A New Space-Time Theory Unravels the Origins of Classical Mechanics for the Dirac Equation

Abstract

The Feynman path integral plays a central role in quantum mechanics, linking classical action to propagators and relating quantum electrodynamics (QED) to Feynman diagrams. However, the path-integral formulations used in non-relativistic quantum mechanics and in QED are neither unified nor directly connected. This suggests the existence of a missing path integral that bridges relativistic action and the Dirac equation at the single-particle level. In this work, we analyze the consistency and completeness of existing path-integral theories and identify a spinor path integral that fills this gap. Starting from a relativistic action written in spinor form, we construct a spacetime path integral whose kernel reproduces the Dirac Hamiltonian. The resulting formulation provides a direct link between the relativistic classical action and the Dirac equation, and it naturally extends the scalar relativistic path integral developed in our earlier work. Beyond establishing this structural connection, the spinor path integral offers a new way to interpret the origin of classical mechanics for the Dirac equation and suggests a spacetime mechanism for spin and quantum nonlocal correlations. These features indicate that the spinor path integral can serve as a unifying framework for existing path-integral approaches and as a starting point for further investigations into the spacetime structure of quantum mechanics.

1. Introduction

The Feynman path integral theory, a significant concept in quantum mechanics, has two forms. One originated from Feynman’s 1948 see paper [1], which established a crucial link between quantum mechanics and classical analytical mechanics. This work explanied the influence of particle trajectories on propagators, indirectly supporting the principle of least action in classical mechanics. It played a pivotal role in connecting the classical action principle with the Schrödinger equation in non-relativistic quantum mechanics. The other form emerged in 1949 when Feynman extended this approach to quantum electrodynamics (QED) with his paper [2], introduced a method to calculate Dyson series. This laid the foundation for the Feynman diagram technique within the QED framework, simplifying the computation and understanding of intricate quantum field theory processes. Over the years, the Feynman path integral formulation has become indispensable in modern physics, aiding physicists in intuitively comprehending and calculating the evolution, interactions and classical limits of quantum systems.

Although both theories being labeled as “space-time approaches”, a significant disconnect exists between the non-relativistic path integral theory and the path integral theory in QED. The former cannot be viewed as a low-energy approximation of the latter, nor can the latter be seen merely as a relativistic extension of the former [3,4]. Additionally, while the non-relativistic path integral cleverly links the classical action $S_c$ with the Schrödinger equation, it lacks a parallel formulation for relating the relativistic classical action $S_r$ to the Dirac equation in relativistic quantum mechanics. Conversely, although the QED formulation provides a general description of action in a field, it does not correspond to classical mechanics. Despite extensive efforts to reconcile quantum field theory with classical interpretive frameworks, significant theoretical discrepancies persist [5,6,7,8,9,10,11,12,13].

Given the widespread applicability of path integrals in both non-relativistic quantum mechanics and QED, the disconnect between these formulations suggests that the conventional path-integral framework may still conceal a more general structure, one that consistently relates relativistic action, single-particle dynamics, and quantum field–theoretic descriptions.

Historically, researchers have attempted to uncover this hidden path integral formula. The Feynman checkerboard model is one of the most extensively studied theoretical models. It offers an elegant method to relate the Dirac equation with particle stochastic trajectories [14]. This model ingeniously derives the one-dimensional Dirac equation from the zigzag motion of particles at the speed of light [10,15,16]. However, extending this model to higher dimensions encounters several obstacles, including computational complexity, divergences and maintaining gauge invariance. Researchers have employed alternative summation rules, transfer matrix methods [17], and Monte Carlo methods [18] to tackle these issues, but a general theory applicable to any dimension has yet to emerge. Gaveau and Quezada attempted to link higher-dimensional Dirac equations with stochastic paths in momentum space, necessitating further clarification of the physical significance and interpretation [19,20]. To advance this line of research, several authors have developed worldline path-integral representations for the Dirac propagator in external electromagnetic fields, starting from the relativistic particle action and introducing Grassmann degrees of freedom for spin [21,22,23,24]. These approaches provide elegant formulae for the Dirac propagator and clarify the role of the spin factor in arbitrary electromagnetic backgrounds and dimensions [24,25]. Further developments have extended the worldline formalism to massive Dirac propagators in four-dimensional space–time and beyond [26,27]. However, extending these methods to completely general external field configurations while maintaining a transparent single-particle interpretation and a direct link to the classical relativistic action remains nontrivial. Moreover, these worldline constructions are conceptually closer to quantum-field-theoretic path integrals than to Feynman’s original non-relativistic path integral, and taken together they still do not provide a single unified framework that simultaneously covers the Schrödinger path integral, relativistic scalar particles and the Dirac equation.

Despite these significant efforts, the development of a relativistic path integral theory for single-particle evolution has remained elusive. This ongoing absence has led some to conclude that such a theory may not exist and that the QED path integral formulation represents the sole relativistic variant. However, this conclusion might be premature. The challenges encountered in quantum mechanics over the past two decades suggest a need to reassess whether current quantum mechanical theories truly provide a complete and accurate description of the physical world. Today, quantum mechanics still struggle to fully explain the spacetime mechanisms underlying quantum nonlocal properties, limiting our capacity to harness and manipulate quantum nonlocal correlations effectively. Discovering a spacetime method to elucidate these physical issues could develop new ways to manipulate quantum nonlocal correlations and quantum states, potentially transforming the current landscape of quantum information and quantum computing. This could even lead to new physical conclusions, addressing the origins of spin and mechanisms of superconductivity, thereby expanding our understanding of the quantum world.

In this paper, we propose a spinor form of the relativistic path integral that fills this gap. Starting from a relativistic classical action written in spinor form, we construct a short–time kernel whose continuum limit yields the Dirac equation with minimal coupling. The construction is carried out in an arbitrary number of spatial dimensions and is expressed in terms of boost–like spinor transformations and Tricomi functions. The resulting path integral extends our previous scalar relativistic path integral and provides a unified spacetime framework that connects classical relativistic mechanics, the Dirac equation and standard QED techniques. We also discuss how this spinor formulation sheds light on the origin of classical mechanics for the Dirac equation and on the spacetime structure underlying spin and quantum nonlocal correlations.

2. The Missing Path Integral and Its Potential Form

The existence of a missing content in current path integrals is evidenced by their formal logical inconsistencies and incompleteness, which fail to encompass all aspects of quantum mechanics. We will discuss these points in the following text.

2.1. Inconsistency in Classical Path Integral Form

The classical Feynman path integral theory, or space-time approach to non-relativistic quantum mechanics, describes the evolution of single particles. In the opinion of this theory, the classical action associated with various particle trajectories is introduced into quantum mechanics as a phase factor. The propagator in quantum mechanics can be represented as a superposition of all these phase factors:

$$K(\mathit{r},t;{\mathit{r}}_0,t_0)=C_0^{{\displaystyle \frac{n}{2}}}\sum _{{\wp }_k}exp(\mathrm{i}{S}_c\left({\wp }_k\right)/\hslash ),$$

(1)

where $S_c$ is the classical action and $C_0=m_0/2\mathrm{i}\pi (t-t_0)$ is a path-independent normalization factor [1]. ${\wp }_k$ here is used to denote the function of the k-th path. However, this formulation presents challenges. The action $S_c$ in Equation (1) is classical, yet the paths considered are arbitrary, encompassing those approaching or surpassing light speed. This raises a critical question regarding the consistency of the theory: is it feasible to exclude these non-classical paths? The answer is negative. Eliminating these non-classical paths would lead to the inability to derive the Schrödinger equation and would detach the path integral from its fundamental connection to quantum mechanics [28,29].

Interestingly, contrary to intuitive understanding, within a time-sliced evolution process, which is a method to analyze quantum processes step-by-step in time, the contribution of classical path actions is marginal, with non-classical paths assuming a dominant role. This leads to another critical question: why rely on a classical action when these non-classical paths are so crucial? One might consider using a relativistic action in Equation (1) as an alternative. However, substituting $S_c$ with a relativistic action would cause the formula to diverge, thereby rendering the path integral ineffective [28].

This dilemma exposes a fundamental inconsistency within the classical path integral paradigm: the path selection is rooted in non-classical trajectories, yet the action is constrained within a classical mechanical quantity. This inconsistency results in the path integral lacking self-consistency and also implies that the classical path integral theory may simply be a projection of a more fundamental theory at low energies. It is an approximate theory.

