Kelvin Waves and Dynamic Knots on Vortex-Membrane

Quantum mechanics is a successful theory that describes the behavior of photons, electrons, and other atomic- and molecular-scale objects. However, it is far from being well understood. In this paper, a new theory - knot physics for entangled vortex-membranes is developed to explore the underline physics of quantum mechanics. From point view of information, three dimensional quantum Dirac model is developed to effectively describe the fluctuations of local deformation of vortex-membranes: The elementary excitations are knots with information unit; The physics quality to describe local deformation is knot density that is defined by the density of zeros between two projected vortex-membranes; The Biot-Savart equation for Kelvin waves becomes Schrodinger equation for probability waves of knots; the classical functions for perturbative Kelvin waves become wave functions for knots; The probability of quantum mechanics comes from dynamic projection for inner observers owing to"fast clock"effect; The angular frequency for leapfrogging motion plays the role of the mass of knots; etc. This work would help people to understand the mystery of quantum mechanics.


I. INTRODUCTION
One hundred years ago, Kelvin (Sir W. Thomson) studied the physical properties of vortex-lines that consist of the rotating motion of fluid around a common centerline. In an incompressible fluid, the vorticity of vortex-lines manifests itself in the circulation v · dl = κ where κ is a constant. For classical hydrodynamic vortex-lines, Kelvin found a transverse and circularly polarized wave[1] (called Kelvin wave), in which the vortex-lines twist around their equilibrium position forming a helical structure [2]. For two vortex-lines, owing to the nonlocal interaction, the leapfrogging motion has been predicted in classical fluids from the works of Helmholtz and Kelvin [3][4][5][6][7]. Owing to Kolmogorov-like turbulence [8,9], a variety of approaches have been used to study this phenomenon. In experiments, Kelvin waves has been observed in uniformly rotating 4 He superfluid (SF) [10] and Bose-Einstein condensates [11].
In addition, Kelvin and Toit tried to develop an early atomic theory that is linked to the existence and dynamics of knotted vortex-rings in ether. However, they failed -the fundamental structure of atoms is irrelevant to knots. Today, the failure reason is clear. The elementary particles in our universe are not classical knots but quantum objects obeying quantum mechanics and Einstein's relativity. Quantum mechanics (also known as quantum physics or quantum theory) is a fundamental branch of modern physics that was proposed from Planck's solution in 1900 to the black-body radiation problem and Einstein's 1905 paper which offered a quantum-based theory to explain the photoelectric effect. After several decades, quantum mechanics is established and becomes a successful theory that agrees very well with experiments and provides an accurate description of the dynamic behavior of microcosmic objects. In quantum mechanics, the energy is quantized and can only change by discrete amounts, i.e. E = ℏω where ℏ is Planck constant. There are several fundamental principles in quantum mechanics: wave-particle duality, uncertainty principle, and superposition principle. The Schrödinger equation describes how wave-functions evolve, playing a role similar to Newton's second law in classical mechanics. In quantum mechanics, when considering special relativity, the Schrödinger equation is replaced by the Dirac equation.
However, quantum mechanics is far from being well understood. There are a lot of unsolved mysteries in quantum mechanics including quantum entanglement problem and quantum measurement problem. Einstein said, "Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He does not throw dice." The exploration of the underlying physics of quantum mechanics is going on since its establishment [12]. There are a lot of attempts, such as De Broglie's pivot-wave theory [13], the Bohmian mechanics [14], the manyworld theory [15], the Nelsonian Mechanics [16] and the idea of primary state diffusion [17]. However, for all these interpretations of quantum mechanics, people focus on the issue of quantum motion but miss another important issue, what is matter (or reality)?
In this paper, we develop a new theory towards understanding quantum mechanics based on three-dimensional (3D) leapfrogging vortex-membranes in five-dimensional (5D) incompressible fluid. We call the new theory -knot physics. According to knot physics, it is the 3D quantum Dirac model that characterizes the knot dynamics of leapfrogging vortex-membranes. The knot physics gives a complete interpretation on quantum mechanics. Fig.1 is an illustration of the framework structure of knot physics.
The paper is organized as below. In Sec. II, we review Biot-Savart mechanics for 3D vortex-membrane in a 5D incompressible fluid. In Sec. III, the Kelvin waves of 3D vortex-membrane are studied. In this section, the Biot-Savart mechanics for helical vortex-membranes is mapped onto "pseudo-quantum mechanics" for a particle, by which we apply to the calculation of the evolution of a deformed helical vortex-membrane. In Sec. IV, we develop the Biot-Savart mechanics for entangled vortex-membranes. In addition, we introduce tensor representation to describe the Kelvin waves. In Sec. V, we develop an emergent quantum mechanics to describe the knot dynamics on knot-crystal. In particular, a 3D massive Dirac model is obtained to characterize the knot dynamics on the entangled vortexmembranes. We also address the measurement theory and quantum entanglement of emergent quantum mechanics. The emergent quantum mechanics help us understand the mysteries of quantum mechanics. Finally, the conclusions are drawn in Sec. VI.

A. The Euler equation on vorticity
In this paper, we would study the vortex in an inviscid incompressible fluid. In condensed matter physics, we may consider superfluid (SF) as an inviscid incompressible fluid. For a divergence-free inviscid incompressible fluid (∇ · v = 0), the fluid motion is described by the classical Euler equation: where v is the fluid velocity field, ρ is the uniform density and p is the pressure function. ∇ v v is Riemannian covariant derivative of the field v in the direction of itself. On the other hand, to characterize vorticity, the Helmholtz form of the Euler equation is written as where L v is the Lie derivative along the velocity field v and Ω = ∇ × v is the vorticity field.
Vortices are extended objects with singular vorticity in an inviscid incompressible fluid that can be regarded as a closed oriented embedded subvortex-membrane with Marsden-Weinstein (MW) symplectic structure. For example, for two dimensional (2D) inviscid incompressible fluid, we have 0-dimensional point-vortices; For 3D case, we have one dimensional (1D) vortex-lines; For 4D case, we have 2D vortex-surfaces; For 5D case, we have 3D vortex-membranes. See the illustration of vortex-line in 3D space in Fig.2. In this paper, we focus on 3D vortex-membranes in the 5D inviscid incompressible fluid.
where the complex value z j = ξ j + iη j denotes the position of the j-th point-vortex and κ j denotes the constant vorticity of j-th point-vortex. So we have a topological condition with loop integral along a closed path C. According to Kirchhoff's theorem, the Euler equation for point-vortices is where H Kirchhoff is the Kirchhoff Hamiltonian given by The Kirchhoff Hamiltonian is really potential energy that characterizes the long range interaction between two pointvortices: for two vortices with vorticities of same signs, we have repulsive interaction; For two vortices with vorticities of opposite signs, we have attractive interaction. In addition, above Euler equation can be changed in term of Poisson bracket {, } = N j=1 1 κj ( ∂ ∂ξj ∂ ∂ηj − ∂ ∂ηj ∂ ∂ξj ) to be [18,19] ∂ t Ω = {H Kirchhoff , Ω}.

Vortex-lines in 3D inviscid incompressible fluid
Secondly, we review the dynamics of vortex-lines in 3D inviscid incompressible fluid. a. Biot-Savart equation vortex-lines are 1D topological objects in 3D inviscid incompressible fluid with ∇ · v ≡ 0. For a fluid with a vortex-line, a rotationless superfluid component flow ∇ × v = 0 is violated on 1D singularities s(ζ, t), which depends on the variables -arc length ζ and the time t. Away from the singularities, the velocity increases to infinity so that the circulation of the fluid velocity remains constant, v · dl = κ (9) where v =ṙ. As a result, the vortex filament could be described by the Biot-Savart equatioṅ where κ is the circulation and s is the vector that denotes the position of vortex filament. For a 3D superfluid, κ = h/m is the discreteness of the circulation owing to its quantum nature. h is Planck constant and m is particle mass of SF. b. Impulse and angular impulse In fluid with a vortex-line, the conventional momentum of the fluid motion cannot be well defined. Instead, the hydrodynamic impulse (the Lamb impulse) plays the role of the effective momentum that denotes the total mechanical impulse of the non-conservative body force applied to a limited fluid volume to generate instantaneously from rest the given motion of the whole of the fluid at time t. In general, the (effective) momentum for a vortex-line from Lamb impulse density is defined by where dV 3D is an infinitesimal volume in 3D fluid and ρ 0 is the superfluid mass density. For a vortex-line with a global velocity U, we have δH − U · δP Lamb = 0.
As a result, the Lamb impulse becomes the right physical quantity describing the momentum of the vortex filament. By using the definition of a vortex-line [20][21][22][23][24]] we have the (Lamb impulse) momentum and the (Lamb impulse) angular momentum for a vortex-line It is obvious that P Lamb and J Lamb are conserved quantities, i.e., dP Lamb dt ≡ 0 (16) and c. Geometric Biot-Savart mechanics for vortex-lines with 1D Marsden-Weinstein symplectic structure under localized induction approximation Localized induction approximation (LIA) of the vorticity motion is to keep the local terms in the vorticity Euler equation. The Biot-Savart mechanics under localized induction approximation for 1D vortex-lines is reduced to a special classical mechanics -geometric Biot-Savart mechanics for geometric objects with 1D Marsden-Weinstein symplectic structure. The corresponding evolution becomes the vortex filament equation where ǫ = ℓ a0 , ℓ is the length of the order of the curvature radius (or inter-vortex distance when the considered vortex filament is a part of a vortex tangle) and a 0 denotes the vortex filament radius which is much smaller than any other characteristic size in the system. For an arc-length parametrization, the tangent vectors t = ∂s/∂ζ = s ′ have unit length and the acceleration vectors are s ′′ = ∂t/∂ζ = κ · n, i.e., ∂ t s = ( κ 4π ln ǫ)(s ′ × s ′′ ) becomes where κ and b = t × n denote the curvature value and bi-normal unit vector of the curve s at the corresponding point, respectively. See the illustration in Fig.2.
We introduce a complex description on the vortex-line, z = ξ + iη = |z| e iφ where |z| is the amplitude and φ is the angle in the complex plane. Under LIA [25] and a simple geometrical constraint z ′ ≪ 1, in terms of the complex canonical coordinate z(x), the Biot-Savart equation becomes [8] When the vortex-line is rotating dz dt = 0, it becomes longer, ∆H length (z) = 0.

