The relationship between modern physics and knot physics.

## 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

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

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

where

For the SOC knot-crystal, along *x*-direction, the plane Kelvin wave becomes *y*-direction, the plane Kelvin wave becomes *z*-direction, the plane Kelvin wave becomes

For a knot-crystal, another important property is generalized spatial translation symmetry that is defined by the translation operation

Here

### 2.2. Winding space and geometric space

For a knot-crystal, we can study it properties on a 3D space (*geometric space*. According to the generalized spatial translation symmetry, each spatial point (*winding space*. See the illustration in Figure 1(d) .

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

where

where the coordinates on winding space are

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

where

### 2.3. Zero-lattice

Before introduce zero-lattice, we firstly review the projection between two entangled vortex-membranes

where

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

According to the knot-equation

where

For a 3D leapfrogging SOC knot-crystal described by *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

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

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 *sublattice-representation*.

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

where

and

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

where

and

#### 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

We define operator of knot states by the region of the phase angle of a knot: for the case of

To characterize the energy cost from global winding, we use an effective Hamiltonian to describe the coupling between two-knot states along

with the annihilation operator of knots at the site

and

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

where

and

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

where

where

In continuum limit, we have

where the dispersion of knots is

where

Next, we consider the mass term from leapfrogging motion, of which the angular frequency

At

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

From the global rotating motion denoted

From above equation, in the limit

where

We then re-write the effective Hamiltonian to be

and

where

Due to Lorentz invariance (see below discussion), the geometric space becomes geometric space-time, i.e., *entanglement matrices* along spatial and tempo direction in winding space-time, respectively. A complete set of entanglement matrices *entanglement pattern*. The coordinate transformation along *x*/*y*/*z*/*t*-direction is characterize by

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

where

and

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

where *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

(

For a knot state with a global velocity

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

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

Under Lorentz transformation with small velocity

where

Noninertial system can be obtained by considering non-uniformly velocities, i.e.,

We can also do non-uniform Lorentz transformation

where the new wave-functions of all quantum states change following the non-uniform Lorentz transformation

### 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,

and

with

Therefore, on winding space, the effective Hamiltonian turns into

where

We introduce *3 + 1D winding space-time* by defining four coordinates on winding space, *x*/*y*/*z*/*t*-direction on winding space-time is given by

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,

The lattice constant of the winding space-time is always *quantized* space-time. Because of

where

## 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

where

Here,

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

where *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

For zero-lattice,

These equations also imply a curved space-time: the lattice constants of the 3 + 1D zero-lattice (the size of a lattice constant with

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

and

where

However, for deformed zero-lattice, the information of knots in projected space is invariant: when the lattice-distance of zero-lattice changes *diffeomorphism invariance*, i.e.,

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

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 *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

where

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

where

In general, an SO(3,1) Lorentz transformation

In physics, under a Lorentz transformation, a distribution of knot-pieces changes into another distribution of knot-pieces. For this reason, the velocity

and

### 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.,

where

#### 4.3.2. Gauge description for deformed tempo entanglement matrix

Firstly, we study the unit SO(4) vector-field of deformed tempo entanglement matrix

and

Under this definition (

However, the effect of deformed zero-lattice from tempo entanglement transformation

We introduce an SO(4) transformation *x*/*y*/*z*-direction)

Here,

where

In general, the SO(4) transformation is defined by

In particular,

The correspondence between index of

We denote this correspondence to be

where

As a result, we can introduce an auxiliary gauge field

and SO(4)/SO(3) parts

The total field strength

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

or

Finally, we emphasize the equivalence between

#### 4.3.3. Gauge description for deformed spatial entanglement matrix

Next, we study the unit SO(4) vector-field of deformed spatial entanglement matrix

and

Here,

However, the effect of deformed zero-lattice from spatial entanglement transformation

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

and

The correspondence between index of

We denote this correspondence to be

Now, the SO(4) transformation *x*/*y*/*z*-direction. However, for the case of *y*/*z*/*t*-direction. The unit SO(4) vector-field on each lattice site becomes

where

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

Finally, we emphasize the equivalence between

#### 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

with

Under this description, we can study the entanglement deformation along orthotropic spatial/tempo directions to

### 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,

by using gauge description. There exists intrinsic relationship between the geometry fields

For a non-uniform zero-lattice, we have

On deformed zero-lattice, the “lattice distances” become dynamic vector fields. We define the vierbein fields

For the smoothly deformed vector-fields

Thus, the relationship between

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

On the other hand, within the representation of

and

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

## 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,

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

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

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

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

In mathematics, to generate a knot at

with

with

with

with

#### 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

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

Consequently, along given direction (for example

Because we have similar result along *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

where

and

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.,

where

#### 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

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

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

where

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

where

The upper indices of

In general, due to the hidden SO(4) invariant, for other gauge descriptions

### 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

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,

or

where

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

where

From

Next, we use Lagrangian approach to characterize the deformation of a knot (an SO(3)/SO(2) topological defect) on 2 + 1D space-time,

where

The upper index of

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

By considering the SO(3,1) Lorentz invariance, the topological mutual BF term

where

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

where

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 *the deformation of entanglement pattern leads to the deformation of space-time*.

In addition, there exist a natural energy cutoff

## 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

Modern physics | Knot physics |
---|---|

Matter | Knot: a topological defect of 3 + 1 D zero-lattice |

Motion | Changing of the distribution of knot-pieces |

Mass | Angular frequency for leapfrogging motion |

Inertial reference system | A knot under Lorentz boosting |

Coordinate translation | Entanglement transformation |

Space-time | 3 + 1D zero-lattice of projected entangled vortex-membranes |

Curved space-time | Deformed 3 + 1D zero-lattice |

Gravity | Topological interplay between 3 + 1D zero-lattice and knots |

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.,

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 ^{4}He 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 ^{4}He superfluid. However, the curved space-time could be simulated.

Firstly, we consider two straight vortex-lines in ^{4}He superfluid between opposite points on the system. Then, we rotate one vortex line around another by a rotating velocity ^{4}He superfluid, ^{2}/s. The length of the half pitch of the windings

However, at finite temperature, there exist mutual friction and phonon radiation for Kelvin waves on quantized vortex-lines in ^{4}He 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.