Open access peer-reviewed chapter

Topological Interplay between Knots and Entangled Vortex-Membranes

By Su-Peng Kou

Submitted: October 10th 2017Reviewed: November 29th 2017Published: May 30th 2018

DOI: 10.5772/intechopen.72809

Downloaded: 207

Abstract

In this paper, the Kelvin wave and knot dynamics are studied on three dimensional smoothly deformed entangled vortex-membranes in five dimensional space. Owing to the existence of local Lorentz invariance and diffeomorphism invariance, in continuum limit gravity becomes an emergent phenomenon on 3 + 1 dimensional zero-lattice (a lattice of projected zeroes): on the one hand, the deformed zero-lattice can be denoted by curved space-time for knots; on the other hand, the knots as topological defect of 3 + 1 dimensional zero-lattice indicates matter may curve space-time. This work would help researchers to understand the mystery in gravity.

Keywords

  • vortex-membrane
  • knot
  • gravity

1. Introduction

A vortex (point-vortex, vortex-line, vortex-membrane) consists of the rotating motion of fluid around a common centerline. It is defined by the vorticity in the fluid, which measures the rate of local fluid rotation. In three dimensional (3D) superfluid (SF), the quantization of the vorticity manifests itself in the quantized circulation vdl=hmwhere his Planck constant and mis atom mass of SF. Vortex-lines can twist around its equilibrium position (common centerline) forming a transverse and circularly polarized wave (Kelvin wave) [1, 2]. Because Kelvin waves are relevant to Kolmogorov-like turbulence [3, 4], a variety of approaches have been used to study this phenomenon. For two vortex-lines, owing to the interaction, the leapfrogging motion has been predicted in classical fluids from the works of Helmholtz and Kelvin [5, 6, 7, 8, 9, 10]. Another interesting issue is entanglement between two vortex-lines. In mathematics, vortex-line-entanglement can be characterized by knots with different linking numbers. The study of knotted vortex-lines and their dynamics has attracted scientists from diverse settings, including classical fluid dynamics and superfluid dynamics [11, 12].

In the paper [13], the Kelvin wave and knot dynamics in high dimensional vortex-membranes were studied, including the leapfrogging motion and the entanglement between two vortex-membranes. A new theory—knot physics is developed to characterize the entanglement evolution of 3D leapfrogging vortex-membranes in five-dimensional (5D) inviscid incompressible fluid [13, 14]. According to knot physics, it is the 3D quantum Dirac model that describes the knot dynamics of leapfrogging vortex-membranes (we have called it knot-crystal, that is really plane Kelvin-waves with fixed wave-length). The knot physics may give a complete interpretation on quantum mechanics.

In this paper, we will study the Kelvin wave and knot dynamics on 3D deformed knot-crystal, particularly the topological interplay between knots and the lattice of projected zeroes (we call it zero-lattice). Owing to the existence of local Lorentz invariance and diffeomorphism invariance, the gravitational interaction emerges: on the one hand, the deformed zero-lattice can be denoted by curved space-time; on the other hand, the knots deform the zero-lattice that indicates matter may curve space-time (see below discussion).

The paper is organized as below. In Section 2, we introduce the concept of “zero-lattice” from projecting a knot-crystal. In addition, to characterize the entangled vortex-membranes, we introduce geometric space and winding space. In Section 3, we derive the massive Dirac model in the vortex-representation of knot states on geometric space and that on winding space. In Section 4, we consider the deformed knot-crystal as a background and map the problem onto Dirac fermions on a curved space-time. In Section 5, the gravity in knot physics emerges as a topological interplay between zero-lattice and knots and the knot dynamics on deformed knot-crystal is described by Einstein’s general relativity. Finally, the conclusions are drawn in Section 6.

2. Knot-crystal and the corresponding zero-lattice

2.1. Knot-crystal

Knot-crystal is a system of two periodically entangled vortex-membranes that is described by a special pure state of Kelvin waves with fixed wave length Zknotcrystalxt[13, 14]. In emergent quantum mechanics, we consider knot-crystal as a ground state for excited knot states, i.e.,

Zknotcrystalxt=zAxtzBxtvacuum.E1

On the one hand, a knot is a piece of knot-crystal and becomes a topological excitation on it; 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.

Because a knot-crystal is a plane Kelvin wave with fixed wave vector k0, we can use the tensor representation to characterize knot-crystals [13],

Γ˜knotcrystalI=nσIσInττ+1τ0E2

where 1=1001and σI, τIare 2×2Pauli matrices for helical and vortex degrees of freedom, respectively. For example, a particular knot-crystal is called SOC knot-crystal Zknotcrystalx[13], of which the tensor state is given by

σX1=nσX=1,0,0,σY1=nσY=0,1,0,σZ1=nσZ=0,0,1.E3

For the SOC knot-crystal, along x-direction, the plane Kelvin wave becomes zx=2r0cosk0x; along y-direction, the plane Kelvin wave becomes zy=12r0eiky+ieiky; along z-direction, the plane Kelvin wave becomes zz=r0eikz.

For a knot-crystal, another important property is generalized spatial translation symmetry that is defined by the translation operation TΔxI=eik̂0IΔxIΓ˜knotcrystalI

ZxItTΔxIZxit=eik̂0IΔxiΓ˜knotcrystalIZxit.E4

Here k̂Iis iddxII=xyz. For example, for the knot states on 3D SOC knot-crystal, the translation operation along xI-direction becomes

TΔxI=eik̂IΔxIσI1.E5

2.2. Winding space and geometric space

For a knot-crystal, we can study it properties on a 3D space (x,y,z). In the following part, we call the space of (x,y,z) geometric space. According to the generalized spatial translation symmetry, each spatial point (x,y,z) in geometric space corresponds to a point denoted by three winding angles ΦxxΦyyΦzzwhere ΦxIxIis the winding angle along xI-direction. As a result, we may use the winding angles along different directions to denote a given point Φx=ΦxxΦyyΦzz. We call the space of winding angles ΦxxΦyyΦzzwinding space. See the illustration in Figure 1(d) .

Figure 1.

(a) An illustration of a 1D knot-crystal; (b) the relationship between winding angle Φ and coordinate position x . The red dots consist of a 1D zero-lattice in geometric space and the blue dots consist of a zero-lattice in winding space; (c) an illustration of a 3D uniform zero-lattice in geometric space; and (d) an illustration of a 3D uniform zero-lattice in winding space.

For a 1D leapfrogging knot-crystal that describes two entangled vortex-lines with leapfrogging motion, the function is given by

Zxt=r0cosωt2isinωt2eiπaxeiω0t+iωt/2,E6

where ωis angular frequency of leapfrogging motion. For the 1D σz-knot-crystal, the coordinate on winding space is Φx=πax. Another example is 3D SOC knot-crystal [10], of which the function is given by

ZKCxt=zKC,AxtzKC,Bxt=r0cosωt2isinωt2eiω0t+iωt/22r0cosΦxx12r0eiΦyy+ieiΦyyeiΦzz,E7

where the coordinates on winding space are Φxx=πax, Φyy=πay, Φzz=πaz, respectively.

