Probabilistic Basis of Parametrized Relativistic Quantum Theory in Curved Spacetime

Abstract

A probabilistic basis for Parametrized Relativistic Quantum Theory (PRQT) has been developed for multicomponent eigenvectors and many-body applications in flat spacetime. This paper presents an extension of the multicomponent, many-body PRQT formalism to curved spacetime.

1. Introduction

The quark model of hadrons has been successful as a means of classifying known particles and understanding scattering experiments, but it is difficult to apply to bound state problems. An alternative approach for solving bound state problems is Relativistic Dynamics. The theory was pioneered by Stueckelberg [1,2,3] and is a manifestly covariant quantum theory with an invariant evolution parameter. Introductions to Relativistic Dynamics are presented by Horwitz [4,5], Fanchi [6,7], and Pavšič [8,9]. The SHP (Stueckelberg–Horwitz–Piron) formulation described by Horwitz [4,5] is an axiomatic formulation based on Stueckelberg’s work and the 1973 paper by Horwitz and Piron [10]. A review of relativistic classical mechanics and electrodynamics in the SHP framework is given by Land and Horwitz [11]. Pavšič [8,9] expressed relativistic dynamics in terms of Clifford algebra. Fanchi [6,7] presented a formulation called Parametrized Relativistic Quantum Theory (PRQT) that is distinguished by its derivation from probabilistic concepts.

The purpose of this paper is to extend the PRQT formalism to multicomponent, many-body systems in curved spacetime. The resulting theoretical framework can be used to study the relationship between quantum theory and general relativity. The PRQT formalism in curved spacetime is a new approach to quantum gravity, which is the unification of quantum theory and general relativity.

An extensive literature exists that presents attempts to develop a theory of quantum gravity. DeWitt [12] and others [13,14,15] presented a historical introduction to the quantum theory of gravity. They considered Rosenfeld [16,17] a pioneer in applying quantum mechanics to the understanding of gravity. Rosenfeld introduced a wave function on a 5-dimensional metric in a constrained Hamiltonian framework. He was able to incorporate a “local gauge, such as those of linearized gravity and electromagnetism, within a Hamiltonian” [13] (Introduction). Zee [18] and Kaku [19] provide a more modern view of gravity as a field theory and discuss gravity in the context of a 5-dimensional Kaluza–Klein action. Pavšič [20] introduced a 6-dimensional theory that led to a parametrized Stueckelberg-like extension of general relativity.

Another approach to quantum gravity is to treat gravity as a field within the framework of quantum field theory (QFT). Scadron [21] discussed lowest-order gravitational interactions in the context of Feynman diagrams. Wald [22] considered the connection between QFT in curved spacetime and black hole thermodynamics. A field theory approach that treats general relativity as an effective field theory is discussed by Cao and Schweber [23], Rocci and Riet [24], and Zee [25]. This approach uses a renormalization group analogous to condensed matter physics. Effective field theories are considered low-energy approximations.

String theory treats the excitation of a 1-dimensional, spin-2 string as a quantum of gravity, the graviton. The string theory approach is discussed by Rocci and Riet [24], Kaku [26], and Zwiebach [27].

Loop quantum gravity (LQG) is a non-perturbative approach to quantizing gravity. It uses canonical quantization of spacetime rather than treating gravity as a field. Canonical quantization is a procedure for transitioning from a classical mechanical system to a quantum mechanical system. Classical variables such as position and momentum are replaced with quantum operators that satisfy commutation relations. The uncertainty principle is represented by non-zero commutation relations. A relatively recent review of LQG was presented by Rovelli [28].

The approaches discussed above are candidates for a theory of quantum gravity. Other approaches have been suggested, such as asymptotically safe quantum gravity [29], which hypotheses that the description of gravity is achieved using a renormalizable QFT with a non-perturbative ultraviolet fixed point. Valentini [30] used the de Broglie–Bohm pilot-wave formulation of quantum physics to argue that the Born rule does not apply at the fundamental level of quantum gravity.

Each of the approaches has strengths and weaknesses. The PRQT multicomponent, many-body formalism with curved spacetime is a new approach for unifying quantum theory and general relativity. Section 2 discusses relevant probabilistic concepts. The derivation of a multicomponent, many-body formalism in curved spacetime is presented in Section 3. A few applications of the approach are discussed in Section 4, and concluding remarks are given in Section 5. This work assumes that the metric tensor does not depend on the invariant evolution parameter.

2. Probabilistic Concepts

The probabilistic foundation for a parametrized quantum theory of a single relativistic spinless particle [31,32] has been extended to multicomponent eigenvectors [33] and many-body interactions in flat spacetime [34]. The single spinless particle formalism has recently been extended to curved spacetime [35]. This section introduces probabilistic concepts with an invariant evolution parameter that is needed to extend the formalism.

