Concept for hadrons

Hadrons, QCD, strong interaction

Idea: It is proposed to treat quarks as a four dimensional analogon to the well-studied phenomenon of partial dislocations. A model to treat the burgers vector and the dislocation core of these partial dislocations as vector fields, generating a combined vector field with $SU(3)$-symmetry, is suggested.

 

Approach

This article does not (yet) provide a complete theory for the modeling of hadrons in the framework of the elastic universe theory. Only the basic concepts are presented. These are:

• A $SU(3)$-symmetry is the symmetry group of of two four-vectors rotating around each other. A locally gauged field with this symmetry is describing a field of two four vectors.
• Partial dislocations are characterized by two vectors, the Burgers vector $\mathbf {b}$ and the partial core vector $\mathbf {d}$. A four dimensional analogon to partial dislocations is therefore described by two four vectors. For multiple partial dislocations, the characterization is achieved by a field of two four vectors and therefore a $SU(3)$-locally gauged field.
• The effects of confinement and asymptotic freedom are well-known for partial dislocations and can serve as a template for further modelling of hadrons by enhancing the known concepts from three to four dimensions.

 

$SU(3)$ as the symmetry of two rotating four-vectors

The situation of two unit vectors $\mathbf {v} ,\mathbf {w}$ in a four dimensional vector space $V$ which are rotating around each other, is considered. Each vector spans a three dimensional submanifold of $V$ which is isomporphic to the three-sphere $S_{3}$, which is again isomorphic to the compact, simple Lie-group $SU(2)=A_{1}$ . To clearify the effect of each vector on the other, and therefore to obtain the symmetry of the overall situation, an appropriate way to connect the two separate symmetry groups has to be found. In the next two paragraphs, it is shown by group theoretical and geomtetrical arguments, that the symmetry of two rotating four-vectors is described by the Lie-Group $SU(3)=A_{2}$ .

Group theoretical argument

The group of rotations of one four vector is the group $A_{1}=SU(2)\cong S_{3}$. $A_{1}$ is the notation used in Lie-Group Theory. This group is compact and simple. The property of compactness will remain unchanged if the situation of two rotating four vectors is considered (why?). The newly formed group by two rotating four-vectors, called $G$, will be simple again, since the group of two rotating four vectors does not contain any normal subgroups $N$ (with $gng^{-1}\in N\;{\text{for}}\;\forall g\in G,\;\forall n\in N$), and is irreducible. This condition makes it that the direct product $A_{1}\times A_{1}$ is not a canditate for the newly formed symmetry group. Still it shows that the new group has to be of rank two. The compact, simple Lie-Groups of rank two are then $A_{2}$, $B_{2}$ and $G_{2}$. The corresponding Dykin and Root Diagrams are depicted in <xr id="fig:su3.roots" /> .

<figure id="fig:su3.roots">

<xr id="fig:su3.roots" />: Root Systems and Dynkin Diagrams of the four Lie-Groups of rank two. $A_{1}\times A_{1}$ is reducible, while $A_{2}$, $B_{2}$ and $G_{2}$ are not. In addition, $B_{2}$ and $G_{2}$ are directed, which is illustrated by the arrows in the corresponding Dynkin Diagrams and the roots of inequal length. $A_{2}=SU(3)$ is the symmetry group of two rotating four vectors.

</figure>

$E_{2}$ and $G_{2}$ can be excluded because they do not treat the two objects equivalently, which is expressed in <xr id="fig:su3.roots" /> by the directed connections in the Dykin Diagrams and the lines of inequal length in the Root Diagrams. It rests $A_{2}=SU(3)$ as a group connecting two circles in an irreducible, undirected way. Thus, $SU(3)$ is the symmetry group describing two rotating four vectors.

Geometrical argument

A less general, but more descriptive treatment can be given by the construction of the Lie-Algebra $su(3)$ from two of its $su(2)$ subalgebras.

