M reveal ~ ~ entanglement. At macro-cosmos level, M | M  M | 2 |  2A|  2B| (22) means total mass-energy, and the energy-mass conservation is extended to | M  M | 2 |  2A|  2B| (23) energy-mass-momentum-angular momentum joint conservation. For forming complete system, The inner product M | M produces scalar, ~ we define M  M and M  M . outer product | M M |and

Taking an orthonormal pair of basis |↑> and <↓| | M | M | M| M leads to tensor, such as representing right-handed spin and left-handed spin or the “up” and “down” directions     2 2 2 respectively, with using Dirac bra-ket notation, M | M   ( , A, B) A   A  B (24)   we define basic states  B

| M | , A, B | (, A, B) | | M | (12)    ~   2 2 M | M   ( , A, B) A    AA  B (25) M |  , A, B | | (, A, B) | | M | (13)    B ~ ~ | M | , A,B | ( , A,B) | | M | (14)    A  (B) ~     | M M |  A( , A,B)   A AA A(B) ~ ~ M |  , A,B | | ( , A,B) | | M | (15)      B   B BA B(B) http://escipub.com/international-journal-of-natural-science-and-reviews/ 0004 Fred Y. Ye, IJNSR, 2017; 1:4 (26) When we defined linked-energy E as well as

linked-momenta pμ with linking Hamilton      ~     function H and Lagrangian function L, the | M  | M    A   A  (27)     analytical principle keeps Hamilton principle as  B  B follows where ψ2=(φ+iθ)(φ+iθ) or ψψ produces scalar,  d 4 x(L  k)  g  0 ; L  p x  H (34) A2=(V+iU)(V+iU) or AA constructs tensor, and s   2 B or B(-B) introduces spinor. where ΣL is total Lagrangians of all states, Ω is The entangled states are formed between M curvature of space-time, k is constant and g is

~ the determinant of the metric tensor gμν. Totally, and M , similarly to Bell states the mathematical physical equation should be a ~ M |  | M  2 |  2A| (28) balance between left discrete quantum matter and right continuous curvature of space-time, as ~ | M  M | 2 |  2A| (29) follows 1 1 ~  F a F a  B b B b  (i  D  m )  | M| M  (A  A)|  2B| (30)   q  q q 4 4 1 1 ~ L  k  (R  g  R  ) M | M | (A  A)|  2B| (31) 8G 2

Meanwhile, we can define an orthonormal (35) function F and a double complex function Z as in which R is the trace of the Ricci tensor, G is the associated function and the core function of the gravitational constant, and Λ is the M, respectively, as follows cosmological constant. With including 8 F  assoc(M )   jA kB  (  i)  j(V  iU )  kB duplicates (a) in 3 generations, the field strength (32) is F   A   A  gf Aa Aa ; f t c  [t a ,t b ] (36) Z  core(M )  X  jY    i  j(V  iU ) (33)      abc   abc

2 2 2 a where we keep Hamilton symbols i =j =k =ijk= A correspond to the gluon fields (a=1,…,8, as -1. In Eq. (33), X is just ψ and Y just A of Eqs. there are eight kinds of gluon) and the  are (10) and (11). q,c It is expected that we can apply the vortex quark-field spinors for a quark of flavor q and formulation to process both classical and mass mq with a color-index c (c=1,2,3), where quantum physics, including standard model [11], a the t correspond to eight 3 × 3 matrices and EPR paradox [12], teleportation [13], quantum are the generators of the SU(3) group, while the information and computation [14-15]. The interactions are stemmed by group SU(2), with associated function F and double complex color-index c changing to b: function [16] Z may be useful for assisting B  [  (1g / 2)B ]B [  (1g / 2)B ]B  physical analysis.   2    2   Bb  ;b  1,2,3 Global analytical principles  b (37) http://escipub.com/international-journal-of-natural-science-and-reviews/ 0005 Fred Y. Ye, IJNSR, 2017; 1:4