2.1. Many-Body System

A Born-like, positive-definite probability density $\rho$ for a single particle is a joint distribution in space and time coordinates $x$, and may be conditioned by an invariant evolution parameter $s$. If we assume that $\rho$ is conditioned by a single invariant scalar s, the normalization condition in flat space is

$${\int }_D\rho \left(\left.x\right|s\right)d^{4}x=1$$

(1)

where $D$ is the hypervolume of interest.

The derivation of field equations for multicomponent, many-particle systems in curved spacetime that are consistent with probability theory begins with the assumption that a conditional probability density $\mathsf{\rho }\left(\left.\mathrm{y}\right|\mathrm{s}\right)$ exists. The symbol $\mathrm{y}$ denotes a set of $4\mathrm{N}$ coordinates with $\mathrm{N}$ being the number of particles in the hypervolume ${\mathrm{D}}^{\mathrm{N}}$ for which $\mathsf{\rho }\left(\left.\mathrm{y}\right|\mathrm{s}\right)$ has nonzero values. The ${\mathsf{\mu }}^{\mathrm{th}}$ component of the position four-vector of particle $\mathrm{a}$ is written as ${\mathrm{y}}_{\mathrm{a}}^{\mathsf{\mu }}$ where $\mathsf{\mu }=\mathrm{0,1},\mathrm{2,3}$. Indices 1, 2, 3 signify space components, and 0 signifies the geometric time component.

According to probability theory, $\rho \left(y\left|s\right.\right)$ must be positive-definite and normalizable. The Born representation of the positive-definite requirement is

$$\rho (\left.y\right|s)={\Psi }^{*}(y,s)\Psi (y,s)\ge 0$$

(2)

where $\Psi$ is the probability amplitude and ${\Psi }^{*}$ is the complex conjugate of $\Psi$. The probability amplitude is specified to within a gauge transformation

$$\Psi (y,s)=\sqrt{\rho (y\left|s\right.)} e^{i\xi (y,s)}$$

(3)

where $\xi (y,s)$ is a real scalar function.

The normalization condition is

$${\int }_{D^N}\rho \left(\left.y\right|s\right)dy=1$$

(4)

where ${\mathrm{D}}^{\mathrm{N}}$ is the hypervolume.

$$dy={\displaystyle \prod _{a=1}^N}\sqrt{\left|g\left(y_a\right)\right|} d^4{y}_a, d^4{y}_a\equiv d{y}_a^{0}d{y}_a^{1}d{y}_a^{2}d{y}_a^3.$$

(5)

The term $\left|g\left(y_a\right)\right|$ is the absolute value of the determinant of the metric tensor $g_{\mu \nu }$ for particle $\mathrm{a}$. Substituting Equation (2) into Equation (4) gives

$${\int }_{D^N}{\Psi }^{*}(y,s)\Psi (y,s)dy=1$$

(6)

The physical interpretation of probability density can be clarified by assuming that each particle is independent, that is, particles do not interact or couple. In this simplified case, the probability density can be written as the product

$$\rho \left(\left.y\right|s\right)={\displaystyle \prod _{a=1}^N}{\rho }_a\left(\left.y_a\right|s\right)$$

(7)

The normalization condition becomes

$${\int }_D\left[{\displaystyle \prod _{a=1}^N}{\rho }_a\left(\left.y_a\right|s\right)\right]\left[{\displaystyle \prod _{a=1}^N}\sqrt{\left|g\left(y_a\right)\right|} d^4{y}_a\right]=1$$

(8)

Applying the Born representation and factoring terms gives

$${\displaystyle \prod _{a=1}^N}{\int }_D{\Psi }_a^{*}(y_a,s){\Psi }_a(y_a,s)\sqrt{\left|g\left(y_a\right)\right|} d^4{y}_a=1$$

(9)

The normalization condition for particle $a$ is

$${\int }_D{\Psi }_a^{*}(y_a,s){\Psi }_a(y_a,s)\sqrt{\left|g\left(y_a\right)\right|} d^4{y}_a=1$$

(10)

The probability ${\Psi }_a^{*}(y_a,s){\Psi }_a(y_a,s)\sqrt{\left|g\left(y_a\right)\right|} d^4{y}_a$ is the probability that particle $a$ will be found in volume $\sqrt{\left|g\left(y_a\right)\right|} d^4{y}_a$ in the neighborhood of spacetime point $y_a$ at evolution parameter $s$. The normalization condition implies that the particle can be observed somewhere in space at some point in time $y_a^0$; it does not require the particle to exist for all time $y_a^0$.

