1 INTRODUCTION

Conventional quantum field theory (QFT) was developed to describe structureless elementary particles, their interaction with each other and with their environment [1–4]. An example is the most successful theory that accounts for the electromagnetic interaction of electrons with photons, called quantum electrodynamics (QED) [5–7]. However, in its early days, QFT did not succeed in describing the interaction of hadrons even at low energies because they are not elementary. It was later replaced by the more successful quantum chromodynamics (QCD), which is the QFT for quarks and gluons as structureless elementary particles [8, 9]. In one of its representations, QFT is visualized by diagrams known as the Feynman diagrams that consist of points (vertices) connected by lines (propagators) [10, 11]. The lines represent free propagation of elementary particles and the points represent the interaction among particles meeting at those points.

If the elementary particle has a structureFootnote 1 of an infinitesimal size relative to a low energy scale, it is then believed that a faithful QFT should still be used successfully at those low energies. It is only at higher energies that hidden structural effects become significant. Therefore, we adopt the view that a more useful and practical QFT should include particle structure in the formulation since what is believed to be a structureless elementary particle at one energy scale (e.g., the nucleon at KeV energies prior to the 1930s) may turn out to be a composite particle at a higher energy scale. After all, if the particle is indeed structureless for all energies, then that could easily be accommodated in the theory by taking the structure as null. Consequently, we introduce a QFT for particles that may or may not be elementary (i.e., particles that may have internal structure or built from elementary constituents). It will become evident in the text that the proposed theory has a clear and strong algebraic underpinning. However, it is fundamentally and technically different from that which is commonly known in the mathematics/physics literature as Algebraic Quantum Field Theory (AQFT). A brief and recent description of AQFT can be found in [12] and references cited therein. We refer to the theory introduced here as “Structural Algebraic Quantum Field Theory” (SAQFT)Footnote 2 and for simplicity we consider in Section 2 scalar non-elementary particles in 3+1 Minkowski space-time. Moreover, in an appendix, we give a brief mathematical depiction of SAQFT for the Dirac spinor with structure but in 1 + 1 space-time. In Section 4, we give an example of a system with nonlinear coupling and show how to perform scattering calculations by means of a revised version of the rules of Feynman diagrams appropriate for use in SAQFT. The presentation here is elementary and requires basic knowledge in QFT and orthogonal polynomials [13–16].

In the literature, different approaches based on conventional QFT can be found that are rather successful in the description of low energy physics. However, if one strictly applies conventional QFT to describe hadrons at low energies in nuclear physics, then the results obtained would be far off. This is one of the reasons that many articles, books and monographs in nuclear physics introduce and discuss a large number of methods and techniques that are distinct from QFT to handle nucleons and their interactions. The premise here is that SAQFT could be used successfully to treat hadrons as particles whose structure is made up of the constituent quarks. The objective of this brief introductory study is to provide motivation for further investigations and advanced research using this structural algebraic approach to QFT.

2 SCALAR PARTICLES IN SAQFT

In QFT, particle resonances are indispensable objects in the theory and of prime physical significance. These objects decay in time whereas stable particles do not. Now, the temporal development of a physical system is governed by the time evolution operator \({{e}^{{{{ - {\text{i}}tH} \mathord{\left/ {\vphantom {{ - {\text{i}}tH} \hbar }} \right. \kern-0em} \hbar }}}}\) with H being the system Hamiltonian, which is a measure of its energy. Then a simple and clear scheme to account for the decay or stability of these objects is to assign real energies to stable particles whereas resonances are given complex energies with negative imaginary part. Therefore, quantum fields representing particles and resonances must be written/analyzed in the complex energy plane (E-plane) or in the complex momentum plane (k-plane) since E and k are related by the free field equation. Now, in the nonrelativistic theory, the energy-momentum relation for free fields is \(E = {{{{k}^{2}}} \mathord{\left/ {\vphantom {{{{k}^{2}}} {2M}}} \right. \kern-0em} {2M}}\), where M is the rest mass of the particle. Thus, the k-plane is more fundamental than the E-plane since any one point in the former corresponds to two points in the latter that are located on two separate complex “energy sheets.” Hence, one E-plane is not physically unique while one k-plane is. However, in the relativistic theory and using the relativistic units, \(\hbar = c = 1\), the energy-momentum relation for free fields is \({{E}^{2}} = {{k}^{2}} + \) \({{M}^{2}}\). Thus, the two complex planes are physically equivalent and in SAQFT we choose to work in the E-plane rather than the k-plane. Consequently, the zero-spin Klein–Gordon quantum field in 3 + 1 Minkowski space-time is represented in SAQFT by the following Fourier expansion in the energy

$$\Psi (t,\vec {r}) = \int\limits_\Omega {{{e}^{{ - {\text{i}}Et}}}\psi (E,\vec {r})\,a(E)\,dE} + \sum\limits_{j = 0}^N {{{e}^{{ - {\text{i}}{{E}_{j}}t}}}{{\psi }_{j}}(\vec {r})\,{{a}_{j}}} .$$
(1)

The integral represents the continuous energy spectrum of the particle whereas the sum represents the discrete spectrum (i.e., the particle structure resolved in the energy domain). The latter is a new addition to conventional QFT that could have a positive impact on the treatment of complex systems. The structure in (1) consists of \(N + 1\) discrete channels whereas \(\Omega \) consists generally of several disconnected but continuous channels (called energy bands or energy intervals). These channels do not overlap (i.e., their intersection is null). For simplicity, we take \(\Omega \) to stand for the single energy interval \({{E}^{2}} \geqslant {{M}^{2}}\) and take \(0 \leqslant E_{j}^{2} < {{M}^{2}}\). A direct and straightforward implication of such an energy spectrum designation is that massless particles (e.g., the photon) in SAQFT are automatically structureless. The objects \(a(E)\) and \({{a}_{j}}\) are field operators (the vacuum annihilation operators). They satisfy the following conventional commutation relations [1–4]

$$\begin{gathered} \left[ {a(E),{{a}^{\dag }}(E{\kern 1pt} ')} \right]:\, = a(E){{a}^{\dag }}(E{\kern 1pt} ') \\ - {{a}^{\dag }}(E{\kern 1pt} ')\,a(E) = \delta (E - E{\kern 1pt} '),\,\,\,\left[ {{{a}_{i}},a_{j}^{\dag }} \right] = {{\delta }_{{i,j}}}. \\ \end{gathered} $$
(2)

All other commutators among \(a(E)\), \({{a}^{\dag }}(E)\), \({{a}_{j}}\), and \(a_{j}^{\dag }\) vanish.

The continuous Fourier energy components \(\psi (E,\vec {r})\) in (1) has an extended spatial range whereas the discrete component \({{\psi }_{j}}(\vec {r})\) has a short-range and is confined in space to the particle extent. They are written as the following pointwise convergent series

$$\psi (E,\vec {r}) = \sum\limits_{n = 0}^\infty {{{f}_{n}}(E)\,{{\phi }_{n}}(\vec {r})} = {{f}_{0}}(E)\sum\limits_{n = 0}^\infty {{{p}_{n}}(z)\,{{\phi }_{n}}(\vec {r})} ,$$
(3a)
$${{\psi }_{j}}(\vec {r}) = \sum\limits_{n = 0}^\infty {{{g}_{n}}({{E}_{j}})\,{{\phi }_{n}}(\vec {r})} = {{g}_{0}}({{E}_{j}})\sum\limits_{n = 0}^\infty {{{p}_{n}}({{z}_{j}})\,{{\phi }_{n}}(\vec {r})} ,$$
(3b)

where z is a spectral energy parameter (spectral parameter, for short) to be determined and \(\left\{ {{{f}_{n}},{{g}_{n}}} \right\}\) are real expansion coefficients which are written as \({{f}_{n}} = {{f}_{0}}{{p}_{n}}\) and \({{g}_{n}} = {{g}_{0}}{{p}_{n}}\) making \({{p}_{0}} = 1\). \(\left\{ {{{\phi }_{n}}(\vec {r})} \right\}\) is a complete set of functions in configuration space that satisfy the following differential equation

$$ - {{\vec {\nabla }}^{2}}{{\phi }_{n}}(\vec {r}) = {{\alpha }_{n}}{{\phi }_{n}}(\vec {r}) + {{\beta }_{{n - 1}}}{{\phi }_{{n - 1}}}(\vec {r}) + {{\beta }_{n}}{{\phi }_{{n + 1}}}(\vec {r}),$$
(4)

where \({{\vec {\nabla }}^{2}}\) is the three dimensional Laplacian and \(\left\{ {{{\alpha }_{n}},{{\beta }_{n}}} \right\}\) are real constants that are independent of z and such that \({{\beta }_{n}} \ne 0\) for all n. Using Eq. (4), the free Klein-Gordon wave equation, \(\left( {\partial _{t}^{2} - {{{\vec {\nabla }}}^{2}} + {{M}^{2}}} \right)\Psi (t,\vec {r}) = 0\), becomes the following algebraic relation

$$z\,{{p}_{n}}(z) = {{\alpha }_{n}}{{p}_{n}}(z) + {{\beta }_{{n - 1}}}{{p}_{{n - 1}}}(z) + {{\beta }_{n}}{{p}_{{n + 1}}}(z),$$
(5)

for \(n = 1,2,3,...\) and with \(z = {{E}^{2}} - {{M}^{2}}\). This equation along with its consequential and ensuing relations like the orthogonality relation (6) and completeness formulas (8) shown below represent the on-shell conditions for the quantum field (1). Equation (5) is a symmetric three-term recursion relation that makes \(\left\{ {{{p}_{n}}(z)} \right\}\) a sequence of polynomials in z with the two initial values \({{p}_{0}}(z) = 1\) and \({{p}_{1}}(z)\) linear in z. Now, Eq. (5) has two linearly independent polynomial solutions. For scalar particles, we choose the solution with the initial values \({{p}_{0}}(z) = 1\) and \({{p}_{1}}(z) = {{(z - {{\alpha }_{0}})} \mathord{\left/ {\vphantom {{(z - {{\alpha }_{0}})} {{{\beta }_{0}}}}} \right. \kern-0em} {{{\beta }_{0}}}}\). For spinors, both solutions are needed as will be shown below. Due to Favard theorem (a.k.a. the spectral theorem; see Section 2.5 in [16]), the polynomial solutions of Eq. (5) satisfy the following general orthogonality relation [13–16]