Therefore, for global states, the Hamilton     F  J (45) principle keeps as core analytical principle in physics. in which bivector F=E+iH includes both electrical field E and H, and J is probability current and satisfies the continuity Local symmetries and gauge invariances equation In core function Z=X+iY, if the functions X and Y are analytic and the end variables are space s  | |2    J  0 (46) and time t, they will match Cauchy-Riemann t condition If there existed strong links among ψ, A and B X Y   0 (38) s t B    A   A ; A   (47) X Y   0 (39) it leads to a strong linked-field. When all t s matter-energy concentrates on ψ, the system is Now we replace X and Y with unified core completely mastered by Schrödinger equation function Z=X+iY, a local stable system should fit d equations H |   i |  (48) dt Z Z  i  0 (40) where H denotes quantum Hamiltonian, s t describing matter-energy eigenvalues. 2 2  Z  Z The energy E (linking to ψ) and momentum p 2  2  0 (41) s t (linking to A) will construct canonical where Eq.(40) is Cauchy-Riemann equation commutation relations and Eq. (41) Laplace equation that describes  E   0  stable states.       (49)  p  0  k  Generally, different initial and boundary conditions will determine different solutions of in which E   and p  k , and they are equations (40) and (41). combined. The local gauge invariances keep as The energy E and momentum p will be abided   ' ei;  'ei (42) by relativistic relation under Lorentz invariance E 2  p2c2  m2c4 (50) A  A'  A    ; A  A'  A    (43) where c is velocity of light. The Eq. (50) links to When the energy distributes bias A completely, following Dirac equation that unified matter and the M looks wave with fitting quantum field F antimatter when operators matching symmetric group U(1) i / t  1   (E, p)    ~ p  i  are applied L   F F  J A (44)  i 4  

This is just Lagrangian of QED, leading to the  (i    m)  0 (51) Maxwell’s equations http://escipub.com/international-journal-of-natural-science-and-reviews/ 0006 Fred Y. Ye, IJNSR, 2017; 1:4 The vortex-based quantum and unified physics wave-particle duality. For macro-things, the contributes a new view of the world, where mass-energy concentrates and keeps mostly in matter realities are discrete quantum states and ψ or A, so that there is no obviously observed space-time is continuous background. The wave-particle duality. When vortex becomes matter existence and evolution will affect the general methodological approach for both curvature and structure of space-time, while micro- and macro-things, the world can be space-time contains matter. The method can naturally and unitedly interpreted. A visual keep concordance of methodology in image of local wave-particle duality can also be processing micro-particle and macro-galaxy [9]. described by relation of ψ and A. When ψ is projection of matter particle to space and A is projection of wave vector to time, they look like PRACTICAL INTERPRETATIONS Fourier-type transformation between ψ and A. Some important practical issues need to be For any normalized vortex M in Hilbert space interpreted under the framework, including and for any operator X on Hilbert space, when wave-particle duality and uncertainty, = and (ΔX)2= -2, there entanglement and teleportation, as well as exists uncertainty relation physical measures of quantum information, 1 particularly energy and . (X )2 (Y)2  ( 2  f 2 ) (52) 22

where f= -2. If f=0, we have Understanding wave-particle duality and Heisenberg uncertainty relation uncertainty  (X )2 (Y)2  ( )2 (53) In a linked-measure M = (ψ, A, B), ψ=φ+iθ is a 2 complex scalar massive function (particle-bias) In the vortex quantum and unified physics, as and A=V+iU looks a complex vector potential the quantum matter-energy is evolutional, the function (wave-bias), while B characterizes curvature of space-time will be also evolutional, rotation (spin). Meanwhile, M is also the following the matter-energy change. As discrete measure of total matter-energy. matter would distribute in the continuous When the matter-energy distributes bias space-time, all look both discrete and   A  M , the M looks like particle but fitting continuous entities.

to wave equation, i.e. Schrödinger equation or Dirac equation. Since ψ and A are particle-bias Charactering quantum entanglement and function and wave-bias function respectively, teleportation the M will look like particle if ψ >> A; while the M Basically, the entanglement can be the will look like wave if ψ << A. As M =(ψ, A, B) entangled quantum pair of right vortex particle becomes unified mass-energy reality for both and left vortex particle. More generally, when micro-particle and macro-cosmos, all things two quantum particles link to same source, such have wave-particle duality. For micro-particle, ~ as M and M linking M, the entanglement may the energy changes easily between ψ and A, so that the micro-particle looks obvious happen. A typical situation looks like Figure 1. http://escipub.com/international-journal-of-natural-science-and-reviews/ 0007 Fred Y. Ye, IJNSR, 2017; 1:4

Fig.1 A typical entangled quantum pair In teleportation, when Alice wants to send Bob (29), or (30), or (31) and so on. After Bob quantum information via classical channel, she receives the information, he can recover an can send two classical items, in which one is ~ entangled quantum state of M and M , as shown portion of (ψ, A, B) as secret key and another is how M does entanglement based on Eq. (28), or as Figure 2.

