Dynamical symmetry breaking in Yang–Mills geometrodynamics

Abstract

We will analyze through a first order perturbative formulation the local loss of symmetry when a source of non-Abelian Yang–Mills and gravitational fields interacts with an external agent that perturbes the original geometry associated to the source. Then, the symmetry in Abelian and non-Abelian field structures in four-dimensional Lorentzian spacetimes is displayed through the existence of local planes of symmetry that we previously called blades one and two. These orthogonal local planes diagonalize the stress-energy tensor and every vector in these planes is an eigenvector of the stress-energy tensor. The loss of symmetry will be manifested by the tilting of these planes under the influence of the external agent. It was also found already that there is an algorithm to block-diagonalize the Yang–Mills field strength in a local gauge invariant way. The loss of symmetry will also be manifested by the tilting of these planes that block-diagonalize the Yang–Mills field strength under the influence of the external agent. Using perturbative analysis from a previous manuscript dealing with the Abelian case, we will demonstrate how to develop an algorithm for constructing local conserved currents inside both local orthogonal planes of stress-energy diagonalization. As the interaction proceeds, the planes will tilt perturbatively, and in this strict sense the original local symmetry will be lost. But we will prove that the new orthogonal planes or blades at the same point will correspond after the tilting generated by perturbation to a new symmetry, with associated new local currents, both on each new local planes of symmetry. Old symmetries will be broken, new symmetries will arise. There will be a local symmetry evolution in the non-Abelian case as well. This result will produce a new theorem on dynamic symmetry evolution. This new classical model will be useful in order to better understand anomalies in quantum field theories.

Appendices

Appendix I

This appendix is introducing the object $\Sigma ^{\alpha \beta }$. This object according to the matrix definitions introduced in the references is Hermitic. The use of this object in the construction of our tetrads allows for the local SU(2) gauge transformations S, to get in turn transformed into purely geometrical transformations. That is, local rotations of the U(1) electromagnetic tetrads $E_{\alpha }^{\,\,\rho }$ included in the definitions of $X^{\sigma } = Y^{\sigma } = Tr[\Sigma ^{\alpha \beta }\,E_{\alpha }^{\,\,\rho }\, E_{\beta }^{\,\,\lambda }\,*\xi _{\rho }^{\,\,\sigma }\,*\xi _{\lambda \tau }\,A^{\tau }]$ in Sect. 3. The object $\sigma ^{\alpha \beta }$ is defined as $\sigma ^{\alpha \beta } = \sigma _{+}^{\alpha }\,\sigma _{-}^{\beta }-\sigma _{+}^{\beta }\,\sigma _{-}^{\alpha }$, [44, 45]. The object $\sigma _{\pm }^{\alpha }$ arises when building the Weyl representation for left handed and right handed spinors. According to [45], it is defined as $\sigma _{\pm }^{\alpha } = (\mathbf{1},\pm \sigma ^{i})$, where $\sigma ^{i}$ are the Pauli matrices for $i = 1\cdots 3$. Under the $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$ spinor representations of the Lorentz group it transforms as,

$$\begin{aligned} S_{(1/2)}^{-1}\,\sigma _{\pm }^{\alpha }\,S_{(1/2)} = \Lambda ^{\alpha }_{\,\,\,\gamma }\,\sigma _{\pm }^{\gamma }. \end{aligned}$$

(103)

Equation (103) means that under the spinor representation of the Lorentz group, $\sigma _{\pm }^{\alpha }$ transform as vectors. In (103), the matrices $S_{(1/2)}$ are local, as well as $\Lambda ^{\alpha }_{\,\,\,\gamma }$ [45]. The SU(2) elements can be considered to belong to the Weyl spinor representation of the Lorentz group. Since the group SU(2) is homomorphic to SO(3), they just represent local space rotations. It is also possible to define the object $\sigma ^{\dagger \alpha \beta } = \sigma _{-}^{\alpha }\,\sigma _{+}^{\beta }-\sigma _{-}^{\beta }\,\sigma _{+}^{\alpha }$, analogously. Then, we have,

$$\begin{aligned} \imath \, \left( \sigma ^{\alpha \beta } + \sigma ^{\dagger \alpha \beta } \right)= & {} \left\{ \begin{array}{ll} 0 \,\,\,\,\, \text{ if } \alpha = 0\hbox { and }\beta = i\\ 4\,\epsilon ^{ijk}\,\sigma ^{k} \,\,\,\,\, \text{ if } \alpha = i\hbox { and } \beta = j, \end{array} \right. \\ \sigma ^{\alpha \beta } - \sigma ^{\dagger \alpha \beta }= & {} \left\{ \begin{array}{ll} -4\,\sigma ^{i} \,\,\,\,\, \text{ if } \alpha = 0\hbox { and }\beta = i\\ 0 \,\,\,\,\, \text{ if } \alpha = i\hbox { and }\beta = j . \end{array} \right. \end{aligned}$$