2.2. Incompleteness of Path Integral Forms

In modern quantum mechanics, the path integral theory serves as a deeply influential conceptual framework, linking the dynamical evolution of quantum states with the various possible processes that a system may undergo. In non-relativistic quantum mechanics, this theory connects the classical action of a single particle with the Schrödinger equation, providing a novel approach to bridging classical mechanics and quantum mechanics. In quantum field theory, the introduction of Feynman diagrams corresponding to Dyson series has significantly advanced our understanding of particle interactions at high energies, offering profound insights into processes such as particle scattering and quantum field interactions.

Despite these significant advancements, a critical gap persists: there is currently no path integral formulation that directly corresponds to the Dirac equation, a cornerstone of relativistic quantum mechanics. The Dirac equation not only unifies quantum mechanics with special relativity but also accurately describes spin-1/2 particles, such as electrons and positrons. It predicts the existence of antimatter and explains phenomena such as the fine structure of atomic spectra and the Landé g-factor, which are fundamental to our understanding of atomic and subatomic processes. The expectation for a comprehensive path integral theory to illuminate these quantum mechanical nuances is both necessary and pressing. Yet, the current theoretical framework lacks a path integral model that seamlessly integrates the Dirac equation with relativistic action, thus limiting our understanding of these phenomena from a spacetime perspective.

It is believed that a more fundamental and comprehensive path integral theory should exist—one that not only connects classical mechanics with non-relativistic quantum mechanics but also bridges classical relativistic mechanics with the Dirac equation and quantum electrodynamics (QED). Such a theory would provide a unified framework for understanding quantum phenomena across all energy scales. This framework would not only enable deeper exploration into quantum mechanics, such as the precise modeling of relativistic quantum systems and the behavior of particles at the intersection of quantum mechanics and general relativity, but it could also potentially resolve unresolved issues in quantum mechanics. These issues include a complete understanding of spin, the origin of quantum entanglement in relativistic systems, and the accurate description of high-energy processes like those occurring near black holes or during particle collisions at extremely high energies. Such a theory could revolutionize our understanding of quantum mechanics and reshape the landscape of quantum physics and its practical applications, from quantum computing to high-energy particle physics.

2.3. Formulating the Missing Path Integral

The above considerations indicate that, within the conventional framework of quantum mechanics, there is a missing path integral that would satisfy the following requirements:

(1)

It reduces to the standard Feynman path integral and reproduces the Schrödinger equation in the appropriate non-relativistic limit.

(2)

It connects the relativistic classical action to the Dirac equation for spin-$1/2$ particles, providing a single-particle relativistic spacetime description.

(3)

It yields a mathematically well-defined and convergent functional integral.

• In other words, we seek a path-integral formulation that unifies the roles of $S_c$ and $S_r$ and bridges non-relativistic and relativistic quantum mechanics within a single spacetime framework.

Considering an m-dimensional spinor wave function, denoted as $\Psi ={({\psi }_1,{\psi }_2,\dots ,{\psi }_m)}^T$, each component has a corresponding path integral representation, i.e., $\langle \mathit{r},t|{\psi }_j\rangle ={\int }_{-\infty }^{\infty }{K}^{jj}\langle {\mathit{r}}_0,t|{\psi }_j\rangle {\mathrm{d}}^n\mathit{r}$, where $K^{jj}={\left(C_0^j\right)}^{n/2}{\sum }_{{\wp }_k}{e}^{\mathrm{i}{S}_r^j\left({\wp }_k\right)/\hslash }$. $S_r$ is the scalar relativistic action. The main reason for writing the action in the form of $S_r^j$ here is to take into account that the nature of the dynamics may be different in different subspaces. In this way, the time evolution of $|\Psi \rangle$ can be expressed as:

$$\begin{array}{cc}\hfill & \Psi (\mathit{r},t)={\left(\begin{array}{cccc}\langle \mathit{r},t|{\psi }_1\rangle ,& \langle \mathit{r},t|{\psi }_2\rangle ,& \dots ,& \langle \mathit{r},t|{\psi }_m\rangle \end{array}\right)}^T\hfill \\ \hfill & ={\int }_{-\infty }^{\infty }\left[\mathrm{Diag}{\left(\begin{array}{c}\langle \mathit{r},t|{\mathit{r}}_0,t_0\rangle \\ \langle \mathit{r},t|{\mathit{r}}_0,t_0\rangle \\ ⋮\\ \langle \mathit{r},t|{\mathit{r}}_0,t_0\rangle \end{array}\right)}^T\right]\left(\begin{array}{c}\langle {\mathit{r}}_0,t_0|{\psi }_1\rangle \\ \langle {\mathit{r}}_0,t_0|{\psi }_2\rangle \\ ⋮\\ \langle {\mathit{r}}_0,t_0|{\psi }_m\rangle \end{array}\right){\mathrm{d}}^3{\mathit{r}}_0\hfill \\ \hfill & ={\int }_{-\infty }^{\infty }\left(\begin{array}{ccccc}& K^{11}& 0& \dots & 0\\ & 0& K^{22}& \dots & 0\\ & ⋮& ⋮& \dots & ⋮\\ & 0& 0& 0& K^{mm}\end{array}\right)\left(\begin{array}{c}\langle {\mathit{r}}_0,t_0|{\psi }_1\rangle \\ \langle {\mathit{r}}_0,t_0|{\psi }_2\rangle \\ ⋮\\ \langle {\mathit{r}}_0,t_0|{\psi }_m\rangle \end{array}\right){\mathrm{d}}^3{\mathit{r}}_0\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ={\int }_{-\infty }^{\infty }{K}_{diag}(\mathit{r},t;{\mathit{r}}_0,t_0)\Psi ({\mathit{r}}_0,t_0){\mathrm{d}}^3{\mathit{r}}_0\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & ={\int }_{-\infty }^{\infty }{K}_{diag}\left[\widehat{M}({\mathit{r}}_0,t_0)\Phi ({\mathit{r}}_0,t_0)\right]{\mathrm{d}}^3{\mathit{r}}_0,\hfill \end{array}$$

(3)

Here, $K_{diag}$ represents the diagonal propagator and $\mathrm{Diag}(a_1,a_2,\dots ,a_n)$ denotes the diagonal matrix with diagonal elements $a_1,a_2,\dots ,a_n$. The unitary transformation matrix $\widehat{M}$ allows us to transition from $\Psi ({\mathit{r}}_0,t_0)$ to $\Phi ({\mathit{r}}_0,t_0)$, so $\Psi ({\mathit{r}}_0,t_0)=\widehat{M}({\mathit{r}}_0,t_0)\Phi ({\mathit{r}}_0,t_0)$. The matrix $\widehat{M}$ is written in operator form because it performs two roles: first, it carries out rotational transformations in spinor space, changing the basis from $\left(\right|{\psi }_1\rangle ,|{\psi }_2\rangle ,\dots ,|{\psi }_n\rangle )$ to $\left(\right|{\varphi }_1\rangle ,|{\varphi }_2\rangle ,\dots ,|{\varphi }_n\rangle )$; second, it transforms the Hilbert space components of the spinor, as expressed by $\widehat{M}|{\psi }_n^j\rangle ={\sum }_m{a}_m|{\psi }_n^m\rangle$. It is worth emphasizing that the ${\psi }_m$ and ${\varphi }_m$ are the m-th spatial components in the n-dimensional spinor space and their j-th eigenstates are expressed as ${\varphi }_m^j$ and ${\psi }_m^j$ respectively. Under these notations, we have ${\psi }_m={\sum }_j{a}_j{\psi }_m^j$ and ${\varphi }_m={\sum }_j{b}_j{\varphi }_m^j$. It is important to note that the key difference between $\left\{\right|{\psi }_m^j\rangle \}$ and $\left\{\right|{\varphi }_m^j\rangle \}$ lies in their behavior under the action of the evolution operator $\widehat{U}(t,t_0)$. Specially, $\left\{\right|{\psi }_m^j\rangle \}$ is closed under the action of $\widehat{U}(t,t_0)$, meaning it remains within the same state space, while $\left\{\right|{\varphi }_m^j\rangle \}$ is not closed under $\widehat{U}(t,t_0)$, indicating that it transitions out of its original state space. Therefore, $|\Psi \rangle$ is a special state where the dynamical evolution of its components is independent and non-interfering, whereas $|\Phi \rangle$ is a general state. The purpose of this derivation’s final step is to extend the path integral theory, which applies to the special state $|\Psi \rangle$, to the general state $|\Phi \rangle$.