Fig.2 A mechanism of teleportation

The entanglement emerges between M and , Setting quantum information and and teleportation happens when ones transfer computation quantum information via classical channel. If a macro-system contains many micro-states ~ As M  M , M and are relative. For example, Mi, its entropy S can be defined as Shannon we all see that there are two entangled entropy [17] for non-quantum states and von electrons in molecule hydrogen, but we cannot Neumann entropy for quantum states [14-15, 18] and then the information is negative entropy as differ which electron is M or . When one is M,

~ another is M .

 k M log M ;non  quantum Therefore, both I and S are measures of   i i I  S   i (54) quantum or non-quantum states, while the  k tr(M log M ); quantum   i i linked-measure M = (ψ, A, B) can be applied to  i measure both micro- and macro- physical http://escipub.com/international-journal-of-natural-science-and-reviews/ 0008 Fred Y. Ye, IJNSR, 2017; 1:4 objects. light spiral vortices with double spiral split, for For a quantum channel, it is a communication determining the foundations of vortex-based channel that can transmit both quantum physics, via observing the vortices with their information (such as the state of a qubit) and interactions. classical information (such as a bit). Both In the experiment of water double vortices, it is classical information and quantum information expected to observe the phenomena of can be transmitted by quantum channel, but the strengthened and weakened vortices, as shown capacity of the channel would follow as Figure 3.

Q(t)  C(t) (55) Differentiating from common split flows, the experiment of water spiral vortices could where Q(t) is entangled quantum capacity, contribute the physical foundations of vortex enhanced by entanglement, and C(t) fits to formulation, which might set up real classical Shannon coding theorem. representations of superposition states. It is For non-entangled quantum suggested to verify and correct the Eqs. (18-23) state | M   f ( j) | j , quantum Fourier of the superposition states based on the j experiment. transformation could help quantum computation In the experiment of light double vortices, it is ~ N 1 ~ expected to observe the phenomena of light | M   F | M    f (k) | k (56) k0 vortices’ interference, as shown in Figure 4, with a little difference from normal double split ~ 1 N 1 jk experiment. in which f (k)  exp(2i ) f ( j) and N j0 N Differentiating from common double split N=2n contains n qubits. experiment, the experiment of light spiral The situation means that the non-entanglement vortices could contribute the real physical could help computation while entanglement foundations of quantum formulation, via which might enhance communication. we might find various quantum states and their interactions. Therefore, it is suggested to verify

and correct the various quantum states based EXPERIMENTAL DESIGN: suggestions on the experiment. The vortex-based quantum and unified physics Meanwhile, on the basis of the experimental certainly need experimental foundations, which results, the various states and physical should be shown by both macro- and micro- principles will have their real foundations. Also, phenomena. So the vortex physics would be above vortex formulation of quantum physics founded by two designed fundamental can be corrected based on the experiments. experiments, in which one is water vortex and In future, more experiments can be designed for another is . quantum entanglement and teleportation, as It is suggested to test water spiral vortices and well as quantum information and computation.

http://escipub.com/international-journal-of-natural-science-and-reviews/ 0009 Fred Y. Ye, IJNSR, 2017; 1:4

Fig.3 Designed experiment 1: water spiral vortices

Fig.4 Designed experiment 2: light spiral vortices DISCUSSION: predictions and limitations non-commutation relation. According to above vortex quantum physics, The vortex-based physics also gave an obvious following predictions are waiting for correction answer to the EPR paradox about the quantum 1. Entangled states happen between M world: it is reality, but it is no-locality. ~ Furthermore, setting discrete quantum states and M . The entanglement might enhance into continuous space-time looks the quantum communication by quantum channel. Fourier transform, as if this is a quantum 2. As antimatter is rare in the analogue of the discrete Fourier transform, universe, M  M . So the antimatter which may be used in several quantum seldom exist independently. algorithms and can be efficiently implemented 3. Uncertainty is a necessary result of on a quantum computer using only a polynomial wave-particle duality and number of quantum gates.