In addition, there exists generalized spatial translation symmetry on winding space. On winding space, the translation operation TΔΦIbecomes

TΔΦI=eiiΔΦIΓ˜knotcrystalIE8

where ΔΦIdenotes the distance on winding space.

2.3. Zero-lattice

Before introduce zero-lattice, we firstly review the projection between two entangled vortex-membranes zA/Bxt=ξA/Bxt+iηA/Bxtalong a given direction θin 5D space by

P̂θξA/BxtηA/Bxt=ξA/B,θxtηA/B,θxt0E9

where ξA/B,θxt=ξA/Bxtcosθ+ηA/Bxtsinθis variable and ηA/B,θxt0=ξA/BxtsinθηA/Bxtcosθis constant. So the projected vortex-membrane is described by the function ξA/B,θxt. For two projected vortex-membranes described by ξA,θxtand ξB,θxt, a zero is solution of the equation

P̂θzAxtξA,θxt=P̂θzBxtξB,θxt.E10

After projection, the knot-crystal becomes a zero lattice. For example, a 1D leapfrogging knot-crystal is described by

ZKCxt=r0cosωt2isinωt2eiπaxeiω0t+iωt/2.E11

According to the knot-equation P̂θzKC,Ax=P̂θzKC,Bx, we have

x¯0=aX+aπω0tE12

where θ=π2and x¯0is the position of zero. As a result, we have a periodic distribution of zeroes (knots).

For a 3D leapfrogging SOC knot-crystal described by ZKCxt=zKC,AxtzKC,Bxt, we have similar situation—the solution of zeroes does not change when the tensor order changes, i.e., σ1=nσ=0,0,1nσ=nxnynxwith nσ=1[13]. We call the periodic distribution of zeroes to be zero-lattice. See the illustration of a 1D zero-lattice in Figure 1(b) and 3D zero-lattice in Figure 1(c) .

Along a given direction e, after shifting the distance a, the phase angle of vortex-membranes in knot-crystal changes π, i.e.,

ΦxtΦx+aet=Φxt+π.E13

Thus, on the winding space, we have a corresponding “zero-lattice” of discrete lattice sites described by the three integer numbers

X=XYZ=1πΦ1πΦmodπ.E14

See the illustration of a 1D zero-lattice in Figure 1(b) and 3D zero-lattice in Figure 1(d) .

3. Dirac model for knot on zero-lattice

3.1. Dirac model on geometric space

3.1.1. Dirac model in sublattice-representation on geometric space

It was known that in emergent quantum mechanics, a 3D SOC knot-crystal becomes multi-knot system, 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 from global winding and the mass term from leapfrogging motion. Based on a representation of projected state, a 3D SOC knot-crystal is reduced into a “two-sublattice” model with discrete spatial translation symmetry, of which the knot states are described by Land R(or the Wannier states cL,ivacuumand cR,jvacuum). We call it the Dirac model in sublattice-representation.

In sublattice-representation on geometric space, the equation of motion of knots is determined by the Schrödinger equation with the Hamiltonian

Hknot=ψĤknotψd3x,Ĥknot=ceffΓpknot+mknotceff2Γ5,E15

where ψtxis an four-component fermion field as ψtx=ψLtxψR(tx)ψL(tx)ψR(tx). Here, L,Rlabel two chiral-degrees of freedom that denote the two possible sub-lattices, ,label two spin degrees of freedom that denote the two possible winding directions. We have

Γ5=1ιx,E16

and

Γ1=σxιy,Γ2=σyιy,Γ3=σzιy.E17

pknot=knotkis the momentum operator. mknotceff2=2knotωplays role of the mass of knots and ceff=aJknot=2aω0play the role of light speed where ais a fixed length that denotes the half pitch of the windings on the knot-crystal.

In addition, the low energy effective Lagrangian of knots on 3D SOC knot-crystal is obtained as

L3D=ψ¯iγμ̂μmknotψE18

where ψ¯=ψγ0, γμare the reduced Gamma matrices,

γ1=γ0Γ1,γ2=γ0Γ2,γ3=γ0Γ3,E19

and

γ0=Γ5,γ5=iγ0γ1γ2γ3.E20

3.1.2. Dirac model in vortex-representation on geometric space

In this paper, we derive the effective Dirac model for a knot-crystal based on a representation of vortex degrees of freedom. We call it vortex-representation.

In Ref. [13], it was known that a knot 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, Aor B. The basis to define the microscopic structure of a knot is given by A, B, A, B.

We define operator of knot states by the region of the phase angle of a knot: for the case of ϕ0mod2ππ0, we have c0; for the case of ϕ0mod2π0π, we have c0. As shown in Figure 2 , we label the knots by Wannier state iA, i+1A, i+2A, i+3A….

Figure 2.

An illustration of knot states in vortex-representation: A and B denote two 1D vortex-lines. Here B* denotes conjugate representation of vortex-line-B. The curves with blue dots denote knots on the knot-crystal—the curves with blue dot above the line are denoted by c i † 0 and the curves with blue dot below the line are denoted by c i † 0 † .

To characterize the energy cost from global winding, we use an effective Hamiltonian to describe the coupling between two-knot states along xI-direction on 3D SOC knot-crystal

JcA/BiTA/B,A/BIcA/B,i+eIE21

with the annihilation operator of knots at the site i, cA/B,i=cA/B,,icA/B,,i. Jis the coupling constant between two nearest-neighbor knots. According to the generalized translation symmetry, the transfer matrices TA/B,A/BIalong xI-direction are defined by

TA,AI=TB,BI=eiak̂IσIE22

and

TA,BI=TB,AI=0.E23

After considering the spin rotation symmetry and the symmetry of vortex-membrane-A and vortex-membrane-B, the effective Hamiltonian from global winding energy can be described by a familiar formulation

Hcoupling=Ĥcoupling,B+Ĥcoupling,AE24

where

Ĥcoupling,A=Ji,IcA,ieiak̂IσIcA,i+eI+h.c.E25

and

Ĥcoupling,B=Ji,IcB,ieiak̂IσIcB,i+eI+h.c.E26

We then use path-integral formulation to characterize the effective Hamiltonian for a knot-crystal as

DψtxDψteiS/E27

where S=Ldtand L=iiψitψiHcoupling. To describe the knot states on 3D knot-crystal, we have introduced a four-component fermion field to be

ψx=ψA,txψB,txψA,txψB,txE28

where A,Blabel vortex degrees of freedom and ,label two spin degrees of freedom that denote the two possible winding directions along a given direction e.