We might then call $\Sigma _{ROT}^{\alpha \beta } = \imath \, \left( \sigma ^{\alpha \beta } + \sigma ^{\dagger \alpha \beta } \right)$, and $\Sigma _{BOOST}^{\alpha \beta } = \imath \, \left( \sigma ^{\alpha \beta } - \sigma ^{\dagger \alpha \beta } \right)$. Therefore, a possible choice for the object $\Sigma ^{\alpha \beta }$ could be for instance $\Sigma ^{\alpha \beta } = \Sigma _{ROT}^{\alpha \beta } + \Sigma _{BOOST}^{\alpha \beta }$. This is a particularly suitable choice when we consider proper Lorentz transformations of the tetrad vectors nested within the structure of the gauge vectors $X^{\mu }$ and $Y^{\mu }$. For spatial, that is, rotations of the U(1) electromagnetic tetrad vectors which in turn are nested within the structure of the two gauge vectors $X^{\mu }$ and $Y^{\mu }$, as is the case under study in this paper, we can simply consider $\Sigma ^{\alpha \beta } = \Sigma _{ROT}^{\alpha \beta }$. These possible choices also ensure the Hermiticity of gauge vectors. Since in the definition of the gauge vectors $X^{\mu }$ and $Y^{\mu }$ we are taking the trace, then $X^{\mu }$ and $Y^{\mu }$ are real.

Appendix II

In order to compare local currents conservation laws we will need the first order perturbed covariant derivative of a vector. Therefore in this section we display the main steps in these calculations. We can start with the standard expression for the covariant derivative of a vector,

$$\begin{aligned} V^{\lambda }_{\,\,\,\,;\mu } = {\partial V^{\lambda } \over \partial x^{\mu } } + \Gamma ^{\lambda }_{\mu \nu }\,V^{\nu }, \end{aligned}$$

(104)

where the expression for the affine connection is the usual,

$$\begin{aligned} \Gamma ^{\lambda }_{\mu \nu } = {1 \over 2}\,g^{\lambda \sigma }\,\left( {\partial g_{\mu \sigma } \over \partial x^{\nu }} + {\partial g_{\nu \sigma } \over \partial x^{\mu }} - {\partial g_{\mu \nu } \over \partial x^{\sigma }} \right) . \end{aligned}$$

(105)

Following the literature in perturbative schemes, see [31, 46,47,48,49] and references therein as examples, we can write the first order perturbed affine connection as,

$$\begin{aligned} {\tilde{\Gamma }}^{\lambda }_{\mu \nu } = {1 \over 2}\,g^{\lambda \sigma }\,\left( h_{\mu \sigma \,;\nu } + h_{\nu \sigma \,;\mu } - h_{\mu \nu \,;\sigma } \right) , \end{aligned}$$

(106)

where the covariant derivatives in (106) are calculated with the unperturbed (105) affine connection and the perturbations to the metric tensor have been introduced in Eq. (81). We proceed then to write to first order the perturbed covariant derivative of a perturbed contravariant vector,

$$\begin{aligned} {\tilde{\nabla }}_{\mu }\,{\tilde{V}}^{\lambda } = {\partial {\tilde{V}}^{\lambda } \over \partial x^{\mu } } + \Gamma ^{\lambda }_{\mu \nu }\,{\tilde{V}}^{\nu } + \varepsilon \,{\tilde{\Gamma }}^{\lambda }_{\mu \nu }\,V^{\nu }, \end{aligned}$$

(107)

where we have used now the operator $\nabla$ to indicate covariant derivative for notational convenience since we can write a tilde above it. The perturbed vector can be written ${\tilde{V}}^{\lambda } = V^{\lambda } + \varepsilon \, \psi ^{\lambda }$, where $\psi ^{\lambda }$ is a local vector field. When we think of $V^{\lambda }$ in a concrete example in this manuscript, we will be thinking of the local currents $J^{\lambda }$. It is important to stress that we are studying genuine physical perturbations to the gravitational, Abelian and non-Abelian fields by external agents to the preexisting source. We are not introducing first order coordinate transformations of the kind ${\tilde{x}}^{\alpha } = x^{\alpha } + \,\varepsilon \,\,\, \zeta ^{\alpha }$, where the local vector field $\zeta ^{\alpha }(x^{\sigma })$ is associated to a first order infinitesimal local coordinate transformation scheme [31].