2.2. Internal Variables

The many-body system in Section 2.1 can be extended to include $\mathrm{L}$ internal variables with discrete parameter values $\left\{{\mathrm{l}}_{\mathrm{i}}\right\}$. Examples of internal variables include spin and isospin. The term $\rho \left(\left.y\right|s\right)\hspace{0.33em}dy$ can be written as

$$\rho \left(\left.y\right|s\right)\hspace{0.33em}dy=\sum _{l_1}\sum _{l_2}\cdots \sum _{l_L}\rho \left(y,l_1,l_2,\dots ,l_L\left|s\right.\right)dy$$

(11)

where the $i^{\mathrm{th}}$ sum extends over the entire range of allowed values of the $i^{\mathrm{th}}$ parameter $l_i$. The term $\rho (y,l_1,\dots ,l_L\left|s\right.)dy$ is the probability of observing a system having discrete parameter values $l_1,\dots ,l_L$ in the configuration $y$ within an infinitesimal $4N$-volume $dy$ at historical time $s$.

The number of terms $\mathsf{\Lambda }$ in the $\mathrm{L}$ sums of Equation (11) is the product of the number ${\mathrm{L}}_{\mathrm{i}}$ of allowed values of the ${\mathrm{i}}^{\mathrm{th}}$ discrete parameter ${\mathrm{l}}_{\mathrm{i}}$, that is,

$$\Lambda ={\displaystyle \prod _{i=1}^L}{L}_i$$

(12)

The $L$ sums can be replaced by a single sum over the range $1\le \lambda \le \Lambda$. This requires assigning a one-to-one correspondence between $\lambda$ and each term of the $L$ sums. Doing so gives

$$\rho \left(\left.y\right|s\right)dy={\displaystyle \sum _{\lambda =1}^{\Lambda }}{\rho }_{\lambda }\left(\left.y\right|s\right)dy, {\rho }_{\lambda }\left(\left.y\right|s\right)\equiv \rho \left(y,l_1,\dots ,l_L\left|s\right.\right)$$

(13)

where the subscript $\lambda$ of ${\rho }_{\lambda }(\left.y\right|s)$ represents a set of values of the discrete parameters $\left\{l_i\right\}$.

The Born representation of ${\rho }_{\lambda }\left(\left.y\right|s\right)$ is

$${\rho }_{\lambda }\left(\left.y\right|s\right)={\psi }_{\lambda }^{*}\left(y,s\right){\psi }_{\lambda }\left(y,s\right)$$

(14)

Equation (14) implies the scalar eigenfunctions ${\psi }_{\lambda }$ are specified within a gauge transformation of the first kind. Consequently, ${\psi }_{\lambda }$ can be written as

$${\psi }_{\lambda }\left(y,s\right)={\left[{\rho }_{\lambda }\left(\left.y\right|s\right)\right]}^{1/2}\mathit{exp}\left[i\xi \left(y,s\right)\right]$$

(15)

where it is assumed that the real scalar function $\xi \left(y,s\right)$ does not depend on discrete parameters. This provision is realistic if the parameters $\left\{l_i\right\}$ characterize mutually independent events. An eigenfunction with the product form $\psi \left(\text{space-time}\right)\otimes \psi \left(\mathrm{spin}\right)\otimes \psi \left(\mathrm{isospin}\right)\dots$ fulfills the independence requirement.

2.3. Parametrized Continuity Equation in Curved Spacetime

The parametrized continuity equation in flat spacetime has been used to construct the Stueckelberg equation in flat spacetime [6,7]. Flat spacetime refers to a Minkowski metric with signature −2. The equation is extended to curved spacetime by replacing the divergence in flat spacetime with the divergence in curved spacetime:

$$\frac{𝜕\rho }{𝜕s}+{\displaystyle \sum _{a=1}^N}{\left(\rho V_a^{\mu }\right);}_{\mu }=0$$

(16)

Einstein’s summation convention for repeated Greek indices is assumed to apply unless otherwise stated. The term $V_a^{\mu }$ is the ${\mu }^{th}$ component of the contravariant four-velocity of particle $a$, and $\rho V_a^{\mu }$ is probability flux. The divergence of the contravariant four-vector ${\left(\rho V_a^{\mu }\right);}_{\mu }$ is

$${\left(\rho V_a^{\mu }\right);}_{\mu }=\frac{𝜕}{𝜕y_a^{\mu }}\left(\rho V_a^{\mu }\right)+{\mathsf{\Gamma }}_{\nu \mu }^{\mu }\left(\rho V_a^{\nu }\right)$$

(17)

Equation (17) is the contracted form of the covariant derivative of the contravariant four-vector

$${\left(\rho V_a^{\sigma }\right);}_{\mu }=\frac{𝜕}{𝜕y_a^{\mu }}\left(\rho V_a^{\sigma }\right)+{\mathsf{\Gamma }}_{\nu \mu }^{\sigma }\left(\rho V_a^{\sigma }\right)$$

(18)

where contraction occurs by setting $\sigma$ equal to $\mu$.