3D vortex-membranes in 5D inviscid incompressible fluid
Thirdly, we develop the physics of 3D vortex-membranes in 5D inviscid incompressible fluid. The Biot-Savart mechanics under localized induction approximation for 3D vortex-membranes is reduced to 3D geometric Biot-Savart mechanics for geometric objects with 3D Marsden-Weinstein symplectic structure.
For 5D case, we have 3D vortex-membranes with MW symplectic structure [19]. The 3D vortex-membrane is defined by a given singular vorticity where the singular δ-type vorticity denotes the sub-manifold P in 5D space, and κ is the constant circulation strength. a. Generalized Biot-Savart equation in 5D fluid A generalized Biot-Savart equation in 5D fluid is given by [19] v where Proj N A is the orthogonal projection of A to the fiber N pP of the normal bundle to P at p ∈ P , and the operatorĴ is the positive rotation around p by π/2 in extra dimensional space N pP . G(q, p) = α |q−p| −3 is the Green's function for the Laplace operator, i.e.
b. Impulse and angular impulse We then consider the (effective) momentum for a vortex-membrane from Lamb impulse density Here, "[×] 5D " denotes positive rotation around p after vector product in 5D space. On the other hand, the (Lamb impulse) angular momentum for a vortex-membrane is defined by P Lamb and J Lamb are also conserved quantities, i.e., dP Lamb dt ≡ 0 and dJ Lamb dt ≡ 0. c. Geometric Biot-Savart mechanics for vortex-membranes with 3D Marsden-Weinstein symplectic structure under localized induction approximation Under LIA, the generalized Biot-Savart equation for a 3D vortex-piece is reduced into [19] v(q) ≃ (κα ln ǫ) · J(M C (p)) (29) where M C (p) is the mean curvature vector to P at the point p. The mean curvature vector M C (p) ∈ N pP is the mean value of the curvature vectors of geodesics in P passing through the point p when we average over the sphere S 4 of all possible unit tangent vectors in T pP for these geodesics. Thus, the generalized Biot-Savart equation under LIA is given by the skew-mean-curvature flow According to the fact that the mean curvature vector field is the gradient to the volume functional, the generalized Biot-Savart equation for a 3D vortex-piece under LIA can be described by Hamiltonian formula and the Hamiltonian on the vortex-membranes is just 3-volume H volume (P ) = (κα ln ǫ) · volume(P ) (31) with volume(P ) = P dV P . In term of the Hamiltonian H(P ) for 3D vortex-membrane, the generalized Biot-Savart equation becomes

FIG. 3: Comparision between classical mechanics for mass-point and (geometric) Biot-Savart mechanics for vortices
The volume of the subvortex-membrane P becomes a conserved quantity that plays the role of Hamiltonian function of the corresponding dynamics. In addition, the generalized Biot-Savart equation becomeṡ In complex representation, z = ξ + iη, Eq.(33) can also be written into [19] i dz dt = δH volume (P ) δz * .
In particular, we emphasize that the evolution equation of the subvortex-membrane P is really mechanics of geometric objects -the Hamiltonian H volume (P ) is 3-volume volume(P ) = P dV P that is a geometric quantity and the dynamic variables (ξ(t), η(t)) or z(t) denote the position in extra dimensions that are also geometric quantities. The generalized Biot-Savart equation is to minimize the 3-volume volume(P ) = P dV P . As a result, the vortex-membranes always do the skew-mean-curvature flow that differs by the π/2-rotation from the mean-curvature vector. The skew-mean-curvature flow does not stretch the subvortex-membrane while moving its points orthogonally to the mean curvatures. In Fig.3, we compare Newton mechanics for mass-point with (geometric) Biot-Savart mechanics for vortices.

III. BIOT-SAVART MECHANICS FOR A VORTEX-MEMBRANE: PSEUDO-QUANTUM MECHANICS FOR SINGLE PARTICLE
The issue of Biot-Savart mechanics for a vortex-membrane is about the dynamic evolution of windings for a vortexmembrane. We may ask a question -how to characterize the evolution of a non-uniform helical vortex-membrane? For example, a non-uniform helical vortex-line is shown in Fig.4.(b). We point out that the corresponding Biot-Savart mechanics is reduced to pseudo-quantum mechanics for single particle. Here, "pseudo" indicates that this mechanics just has similar structure to the traditional quantum mechanics but is different. For 1D vortex-lines, there exists Kelvin wave, a transverse and circularly polarized wave, in which the vortex-line twists around its equilibrium position forming a helical structure. The plane-wave ansatz of Kelvin waves is described by a complex field [25,26], where r 0 is the winding radius of vortex-line, k = π a > 0 and a is a fixed length that denotes the half pitch of the windings. φ 0 is a constant angle. ± denotes two possible winding directions: left-hand with clockwise winding, or right-hand with counterclockwise winding. For Kelvin waves on vortex-lines, under LIA, the dispersion is obtained as where ǫ ∼ 1 ka0 . In mathematics, we can generate a Kelvin wave by an operatorÛ(x, t) on a flat vortex-line whereÛ Here, z 0 = 0 denotes a vortex-membrane with is an expanding operator by shifting radius from 0 to r 0 on the membrane, i.e.,̥(r 0 ) · 0 = r 0 . So ̥ (r 0 ) −1 is a shrinking operator by shifting radius from r 0 to 0 on the membrane, i.e., We then calculate the physical quantities of the Kelvin waves along winding direction (x-direction), i.e., the projected angular momentum J x and the projected momentum p x . The projected (Lamb impulse) momentum along the xdirection p x of a vortex-line with a plane Kelvin wave is obtained as [26] that leads to the projected (Lamb impulse) momentum density ρ px = 1 2 ρ 0 κr 2 0 · k. l is the length of the system in x-direction. The projected (Lamb impulse) angular momentum along x-direction of a vortex-line with a plane Kelvin wave is given by [26] that leads to the projected (Lamb impulse) angular momentum density along x-direction ρ Jx = Jx l = ρ 0 κ r 2 0 2 . Here, we use a mathematic result, with a (uniform or non-uniform) helical curve on a cylinder with volume V cylinder = r 2 0 · l (the symmetric axis is along x-direction, the radius of cross-section is r, the length along x-direction is l).
For Kelvin waves on a 3D helical vortex-membrane, the plane-wave ansatz is described by a complex field, where k is the winding wave vector on 3D vortex-membrane with k = π a and a is a fixed length that denotes the half pitch of the windings.
For the plane Kelvin waves on a 3D helical vortex-membrane, we have the Hamiltonian H volume (P ) = (κα ln ǫ)volume(P ) = (κα ln ǫ) where where V p is the total volume of the vortex-membrane. The (Lamb impulse) angular momentum of a plane Kelvin wave alongẽ-direction is given by The projected (Lamb impulse) angular momentum is a constant on the vortex-membrane.

B. Mapping to pseudo-quantum mechanics
We then map the Biot-Savart mechanics for a (constraint) helical vortex-membrane to a pseudo-quantum mechanics. Firstly, we define the state-vector z k ( x, t) to denote the helical vortex-membrane as With the help of the operatorÛ( x, t) = e i [φ( x,t)·K]dx ·̥(r 0 ), the Kelvin waves can also be generated from a straight one, The Hilbert space of pseudo-quantum mechanics for a helical vortex-membrane consists of different states z k ( x, t). We use the state vector k to denote the state of Kelvin waves of vortex-membranes and the state vector k to denote the state of knots in the following parts. In general, under LIA, according to superposition principle, we construct the Kelvin wave as where r k is the radius of a partial wave exp(−iω · t + i k · x). There exists a normalized volume condition, We may consider the value V P · r 2 0 to be the fixed volume of the system. For example, a 1D standing Kelvin wave denoted by 1 is a superposition Kelvin wave of two plane Kelvin waves with opposite wave vectors For a plane wave, z k ( x, t) = r 0 exp(−iω · t + i k · x), the (Lamb impulse) momentum is where the (Lamb impulse) angular momentum J Lamb is obtained as the effective Planck constant eff , The effective Planck constant is proportional to the total volume of the system. The effective energy of a Kelvin wave is As a result, we define the "wave-function" of a plane Kelvin wave as and a generalized constraint Kelvin waves as where a k = 1 VP r 2 0 r k .
For the average value of the effective energy E Lamb , we have This result indicates that the effective energy becomes an operator Using the similar approach, we derive As a result, the (Lamb impulse) momentum also becomes an operator From the dispersion of Kelvin waves, we have The energy-momentum relationship E Lamb = H( p Lamb ) = (ακ ln ǫ) 2 eff p 2 Lamb determines the equation of motion for vortexmembranes that becomes the "Schrödinger equation", with an effective mass For the eigenstate with eigenvalue E Lamb , the wave-function is given by where f ( x) is spatial function. This corresponds to a twisting motion with fixed twisting angular velocity, ω = E Lamb eff . That means the excitations of the quantum state must have the quantized projected (Lamb impulse) energy, where n is a positive integer number.
In pseudo-quantum mechanics, there are three conserved physical quantities for helical vortex-membranes: the energy H( p Lamb ) that is proportional to the volume of the vortex-membrane V P = volume(P ); the momentum p Lamb that is proportional to the winding number along given direction w 1D (see below discussion); and the (Lamb impulse) angular momentum (the effective Planck constant eff ) that is proportional to the volume of the vortex-membrane in the 5D fluid V P · r 2 0 . As a result, the pseudo-quantum mechanics describes the dynamics of a vortex-membrane with fixed volume in the 5D fluid. Under the constraint, the total degree of freedom of a vortex-membrane is reduced to 1. In Fig.5, we show the comparison between pseudo-quantum mechanics for helical vortex-membrane and quantum mechanics for single quantum particle. The quantization of (constraint) Kelvin waves is similar to quantization of a matter wave in quantum mechanics: a constant J Lamb plays the role of the Planck constant eff , m pseudo plays the role of mass in Schrödinger equation, and so on. However, they are different. The Kelvin waves are classical waves that obey deterministic classical mechanics, not probabilistic quantum mechanics. This is why we call it pseudo-quantum mechanics. In the following parts, we will show that the knots rather than vortex-membranes obey true quantum mechanics.

C. Winding number and winding-number density
In above part, we derive the equation of motion of Kelvin waves. In principle, the evolution of the system can be solved. We then introduce winding number and winding-number density to describe the deformed helical vortexmembrane.
The winding number w 1D of two 1D vortex-lines is defined as To locally characterize the winding behavior, we define the winding-number density ρ wind , ρ wind = ∆w1D ∆x . For example, for a vortex-line described by a plane Kelvin wave z(x, t) = r 0 e −iω·t+ik·x , the winding number w 1D = k 2π l is proportional to the length of the system l along the Kelvin wave and the density of winding number ρ wind is uniform, i.e., ρ wind = w1D l = k 2π . From this equation, the momentum p Lamb is proportional of the winding number w 1D .
According to pseudo-quantum mechanics, the local winding can be described by the density operator of winding number, ρ wind →ρ wind =k 2π (68) For a vortex-line described by an arbitrary plane Kelvin wave z(x, t) = k c k e −iω·t+ik·x , the total 1D winding number is obtained by the following equation, The winding-number density becomes For example, a local winding at x = x 0 is defined by ρ wind (x, t) = δ(x − x 0 ) that corresponds to a sharp changing of vortex-line; for the case of z(x, t) = constant, there is no winding at all ρ wind (x, t) = 0. In 5D space, we also define the winding number and the winding-number density for a helical vortex-membrane along different directions.
Along the given direction e I , winding number of the vortex-membrane (r 0 is a constant) described by plane Kelvin wave z(x I , t) = r 0 e −iω·t+ik I ·x I is well defined. Because an arbitrary Kelvin wave can be described by z( x, t) = k c k e −iω·t+i k· x , we can use a vector of winding-number density to describe local winding of a vortex-membrane z( x, t). So In 3D space, there are three winding-numbers w I 1D (I = X, Y, Z) along different directions as where C I denotes the closed path around a flat membrane along different directions. For a vortex-membrane described by a plane Kelvin wave z( x, t) = r 0 e −iω·t+i k· x , the winding number along direction e I is given by that is proportional to the length of the system L I along e I -direction and the winding-number density is ρ I wind = k I 2π . Thus, the local winding along direction e I can be described by the operator of winding-number density, whereρ wind =ρ wind · e I . With the help ofρ wind , the winding number of a vortex-membrane is obtained as The vector of winding-number density becomes Therefore, we answer the question -how to characterize the evolution of a non-uniform helical vortex-membrane? We consider a deformed vortex-membrane z( x) as the initial condition. We expand z( x) by the eigenstates of Kelvin-waves as z( x) = k c k e i k· x . Under time evolution, the function of vortex-membranes becomes z( x, t) = k c k e i k· x e −iω·t . Finally, the vector of winding-number density for the deformed vortex-membrane is obtained as We emphasize that under time evolution, the energy H( p Lamb ), the momentum p Lamb (the total winding number along given direction w 1D ) and the (Lamb impulse) angular momentum (the effective Planck constant eff ) are all conserved.
In particular, we have the topological representation on action S, i.e., From the point view of topology, we explain the Sommerfeld quantization condition where h eff = 2π eff and n is an integer number. The Sommerfeld quantization condition is just the topological condition of the winding number, i.e., where w 1D is the winding number of the helical vortex-membrane.