As an illustrative example, the construction of the Lie-Algebra of rotations in three dimensions $so(3)$ in a cartesian space from two of its $so(2)$ subalgebras corresponding to rotations in different planes is examined. Without loss of generality, let two $so(2)$ Lie algebras denote rotations in the $xy$ and $yz$ planes as depicted in <xr id="fig:su3.sphere" />.

These two algebras act both on the coordinate $y$. Therefore, for the construction of rotations in three dimensions, they are not connected in the form of the tensor product $so(2)\times so(2)$, but in matrix notation:

<equation id="eqn:su3.01" shownumber> $\sigma _{xy}=\left({\begin{array}{cc}{\begin{array}{|cc|}\hline 0&-1\\1&0\\\hline \end{array}}&{\begin{array}{c}0\\0\\\end{array}}\\{\begin{array}{cc}0&0\\\end{array}}&0\\\end{array}}\right),\qquad \sigma _{yz}=\left({\begin{array}{cc}0&{\begin{array}{cc}0&0\\\end{array}}\\{\begin{array}{c}0\\0\\\end{array}}&{\begin{array}{|cc|}\hline 0&-1\\1&0\\\hline \end{array}}\end{array}}\right)$ </equation>

acting on a cartesian space of vectors $\mathbf {v} =(x,y,z)^{T}$. $\sigma$ denotes the generator of the Lie algebra $so(2)$ in two dimensions. With this notation it is ensured, that both subalgebras act on their corresponding coordinates, sharing the $y$ direction. It gets immediately clear, that by combined rotations of $\sigma _{xy}$ and $\sigma _{yz}$ the whole rest of $S_{2}$ can be reached. On the level of Lie-Algebras this reads:

<equation id="eqn:su3.02" shownumber> Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_{xz}=[\sigma_{xy},\sigma_{yz}]= \begin{pmatrix}\fbox{0}&0&\fbox{1}\\0&0&0\\\fbox{-1}&0&\fbox{0}\end{pmatrix} } </equation>

And the third $so(2)$ subalgebra of $so(3)$ appears as the rotations in $xz$-plane. This subgroup makes $so(3)$ irreducible.

<figure id="fig:su3.sphere">

<xr id="fig:su3.sphere" />: Construction of $SO(3)$ by two $SO(2)$ submanifolds acting on different planes in a cartesian space. Put together, they span the sphere </math>S_2\cong SO(3)</math>. In the new formed situation, the third $SO(2)$ subgroup appears, making the symmetry group of the sphere irreducible.

</figure>

In analogy, the treatment can be made for two $su(2)$ Lie-Algebras corresponding to two rotating four vectors: Rotations generally apply on a plane and therefore on at least two coordinates. In a four dimensional space, this means, that for each rotation of one vector, two components of the other vector are rotated too, where two are left invariant. If the operations are acting on a complex vector space, this corresponds to an alignement of the two subgroups like:

<equation id="eqn:su3.03" shownumber> $\lambda _{1,2,3}=\left({\begin{array}{cc}{\begin{array}{|cc|}\hline \;\;\sigma _{i}\;\;\\\;\\\hline \end{array}}&{\begin{array}{c}0\\0\\\end{array}}\\{\begin{array}{cc}0&0\\\end{array}}&0\\\end{array}}\right),\qquad \lambda _{4,5,6}=\left({\begin{array}{cc}0&{\begin{array}{cc}0&0\\\end{array}}\\{\begin{array}{c}0\\0\\\end{array}}&{\begin{array}{|cc|}\hline \;\;\sigma _{i}\;\;\\\;\\\hline \end{array}}\end{array}}\right)$ </equation>

Where the $\sigma _{i}$ denote the Pauli-Matrices. Again, subsequent application of elements of both Lie Subalgebras constructs the third Lie Subalgebra, and by linear combinations the Gell-Mann Matrices are obtained, constructing the Lie-Algebra $su(3)$ and making it irreducible. Thus, $SU(3)$ is again the symmetry group of rotations of two four vectors.