In continuum limit, we have

Hcoupling=Ĥcoupling,B+Ĥcoupling,A=2aJkψA,kσxcoskx+σycosky+σzcoskzψA,k+2aJkψB,kσxcoskx+σycosky+σzcoskzψB,kE29

where the dispersion of knots is

EA/B,kceffkk0σ,E30

where k0=π2π2π2and ceff=2aJis the velocity. In the following part we ignore k0.

Next, we consider the mass term from leapfrogging motion, of which the angular frequency ω. For leapfrogging motion obtained by [10], the function of the two entangled vortex-membranes at a given point in geometric space is simplified by

zAx=0tzBx=0t=r021+eiωt1eiωt.E31

At t=0, we have zAxtzBxt=10; at t=πω, we have zAxtzBxt=01. The leapfrogging knot-crystal leads to periodic varied knot states, i.e., at t=0we have a knot on vortex-membrane-A that is denoted by σA; at t=πωwe have a knot on vortex-membrane-B denoted by σB. As a result, the leapfrogging motion becomes a global winding along time direction, tA, t+πωB, t+2πωA, t+3πωB, … See the illustration of vortex-representation of knot states for knot-crystal in Figure 2(c) . After a time period t=πω, a knot state ϕAmod2ππ0turns into a knot state ϕBmod2ππ0. Thus, we use the following formulation to characterize the leapfrogging process,

ψAψB.E32

After considering the energy from the leapfrogging process, a corresponding term is given by

2knotωψAψB+h.c.E33

From the global rotating motion denoted eiω0t, the winding states also change periodically. Because the contribution from global rotating motion eiω0tis always canceled by shifting the chemical potential, we do not consider its effect.

From above equation, in the limit k0we derive low energy effective Hamiltonian as

H3D2aJkψA,kσkψA,k+2aJkψB,kσkψB,k+2knotωk,σψA,σ,kψB,σ,kE34
=ceffΨTzσk̂Ψd3x+mknotceff2Ψτx1Ψd3x.E35

where

Ψx=ψA,txψB,txψA,txψB,tx.E36

We then re-write the effective Hamiltonian to be

H3D=ΨĤ3DΨd3xE37

and

Ĥ3D=ceffΓpknot+mknotceff2Γ5E38

where

Γ5=τx1,Γ1=τzσx,E39
Γ2=τzσy,Γ3=τzσz.

p=knotkis the momentum operator. Ψ=ψA,ψB,ψA,ψB,is the annihilation operator of four-component fermions. mknotceff2=2knotωplays role of the mass of knots and ceff=2aJknotplay the role of light speed where ais a fixed length that denotes the half pitch of the windings on the knot-crystal. In the following parts, we set knot=1and ceff=1.

Due to Lorentz invariance (see below discussion), the geometric space becomes geometric space-time, i.e., xyzxyzt. Here, we may consider Γand Γ5to be entanglement matrices along spatial and tempo direction in winding space-time, respectively. A complete set of entanglement matrices ΓΓ5is called entanglement pattern. The coordinate transformation along x/y/z/t-direction is characterize by eiΓk̂xand eiΓ5ω̂t, respectively. Now, the knot becomes topological defect of 3 + 1D entanglement—a knot is not only anti-phase changing along arbitrary spatial direction ebut also becomes anti-phase changing along tempo direction (along tempo direction, a knot switches a knot state A/Bto another knot state B/A).

Finally, the low energy effective Lagrangian of 3D SOC knot-crystal is obtained as

L3D=iΨtΨH3D=Ψ¯iγμ̂μmknotΨE40

where Ψ¯=Ψγ0, γμare the reduced Gamma matrices,

γ1=γ0Γ1,γ2=γ0Γ2,γ3=γ0Γ3,E41

and

γ0=Γ5=τx1,γ5=iγ0γ1γ2γ3.E42

In addition, we point out that there exists intrinsic relationship between the knot states of sublattice-representation and the knot states of vortex-representation

AB=ULRE43

where U=exp0ii0. From the sublattice-representation of knot states, the knot-crystal becomes an object with staggered R/L zeroes along x/y/z spatial directions and time direction; From the vortex-representation of knot states, the knot-crystal becomes an object with global winding along x/y/z spatial directions and time direction. See the illustration of knot states of vortex-representation on a knot-crystal in Figure 2 .

3.1.3. Emergent Lorentz-invariance

We discuss the emergent Lorentz-invariance for knot states on a knot-crystal.

Since the Fermi-velocity ceffonly depends on the microscopic parameter Jand a, we may regard ceffto be “light-velocity” and the invariance of light-velocity becomes an fundamental principle for the knot physics. The Lagrangian for massive Dirac fermions indicates emergent SO(3,1) Lorentz-invariance. The SO(3,1) Lorentz transformations SLoris defined by

SLorγμSLor1=γμE44

(μ=0,1,2,3) and

SLorγ5SLor1=γ5.E45

For a knot state with a global velocity v, due to SO(3,1) Lorentz-invariance, we can do Lorentz boosting on xtby considering the velocity of a knot,

tt=txv1v2,xx=xvt1v2.E46

We can do non-uniform Lorentz transformation SLorxton knot states Ψxt. The inertial reference-frame for quantum states of the knot is defined under Lorentz boost, i.e.,

ΨxtΨxt=SLorΨxt.E47

For a particle-like knot, a uniform wave-function of knot states ψtis

ψt=1Vei2ωt.E48

Under Lorentz transformation with small velocity v, this wave-function ψtis transformed into

ψt=1Vei2ωtψ=1Vei2ωt1Vei2ωtexpiEknottpknotxE49

where Eknotpknot22mknot, pknotωvand mknotc2=2ω. As a result, we derive a new distribution of knot-pieces by doing Lorentz transformation, that are described by the plane-wave wave-function 1Vei2ωtexpiEknottpknotx. The new wave-function 1VexpiEknottpknotxcomes from the Lorentz boosting SLor.

Noninertial system can be obtained by considering non-uniformly velocities, i.e., vΔvxt. According to the linear dispersion for knots, we can do local Lorentz transformation on xti.e.,

ttxt=txΔv1Δv2,xxxt=xΔvt1Δv2.E50

We can also do non-uniform Lorentz transformation SLorxton knot states Ψxt, i.e.,

ΨxtΨxxttxt=SLorxtΨxtE51

where the new wave-functions of all quantum states change following the non-uniform Lorentz transformation SLorxt. It is obvious that there exists intrinsic relationship between noninertial system and curved space-time.

3.2. Dirac model on winding space

In this part, we show the effective Dirac model of knot states on winding space.

The coordinate measurement of zero-lattice on winding space is the winding angles, Φ. Along a given direction e, after shifting the distance a, the winding angle changes π. The position is determined by two kinds of values: Xare integer numbers

X=XYZ=1πΦ1πΦmodπE52

and ϕdenote internal winding angles

ϕ=ϕxϕyϕz=ΦmodπE53

with ϕx,ϕy,ϕz0π.