http://escipub.com/international-journal-of-natural-science-and-reviews/ 0010 Fred Y. Ye, IJNSR, 2017; 1:4 Actually, vortex looks a really universal with a plane wave of light reveals the spiral phenomenon [19], and there are two terms phase as concentric spirals. The number of related to our discussed topics in physics, one is arms in the spiral equals the topological charge. “quantum vortex”, and another is “optical The knowledge is expected to apply into above vortex”. designed experiment 2. A quantum vortex represents a quantized flux The vortex formulation may contribute a unified circulation of some physical quantity. In most math-physical foundation for quantum physics, cases quantum vortices are a type of on which theoretical progresses are expected. exhibited in superfluids and Also, since 1990s, experimental progresses had superconductors, which were observed been developing, from entangled photo pair [20], experimentally in Type-II superconductors, five- entanglement [21], six-photon liquid , as well as Bose–Einstein entanglement [22], eight-photon entanglement [23], condensate and so on. In a superfluid, a to the experiment by quantum satellite [24], quantum vortex "carries" quantized orbital promoting new findings. It seems really true that angular momentum, thus allowing the superfluid we face the second quantum revolution. The to rotate. In a superconductor, the vortex carries quantum experiments and theories [25] are quantized magnetic flux. The knowledge is pushing quantum physics, quantum information expected to apply into above designed and computation to quicken progresses. experiment 1. At last we also discuss the limitations. As An optical vortex, or a photonic quantum vortex, space-time is continuous and matter realities is a corkscrew of light with darkness at the are discrete, it constructs a basic contradiction. center, which is just quantum vortex in photon There are two possibilities: (1) the space-time is fields. In an optical vortex, light is twisted like a the existed background of matter realities, so all corkscrew around its axis of travel. Because of are included in space-time. In this case, there is the twisting, the light waves at the axis itself no space-time singularity. All realities came cancel each other out. When projected onto a from original matter. (2) the space-time and flat surface, an optical vortex looks like a ring of matter realities generated from “big-bang”, so light, with a dark hole in the center. The optical space-time will be evolutional with matter. In this vortex can be given a number, called the case, there is a space-time singularity. What is topological charge, according to how many truth? twists the light does in one wavelength. The This is an issue concerning the structure and number is always an integer, and can be evolution of space-time. The issue is a key to positive or negative, depending on the direction understand our universe. Also, it is an important of the twist. The higher the number of the twist, topic in physics from ancient to modern age. the faster the light is spinning around the axis. Newton believed absolute space and time, This spinning carries orbital angular momentum where motion takes place against the backdrop with the wave train, and will induce torque on an of a rigid Euclidean reference frame that electric dipole. This orbital angular momentum extends throughout all space and all time, and of light can be observed in the orbiting motion of gravity is mediated by a mysterious force, acting trapped particles. Interfering an optical vortex http://escipub.com/international-journal-of-natural-science-and-reviews/ 0011 Fred Y. Ye, IJNSR, 2017; 1:4 instantaneously across a distance, without Dirac equation. depending on the intervening space. Einstein 3. Quantum entanglement happens denied that there is any background Euclidean ~ between M and M . While entanglement reference frame that extends throughout space, nor is there any such thing as a force of might enhance communication via gravitation, in his general relativity. Now it is well quantum channel, the non-entanglement known that Newtonian gravitation is a theory of could help computation. curved time, and general relativity is a theory of Totally, quantum entanglement and curved time and curved space where the teleportation, matter and antimatter, curvature of space-time is equivalent to wave-particle duality can be united smoothly. gravitation. Two experiments are suggested to correct and Another newfangled concept is time crystal or verify the foundations for setting discrete space-time crystal, which is a structure that quantum states into continuous space-time via repeats periodically in time, as well as in space. vortex. The view contributes a unified A time crystal is a type of non-equilibrium matter, framework for interpreting and understanding which never reaches thermal equilibrium. Now it quantum physics, on which further studies are has been observed [26]. expected. Totally, the issue of space-time keeps both scientific and philosophical open question. Meanwhile, another extended question References concerns information and knowledge. While 1. Styer, D. F.; Balkin, M. S.; Becker, K. M. et al., information is physical and objective [27]. Knowledge is psychological and subjective, as Nine formulations of quantum mechanics. knowledge is different person by person. When American Journal of Physics, 70 (3), 288-297 we set up measurable link between information (2002) and knowledge [28], it is a non-quantum 2. Schrödinger, E. Probability relations between metrics, beyond the scope of discussion here. separated systems. Mathematical Proceedings of the Cambridge Philosophical Society, 32 (3), CONCLUSION 446–452 (1936)