$$\int\limits_\Omega {\rho (z){{p}_{n}}(z){{p}_{m}}(z)\,dz} + \sum\limits_{j = 0}^N {\xi ({{z}_{j}}){{p}_{n}}({{z}_{j}}){{p}_{m}}({{z}_{j}})} = {{\delta }_{{n,m}}},$$
(6)

where \(\rho (z)\) is the continuous component of the weight function and \(\xi ({{z}_{j}})\) is the discrete component. These weight functions are positive definite and will be determined below in terms of \({{f}_{0}}(E)\) and \({{g}_{0}}({{E}_{j}})\), respectively. The fundamental algebraic relation (5), which is equivalent to the Klein-Gordon wave equation, is the reason behind the algebraic setup of the theory and for which we qualify this QFT as algebraic. In fact, postulating the three-term recursion relation (5) eliminates the need for specifying a free field wave equation. Furthermore, once the set of orthogonal polynomials \(\left\{ {{{p}_{n}}(z)} \right\}\) is given then all physical properties of the corresponding particle are determined. Thus, a physical process in SAQFT is equivalent to calculating the change from the set \(\left\{ {{{p}_{n}}(z)} \right\}\) to another set \(\left\{ {p_{n}^{'}(z{\kern 1pt} ')} \right\}\) due to this process. For elastic scattering, the identity of the incoming particles is retained and \(\left\{ {{{p}_{n}}(z)} \right\} \mapsto \left\{ {p_{n}^{'}(z{\kern 1pt} ')} \right\}\). However, for inelastic scattering, \(\left\{ {{{p}_{n}}(z)} \right\} \mapsto \left\{ {q_{n}^{'}(z{\kern 1pt} ')} \right\}\), where \(\left\{ {{{q}_{n}}(z)} \right\}\) is a different set of polynomials with its own recursion coefficients \(\left\{ {\alpha _{n}^{'},\beta _{n}^{'}} \right\}\) and orthogonality corresponding to different outgoing particles.

In conventional QFT, the quantum field (1) is expressed as Fourier expansion in the linear momentum \(\vec {k}\)-space not in the energy space. That is, \(\Psi (t,\vec {r})\) is written as the integral \(\int {{{e}^{{ - {\text{i}}Et + {\text{i}}\vec {k} \cdot \vec {r}}}}a(\vec {k})\tfrac{{{{d}^{3}}k}}{{\sqrt {{{{(2\pi )}}^{3}}2E} }}} \), where \({{\vec {k}}^{2}} = {{E}^{2}} - {{M}^{2}}\) and giving \(\psi (E,\vec {r}) \propto {{e}^{{{\text{i}}\vec {k} \cdot \vec {r}}}}\). One can show that \({{e}^{{{\text{i}}\vec {k} \cdot \vec {r}}}}\) could be written as an infinite series having the same SAQFT form (3a) by using the relation

$${{e}^{{{\text{i}}kx}}} = \sqrt 2 \,{{e}^{{ - {{{{z}^{2}}} \mathord{\left/ {\vphantom {{{{z}^{2}}} 2}} \right. \kern-0em} 2}}}}{{e}^{{ - {{{{\lambda }^{2}}{{x}^{2}}} \mathord{\left/ {\vphantom {{{{\lambda }^{2}}{{x}^{2}}} 2}} \right. \kern-0em} 2}}}}\sum\limits_{n = 0}^\infty {\frac{{{{{\text{i}}}^{n}}}}{{{{2}^{n}}n!}}{{H}_{n}}(z){{H}_{n}}(\lambda x)} ,$$
(7a)

where \(z = {k \mathord{\left/ {\vphantom {k \lambda }} \right. \kern-0em} \lambda }\), \({{H}_{n}}(y)\) is the Hermite polynomial, λ is a real scale parameter, and \({{\phi }_{n}}(x) \propto \) \(\left\{ {{{e}^{{ - {{{{\lambda }^{2}}{{x}^{2}}} \mathord{\left/ {\vphantom {{{{\lambda }^{2}}{{x}^{2}}} 2}} \right. \kern-0em} 2}}}}\left[ {{{{{i}^{n}}{{H}_{n}}(\lambda x)} \mathord{\left/ {\vphantom {{{{i}^{n}}{{H}_{n}}(\lambda x)} {\sqrt {{{2}^{n}}n!} }}} \right. \kern-0em} {\sqrt {{{2}^{n}}n!} }}} \right]} \right\}\). Therefore, \({{p}_{n}}(z) = \) \({{{{H}_{n}}(z)} \mathord{\left/ {\vphantom {{{{H}_{n}}(z)} {\sqrt {{{2}^{n}}n!} }}} \right. \kern-0em} {\sqrt {{{2}^{n}}n!} }}\) with \({{\alpha }_{n}} = 0\), \({{\beta }_{n}} = \sqrt {{{(n + 1)} \mathord{\left/ {\vphantom {{(n + 1)} 2}} \right. \kern-0em} 2}} \), and \({{f}_{0}}(E) = {{\pi }^{{ - \tfrac{1}{4}}}}{{e}^{{ - {{{{z}^{2}}} \mathord{\left/ {\vphantom {{{{z}^{2}}} 2}} \right. \kern-0em} 2}}}}\). An equivalent expansion could also be written in terms of products of the Gegenbauer (ultra-spherical) polynomials \(\left\{ {C_{n}^{\nu }(z)} \right\}\) and the Bessel function with discrete index \({{J}_{{n + \nu }}}(\lambda x)\) as follows, (see Eq. (4.8.3) in [16])

$${{e}^{{{\text{i}}kx}}} = {{2}^{\nu }}\Gamma (\nu ){{\left( {\lambda x} \right)}^{{ - \nu }}}\sum\limits_{n = 0}^\infty {{{{\text{i}}}^{n}}(n + \nu )C_{n}^{\nu }(z){{J}_{{n + \nu }}}(\lambda x)} ,$$
(7b)

giving \({{p}_{n}}(z) \propto C_{n}^{\nu }(z)\) and \({{\phi }_{n}}(x) \propto \) \({{{\text{i}}}^{n}}(n + \nu ){{\left( {\lambda x} \right)}^{{ - \nu }}}{{J}_{{n + \nu }}}(\lambda x)\).

Now, the conjugate quantum field \(\bar {\Psi }(t,\vec {r})\) in SAQFT is obtained from (1) by complex conjugation and the replacement \({{\phi }_{n}}(\vec {r}) \mapsto {{\bar {\phi }}_{n}}(\vec {r})\) where

$$\left\langle {{{{\phi }_{n}}(\vec {r})}} \mathrel{\left | {\vphantom {{{{\phi }_{n}}(\vec {r})} {{{{\bar {\phi }}}_{m}}(\vec {r})}}} \right. \kern-0em} {{{{{\bar {\phi }}}_{m}}(\vec {r})}} \right\rangle = \left\langle {{{{{\bar {\phi }}}_{n}}(\vec {r})}} \mathrel{\left | {\vphantom {{{{{\bar {\phi }}}_{n}}(\vec {r})} {{{\phi }_{m}}(\vec {r})}}} \right. \kern-0em} {{{{\phi }_{m}}(\vec {r})}} \right\rangle = {{\delta }_{{n,m}}},$$
(8a)
$$\sum\limits_{n = 0}^\infty {{{\phi }_{n}}(\vec {r})\,{{{\bar {\phi }}}_{n}}(\vec {r}{\kern 1pt} ')} = \sum\limits_{n = 0}^\infty {{{{\bar {\phi }}}_{n}}(\vec {r})\,{{\phi }_{n}}(\vec {r}{\kern 1pt} ')} = {{\delta }^{3}}(\vec {r} - \vec {r}{\kern 1pt} ').$$
(8b)

The first is the orthogonality relation and the second is the completeness statement. Therefore, we write \(\bar {\Psi }(t,\vec {r})\) as follows

$$\bar {\Psi }(t,\vec {r}) = \int\limits_\Omega {{{e}^{{{\text{i}}Et}}}\bar {\psi }(E,\vec {r})\,{{a}^{\dag }}(E)\,dE} + \sum\limits_{j = 0}^N {{{e}^{{{\text{i}}{{E}_{j}}t}}}{{{\bar {\psi }}}_{j}}(\vec {r})\,a_{j}^{\dag }} .$$
(9)

where the components \(\bar {\psi }(E,\vec {r})\) and \({{\bar {\psi }}_{j}}(\vec {r})\) are identical to (3) but with \({{\phi }_{n}}(\vec {r}) \mapsto {{\bar {\phi }}_{n}}(\vec {r})\). Using the commutators (2) of the field operators \(a(E)\) and \({{a}_{j}}\), we can write

$$\begin{gathered} \left[ {\Psi (t,\vec {r}),\bar {\Psi }(t{\kern 1pt} ',\vec {r}{\kern 1pt} ')} \right] = \sum\limits_{n,m = 0}^\infty {{{\phi }_{n}}(\vec {r})\,{{{\bar {\phi }}}_{m}}(\vec {r}{\kern 1pt} ')} \\ \times \,\,\left[ {\int\limits_\Omega {{{e}^{{ - {\text{i}}E(t - t{\kern 1pt} ')}}}f_{0}^{2}(E){{p}_{n}}(z){{p}_{m}}(z)\,dE} } \right. \\ \left. { + \,\,\sum\limits_{j = 0}^N {{{e}^{{ - {\text{i}}{{E}_{j}}(t - t{\kern 1pt} ')}}}g_{0}^{2}({{E}_{j}}){{p}_{n}}({{z}_{j}}){{p}_{m}}({{z}_{j}})} } \right]. \\ \end{gathered} $$
(10)