To isolate $\Phi (\mathit{r},t)$, we apply the inverse of the transformation matrix $\widehat{M}$:

$$\begin{array}{cc}\hfill \Phi (\mathit{r},t)& ={\int }_{-\infty }^{\infty }\left[{\widehat{M}}^{-1}(\mathit{r},t)K_{diag}\widehat{M}({\mathit{r}}_0,t_0)\right]\Phi ({\mathit{r}}_0,t_0){\mathrm{d}}^3{\mathit{r}}_0\hfill \\ \hfill & ={\int }_{-\infty }^{\infty }{\widehat{K}}_{non-diag}\Phi ({\mathit{r}}_0,t_0){\mathrm{d}}^3{\mathit{r}}_0.\hfill \end{array}$$

(4)

Equation (4) reveals that to extend path integral theory into a general spinor space, it must be expressed in the form of an operator rather than a simple function. In particular, the unitary transformation $\widehat{M}$ is not an arbitrary change of basis. As will be shown in Section 3.1, when the relativistic action is expressed in terms of the classical velocity the same mapping is realized explicitly by the Lorentz–boost matrix $L_0$. In this way, the abstract construction in Equations (2)–(4) is later identified with the relativistic boost that diagonalizes the Dirac Hamiltonian, and the operator-valued path integral kernel ${\widehat{K}}_{\mathrm{non}\text{-}\mathrm{diag}}$ acquires a concrete relativistic interpretation.

From our analysis, we observe that if a universal scalar form of the path integral exists within a subspace $|{\psi }_j\rangle$, represented as $K^{jj}={\left(C_0^j\right)}^{n/2}{\sum }_{{\wp }_k}exp(\mathrm{i}{S}_r^j\left({\wp }_k\right)/\hslash )$, the differing dynamical properties across subspaces imply that the actions $S^j$ will differ accordingly. Consequently, the normalization factors $C_0^j$ will also vary. Under the action of $\widehat{M}$, the wave functions of different subspaces will be recombined, transforming the corresponding propagator ${\widehat{K}}_{non-diag}$ from a simple function into an operator form. This transformation is essential to accurately link the relativistic action with the Dirac equation in spinor form. Therefore, we must shift our perspective from viewing the path integral theory as a mere scalar function to constructing its general form from the standpoint of a functional operator. This shift is not merely a mathematical formality but a necessary step toward developing a more comprehensive and accurate quantum theory that accounts for the complexities of relativistic spinor fields.

Since this path integral form needs to include the relativistic action and might be an operator, we can express this path integral in a general form:

$${\widehat{K}}_R={\widehat{K}}_{non-diag}=\widehat{R}\sum _{{\wp }_k}{e}^{\mathrm{i}{S}_R\left({\wp }_k\right)/\hslash },$$

(5)

where ${\widehat{K}}_R$ represents the spinor form of the relativistic path integral, $S_R$ is the relativistic action in spinor form, which is different from the scalar relativistic action $S_r={\int }_{t_0}^t(-{\gamma }_L{m}_0{c}^2-q\mathit{A}+qV)\mathrm{d}{t}_1={\int }_{t_0}^t{\mathcal{L}}_r\mathrm{d}{t}_1$. ${\gamma }_L$ represents the Lorentz factor, equaling $1/\sqrt{1-v^2/c^2}$. $\widehat{R}$ is an operator that is independent of the path and ${\wp }_k$ denotes the function of the k-th path.

There might be reservations about $\widehat{R}$ being in an operator form. However, transitioning $\widehat{R}$ to an operator form is not merely an inevitable result of evolving from diagonalized propagators to non-diagonalized spinor path integrals; it also signifies our mathematical requirement to move from handling the evolution of plane waves to accommodating general spinor wave functions (as we will see in subsequent sections). This insight further stimulates our contemplation on the path integral form. In the context of Feynman’s path integral theory, the normalization factor $C_0$ is traditionally considered path-independent, a premise that, while widely accepted, lacks rigorous theoretical or empirical validation. This raises a possibility that the constancy of $C_0$ might be an effective approximation within a more generalized path integral framework. Such a perspective points to a new clue for us to explore a path integral that transcends the limitations of current formulations, particularly by ensuring that $\widehat{R}$ smoothly converges to $C_0$ in the low-energy regime, thereby maintaining compatibility with the Feynman path integral. Importantly, despite the spinor form of the path integral described in Equation (5) losing its identity as a propagator, it still functions as a spacetime entity that delineates the evolution of the wave function, thereby establishing a link between the action and the quantum mechanics evolution equation.

3. Construction of Path Integral Theory in Spinor Form

3.1. Tricomi Function and the Spinor Path Integral

The key to constructing the path integral in spinor form is to determine the expressions for $\widehat{R}$ and $S_R$. The form of $S_R$ is crucial; choosing an appropriate expression for $S_R$ can simplify the expression of $\widehat{R}$ and make its physical meaning more apparent.

The core issue in determining $S_R$ is how to write ${\gamma }_L{m}_0{c}^2$ in spinor form. Basing on Dirac’s reformulation of the Klein–Gordon equation, one might think that $\sqrt{1-v^2/c^2}=\beta \pm \mathit{\alpha }·v/c$. Here $\mathit{\alpha }$ and $\beta$ use the Dirac representation [30]. However, this spinor expression is not appropriate. Due to the specificity of spinor forms, $\beta m_0{c}^2-m_{0}c\mathit{\alpha }·v$ has different diagonalization matrices for different $v$ values, making it difficult to construct the connection $\beta m_0{c}^2\pm m_{0}c\mathit{\alpha }·v\to \beta m_0{c}^2+\mathit{\alpha }·\widehat{\mathit{p}}c$ in the path integral process.

To establish the connection between the relativistic action and the Dirac equation, it is required that $\sqrt{1-v^2/c^2}=\beta -\beta \mathit{\alpha }·v/c$. We must note that this spinor form belongs to $C{l}_{3,1}$, making $\left|v\right|<c$ and $\left|v\right|>c$ belong to different branches. Consequently,

$$\begin{array}{cc}\hfill S_R& ={\int }_{t_0}^t(\beta m_0{c}^2({\displaystyle \frac{\mathit{\alpha }}{c}}·v-1)-qU)\mathrm{d}{t}_1={\int }_{t_0}^t{\mathcal{L}}_R\mathrm{d}{t}_1.\hfill \end{array}$$

(6)

Here and in what follows, $\beta$ and $\mathit{\alpha }$ are understood as elements of the real Clifford algebra ${\mathrm{Cl}}_{3,0}$ represented in the standard Dirac matrix representation. The symbol $\mathit{\alpha }$ denotes a spatial vector in ${\mathrm{Cl}}_{3,0}$ whose components are Dirac matrices, and is therefore written with an arrow to emphasize its vector character, whereas $\beta$ is a scalar element of the algebra and is written without an arrow. Scalar quantities such as the potential $U=V-\mathit{A}·v$ enter the Lagrangian as multiples of the unit Clifford element, which corresponds to the $4\times 4$ identity matrix in the Dirac representation. V and $\mathit{A}$ here are used to denote the scalar and vector potential respectively. In this new expression, the diagonalization matrix $L_0$ of $S_R$ can be obtained:

$$\begin{array}{cc}\hfill L_0\left(v\right)& =e^{-{\displaystyle \frac{1}{2}}\mathrm{arctanh}(\mathit{\alpha }·v/c)}\hfill \\ \hfill & =\sqrt{{\displaystyle \frac{{\gamma }_L+1}{2}}}-\sqrt{{\displaystyle \frac{{\gamma }_L-1}{2}}}\mathit{\alpha }·{\mathit{n}}_v\hfill \\ \hfill & =\left(\begin{array}{cc}{L}_0^{11}\left(v\right)& -L_0^{12}\left(v\right)\\ -L_0^{21}\left(v\right)& L_0^{22}\left(v\right)\end{array}\right),\hfill \end{array}$$

(7)

where ${\mathit{n}}_v=v/\left|v\right|$ is the unit vector of $v$, and $\mathbf{\sigma }$ signifies the Pauli matrices. $L_0^{ij}\left(v\right)=\sqrt{{\displaystyle \frac{{\gamma }_L\left(v\right)+{(-1)}^{i+j}}{2}}}{(\mathbf{\sigma }·{\mathit{n}}_v)}^{i+j}$ Although the expression of $L_0$ is not often seen in physics, it is in fact a Lorentz transformation under the structure of the Clifford algebra. Under the Clifford algebra structure, spacetime can be expressed as $\mathbb{R}=ct+\mathit{\alpha }·\mathit{r}$, and under this expression, the spacetime transformation (Lorentz boost) can be written as: ${\mathbb{R}}^{\prime }=L_0^{-1}\mathbb{R}{L}_0$.