D. Path integral formulation for winding evolution
In 1948, Feynman derived path integral formulation of quantum mechanics, based on the fact that the propagator can be written as a sum over all possible paths between the initial point and the final point. With the help of the path integral, people can calculate the probability amplitude K( x ′ , t f ; x, t i ) from an initial position x at time t = t i (that is described by a state |t i , x ) to position x ′ at a later time t = t f (|t f , x ′ ).
Using similar approach, we derive the path integral formulation for pseudo-quantum mechanics. A state [t, x in pseudo-quantum mechanics denotes a local winding at point x and time t. Letting the state evolve with time and project on the state [ x ′ , we have the transition amplitude of the process as Lamb 2m pseudo Each path contributes e iSn/ eff where S n is the n-th classical action i.e., n e iSn/ eff . Let us discuss the implication of path-integral formulation in pseudo-quantum mechanics. We firstly consider a state with a local winding at the position ( x i , φ 0 ) and time t = t i where x i denotes the original position and φ 0 the phase angle. To calculate the transition amplitude K( x ′ , t f ; x, t i ) = t f , x ′ ] [t i , x , the winding splits into pieces. For a winding-piece, the action is denoted by S n for an arbitrary possible classical path from ( x i , φ 0 ) to ( x f , φ n ) within time T = t f − t i . In particular, the phase angle φ may be different for a winding-piece moving along different classical pathes. The phase-changing of a possible classical path is given by Because the path may be not closed, Sn eff may be not an integer number. As a result, we have the transition amplitude as e iSn/ eff . We summarize the contribution from all windings and get the final transition amplitude as n e i∆φn = n e iSn/ eff (81) that is just the Feynman's path integral formulation. In summary, pseudo-quantum mechanics becomes a toy mechanics to learn the mechanism of true quantum mechanics.

IV. BIOT-SAVART MECHANICS FOR TWO ENTANGLED VORTEX-MEMBRANES: PSEUDO-QUANTUM MECHANICS FOR A TWO-COMPONENT PARTICLE
In this section, we discuss the Biot-Savart mechanics for two entangled vortex-membranes. The issue of Biot-Savart mechanics for two entangled vortex-membranes is about the dynamic evolution of entanglement between them. Here, we ask a question -how to characterize the evolution of non-uniform entangled vortex-membranes? For example, a non-uniform entangled vortex-line is shown in Fig.6.(b). We point out that to characterize the two entangled vortexmembranes, the corresponding Biot-Savart mechanics is mapped to pseudo-quantum mechanics for a two-component particle.

A. Biot-Savart mechanics for two entangled vortex-membranes with leapfrogging motion
In this part, we discuss the Biot-Savart mechanics for two entangled vortex-membranes. In five dimensional space { x, ξ, η} ( x = (x, y, z)), we introduce a complex description on ξ-η complex plane, ≪ 1, the generalized Biot-Savart equation can be written into where the Hamiltonian is given by [27] H = Kinetic term + interaction term Here r 0 is the inter-vortex distance and a 0 denotes the vortex filament radius, i.e., r 0 ≫ a 0 . Because the vortex-membranes are almost straight, the nonlocal interactions, varying on vortex-membranes, are approximated to be similar to those of two straight vortex-membranes that is described by Kirchhoff Hamiltonian, H Kirchhoff . Now, the Biot-Savart equation of motion for each vortex-membrane is given by [27] i where By introducing z A = zu+zv 2 and z B = zv−zu 2 , above equations are transform into [27] ∂z u ∂t = i (ακ ln ǫ) 2 The plane Kelvin wave solutions are obtained as where φ u and φ v are constant phase angles, and the angular frequencies are and In this paper, we set φ u = 0 and φ v = 0. For two entangled vortex-membranes, the nonlocal interaction leads to leapfrogging motion. Above solutions of the Biot-Savart equation can be written into where the winding radii of two vortex-membranes are r A = r 0 cos( ω * t 2 ) and denote (global) rotating velocity frequency and (internal) leapfrogging angular frequency, respectively. From above solutions, we have a constraint condition of total volume, i.e., For global rotating motion with finite ω 0 , the entangled vortex-membranes are described by plane waves e i k· x−iω0t . For leapfrogging motion with finite ω * , the entangled vortex-membranes exchange energy in a periodic fashion. The winding radii of two vortex-membranes oscillate with a period T = 2π ω * : At t = 0, helical vortex-membrane-A winds around straight vortex-membrane-B clockwise (the state [A ); At t = T 4 , the system becomes a symmetric double helix vortex-membranes (the state During leapfrogging process, the Lamb impulse (momentum) and the angular Lamb impulse (angular momentum) are conserved. The effective Planck constant is obtained as projected (Lamb impulse) angular momentum as eff = J Lamb = 1 2 ρ 0 κr 2 0 · V P where V P is the total volume of the system. We then map the Biot-Savart mechanics to the pseudo-quantum mechanics. The system with two entangled vortexmembranes is mapped to a two-component particle (not a two-particle case), i.e., Here, τ x = ( 0 1 1 0 ) is Pauli matrix and 1 = ( 1 0 0 1 ). The leapfrogging process is characterized by eff ω * 2 · (τ x − 1). As a result, the pseudo-quantum mechanics describes the dynamics of two entangled vortex-membranes with fixed volume in the 5D fluid. Under the constraint, the total degree of freedom of a vortex-membrane is reduced to 2.
In pseudo-quantum mechanics, there are also three conserved physical quantities for entangled vortex-membranes: the total energy H( p Lamb ) that is proportional to the total volume of the two vortex-membranes V P = volume(P ); the total momentum p Lamb that is proportional to the linking number between two vortex-membranes (see below discussion); and the total (Lamb impulse) angular momentum (the effective Planck constant eff ) that is proportional to the total volume of the two vortex-membranes in the 5D fluid V P · r 2 0 .

B. Linking number and linking-number density
In above part, we derive the equation of motion of two entangled vortex-membranes. However, due to the leapfrogging process, the winding number and winding-number density are not well defined. Instead, it is the linking number and linking-number density that characterize the entanglement between two vortex-membranes.
Firstly, we introduce the linking number to characterize entanglement between two vortex-membranes. The (Gauss) linking-number ζ 1D for two 1D vortex-lines is a topological invariable to characterize the entanglement that is defined as [28] The 1D linking number is also integer number. The system with different linking numbers has different entanglement patterns between two vortex-lines.
To locally characterize the entanglement, we define the density of linking-number ρ link , For the entangled vortex-membranes, owing to the intrinsic reversal symmetry between vortex-membrane-A and vortex-membrane-B, we can obtain the linking-number density from the winding-number density at special time of the leapfrogging motion, t = 0, i.e., At this time, the winding-number density of vortex-membrane-A is ρ wind,A = k 2π . The linking-number density is equal to ρ wind,A , Because the linking number is a conserved quantity and doesn't change during leapfrogging motion, the corresponding operator of linking-number density is given by For two entangled 1D vortex-line, the linking-number is defined by . Thus, the linking-number density that is proportional to the total momentum p Lamb . is In 5D space, the 2D Gauss linking number can only be defined between two 2D closed, oriented, submanifolds given by [29] where Ω k,ℓ (α) = π θ=α sin k (θ − α) sin ℓ θ dθ. Here α(x, y) is the angle between x ∈ X and y ∈ Y , thought of as vectors in 5D space. So there is no 3D linking number to characterize the entanglement between two 3D vortex-membranes in 5D fluid.
Instead, we discuss entanglement between two 3D vortex-membranes via (projected) 1D linking number. There are three 1D linking-numbers ζ I 1D (I = X, Y, Z) along different directions that are where s I = r · e I . The vector of linking number is defined as . We define the vector of linking-number densities and linking-number density operators for 3D vortex-membranes, respectively. For a vortex-membrane described by a plane Kelvin wave z A (x I , t) = r 0 e −iω·t+ik I ·x I and a constant vortex-line described by z B (x I , t) ≡ 0, the 1D linking-number along direction e I is given by ζ I 1D = |k I | 2π L I and the density of linking-number ρ link is ρ I link = ζ I

1D
LI . As a result, we answer the question -how to characterize the evolution of non-uniform entangled vortex-membranes? For a given initial state Z( x), the linking-number density for the local deformation of entangled vortex-membranes is obtained as ρ I linking ( x, t) = 1 From the point view of topology, the Sommerfeld quantization condition for two-component particle is the topological condition of the linking number, i.e., where ζ 1D is the linking number of the two entangled vortex-membranes.
C. Tensor representation for entangled vortex-membranes

Tensor states for plane Kelvin waves
In this part, we introduce the tensor representation for Kelvin waves by separating the wave vector along a given direction into positive part and negative part, ± k.
For a plane Kelvin wave with fixed wavelength 2a = 2π | k| along e-direction, there are two helical degrees of freedom: z k ( x, t) at k or z − k ( x, t) at − k. After considering the vortex degrees of freedom A or B that characterize the Kelvin waves on different vortex-membranes, there are four degenerate states to characterize a plane Kelvin wave with fixed wave-length 2a, i.e., the basis is Here, k, A denotes a plane Kelvin-wave with wave vector k on vortex-membrane-A, k, B denotes a plane Kelvin-wave with wave vector k on vortex-membrane-B, − k, A denotes a plane Kelvin-wave with wave vector − k on vortex-membrane-A, See the illustration in Fig.7. Thus, an arbitrary plane Kelvin wave can be characterized by a superposed state of the basis along different spatial directions. We introduce a tensor representation to define the plane Kelvin waves with fixed wave-length, where I denotes the spatial indices, X, Y , Z, ...; i = 1, −1 denotes helical degrees of freedom k or − k; j = 1, −1 denotes the vortex-degrees of freedom A or B. In general, there are d × 2 × 2 elements α I i,j . Each element α I i,j is proportional to the winding radius of the corresponding plane Kelvin waves. According to the geometric constraint condition, we have So the resulting function for a Kelvin wave with fixed wave-length is given by Owing to the fact of the energy degeneracy, all superposed states that consist of four degenerate states along different spatial directions denoted by different weights α I i,j have the same energy.

Tensor operators and tensor order
The tensor states of a plane Kelvin wave can be classified by operator representation. We define the d × 3 × 4 tensor operatorsΓ I i,j = σ I i ⊗ τ I j (I = X, Y, Z, ...; i = x, y, z for helical degrees of freedom; j = 0, x, y, z for the vortex-degrees of freedom) to bê whereΓ I = σ I ⊗ (τ I , 1) and σ I , τ I are 2 × 2 Pauli matrices for helical and vortex degrees of freedom, respectively. On the basis of pseudo-spin base (τ I x , τ I y , τ I z , 1), the tensor states are determined by the "direction" of pseudo-spin order for helical degrees of freedom, In general, we have n I τ 2 + |τ | 2 = 1. On the basis of pseudo-spin base (σ I x , σ I y , σ I z ), the tensor states are determined by the "direction" of pseudo-spin order for helical degrees of freedom, In general, we have n I σ 2 = 1. For example, n X σ = (1, 0, 0) is a tensor state along spatial x-direction and helical x-direction, σ X x ⊗ 1 = 1, σ X y ⊗ 1 = 0, σ X z ⊗ 1 = 0. Because the vortex-degrees of freedom are always trivial, the vortex-vector n I τ is a constant vector along different directions, i.e., n X τ = n Y τ = n Z τ = n τ and τ = τ 0 . Therefore, in the following parts, we focus on the tensor states n I σ for helical degrees of freedom.