The general orthogonality (6) and the completeness (8b) turn Eq. (10) with \(t = t{\kern 1pt} '\) into

$$\left[ {\Psi (t,\vec {r}),\bar {\Psi }(t,\vec {r}{\kern 1pt} ')} \right] = {{\delta }^{3}}(\vec {r} - \vec {r}{\kern 1pt} '),$$
(11)

provided that we take \(f_{0}^{2}(E)dE = \rho (z)dz\) and \(g_{0}^{2}({{E}_{j}}) = \xi ({{z}_{j}})\), which also imply positivity of the two weight functions with \(\tfrac{{dz}}{{dE}} > 0\) for \(E \in \Omega \). Moreover, it is straightforward to write

$$\left[ {\Psi (t,\vec {r}),\Psi (t,\vec {r}{\kern 1pt} ')} \right] = \left[ {\bar {\Psi }(t,\vec {r}),\bar {\Psi }(t,\vec {r}{\kern 1pt} ')} \right] = 0.$$
(12)

In the canonical quantization of fields, the canonical conjugate to \(\Psi (t,\vec {r})\) is written as \(\Pi (t,\vec {r})\) and they satisfy the following equal time commutation relations [1–4]

$$\left[ {\Psi (t,\vec {r}),\Psi (t,\vec {r}{\kern 1pt} ')} \right] = \left[ {\Pi (t,\vec {r}),\Pi (t,\vec {r}{\kern 1pt} ')} \right] = 0,$$
(13a)
$$\left[ {\Psi (t,\vec {r}),\Pi (t,\vec {r}{\kern 1pt} ')} \right] = {\text{i}}{{\delta }^{3}}(\vec {r} - \vec {r}{\kern 1pt} ').$$
(13b)

Therefore, we obtain the following identification: \(\Pi (t,\vec {r}) = {\text{i}}\bar {\Psi }(t,\vec {r})\). Moreover, in analogy with conventional QFT [1–4], we can write Eq. (10) as

$$\left[ {\Psi (t,\vec {r}),\bar {\Psi }(t{\kern 1pt} ',\vec {r}{\kern 1pt} ')} \right] = \Delta (t - t{\kern 1pt} ',\vec {r} - \vec {r}{\kern 1pt} '),$$
(14)

where the singular function \(\Delta (t - t{\kern 1pt} ',\vec {r} - \vec {r}{\kern 1pt} ')\) in SAQFT reads as follows

$$\begin{gathered} \Delta (t - t{\kern 1pt} ',\vec {r} - \vec {r}{\kern 1pt} ') \\ = \sum\limits_{n,m = 0}^\infty {{{\phi }_{n}}(\vec {r})\,{{{\bar {\phi }}}_{m}}(\vec {r}{\kern 1pt} ')\left[ {\int\limits_\Omega {{{e}^{{ - {\text{i}}E(t - t{\kern 1pt} ')}}}\rho (z){{p}_{n}}(z){{p}_{m}}(z)\,dz} } \right.} \\ \left. { + \,\,\sum\limits_{j = 0}^N {{{e}^{{ - {\text{i}}{{E}_{j}}(t - t{\kern 1pt} ')}}}\xi ({{z}_{j}}){{p}_{n}}({{z}_{j}}){{p}_{m}}({{z}_{j}})} } \right]. \\ \end{gathered} $$
(15)

Moreover, Eqs. (11) and (14) give \(\Delta (0,\vec {r} - \vec {r}{\kern 1pt} ') = {{\delta }^{3}}(\vec {r} - \vec {r}{\kern 1pt} ')\).

Now, we can define the real (neutral) scalar non-elementary particle by the quantum field \(\Phi (t,\vec {r}) = \frac{1}{{\sqrt 2 }}\left[ {\Psi (t,\vec {r}) + \bar {\Psi }(t,\vec {r})} \right]\) with \(\,{{\bar {\phi }}_{n}}(\vec {r}) = \phi _{n}^{\dag }(\vec {r})\). On the other hand, the complex (charged) scalar non-elementary particle is defined by the positive-energy quantum field

$$\Phi (t,\vec {r}) = \frac{1}{{\sqrt 2 }}\left[ {{{\Psi }_{ + }}(t,\vec {r}) + \Psi _{ - }^{\dag }(t,\vec {r})} \right].$$
(16a)

\({{\Psi }_{ \pm }}(t,\vec {r})\) is identical to (1) but with \(a(E) \mapsto {{a}_{ \pm }}(E)\) and \({{a}_{j}} \mapsto a_{ \pm }^{j}\) such that \(\left[ {{{a}_{r}}(E),a_{{r{\kern 1pt} '}}^{\dag }(E{\kern 1pt} ')} \right] = \) \({{\delta }_{{r,r{\kern 1pt} '}}}\,\delta (E - E{\kern 1pt} ')\) and \(\left[ {a_{r}^{i},{{{(a_{{r{\kern 1pt} '}}^{j})}}^{\dag }}} \right] = {{\delta }_{{r,{\kern 1pt} r{\kern 1pt} '}}}\,{{\delta }^{{i,j}}}\) where r and \(r{\kern 1pt} '\) stand for ±. The corresponding charged scalar antiparticle is represented by the following negative-energy quantum field

$$\bar {\Phi }(t,\vec {r}) = \frac{1}{{\sqrt 2 }}\left[ {{{{\bar {\Psi }}}_{ + }}(t,\vec {r}) + \bar {\Psi }_{ - }^{\dag }(t,\vec {r})} \right].$$
(16b)

For this scalar particle, the Feynman propagator \({{\Delta }_{{\text{F}}}}(t{\kern 1pt} '\,\, - t,\vec {r}{\kern 1pt} '\,\, - \vec {r})\) between the two space-time points \((t,\vec {r})\) and \((t{\kern 1pt} ',\vec {r}{\kern 1pt} ')\) is constructed by combining the following two processes [1–4]:

(1) The creation of a particle from the vacuum \(\left| 0 \right\rangle \) at \((t,\vec {r})\) and annihilating it later (\({\kern 1pt} t{\kern 1pt} ' > t\)) back into the vacuum at \((t{\kern 1pt} ',\vec {r}{\kern 1pt} ')\).

(2) The conjugate process of creating an antiparticle from the vacuum at \((t{\kern 1pt} ',\vec {r}{\kern 1pt} ')\) then annihilating it later (\(t > t{\kern 1pt} '\)) at \((t,\vec {r})\).

That is,

$$\begin{gathered} {{\Delta }_{F}}(t{\kern 1pt} '\,\, - t,\vec {r}{\kern 1pt} '\,\, - \vec {r}) = \left\langle 0 \right|T\left( {\bar {\Phi }(t{\kern 1pt} ',\vec {r}{\kern 1pt} '),\Phi (t,\vec {r})} \right)\left| 0 \right\rangle \\ = \left\langle 0 \right|\bar {\Phi }(t{\kern 1pt} ',\vec {r}{\kern 1pt} ')\,\Phi (t,\vec {r})\left| 0 \right\rangle \theta (t{\kern 1pt} ' - t) \\ + \,\,\left\langle 0 \right|\Phi (t,\vec {r})\,\bar {\Phi }(t{\kern 1pt} ',\vec {r}{\kern 1pt} ')\left| 0 \right\rangle \theta (t - t{\kern 1pt} '), \\ \end{gathered} $$
(17)

where T is the time ordering operator and \(\theta (x) = \left\{ {_{0}^{1}{}_{{,x < 0}}^{{,x > 0}}} \right.\). With the free propagator being determined, one needs to identify the type of interaction to account for the behavior of the scalar particle when coupled to its environment. Without such an interaction, the internal structure of the non-elementary particle (summation part of the quantum field) has no bearing on its free motion. Only in the presence of interaction will we observe the added effect of the internal structure. Moreover, the type and extent of such an effect will certainly depend on the nature of the interaction (electromagnetic, nuclear, gravitational, etc.). One way to incorporate the interaction of the particle with an external field is by using the gauge invariant minimal coupling scheme where the 4-gradient \(\left( {{{\partial }_{0}},\vec {\nabla }} \right)\) in the wave equation is replaced by \(\left( {{{\partial }_{0}} + {\text{i}}q{{A}_{0}},\vec {\nabla } + {\text{i}}q\vec {A}} \right)\) with q being the coupling parameter and \(({{A}_{0}},\vec {A})\) the external 4-vector field.

In the Dirac–Coulomb problem, it was shown elsewhere [17, 18] that electromagnetic interaction of electrons, of which the Hydrogen atom is an example, is associated with the two-parameter Meixner–Pollaczek polynomial \(P_{n}^{\mu }(z,\theta )\) where \(\mu > 0\) and \(0 < \theta < \pi \). This polynomial is known to have only a continuous spectrum. That is, the summation part in the orthogonality (6) is absent. Consequently, the internal structure is null and in this case, SAQFT is equivalent to the conventional QFT; both leading to QED. In the following section, we present a simple nontrivial example where the particle structure will have an effect on the outcome.

3 SCALAR SAQFT EXAMPLE

As a simple but nontrivial example of scalar SAQFT, we consider a non-elementary scalar particle in 1 + 1 Minkowski space-time whose structure is associated with the three-parameter continuous dual Hahn polynomial \(S_{n}^{\mu }(z;\sigma ,\tau )\). The properties of this orthogonal polynomial that are relevant to our study are given in Appendix A. We consider here the special case where \(\mu < 0\) and \(\sigma = \tau > - \mu \). From the formulation given above, it is obvious that the particle structure is determined by the recursion relation (5) vis-à-vis its recursion coefficients \(\left\{ {{{\alpha }_{n}},{{\beta }_{n}}} \right\}.\) The symmetric three-term recursion relation for the orthonormal version of the polynomial \(S_{n}^{\mu }(z;\sigma ,\sigma )\) is shown as Eq. (A.6) in Appendix A giving

$${{\alpha }_{n}} = {{\left( {n + \mu + \sigma } \right)}^{2}} + n\left( {n + 2\sigma - 1} \right) - {{\mu }^{2}},$$
(18a)
$${{\beta }_{n}} = - \left( {n + \mu + \sigma } \right)\sqrt {\left( {n + 1} \right)\left( {n + 2\sigma } \right)} .$$
(18b)