The tensor ${\mathsf{\Gamma }}_{\nu \mu }^{\mu }$ is the contracted form of the Christoffel symbol of the second kind

$${\mathsf{\Gamma }}_{\nu \mu }^{\sigma }=\frac{1}{2}{g}^{\sigma \kappa }\left(\frac{𝜕g_{\kappa \nu }}{𝜕y_a^{\mu }}+\frac{𝜕g_{\mu \kappa }}{𝜕y_a^{\nu }}-\frac{𝜕g_{\nu \mu }}{𝜕y_a^{\kappa }}\right)$$

(19)

for metric tensor $g_{\nu \mu }$. A similar continuity equation was discussed by Pavšič [8] (Chapter 1). The Christoffel symbol of the second kind ${\mathsf{\Gamma }}_{\nu \mu }^{\sigma }$ is also known as the affine connection. The contracted form of ${\mathsf{\Gamma }}_{\nu \mu }^{\sigma }$ is

$${\mathsf{\Gamma }}_{\nu \mu }^{\mu }=\frac{1}{2}{g}^{\mu \kappa }\frac{𝜕g_{\mu \kappa }}{𝜕y_a^{\nu }}$$

(20)

3. The Multicomponent, Many-Body Formalism in Curved Spacetime

The derivation of a multicomponent, many-body formalism in curved spacetime begins by writing the four-velocity $V_a^{\mu }$ in the form

$$V_a^{\mu }\left(y_a,s\right)={\epsilon }_a\left[{\epsilon }^{\prime }\frac{𝜕\xi \left(y_a,s\right)}{𝜕y_{a\mu }}+{\epsilon }^{″}{A}^{\mu }(y_a,s)\right]$$

(21)

where ${\epsilon }_a, {\epsilon }^{\prime }, {\epsilon }^{″}$ are real, scalar constants, $\xi$ is a real, scalar function, and $A^{\mu }$ is a contravariant four-vector. The physical interpretation of the terms ${\epsilon }_a, {\epsilon }^{\prime }, {\epsilon }^{″}, \xi , A^{\mu }$ is provided later.

The term $\frac{𝜕\xi \left(y_a,s\right)}{𝜕y_{a\mu }}$ can be obtained by combining the Born representation of probability density in Equation (14) with Equation (15) to write

$$\frac{𝜕\xi \left(y_a,s\right)}{𝜕y_{a\mu }}=\frac{-i}{2{\rho }_{\lambda }}\left[{\psi }_{\lambda }^{*}\left(y,s\right)\frac{𝜕{\psi }_{\lambda }\left(y,s\right)}{𝜕y_{a\mu }}-{\psi }_{\lambda }\left(y,s\right)\frac{𝜕{\psi }_{\lambda }^{*}\left(y,s\right)}{𝜕y_{a\mu }}\right]$$

(22)

where there is no sum over the repeated index $\lambda$. The derivation of Equation (22) is presented in Appendix A.

Combining Equations (13), (14), (21) and (22) gives

$$\rho V_a^{\mu }={\displaystyle \sum _{\lambda =1}^{\Lambda }}{\epsilon }_a\left\{\frac{-i{\epsilon }^{\prime }}{2}\left[{\psi }_{\lambda }^{*}\frac{𝜕{\psi }_{\lambda }}{𝜕y_{a\mu }}-{\psi }_{\lambda }\frac{𝜕{\psi }_{\lambda }^{*}}{𝜕y_{a\mu }}\right]+{\epsilon }^{″}{A}_a^{\mu }{\psi }_{\lambda }^{*}{\psi }_{\lambda }\right\}$$

(23)

The derivative $\frac{𝜕\rho V_a^{\mu }}{𝜕y_a^{\mu }}$ is

$$\begin{gathered} \frac{𝜕\rho V_a^{\mu }}{𝜕y_a^{\mu }}=\frac{𝜕}{𝜕y_a^{\mu }}{\displaystyle \sum _{\lambda =1}^{\Lambda }}{\epsilon }_a\left\{\frac{-i{\epsilon }^{\prime }}{2}\left[{\psi }_{\lambda }^{*}\frac{𝜕{\psi }_{\lambda }}{𝜕y_{a\mu }}-{\psi }_{\lambda }\frac{𝜕{\psi }_{\lambda }^{*}}{𝜕y_{a\mu }}\right]+{\epsilon }^{″}{A}_a^{\mu }{\psi }_{\lambda }^{*}{\psi }_{\lambda }\right\} \\ ={\displaystyle \sum _{\lambda =1}^{\Lambda }}\left\{\frac{-i{\epsilon }_a{\epsilon }^{\prime }}{2}\frac{𝜕}{𝜕y_a^{\mu }}\left[{\psi }_{\lambda }^{*}\frac{𝜕{\psi }_{\lambda }}{𝜕y_{a\mu }}-{\psi }_{\lambda }\frac{𝜕{\psi }_{\lambda }^{*}}{𝜕y_{a\mu }}\right]\right\} \\ +{\displaystyle \sum _{\lambda =1}^{\Lambda }}\left\{{\epsilon }_a{\epsilon }^{″}\frac{𝜕}{𝜕y_a^{\mu }}\left[A_a^{\mu }{\psi }_{\lambda }^{*}{\psi }_{\lambda }\right]\right\} \end{gathered}$$

