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 Physics 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 (wave 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 angular momentum 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.