The physical effect of this internal structure becomes more evident if we write the differential equation satisfied by \({{\phi }_{n}}(x)\). We choose

$${{\phi }_{n}}(x) = \sqrt {\frac{{\lambda \,\Gamma (n + 1)}}{{\Gamma (n + 2\sigma )}}} \,{{y}^{\sigma }}{{e}^{{{{ - y} \mathord{\left/ {\vphantom {{ - y} 2}} \right. \kern-0em} 2}}}}L_{n}^{{2\sigma - 1}}(y),$$
(19)

where \(y = {{e}^{{ - \lambda x}}}\) and \(L_{n}^{{2\sigma - 1}}(y)\) is the Laguerre polynomial. The scale parameter λ is real and positive with inverse length dimension. It could be considered as measure of the size of the structure. The orthogonality of the Laguerre polynomials shows that \(\left\{ {{{\phi }_{n}}(x)} \right\}\) is an orthonormal set: \({{\bar {\phi }}_{n}}(x) = {{\phi }_{n}}(x)\). Using the differential equation and recursion relation of the Laguerre polynomials, we obtain the free wave equation associated with this non-elementary scalar particle that replaces Eq. (4). It reads

$$\begin{gathered} \left[ {\frac{{ - 1}}{{{{\lambda }^{2}}}}\frac{{{{d}^{2}}}}{{d{{x}^{2}}}} + W(x)} \right]{{\phi }_{n}}(x) \\ = {{\alpha }_{n}}{{\phi }_{n}}(x) + {{\beta }_{{n - 1}}}{{\phi }_{{n - 1}}}(x) + {{\beta }_{n}}{{\phi }_{{n + 1}}}(x), \\ \end{gathered} $$
(20)

where \(W(x)\) is a manifestation of the structure of the particle corresponding to \(S_{n}^{\mu }(z;\sigma ,\sigma )\) and it reads

$$W(x) = \frac{1}{4}{{e}^{{ - 2\lambda x}}} + \left( {\mu - \frac{1}{2}} \right){{e}^{{ - \lambda x}}}.$$
(21)

Moreover, \(z = {{({{E}^{2}} - {{M}^{2}})} \mathord{\left/ {\vphantom {{({{E}^{2}} - {{M}^{2}})} {{{\lambda }^{2}}}}} \right. \kern-0em} {{{\lambda }^{2}}}}\) and the size of the structure is equal to \(N + 1\), where N is the largest integer less than \( - \mu \). The rest of the objects needed to determine the quantum fields and propagators are the continuous and discrete components of the weight functions \(\rho (z)\) and \(\xi ({{z}_{j}})\) in addition to the spectrum \(\{ {{z}_{j}}\} \) of the discrete structure. These are given in Appendix A by Eqs. (A.2), (A.4) and (A.5), respectively.

The interaction in this system is a physical process whereby the set of parameters \(\left\{ {E,\lambda ,\mu ,\sigma } \right\}\) that defines the particle is changed to \(\left\{ {E{\kern 1pt} ',\lambda {\kern 1pt} ',\mu {\kern 1pt} ',\sigma {\kern 1pt} '} \right\}\). That is, \(S_{n}^{\mu }(z;\sigma ,\sigma ) \mapsto \) \(S_{n}^{{\mu {\kern 1pt} '}}(z{\kern 1pt} ';\sigma {\kern 1pt} ',\sigma {\kern 1pt} ')\) which, for inelastic scattering, could alter the size and nature of the particle’s structure. For example, if \(\mu = - 3.7 \mapsto \mu {\kern 1pt} ' = - 3.2\) while λ remains the same then the structure will maintain its size \(\left( {N + 1 = 4} \right)\) but the levels of the structure will change from \(E_{j}^{2} = {{M}^{2}} - \) \({{\lambda }^{2}}{{\left( {j - 3.7} \right)}^{2}}\) to \(E_{j}^{{'2}} = {{M}^{2}} - {{\lambda }^{2}}{{\left( {j - 3.2} \right)}^{2}}\) where \(j = 0,1,2,3\). However, if μ changes to, say \(\mu {\kern 1pt} ' = - 1.5\), then not only the levels will change but also the size of the structure (i.e., nature of the particle itself) changes from 4 to 2. Furthermore, if \(\mu {\kern 1pt} '\) becomes positive then the entire structure disappears signifying a total decay of the particle’s constituents or structure due to the process.

4 SCATTERING EXAMPLE IN SCALAR SAQFT

In this section, we outline a procedure to account for scattering in SAQFT with scalar particles (e.g., scalar mesons). As example, we consider the nonlinear interaction Lagrangian model \({{\mathcal{L}}_{I}} = g{{[\Phi (t,\vec {r})]}^{3}}\), where g is a coupling parameter.

In conventional QFT, a physical process could be accounted for by summing all Feynman diagrams occurring within the process up to a given order. In SAQFT, however, the set of orthogonal polynomials \(\left\{ {{{p}_{n}}(z)} \right\}\) associated with a given quantum field determines all physical properties of the corresponding particle. Thus, a scattering process in SAQFT is equivalent to the evaluation of the change in these polynomials. Now, the wave equation for the quantum field \(\Phi (t,\vec {r})\) in the presence of interaction results in a three-term recursion relation that differs from that of the free field relation (5) and reads

$$\begin{gathered} z\,{{p}_{n}}(z) = {{\alpha }_{n}}{{p}_{n}}(z) + {{\beta }_{{n - 1}}}{{p}_{{n - 1}}}(z) \\ + \,\,{{\beta }_{n}}{{p}_{{n + 1}}}(z) + g\Delta {{p}_{n}}(z), \\ \end{gathered} $$
(22)

which is to be solved for the modified orthogonal polynomial. Henceforth, our task in this section is to calculate the added component \(\Delta {{p}_{n}}(z)\) for the proposed model. For that, we use the Feynman diagrams to compute these changes up to second order in the coupling. In other scenarios, such change could also be evaluated perturbatively in powers of physical parameters that appear in the recursion coefficients \(\left\{ {{{\alpha }_{n}},{{\beta }_{n}}} \right\}\) or in z. For example, in Section 3, we could have \(\left| {{\lambda \mathord{\left/ {\vphantom {\lambda M}} \right. \kern-0em} M}} \right| \ll 1\) or \(\sigma + \mu \ll 1\), and so on. Ideally, if we could evaluate \(\Delta {{p}_{n}}(z)\) to all orders then we should obtain \(\Delta {{p}_{n}}(z) = \Delta {{\alpha }_{n}}{{p}_{n}}(z) + \) \(\Delta {{\beta }_{{n - 1}}}{{p}_{{n - 1}}}(z) + \Delta {{\beta }_{n}}{{p}_{{n + 1}}}(z)\) and end up with one of two situations. If the scattering is elastic then the particle will maintain its identity, which means that the modified polynomial will satisfy the same recursion relation but with possibly different initial values \({{p}_{0}}(z)\) and \({{p}_{1}}(z)\) (i.e., different spin, polarization, etc.). This means that for elastic scattering, \(\Delta {{p}_{n}}(z)\) is a linear function of z multiplied by \({{\delta }_{{n,0}}}\). However, if the scattering is inelastic then the particle changes its identity, which means that the modified polynomial will satisfy a different recursion relation with the modified coefficients \(\left\{ {\alpha _{n}^{'},\beta _{n}^{'}} \right\} = \) \(\left. {\left\{ {{{\alpha }_{n}} + g\Delta {{\alpha }_{n}},} \right.{{\beta }_{n}} + g\Delta {{\beta }_{n}}} \right\}\). Below, we outlined a procedure to calculate \(\Delta {{p}_{n}}(z)\) for the proposed nonlinear interacting model up to second order by following the rules of Feynman diagrams. However, the rules used in conventional QFT need some amendments to be suitable for application in SAQFT.

Figure 1 gives a graphical representation for the evaluation of \(\Delta {{p}_{n}}(z)\) to second order by following the rules of Feynman diagrams for this specific model. However, these rules which are well established in conventional QFT, need few alterations to be suitable for use in SAQFT. For example, propagators in the diagrams are associated, in conventional QFT, with linear momentum vectors making direction and vector addition very critical in closed loops. In SAQFT, however, spectral parameters (the polynomial arguments, which are scalar quantities) and polynomial degrees are assigned to propagators instead, making parameter addition and subtraction in closed loops critical. Adding and subtracting these parameters means adding and subtracting their corresponding energies making this operation nontrivial, especially for scalar particles. Shortly, we will show how this is done. Other alterations of the rules might be necessary as the theory becomes more developed.

Fig. 1.
figure 1

Feynman diagrams contributing to the evaluation of \(\Delta {{p}_{n}}(z)\) in the model up to order \({{g}^{2}}\).

Full size image

As stated above, if we could evaluate \(\Delta {{p}_{n}}(z)\) to all orders then it should come out to be a polynomial in z of degree \(n + 1\). Specifically, \(\Delta {{p}_{n}}(z) = \) \(\Delta {{\alpha }_{n}}{{p}_{n}}(z) + \Delta {{\beta }_{{n - 1}}}{{p}_{{n - 1}}}(z)\) \( + \Delta {{\beta }_{n}}{{p}_{{n + 1}}}(z)\) where \(\left\{ {\Delta {{\alpha }_{n}},\Delta {{\beta }_{n}}} \right\}\) are constant parameters. In Fig. 1, the pair \((a,l)\) on a Feynman propagator (or at one end for incoming/outgoing particles) stands for the associated orthogonal polynomial \({{p}_{l}}(a)\), with a being the spectral parameter (continuous and/or discrete). The two incoming particles are associated with the polynomials \({{p}_{m}}(x)\) and \({{p}_{k}}(y)\). The outgoing particle is associated with the polynomial \({{p}_{n}}(z)\). Therefore, in this scattering process, the degrees \(\left\{ {m,k,n} \right\}\) and corresponding spectral parameters \(\left\{ {x,y,z} \right\}\) are fixed. At each vertex, the energy-momentum vector is conserved. For example, in the first diagram on the second line of the equation and with the choice of a counterclockwise loop, the three energy conservation equations are: \(E(x) = E(w) - E(x{\kern 1pt} ')\), \(E(y) = E(y{\kern 1pt} ') - E(w)\), and \(E(z) = \) \(E(y{\kern 1pt} ') - E(x{\kern 1pt} ') = E(x)\) \( + E(y)\). This leads to a special rule for adding and subtracting spectral parameters in the Feynman diagrams that goes as follows. We define \(x \oplus y\) as the quantity z that makes \(E(x) + E(y) = E(z)\). Similarly, \(x \ominus y: = z\) means that \(E(x) - E(y) = E(z)\). Alternatively, we can write \(x \oplus y = {{E}^{{ - 1}}}\left( {E(x) + E(y)} \right)\) and \(x \ominus y = {{E}^{{ - 1}}}\left( {E(x) - E(y)} \right)\). For scalars with \({{[E(z)]}^{2}} = \) \(z + {{M}^{2}}\), this operation is nontrivial but becomes simpler for massless particles where \(x \oplus y = \) \(x + y + 2\operatorname{sgn} \sqrt {xy} \) and \(x \ominus y = x + y - 2\operatorname{sgn} \sqrt {xy} \) with the sign being that of \(E(x)E(y)\). If both x and y are positive (i.e., in the continuous spectrum Ω) then \(x \oplus y\) is either:

(i) Positive (belonging to Ω) and greater than or equal to \(3{{M}^{2}}\) if the particles associated with x and y are both with positive or both with negative energies, or

(ii) Having indefinite sign (but, of course, always greater than or equal to \( - {{M}^{2}}\)) if the particles associated with x and y have energies of different signs.