Therefore, on winding space, the effective Hamiltonian turns into

Ĥ3D=Γpknot+mknotΓ5=ΓpX,knot+Γpϕ,knot+mknotΓ5E54

where pX=1aiddXand pϕ=1aiddϕ. Because of ϕj0π, quantum number of pϕis angular momentum Lϕand the energy spectra are 1aLϕ. If we focus on the low energy physics E1a(or Lϕ=0), we may get the low energy effective Hamiltonian as

Ĥ3DΓpX,knot+mknotΓ5.E55

We introduce 3 + 1D winding space-time by defining four coordinates on winding space, Φ=ΦΦtwhere Φtis phase changing under time evolution. For a fixed entanglement pattern ΓΓ5, the coordinate transformation along x/y/z/t-direction on winding space-time is given by eiΓΦ̂and eiΓ5Φ̂t, respectively.

For low energy physics, the position in 3 + 1D winding space-time is 3 + 1D zero-lattice of winding space-time labeled by four integer numbers, X=XX0where

X=1πΦ1πΦmodπ,X0=1πΦt1πΦtmodπ.E56

The lattice constant of the winding space-time is always πthat will never be changed. As a result, the winding space-time becomes an effective quantized space-time. Because of xμ=aXμ, the effective action on 3 + 1D winding space-time becomes

S3Da4X,Y,Z,X0L3DE57

where

L3D=Ψ¯i1aγμ̂μmknotΨ.E58

4. Deformed zero-lattice as curved space-time

In this section, we discuss the knot dynamics on smoothly deformed knot-crystal (or deformed zero-lattice). We point out that to characterize the entanglement evolution, the corresponding Biot-Savart mechanics for a knot on smoothly deformed zero-lattice is mapped to that in quantum mechanics on a curved space-time.

4.1. Entanglement transformation

Firstly, based on a uniform 3D knot-crystal (uniform entangled vortex-membranes), we introduce the concept of “entanglement transformation (ET)”.

Under global entanglement transformation, we have

ΨxtΨxt=ÛETxtΨxtE59

where

ÛETxt=eΦΓeΦtΓ5.E60

Here, δΦand δΦtare constant winding angles along spatial Φ-direction and that along tempo direction on geometric space-time, respectively. The dispersion of the excitation changes under global entanglement transformation.

In general, we may define (local) entanglement transformation, i.e.,

ÛETxt=eΦx.tΓeΦtx.tΓ5E61

where δΦxtand δΦtxtare not constant. We call a system with smoothly varied-(δΦxt, δΦtxt) deformed knot-crystal and its projected zero-lattice deformed (3 + 1D) zero-lattice.

4.2. Geometric description for deformed zero-lattice: curved space-time

For knots on a deformed zero-lattice, there exists an intrinsic correspondence between an entanglement transformation ÛETxtand a local coordinate transformation that becomes a fundamental principle for emergent gravity theory in knot physics.

For zero-lattice, ÛETxtchanges the winding degrees of freedom that is denoted by the local coordination transformation, i.e.,

ΦxtΦxt=Φxt+δΦxt,ΦtxtΦtxt=Φtxt+δΦtxt.E62

These equations also imply a curved space-time: the lattice constants of the 3 + 1D zero-lattice (the size of a lattice constant with 2πangle changing) are not fixed to be 2a, i.e.,

2a2aeffxtE63

The distance between two nearest-neighbor “lattice sites” on the spatial/tempo coordinate changes, i.e.,

Δx=x+exx=ex,Δx=x+exx=exxtE64

and

Δt=t+e0t=e0,Δt=t+e0t=e0xtE65

where eaa=0,1,2,3and eaxtare the unit-vectors of the original frame and the deformed frame, respectively. See the illustration of a 1 + 1D deformed zero-lattice on winding space-time with a non-uniform distribution of zeroes in Figure 3(d) .

Figure 3.

(a) An illustration of deformed knot-crystal; (b) an illustration of smoothly deformed relationship between winding angle Φ and spatial coordinate x . The zero-lattice in winding space is still uniform; while the zero-lattice in geometric space is deformed; (c) an illustration of a uniform 1 + 1D zero-lattice in geometric space-time; and (d) an illustration of a deformed 1 + 1D zero-lattice in geometric space-time.

However, for deformed zero-lattice, the information of knots in projected space is invariant: when the lattice-distance of zero-lattice changes aaeffxt, the size of the knots correspondingly changes aaeffxt. Therefore, due to the invariance of a knot, the deformation of zero-lattice does not change the formula of the low energy effective model for knots on winding space-time. Because one may smoothly deform the zero-lattice and get the same low energy effective model for knots on winding space-time, there exists diffeomorphism invariance, i.e.,

Knotinvarianceonwinding spacetimeDiffeomorphism invariance.E66

Therefore, from the view of mathematics, the physics on winding space-time is never changed! The invariance of the effective model for knots on winding space-time indicates the diffeomorphism invariance

Szerolatticea4X,Y,Z,X0Ψ¯i1aγμ̂μXmknotΨ.E67

On the other hand, the condition of very smoothly entanglement transformation guarantees a (local) Lorentz invariance in long wave-length limit. Under local Lorentz invariance, the knot-pieces of a given knot are determined by local Lorentz transformations.

According to the local coordinate transformation, the deformed zero-lattice becomes a curved space-time for the knots. In continuum limit Δka1and Δωω0, the diffeomorphism invariance and (local) Lorentz invariance emerge together. E. Witten had made a strong claim about emergent gravity, “whatever we do, we are not going to start with a conventional theory of non-gravitational fields in Minkowski space-time and generate Einstein gravity as an emergent phenomenon.” He pointed out that gravity could be emergent only if the notion on the space-time on which diffeomorphism invariance is simultaneously emergent. For the emergent quantum gravity in knot physics, diffeomorphism invariance and Lorentz invariance are simultaneously emergent. In particular, the diffeomorphism invariance comes from information invariance of knots on winding space-time—when the lattice-distance of zero-lattice changes, the size of the knots correspondingly changes.