According to Equation (6), we can see the eigenvalues of $S_R$ contain terms $\pm m_0{c}^2\sqrt{1-v^2/c^2}$. Thus, using the path integral in spinor form to calculate the evolution of the wave function will inevitably involve the following integral form:

$${\int }_{-\infty }^{\infty }f\left(v\right)e^{\pm \mathrm{i}\chi \sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\varphi (x_0+v\epsilon ,t)\mathrm{d}\left(v\epsilon \right)$$

(8)

where $\chi =m_0{c}^2\epsilon /\hslash$ is dimensionless constant and $f\left(v\right)$ is an integral function brought about by $\widehat{R}$ and the diagonalization matrix $L_0$. Here $\chi =m_0{c}^2\epsilon /\hslash$ is a dimensionless constant, and $f\left(v\right)$ is an integral function brought about by $\widehat{R}$ and the diagonalization matrix $L_0$. This type of integral will face convergence issues. To ensure convergence, we require $e^{\pm \mathrm{i}\chi \sqrt{1-v^2/c^2}}\to 0$ as $\left|v\right|\to \infty$, meaning that when $\left|v\right|>0$, ${\mathcal{L}}_R=\mathrm{i}{m}_0{c}^2\sqrt{v^2/c^2-1}-qU$. Only under this assumption can we establish a single-particle path integral theory corresponding to the Dirac equation.

Even if we ensure the convergence of the integral above, we still face the complexity of solving such integrals, which often do not have an analytical expression. However, we notice that when $f\left(v\right)$ takes certain specific functions, these integrals can be expressed analytically using the Tricomi function:

$$\begin{array}{cc}\hfill & {\int }_{-\infty }^{\infty }{\gamma }_L^{1/2}\sqrt{{\displaystyle \frac{{\gamma }_L\pm 1}{2}}}{e}^{\pm \mathrm{i}\chi \sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}{e}^{\mathrm{i}pv\epsilon }\mathrm{d}v\hfill \\ \hfill \sim & \sum _k{a}_k\left(p\right)U(-k,{\displaystyle \frac{1}{2}}-2k,\mp 2\mathrm{i}\chi ).\hfill \end{array}$$

(9)

The function $U(k,m,x)$ is known as the Tricomi function. When k is a non-positive integer, it is closely related to the spherical Bessel functions $j_k(1/x)$ and $y_k(1/x)$ [31]. It is known that the spherical Hankel functions $h_k^{\pm }\left(z\right)=j_k\left(z\right)\pm \mathrm{i}{y}_k\left(z\right)$ [32] have the generating function ${\displaystyle \frac{1}{z}}{e}^{\pm \mathrm{i}\sqrt{z^2-2zt}}$ [33]. This means that, if we carefully choose a suitable expression for $f\left(v\right)$, it will be possible to establish the following relationships:

$$\begin{array}{cc}\hfill & \sum _k{a}_k\left(p\right)U(-k,{\displaystyle \frac{1}{2}}-2k,\pm 2\mathrm{i}\chi )\hfill \\ \hfill \sim & \sum _k{b}_k\left(p\right)h_k^{\pm }\left(\chi \right)\sim e^{\pm \mathrm{i}\chi \sqrt{1+{\left({\displaystyle \frac{p}{p_0}}\right)}^2}}.\hfill \end{array}$$

(10)

These relationships are crucial for the construction of relativistic path integrals, implying that the integral with a core of $e^{\mathrm{i}{\mathcal{L}}_r\epsilon /\hslash }$ can be transmuted $e^{-\mathrm{i}{\widehat{H}}_r\epsilon /\hslash }$ by recognizing that $\chi \sqrt{1+{(p/p_0)}^2}=\sqrt{m_0^2{c}^4+p^2{c}^2}\epsilon /\hslash$, thus making it possible for us to establish the linkage between relativistic classical mechanics and relativistic quantum mechanics. The scalar Hamiltonian here is ${\widehat{H}}_r=\sqrt{m_0^2{c}^4+{\widehat{\mathit{p}}}^2{c}^2}+qU$. The key to this linkage, namely the expression for $\widehat{R}$, can be derived through the relationship between Tricomi functions and the generating functions of Hankel functions. After some calculations, we come to the following strong conclusions (See Appendix A and Appendix B).

$$\begin{array}{cc}\hfill & {\int }_{-\infty }^{\infty }{L}_n^{ij}\left(v\right)e^{\mathrm{i}{\mathcal{L}}_r\epsilon /\hslash }\psi ({\mathit{r}}_0,t_0){\mathrm{d}}^n{\mathit{r}}_0\hfill \\ \hfill & ={(-1)}^{i-j}{C}_0^{-{\displaystyle \frac{n}{2}}}{L}_n^{ij}\left(\widehat{v}\right)e^{-\mathrm{i}{\widehat{H}}_r\epsilon /\hslash }\psi (\mathit{r},t_0);\hfill \\ \hfill & {\int }_{-\infty }^{\infty }{L}_n^{ij}\left(v\right)e^{-\mathrm{i}{\mathcal{L}}_r\epsilon /\hslash }\psi ({\mathit{r}}_0,t_0){\mathrm{d}}^n{\mathit{r}}_0\hfill \\ \hfill & ={\mathrm{i}}^n{C}_0^{-{\displaystyle \frac{n}{2}}}{L}_n^{ij}\left(\widehat{v}\right)e^{\mathrm{i}{\widehat{H}}_r\epsilon /\hslash }\psi (\mathit{r},t_0).\hfill \end{array}$$

(11)

where, $\widehat{v}=(\widehat{\mathit{p}}-q\mathit{A})/\left(\beta m_0\right)$ and the scalar relativity Hamiltonian ${\widehat{H}}_r=\sqrt{m_0^2{c}^4+{(\widehat{\mathit{p}}-q\mathit{A})}^2}+qV$. ${\mathrm{d}}^n\mathit{r}$ is the abbreviation of the form $\mathrm{d}{x}_1\mathrm{d}{x}_2\dots \mathrm{d}{x}_n$ and $L_n^{ij}$ is the $(i,j)$-th element of the matrix $L_n\left(v\right)$, which takes the form

$$L_n\left(\mathit{r}\right)={\gamma }_L^{{\displaystyle \frac{n}{2}}}{\left({\displaystyle \frac{2}{1+{\gamma }_L}}\right)}^{{\displaystyle \frac{n-1}{2}}}{L}_0.$$

(12)

The significance of Equation (11) lies in its applicability to any dimensionality $n\in \mathbb{N}$. In classical Feynman path integrals, the reason for using the classical Lagrangian $L_c(\mathit{r},\dot{\mathit{r}})=m_0{\dot{\mathit{r}}}^2/2-V\left(\mathit{r}\right)$ is that, when it acts as a phase factor in the propagator, the wave function’s evolution over a time slice reduces to solving the integral ${\int }_{-\infty }^{\infty }{e}^{\mathrm{i}\chi v^2/c^2}{e}^{\mathrm{i}\mathit{p}·v}{\mathrm{d}}^{n}v$, which can be separated into a product of n-independent one-dimensional integrals ${\int }_{-\infty }^{\infty }{e}^{\mathrm{i}\chi v_x^2/c^2}{e}^{\mathrm{i}{p}_x{v}_x}\mathrm{d}{v}_x{\int }_{-\infty }^{\infty }{e}^{\mathrm{i}\chi v_y^2/c^2}{e}^{\mathrm{i}{p}_y{v}_y}\mathrm{d}{v}_y\dots$. Thanks to this separability, solving the one-dimensional case automatically extends to the n-dimensional case in classical path integrals. However, in the relativistic case, the presence of the term $\sqrt{1-v^2/c^2}$ introduces a non-linearity that couples the velocity components, preventing the dimensional separation of variables. Thus, solving the one-dimensional relativistic case does not easily generalize to higher dimensions. Nonetheless, Equation (11) presented above provide a method that can indeed generalize to arbitrary dimensions. Hence, these results are powerful and significant conclusions.