The reverse is true for \(x \ominus y\). Table 1 shows all possible range of values of \(x \oplus y\) and \(x \ominus y\). A remarkable byproduct of working with the spectral parameters in scalar SAQFT is a novel algebraic system [19].

Table 1.   Range of values of \(z = x \oplus y\). For \(z = x \ominus y\), the ± in the first column must be interchanged
Full size table

If we combine \({\text{the}}\) continuous and discrete weight functions in the orthogonality (6) as \(\tilde {\rho }(z): = \) \(\rho (z) + \xi (z)\sum\nolimits_{j = 0}^N {\delta (z - {{z}_{j}})} \), then the orthogonality takes the following compact form

$$\int\limits_{\tilde {\Omega }} {\tilde {\rho }(z){{p}_{n}}(z){{p}_{m}}(z)\,dz} = {{\delta }_{{n,m}}},$$
(23)

where \(\tilde {\Omega } = \Omega \cup \left[ {{{E}_{0}},{{E}_{N}}} \right]\). Now, the quantum field representation (1) means that each propagator in the Feynman diagrams with the pair \((a,l)\) is associated not just with \({{p}_{l}}(a)\) but with \({{f}_{0}}\left( {E(a)} \right){{p}_{l}}(a)\) for the continuous spectrum and with \({{g}_{0}}\left( {E({{a}_{j}})} \right){{p}_{l}}({{a}_{j}})\) for the discrete, which we write collectively as \(\sqrt {\tilde {\rho }(a)} {{p}_{l}}(a)\). If the interaction vertex parameters are \(g\left\{ {\eta _{n}^{{m,k}}} \right\}\), then the first line of the equation in Fig. 1 gives \(g\Delta {{p}_{n}}(z)\) to first order as \(g\Delta {{p}_{n}}(z) = g\,\eta _{n}^{{m,k}}\). On the other hand, the first diagram on the second line of the equation with the choice of counterclockwise loop gives the following second order contribution to \(g\Delta {{p}_{n}}(z)\)

$$\begin{gathered} \frac{{{{g}^{3}}}}{{3!}}\sqrt {\tilde {\rho }(w \ominus x)\tilde {\rho }(y \oplus w)\tilde {\rho }(w \ominus x)\tilde {\rho }(w)\tilde {\rho }(y \oplus w)\tilde {\rho }(w)} \\ \times \,\,\sum\limits_{i,j,l = 0}^\infty {\eta _{n}^{{i,j}}\eta _{m}^{{i,l}}\eta _{k}^{{j,l}}{{p}_{i}}(w \ominus x){{p}_{j}}(y \oplus w){{p}_{i}}(w \ominus x){{p}_{l}}(w){{p}_{j}}(y \oplus w){{p}_{l}}(w)} . \\ \end{gathered} $$
(24)

Using the property \(y \oplus w = w \oplus y\) and collecting terms, we can rewrite (24) as follows

$$\begin{gathered} \frac{{{{g}^{3}}}}{{3!}}\tilde {\rho }(w \ominus x)\tilde {\rho }(w \oplus y)\tilde {\rho }(w) \\ \times \,\,\sum\limits_{i,j,l = 0}^\infty {\eta _{n}^{{i,j}}\eta _{m}^{{i,l}}\eta _{k}^{{j,l}}p_{i}^{2}(w \ominus x)p_{j}^{2}(w \oplus y)p_{l}^{2}(w)} . \\ \end{gathered} $$
(25)

The second diagram on the second line of the equation in the figure with a counterclockwise loop gives the following second order contribution to \(g\Delta {{p}_{n}}(z)\)

$$\begin{gathered} \frac{{{{g}^{3}}}}{{3!}}\tilde {\rho }(w \oplus z)\tilde {\rho }(w \oplus x)\tilde {\rho }(w) \\ \times \,\,\sum\limits_{i,j,l = 0}^\infty {\eta _{n}^{{i,l}}\eta _{m}^{{j,l}}\eta _{k}^{{i,j}}p_{i}^{2}(w \oplus z)p_{j}^{2}(w \oplus x)p_{l}^{2}(w)} . \\ \end{gathered} $$
(26)

The third diagram on the second line of the equation is topologically equivalent to the second giving a contribution identical to (26) but with \(x \leftrightarrow y\) and \(m \leftrightarrow k\). The fourth diagram, on the other hand, gives the following contribution to \(g\Delta {{p}_{n}}(z)\) with a counterclockwise loop

$$\frac{{{{g}^{3}}}}{{3!}}\tilde {\rho }(w)\tilde {\rho }(w \oplus z)\sum\limits_{i,j,l = 0}^\infty {\eta _{n}^{{i,j}}\eta _{l}^{{i,j}}\eta _{l}^{{m,k}}p_{i}^{2}(w)p_{j}^{2}(w \oplus z)} ,$$
(27)

where we have taken \(w\,:\, = z{\kern 1pt} '{\kern 1pt} \). The fifth diagram on the second line of the equation in the figure gives the following second order contribution to \(g\Delta {{p}_{n}}(z)\)

$$\frac{{{{g}^{3}}}}{{3!}}\tilde {\rho }(w)\tilde {\rho }(w \oplus x)\sum\limits_{i,j,l = 0}^\infty {\eta _{m}^{{i,j}}\eta _{l}^{{i,j}}\eta _{l}^{{n,k}}p_{j}^{2}(w)p_{i}^{2}(w \oplus x)} ,$$
(28)

where we have taken \(w: = x''\). The last diagram on the second line of the equation is topologically equivalent to the fifth and gives a contribution identical to (28) but with \(x \leftrightarrow y\) and \(m \leftrightarrow k\).

As the two incoming particles being already selected and prepared in the lab (e.g., in a monochrome scattering experiment) then their corresponding energies \(\left\{ {E(x),E(y)} \right\}\), spectral parameters \(\left\{ {x,y} \right\}\) and polynomial degrees \(\left\{ {m,k} \right\}\) are fixed. That is, \({{p}_{m}}(x)\) and \({{p}_{k}}(y)\) are fixed. Likewise, the output channel for this scattering process has also been determined to correspond to \({{p}_{n}}(z)\), where \(E(z) = E(x) + E(y)\). Hence, the spectral parameter z and corresponding polynomial degree n are also fixed. Therefore, w and the degrees \(\left\{ {i,j,l} \right\}\) become the only variables in Fig. 1. However, since the infinite summations in (25) to (28) have already accounted for all possible degrees \(\left\{ {i,j,l} \right\}\), we only need to integrate over the whole possible range of values of w (continuous and discrete). That is, integrating w over the complete energy domain \(\tilde {\Omega }\) (i.e., integration over Ω and summation over \(\left\{ {{{w}_{i}}} \right\}_{{i = 0}}^{N}\)). Therefore, at this stage in the process we need to perform this task in order to write down the one-loop vertex correction shown in Fig. 1. We start by defining the following fundamental SAQFT integrals:

$$\begin{gathered} {{\zeta }_{n}}: = \int\limits_{\tilde {\Omega }} {\tilde {\rho }(w)p_{n}^{2}(w)\,dw} \\ = \int\limits_\Omega {\rho (w)p_{n}^{2}(w)\,dw} + \sum\limits_{j = 0}^N {\xi ({{w}_{j}})p_{n}^{2}({{w}_{j}})} = 1, \\ \end{gathered} $$
(29a)
$$\begin{gathered} {{\zeta }_{{n,m}}}(a): = \int\limits_{\tilde {\Omega }} {\tilde {\rho }(w)\tilde {\rho }(w \circledast a)p_{n}^{2}(w)p_{m}^{2}(w \circledast a)\,dw} \\ = \int\limits_\Omega {\rho (w)\rho (w \circledast a)p_{n}^{2}(w)p_{m}^{2}(w \circledast a)\,dw} \\ + \,\,\sum\limits_{j = 0}^N {\xi ({{w}_{j}})\xi ({{w}_{j}} \circledast a)p_{n}^{2}({{w}_{j}})p_{m}^{2}({{w}_{j}} \circledast a)} , \\ \end{gathered} $$
(29b)
$$\begin{gathered} {{\zeta }_{{n,m,k}}}(a,b): = \int\limits_\Omega {\rho (w)\rho (w \circledast a)\rho (w \circledast b)p_{n}^{2}(w)p_{m}^{2}(w \circledast a)p_{k}^{2}(w \circledast b)\,dw} \\ + \sum\limits_{j = 0}^N {\xi ({{w}_{j}})\xi ({{w}_{j}} \circledast a)\xi ({{w}_{j}} \circledast b)p_{n}^{2}({{w}_{j}})p_{m}^{2}({{w}_{j}} \circledast a)p_{k}^{2}({{w}_{j}} \circledast b)} , \\ \end{gathered} $$
(29c)