To characterize the deformed 3 + 1D zero-lattice xxtt(xt), we introduce a geometric description. In addition to the existence of a set of vierbein fields ea, the space metric is defined by ηabeαaeβb=gαβwhere ηis the internal space metric tensor. The geometry fields (vierbein fields eaxtand spin connections ωabxt) are determined by the non-uniform local coordinates xxtt(xt). Furthermore, one needs to introduce spin connections ωabxtand the Riemann curvature two-form as

Rba=dωba+ωcaωbc=12Rbμνadxμdxν,E68

where RbμνaeαaebβRβμναare the components of the usual Riemann tensor projection on the tangent space. The deformation of the zero-lattice is characterized by

Rab=dωab+ωacωcb.E69

So the low energy physics for knots on the deformed zero-lattice turns into that for Dirac fermions on curved space-time

ScurvedST=gΨ¯eaμγaîμ+iωμmknotΨd4xE70

where ωμ=ωμ0iγ0i/2ωμijγij/2ij=1,2,3and γab=14γaγbab=0,1,2,3[15]. This model described by ScurvedSTis invariant under local (non-compact) SO(3,1) Lorentz transformation Sxt=eθabxtγabas

ΨxtΨxt=SxtΨxt,γμγμxt=SxtγμSxt1,ωμωμxt=SxtωμxtSxt1+SxtμSxt1.E71

γ5is invariant under local SO(3,1) Lorentz symmetry as

γ5γ5=Sxtγ5Sxt1=γ5.E72

In general, an SO(3,1) Lorentz transformation Sxtis a combination of spin rotation transformation R̂xt=R̂spinxtR̂spacextand Lorentz boosting SLorxt.

In physics, under a Lorentz transformation, a distribution of knot-pieces changes into another distribution of knot-pieces. For this reason, the velocity ceffand the total number of zeroes Nknotare invariant,

ceffceffceffE73

and

NknotNknotNknot.E74

4.3. Gauge description for deformed zero-lattice

4.3.1. Deformed entanglement matrices and deformed entanglement pattern

The deformation of the zero-lattice leads to deformation of entanglement pattern, i.e.,

ΓΓ5ΓxΓ5xE75

where

Γx=ÛETxΓÛETx1,Γ5x=ÛETxtΓ5ÛETx1.E76

xdenotes the space-time position of a site of zero-lattice, xt. Each entanglement matrix becomes a unit SO(4) vector-field on each lattice site. The deformed zero-lattice induced by local entanglement transformation ÛETxis characterized by four SO(4) vector-fields (four entanglement matrices) ΓxΓ5x. See the illustration of a 2D deformed zero-lattice in Figure 4(d) , in which the arrows denote deformed entanglement matrix Γ5x.

Figure 4.

(a) An illustration of the effect of an extra knot on a 1D knot-crystal along spatial direction; (b) an illustration of the effect of an extra knot on a 1D knot-crystal along tempo direction. Here A ∗ /B ∗ denotes conjugate representation of vortex-line-A/B; (c) the entanglement pattern for a uniform knot-crystal. The arrows denote the directions of entanglement matrices; and (d) the entanglement pattern for a knot-crystal with an extra knot at center. The purple spot denotes the knot. The red arrows that denote local tangential entanglement matrices have vortex-like configuration on 2D projected space.

4.3.2. Gauge description for deformed tempo entanglement matrix

Firstly, we study the unit SO(4) vector-field of deformed tempo entanglement matrix Γ5x. To characterize Γ5x, the reduced Gamma matrices γμis defined as

γ1=γ0Γ1,γ2=γ0Γ2,γ3=γ0Γ3,E77

and

γ0=Γ5=τx1,γ5=iγ0γ1γ2γ3.E78

Under this definition (γ0=Γ5), the effect of deformed zero-lattice from spatial entanglement transformation eiΓ1ΔΦx, eiΓ2ΔΦy, eiΓ3ΔΦzcan be studied due to

Γ5Γ5x=ÛETx/y/zxtΓ5ÛETx/y/zx1Γ5.E79

However, the effect of deformed zero-lattice from tempo entanglement transformation eΦtΓ5cannot be well defined due to

Γ5Γ5x=ÛETtxtΓ5ÛETtx1=Γ5.E80

We introduce an SO(4) transformation Ûxtthat is a combination of spin rotation transformation R̂xand spatial entanglement transformation (entanglement transformation along x/y/z-direction) ÛETx/y/zx=eΦxΓ, i.e.,

Ûx=R̂xÛETx/y/zx.E81

Here, denotes operation combination. Under a non-uniform SO(4) transformation Ûx, we have

γ0Ûxγ0Ûx1=γ0x=aγanaxE82

where n=n1n2n3ϕ00=nϕ00is a unit SO(4) vector-field. For the deformed zero-lattice, according to γ0xγ0, the entanglement matrix Γ5=γ0along tempo direction is varied, Γ5Γ5xΓ5.

In general, the SO(4) transformation is defined by Ûx=eΦabxγabγab=14γaγb. Under the SO(4) transformation, we have

γμγμx=ÛxγμÛx1,AμAμxt=ÛxtAμxÛx1+ÛxμÛx1.E83

In particular, γ5is invariant under the SO(4) transformation as

γ5γ5=Ûxγ5Ûx1=γ5.E84

The correspondence between index of γaand index of space-time xais

γ1x,γ2y,γ3z,γ0t.E85

We denote this correspondence to be

1,2,3,0ET1,2,3,0STE86

where 1,2,3,0ETdenotes the index order of γaand 1,2,3,0STdenotes the index order of space-time xa.

As a result, we can introduce an auxiliary gauge field Aμabxand use a gauge description to characterize the deformation of the zero-lattice. The auxiliary gauge field Aμabxis written into two parts [15]: SO(3) parts

Aijx=trγijÛxdÛx1E87

and SO(4)/SO(3) parts

Ai0x=trγi0ÛxdÛx1)=γ0dγix=γidγ0x.E88

The total field strength Fijxof i,j=1,2,3components can be divided into two parts

Fijx=Fij+Ai0Aj0.E89

According to pure gauge condition, we have Maurer-Cartan equation,

Fijx=Fij+Ai0Aj00E90

or

Fij=dAij+AikAkjAi0Aj0.E91

Finally, we emphasize the equivalence between γ0iand Γi, i.e., γ0iΓi.

4.3.3. Gauge description for deformed spatial entanglement matrix

Next, we study the unit SO(4) vector-field of deformed spatial entanglement matrix Γix. To characterize Γix, the reduced Gamma matrices γμis defined as

γ1=γ0Γj,γ2=γ0Γk,γ3=γ0Γ5,E92

and

γ0=Γi=τzσi,γ5=iγ0γ1γ2γ3.E93

Here, Γi, Γj, and Γkare three orthotropic spatial entanglement matrices. Under this definition (γ0=Γi), the effect of deformed zero-lattice from partial spatial/tempo entanglement transformation eiΓjΔΦj, eiΓkΔΦk, eiΓ5ΔΦtcan be studied due to

ΓiΓix=ÛETxj/xk/txtΓiÛETxj/xk/tx1Γi.E94

However, the effect of deformed zero-lattice from spatial entanglement transformation eΦtΓ5cannot be well defined due to

ΓiΓix=ÛETxixtΓiÛETxix1=Γi.E95

We use similar approach to introduce the gauge description. We can also define the reduced Gamma matrices γ˜μas

