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Excited Particles

Consider a particle P described by some repetitive chain of events written as

\Psi^{\mathsf{P}} = \left( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2}, \, \mathsf{\Omega}_{3} \; \ldots  \;       \right)

where each repeated cycle is a bundle of quarks

\mathsf{\Omega}^{\mathsf{P}} = \left( \mathsf{q}_{1}, \, \mathsf{q}_{2},  \;   \ldots \; \mathsf{q}_{i}  \;   \ldots   \; \mathsf{q}_{N}     \right)

and each quark is described by its phase  \delta_{\theta}. Use this phase to sort quarks into a pair of sets,  \mathsf{P}_{\mdsmwhtcircle} and  \mathsf{P}_{\mdsmblkcircle} so that all quarks of the same phase are in the same set. Then  \mathsf{P}_{\mdsmwhtcircle} and  \mathsf{P}_{\mdsmblkcircle} are called phase components of P, and they are out of phase with each other. We write

\mathsf{\Omega}^{\mathsf{P}} = \left\{ \mathsf{P}_{\mdsmblkcircle}, \, \mathsf{P}_{\mdsmwhtcircle} \rule{0px}{9px} \right\}

and

\delta_{\theta} \left( \mathsf{P}_{\mdsmblkcircle} \right) = - \, \delta_{\theta} \left( \mathsf{P}_{\mdsmwhtcircle} \right)

Some sub-set of the quarks in  \mathsf{P}_{\mdsmwhtcircle} may be matched with quarks of the same type in  \mathsf{P}_{\mdsmblkcircle}. These quarks have phase symmetry with each other, so we use  \mathcal{S}_{\mdsmwhtcircle} and  \mathcal{S}_{\mdsmblkcircle} to symbolize these sub-sets. A different sub-set of quarks might be matched with anti-quarks of the same type. These quarks have phase anti-symmetry with each other, so we note them as  \mathcal{A}_{\mdsmwhtcircle} and  \mathcal{A}_{\mdsmblkcircle}. Some quarks in  \mathsf{P}_{\mdsmwhtcircle} might not correspond with any quarks in  \mathsf{P}_{\mdsmblkcircle} and vice versa. But such lopsided possibilities seem to be superfluous so we do not consider them further. Thus P is represented by the union of two entirely symmetric, and two purely anti-symmetric components. Quarks in the symmetric sets may vary independently of the quarks in the anti-symmetric sets. This is expressed mathematically as

\mathsf{\Omega}^{\mathsf{P}}  =  \left\{   \left\{\mathcal{S}_{\mdsmblkcircle}, \,   \mathcal{A}_{\mdsmblkcircle} \right\}, \ \left\{  \mathcal{S}_{\mdsmwhtcircle}, \,  \mathcal{A}_{\mdsmwhtcircle} \right\} \rule{0px}{11px} \right\}

where

\mathcal{S}_{\mdsmblkcircle} =  \mathcal{S}_{\mdsmwhtcircle}

\mathcal{A}_{\mdsmblkcircle} =  \overline{\mathcal{A}  _{\mdsmwhtcircle}  }

\delta_{\theta}^{ \mathcal{S}_{\mdsmblkcircle} }  =- \, \delta_{\theta}^{ \mathcal{S}_{\mdsmwhtcircle} }

and

\delta_{\theta}^{ \mathcal{A}_{\mdsmblkcircle} }  =- \, \delta_{\theta}^{ \mathcal{A}_{\mdsmwhtcircle} }

This arrangement provides a general way of describing particles that are moved or excited by the absorption of additional quarks. P is defined as an excited particle, or said to be in an excited-state, if it contains at least one anti-symmetric pair of quarks

\mathcal{A}_{\mdsmblkcircle} = \overline{ \mathcal{A}_{\mdsmwhtcircle} }    \ne    \left\{   \varnothing   \right\}

These anti-symmetric quark-pairs may be due to the absorption of a photon. Or more generally, to interactions with any field quanta.