Example
For a 1D travelling Kelvin wave or of which the tensor state is denoted by We call it plane σ z -Kelvin wave.
For a standing Kelvin wave described by , the tensor state is represented by or of which the tensor state is denoted by We call it σ x -Kelvin wave. Next, we discuss the 2D plane Kelvin waves.
For a 2D σ z -Kelvin wave described by , the tensor state can be represented by The tensor state is denoted by Another 2D plane Kelvin wave with fixed wave-length is standing Kelvin-wave. Along x-direction, the function of plane Kelvin wave becomes along y-direction, the function of the plane Kelvin wave becomes The tensor state is represented by The tensor state is denoted by For the plane Kelvin wave along x-direction, we have, For the plane Kelvin wave along y-direction, we have, Thirdly, we discuss 3D plane Kelvin waves with fixed wave-length.
For a 3D plane σ z -Kelvin waves with fixed wave-length described by , the tensor state is represented by or of which the tensor state is also denoted by Another 3D plane Kelvin waves with fixed wave-length is described by the following tensor state, Along x-direction, the function of the plane Kelvin wave becomes along y-direction, the function of the plane Kelvin wave becomes along z-direction, the function of the plane Kelvin wave becomes For the tensor state along x-direction, we have, For the tensor state along y-direction, we have, For the tensor state along z-direction, we have,

Generation operator
In this part, we define generator operators for Kelvin waves of different tensor states. Firstly, we generate the four degenerate states along a given direction k, A , k, B , − k, A , − k, B by four operators where̥(r 0 ) is an expanding operator by shifting radius from 0 to r 0 on the membrane, For different Kelvin waves with fixed wave-length, the generation operators can be changed from a basis  S = IŜ I withŜ I the SU(2) operation on helical degrees of freedom andV = IV I withV I the SU(2) operation on vortex degrees of freedom In this paper, we consider a trivialV operation,V ≡ 1 and focus on the effect fromŜ I , i.e., By doingŜ I operation, we haveσ orΓ By doing SU(2)⊗SU(2) rotation operationŜ, the generation operator rotates aŝ As a result, after SU(2)⊗SU(2) rotation operation, a tensor state described by n I σ changes to another described by n I σ ′ . Thus, a new basis becomes For example, by using the SU(2)⊗SU(2) rotation operationŜ, we can change a 3D σ z -Kelvin waves to another standing Kelvin waves by the rotating generation operatorŝ S Y = e −i π 2 σx⊗ 1 , S Z = 1.

Generalized spatial translation symmetry
We discuss the spatial translation symmetry of plane Kelvin waves with fixed wave-length. Different plane Kelvin waves of different tensor states Γ I = Γ I have different translation symmetries. We define For the case of the system has translation symmetry.
The generalized translation symmetry of the plane Kelvin waves is characterized by three translation operators, whereΓ I is a four-by-four matrix,Γ is wave vector operator. For the plane Kelvin waves described by tensor state n I σ and τ 0 = 1, we have It looks like all plane Kelvin waves with different tensor states Γ I have the same translation symmetries aŝ However, in addition to the translation symmetry, there exists generalized translation symmetry by doing a translation operation T (∆x) = e i∆x(k I ·Γ I ) Thus, an arbitrary continuous spatial translation operation is combination of a discrete spatial translation operation and a global gauge transformation operation where ∆φ = ik · [2a[(∆x) mod 2a].

Biot-Savart equation in tensor representation
Then we derive the Biot-Savart equation in tensor representation.
For perturbative Kelvin waves, the Biot-Savart equation i we obtain the total Hamiltonian aŝ Thus, in tensor representation, the Biot-Savart equation is changed into

Zeros and zero-density
In the previous section, we have defined linking number and linking-number density to characterize the deformed two vortex-membranes. However, the winding number and winding-number density are not well defined for standing waves. To locally characterize different Kelvin waves, we introduce the zeros between two projected entangled vortexmembranes. Thus, it is the zero number and zero-number density that characterize the deformation of Kelvin waves in tensor representation.
. We then introduce the concept of projection: a projection of a vortex-membrane along a given direction θ on {ξ, η} space. In mathematics, the projection is defined bŷ In the following parts we useP to denote the projection operators.
Because the projection direction out of vortex-membrane is characterized by an angle θ in {ξ, η} space, we have where θ is angle, i.e. θ mod 2π = 0. So the projected vortex-membrane is described by the function b. Zeros between two projected entangled vortex-lines We then define the projection between two entangled vortexlines {ξ A/B (x, t), η A/B (x, t)} along a given direction θ in 3D space bŷ For two projected vortex-lines described by ξ A,θ (x, t) and ξ B,θ (x, t), a zero is solution of the equation We call the equation to be zero-equation and its solutions to be zero-solution (See the below discussion).
For vortex-lines described by plane wave Z(x, t) = r 0 e −iω·t+ik·x 0 , there exist periodic zeros. From the zeroequation ξ 0,A,θ (x, t) = ξ 0,B,θ (x, t) or cos(kx − ωt − θ) = 0, we get the zero-solutions to be where n is an integer number and θ = − π 2 . From the projection, we have a 1D crystal of zeros, of which the zero density ρ zero is Fig.8 shows zeros between two projected vortex-lines. In addition, for this case with z B = 0, the zero-number density is twice as the linking-number density between two entangled vortex-membranes that is also equal to the winding-number density for vortex-membrane-A, i.e., The zeros from the zero-solution don't change for a plane Kelvin wave with different tensor states. For a plane Kelvin wave with clockwise winding characterized by k and another plane Kelvin wave with counterclockwise winding characterized by − k , the functions for plane Kelvin waves are r 0 e −iω·t+ik·x and r 0 e −iω·t−ik·x , respectively.
For both cases with fixed time t = 0, up to a constant phase factor, we have the same zero-equation cos(kx) = 0 and the same zero-solution x(t) = a · n, respectively. For different plane Kelvin waves with different helical degrees of freedom, we have [Z = α k + β − k with |α| 2 + |β| 2 = 1. As a result, up to a constant phase θ 0 , a plane Kelvin wave with the same wave-length but different tensor orders has the same zero-equation and the same zero-solution respectively. Next, we define the zeros from the projection of 3D vortex-membranes in 5D space. a 3D vortex-membrane in 5D space { x, ξ( x), η( x)} ( x = (x, y, z)) can be described by the two-component function . Along e-direction, for the helical vortex-membrane described by z(x, t) = r 0 e −iω·t+i k· x , there exist periodic zeros between them. From the zero-equation ξ 0,A,θ ( x, t) = ξ 0,B,θ ( x, t) or cos( k · x − ωt − θ) = 0, we get the zero-solutions to be where n is an integer number and θ = − π 2 . So there exist the vector of zero density ρ zero = (ρ x,zero , ρ y,zero , ρ z,zero ) along three different directions ( e x e y , e z ).
c. Zero-density operator We define zero-density operator to characterize two vortex-membranes. According to pseudo-quantum mechanics, for each plane Kelvin wave e −iω·t+ik·x , the linking-number density is given by ρ zero = 2ρ linking = k π . Thus, the corresponding zero-density operator is given by In tensor representation, we can classify different types of zeros. For two entangled 1D vortex-line, the zero-number along σ z -internal direction is given by The zero-number density is given by Using similar approach, we define the zero density for standing wave along σ x -internal direction The corresponding zero density along σ x -internal direction is given by In general, the zero density and zero-density operator are given by We also define the tensor of zero densities ρ I, σ zero (x I , t) and its operatorsρ I, σ zero for 3D vortex-membranes, i.e., respectively. Finally, we could use the zero-density tensor ρ I, σ zero (x I , t) to describe the time-evolution of two non-uniform entangled vortex-membranes: for two entangled vortex-membranes with given initial stateZ( x), the time-dependent zero-density tensor ρ I, σ zero (x I , t) is obtained as ρ I, σ zero (x I , t) =Z * (x I , t)[( σ ⊗ 1) · (−i 1 From the point view of information (see below discussion), the Sommerfeld quantization condition for fourcomponent particle is the condition of the zero number, i.e.,

V. EMERGENT QUANTUM MECHANICS IN KNOT PHYSICS
The issue of emergent quantum mechanics is about the dynamic evolution for an object of a zero (a knot) with volume-changing, another types of local deformation of two entangled vortex-membranes. Here, we ask another question -how to characterize the evolution of the zeros with volume-changing?
In last section, we found that the deformed knot-crystal is described by the pseudo-quantum mechanics, of which two entangled vortex-membranes with fixed volume are mapped onto a particle with two internal degrees of freedom in tensor representation. It is obvious that the physical degrees of freedom for a vortex-membrane (the size V P → ∞) are seriously reduced. Because a vortex-membrane is an extended object, in general, each vortex-piece (vortex filament) with infinitesimal size (∆V P → 0) has a degree of freedom. So the pseudo-quantum mechanics cannot characterize the zero fluctuations from volume-changing on entangled vortex-membranes. In the following parts, from point view of information, we consider the smallest volume-changing on entangled vortex-membranes as an object with a single zero (∆V knot ≡ a 3 r 2 0 ). We call such object a knot. To characterize knot dynamics, the Biot-Savart mechanics becomes emergent quantum mechanics, the true quantum mechanics.

A. Basic theory for knots
Before developing emergent quantum mechanics for knots, we review the concept of "information". A generalized definition of the concept "information" is "An answer to a specific question". In general, the answer is related to data that represents values attributed to parameters. In other words, information can be interpreted as a data pattern or an ordered sequence of symbols from an alphabet. Take DNA as an example. The sequence of nucleotides is a pattern that influences the formation and development of an organism. In addition, information processing consists of an input-output function that maps any input sequence into an output sequence. Therefore, information reduces uncertainty. The uncertainty of an event is measured by its probability of occurrence. The more uncertain an event, the more information is required to resolve uncertainty of that event.
To represent information, information unit must be defined. The bit is a typical unit of information. Example: information in one "fair" coin flip: log 2 ( 2 1 ) = 1 bit, and in two fair coin flips is log 2 ( 4 1 ) = 2 bits. In summary, the information is defined by the following equation Information = a data pattern of information unit.
The use of knotted strings for record-keeping purposes in China dates back to ancient times. In 2003 J. D. Bekenstein claimed that a growing trend in physics was to define the physical world as being made up of information itself (and thus information is defined in this way).
By using projected representation, we show that different entanglement patterns between two vortex-membranes correspond to different patterns of zeros that are obtained from different zero-solutions, i.e., Information = a pattern of zeros between projected vortex-membranes.
In physics, the information is characterized by the distribution of zeros. Here, the information unit is a zero between projected vortex-membranes, Information unit = A zero between two projected vortex-membranes.
For the case of an extra zero, we have a knot; for the case of missing zero, we have an anti-knot. See the illustration in Fig.8. The colors of the dots in Fig.8 denotes L and R types of zeros.

Definition of a unified knot
Each zero corresponds to a piece of vortex-membranes. We call the object to be a knot, a piece of plane Kelvinwave with fixed wave-length. As a result, the system (plane Kelvin-wave with fixed wave-length) can be regarded as a crystal of knots. I call the system knot-crystal. In this section, we will study knot and show its properties on a knot-crystal.
We give the mathematic definition of knot. From point view of information, a knot is an information unit with fixed geometric properties. For example, for 1D knot, people divide the knot-crystal into N identical pieces, each of which is just a knot. As a result, some properties of knot can be obtained by scaling the properties of a knot-crystal; From point view of a classical field theory, a knot is elementary topological defect of a knot-crystal that is always anti-phase changing along arbitrary direction e, Z( x, t) → Z knot ( x, t), i.e., Let us show the fact that an information unit must be an anti-phase changing on Kelvin waves. Based on the projected vortex-membranes, we define a knot by a monotonic function where x denotes the position along the given direction e. A knot can be regarded as domain wall of π-phase shifting. When there exists a knot, the periodic boundary condition of Kelvin waves along arbitrary direction is changed into anti-periodic boundary condition. So the sign-switching character can be labeled by winding number w 1D . The winding number w 1D for a knot along given direction is ± 1 2 , i.e., where x is the coordinate along the given direction. Because the zero number is twice of winding-number, each zero ∆N zero = ±1 corresponds to a knot with half winding-number, ∆w 1D = ± 1 2 . A knot is an object with a fractional winding-number. As a result, the quantum number for a 3D knot can be denoted by On the other hand, from the topological character of a knot, there must exist a point, at which ξ A (x) is equal to ξ B (x). The position of the point x is determined by a local solution of the zero-equation So each knot corresponds to a zero between two vortex-membranes along the given direction.
In summary, for a knot, there are three properties: 1) Conservation condition: The conservation condition indicates that as an information unit, the existence of a zero for a knot is independent on the directions of projection angle θ. When one gets a zero-solution along a given direction θ, it will never split or disappear whatever changing the projection direction, θ → θ ′ . The type of zeros is switched L ↔ R by rotating the projection angle θ → θ + π (see below discussion); 2) Isotropic property: According to isotropic condition, there exists a single knot solution along arbitrary direction e. During rotation operation e → e ′ = R e, the zero of the knot still exists but its spin degrees of freedom is rotating synchronously, σ → σ ′ = Rσ; 3) Z2 topological property -each knot is π-phase changing along arbitrary direction e. When there exists a knot, for all Kelvin waves the periodic boundary condition along the given direction is changed into anti-periodic boundary condition.
For a knot, there are three conserved physical quantities: the energy of a (static) knot that is proportional to the volume of the knot on vortex-membrane V P = a 3 where a is a characteristic winding length along different directions; the (Lamb impulse) angular momentum (the effective Planck constant knot ) that is proportional to the volume of the knot in the 5D fluid V knot = V P · r 2 0 = a 3 · r 2 0 where r 0 is a characteristic winding radius; the winding number |∆w 1D | = 1 2 or the zero number |∆N zero | = 1. According to the geometric character of the three conserved physical quantities, the shape of knot will never be changed. However, the knot can split and the three physical quantities are conserved for all knot-pieces. Quantum mechanics is a mechanics to determine the distribution of knot-pieces.