(24)

The covariant derivative ${\displaystyle {\sum }_{a=1}^N}{\left(\rho V_a^{\mu }\right);}_{\mu }$ in Equation (16) is

$$\begin{gathered} {\displaystyle \sum _{a=1}^N}{\left(\rho V_a^{\mu }\right);}_{\mu }={\displaystyle \sum _{a=1}^N}\left\{\frac{𝜕}{𝜕y_a^{\mu }}\left(\rho V_a^{\sigma }\right)+{\mathsf{\Gamma }}_{\nu \mu }^{\mu }\left(\rho V_a^{\nu }\right)\right\} \\ ={\displaystyle \sum _{a=1}^N}\left\{{\displaystyle \sum _{\lambda =1}^{\Lambda }}\left[\frac{-i{\epsilon }_a{\epsilon }^{\prime }}{2}\frac{𝜕}{𝜕y_a^{\mu }}\left({\psi }_{\lambda }^{*}\frac{𝜕{\psi }_{\lambda }}{𝜕y_{a\mu }}-{\psi }_{\lambda }\frac{𝜕{\psi }_{\lambda }^{*}}{𝜕y_{a\mu }}\right)+{\epsilon }_a{\epsilon }^{″}\frac{𝜕}{𝜕y_a^{\mu }}\left(A_a^{\mu }{\psi }_{\lambda }^{*}{\psi }_{\lambda }\right)\right] \\ +{\mathsf{\Gamma }}_{\nu \mu }^{\mu }\left\{{\displaystyle \sum _{\lambda =1}^{\Lambda }}\left[\frac{-i{\epsilon }_a{\epsilon }^{\prime }}{2}\left({\psi }_{\lambda }^{*}\frac{𝜕{\psi }_{\lambda }}{𝜕y_{a\nu }}-{\psi }_{\lambda }\frac{𝜕{\psi }_{\lambda }^{*}}{𝜕y_{a\nu }}\right)+{\epsilon }_a{\epsilon }^{″}\left(A_a^{\nu }{\psi }_{\lambda }^{*}{\psi }_{\lambda }\right)\right]\right\}\right\} \end{gathered}$$

(25)

The derivative $\frac{𝜕\rho }{𝜕s}$ in Equation (16) is

$$\frac{𝜕\rho }{𝜕s}={\displaystyle \sum _{\lambda =1}^{\Lambda }}\frac{𝜕{\psi }_{\lambda }^{*}{\psi }_{\lambda }}{𝜕s}={\displaystyle \sum _{\lambda =1}^{\Lambda }}\left(\frac{𝜕{\psi }_{\lambda }^{*}}{𝜕s}{\psi }_{\lambda }+{\psi }_{\lambda }^{*}\frac{𝜕{\psi }_{\lambda }}{𝜕s}\right)$$

(26)

Substituting Equations (25) and (26) into Equation (16) gives

$$\begin{array}{cc}\hfill {\displaystyle \sum _{\lambda =1}^{\Lambda }}\left({\displaystyle \frac{𝜕{\psi }_{\lambda }^{*}}{𝜕s}}{\psi }_{\lambda }+{\psi }_{\lambda }^{*}{\displaystyle \frac{𝜕{\psi }_{\lambda }}{𝜕s}}\right)& \\ & +{\displaystyle \sum _{a=1}^N}\left\{{\displaystyle \sum _{\lambda =1}^{\Lambda }}\right[{\displaystyle \frac{-i{\epsilon }_a{\epsilon }^{\prime }}{2}}{\displaystyle \frac{𝜕}{𝜕y_a^{\mu }}}\left({\psi }_{\lambda }^{*}{\displaystyle \frac{𝜕{\psi }_{\lambda }}{𝜕y_{a\mu }}}-{\psi }_{\lambda }{\displaystyle \frac{𝜕{\psi }_{\lambda }^{*}}{𝜕y_{a\mu }}}\right)\hfill \\ & +{\epsilon }_a{\epsilon }^{″}{\displaystyle \frac{𝜕}{𝜕y_a^{\mu }}}\left(A_a^{\mu }{\psi }_{\lambda }^{*}{\psi }_{\lambda }\right)]\hfill \\ & +{\mathsf{\Gamma }}_{\nu \mu }^{\mu }\left\{{\displaystyle \sum _{\lambda =1}^{\Lambda }}\left[{\displaystyle \frac{-i{\epsilon }_a{\epsilon }^{\prime }}{2}}\left({\psi }_{\lambda }^{*}{\displaystyle \frac{𝜕{\psi }_{\lambda }}{𝜕y_{a\nu }}}-{\psi }_{\lambda }{\displaystyle \frac{𝜕{\psi }_{\lambda }^{*}}{𝜕y_{a\nu }}}\right)+{\epsilon }_a{\epsilon }^{″}\left(A_a^{\nu }{\psi }_{\lambda }^{*}{\psi }_{\lambda }\right)\right]\right\}\}\hfill \\ & =0\hfill \end{array}$$