 

Characterisation of partial dislocations

Following the treatment in material sciences textbooks (e.g. ), a dislocation in a crystal is a one dimensional topological defect, which can be parametrised by a dislocation core $\mathbf {d}$ and a Burgers Vector $\mathbf {b}$, as depicted in <xr id="fig:su3.dislocations" /> on the left. The Burgers Vector can be defined by introducing a Burgers Vector Density $d\mathbf {b}$ with the property:

<equation id="eqn:su3.04" shownumber> $\displaystyle \oint d\mathbf {b} =\mathbf {b}$ </equation>

If the integration path leads around the dislocation core. It can be shown, that the energy $E_{0}$ per dislocation line element is proportional to the square of the length of the Burgers Vector. This leads to an interesting phenomenon in real crystals: if the dislocation core is splitted into several parts with Burgers Vectors

<equation id="eqn:su3.05" shownumber> $\left|\mathbf {b} _{i}\right|={\left|\mathbf {b} \right| \over n}\qquad n\in \mathbb {N} ,\qquad \sum \mathbf {b} _{i}=\mathbf {b}$ </equation>

being fractions of the total Burgers Vector, the energy may be reduced:

<equation id="eqn:su3.06" shownumber> $E'=\sum k\left|\mathbf {b_{i}} \right|^{2}\leq E_{0}=k\left|\mathbf {b} \right|^{2}$ </equation>

Since the $\mathbf {b} _{i}$ are shorter than one unit length of the crystal lattice, neccessarily a stacking fault is introduced inbetween the partial dislocation cores (<xr id="fig:su3.dislocations" />). The energy of this fault increases proportional with discance $\mathbf {r}$ between the partial dislocations. Overall, this leads to a potential of the approximate form

<equation id="eqn:su3.07" shownumber> ${\begin{aligned}E&=E_{0}&\mathrm {if} \;\;E'+q\cdot r\leq E_{0}\\E&=E'+q\cdot r&\mathrm {if} \;\;E'+q\cdot r>E_{0}\end{aligned}}$ </equation>

introducing asymptotic freedom for small distances and a confinement distance, after which two new partial dislocation cores are formed inbetween.

<figure id="fig:su3.dislocations">

<xr id="fig:su3.dislocations" />: Illustration of a single dislocation core (left) and the core splitted into two partial dislocations, introducing a stacking fault in the lattice (right). The potential between the partial dislocation cores as a function of the distance $r$ is depicted on the bottom right.

</figure>

If one wants to develop a general theory for dislocations, it is common to replace the Burgers Vector $\mathbf {b}$ by a Burgers Vector Density as a vector field . In addition, for a complete theory of partial dislocations, the dislocation core can be spread out, and the line element can be replaced by a core density as a vector field in the area of the dislocation core.

For ensembles of partial dislocations, the Lagrangian can be replaced by a Lagrange Density and the invariance under rotations of $\mathbf {b}$ and $\mathbf {d}$ is replaced by a local gauge of the symmetry of rotations of two vectors. If extended to four dimensions, the two vectors become four-vectors, and the invariance under rotations becomes a $SU(3)$ symmetry.

 

Conclusion

It was shown that the symmetry group describing two four vectors rotating around each other is $A_{2}=SU(3)$, and the description as a local symmetry leads to a local $SU(3)$ gauge.

On the other hand, it was shown that partial dislocations are characterised by two vector fields: the Burgers vector field $\mathbf {b} (r)$ and the partial core vector field $\mathbf {d} (r)$. They are allowed to rotate locally around each other without change in the Lagrange Density. In this way partial dislocation-like defects build additional internal degrees of freedom.

The conclusion is, that partial dislocation-like defects in four dimensions can be described by the Lagrange Field describing general defects in four dimensions as found in chapters 2 & 3, and apply an additional local $SU(3)$ gauge. Overall, this treatment leads exactly to the well known equations of the $SU(3)$ Local Gauge Theory for the strong interaction as used in the Standard Model. Quarks can be interpreted as partial-dislocation like defects in a four dimensional space.

 

References

 

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