Degrees of freedom of a knot
For a knot, there exists a zero satisfying conservation condition and Z2 topological property. A knot (a zero) has four degrees of freedom: two spin degrees of freedom ↑ or ↓ from the helicity degrees of freedom, the other two vortex degrees of freedom from the vortex degrees of freedom that characterize the vortex-membranes, A or B. The basis to define the microscopic structure of a knot is given by where A/B denotes the degrees of freedom of vortex-membranes; ↑/↓ denotes the degrees of freedom of (pseudo-) spin. z knot,↑,A/B (x, t) is the knot-function for a knot on vortex-membrane A/B with clockwise winding; z knot,↓,A/B (x, t) is the knot-function for a knot on vortex-membrane A/B with counterclockwise winding.
Firstly, we discuss spin degrees of freedom of knots. We have shown that there exist two types of winding directions for a knot A knot with the eigenstates of σ z described by |↑ or |↓ is really a piece of traveling Kelvin waves; A knot with the eigenstates of σ x described by |→ = 1 √ 2 (|↑ + |↓ ) or |← = 1 √ 2 (|↑ − |↓ ) is a piece of standing Kelvin waves; A knot with the eigenstates of σ y described by 1 √ 2 (|↑ + i |↓ ) or 1 √ 2 (|↑ − i |↓ ) is a piece of another type of standing Kelvin waves. An important property of spin degrees of freedom is time-reversal symmetry. In physics, under time-reversal operation, a clockwise winding knot (spin-↑ particle) turns into a counterclockwise winding knot (spin-↓ particle). An important property of spin degrees of freedom is time-reversal symmetry. In physics, under time-reversal operation, a clockwise winding knot (spin-↑ particle) turns into a counterclockwise winding knot (spin-↓ particle). In addition, in next sections we will show that owing to the existence of spin-orbital coupling (SOC) in the Dirac model, the degrees of freedom ↑/↓ indeed play the role of spin degrees of freedom as those in quantum mechanics.
Next, we discuss the vortex degrees of freedom of knots. The vortex-degrees of freedom of knots is denoted by For knots, we identify z knot,A (x, t) the function of A-vortex-membrane to be the wave-function of |A , z knot,B (x, t) the function of an B-vortex-membrane to be the wave-function of |B . Let us give the functions of four 1D knot states: 1) The up-spin knot on vortex-membrane-A, |↑, A is defined by where φ knot,↑,A (x) is a monotonic function changing π-phase. For unified knot, we have where x is the coordination on the axis and φ 0 is an arbitrary constant angle, k 0 = π a . The center of the unified knot is at x = x 0 + a 2 . For fragmentized knot, we have We will give the definition of φ fragmentized,↑,A (x) in the following section. For different types of knots, there exists a linear relationship between φ(x) and x, i.e., dφ(x) dx ∝ k 0 in the phase-changing region of x 0 < x ≤ x 0 + a. The linear character comes from the minimization of the volume volume(P ) of vortex-membrane. See the illustration of unified knots in Fig.9. In the limit of x → −∞, we have φ(x) = φ 0 − π 2 ; in the limit of x → ∞, we have φ(x) = φ 0 + π 2 . So we obtain and 2) The up-spin knot on vortex-membrane-B, |↑, B is defined by where Thus, we have and 3) The down-spin knot on vortex-membrane-A, |↓, A is defined by where Thus, we have and 4) The down-spin knot on vortex-membrane-B, |↓, B is defined by where Thus, we have and The knots with different internal degrees of freedom are described by different superposed states of the basis Along arbitrary direction e, due to sign-switching character for each of element states |↑, A , |↑, B , |↓, A , |↓, B , all superposed states |ψ = α |↑, A + β |↑, B + γ |↓, A + κ |↓, B change sign as On the other hand, for each of element states |↑, A , |↑, B , |↓, A , |↓, B along a given projection direction e and a fixed θ, we get a zero-equation = ± π 2 + θ.
The total volume of the knot (volume-changing of two entangled vortex-membranes) V knot = l · r 2 0 is always fixed. In addition to the four degrees of freedom, we have another degrees of freedom -chiral degrees of freedom by projecting a knot. Under projection with given projected angle θ, each projected knot state corresponds to two possible zeros -one zero with sgn[Υ] = 1 is denoted by |L ; the other with sgn[Υ] = −1 is denoted by |R (Υ = η A,θ (x, t) − η B,θ (x, t)).

Tensor states for a knot
For a 3D knot with four internal degrees of freedom we introduce a tensor representation to define a knot, For β I i,j , I denotes the spatial indices X, Y , Z; i = 1, −1 denotes helical degrees of freedom; j = 1, −1 denotes the vortex-degrees of freedom. According to the constraint condition, we have So the function for a 3D knot is given by The tensor states for a knot can also be classified by operator representation. We define the 3 × 3 × 3 tensor operatorsΓ I knot,i,j = σ I knot,i ⊗ (τ I knot,j , 1) (I = X, Y, Z for different spatial directions; i = x, y, z for spin degrees of freedom; j = x, y, z, 0 for the vortex-degrees of freedom). Different tensor states are described by = Γ I knot = n I knot,σ ⊗ ( n τ τ + 1τ 0 ).

Tensor-network state for a knot-crystal
Periodic entangled vortex-membranes form knot-crystal. In pseudo-quantum mechanics, a knot-crystal corresponds to two entangled vortex-membranes described by a special pure state of Kelvin waves with fixed wave length. In pseudo-quantum mechanics, we consider flat vortex-membrane as the ground state, i.e., A Kelvin wave becomes excited state around [vacuum ; in emergent quantum mechanics, we consider knot-crystal as a ground state, the knots become topological excitations on it. Because a knot-crystal is a plane Kelvin wave with fixed wave vector k 0 , we can use the tensor representation to characterize knot-crystals,Γ I knot−crystal = ( n I σ σ I ) ⊗ ( n τ τ + 1τ 0 ) and define corresponding generation operator U knot−crystal ( k 0 ,Γ I knot−crystal ) for it. Firstly, we define generate operator by knot-crystal where̥(r 0 ) is an expanding operator by shifting radius from 0 to r 0 on the membrane,K = −i d dφ , φ(x) = ±kx, and x is coordinate along x direction. Here [0 denotes a vortex-membrane with r 0 = 0. The tensor state of the knot-crystal is determined byΓ I knot−crystal = ( n I σ σ I ) ⊗ 1. For example, a particular knot-crystal is called SOC knot-crystal that is described by a 3D plane Kelvin waves with fixed wave-length Z knot−crystal ( x). The tensor state of it is On the one hand, a knot is a piece of knot-crystal; On the other hand, a knot-crystal can be regarded as a composite system with multi-knot, each of which is described by same tensor state. After projection, the knot-crystal becomes a zero lattice, of which there are two sublattices R/L and a tensor-network state Γ Tensor−network knot−crystal describes the entanglement relationship between two nearest-neighbor zeros. The tensor-network state comes from the product of the basis of all knots The summation of x 0 is obtained by summing the zeros between projected vortex-membranes. The generalized translation symmetry for a knot-crystal becomes where ι ± is an operator denoting switching of the chiral (sub-lattice) degrees of freedom as There exist topological defects of the tensor-network states that are just extra zeros (the knots). The chiral degrees of freedom become the sublattice degrees of freedom for knots on knot-crystal! In particular, the emergent Lorentz invariance for knots comes from the two-sublattice structure. By considering the local degrees of freedom, the state space (the Hilbert space) becomes suddenly enlarged from 2 in pseudo-quantum mechanics to 4 N knot in emergent quantum mechanics. The chiral degrees of freedom become the freedom of "lattice" and label the position of the sublattices.

Knot-operation and knot-number operator
To characterize knots (the elementary volume-changing of two entangled vortex-membranes), we introduce its operator-representation.
For 1D unified knot in Dirac representation, we define it by whereÛ knot is a generation operator for a knot that is an eigenstate of spin operator σ z , where̥ knot (r 0 ) is an expanding operator by shifting 0 to r 0 in the winding region of a knot (for example, x ∈ (x 0 , x 0 + a]).Û knot−crystal denotes the generation operator for a 1D knot-crystal that is the plane Kelvin wave with fixed wave-length and becomes the same tensor state for a knot. We define the generation operator for a 3D knot by doing a knot-operation For example, 3D SOC knot is defined bŷ In addition, the knot-number of a 3D knot is defined by

B. Emergent quantum mechanics
In above section, we show that a knot (the elementary volume-changing of two entangled vortex-membranes) is an elementary topological defect of knot-crystal and also fundamental information carrier. In this section, we will show that knots obey the emergent quantum mechanics.

Information scaling between knot and knot-crystal
From the point view of pseudo-quantum mechanics, knot-crystal is a special entangled vortex-membranes that is described by a tensor state. The total degrees of freedom for a knot-crystal is 2. From the point view of emergent quantum mechanics, a knot-crystal is periodic zeros between projected vortex-membranes that is described by tensornetwork state for knots. The total degrees of freedom of a knot-crystal becomes 4 N knot . In particular, there exists a holographic principle: the pseudo-spin-tensor state for a knot on a knot-crystal is same to the tensor state for the knot-crystal in emergent quantum mechanics.
On the other hand, a knot corresponds to the changing of one zero on a knot-crystal. When a knot is generated on a knot-crystal, the boundary of all Kelvin waves changes -periodic boundary condition changes into anti-periodic boundary condition, vice versa. Because the Hilbert space is never changed during the changing the boundary condition, the Hilbert space of knots According to the correspondence, several quantum quantities of knots (momentum and angular momentum) can be easily obtained by using the information scaling, where knot is the Planck constant for knots in emergent quantum mechanics and eff is the Planck constant for plane Kelvin wave with fixed wave-length in pseudo-quantum mechanics. N knot is the total knot number in the knot-crystal.