(27)

Appendix B shows how Equation (27) can be written in the form

$${\displaystyle \sum _{\lambda =1}^{\Lambda }}{F}_{\lambda }^{*}{\psi }_{\lambda }={\displaystyle \sum _{\lambda =1}^{\Lambda }}{\psi }_{\lambda }^{*}{F}_{\lambda }$$

(28)

where

$$\begin{gathered} F_{\lambda }=i{\epsilon }^{\prime }\frac{𝜕{\psi }_{\lambda }}{𝜕s}+{\displaystyle \sum _{a=1}^N}[\frac{{\epsilon }_a{{\epsilon }^{\prime }}^2}{2}\frac{𝜕}{𝜕y_a^{\mu }}\frac{𝜕{\psi }_{\lambda }}{𝜕y_{a\mu }}+{i\epsilon }_a{\epsilon }^{\prime }{\epsilon }^{″}\left(\frac{1}{2}\frac{𝜕A_a^{\mu }}{𝜕y_a^{\mu }}{\psi }_{\lambda }+A_a^{\mu }\frac{𝜕{\psi }_{\lambda }}{𝜕y_a^{\mu }}\right) \\ +{\mathsf{\Gamma }}_{\nu \mu }^{\mu }\left[\left(+\frac{{\epsilon }_a{{\epsilon }^{\prime }}^2}{2}\frac{𝜕{\psi }_{\lambda }}{𝜕y_{a\nu }}+\frac{{i\epsilon }_a{\epsilon }^{\prime }{\epsilon }^{″}}{2}{A}_a^{\nu }{\psi }_{\lambda }\right)\right]] \end{gathered}$$

(29)

Equation (28) can be written in matrix notation as

$${\underset{\_}{\Psi }}^{+}\underset{\_}{F}={\underset{\_}{F}}^{+}\underset{\_}{\Psi }$$

(30)

where $+$ denotes conjugate transpose and we have assumed that $A_a^{\mu }$ is a real, scalar function. The ${\lambda }^{\mathrm{th}}$ element of the column vector $\underset{\_}{F}$ is $F_{\lambda }$, and $\underset{\_}{\Psi }$ is the column vector

$$\underset{\_}{\Psi }=\left[\begin{array}{c}{\psi }_1\\ ⋮\\ {\psi }_{\Lambda }\end{array}\right].$$

(31)

Equation (28) is decoupled by observing that the product ${\underset{\_}{\Psi }}^{+}\underset{\_}{F}$ is equal to its conjugate transpose ${\left({\underset{\_}{\Psi }}^{+}\underset{\_}{F}\right)}^{+}$ and must be real since

$${\underset{\_}{\Psi }}^{+}\underset{\_}{F}={\underset{\_}{F}}^{+}\underset{\_}{\Psi }={\left({\underset{\_}{\Psi }}^{+}\underset{\_}{F}\right)}^{+}.$$

(32)

This constraint is generally satisfied by introducing a $\Lambda \times \Lambda$ Hermitian, or self-adjoint, matrix $\underset{\_}{\underset{\_}{V^{\prime }}}$ and writing

$$\underset{\_}{F}=\underset{\_}{\underset{\_}{V^{\prime }}}\hspace{0.33em}\underset{\_}{\Psi }.$$