Considering the following expression

$$e^{\left({\displaystyle \frac{\mathrm{i}\epsilon }{\hslash }}\beta {\mathcal{L}}_r\right)}=\left(\begin{array}{cc}{e}^{\mathrm{i}{\mathcal{L}}_r\epsilon /\hslash },& 0\\ 0,& e^{-\mathrm{i}{\mathcal{L}}_r\epsilon /\hslash }\end{array}\right),$$

(13)

we can further transform Equation (11) as follows:

$$L_0^{-1}{e}^{\left({\displaystyle \frac{\mathrm{i}\epsilon }{\hslash }}\beta {\mathcal{L}}_r\right)}{L}_0=e^{{\displaystyle \frac{\mathrm{i}\epsilon }{\hslash }}{\mathcal{L}}_R}.$$

(14)

Noting that $L_0^{-1}{e}^{\left({\displaystyle \frac{\mathrm{i}\epsilon }{\hslash }}\beta {\mathcal{L}}_r\right)}{L}_0=e^{{\displaystyle \frac{\mathrm{i}\epsilon }{\hslash }}{\mathcal{L}}_R}$, Equation (14) can be further simplified to

$$\begin{array}{cc}\hfill & {\int }_{-\infty }^{\infty }{L}_n\left(v\right)e^{\left({\displaystyle \frac{\mathrm{i}\epsilon }{\hslash }}{\mathcal{L}}_R\right)}\Psi ({\mathit{r}}_0,t_0){\mathrm{d}}^n{\mathit{r}}_0\hfill \\ \hfill =& {\left(C_0\beta \right)}^{-{\displaystyle \frac{n}{2}}}{L}_n\left(\widehat{v}\right)e^{-\mathrm{i}{\widehat{H}}_R\epsilon /\hslash }\Psi (\mathit{r},t_0).\hfill \end{array}$$

(15)

Here, ${\widehat{H}}_R=\beta m_0{c}^2+\mathit{\alpha }·(\widehat{\mathit{p}}-q\mathit{A})c+qU(\mathit{r},t)$, which is satisfied this transformation ${\widehat{H}}_R=L_n^{-1}\left(\widehat{v}\right)\left(\beta {\widehat{H}}_r\right)L_n\left(\widehat{v}\right)$. Thus, in Equation (5), we can construct the relativistic single-particle path integral in spinor form for arbitrary dimensions by defining

$$\widehat{R}(\mathit{r},{\mathit{r}}_0;t,t_0)=L_n^{-1}\left(\widehat{v}\right){\left(C_0\beta \right)}^{{\displaystyle \frac{n}{2}}}{L}_n\left(v\right).$$

(16)

Here, $v=(\mathit{r}-{\mathit{r}}_0)/(t-t_0)$. This kind of $\widehat{R}$ will leads Equation (15) to

$$\begin{array}{cc}\hfill & {\int }_{-\infty }^{\infty }\widehat{R}(\mathit{r},{\mathit{r}}_0;t,t_0)e^{\left({\displaystyle \frac{\mathrm{i}\epsilon }{\hslash }}{\mathcal{L}}_R\right)}\Psi ({\mathit{r}}_0,t_0){\mathrm{d}}^n{\mathit{r}}_0\hfill \\ \hfill =& L_n\left(\widehat{v}\right){\left(C_0\beta \right)}^{{\displaystyle \frac{n}{2}}}{\int }_{-\infty }^{\infty }{L}_n\left(v\right)e^{\left({\displaystyle \frac{\mathrm{i}\epsilon }{\hslash }}{\mathcal{L}}_R\right)}\Psi ({\mathit{r}}_0,t_0){\mathrm{d}}^n{\mathit{r}}_0\hfill \\ \hfill =& e^{-\mathrm{i}\widehat{H}\epsilon /\hslash }\Psi (\mathit{r},t_0).\hfill \end{array}$$

(17)

It should be emphasized that the expression for $\widehat{R}$ is unique. As can be seen from Equation (9), changing the expression of $\widehat{R}$ will change the value of $a_k\left(p\right)$, which will result in the resulting expression for the path integral being linearly independent of $e^{\pm \mathrm{i}\chi \sqrt{1+{(p/p_0)}^2}}$, and thus it will be impossible to construct a relationship between $S_r$ and ${\widehat{H}}_r$.

Using $\widehat{\mathcal{T}}$ to denote the time-ordering operator and incorporating the expressions for $S_R$ in Equation (6) and $\widehat{R}$ in Equation (16), we have

$$\begin{array}{cc}\hfill & \left\{\begin{array}{c}{S}_R={\int }_{\wp }(-\beta m_0{c}^2(1-{\displaystyle \frac{\mathit{\alpha }·v}{c}})-qU)\mathrm{d}{t}_1\hfill \\ {\widehat{K}}_R\left(\mathit{r},t;{\mathit{r}}_0,t_0\right)=\widehat{R}\sum _{{\wp }_k}exp\left({\displaystyle \frac{\mathrm{i}{S}_R\left({\wp }_k\right)}{\hslash }}\right)\hfill \\ \Psi (\mathit{r},t)={\int }_{-\infty }^{\infty }{\widehat{K}}_R(\mathit{r},t;{\mathit{r}}_0,t_0)\Psi ({\mathit{r}}_0,t_0){\mathrm{d}}^n{\mathit{r}}_0\hfill \end{array}\right.\hfill \end{array}$$

(18)

$$\begin{array}{cc}\Rightarrow \hfill & \left\{\begin{array}{c}{\widehat{H}}_R=\beta m_0{c}^2+\mathit{\alpha }·(\widehat{\mathit{p}}-q\mathit{A})c+qU(\mathit{r},t)\hfill \\ \Psi (\mathit{r},t)=\widehat{\mathcal{T}}{e}^{-{\displaystyle \frac{\mathrm{i}}{\hslash }}{\int }_{t_0}^t{\widehat{H}}_R\mathrm{d}{t}_1}\Psi (\mathit{r},t_0)\hfill \\ \mathrm{i}\hslash {\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }t}}\Psi (\mathit{r},t)={\widehat{H}}_R\Psi (\mathit{r},t).\hfill \end{array}\right.\hfill \end{array}$$

(19)

It is the Dirac equation.

3.2. Exposition on the Spinor Form of Path Integral Expression

From the analysis above, we can see that the expression for $\widehat{R}$ is the result of mathematical operations. It may seem complicated, but holds substantial physical significance. Specifically, in the expression for $L_n$, $L_0=e^{-{\displaystyle \frac{1}{2}}\mathrm{arctanh}(v/c)}$ represents the Lorentz transformation, which is a physical quantity that must appear in the transition from a diagonal propagator to a spinor form of the path integral. Moreover, the occurrence of the term ${\gamma }_L^{n/2}$ in Equation (12) is notably deliberate, mirroring the relativistic adjustment of the original coefficient $C_0$ to accommodate proper time $\Delta \tau =(t-t_0)/{\gamma }_L$, a Lorentz invariant. This adaptation emphasizes the relativistic covariance of the path integral formulation. As for the term $\sqrt{(1+\widehat{{\gamma }_L})/2}$, though less common in physics, aligns with the normalization factor $\sqrt{(H_r+m_0{c}^2)/2H_r}$ for Dirac plane waves, underscoring its relevance in relativistic corrections.