Ground State Particles

We say that P is in its ground state if it has perfect phase symmetry. Then P has no anti-symmetric quark-pairs

\mathcal{A}_{\mdsmblkcircle} = \overline{ \mathcal{A}_{\mdsmwhtcircle} } = \left\{   \varnothing   \right\}

This definition constrains particle-models so that quark-coefficients must all be integer multiples of two when in the ground-state. It is why many nuclear particle models show patterns like 2-4-6 instead of 1-2-3. We can also evaluate the wavevector for a free particle, which is given by the sum

\displaystyle \overline{\kappa }^{\, \mathsf{P}}  = \dfrac{k_{\mathsf{F}}}{hc}   \sum_{i=1}^{N} \delta_{\theta}^{\,i} \, \overline{\rho}^{i}

Since the asymmetric sets are empty, the sum is taken over just the symmetric sets, so

    \begin{align*} \overline{\kappa }^{\, \mathsf{P}}  &= \dfrac{k_{\mathsf{F}}}{hc} \left( \delta_{\theta}^{\, \mathcal{S}_{\mdsmwhtcircle} } \hspace{-5px} \sum_{\mathsf{q} \, \in \, \mathcal{S}_{\mdsmwhtcircle} }  \! \overline{\rho}^{\, \mathsf{q}} \; + \;  \delta_{\theta}^{\, \mathcal{S}_{\mdsmblkcircle} }  \hspace{-5px}  \sum_{\mathsf{q} \, \in \, \mathcal{S}_{\mdsmblkcircle} } \!  \overline{\rho}^{\, \mathsf{q}}  \right) \\ &= \dfrac{k_{\mathsf{F}}}{hc} \left( \delta_{\theta}^{ \mathcal{S}_{\mdsmwhtcircle} } \, \overline{\rho}^{ \mathcal{S}_{\mdsmwhtcircle} } \; + \; \delta_{\theta}^{    \mathcal{S}_{\mdsmblkcircle} } \, \overline{\rho}^{ \mathcal{S}_{\mdsmblkcircle} } \right) \\ &= \dfrac{k_{\mathsf{F}}}{hc} \left( \delta_{\theta}^{ \mathcal{S}_{\mdsmwhtcircle} } + \delta_{\theta}^{    \mathcal{S}_{\mdsmblkcircle} }  \right) \, \overline{\rho}^{ \mathcal{S}_{\mdsmwhtcircle} } \ \ \mathsf{\text{because}} \ \ \overline{\rho}^{ \mathcal{S}_{\mdsmwhtcircle} } =  \overline{\rho}^{    \mathcal{S}_{\mdsmblkcircle} } \\ &= (0, 0, 0)   \ \ \ \   \mathsf{\text{because}}   \ \  \ \   \delta_{\theta}^{\mathcal{S}_{\mdsmwhtcircle}} = - \, \delta_{\theta}^{\mathcal{S}_{\mdsmblkcircle}} \rule{0px}{16px} \end{align*}

Thus the wavenumber of a free particle in its ground state is always zero

\kappa^{\mathsf{P}}       \equiv \left\| \, \overline{\kappa}^{\mathsf{P}} \right\| = 0

The Principal Quantum Number

Whenever P interacts by absorbing or emitting a photon, its degree or level of excitation changes too because photons contain anti-symmetric quark-pairs. This relationship is quantified by defining the principal quantum number as

\mathrm{n} \equiv  \dfrac{ n^{\mathsf{d}} }{4}

where  n^{\mathsf{d}} is the number of ordinary down quarks in P. Note that the italic letter  n is employed for quark coefficients, whereas the upright font  \mathrm{n} is reserved for the new quantum number. We use  \mathrm{n} to describe the level of excitation, and so down-quarks retain an important role despite having almost no internal energy. Remember that n^{\mathsf{d}} \ge  0 so the principal quantum number is never negative. And recall that quarks are conserved, so when P changes from some initial state  i, to some final state  f, by emitting a photon  \boldsymbol{\gamma}, we can write

\mathsf{P}_{i} \to \mathsf{P}_{f} + \boldsymbol{\gamma}

and

{\mathrm{n}} \! \left( \mathsf{P}_{i} \right)  = {\mathrm{n}} \! \left( \mathsf{P}_{f} \right) +  {\mathrm{n}} \! \left( \boldsymbol{\gamma} \right)

{\rm{n}} \! \left( {\mathsf{P}}_{i} \right)  =  \dfrac{N^{\mathsf{D}}\left( \boldsymbol{\gamma} \right) }{8}

and

{\rm{n}} \! \left( {\mathsf{P}}_{f} \right)  =  \dfrac{\Delta n^{\mathsf{D}}\left( \boldsymbol{\gamma} \right) }{8}

Next we take a closer look at the fields and photons that cause particle excitations.

Excited states are a mix of symmetric and asymmetric patterns, like this bead panel from Kalimantan.
Baby Carrier panel, Kayan or Kenyah people. Borneo 20th century, 27 x 27 cm. Photograph by D Dunlop.