where \( \circledast \) stands for either \( \oplus \) or \( \ominus \). These are the lowest three in the series of fundamental SAQFT integrals corresponding to closed loops with one, two, and three vertices, respectively. The orthogonality (23) shows that the first integral is, in fact, trivial. It corresponds to the bare vertex in the first diagram of Fig. 1. The integrals in (29b) and (29c) are valid only if \(w \circledast a\) and \(w \circledast b\) are elements of the continuous spectrum Ω. Table 1 is useful in making such validity determination. For example, with \(\left\{ {x,y,z,w} \right\}\) belonging to the continuous spectrum Ω and thus positive, the quantity \(w \ominus x\) in the integral with \(E(x) \gtrless 0\) becomes positive definite (in fact, greater than or equal to \(3{{M}^{2}}\)) by choosing the (arbitrary) positive variable w such that \(E(w) \lessgtr 0\), respectively. The summations, on the other hand, are valid only if \({{w}_{i}} \circledast {{a}_{j}}\) and \({{w}_{i}} \circledast {{b}_{j}}\) are elements of the discrete spectrum set \(\left\{ {{{w}_{i}}} \right\}_{{i = 0}}^{N}\). On the other hand, for higher order loops with multiple integrals over several arbitrary energy variables (e.g., \(w,u,v,..,\) etc.), it is always possible to choose, say u and w, such that \(w \circledast u \in \Omega \), \({{w}_{i}} \circledast {{u}_{j}} \in \left\{ {{{w}_{i}}} \right\}_{{i = 0}}^{N}\) and \({{u}_{i}} \circledast {{w}_{j}} \in \) \(\left\{ {{{u}_{i}}} \right\}_{{i = 0}}^{N}\). Now, the orthogonality relation (6), or equivalently (23), guarantees that the integrals in (29b) and (29c) are finite provided that the range of values of w is such that \(w \circledast a\) and \(w \circledast b\) belong to Ω [20]. Moreover, one can show that the values of \({{\zeta }_{{n,m}}}(a)\) and \({{\zeta }_{{n,m,k}}}(a,b)\) are not just finite but, in fact, fall within the interval \(\left[ {0, + 1} \right]\). Additionally, in the limit as n, m, and/or k go to infinity, \({{\zeta }_{{n,m}}}(a)\) and \({{\zeta }_{{n,m,k}}}(a,b)\) go to zero. These properties will shortly be demonstrated numerically below. Moreover, the symmetry \({{\zeta }_{{n,m,k}}}(a,b) = {{\zeta }_{{n,k,m}}}(b,a)\) is also evident. Finally, after the integration and summation in (25) to (28), we obtain the following correction (to second order) of the vertex diagram shown in Fig. 1

$$\begin{gathered} g\Delta {{p}_{n}}(z) \approx g\,\eta _{n}^{{m,k}} \\ + \,\,\frac{{{{g}^{3}}}}{{3!}}\left[ {{{\eta }_{{n,m,k}}}(x,y) + {{\eta }_{{k,m,n}}}(x,z) + {{\eta }_{{m,k,n}}}(y,z)} \right. \\ \left. { + \,\,{{{\tilde {\eta }}}_{{n,m,k}}}(z) + {{{\tilde {\eta }}}_{{m,n,k}}}(x) + {{{\tilde {\eta }}}_{{k,n,m}}}(y)} \right], \\ \end{gathered} $$
(30)

where \({{\eta }_{{n,m,k}}}(a,b)\,:\, = \sum\nolimits_{i,j,l = 0}^\infty {\eta _{n}^{{i,j}}\eta _{m}^{{i,l}}\eta _{k}^{{j,l}}{{\zeta }_{{l,i,j}}}(a,b)} \), \({{\tilde {\eta }}_{{n,m,k}}}(a)\,:\, = \sum\nolimits_{i,j,l = 0}^\infty {\eta _{n}^{{i,j}}\eta _{l}^{{i,j}}\eta _{l}^{{m,k}}{{\zeta }_{{i,j}}}(a)} \) and we assumed that \(\eta _{n}^{{m,k}} = \eta _{n}^{{k,m}}\). These infinite sums converge because the vertex parameters \(g\left\{ {\eta _{n}^{{m,k}}} \right\}\) are finite and \({{\zeta }_{{i,j}}}(a)\) as well as \({{\zeta }_{{l,i,j}}}(a,b)\) go to zero fast enough as their indices tend to infinity.

Evidently, the evaluation of the integrals of the type shown in (29) is of prime importance in the calculation of the Feynman diagrams in SAQFT. We find that Gauss quadrature integral approximation associated with the polynomials \(\left\{ {{{p}_{n}}(w)} \right\}\) produces highly accurate results. For a brief description of Gauss quadrature, one may consult, for example, [21, 22]. In summary, it goes as follows. Let K be a large-enough integer (called the order of the quadrature) and let J be the \(K \times K\) truncated version of the tridiagonal symmetric matrix Q shown below in Eq. (39), which is the Jacobi matrix associated with the polynomials \(\left\{ {{{p}_{n}}(w)} \right\}\). We designate the K distinct real eigenvalues of J as the set \(\left\{ {{{\lambda }_{k}}} \right\}_{{k = 0}}^{{K - 1}}\) whose corresponding normalized eigenvectors are \(\left\{ {{{\Lambda }_{{i,k}}}} \right\}_{{i = 0}}^{{K - 1}}\). In this setting, Gauss quadrature gives the following approximation for the integrals in \({{\zeta }_{{n,m}}}(a)\) and \({{\zeta }_{{n,m,l}}}(a,b)\)

$${{\zeta }_{{n,m}}}(a) \cong \sum\limits_{k = 0}^{K - 1} {\Lambda _{{n,k}}^{2}\rho ({{\lambda }_{k}} \circledast a)p_{m}^{2}({{\lambda }_{k}} \circledast a)} ,$$
(31a)
$$\begin{gathered} {{\zeta }_{{n,m,l}}}(a,b) \\ \cong \,\sum\limits_{k = 0}^{K - 1} {\Lambda _{{n,k}}^{2}\rho ({{\lambda }_{k}}\, \circledast \,a)\rho ({{\lambda }_{k}}\, \circledast \,b)p_{m}^{2}({{\lambda }_{k}}\, \circledast \,a)p_{l}^{2}({{\lambda }_{k}}\, \circledast \,b)} . \\ \end{gathered} $$
(31b)

The approximation improves with the order of the quadrature, K. Figure 2 is a sample plot of \({{\zeta }_{{n,m}}}(a)\), with \({{p}_{n}}(w)\) being the normalized version of the Laguerre polynomial \(L_{n}^{\nu }(w)\) and \( \circledast \mapsto \oplus \), \(E(w)E(a) > 0\). The figure demonstrates the diminishing value of \({{\zeta }_{{n,m}}}(a)\) with increasing n and m. Obtaining finite values for these fundamental SAQFT integrals that represent closed loops in the Feynman diagrams is an amazing property of the theory that will certainly have a positive impact on renormalizationFootnote 3.

Fig. 2.
figure 2

Plot of \({{\zeta }_{{n,m}}}(a)\) with \({{p}_{n}}(w)\) being the normalized Laguerre polynomial \(L_{n}^{\nu }(w)\), \( \circledast \mapsto \oplus \), and \(E(w)E(a) > 0\). We took the mass \(M = 1.0\), \(a = 2.0\), \(\nu = 1.5\) and the quadrature order \(K = 200\).

Full size image

For completeness, we end this section by calculating the first order correction to the propagator (self-energy) in this model where the corresponding Feynman diagram is shown as Fig. 3. Writing \(w\,:\, = x{\kern 1pt} {''}\), we obtain the following first order contribution to \(g\Delta {{p}_{n}}(x)\)

$$\frac{{{{g}^{2}}}}{{2!}}\tilde {\rho }(w)\tilde {\rho }(w \ominus x)\sum\limits_{i,j = 0}^\infty {\eta _{n}^{{i,j}}\eta _{m}^{{i,j}}\,p_{i}^{2}(w)p_{j}^{2}(w \ominus x)} .$$
(32)
Fig. 3.
figure 3

Self-energy correction diagram to order g in \(\Delta {{p}_{n}}(x)\).

Full size image

After integrating the arbitrary energy variable w over \(\tilde {\Omega }\), we obtain the following self-energy correction to first order as shown in Fig. 3

$$g\Delta {{p}_{n}}(x) \approx \frac{{{{g}^{2}}}}{{2!}}\sum\limits_{i,j = 0}^\infty {\eta _{n}^{{i,j}}\eta _{m}^{{i,j}}{{\zeta }_{{i,j}}}(x)} .$$
(33)

5 CONCLUSIONS AND COMMENTS

In this work, we posed the critical question: Why doesn’t conventional QFT work at low energies for particles with structure (e.g., in nuclear physics with hadrons)? Instead of laying blame on strong coupling and relying on “asymptotic freedom” at high energy, we suggested a simple answer that it is because of a missing piece; the particle structure itself. Therefore, we adopted the view that instead of assuming structureless particles at a given energy only to discover at higher energy that they do have structure, is to develop a practical QFT with structure already built-in. Consequently, we introduced an algebraic version of QFT that accommodates particles with structure, which is resolved in the energy domain. The theory of orthogonal polynomials plays a critical role in the formulation. It is hoped that the brief introduction of the theory given here will motivate further studies using this approach towards a more effective and generalized QFT. Incorporating the particle structure may bring new elements into the theory that could be exploited to tackle some of the persistent difficulties in conventional QFT. We believe that these new elements may have a positive impact on the renormalization program (e.g., the size of the structure provides a natural cutoff of ultraviolet divergences). Moreover, energy integration (both continuous and discrete) over closed loops in the Feynman diagrams will utilize the continuous and discrete integration measures \(\rho (z)dz\) and \(\xi ({{z}_{j}})\) that could render such integrals finite by exploiting the orthogonality relation (6) as demonstrated in the interaction model of the previous section.

In Appendix B, we give a rough and brief mathematical presentation of SAQFT for spinor particles with structure in 1 + 1 Minkowski space-time. It is understood that spinors in such space-time do not carry physical significance. However, the aim is to provide the framework and tools for further rigorous treatment. For completeness, one should also provide the massless vector field (e.g., the electromagnetic potential) and massive vector meson in the SAQFT formulation.

One of the remaining tasks in SAQFT (as presented here) is to establish the relativistic invariance of its physical objects and covariance of its elements under the space-time Poincaré transformation (Lorentz rotation + translation). It is conceivable that results from such an endeavor may alter and/or improve on the development of the theory as presented in this work.

Finally, we conclude with the following ten-point remarks:

• The SAQFT presented in this work does not require a free field wave equation to be specified. In fact, the formulation of the theory is built upon four defining postulates:

(i) The quantum field and its conjugate resolved in the energy domain, which for scalar particles are given by Eqs. (1) and (9), respectively.