It is important to note that since the conclusion

$${\int }_{t_0}^t\left(1-\mathit{\alpha }·{\displaystyle \frac{v}{c}}\right)\mathrm{d}{t}_1=\left(t-t_0-\mathit{\alpha }·{\displaystyle \frac{\mathit{r}-{\mathit{r}}_0}{c}}\right)$$

(20)

is path-independent, ${\widehat{K}}_R$ in Equation (18) can be further simplified as:

$$\begin{array}{cc}\hfill {\widehat{K}}_R(\mathit{r},{\mathit{r}}_0;t,t_0)=& \widehat{R}{e}^{{\displaystyle \frac{\mathrm{i}}{\hslash }}(\beta m_0{c}^2(t-t_0)+\beta m_{0}c\mathit{\alpha }·(\mathit{r}-{\mathit{r}}_0))}\hfill \\ \hfill & \sum _{{\wp }_k}{e}^{{\displaystyle \frac{\mathrm{i}}{\hslash }}{\int }_{{\wp }_k}(q\mathit{A}·v-qV)\mathrm{d}{t}_1}\hfill \\ \hfill =& K_0(\mathit{r},{\mathit{r}}_0;t,t_0)\sum _{{\wp }_k}{e}^{{\displaystyle \frac{\mathrm{i}}{\hslash }}{\int }_{{\wp }_k}(q\mathit{A}·v-qV)\mathrm{d}{t}_1}.\hfill \end{array}$$

Here, we use ${\widehat{K}}_0$ to represent the path integral in spinor form without potential energy. The path independence of the integration involving $\beta m_0{c}^2(1-\mathit{\alpha }·{\displaystyle \frac{v}{c}})$ across time leads to a significant simplification in the expression for ${\widehat{K}}_R$. This simplification reveals that the path integral’s dependence on particle paths, in the absence of potential energy boils down to the contributions from vector and scalar potentials.

The integration of the vector potential $\mathit{A}$ across different paths affects the kernel K, but intriguingly, this does not alter the wave function’s evolution. Under different paths, the contribution of $\mathit{A}$ to K is expressed as: $e^{{\displaystyle \frac{\mathrm{i}}{\hslash }}{\int }_{{\wp }_k}q\mathit{A}·v\mathrm{d}{t}_1}=e^{{\displaystyle \frac{\mathrm{i}}{\hslash }}(F_k\left(\mathit{r}\right)-F_k\left({\mathit{r}}_0\right))}$. Here, $F_k$, depending on path function ${\wp }_k$, satisfies $\nabla F_k\left(\mathit{r}\right)=q\mathit{A}$ and leads to $e^{\mathrm{i}{F}_k\left(\mathit{r}\right)/\hslash }\widehat{\mathit{p}}{e}^{-\mathrm{i}{F}_k\left(\mathit{r}\right)/\hslash }=\widehat{\mathit{p}}-q\mathit{A}$. When using Equation (18) to calculate the evolution of the state function, $e^{-\mathrm{i}{F}_k\left({\mathit{r}}_0\right)/\hslash }$ will be included in the integration and have the following relations

$$\begin{array}{cc}\hfill & \Psi (\mathit{r},t)={\int }_{-\infty }^{\infty }{\widehat{K}}_R(\mathit{r},t;{\mathit{r}}_0,t_0)\Psi ({\mathit{r}}_0,t_0){\mathrm{d}}^n{\mathit{r}}_0\hfill \\ \hfill & ={\int }_{-\infty }^{\infty }{\widehat{K}}_0(\mathit{r},t;{\mathit{r}}_0,t_0)\sum _k^N{e}^{\mathrm{i}{F}_k\left(\mathit{r}\right)}{e}^{-\mathrm{i}{F}_k\left({\mathit{r}}_0\right)}\Psi ({\mathit{r}}_0,t_0){\mathrm{d}}^n{\mathit{r}}_0\hfill \\ \hfill & =\sum _k^N{e}^{\mathrm{i}{F}_k\left(\mathit{r}\right)}{\int }_{-\infty }^{\infty }{\widehat{K}}_0^{\prime }(\mathit{r},t;{\mathit{r}}_0,t_0)\left[e^{-\mathrm{i}{F}_k\left({\mathit{r}}_0\right)}\Psi ({\mathit{r}}_0,t_0)\right]{\mathrm{d}}^n{\mathit{r}}_0\hfill \\ \hfill & ={\displaystyle \frac{1}{N}}\sum _k^N{e}^{\mathrm{i}{F}_k\left(\mathit{r}\right)}\widehat{\mathcal{T}}{e}^{-{\displaystyle \frac{\mathrm{i}}{\hslash }}{\int }_{t_0}^t{\widehat{H}}_R\mathrm{d}{t}_1}\left(e^{-\mathrm{i}{F}_k\left(\mathit{r}\right)}\Psi (\mathit{r},t_0)\right)\hfill \\ \hfill & =\widehat{\mathcal{T}}{e}^{-{\displaystyle \frac{\mathrm{i}}{\hslash }}{\int }_{t_0}^t{\widehat{H}}_R^{\prime }\mathrm{d}{t}_1}\Psi (\mathit{r},t_0),\hfill \end{array}$$

where ${\widehat{K}}_0^{\prime }=e^{\mathrm{i}{F}_k\left(\mathit{r}\right)/\hslash }{\widehat{K}}_0{e}^{-\mathrm{i}{F}_k\left(\mathit{r}\right)/\hslash }$. As we can see, after integration, $e^{-\mathrm{i}{F}_k\left({\mathit{r}}_0\right)/\hslash }$ will become $e^{-\mathrm{i}{F}_k\left(\mathit{r}\right)/\hslash }$, forming the structure $e^{\mathrm{i}{F}_k\left(\mathit{r}\right)/\hslash }\widehat{U}{e}^{-\mathrm{i}{F}_k\left(\mathit{r}\right)/\hslash }$ with the evolution operator $\widehat{U}=\widehat{\mathcal{T}}{e}^{{\displaystyle \frac{\mathrm{i}}{\hslash }}{\int }_{t_0}^t{\widehat{H}}_R\mathrm{d}{t}_1}$, eventually leading to ${\widehat{H}}_R^{\prime }\left(\widehat{p}\right)={\widehat{H}}_R(\widehat{p}-q\mathit{A})$ and thus

$$\mathrm{i}\hslash {\mathit{\partial }}_t\Psi (\mathit{r},t)=(\beta m_0{c}^2+\mathit{\alpha }·(\widehat{\mathit{p}}-q\mathit{A})c)\Psi (\mathit{r},t).$$

(21)

This analysis culminates in the realization that, despite the vector potential’s variable contribution under different paths to ${\widehat{K}}_R$, its effect on the state function’s evolution remains uniform. This uniformity, encapsulated in the transformation $e^{\mathrm{i}{F}_k\left(\mathit{r}\right)/\hslash }\widehat{\mathit{p}}{e}^{-\mathrm{i}{F}_k\left(\mathit{r}\right)/\hslash }=\widehat{\mathit{p}}-q\mathit{A}$, ensures that the vector potential’s integral translates identically across varying paths in the evolution of the state function.

In contrast, the scalar potential $V(\mathit{r},t)$ exhibits a true path dependency within the spinor form of the path integral. This dependency complicates the application of path integrals for analyzing the wave function’s long-term evolution, confining our analysis to within a temporal slice $\epsilon$. Within such a time slice, as $\epsilon \to 0$, the sum ${\sum }_{{\wp }_k}{e}^{{\displaystyle \frac{\mathrm{i}}{\hslash }}{\int }_{{\wp }_k}qV\mathrm{d}{t}_1}$ becomes proportional to $e^{{\displaystyle \frac{\mathrm{i}}{\hslash }}qV(\mathit{r},t)\epsilon }$. Furthermore, as $\epsilon \to 0$, the operator $\widehat{\mathit{p}}{e}^{{\displaystyle \frac{\mathrm{i}}{\hslash }}qV(\mathit{r},t)\epsilon }$ approaches $e^{{\displaystyle \frac{\mathrm{i}}{\hslash }}qV(\mathit{r},t)\epsilon }\widehat{\mathit{p}}$. Then, let $t-t_0=\epsilon$, we have

$$\begin{array}{cc}\hfill & \Psi (\mathit{r},t)={\int }_{-\infty }^{\infty }{\widehat{K}}_0(\mathit{r},{\mathit{r}}_0;t,t_0)e^{-{\displaystyle \frac{\mathrm{i}}{\hslash }}qV\epsilon }\Psi ({\mathit{r}}_0,t_0){\mathrm{d}}^n{\mathit{r}}_0\hfill \\ \hfill \Rightarrow & e^{\epsilon {\mathit{\partial }}_{t_0}}\Psi (\mathit{r},t_0)=e^{-\epsilon {\displaystyle \frac{\mathrm{i}}{\hslash }}(\beta m_0{c}^2+\mathit{\alpha }·\widehat{\mathit{p}}c)}{e}^{-{\displaystyle \frac{\mathrm{i}}{\hslash }}qV\epsilon }\Psi (\mathit{r},t_0)\hfill \\ \hfill \Rightarrow & e^{-{\displaystyle \frac{\epsilon \mathrm{i}}{\hslash }}(\mathrm{i}\hslash {\mathit{\partial }}_{t_0}-V)}\Psi (\mathit{r},t_0)=e^{-{\displaystyle \frac{\epsilon \mathrm{i}}{\hslash }}(\beta m_0{c}^2+\mathit{\alpha }·\widehat{\mathit{p}}c)}\Psi (\mathit{r},t_0).\hfill \end{array}$$