Fragmentized knot
According to the geometric character of the three conserved physical quantities, the shape of knot will never be changed. However, the knot can split and the three physical quantities are conserved for all knot-pieces. To characterize this property, we introduce the concept of "fragmentized knot". Quantum mechanics is a mechanics to determine the distribution of knot-pieces.
We take 1D knot with fixed spin and vortex degrees of freedom as an example to show the concept of "fragmentized knot" (σ →↑, τ → B). Before introducing fragmentized knot, we give the definition of unified knot that is described by where x is coordinate along an arbitrary direction e and φ knot (x) is the corresponding phase angle. We split a unified knot into two pieces, each of them is a half-knot with π 2 phase-changing. As shown in Fig.10, we split the knot into two pieces. The knot-function of two-piece fragmentized knot is z(φ knot (x)) where and It is obvious that for the fragmentized knot, there exists only single zero-solution due to topological condition. And we have 50% probability to find a zero-solution in the region of x 0 < x ≤ x 0 + a 2 and 50% probability to find a zero-solution in the region of We had defined a knot with half-winding to be of which the knot is at x 0 . As a result, The knot-function of two-piece fragmentized knot is defined by of which the two half-knots are at x 0 and x 0 + x ′ 0 , respectively. Similarly, we may split a knot into N pieces, each of which is an identical knot-piece with π N phase-changing. The knot-function of N knot-pieces is For N knot-pieces, there also exists a zero and the knot-number is conserved. We have 1 N probability to find a zero in a knot-piece. By using similar argument, an arbitrary superposed states of knots can be fragmentized. Fig.10.(d) is an illustration of a fragmentized knot.
The generation operator for a fragmentized knot with uniform distribution of knot-pieces is defined as fragment denotes a unform distributed knot-pieces aŝ is an operator by shifting 0 to r 0 in the winding region of x ∈ ((x 0 ) i , (x 0 ) i + a N ], ̥ 1 N -knot (r 0 , (y 0 ) i ) is an operator by shifting 0 to r 0 in the winding region of y ∈ ((y 0 ) i , (y 0 ) i + a N ],̥ 1 N -knot (r 0 , (z 0 ) i ) is an operator by shifting 0 to r 0 in the winding region of z ∈ ((z 0 ) i , (z 0 ) i + a N ], and where k 0 = π a . The center of the knot-piece with π N phase-changing is at (x 0 , y 0 , z 0 ) i . For this case, we assume the distribution of the knot-pieces is same as the zeros of the knot-crystal, An important issue is to obtain the spatial and temporal distribution of the knot-pieces that has the information of a knot. In next part, we will show that the quantum states of a knot -the wave-functions describe the spatial and temporal distribution of the knot-pieces determined by the perturbative Kelvin waves on a knot-crystal.

Quantum states of fragmentized knots
Firstly, we consider a knot with uniform knot-pieces, of which the state is denoted bŷ where c † k=0 means that the knot has the same wave vectors to those of its background. The density of knot-pieces is given by where V P is volume of the system, L x /L y /L z denotes the size of vortex-membrane along X/Y /X direction. It is obvious that the function for a knot with k = 0 is same as that of a knot-crystal by scaling the linking number along given direction. We take 1D knot with fixed spin and vortex degrees of freedom (for example, σ =↑, τ = B) as an example to prove the knot density ρ knot = 1 Lx for a uniform distribution of knot-pieces. For 1D knot, the knot-number is defined as To calculate the knot-number, we introduce two types of integrals: one is on-vortex-membrane integral with a sorting of knot-positions in on vortex-membrane, the other is out-vortex-membrane integral with a sorting of knot-phases in extra space (ξ, η).
We can label a knot-piece by its position (x 0 ) ix on vortex-membrane space or label a knot-piece by its position (x 0 ) i φ in (ξ, η) space. Here i φ denotes a sorting of ordering of phase φ from small to bigger and i x denotes a sorting of coordination x with a given order. Each i φ corresponds to an i x . As a result, there exists an integral-transformation by reorganizing all knot-pieces with different rules as By transforming an out-vortex-membrane integral to on-vortex-membrane integral, the knot-number is obtained as As a result, we get In the limit of N → ∞, we indeed have a uniform distribution of the N identical knot-pieces on vortex-membrane, Next, we consider a plane Kelvin wave with an extra fragmentized knot. Now, the system is described by Here, for simplify, we take a fragmentized knot with fixed spin (for example, ↑) and vortex degrees of freedom (for example, B) as an example. Then we consider the effect of an extra knot, where ∆φ( x, t) = k · x − ω · t. We may also denote the state by The wave-function ψ k ( x, t) and the function ψ k ( x, t) correspond to anti-periodic boundary condition and periodic boundary condition for the plane Kelvin waves.Û k is the operator changing the boundary condition. So the number of zeros is changed by ±1. We say that ψ k ( x, t) becomes plane waves for knots. From the superposition principle of Kelvin waves, an arbitrary quantum state for a knot is given by where r σ,τ, k is the amplitude of given plane wave k and σ =↑/↓ denotes spin degrees of freedom, τ = A/B denotes vortex degrees of freedom. The results illustrate superposition principle for fragmentized knot. We may denote the equation by For simplify, in this part we discuss the properties of quantum states for knots by fixing the spin and vortex degrees of freedom, (for example, σ →↑, τ → B). We point out that the function of a Kelvin wave with an extra fragmentized knot describes the distribution of the N identical knot-pieces and plays the role of the wave-function in quantum mechanics as and where the function of Kelvin wave with a fragmentized knot z( x,t) √ VP r0 becomes the wave-function ψ( x, t) in emergent quantum mechanics, the angle ∆φ( x, t) becomes the quantum phase angle of wave-function, the knot density ρ knot = ∆K ∆VP becomes the density of knot-pieces n knot ( x). Thus, the density of knot-pieces in a given region is obtained as We then prove that the wave-function ψ( x, t) = ρ knot (x, t)e iφ( x,t) is the function of Kelvin wave of knots (elementary particles). The knot density ρ knot = ∆K ∆VP is equal to For this case, we have set the bare density of finding a knot to be zero ρ knot ( x, t) ≡ 0. As a result, ψ( x, t) indeed plays the role of wave-function in quantum mechanics, ρ knot (x, t) is the knot density and φ( x, t) is the phase (the angle in extra space). For a single knot, we have (volume) normalization condition ρ knot ( x, t)dV P = 1. In the following part, we will show the probability interpretation for wave-functions according to the dynamic projection with fast-clock effect. Thus, the quantum state is invariant under a global gauge transformation, i.e., where φ 0 is constant. Such invariant comes from a global rotation symmetry in (ξ, η) space.

Momentum operator and energy operator for knots
For a plane wave, ψ( x, t) = 1 √ VP e −iω·t+i k· x (for simplify, we have fixed the spin and vortex degrees of freedom, σ →↑, τ → B), the projected (Lamb impulse) energy of a knot is and the projected (Lamb impulse) momentum of a knot is where the effective Planck constant knot is obtained as projected (Lamb impulse) angular momentum of a knot (the elementary volume-changing of two entangled vortex-membranes) As a result, we have From the fact of superposition principle of Kelvin waves, a generalized wave-function can be ).
Next, we calculate the expect values of the projected (Lamb impulse) energy E knot and the projected (Lamb impulse) momentum p knot , respectively.
From the wave-function ψ( This result indicates that the projected (Lamb impulse) energy becomes operator Using the similar approach, we derive As a result, the projected (Lamb impulse) momentum also becomes operators (287)

The Schrödinger equation
For deriving the Hamiltonian of knots, we treat the Hamiltonian of perturbative Kelvin waves around the knotcrystal. For knot-crystal of two entangled vortex-membranes, the Biot-Savart equation i turns into the Schrödinger equation for constraint Kelvin waves where Z knot−crystal ( x, t) denote perturbative Kelvin waves around the knot-crystal (a special plane Kelvin wave) and the Hamiltonian density is given bŷ where m pseudo = eff ακ ln ǫ andp I lamb = −i eff For a knot, by scaling the Planck constant, we derive the Schrödinger equation for knots as    denotes quantum states of a knot with four degrees of freedom andĤ knot is the Hamiltonian of knots. On flat vortex-membranes, the Hamiltonian of SOC knots is obtained by scaling that of plane Kelvin wave with same tensor state aŝ where m pseudo = knot ακ ln ǫ andp I knot = −i knot d dx I . In next section, we will derive the knot HamiltonianĤ knot on the SOC knot-crystal that is different from above Hamiltonian.
For the eigenstate with eigenvalue E knot ,Ĥ the wave-function becomes where f ( x) is spatial function. This state corresponds to a vortex-membrane with fixed angular velocity, ω = E knot knot . As a result, the quantized condition for knot is due to the conservation of angular momentum in extra space (the volume of the knot in 5D space) for information-unit, knot .
In particular, we have the topological representation on action S for knots, i.e., From the point view of topology, we explain the Sommerfeld quantization condition for knot-pieces where h knot = 2π knot and n is an integer number. The Sommerfeld quantization condition is just the topological condition of the winding number for a process of "static" knot-pieces, i.e., where w 1D is the winding number of the process that can be regarded as the winding number for corresponding Kelvin waves.

C. Emergent quantum field theory
In this part, we regard knot-crystal as a multi-knot system that is described by quantum field theory and a knot (the elementary volume-changing of two entangled vortex-membranes) becomes elementary fermionic particle.

Fermionic statistics
To derive the formulation of quantum field theory for knot-crystal, we consider the knot-crystal as a many-knot system. We may introduce N -wave-function to describe the motions of N -knots, Ψ( x 1 , x 2 , ..., x N ). For this many-body system, an important feature is the statistics. In quantum mechanics, an assumption is spin-statistics of fermions. An electron is a fermion with half spin. The wave-function of a system of identical spin-1/2 particles changes sign when two particles exchange. Particles with wave-functions antisymmetric under exchange are called fermions.
To distinguish the statistics for the knots, we consider two knots. We show that the wave-functions antisymmetric by exchanging knots is due to π-phase changing nature. When we exchange two 1D knots, the angle in extra space (that is just the phase of wave-function) is π. According to the definition of knots, we have two static unified knots and get knot-functions Due toÛ * after exchanging two knots, we get The results are independent of the dimensions. As a result, the knot obeys fermionic statistics in different dimensions. We then introduce second quantization representation for knots by defining fermionic operator c † The knot-crystal without extra knots corresponds to the vacuum state |vacuum Then after exchanging two identical particles, the many-body wave-function changes by a phase to be After considering the four internal degrees of freedom, we have We identify c † ↑,A/B ( x, t) (c ↑,A/B ( x, t)) the creation (or annihilation) operator of a spin-↑ knot on vortex-membrane-A/B; c † ↓,A/B ( x, t) (c ↓,A/B ( x, t)) the creation (or annihilation) operator of a spin-↓ knot on vortex-membrane-A/B. According to the fermionic statistics, there exists anti-commutation relation In addition, we can introduce the field representation via Grassmann number Path integral formulation for quantum mechanics is another formulation describing the dynamic evolution of the distribution of knot-pieces. By considering information scaling the effective Planck constant of knot-crystal to that of knots, the probability amplitude K knot ( x ′ , t f ; x, t i ) for a knot (a knot-piece) from an initial position x at time t = t i (that is described by a state |t i , x ) to position x ′ at a later time t = t f (|t f , x ′ ) is obtained as, where Each knot-piece's path contributes e iSn/ knot where S n is the n-th classical action i.e., n e iSn/ knot .
With the help of particle operators, we define the many-body Hamiltonian of the system with multi-knot aŝ whereĤ knot is the Hamiltonian operator of single knot and ψ( wheren p = c † p c p . The definition of total HamiltonianĤ means that for a given Kelvin wave there are n p knots. Now, we consider the path integral formulation of multi-knot. The probability amplitude becomes a multi-variable function where x ′ j and x j denote the final position and initial position of j-th knot, respectively. For a multi-knot system, quantum processes are described by where with and The symbol n denotes the summation of different pathes and the symbol j denotes the product of different volume changing with different zeros.
3. Discrete spatial translation symmetry for knots on knot-crystal Symmetry and symmetry breaking play profound roles in particle physics for quantum field theory and quantum many-body physics for solid physics.
In solid physics, a crystal is the system of periodically arranged atoms with spontaneous breaking of continuous spatial translation symmetry to discrete spatial translation symmetry. On a crystal, due to spontaneous breaking of continuous spatial translation symmetry to discrete spatial translation symmetry, the eigenstates are quantum states k . The Bloch vector k is defined as a linear superposition of the localized states | R, a j in the unit cell R at position a j , that depends on Bloch momentum k and type of orbital j in the unit cell. One cannot do a translation operation T (∆ x) with ∆ x = α R where α is a non-integer real number. The Bloch theorem is very useful in description of a quasi-particle in crystals that states that the basis vector k translated for the Bravais vector R changes as where T ( R) is translation operation. Due to the existence of Brillouin zone (BZ), we have k = k + Q where Q is reciprocal lattice vector. For knot-crystal, the eigenstates of an arbitrary quantum state in pseudo-quantum mechanics have generalized spatial translation symmetry k where k is wave-vector. By defining a translation operation T (∆ x), we have where ∆ x denotes the distance from generalized spatial translation operation. However, for emergent quantum mechanics, the situation changes. If we consider the vortex-piece with a zero as the elementary physics object, there exists a generalization of the Bloch theorem that incorporates generalized translational symmetries for quantum states of knots on a knot-crystal in emergent quantum mechanics.
For a plane wave state k L/R of zeros on L/R-sublattice along the direction k, we do a translation operation T (∆ x) (that corresponds to T (∆ x) on knot-crystal) and get where (∆ x) = 2a.
For the case of (∆ x) = a, we haveT that is reduced to the results in solid state physics. And we have a generalized BZ along a given direction e as where k 0 = π a e is reciprocal lattice vector along the given direction e. The generalized Bloch theorem constrains the formulation of the Hamiltonian which becomes manifestly invariant under additional spatial rotating symmetry. For the case of (∆ x) mod a = 0, we can recover the continuous spatial translation symmetry by changing projection angle (323)