Appendix III

We will establish the relationship between the Yang–Mills stress-energy tensor $T^{(ym)}_{\mu \nu }$ in Eq. (36) and the local gauge invariants of the type ${\overline{f}}_{\mu \nu }$ which is nothing but compact notation for $Tr[\vec {n}\, \cdot \, f_{\mu \nu }]$ a local SU(2) gauge invariant object. Therefore, let us introduce three orthogonal unit local vectors in isospace $\vec {n_{1}} = n_{1}^{a}\,\sigma ^{a}$, $\vec {n_{2}} = n_{2}^{a}\,\sigma ^{a}$ and $\vec {n_{3}} = n_{3}^{a}\,\sigma ^{a}$ described in terms of local cosines as in expressions (60–61). Let us block-diagonalize the field strength independently in all three isospace directions, see Ref. [27]. For the sake of notational simplicity we write,

$$\begin{aligned} Tr[\vec {n_{1}}\, \cdot \, f_{\mu \nu }]= & {} -2\,\sqrt{-Q_{n_{1}}/2}\,\,A_{n_{1}}\,\,\,{\overline{S}}_{(o)[\mu }\,{\overline{S}}_{(1)\nu ]}\nonumber \\&+ 2\,\sqrt{-Q_{n_{1}}/2}\,\,B_{n_{1}}\,\,{\overline{S}}_{(2)[\mu }\,{\overline{S}}_{(3)\nu ]} \end{aligned}$$

(108)

$$\begin{aligned} Tr[\vec {n_{2}}\, \cdot \, f_{\mu \nu }]= & {} -2\,\sqrt{-Q_{n_{2}}/2}\,\,A_{n_{2}}\,\,\,{\overline{T}}_{(o)[\mu }\,{\overline{T}}_{(1)\nu ]}\nonumber \\&+ 2\,\sqrt{-Q_{n_{2}}/2}\,\,B_{n_{2}}\,\,{\overline{T}}_{(2)[\mu }\,{\overline{T}}_{(3)\nu ]} \end{aligned}$$

(109)

$$\begin{aligned} Tr[\vec {n_{3}}\, \cdot \, f_{\mu \nu }]= & {} -2\,\sqrt{-Q_{n_{3}}/2}\,\,A_{n_{3}}\,\,\,{\overline{U}}_{(o)[\mu }\,{\overline{U}}_{(1)\nu ]}\nonumber \\&+ 2\,\sqrt{-Q_{n_{3}}/2}\,\,B_{n_{3}}\,\,{\overline{U}}_{(2)[\mu }\,{\overline{U}}_{(3)\nu ]} . \end{aligned}$$

(110)

It is clear then, that it is possible to express the field strength in general as,

$$\begin{aligned} f_{\mu \nu }= & {} \vec {n_{1}}\,\,Tr[\vec {n_{1}}\, \cdot \, f_{\mu \nu }] + \vec {n_{2}}\,\,Tr[\vec {n_{2}}\, \cdot \, f_{\mu \nu }] + \vec {n_{3}}\,\,Tr[\vec {n_{3}}\, \cdot \, f_{\mu \nu }] . \end{aligned}$$

(111)

This expression is general, since the three unit orthogonal $Tr[\vec {n_{i}}\, \cdot \,\vec {n_{j}}] = \delta _{ij}$ local isovectors $\vec {n_{1}}$, $\vec {n_{2}}$ and $\vec {n_{3}}$ are arbitrary, it just suffices for them not to be trivial and be described through equations like (60–61). The important point here is that the three new local objects given by $Tr[\vec {n_{1}}\, \cdot \, f_{\mu \nu }]$, $Tr[\vec {n_{2}}\, \cdot \, f_{\mu \nu }]$ and $Tr[\vec {n_{3}}\, \cdot \, f_{\mu \nu }]$ are invariant under local spatial three dimensional rotations in isospace. Which is equivalent to say, under local non-Abelian SU(2) gauge transformations in the local unit iso-surface, see Ref. [27]. Therefore, we can establish six local observables. They are the three pairs ($\sqrt{-Q_{n_{1}}/2}\,\,A_{n_{1}}, \sqrt{-Q_{n_{1}}/2}\,\,B_{n_{1}}$), ($\sqrt{-Q_{n_{2}}/2}\,\,A_{n_{2}}, \sqrt{-Q_{n_{2}}/2}\,\,B_{n_{2}}$) and ($\sqrt{-Q_{n_{3}}/2}\,\,A_{n_{3}}, \sqrt{-Q_{n_{3}}/2}\,\,B_{n_{3}}$). These three pairs of local objects have the following properties. First $(A_{n_{j}})^{2}+(B_{n_{j}})^{2}=1$ separately for $j=1\ldots 3$. They are locally invariant under general coordinate transformations. They are locally invariant under $SU(2) \times U(1)$ gauge transformations, see Ref. [23]. Therefore, they are local observables. Next, we create one of the type of terms in the Yang–Mills stress-energy tensor $T^{(ym)}_{\mu \nu }$ as in Eq. (36).