(33)

so that ${\underset{\_}{\Psi }}^{+}\underset{\_}{F}={\underset{\_}{\Psi }}^{+}\underset{\_}{\underset{\_}{V^{\prime }}}\hspace{0.33em}\underset{\_}{\Psi }={\left({\underset{\_}{\Psi }}^{+}\underset{\_}{\underset{\_}{V^{\prime }}}\hspace{0.33em}\underset{\_}{\Psi }\right)}^{+}$ is a real scalar. The general form of the subsequent field equation is

$$\begin{gathered} i{\epsilon }^{\prime }\hspace{0.33em}\frac{𝜕\underset{\_}{\Psi }}{𝜕s}+{\displaystyle \sum _{a=1}^N}{\epsilon }_a\left[\frac{{\epsilon }^{\prime 2}}{2}\hspace{0.33em}\frac{{𝜕}^2\underset{\_}{\Psi }}{𝜕y_a^{\mu }𝜕y_{a\mu }}\right. \\ \left.+i\hspace{0.33em}\frac{{\epsilon }^{\prime }{\epsilon }^{″}}{2}\left(A_a^{\mu }\frac{𝜕\underset{\_}{\Psi }}{𝜕y_a^{\mu }}+\frac{𝜕A_a^{\mu }\underset{\_}{\Psi }}{𝜕y_a^{\mu }}\right)\right]-\underset{\_}{\underset{\_}{V^{\prime }}}\hspace{0.33em}\underset{\_}{\Psi }=0. \end{gathered}$$

(34)

4. Discussion

The interpretation of field equations derived above is supported by a range of applications. A few applications are outlined here to illustrate the approach.

4.1. Physical Correspondence [6,7]

Equation (34) may be put in a more suggestive form by replacing the Hermitian matrix $\underset{\_}{\underset{\_}{V^{\prime }}}$ with

$$\underset{\_}{\underset{\_}{V^{\prime }}}={\displaystyle \sum _{a=1}^N}\frac{{\epsilon }_a{\epsilon }^{″2}}{2}{A}_a^{\mu }{A}_{a\mu }\underset{\_}{\underset{\_}{I}}+\underset{\_}{\underset{\_}{V}}$$

(35)

where $\underset{\_}{\underset{\_}{I}}$ is the identity matrix. The first term on the right-hand side of Equation (35) is real and therefore Hermitian. To retain the Hermiticity of $\underset{\_}{\underset{\_}{V^{\prime }}}$, we require $\underset{\_}{\underset{\_}{V}}$ to be Hermitian. The resulting field equation is

$$i{\epsilon }^{\prime }\frac{𝜕\underset{\_}{\Psi }}{𝜕s}={\displaystyle \sum _{a=1}^N}\hspace{0.33em}\frac{{\epsilon }_a}{2}{\pi }_a^{\mu }{\pi }_{a\mu }\underset{\_}{\Psi }+\underset{\_}{\underset{\_}{V}}\hspace{0.33em}\underset{\_}{\Psi }$$

(36)

The operators ${\pi }_a^{\mu }$ and $p_a^{\mu }$ are

$${\pi }_a^{\mu }\equiv p_a^{\mu }+{\epsilon }^{″}{A}_a^{\mu }$$

(37)

and

$$p_a^{\mu }\equiv \frac{{\epsilon }^{\prime }}{i}\hspace{0.33em}\frac{𝜕}{𝜕y_{a\mu }}.$$

(38)

Equations (36)–(38) are field equations for an $N$-body system of multicomponent, interacting particles exhibiting a “minimal coupling” electromagnetic effect if the real, scalar constants ${\epsilon }_a,{\epsilon }^{\prime },{\epsilon }^{″}$ are linked to physics by identifying

$${\epsilon }_a=\frac{1}{m_a}, {\epsilon }^{\prime }=\mathrm{\hslash }, {\epsilon }^{″}=-\frac{e}{c}$$

(39)

where $m_a$ for particle $a$ has mass units, $\mathrm{\hslash }$ is Planck’s constant divided by $2\pi$, $e$ is the elementary charge, and $c$ is the speed of light in vacuum. Equation (36) is called here the Stueckelberg equation because Stueckelberg was the first to develop a parametrized quantum theory that included a new normalization condition, a new interpretation of the probability function, and a parametrized wave equation.

4.2. Interpretation of Mass [6,7]

The interpretation of $m_a$ as a mass can be shown by considering the field equation for a single noninteracting spinless particle with wave function ${\psi }_f$:

$$i\hslash \frac{𝜕{\psi }_f}{𝜕s}=-\frac{{\hslash }^2}{2m_f}{𝜕}_{\mu }{𝜕}^{\mu }{\psi }_f$$

(40)

The general solution of Equation (40) is

$$\begin{gathered} {\Psi }_f\left(x,s\right)=\int {\psi }_{f\kappa }\left(x,s\right)d{k}_f \\ =\int {\eta }_{f\kappa }\mathit{exp}\left[i{\kappa }_f\left(k_f\right)s+i{k}_{f\mu }{x}^{\mu }\right]d{k}_f \end{gathered}$$

(41)

where the integral is over energy-momentum $k_f$, ${\eta }_{f\kappa }$ denotes normalization coefficients, and

$${\kappa }_f\left(k_f\right)=-\frac{{\hslash }^2}{2m_f}{k}_{f\mu }{k}_f^{\mu }$$