γ˜1=γ˜0Γ2,γ˜2=γ˜0Γ3,γ˜3=γ˜0Γ5,E96

and

γ˜0=Γi=τzσx,γ˜5=iγ˜0γ˜1γ˜2γ˜3.E97

The correspondence between index of γ˜aand index of space-time xais

γ˜1y,γ˜2z,γ˜3t,γ˜0x.E98

We denote this correspondence to be

1,2,3,0ET2,3,0,1ST.E99

Now, the SO(4) transformation U˜xt=eΦabxtγ˜abγ˜ab=14γ˜aγ˜bis not a combination of spin rotation symmetry and entanglement transformation along x/y/z-direction. However, for the case of aor bto be 0, U˜xt=eΦa0xtγ˜a0denotes the entanglement transformation along y/z/t-direction. The unit SO(4) vector-field on each lattice site becomes

U˜xγ˜0U˜x1=γ˜0x=aγ˜an˜axE100

where n˜=n˜1n˜2n˜3ϕ˜00is a unit vector-field. The auxiliary gauge field A˜abxare defined by

A˜abx=trγ˜ijU˜xdU˜x1.E101

According to pure gauge condition, we also have the following Maurer-Cartan equation,

F˜ij=dA˜ij+A˜ikA˜kjA˜i0A˜j0.E102

Finally, we emphasize the equivalence between γ˜0iand Γa, i.e., γ˜01Γ2, γ˜02Γ3, γ˜03Γ5.

4.3.4. Hidden SO(4) invariant for gauge description

In addition, there exists a hidden global SO(4) invariant for entanglement matrices along different directions in 3 + 1D (winding) space-time ΓΓ5Γ'Γ5'. To show the hidden SO(4) invariant, we define the reduced Gamma matrices γ˜μas

γ˜1=γ˜0Γ2,γ˜2=γ˜0Γ3,γ˜3=γ˜0Γ5,γ˜0=αΓ1+βΓ2+γΓ3+δΓ5,γ˜5=iγ˜0γ˜1γ˜2γ˜3E103

with α2+β2+γ2+δ2=1. Here, α, β, γ, δare constant.

Under this description, we can study the entanglement deformation along orthotropic spatial/tempo directions to x=αx+βy+γz+δt.

4.4. Relationship between geometric description and gauge description for deformed zero-lattice

Due to the generalized spatial translation symmetry there exists an intrinsic relationship between gauge description for entanglement deformation between two vortex-membranes and geometric description for global coordinate transformation of the same deformed zero-lattice.

On the one hand, to characterize the changes of the positions of zeroes, we must consider a curved space-time by using geometric description, x=xtx=xt. On the other hand, we need to consider a varied vector-field

γ0x=Ûxγ0Ûx1=aγanaxE104

by using gauge description. There exists intrinsic relationship between the geometry fields eaxa=1,2,3,0and the auxiliary gauge fields Aa0x.

For a non-uniform zero-lattice, we have

ΦxtΦxt=Φxt+δΦxt,ΦtxtΦtxt=Φtxt+δΦtxt.E105

On deformed zero-lattice, the “lattice distances” become dynamic vector fields. We define the vierbein fields eaxthat are supposed to transform homogeneously under the local symmetry, and to behave as ordinary vectors under local entanglement transformation along xa-direction,

eax=dxax=aπdΦax.E106

For the smoothly deformed vector-fields nix1, within the representation of Γ5=γ0we have

dΦix2π=nix=trγ0dγix=Ai0x,i=1,2,3.E107

Thus, the relationship between eixand Ai0xis obtained as

eix2aAi0x.E108

According to this relationship, the changing of entanglement of the vortex-membranes curves the 3D space.

On the other hand, within the representation of Γi=γ˜0we have

dΦax2π=n˜ax=trγ˜0dγ˜ax=A˜i0x,i=j,k,0,E109

and

e0x=dtx=aπdΦtx=2aA˜30x.E110

According to this relationship, the changing of entanglement of the vortex-membranes curves the 4D space-time.

In addition, we point out that for different representation of reduced Gamma matrix, there exists intrinsic relationships between the gauge fields Axand A˜x. After considering these relationships, we have a complete description of the deformed zero-lattice on the geometric space-time,

5. Emergent gravity

Gravity is a natural phenomenon by which all objects attract one another including galaxies, stars, human-being and even elementary particles. Hundreds of years ago, Newton discovered the inverse-square law of universal gravitation, F=GMmr2where Gis the Newton constant, ris the distance, and Mand mare the masses for two objects. One hundred years ago, the establishment of general relativity by Einstein is a milestone to learn the underlying physics of gravity that provides a unified description of gravity as a geometric property of space-time. From Einstein’s equations Rμν12Rgμν=8πGTμν, the gravitational force is really an effect of curved space-time. Here Rμνis the 2nd rank Ricci tensor, Ris the curvature scalar, gμνis the metric tensor, and Tμνis the energy-momentum tensor of matter.

In this section, we point out that there exists emergent gravity for knots on zero-lattice.

5.1. Knots as topological defects

5.1.1. Knot as SO(4)/SO(3) topological defect in 3 + 1D space-time

A knot corresponds to an elementary object of a knot-crystal; a knot-crystal can be regarded as composite system of multi-knot. For example, for 1D knot, people divide the knot-crystal into Nidentical pieces, each of which is just a knot.

From point view of information, each knot corresponds to a zero between two vortex-membranes along the given direction. For a knot, there must exist a zero point, at which ξAxis equal to ξBx. The position of the zero is determined by a local solution of the zero-equation, Fθx=0or ξA,θx=ξB,θx.

From point view of geometry, a knot (an anti-knot) removes (or adds) a projected zero of zero-lattice that corresponds to removes (or adds) half of “lattice unit” on the zero-lattice according to

Δxi=±aeffxt±a.E111

As a result, a knot looks like a special type of edge dislocation on 3 + 1D zero-lattice. The zero-lattice is deformed and becomes mismatch with an additional knot.

From point view of entanglement, a knot becomes topological defect of 3 + 1D winding space-time: along x-direction, knot is anti-phase changing denoted by eiΓ1ΔΦx, ΔΦx=π;along y-direction, knot is anti-phase changing denoted by eiΓ2ΔΦy, ΔΦy=π; along z-direction, knot is anti-phase changing denoted by eiΓ3ΔΦz, ΔΦz=π; along t-direction, knot is anti-phase changing denoted by eiΓ5ΔΦt, ΔΦt=π. Figure 4(a) and (b) shows an illustration a 1D knot.

In mathematics, to generate a knot at x0y0z0t0, we do global topological operation on the knot-crystal, i.e.,

eiΓ1ΔΦxx0E112

with ΔΦx=0,x<x0and ΔΦx=π,xx0;

eiΓ2ΔΦyx0E113

with ΔΦy=0,y<y0and ΔΦy=π,yy0;

eiΓ3ΔΦzx0E114

with ΔΦz=0,z<z0and ΔΦz=π,zx0;

eiΓ5ΔΦtx0E115

with ΔΦt=0,t<t0and ΔΦt=π,tt0. As a result, due to the rotation symmetry in 3 + 1D space-time, a knot becomes SO(4)/SO(3) topological defect. Along arbitrary direction, the local entanglement matrices around a knot at center are switched on the tangential sub-space-time.