Quantum field theory for knot-crystal
In emergent quantum mechanics, a knot-crystal becomes multi-knot system described by a tensor-network state, of which the effective theory becomes a Dirac model in quantum field theory.
In emergent quantum mechanics, the Hamiltonian for a 3D SOC knot-crystal has two terms -the kinetic term and the mass term from leapfrogging motion. In Fig.11.(a), we consider a knot-crystal to be a "two-sublattice" model with discrete spatial translation symmetry. We then label the knots by Wannier state |i, L , |i + 1, R , |i + 2, L , ...According to the definition of knot states |L and |R , we use the Wannier states c † L,i |vacuum and c † R,j |vacuum to describe them.
We use the tight-binding formula to characterize the Hamiltonian that couples two nearest neighbor states |i, L and |i + e I , R (I = X, Y, Z), where i denotes a given knot and J denotes the entanglement energy of vortex-membranes that is equal to the global rotating energy of the two entangled vortex-membranes. Then the total kinetic term is obtained as where i, j denotes the nearest-neighbor knots. With help ofT (a) and ι ± , we have |i + e I , L =T (a) · ι + |i, R that is denoted by |R . From the global rotating motion denoted e −iω0t , the projected states also change periodically.
As a result, we use the following formulation to characterize the global rotating process and the leapfrogging process, After considering the energy from the global rotating process and the leapfrogging process, a corresponding term is given by We write down the total Lagrangian of a 3D leapfrogging knot-crystal in tight-binding formula as From the fact that the energy of vortex-membranes is equal to the global rotating energy, J − 2 knot ω 0 = 0, we derive the value of J, i.e., J = 2 · knot ω 0 . Here, the factor "2" comes from the global rotating energy of the two entangled vortex-membranes. Finally, in long wave-length limit ∆k → 0 we obtain the low energy effective Hamiltonian as where c eff = a·J knot = 2aω 0 play the role of light speed and m knot c 2 eff = 2 knot ω * plays role of the mass of knots. p knot = knot k is the momentum operator. We then re-write the effective Hamiltonian to be It is obvious that the total Hamiltonian H has translation symmetry and rotation symmetry and R ,Ĥ knot = 0, respectively. Here,T (∆x) = e ia(k·Γ) ·Ŝ is translation operator. That means we have generalized translation symmetry. Another important property of the Dirac model for knot-crystal is spin-orbital coupling. The global spatial rotation operator is defined byR whereR spin is SO(3) spin rotation operatorR spinΓR −1 spin =Γ ′ , andR space is SO(3) spatial rotation operator, After doing a global spatial rotation operation, the motion direction changes from e to e ′ . That means we have generalized rotation symmetry.
The ground state of knot-crystal is a filled state of knots, of which the total number of knots is N knot . The excitation is a knot, of which the energy dispersion is where k 2 = k 2 x +k 2 y +k 2 z and m knot is the mass of knot. There are four energy bands for Dirac states. For a particle-like excitation, the energy is 2 knot k 2 c 2 eff + m 2 c 4 eff ; for a hole-like excitation, the energy is − 2 knot k 2 c 2 eff + m 2 knot c 4 eff . For a particle-like excitation, an extra knot with a zero is put on the knot-crystal and the total volume of the deformed knot-crystal becomes (N knot + 1) V knot ; for a hole-like excitation, a knot with a zero is taken off from the knot-crystal and the total volume of the deformed knot-crystal is (N knot − 1) V knot . The situation is very similar to the quasiparticles in solid physics. FromĤ 3D , for a particle-like knot, we can easily obtain the Schrödinger equation in low velocity limit, Finally, the low energy effective Lagrangian of 3D SOC knot-crystal is derived as =ψ(iγ µ∂ µ − m knot )ψ whereψ = ψ † γ 0 , γ µ are the reduced Gamma matrices, γ 1 = γ 0 Γ 1 , γ 2 = γ 0 Γ 2 , γ 3 = γ 0 Γ 3 , and γ 0 = Γ 5 = 1 ⊗ ι x , γ 5 = iγ 0 γ 1 γ 2 γ 3 . This is Lagrangian for massive Dirac fermions with emergent SO(3,1) Lorentz-invariance. Since the velocity c eff only depends on the microscopic parameter k 0 and κ, we may regard c eff to be "light-velocity" and the invariance of light-velocity becomes a fundamental principle for the knot physics. In above equation, we have set c eff = 1. As a result, we answer the question -how to characterize the evolution of the object with a zero? The pseudoquantum mechanics describes the entanglement-fluctuation of two entangled vortex-membrane with fixed volume in the 5D fluid; The emergent quantum field theory describes the volume-changing entanglement-fluctuation of two entangled vortex-membrane in the 5D fluid. From point view of information, the elementary volume-changing with a zero is a knot. We consider a given pattern of knot-pieces ψ( x) as the initial condition. We expand ψ( x) by the eigenstates of plane waves as ψ( x) = k c k e i p knot · x/ knot . Thus, under time evolution, the function of vortexmembranes becomes ψ( x, t) = k c k e i p knot · x/ knot e −iE knot ·t/ knot . Finally, the density of knot-pieces is obtained as We point out that under time evolution, the energy H( p knot ), the momentum p knot (the Lamb impulse) and the (Lamb impulse) angular momentum (the effective Planck constant knot ) are all conserved. And owning to the the conservation conditions of the volume of the knot in 5D space, the shape of knot is never changed and the corresponding Kelvin waves of knots cannot evolve smoothly. Instead, the knot can only split and knot-pieces evolves following the equation of motion of Biot-Savart equation (Schrödinger equation). This is the fundamental principle of quantum mechanics.
So for 3D SOC knot, there exist two types of Hamiltonian -one type on flat vortex-membranes with continuum spatial translation symmetry, the other on knot-crystal with lattice translation symmetry. For, a knot on flat vortexmembranes, the equation of motion is determined by the Schrödinger equation with the Hamiltonian There doesn't exist Lorentz invariance; For the knots on 3D SOC knot-crystal, the equation of motion is determined by the Schrödinger equation with the Hamiltonian

Possible physical realization
We address the issue of the physical realization of a 1D knot-crystal based on entangled vortex-lines in 4 He superfluid.
We consider two rectilinear vortex-lines in SF that is stretched between opposite points on the system. Then we rotate one vortex-line around the other with a rotating velocity ω 0 . According to the theory of Kelvin waves, the winding vortex-line becomes a helical one that is described by r 0 e ik0·x−iω0t+iφ0 where ω 0 ≃ ( κ 4π ln 1 k0a0 )k 2 0 . As a result, in principle, we realize a knot-crystal.
We estimate the two physical constants, knot and c eff . The emergent Planck constant knot is defined by knot = 2k0 . We set the length of the half pitch of the windings a = π k0 to be 10 −5 cm, and the distance between two vortex-lines r 0 as 10 −6 cm. For 4 He superfluid, κ is about 10 −3 cm 2 /s, Then the effective Planck constant is estimated to be knot = N He4 · ∼ 10 5 where N He4 = πr 2 0 a · ρ * is the atom's number inside a knot. ρ * is the superfluid density. Another important parameter is the effective light speed c eff , which is defined by c eff = κk0 2 (ln 1 k0a0 − 1 2 ). The effective light speed is estimated to be c eff ∼ 12m/s.

D. Measurement theory
Measurement is an important issue of quantum mechanics. P.A.M. Dirac (1958) in "The Principles of Quantum Mechanics said, "A measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured, the eigenvalue this eigenstate belongs to being equal to the result of the measurement." According to the Copenhagen interpretation, the state of a system is assumed to "collapse" into an eigenstate of the operator during measurement. That is the so-called measurement process of "wave-function collapse". The wave-function collapse is random and indeterministic and the predicted value of the measurement is described by a probability distribution. The wave-function collapse raises "the measurement problem", as well as questions of determinism and locality, as demonstrated in the Einstein-Podolsky-Rosen (EPR) paradox [30]. In this section, we show measurement theory in emergent quantum mechanics.

Information measurement and observers
In physics, people try to know the information of patterns for perturbative Kelvin wave of knots described by Z knot ( x, t), i.e., In mathematics, the information is the distribution of zeros between projected vortex-membranes that is determined by knot density for a given quantum state described by |ψ( x, t) , i.e., That means to identify the entanglement pattern of vortex-membranes Z knot ( x, t), people detect the distribution function of knot-pieces ρ knot ( x, t), i.e., Measurement = to detect Z knot ( x, t) (352) → to detect ρ knot ( x, t). To do measurement, there exist observers. To detect the distribution of knot-pieces, there are two types of observers: one type is the inner observers that are special multi-knot systems on vortex-membranes, the other is outer observers that are objects out of vortex-membranes (or even out of the inviscid incompressible fluid). For the outer observers, the rulers and clocks are independent on the physical properties of the knot-crystal. In principle, the outer observers have the ability to detect the every detail of the perturbative Kelvin waves on the knot-crystal. As a result, the theory for outer observers is Biot-Savart mechanics. However, for the inner observers, the rulers and clocks depend on the physical properties of the knot-crystal. In general, the inner observers have very very slow clocks and very very large rules and have no the ability to detect the every detail of the perturbative Kelvin waves on the knot-crystal. By using a huge ruler, the inner observers obtain a coarse-graining picture of a given system of multi-knots; By using a slow clock, the inner observers obtain average results of a dynamic system of multi-knots via time. So the inner observers will never obtain the complete information. In particular, the information of absolute phase factor cannot be observed for inner observers owing to a slow clock and a huge rule with very large scale-distance. In this paper, we consider the measurement theory for inner observers by considering slow clock. The situation for huge rule with very large scale-distance is similar.