(ii) The creation/annihilation operators, which for scalar particles satisfy the algebra given by the commutation relations (2).

(iii) The orthogonal energy polynomials satisfying the three-term recursion relation (5) and orthogonality (6).

(iv) The completeness of spatial functions used in the expansion series of the continuous and discrete Fourier kernels, which for scalar particles is given by Eq. (8).

The corresponding postulates in the case of spinors are shown in Appendix B by Eq. (B.2) and Eq. (B.7) for the quantum fields, and by the anti-commutation algebra (B.3) for the creation/annihilation operators. In their classic book on QFT [1], Bjorken and Drell posed a fundamental question at the beginning of Section 11.1 on page 4: “…why local field theories, that is, theories of fields which can be described by differential laws of wave propagation, have been so extensively used and accepted”. The answer was given therein “there exist no convincing form of a theory which avoids differential field equations.” It is expected that the theory proposed here where the differential wave equation is replaced by the equivalent algebraic three-term recursion relation constitutes a viable alternative.

• For consistency, it is fruitful and possibly necessary to do dimensional analysis of the main objects in SAQFT. In the relativistic units \(\hbar = c = 1\), physical quantities like space, time, mass, field operators, etc. are measured in units of the energy; say \(\mathcal{E}\). For example, dimensional analysis shows that the annihilation/creation operators \(a(\vec {k})\) and \({{a}^{\dag }}(\vec {k})\) in conventional QFT are measured in units of \({{\mathcal{E}}^{{ - {3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}\) whereas the field operators \(\Psi (t,\vec {r})\) are measured in units of \(\mathcal{E}\). On the other hand, the commutation relation (2) shows that the annihilation/creation operators \(a(E)\) and \({{a}^{\dag }}(E)\) are measured in units of \({{\mathcal{E}}^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}\) whereas \({{a}_{j}}\) and \(a_{j}^{\dag }\) are measured in units of \({{\mathcal{E}}^{0}}\). Moreover, equations (1) to (10) result in the following energy units for measuring the following objects in SAQFT: \({{\phi }_{n}}(\vec {r})\sim {{\mathcal{E}}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}\), \({{f}_{n}}(E)\sim {{\mathcal{E}}^{{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}\), \({{g}_{n}}({{E}_{j}})\sim {{\mathcal{E}}^{0}}\), \({{p}_{n}}(z)\sim {{\mathcal{E}}^{0}}\), \(\psi (E,\vec {r})\sim \mathcal{E}\), and \({{\psi }_{j}}(\vec {r})\sim {{\mathcal{E}}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}\). Therefore, the field operator \(\Psi (t,\vec {r})\) in SAQFT is measured in units of \({{\mathcal{E}}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}\).

• One may suggest a modification of the conventional QFT to incorporate the particle structure in a simpler manner without going through the sumptuous construction of SAQFT involving orthogonal polynomials. That is, the conventional quantum field for scalar particles could be redefined as follows

$$\tilde {\Psi }(t,\vec {r}) = \int {{{e}^{{ - {\text{i}}Et + {\text{i}}\vec {k} \cdot \vec {r}}}}a(\vec {k})\tfrac{{{{d}^{3}}k}}{{\sqrt {{{{(2\pi )}}^{3}}2E} }}} + \sum\limits_{j = 0}^N {{{e}^{{ - {\text{i}}{{E}_{j}}t + {\text{i}}{{{\vec {k}}}_{j}} \cdot \vec {r}}}}{{a}_{j}}} .$$
(34)

However, this will result in a quantum field that is “over-complete”. Namely, \(\left\| {\tilde {\Psi }(t,\vec {r})} \right\| > 1\) because the continuous spectrum is already complete since \(\left\| {{{e}^{{{\text{i}}\vec {k} \cdot \vec {r}}}}} \right\| = 1\). On the other hand, the orthogonality (6) of the polynomials in SAQFT gives the completeness of the quantum field \(\Psi (t,\vec {r})\) as a sum of the continuous and discrete spectra that could be written symbolically as follows

$$\left\| {\Psi (t,\vec {r})} \right\| = \left\| {\psi (E,\vec {r})} \right\| + \sum\limits_{j = 0}^N {\left\| {{{\psi }_{j}}(\vec {r})} \right\|} = 1.$$
(35)

In fact, even for very small structures, where for example \(N = 0\) or 1, it may happen that the contribution of the discrete spectrum (the sum in Eq. (35)) could dominate completeness. Another way to show the inadequacy of the representation (34) is to note that, in the language of SAQFT, the expansion of \({{e}^{{{\text{i}}\vec {k} \cdot \vec {r}}}}\) is associated with the Hermite polynomial (or the Gegenbauer polynomial) whose entire spectrum is continuous leaving no room for including an internal discrete structure; cf. Eq. (7).

• If the set of orthogonal polynomials \(\left\{ {{{p}_{n}}(z)} \right\}\) is endowed only with a discrete spectrum, then the continuous integral in the definition of the quantum field (1) does not appear and the orthogonality (6) consists only of the summation part. Examples of such polynomials include the Meixner, Charlier, dual Hahn, Krawtchouk and the Racah polynomials [23]. We conjecture that such systems might constitute a feasible alternative to the traditional approach that accounts for the confinement of particles like the quarks. They could also be used in describing point contact (zero range) interactions (those where massless gauge fields that mediate coupling in the interaction are absent). If such discrete polynomials are finite and with a finite spectrum, i.e., \(\{ {{p}_{n}}({{z}_{j}})\} _{{n,j = 0}}^{N}\), then the associated configuration space functions must form a complete finite discrete set, \(\{ {{\phi }_{n}}({{\vec {r}}_{j}})\} _{{n,j = 0}}^{N}\), over the space lattice \(\{ {{\vec {r}}_{j}}\} _{{j = 0}}^{N}\) in the sense that \(\sum\nolimits_{n = 0}^N {{{\phi }_{n}}({{{\vec {r}}}_{i}}){{{\bar {\phi }}}_{n}}({{{\vec {r}}}_{j}})} \) = \(\sum\nolimits_{n = 0}^N {{{{\bar {\phi }}}_{n}}({{{\vec {r}}}_{i}}){{\phi }_{n}}({{{\vec {r}}}_{j}})} = {{l}^{{ - 3}}}\,{{\delta }_{{i,j}}}\) and \(\sum\nolimits_{j = 0}^N {{{\varsigma }_{j}}{{\phi }_{n}}({{{\vec {r}}}_{j}}){{{\bar {\phi }}}_{m}}({{{\vec {r}}}_{j}})} \) = \(\sum\nolimits_{j = 0}^N {{{\varsigma }_{j}}{{{\bar {\phi }}}_{n}}({{{\vec {r}}}_{j}}){{\phi }_{m}}({{{\vec {r}}}_{j}})} = {{\delta }_{{n,m}}}\) for some positive discrete weight \({{\varsigma }_{j}}\) and lattice size \(l\). For example, in 1+1 space-time with a linear lattice of total length L and \(l = {L \mathord{\left/ {\vphantom {L N}} \right. \kern-0em} N}\), we could take

$$\begin{gathered} {{\phi }_{n}}({{x}_{j}}) = {{{\bar {\phi }}}_{n}}({{x}_{j}}) \\ = \left( {{{N!} \mathord{\left/ {\vphantom {{N!} {\sqrt l }}} \right. \kern-0em} {\sqrt l }}} \right)\sqrt {\tfrac{{{{\vartheta }^{{n + j}}}{{{(1 - \vartheta )}}^{{N - n - j}}}}}{{n!j!(N - n)!(N - j)!}}} \,{}_{2}{{F}_{1}}\left( {\left. {_{{{\text{ }} - N}}^{{ - n, - j}}} \right|{{\vartheta }^{{ - 1}}}} \right), \\ \end{gathered} $$
(36)

where \({{x}_{j}} = jl\), \(0 < \vartheta < 1\), and \({{\varsigma }_{j}} = l\). For a proof of completeness and orthogonality of these discrete functions, see Appendix A in [24] for the Krawtchouk polynomials.

• All physically relevant orthogonal polynomials with a continuous spectrum that are compatible with SAQFT must have a sinusoidal asymptotic behavior. Specifically, in the limit as \(n \to \infty \), we require that \({{p}_{n}}(z)\) takes the following form

$${{p}_{n}}(z) \approx \frac{1}{{{{n}^{\kappa }}\sqrt {\rho (z)} }}\cos \left[ {{{n}^{\nu }}\varphi (z) + \delta (z)} \right],$$
(37)

where κ and ν are positive real parameters, \(\varphi (z)\) is an entire function, and \(\delta (z)\) is the scattering phase shift. If \(\nu \to 0\) then \({{n}^{\nu }} \to \ln (n)\). Only under this asymptotic condition, will the series (3a) produce oscillatory continuum particle states at the boundaries of space (see, for example, [17, 24–27]). For a rigorous discussion about the connection between the asymptotics of such polynomials and scattering, one may consult [28–30] and references therein. Fortunately, all of the many known hypergeometric orthogonal polynomials that appear abundantly in the physics literature do meet this requirement. They include, but not limited to, the polynomials in the Askey scheme [23] such as the Wilson, continuous Hahn, continuous dual Hahn, Meixner-Pollaczek, Jacobi, Laguerre, Gegenbauer, Chebyshev, Hermite, etc.

• A more general differential equation satisfied by the complete set of functions \(\left\{ {{{\phi }_{n}}(\vec {r})} \right\}\) that maintains the tridiagonal algebraic structure of the theory is

$$\begin{gathered} \mathcal{D}{{\phi }_{n}}(\vec {r}) = \omega \,(\vec {r}) \\ \times \,\,\left[ {\left( {{{\alpha }_{n}} - z} \right){{\phi }_{n}}(\vec {r}) + {{\beta }_{{n - 1}}}{{\phi }_{{n - 1}}}(\vec {r}) + {{\beta }_{n}}{{\phi }_{{n + 1}}}(\vec {r})} \right], \\ \end{gathered} $$
(38)

where \(\mathcal{D}\) is the wave operator (e.g., the Klein–Gordon, Dirac, Dirac–Coulomb, etc.) and \(\omega (\vec {r})\) is an entire function the does not vanish locally. This equation should be considered a generalization of Eq. (4) for scalars or Eq. (B.5) for spinors. For example, in Eq. (20), \(\mathcal{D} = - \tfrac{{{{d}^{2}}}}{{d{{x}^{2}}}} + {{\lambda }^{2}}W(x) + {{M}^{2}} - {{E}^{2}}\) and \(\omega (\vec {r}) = {{\lambda }^{2}}\), which is a generalization of the Klein–Gordon equation where \(W(x) \mapsto 0\).