This relationship holds for any wave function $\Psi (\mathit{r},t_0)$, and thus

$$(\mathrm{i}\hslash {\mathit{\partial }}_{t_0}-V)\Psi (\mathit{r},t_0)=(\beta m_0{c}^2+\mathit{\alpha }·\widehat{\mathit{p}}c)\Psi (\mathit{r},t_0).$$

(22)

It should be emphasized that the derivation of the Dirac equation involving the scalar potential is based on the premise that $\epsilon \to 0$. Without this premise, $\widehat{\mathit{p}}$ would not have a commutation relation with $e^{{\displaystyle \frac{\mathrm{i}}{\hslash }}{\int }_{{\wp }_k}V\mathrm{d}{t}_1}$, causing the operator ${\widehat{p}}_i$ in the Hamiltonian to become ${\widehat{p}}_i+V/v_i$, which is inconsistent with the Dirac equation. It is precisely because of the presence of the scalar potential V that the dynamical evolution equation (Dirac equation) replaces $\Psi (\mathit{r},t)=e^{{\displaystyle \frac{\mathrm{i}}{\hslash }}{\int }_{t_0}^t{\widehat{H}}_R\mathrm{d}{t}_1}\Psi (\mathit{r},t_0)$ as the most fundamental principle in quantum mechanics.

4. Relations to Existing Path-Integral Formulations

4.1. Spinor Form of Path Integrals vs. Scalar Form of Path Integrals

In previous work, we established the path integral in scalar form, constructing the relationship between the relativistic Lagrangian in scalar form, ${\mathcal{L}}_r={\gamma }_L{m}_0{c}^2+q\mathit{A}-qV$, and the scalar form of the Hamiltonian, ${\widehat{H}}_r=\sqrt{m_0^2{c}^4+{(\widehat{\mathit{p}}-q\mathit{A})}^2{c}^2}+qV$. In fact, the path integral in spinor form is a further extension of this work. The diagonal form of the path integral in spinor form is essentially the path integral theory in scalar form. From Equation (3), it is evident that the matrix M functions as the bridge between the diagonal and non-diagonal path integral forms. Our analysis indicates that the matrix M is, in fact, equivalent to $L_0$. When we apply $L_0$ to the path integral form, we obtain

$$\begin{array}{cc}\hfill & L_0{\left(\widehat{v}\right)}^{-1}{\widehat{K}}_R(\mathit{r},t_0+\epsilon ;{\mathit{r}}_0,t_0)L_0\left(v\right)\hfill \\ \hfill =& {\lambda }_n{\left(\widehat{v}\right)}^{-1}{\lambda }_n\left(v\right)\left(\begin{array}{cc}{e}^{\mathrm{i}{L}_r\epsilon /\hslash },& 0\\ 0,& {\left(\mathrm{i}\right)}^{{\displaystyle \frac{n}{2}}}{e}^{-\mathrm{i}{L}_r\epsilon /\hslash }\end{array}\right)\otimes I_{m\times m}\hfill \\ \hfill =& {\widehat{K}}_r(\mathit{r},t_0+\epsilon ;{\mathit{r}}_0,t_0),\hfill \end{array}$$

(23)

where ${\lambda }_n\left(v\right)=L_n\left(v\right)L_0^{-1}\left(v\right)={\gamma }_L^{{\displaystyle \frac{1}{2}}}{\left({\displaystyle \frac{2{\gamma }_L}{1+{\gamma }_L}}\right)}^{{\displaystyle \frac{n-1}{2}}}$. This indicates that the path integral in scalar form is the result of decoupling the positive and negative energy state spaces in the spinor form of path integrals. The scalar form of the path integral, ${\widehat{K}}_r$, plays a crucial role in establishing the quantum mechanical evolution equation in scalar form. It connects the relativistic scalar Lagrangian with the scalar quantum mechanical evolution equation. In particular, the following relationship holds

$$\begin{array}{cc}\hfill & {\int }_{-\infty }^{\infty }{\widehat{K}}_r(\mathit{r},t;{\mathit{r}}_0,t_0)\left(\begin{array}{c}{\psi }_1({\mathit{r}}_0,t_0)\\ {\psi }_2({\mathit{r}}_0,t_0)\\ ⋮\\ {\psi }_{2m}({\mathit{r}}_0,t_0)\end{array}\right){\mathrm{d}}^n{\mathit{r}}_0\hfill \\ \hfill ⟺& \left\{\begin{array}{cc}\mathrm{i}\hslash {\mathit{\partial }}_t{\psi }_j=(\sqrt{m_0^2{c}^4+{\widehat{\mathit{p}}}^2{c}^2}+qU){\psi }_j;& 0<j\le m\\ \mathrm{i}\hslash {\mathit{\partial }}_t{\psi }_j=(-\sqrt{m_0^2{c}^4+{\widehat{\mathit{p}}}^2{c}^2}+qU){\psi }_j;& m<j\le 2m\end{array}\right.\hfill \end{array}$$

This implies that in the $2m$-dimensional spinor space, there are m components corresponding to positive-energy solutions and m components corresponding to negative-energy solutions. If we redefine the spinor wave functions as ${\psi }_j^{+}=({\psi }_j+{\psi }_{j+m})/\sqrt{2}$ and ${\psi }_j^{-}=({\psi }_j-{\psi }_{j+m})/\sqrt{2}$, it can be shown that the newly formed set of $2m$ wave functions, $\left\{ps{i}_j^{-}\right\}$, satisfies the Klein–Gordon equation:

$${\left(\mathrm{i}\hslash {\mathit{\partial }}_t-qV\right)}^2{\psi }_j^{\pm }=(m_0^2{c}^4+{(\mathrm{i}\hslash \nabla -q\mathit{A})}^2{c}^2){\psi }_j^{\pm }.$$

(24)

We observe that the wave functions satisfying the Klein–Gordon equation are superpositions of the positive-energy particle wave functions (the first m components of $\Psi$) and the negative-energy particle wave functions (the next m components of $\Psi$). Thus, from the perspective of path integral theory, one explanation for why the solutions of the Klein–Gordon equation do not obey the conservation of current density is that the wave functions contain components from both positive- and negative-energy particles. Therefore, it is not a single-particle dynamical equation.

If we revisit the theory of path integrals in spinor form, we find that $S_R$ has lost its role in describing the motion of objects; we cannot derive the general laws of motion from the principle of least action. However, in the scalar form of path integral theory, $S_r$ still retains its physical significance as an action, capable of describing the classical motion of spin particles. The transformation from $S_r$ to $S_R$ requires us to introduce $-{\mathcal{L}}_r$. In classical mechanics, we retained ${\mathcal{L}}_r$ and discarded $-{\mathcal{L}}_r$, but in the theory of path integrals, we need to include $-{\mathcal{L}}_r$ into the new theory.

From ${\mathcal{L}}_R=L_0^{-1}\left(\beta {\mathcal{L}}_r\right)L_0$, we know that because $\delta {\mathcal{L}}_R$ involves the variation of $L_0$, $\delta \int {\mathcal{L}}_R\mathrm{d}t=0$ no longer describes the true trajectory of an object. This conclusion underscores that the scalar form of path integrals aligns more closely with Feynman’s conceptualization of path integrals, embodying a more direct approach to quantum dynamics. Conversely, the spinor form, with $S_R$ detached from motion description, evolves into a theory of spacetime correlation. This theory, while not directly elucidating motion, offers a richer tapestry of insights, suggesting that the intertwining of positive and negative energy states underpins the fundamental laws of matter.