5.1.2. Knot as SO(3)/SO(2) magnetic monopole in 3D space

To characterize the topological property of a knot on the 3 + 1D zero-lattice, we use gauge description. We firstly study the tempo entanglement deformation and define Γ5=γ0. Under this gauge description, we can only study the effect of a knot on three spatial zero-lattice.

When there exists a knot, the periodic boundary condition of knot states along arbitrary direction is changed into anti-periodic boundary condition,

ΔΦx=π,ΔΦy=π,ΔΦz=π.E116

Consequently, along given direction (for example x-direction), the local entanglement matrices on the tangential sub-space are switched by eiΓ1ΔΦxΔΦx=π. Along x-direction, in the limit of x, we have the local entanglement matrices on the tangential sub-space as Γ2and Γ3; in the limit of x, we have the local entanglement matrices on the tangential sub-space as eiΓ1ΔΦxΓ2eiΓ1ΔΦx=Γ2and eiΓ1ΔΦxΓ3eiΓ1ΔΦx=Γ3.

Because we have similar result along xi-direction for the system with an extra knot, the system has generalized spatial rotation symmetry. Due to the generalized spatial rotation symmetry, when moving around the knot, the local tangential entanglement matrices (we may use indices j, kto denote the sub space) must rotate synchronously. See the red arrows that denote local tangential entanglement matrices in Figure 4(c) and (d) . In Figure 4(d) , local tangential entanglement matrices induced by an extra (unified) knot shows vortex-like topological configuration in projected 2D space (for example, x-y plane). As a result, local tangential entanglement matrices induced by an extra knot can be exactly mapped onto that of an orientable sphere with fixed chirality.

To characterize the topological property of 3 + 1D zero-lattice with an extra (unified) knot, we apply gauge description and write down the following constraint

ρFdV=14πϵjkϵijkFjkjkdSiE117

where

Fij=dAij+AikAkjAi0Aj0E118

and ρF=gψψ. The upper indices of Fjkjklabel the local entanglement matrices on the tangential sub-space and the lower indices of Fjkjkdenote the spatial direction. The non-zero Gaussian integrate 14πϵjkϵijkFjkjkdSijust indicates the local entanglement matrices on the tangential sub-space Ai0Aj0to be the local frame of an orientable sphere with fixed chirality.

As a result, the entanglement pattern with an extra 3D knot is topologically deformed and the 3D knot becomes SO(3)/SO(2) magnetic monopole. From the point view of gauge description, a knot traps a “magnetic charge” of the auxiliary gauge field, i.e.,

NF=gΨΨd3x=qmE119

where qm=14πϵjkϵijkFjkjkdSiis the “magnetic” charge of auxiliary gauge field Ajk. For single knot NF=1, the “magnetic” charge is qm=1.

5.1.3. Knot as SO(3)/SO(2) magnetic monopole in 2 + 1D space-time

Next, we study the spatial entanglement deformation and define Γi=γ˜0. Under this gauge description, we can only study the effect of a knot on 2D spatial zero-lattice and 1D tempo zero-lattice.

In the 2 + 1D space-time, a knot also leads to π-phase changing,

ΔΦi=π,ΔΦj=π,ΔΦt=π.E120

Due to the spatial-tempo rotation symmetry, the knot also becomes SO(3)/SO(2) magnetic monopole and traps a “magnetic charge” of the auxiliary gauge field A˜jk, i.e.,

NF=gΨΨd3x=q˜mE121

where q˜mis the “magnetic” charge of auxiliary gauge field A˜ij. Remember that the correspondence between index of γ˜iand index of space-time xiis γ˜1y, γ˜2z, γ˜3t.

To characterize the induced magnetic charge, we write down another constraint

ρFdV=14πϵijϵijkF˜jkijdSiE122

where

F˜ij=dA˜ij+A˜ijA˜ijA˜i0A˜j0.E123

The upper indices of F˜ij=dF˜ij+F˜ikF˜kjdenote the local entanglement matrices on the tangential sub-space-time and the lower indices of F˜jkijdenote the spatial direction. Therefore, according to above equation, the 2 + 1D zero-lattice is globally deformed by an extra knot.

In general, due to the hidden SO(4) invariant, for other gauge descriptions γ˜0=αΓ1+βΓ2+γΓ3+δΓ5, a knot also play the role of SO(3)/SO(2) magnetic monopole and traps a “magnetic charge” of the corresponding auxiliary gauge field.

5.2. Einstein-Hilbert action as topological mutual BF term for knots

It is known that for a given gauge description, a knot is an SO(3)/SO(2) magnetic monopole and traps a “magnetic charge” of the corresponding auxiliary gauge field. For a complete basis of entanglement pattern, we must use four orthotropic SO(4) rotors Γ1xΓ2xΓ3xΓ5xand four different gauge descriptions to characterize the deformation of a knot (an SO(4)/SO(3) topological defect) on a 3 + 1D zero-lattice.

Firstly, we use Lagrangian approach to characterize the deformation of a knot (an SO(3)/SO(2) topological defect) on a 3D spatial zero-lattice, NF=qm. The topological constraint in Eq. (117) can be re-written into

i4trgΨ¯γiγ0i/2Ψ=ϵjkϵijk14πD̂iFjkjkE124

or

i4trgΨ¯ϖ00iγiγ0i/2Ψ=iϵ0ijkϵ0ijkϖ00i14πD̂iFjkjkE125

where D̂i=îi+iωiis covariant derivative in 3 + 1D space-time. ϖ0iis a field that plays the role of Lagrangian multiplier. The upper index iof ϖ0idenotes the local radial entanglement matrix around a knot, along which the entanglement matrix does not change. Thus, we use the dual field ϖ0ito enforce the topological constraint in Eq. (117). That is, to denote the upper index of Fjkthat is the local tangential entanglement matrices, we set antisymmetric property of upper index of ϖ0iand that of Fjk. Because ϖ0iand ω0ihave the same SO(3,1) generator γ0i/2, due to SO(3,1) Lorentz invariance we can do Lorentz transformation and absorb the dual field ϖ0iinto ω0i, i.e., ω0iω0i=ω0i+ϖ0i. As a result, the dual field ϖ0iis replaced by ω0i.

