[Paper Review] A Field Theory of Knotted Solitons

Robert J. Finkelstein|ArXiv.org|Jan 12, 2007

Advanced Topics in Algebra1 references4 citations

TL;DR

This paper proposes a field theory unifying local $SU(2)\times U(1)$ electroweak symmetry with global $SU_q(2)$ symmetry of knotted solitons, treating elementary fermions as topological solitons (knotted flux tubes). By expanding quantum fields in $SU_q(2)$-deformed normal modes, the model replaces point particles with knotted structures, yielding fermion masses and gauge boson masses consistent with the standard model when constraints on deformation parameters are imposed.

ABSTRACT

The conjecture that the elementary fermions are knotted flux tubes permit the construction of a phenomenology that is not accessible from the standard electroweak theory. In order to carry these ideas further we have attempted to formulate the elements of a field theory in which local SU(2) x U(1), the symmetry group of standard electroweak theory, is combined with global SU_q(2), the symmetry group of knotted solitons.

Motivation & Objective

• To develop a field theory framework in which elementary fermions are modeled as knotted solitons rather than point particles.
• To unify the local $SU(2)\times U(1)$ symmetry of the standard electroweak theory with the global $SU_q(2)$ symmetry of knotted solitons.
• To construct a Lagrangian invariant under $[\text{local } SU(2)\times U(1)] \otimes [\text{global } U(1)]$ by deforming standard fields using $SU_q(2)$ irreducible representations.
• To reproduce standard model masses for gauge bosons and fermions through constraints on $q$-deformation parameters and vacuum expectation values.

Proposed method

• The standard electroweak Lagrangian is modified by replacing normal modes of gauge and Higgs fields with $SU_q(2)$-deformed matrix elements $D^{j}_{mm'}(a,\bar{a},b,\bar{b})$.
• The $q$-deformation is defined via non-commutative algebra: $ab = qba$, $a\bar{a} + b\bar{b} = 1$, with $q_1 = q^{-1}$, and $\langle n\rangle_1 = (q_1^{2n} - 1)/(q_1^2 - 1)$.
• The theory is constructed to be invariant under $[\text{local } SU(2)\times U(1)] \otimes [\text{global } U(1)]$, with gauge transformations induced on $D^{j}_{mm'}$ via $U_a D^{j}_{mm'} = e^{iQ_a \varphi_a/k} D^{j}_{mm'}$, where $Q_a \propto m + m'$.
• Fermion and gauge boson masses are derived from vacuum expectation values of $\langle 0| \bar{D}_\nu D^\nu |0\rangle$ and $\langle 0| \bar{D}_\nu \bar{\tau}_k \tau_k D_\nu |0\rangle$, with normalization constraints $I_{kk}/I = 1$ for $k = +, -, 3$.
• The Higgs mass term is modified to $\bar{L} \varphi R + \bar{R} \bar{\varphi} L$, with $L$, $R$, and $\varphi$ as tensor products of external standard model representations and internal $SU_q(2)$ soliton states.
• Fermion masses are computed as $m_n(w,r) = \rho(w,r) \langle n| \bar{D}^{3/2}_{w/2, (r+1)/2} D^{3/2}_{w/2, (r+1)/2} |n\rangle$, where $D^{3/2}$ is a $q$-deformed matrix element.

Experimental results

Research questions

• RQ1 Can the standard model's gauge and Higgs sectors be consistently extended to include knotted soliton structures via $SU_q(2)$ deformation?
• RQ2 How do $q$-deformed normal modes of quantum fields affect the mass spectrum of gauge bosons and fermions?
• RQ3 What constraints on the deformation parameter $q$ and vacuum parameter $\beta$ are required to reproduce the observed $W$, $Z$, and fermion masses?
• RQ4 How is gauge invariance preserved when introducing non-commutative $SU_q(2)$-deformed fields into a local $SU(2)\times U(1)$ framework?
• RQ5 Can the Weinberg angle be reproduced from the $q$-deformed structure of the $W^\pm$ and $Z$ couplings?

Key findings

• The gauge boson mass term is reproduced as $\partial_\mu \bar{\rho} \partial^\mu \bar{\rho} + g^2 \bar{\rho}^2 [W_+^\mu W_{+\mu} + W_-^\mu W_{-\mu} + \frac{1}{\cos^2\theta} Z^\mu Z_\mu]$ when normalization constraints $I_{kk}/I = 1$ are imposed.
• The coefficient $|c_-|^{-2} = |\beta|^6 \prod_{t=1}^3 (1 - q_1^{2t} |\beta|^2)$ is derived from the vacuum expectation value $\langle 0| b^3 \bar{b}^3 \bar{a}^3 a^3 |0\rangle / \langle 0| b^3 \bar{b}^3 |0\rangle$, fixing the $W^+$ coupling in terms of $q$ and $\beta$.
• The $W^-$ coupling coefficient is $|c_+|^{-2} = |\beta|^6 \prod_{t=0}^2 (1 - q^{2t} |\beta|^2)$, showing explicit $q$-dependence.
• The $Z$-boson coupling is determined by $|c_3|^{-2} = [f(|\beta|^2)]^2$, where $f(\bar{b}b) = D^3_{oo}$, a $q$-deformed polynomial from the $SU_q(2)$ representation.
• The fermion mass formula $m_n(w,r) = \rho(w,r) \langle n| \bar{D}^{3/2}_{w/2, (r+1)/2} D^{3/2}_{w/2, (r+1)/2} |n\rangle$ is derived, linking mass to soliton quantum numbers and $q$-deformation.
• The Weinberg angle is recovered via $\tan\theta = g_o/g$, consistent with the $q$-deformed structure of the $W^\pm$ and $Z$ couplings.