$$\begin{aligned} Tr[f_{\mu \nu }\, \cdot \,f^{\mu \rho }]= & {} Tr[ \left( \vec {n_{1}}\,\,Tr[\vec {n_{1}}\, \cdot \, f_{\mu \nu }] + \vec {n_{2}}\,\,Tr[\vec {n_{2}}\, \cdot \, f_{\mu \nu }] + \vec {n_{3}}\,\,Tr[\vec {n_{3}}\, \cdot \, f_{\mu \nu }]\right) \, \cdot \,\nonumber \\&\left( \vec {n_{1}}\,\,Tr[\vec {n_{1}}\, \cdot \, f^{\mu \rho }] + \vec {n_{2}}\,\,Tr[\vec {n_{2}}\, \cdot \, f^{\mu \rho }] + \vec {n_{3}}\,\,Tr[\vec {n_{3}}\, \cdot \, f^{\mu \rho }]\right) ] \nonumber \\= & {} \sum _{h=1}^{3} Tr[\vec {n_{h}}\, \cdot \, f_{\mu \nu }]\,Tr[\vec {n_{h}}\, \cdot \, f^{\mu \rho }], \end{aligned}$$

(112)

since $Tr[\vec {n_{i}}\, \cdot \, \vec {n_{j}}] = \delta _{ij}$.

$$\begin{aligned}&\sum _{h=1}^{3} Tr[\vec {n_{h}}\, \cdot \, f_{\mu \nu }]\,Tr[\vec {n_{h}}\, \cdot \, f^{\mu \rho }] \nonumber \\&\quad = (\sqrt{-Q_{n_{1}}/2})^{2}\,\,\left( (A_{n_{1}})^{2}\,\,\,[-{\overline{S}}_{(1)\nu }\,{\overline{S}}_{(1)}^{\rho } + {\overline{S}}_{(o)}^{\rho }\,{\overline{S}}_{(o)\nu }] \right. \nonumber \\&\left. \qquad + (B_{n_{1}})^{2}\,\,[{\overline{S}}_{(2)\nu }\,{\overline{S}}_{(2)}^{\rho } + {\overline{S}}_{(3)}^{\rho }\,{\overline{S}}_{(3)\nu }]\right) \nonumber \\&\qquad + (\sqrt{-Q_{n_{2}}/2})^{2}\,\,\left( (A_{n_{2}})^{2}\,\,\,[-{\overline{T}}_{(1)\nu }\,{\overline{T}}_{(1)}^{\rho } + {\overline{T}}_{(o)}^{\rho }\,{\overline{T}}_{(o)\nu }] \right. \nonumber \\&\left. \qquad + (B_{n_{2}})^{2}\,\,[{\overline{T}}_{(2)\nu }\,{\overline{T}}_{(2)}^{\rho } + {\overline{T}}_{(3)}^{\rho }\,{\overline{T}}_{(3)\nu }]\right) + \nonumber \\&\qquad + (\sqrt{-Q_{n_{3}}/2})^{2}\,\,\left( (A_{n_{3}})^{2}\,\,\,[-{\overline{U}}_{(1)\nu }\,{\overline{U}}_{(1)}^{\rho } + {\overline{U}}_{(o)}^{\rho }\,{\overline{U}}_{(o)\nu }] \right. \nonumber \\&\left. \qquad + (B_{n_{3}})^{2}\,\,[{\overline{U}}_{(2)\nu }\,{\overline{U}}_{(2)}^{\rho } + {\overline{U}}_{(3)}^{\rho }\,{\overline{U}}_{(3)\nu }]\right) \end{aligned}$$

(113)

We can also calculate the other half of the stress-energy tensor $T^{(ym)}_{\mu \rho }=Tr[f_{\mu \nu }\, \cdot \,f^{\mu \rho }]+Tr[*f_{\mu \nu }\, \cdot \,*f^{\mu \rho }]$ with the expression $Tr[*f_{\mu \nu }\, \cdot \,*f^{\mu \rho }] = \sum _{h=1}^{3} Tr[\vec {n_{h}}\, \cdot \,*f_{\mu \nu }]\,Tr[\vec {n_{h}}\, \cdot \, *f^{\mu \rho }]$. Three orthogonal local unit isovectors determine three pair of local orthogonal planes associated to the local, covariant and gauge invariant block-diagonalization of different orthogonal projections of the non-Abelian field strength. The local diagonalization of the stress-energy tensor per se, as a whole tensor $T^{(ym)}_{\mu \rho }$ involves just a pair of local orthogonal planes of symmetry as we learnt in our analysis in Ref. [23].