• The linear momentum \(\vec {k}\) (where \({{\vec {k}}^{2}} = {{E}^{2}} - {{M}^{2}}\)) is an essential variable in the formulation of conventional QFT, which is usually written in the k-space representation. On the other hand, \(\vec {k}\) does not appear explicitly in SAQFT. This is due to the tridiagonal structure of the fundamental differential equation for \({{\phi }_{n}}(\vec {r})\). In fact, using (4) or (20) in the Klein–Gordon wave equation gives \({{\vec {k}}^{2}}\) as one of the eigenvaluesFootnote 4 of the following infinite tridiagonal symmetric matrix

$$\left( {\begin{array}{*{20}{c}} {{{\alpha }_{0}}}&{{{\beta }_{0}}}&{}&{}&{}&{}&{} \\ {{{\beta }_{0}}}&{{{\alpha }_{1}}}&{{{\beta }_{1}}}&{}&{}&{}&{} \\ {}&{{{\beta }_{1}}}&{{{\alpha }_{2}}}&{{{\beta }_{2}}}&{}&{}&{} \\ {}&{}&{{{\beta }_{2}}}&{{{\alpha }_{3}}}&{{{\beta }_{3}}}&{}&{} \\ {}&{}&{}& \times & \times & \times &{} \\ {}&{}&{}&{}& \times & \times & \times \\ {}&{}&{}&{}&{}& \times & \times \end{array}} \right).$$
(39)

• For special systems where the continuous and discrete channels are totally independent, SAQFT formulation could be extended by choosing two distinct sets of functions in the expansion of the continuous channel kernel \(\psi (E,\vec {r})\) and discrete channel kernel \({{\psi }_{j}}(\vec {r})\). That is, we could still expand \(\psi (E,\vec {r})\) in the same set \(\left\{ {{{\phi }_{n}}(\vec {r})} \right\}\) as given by (3a) but expand \({{\psi }_{j}}(\vec {r})\) in another independent set \(\left\{ {{{\chi }_{n}}(\vec {r})} \right\}\) by rewriting (3b) as

$${{\psi }_{j}}(\vec {r}) = \sum\limits_{n = 0}^\infty {{{g}_{n}}({{E}_{j}})\,{{\chi }_{n}}(\vec {r})} = {{g}_{0}}({{E}_{j}})\sum\limits_{n = 0}^\infty {{{q}_{n}}({{z}_{j}})\,{{\chi }_{n}}(\vec {r})} ,$$
(40)

where \(\left\{ {{{q}_{n}}({{z}_{j}})} \right\}\) is a set of orthogonal polynomials having only discrete spectrum, whereas \(\left\{ {{{p}_{n}}(z)} \right\}\) has a continuous spectrum only. Moreover, \(\sum\nolimits_{n = 0}^\infty {{{\chi }_{n}}(\vec {r})\,{{{\bar {\chi }}}_{n}}(\vec {r}{\kern 1pt} ')} = \) \(\sum\nolimits_{n = 0}^\infty {{{{\bar {\chi }}}_{n}}(\vec {r})\,{{\chi }_{n}}(\vec {r}{\kern 1pt} ')} = {{\delta }^{3}}(\vec {r} - \vec {r}{\kern 1pt} ')\) and \(\left\langle {{{{\chi }_{n}}(\vec {r})}} \mathrel{\left | {\vphantom {{{{\chi }_{n}}(\vec {r})} {{{{\bar {\chi }}}_{m}}(\vec {r})}}} \right. \kern-0em} {{{{{\bar {\chi }}}_{m}}(\vec {r})}} \right\rangle \) \( = \left\langle {{{{{\bar {\chi }}}_{n}}(\vec {r})}} \mathrel{\left | {\vphantom {{{{{\bar {\chi }}}_{n}}(\vec {r})} {{{\chi }_{m}}(\vec {r})}}} \right. \kern-0em} {{{{\chi }_{m}}(\vec {r})}} \right\rangle = {{\delta }_{{n,m}}}\). This allows us to use two different set of functions, one is suitable for the continuous channel and the other is more appropriate for the discrete channel (structure). The wave Eq. (4) or its generalization (38) for \(\left\{ {{{\chi }_{n}}(\vec {r})} \right\}\) maintains the same tridiagonal form but with different recursion coefficients \(\left\{ {{{\alpha }_{n}},{{\beta }_{n}}} \right\}\). Furthermore, \(f_{0}^{2}(E)dE = \frac{1}{2}\rho (z)dz\) and \(g_{0}^{2}({{E}_{j}}) = \) \(\frac{1}{2}\xi ({{z}_{j}})\) where we obtain

$$\begin{gathered} \left[ {\Psi (t,\vec {r}),\bar {\Psi }(t{\kern 1pt} ',\vec {r}{\kern 1pt} ')} \right] = \Delta (t - t{\kern 1pt} ',\vec {r} - \vec {r}{\kern 1pt} ') \\ = \frac{1}{2}\sum\limits_{n,m = 0}^\infty {{{\phi }_{n}}(\vec {r})\,{{{\bar {\phi }}}_{m}}(\vec {r}{\kern 1pt} ')\int\limits_\Omega {{{e}^{{ - {\text{i}}E(t - t{\kern 1pt} ')}}}\rho (z){{p}_{n}}(z){{p}_{m}}(z)\,dz} } \\ + \,\,\frac{1}{2}\sum\limits_{n,m = 0}^\infty {{{\chi }_{n}}(\vec {r})\,{{{\bar {\chi }}}_{m}}(\vec {r}{\kern 1pt} ')} \sum\limits_{j = 0}^N {{{e}^{{ - {\text{i}}{{E}_{j}}(t - t{\kern 1pt} ')}}}\xi ({{z}_{j}}){{q}_{n}}({{z}_{j}}){{q}_{m}}({{z}_{j}})} \\ \end{gathered} $$
(41)

giving \(\Delta (0,\vec {r} - \vec {r}{\kern 1pt} ') = {{\delta }^{3}}(\vec {r} - \vec {r}{\kern 1pt} ')\). A remarkable example of such a system with an infinite, but spatially confined, structure is that for which \({{p}_{n}}(z)\) is the Meixner–Pollaczek polynomial \(P_{n}^{\mu }(z,\theta )\) whereas \({{q}_{n}}({{z}_{j}})\) is the Meixner polynomial \(M_{n}^{\mu }\left( {{{z}_{j}},\vartheta } \right)\), [23]

$$P_{n}^{\mu }(z,\theta ) = \sqrt {\tfrac{{\Gamma (n + 2\mu )}}{{\Gamma (n + 1)\Gamma (2\mu )}}} {{e}^{{{\text{i}}n\theta }}}{}_{2}{{F}_{1}}\left( {\left. {_{{{\text{ 2}}\mu }}^{{ - n,\mu + {\text{i}}z}}} \right|1 - {{e}^{{ - 2i\theta }}}} \right),$$
(42a)
$$\begin{gathered} M_{n}^{\mu }\left( {{{z}_{j}},\vartheta } \right) \\ = \sqrt {\tfrac{{\Gamma (n + 2\mu )}}{{\Gamma (n + 1)\Gamma (2\mu )}}} \,\,{{\vartheta }^{{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0em} 2}}}}{}_{2}{{F}_{1}}\left( {\left. {_{{{\text{ }}\,2\mu }}^{{ - n, - j}}} \right|1 - {{\vartheta }^{{ - 1}}}} \right), \\ \end{gathered} $$
(42b)

where \({{z}_{j}} = j + \mu \), \(\mu > 0\), \(0 < \theta < \pi \), \(0 < \vartheta < 1\), and \(j = 0,1,2,..\). Another interesting example, but with a finite structure, is where \({{p}_{n}}(z)\) becomes the continuous dual Hahn polynomial \(S_{n}^{\mu }(z;\sigma ,\tau )\) with \((\mu ,\sigma ,\tau ) > 0\) and \({{q}_{n}}({{z}_{j}})\) is the dual Hahn polynomial \(R_{n}^{N}({{z}_{j}};\sigma ,\tau )\), [23]

$$\begin{gathered} R_{n}^{N}({{z}_{j}};\sigma ,\tau ) \\ = \sqrt {\tfrac{{{{{(\sigma )}}_{n}}{{{(N - n + 1)}}_{n}}}}{{n!{{{(N + \tau - n)}}_{n}}}}} {}_{3}{{F}_{2}}\left( {\left. {_{{\sigma , - N}}^{{ - n, - j,j + \sigma + \tau - 1}}} \right|1} \right), \\ \end{gathered} $$
(43)

where \({{z}_{j}} = {{\left( {j + \frac{{\sigma + \tau - 1}}{2}} \right)}^{2}}\) and \(n,j = 0,1,..,N\).

• By incorporating hadron structure, we believe that SAQFT could be used to develop a QFT for hadrons (baryons and mesons) without the need for color charges. That is, an alternative to QCD for nuclear physics calculations. In that case, the polynomial \({{p}_{n}}(z)\) associated with baryons could be taken as the Wilson polynomial \(W_{n}^{\mu }(z;a,b,c)\) [23] with \( - 3 < \mu < - 2\) whereas the polynomial associated with mesons would be the continuous dual Hahn polynomial \(S_{n}^{\mu }(z;a,b)\) with \( - 2 < \mu < - 1\). Each of the parameters \(\left\{ {a,b,c} \right\}\) can assume one of 6 values corresponding to one of the six flavors of quarks or their conjugate values for the six anti-quarks.

• The tridiagonal representation approach (TRA) is an algebraic method for solving linear ordinary differential equations of the second order [31, 32]. It has been used successfully in the solution of the wave equation in quantum mechanics resulting in a larger class of exactly solvable problems (see, for example, [33] and references therein). The SAQFT presented here could be viewed as an application of the TRA in QFT.