Conclusively, above vortex formulation of 3. Dirac, P.A.M. The Principles of Quantum quantum physics are characterized as follows: Mechanics (4th ed.), Oxford: Oxford University 1. Globally, Hamilton principle keep effective Press (1958) for linked-measure M=(ψ, A, B). The EPR 4. Feynman, R. P. & Hibbs, A. R. Quantum paradox is solved by its reality and Mechanics and Path Integrals. New York: no-locality. McGraw-Hill (1965)

2. Locally, gauge invariances keep effective. 5. Hestenes, D. Spacetime physics with geometric When the mass-energy concentrates on algebra. American Journal of Physics, 71, ψ in a system, the system will be 691-714 (2003)

mastered by Schrödinger equation and 6. Doran, C. J. L. & Lasenby, A. N. Geometric

http://escipub.com/international-journal-of-natural-science-and-reviews/ 0012 Fred Y. Ye, IJNSR, 2017; 1:4 Algebra for Physicists. Cambridge University 17. Shannon, C.E. (1948). A mathematical theory of Press (2003) communication. The Bell System Technical

7. Ye, F. Y. The Linked-measure and linked-field for Journal, 27(3-4): 379-423, 623-656 linking micro-particles to macro-cosmos with 18. Shor, P.W. Equivalence of additivity questions on dispelling dark matter and dark energy. Physical quantum information theory. Communications in Journal, 1, 89-96 (2015) Mathematical Physics, 246, 4334-4340 (2004)

8. Ye, F.Y. The physical linked-measure works as 19. Dominici, L. et al. Vortex and half-vortex dynamics vortex with linking to turbulence. Physical Journal, in a nonlinear spinor quantum fluid. Science 1, 209-215 (2015) Advances, 1(11), e1500807 (2015)

9. Ye, F.Y. A vortex mechanism linking micro-particle 20. Kwiat, P. G. et al. New high-intensity source of to macro-galaxy without supersymmetry. in Fred Y. polarization-entangled photon pairs. Physical Ye. Scientific Metrics: towards analytical and Review Letters, 75, 4337–4341 (1995) quantitative sciences. Springer & Science Press, 21. Zhao, Z. et al. Experimental demonstration of 57-72 (2017) five-photon entanglement and open-destination 10. Penrose, R. The Road to Reality: a complete teleportation. Nature, 430, 54–58 (2004) guide to the laws of the universe. London: 22. Lu, C.-Y. et al. Experimental entanglement of six Jonathan Cape (2004) in graph states. Nature Physics, 3, 91–95 11. Patrignani, C. et al. (Particle Data Group). Review (2007)

of Particle Physics. Chinese Physics C, 40, 23. Yao, X.-C. et al. Observation of eight-photon 100001 (2016) entanglement. Nature Photonics, 6, 225–228 12. Einstein, A.; Podolsky, B. & Rosen, N. Can (2012) quantum-mechanical description of physical 24. Yin, J. et al. Satellite-based entanglement reality be considered complete? Physical Review, distribution over 1200 kilometers. Science, 356, 47 (10), 777–780 (1935). 1140–1144 (2017)

13. Bennett, C.H. et al. Teleporting an unknown 25. Zhang, Y. D. Principles of Quantum Information quantum state via dual classical Physics (Chinese). Beijing: Science Press (2006).

Einstein–Podolsky–Rosen channels. Physical 26. Zhang, J. et al. Observation of a discrete time Review Letters, 70, 1895–1899 (1993) crystal. Nature, 543, 217–220 (2017).

14. Bennett, C.H. & Di Vincenzo, D.P. Quantum 27. Laudauer, R. Information is physical. Physics information and computation. Nature, 404, 247 Today, 44, 5-23 (1991). (2000) 28. Ye, F. Y. Measuring knowledge: a quantitative 15. Bennett, C.H. & Shor, P. W. Quantum information approach to knowledge theory. International theory. IEEE Transactions on Information Theory, Journal of Data Science and Analysis, 2(2), 32-35 44, 2724-2742 (1998) (2016). 16. Zhong, Z-Z. Generation of new solutions of the stationary axisymmetric Einstein equations by a double complex function method. Journal of Mathematical Physics, 26(10): 2589-2595 (1985).

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