4.2. Spinor Form of Path Integrals vs. Feynman Path Integrals

From our analysis, we know that if the positive and negative energy state spaces are decoupled, then the spinor form of path integrals will transition to the scalar form. On this basis, if the quantum system’s momentum $\langle {(\widehat{p}-q\mathit{A})}^2\rangle \ll m_0^2{c}^2$, implying $\langle {\left(\beta \widehat{v}\right)}^2\rangle \ll c^2$, this leads to ${\lambda }_n\left(\widehat{v}\right)\approx 1$ and ${\widehat{H}}_r\approx m_0{c}^2+{\widehat{\mathit{p}}}^2/\left(2m_0\right)+qU$, thus causing the operator ${\widehat{K}}_r$ to degenerate into a non-operator path-independent function and leading to

$$\begin{array}{cc}\hfill & {\widehat{K}}_r(\mathit{r},t;{\mathit{r}}_0,t_0)\underset{\mathrm{Low}\mathrm{energy}}{\overset{\langle {\widehat{\mathit{p}}}^2\rangle \ll m_0^2{c}^2}{⟹}}{K}_c(\mathit{r},t;{\mathit{r}}_0,t_0)\hfill \\ \hfill ⟹& \left(\begin{array}{cc}{C}_0^{{\displaystyle \frac{n}{2}}}\sum _{{\wp }_k}{e}^{{\displaystyle \frac{\mathrm{i}{S}_c}{\hslash }}},& 0\\ 0,& C_0^{{\displaystyle \frac{n}{2}}}\sum _{{\wp }_k}{e}^{-{\displaystyle \frac{\mathrm{i}{S}_c}{\hslash }}}\end{array}\right)\hfill \\ \hfill ⟹& \left\{\begin{array}{cc}\mathrm{i}\hslash {\mathit{\partial }}_t{\psi }_j=(m_0{c}^2+{\displaystyle \frac{{(\widehat{\mathit{p}}-q\mathit{A})}^2}{2m_0}}+qV){\psi }_j;& 0<j\le m\\ \mathrm{i}\hslash {\mathit{\partial }}_t{\psi }_j=-(m_0{c}^2+{\displaystyle \frac{{(\widehat{\mathit{p}}-q\mathit{A})}^2}{2m_0}}-qV){\psi }_j.& m<j\le 2m\end{array}\right.\hfill \end{array}$$

This is the classical Feynman path integral form that includes solutions with negative energy. Here $S_c={\int }_{t_0}^t({\mathit{p}}^2/2m_0+q\mathit{A}-V)\mathrm{d}{t}_1$. This tells us that the Feynman path integral is a low-energy approximation of the spinor form of path integrals when the positive and negative energy state spaces are decoupled.

4.3. Spinor Form of Path Integrals vs. Path Integrals in QED

The path integral in spinor form and the Feynman path integral under QED belong to different levels of path integral theory. The former is a spacetime theory about the evolution of single particles, while the latter is based on the theory of particle creation and annihilation under field quantization. They are connected through the Dirac equation.

Before these two path integrals can be linked, the issue of covariance in the spinor form of path integrals needs to be addressed. It is noted that the Dirac equation derived from Equation (18) is not covariant. In fact, this equation is also the initial equation that Dirac constructed based on the principle of correspondence from the relativistic Hamiltonian. Multiplying both sides of this equation by $\beta$ yields the standard form of the Dirac equation:

$$\begin{array}{cc}\hfill & (\beta (\mathrm{i}\hslash {\mathit{\partial }}_t-qV)-\beta \mathit{\alpha }·(\widehat{\mathit{p}}-q\mathit{A})c-m_0{c}^2)\Psi =0\hfill \\ \hfill \iff & ({\gamma }^{\mu }(\mathrm{i}\hslash {\mathit{\partial }}_u-q{A}_{\mu })-m_{0}c)\Psi =0.\hfill \end{array}$$

(25)

where ${\gamma }^{\mu }$ are Dirac Matrixes [30,34], ${\mathit{\partial }}_0=\mathit{\partial }/(c\mathit{\partial }t)$ and $A_0=V/c$. Mathematically, Equations (25) and (19) are essentially two equivalent equations, but Equation (25) cannot be directly obtained from path integral theory, primarily because, in the single-particle path integral theory, time and space do not have equivalent status. In the single-particle path integral theory, the action involves an integration over time but not over space, while the propagator calculation involves spatial integration but not temporal integration. Therefore, the single-particle path integral theory is necessarily non-covariant. Then, does a covariant form of path integral theory exist? It is believed not to exist. In classical mechanics, a “path” refers to the “trajectory” of mechanical motion, involving “changes in space over time”, which implies time is a variable, and space is a physical quantity, thus their statuses are different. Therefore, as long as the classical concept of “path” is retained, a covariant form of path integral does not exist. In fact, the “path” in Feynman path integrals under QED has deviated from the classical concept of “trajectory”; it refers to “types of scattering”. Precisely for this reason, the Feynman path integrals in QED do not possess a specific expression form but are merely a method for calculating the Dyson series.

5. Conclusions

This work presents a new perspective on path integral theory by addressing the inconsistencies and incompleteness inherent in its current formulation. Path integral theory, as a cornerstone of quantum mechanics, holds profound implications for understanding the foundational principles of quantum phenomena. By analyzing its limitations, we uncovered a more fundamental framework that connects the relativistic action with the Dirac equation. This hidden theory, discovered through rigorous mathematical analysis, is shown to be unique and represents a natural extension of Feynman’s path integrals without requiring path constraints or action modifications.

Central to this development is the spinor form of path integrals, which not only reveals the origins of classical mechanics for the Dirac equation but also sheds light on the origins of quantum spin. The origin of spin has long posed a big mystery in the realm of physics. From a mathematical perspective, the concept of spin space as the $2n$-dimensional inequivalent irreducible representation of the $SU\left(2\right)$ group is well-established [35]. This mathematical formalism provides a solid foundation for describing the properties of spin, such as its quantization and transformation under rotations. However, the fundamental reasons for the emergence of spin and the specific details of its physical genesis remain subjects of considerable intrigue research. Traditionally, the introduction of the $\mathit{\alpha }$ matrices in the Dirac equation is regarded as a crucial step towards understanding spin. The Dirac equation, which combines quantum mechanics and special relativity, successfully describes the behavior of spin-$1/2$ particles, such as electrons. Yet, when examined within a broader algebraic context, the $\mathit{\alpha }$ matrices merely serve as the fundamental vectors in the $C{l}_{3,0}$ space, aligning with the inherent algebraic structure of Maxwell’s equations [36,37]. This observation suggests that the $\mathit{\alpha }$ matrices alone do not fully capture the physical origin of spin.

From the perspective of spinor form path integrals, the key to making spin manifest in quantum mechanics is the introduction of the $\beta$ matrix, which shifts the physical laws from those belonging to the $C{l}_{3,0}$ algebraic space of Maxwell’s equations to the $C{l}_{3,1}$ algebraic space of Dirac’s theory [37]. The $\beta$ matrix, along with the $\mathit{\alpha }$ matrices, forms the complete set of Dirac matrices, which are essential for describing the behavior of spin-1/2 particles. Notably, in our spinor path integral theory, the $\beta$ matrix always appears in conjunction with the mass $m_0$. This observation suggests that mass plays a crucial role in the emergence of spin. The introduction of the concept of negative mass and allowing negative mass to form a spinor space with positive mass lays the foundation for constructing the Dirac equation. This mathematical framework enables the description of particles with both positive and negative energy states, which is a key feature of relativistic quantum mechanics. From this, we can speculate that spin might arise from the coupling of electromagnetic laws with particles of positive and negative mass.

Moreover, this work extends the implications of path integrals to quantum non-local correlations. As spacetime correlation functions, path integrals serve as a fundamental tool for analyzing the correlations across spacetime points during wavefunction evolution. By generalizing the scalar path integral theory to its spinor form, we have provided a framework to explore the mysteries of spinor particle wavefunction collapse, entanglement, and the manipulation of quantum non-local correlations. This marks a significant step forward, building upon earlier works that explained the stochastic and irreversible nature of scalar particle wavefunction collapse through path integral theory. The spinor extension holds promise for uncovering new aspects of quantum measurement and entanglement phenomena.

In summary, the spinor form of path integrals provides a unified framework that addresses critical questions in quantum mechanics, including the origin of spin and the nature of quantum non-local correlations. By offering a coherent and natural extension of Feynman’s path integrals, this work establishes connections to scalar path integrals, quantum electrodynamics and beyond. These advancements open new avenues for research in quantum information science, condensed matter physics and other areas where the principles of quantum mechanics play a pivotal role.