Dynamic projection
In this part, we show the concept of dynamic projection, that is a process to detect ρ knot ( x, t). Knot is topological object that leads to π-phase changing (∆φ = π) and knot-piece is tiny phase-changing (∆φ → 0) with fixed phase angle (φ = φ 0 ). To find a knot (or a knot-piece), people need to detect the phase-changing ∆φ that changes the density of zeros between two vortex-membranes, ρ zero → ρ zero (x, t). So we may project the vortexmembranes and solve the knot-function to look for extra solutions owing to the knot-piece ∆φ. In experiments, people identify phase-changing ∆φ via dynamic projection P dynamic = to capture a knot-piece ∆φ at t = t 0 .
To find a fragmentized knot, we need to do projection Under a projection, a fragmentized (or randomized) knot is fixed at x 0 and time t 0 .
Because a knot is really a sharp phase-changing that could interact with the measuring devices, Thus, according to the different effects, we can decompose the dynamic operator into three parts: whereP (t 0 ) denotes the projection at time t 0 ,P Klevin−wave denotes the projection on a given eigenstate of perturbative Kelvin waves e −i∆ω·t0+i∆ k· x0     with fixed spin and vortex degrees of freedom,P fragment denotes the projection on a given knot-piece for the projected pertubative Kelvin waves. As a result, the dynamic projection operator is defined byP After repeating the measurement processesP dynamic , a lot of knot-pieces is observed and people know their distribution ρ knot ( x, t). Finally, the information of perturbative Kelvin waves on vortex-membrane Z knot ( x, t) is explored. During measurement processes, the phase angle of knot-pieces is fixed to be φ 0 (t 0 ) that depends on the time of measurement. This gives an explanation of "wave-function collapse" in quantum mechanics.

Fast-clock effect
We point out that the probability of emergent quantum mechanics comes from dynamically projecting entangled vortex-membranes via stochastic projected time t 0 that leads to stochastic projected angle.
It is known that the knots are phase-changing of knot-crystal. For a knot crystal, Z knot−crystal ( x, t) ∼ e −i(ω0+∆ω)t+φ0 , the period T clock of the phase angle changing 2π is T clock = 2π ω0+∆ω . So we have an intrinsic clock for knots, of which the phase is just the phase of the wave-function. The clock for inner observers T inner must be very very slower than the intrinsic clock, T inner ≫ T clock . The inner observers can do measurement at the scheduled time ∆t = nT inner (n is an arbitrary integer number), but cannot accurately do measurement at the scheduled time ∆t = nT clock . In the limit of Tinner T clock → ∞, ∆t T clock − mod( ∆t T clock ) is a random value and the projection angles ∆t T clock − mod( ∆t T clock ) ω 0 become stochastic for the different projection processes. As a result, owning to the stochastic projecting timeP (t 0 ), the projecting angle θ is a random value, i.e., = a random value.
We call it "fast-clock" effect.
As shown in Fig.12.(b), the detecting process can be exactly demonstrated by pulse sample figures -people can detect the phase-changing from knots but cannot detect the absolute phase angle.
For stochastic projection process due to "fast-clock" effect, the probability for a knot-piece is defined by the average of the projection angle. If there exists a knot solution atx(θ) for a given projection angleθ, we callθ the permitted projection angle; If there doesn't exist knot solution for a given projected angleθ, we callθ the unpermitted projection angle. Thus, the probability of a knot-piece is obtained as 1 2π dθ = 1 N where dθ = 2π N denotes the summation of all permitted projected anglesθ. For a system with a knot-piece, the probability for finding a knot (a zero between two projected vortex-membranes) is just its knot number, i.e., 1 2π dθ = K = 1 N . As a result, if we ignore the contribution from background (the knot crystal), the probability density for finding a knot n knot ( x, t) = 1 ∆Vp · 1 2π dθ is obtained as This result indicates that the probability density for finding a knot is equal to the knot density ρ knot ( x, t).

Probability interpretation for wave-functions
We point out that the function of a Kelvin wave with a fragmentized knot describes the distribution of the knotpieces and plays the role of the wave-function in quantum mechanics as ψ( x, t) ⇐⇒ ρ knot ( x, t)e i∆φ( x,t) , and ρ knot ( x, t) ⇐⇒ n knot ( x, t) where the function of Kelvin wave with a fragmentized knot becomes the wave-function ψ( x, t) in emergent quantum mechanics, the angle ∆φ B (x, t) becomes the quantum phase angle of wave-function, the knot density ρ knot = ∆K ∆VP becomes the probability density of knots n knot ( x). Thus, the measurement is to find the zero between two projected vortex-membranes that occurs at certain fragmentized knot with probability in a given region, ψ * ( x, t)ψ( x, t)∆V P . In particular, we emphasize knot has holographic property. Although, a knot-piece has a probability of ρ knot = 1 N to find a knot, not only its tensor state are same to that of a knot, but also its mass is m knot rather than m knot /N .

Complementarity principle
In emergent quantum mechanics, complementarity principle comes from complementarity property of knots. On the one hand, a knot piece is phase-changing -a sharp, time-independent, topological phase-changing, ∆φ = 0. To count the knot number, we had introduced knot-number operatorK = −i d dφ ; On the other hand, a knot-piece has a fixed phase angle, φ 0 that is determined by perturbative Kelvin waves on the vortex-membranes. One cannot exactly determine the phase angle of a knot-piece by observing its phase-changing. We call this property to be complementarity principle in emergent quantum mechanics that leads to uncertainty principle. Based on the complementarity principle, we discuss wave-particle duality and uncertainty principle.
Wave-particle duality is the fact that elementary particles exhibit both particle-like behavior and wave-like behavior. As Einstein wrote: "It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do". Here, we point out that wave-particle duality of quantum particles is really "information-motion duality" that is property of complementarity principle. On one hand, a knot is information unit that is a sharp, fragmentized, topological phase-changing in physics and would become a "point" after projection in mathematics. Thus, it shows particle-like behavior; On the other hand, the dynamic, smooth, non-topological phase-changing from perturbative Kelvin waves shows wave-like behavior that is characterized by wave-functions. In other words, the Kelvin waves look like pilot-waves [13], on which knot-pieces stay.
For emergent quantum mechanics, the uncertainty principle is related to the information nature of knots. From the point view of information, the momentum denotes the spatial distribution of knot-pieces (or information), the energy denotes the temporal distribution of knot-pieces (or information). On one hand, a uniform distribution of knot-pieces generated byÛ fragment on knot-crystal with an excited Kelvin wave ψ(x, t) = e −iωt+i k· x is described by a wave-function of a plane wave with a fixed projected momentum p knot = knot k. For this case, we know momentum information of the knot but we do not know its position information; On the other hand, a unified knot generated byÛ Unified−knot is really a special distribution of knot-pieces and can be regarded as a superposition state of ψ(x, t) ∼ k e −iωt+i k· x .
For this case, we know the position information of the knot but we do not know its momentum information. 6. Applications: the Schrödinger's cat paradox and the double-slit experiment From above discussion, we show the answer to the Schrödinger's cat paradox. The stochastic and uncertain come from dynamic projection and fragmentized knot. A cat is a classical object with slow inner clock. The time scale of a cat is the same order of the detecting objects. There is no dynamic projection process and all information of a cat can be obtained. So there is no Schrödinger's cat paradox at all.
We then give an explanation on the double-slit experiment. Before measurement in double-slit experiment, the knot can be regarded as fragmentized knot on a perturbative Kelvin wave, of which the probability distribution is described by the wave-function. It has no classical path. There exists particular interference pattern on the screen that agrees to the prediction from quantum mechanics. However, after measurement, the phase angle of knot-pieces becomes randomized. As a result, the phase coherence is destructed and the interference disappears.

E. Quantum entanglement theory
In the end of the section, we address the issue of quantum entanglement. Quantum entanglement is a physical phenomenon for two knots that the quantum state of each knot cannot be described independently of the others, even when the particles are separated by a large distance. An entangled state for n-body quantum system comes from new type of knots -composite knot with n zeros. We call it (composite) mπ-knots that change the volume of vortex-membranes via ±mV knot . Here, m may be not equal to n. For a fragmentized mπ-knot, the distribution of its pieces is described by its "wave-functions". In general, the quantum states for a fragmentized mπ-knot cannot be reduced into a product state of the wave-function for n knots.
Firstly, we study entangled states for two knots.
We then define the composite knots for each state of the basis |↑ knot,1 ⊗ |↑ knot,2 , |↑ knot,1 ⊗ |↓ knot,2 , |↓ knot,1 ⊗ |↑ knot,2 , |↓ knot,1 ⊗ |↓ knot,2 : In the end, we draw the conclusion. Fig.14 shows the correspondence between the Biot-Savart mechanics for knots and emergent quantum mechanics for particles. Owning to the the conservation conditions of the volume of the knot in 5D space, the shape and the volume of knot is never changed and the knot can only split and knot-pieces evolves following the equation of motion of Schrödinger equation. Three dimensional quantum Dirac model is obtained to effectively describe the deformed vortex-membranes: The elementary excitations are knots with information unit; The physics quality to describe local deformation is knot density that is defined by the density of zeros between two projected vortex-membranes; The Biot-Savart equation for Kelvin waves becomes Schrödinger equation for probability waves of knots; the classical functions for perturbative Kelvin waves become wave-functions for knots; The probability of quantum mechanics comes from dynamic projection for inner observers owing to "fast-clock" effect; The angular frequency for leapfrogging motion plays the role of the mass of knots; etc. This work would help people to understand the mystery of quantum mechanics.
Finally, there are several unsolved problems. In this paper, we only consider the perturbative effect on the Kelvin waves. According to Hasimoto, an important result after considering the nonlinear effect is the existence of soliton solutions by mapping the Biot-Savart equation to 1D non-linear Schrödinger equation via Hasimoto representation [31]. The generalization of Hasimoto representation to 3D knot-crystal may be interesting. In addition, after considering the dissipation effect, the Kelvin waves are subject to Kolmogorov-like turbulence (even in quantum fluid [8,9]). The stability of 3D knot-crystals and the possible cascade of Kelvin waves to turbulence are also open issues. From the point view of emergent quantum field theory, the emergence of gauge fields and gauge interaction on a knot-crystal is a critical problem. In the future, we will study these problems and develop a complete theory for emergent quantum physics. vortex-lines are one-dimensional topological objects in three-dimensional superfluid (SF). For SF with a vortex-line, a rotationless superfluid component flow ∇ × v = 0 is violated on one-dimensional singularities s(, t), which depends on the variables -arc length and the time t. Away from the singularities, the velocity increases to infinity so that the circulation of the superfluid velocity remains constant, v · dl = κ where v =ṡ. As a result, the superfluid vortex filament could be described by the Biot-Savart equatioṅ where κ = h/m 4 (for 4 He) or h/m 3 (for 3 He) is the discreteness of circulation. r is the vector that denotes the position of vortex filament and s denotes the integral variable. This means that the singularity of v(r) comes from integrating around the vicinity part of s 0 .

A. Local induction approximation
In Cartesian coordinate system, the position vector of vortex filament is s = (x, y, z). By using the natural parameterizations, it is convenient to express s in form of s(ξ), in which ξ is the (algebraic) arc length measured from the point s 0 . In the vicinity of s 0 , we can expand the function s(ξ) as s(ξ) = s 0 + s ′ ξ + s ′′ ξ 2 /2 + · · · (389) and ds = s ′ dξ + s ′′ ξdξ + · · · (390) in which s ′ = ∂S ∂ξ and s ′′ = ∂ 2 S ∂ 2 ξ , respectively. From the geometrical point of view, s ′ is the unit vector tangent to vortex filament, s ′′ is the vector normal to vortex filament.
When r → s 0 , we can separate the integral into two parts: one is in the vicinity of s 0 , the other is regular part without singularity, i.e., v(r → s 0 ) = κ 4π where ξ 0 is the length of the order of the curvature radius (or inter-vortex distance when the considered vortex filament is a part of a vortex tangle). By using and a 0 denotes the vortex filament radius that is much smaller than any other characteristic size in the system.

B. Physical quantities of Kelvin waves
Next, we discuss the physical quantities of the Kelvin waves along winding direction (z-direction), i.e., the projected momentum p z and the projected angular momentum J z . For a plane Kelvin wave of a lightly deformed straight vortexline, the function is described by x = a cos(kz − ωt), y = a sin(kz − ωt).
In SF with a vortex-line, the conventional momentum of the superfluid motion cannot be well defined. Instead, the hydrodynamic impulse (the Lamb impulse) plays the role of the effective momentum that denotes the total mechanical impulse of the non-conservative body force applied to a limited fluid volume to generate instantaneously from rest the