In the path-integral formulation, to enforce such topological constraint, we may add a topological mutual BF term SMBFin the action that is

SMBF1=14πϵ0ijkϵ0νλκR0ν0iFλκjkd4x=14πϵ0ijkR0iFjkE126

where

R0i=dω0i+ω0jωji.E127

From FjkAj0Ak0and eiej=2a2Aj0Ak0. The induced topological mutual BF term SMBF1is linear in the conventional strength in R0iand Fjk. This term is becomes

SMBF1=14π2a2ϵ0ijkR0iejek.E128

Next, we use Lagrangian approach to characterize the deformation of a knot (an SO(3)/SO(2) topological defect) on 2 + 1D space-time, NF=q˜m. Using the similar approach, we derive another topological mutual BF term SMBF2in the action that is

SMBF2=14πϵ0ijkϵ0νλκR˜0ν0iF˜λκjkd4x=14πϵ0ijkR˜0iF˜jkE129

where R˜0i=dω˜0i+ω˜0jω˜ji. From F˜k0A˜kjA˜j0and e˜ie˜j=2a2A˜j0A˜k0, this term becomes

SMBF2=14π2a2ϵijk0R˜0ie˜je˜k.E130

The upper index of R˜0idenotes entanglement transformation along given direction in winding space-time. We unify the index order of space-time into 1,2,3,0STand reorganize the upper index. The topological mutual BF term becomes 14π2a2ϵijk0Rijeke0. In Ref. [16, 17, 18, 19], a topological description of Einstein-Hilbert action is proposed by S. W. MacDowell and F. Mansouri. The topological mutual BF term proposed in this paper is quite different from the MacDowell-Mansouri action.

According to the diffeomorphism invariance of winding space-time, there exists symmetry between the entanglement transformation along different directions. Therefore, with the help of a complete set of definition of reduced Gamma matrices γμ, there exist other topological mutual BF terms SMBF,i. For the total topological mutual BF term SMBF=iSMBF,ithat characterizes the deformation of a knot (an SO(4)/SO(3) topological defect) on a 3 + 1D zero-lattice, the upper index of the topological mutual BF term Rijekelmust be symmetric, i.e., i,j,k,l=1,2,3,0.

By considering the SO(3,1) Lorentz invariance, the topological mutual BF term SMBFturns into the Einstein-Hilbert action SEHas

SMBF=SEH=116πa2ϵijklRijekel=116πGgRd4xE131

where Gis the induced Newton constant which is G=a2. The role of the Planck length is played by lp=a, that is the “lattice” constant on the 3 + 1D zero-lattice.

Finally, from above discussion, we derived an effective theory of knots on deformed zero-lattice in continuum limit as

S=Szerolattice+SEH=gxΨ¯eaμγaD̂μmknotΨd4x.+116πGgRd4xE132

where D̂μ=îμ+iωμ. The variation of the action Svia the metric δgμνgives the Einstein’s equations

Rμν12Rgμν=8πGTμν.E133

As a result, in continuum limit a knot-crystal becomes a space-time background like smooth manifold with emergent Lorentz invariance, of which the effective gravity theory turns into topological field theory.

For emergent gravity in knot physics, an important property is topological interplay between zero-lattice and knots. From the Einstein-Hilbert action, we found that the key property is duality between Riemann curvature Rijand strength of auxiliary gauge field Fkl: the deformation of entanglement pattern leads to the deformation of space-time.

In addition, there exist a natural energy cutoff ω0and a natural length cutoff a. In high energy limit (Δωω0) or in short range limit (Δxa), without well-defined 3 + 1D zero-lattice, there does not exist emergent gravity at all.

6. Discussion and conclusion

In this paper, we pointed out that owing to the existence of local Lorentz invariance and diffeomorphism invariance there exists emergent gravity for knots on 3 + 1D zero-lattice. In knot physics, the emergent gravity theory is really a topological theory of entanglement deformation. For emergent gravity theory in knot physics, a topological interplay between 3 + 1D zero-lattice and the knots appears: on the one hand, the deformation of the 3 + 1D zero-lattice leads to the changes of knot-motions that can be denoted by curved space-time; on the other hand, the knots trapping topological defects deform the 3 + 1D zero-lattice that indicates matter may curve space-time. The Einstein-Hilbert action SEHbecomes a topological mutual BF term SMBFthat exactly reproduces the low energy physics of the general relativity. In Table 1 , we emphasize the relationship between modern physics and knot physics.

Modern physicsKnot physics
MatterKnot: a topological defect of 3 + 1 D zero-lattice
MotionChanging of the distribution of knot-pieces
MassAngular frequency for leapfrogging motion
Inertial reference systemA knot under Lorentz boosting
Coordinate translationEntanglement transformation
Space-time3 + 1D zero-lattice of projected entangled vortex-membranes
Curved space-timeDeformed 3 + 1D zero-lattice
GravityTopological interplay between 3 + 1D zero-lattice and knots

Table 1.

The relationship between modern physics and knot physics.

In addition, this work would help researchers to understand the mystery in gravity. In modern physics, matter and space-time are two different fundamental objects and matter may move in (flat or curved) space-time. In knot physics, the most important physics idea for gravity is the unification of matter and space-time, i.e.,

MatterknotsSpacetimezerolattice.E134

One can see that matter (knots) and space-time (zero-lattice) together with motion of matter are unified into a simple phenomenon—entangled vortex-membranes and matter (knots) curves space-time (3 + 1D zero-lattice) via a topological way.

In the end of the paper, we address the possible physical realization of a 1D knot-crystal based on quantized vortex-lines in 4He superfluid. Because the emergent gravity in knot physics is topological interplay between zero-lattice and knots, there is no Einstein gravity on a 1D knot-crystal based on entangled vortex-lines in 4He superfluid. However, the curved space-time could be simulated.

Firstly, we consider two straight vortex-lines in 4He superfluid between opposite points on the system. Then, we rotate one vortex line around another by a rotating velocity ω0. Now, the winding vortex-line becomes a helical one described by r0eik0xiω0t+iϕ0with ω0κ4πln1k0a0k02. As a result, a knot-crystal is realized. For 4He superfluid, κ=h/mis the discreteness of the circulation owing to its quantum nature [2]. his Planck constant and mis atom mass of SF. So κ=h/mis about 103cm2/s. The length of the half pitch of the windings a=πk0is set to be 105cm, and the distance between two vortex lines r0is set to be 106cm. We then estimate the effective light speed ceffthat is defined by ceff=κk02πln1k0a012(a0denotes the vortex filament radius which is much smaller than any other characteristic size in the system). The effective light speed ceffis about 4 m/s. A non-uniform winding length leads to an effective curved 1 + 1D space-time.

However, at finite temperature, there exist mutual friction and phonon radiation for Kelvin waves on quantized vortex-lines in 4He superfluid. After considering these dissipation effects, the Kelvin waves are subject to Kolmogorov-like turbulence (even in quantum fluid [3, 4]).

Acknowledgments

This work is supported by NSFC Grant No. 11674026.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Su-Peng Kou (May 30th 2018). Topological Interplay between Knots and Entangled Vortex-Membranes, Superfluids and Superconductors, Roberto Zivieri, IntechOpen, DOI: 10.5772/intechopen.72809. Available from:

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