(42)

The observable world-line of the free particle satisfies the inner product of the expectation value of

$$\delta ⟨x_f^{\mu }⟩\delta ⟨x_{f\mu }⟩=\frac{⟨p_f^{\mu }⟩⟨p_{f\mu }⟩}{m_f^2}{\left(\delta s\right)}^2$$

(43)

where $⟨p_f^{\mu }⟩$ is the expectation value of four-momentum and $\delta ⟨x_f^{\mu }⟩$ is change in expectation value of spacetime position during the interval from $s$ to $s+\delta s$. Solving for $m_f^2$ gives

$$m_f^2=\frac{⟨p_f^{\mu }⟩⟨p_{f\mu }⟩}{\delta ⟨x_f^{\mu }⟩\delta ⟨x_{f\mu }⟩}{\left(\delta s\right)}^2$$

(44)

4.3. A Relativistic N-Particle System [6,7,34]

The field equation for $N$ spin-0 particles is

$$\left\{i\hslash \frac{𝜕}{𝜕s}-\left[{\displaystyle \sum _{a=1}^N}\frac{{\pi }_a^{\mu }{\pi }_{a\mu }}{2m_a}+V_N\right]\right\}\psi \left(\mathrm{1,2},\dots ,N,s\right)=0$$

(45)

where $V_N$ is a scalar interaction potential representing the interactions of all $N$ particles. The canonical four-momentum with minimal electromagnetic coupling is

$${\pi }_a^{\mu }=\frac{\hslash }{i}\frac{𝜕}{𝜕x_a^{\mu }}-\frac{e}{c}{A}_a^{\mu }$$

(46)

We simplify the example by assuming the 4-vector potentials $A_a^{\mu }$ are zero. Equation (45) becomes

$$\left\{i\hslash \frac{𝜕}{𝜕s}-\left[{\displaystyle \sum _{a=1}^N}\left(-\frac{{\hslash }^2}{2m_a}{𝜕}_a^{\mu }{𝜕}_{a\mu }\right)+V_N\right]\right\}\psi \left(\mathrm{1,2},\dots ,N,s\right)=0$$

(47)

The wave function $\psi \left(\mathrm{1,2},\dots ,N,s\right)$ is written as

$$\psi \left(\mathrm{1,2},\dots ,N,s\right)={\rho }^{1/2}\left(\mathrm{1,2},\dots ,N,s\right)\mathit{exp}\left[i\zeta \left(\mathrm{1,2},\dots ,N,s\right)\right]$$

(48)

with real, scalar functions $\rho$ and $\zeta$. The relativistic continuity equation

$$\frac{𝜕\rho }{𝜕s}+{\displaystyle \sum _{a=1}^N}{𝜕}_{a\mu }\left(\rho v_a^{\mu }\right)=0$$

(49)

for the $N$-particle system with 4-velocity

$$v_a^{\mu }=\frac{1}{m_a}{𝜕}_a^{\mu }S$$

(50)

is derived from Equations (47) and (48). The term $S=\hslash \zeta$ is the relativistic action for the $N$-particle system. It has the relativistic Hamilton–Jacobi equation

$$\frac{𝜕S}{𝜕s}+{\displaystyle \sum _{a=1}^N}\frac{1}{2m_a}\left({𝜕}_{a\mu }S\right)\left({𝜕}_a^{\mu }S\right)+Q_N+V_N=0$$

(51)

with scalar quantum potential

$$Q_N={\displaystyle \sum _{a=1}^N}\frac{{\hslash }^2}{2m_a}\left[\frac{1}{4{\rho }^2}\left({𝜕}_{a\mu }\rho \right)\left({𝜕}_n^{\mu }\rho \right)-\frac{1}{2\rho }{𝜕}_{a\mu }{𝜕}_a^{\mu }\rho \right]$$

(52)

The quantum potential Q_N depends on the joint probability distribution $\rho \left(\mathrm{1,2},\dots ,N,s\right)$ of all $N$ particles in spacetime and evolution parameter s. The quantum potential Q_N changes when the probability distribution changes. Three quantum potential models are presented by Fanchi [36].

4.4. Two-Body Bound State Problem

An example of a bound state problem is the two-body bound state problem for interacting scalar particles. The field equation for $N=2$ spin-0 particles is

$$\left\{i\hslash \frac{𝜕}{𝜕s}-\left[{\displaystyle \sum _{a=1}^2}\frac{{\pi }_a^{\mu }{\pi }_{a\mu }}{2m_a}+V_2\right]\right\}\psi \left(\mathrm{1,2},s\right)=0$$

(53)

where $V_2$ is a scalar interaction potential representing non-electromagnetic interactions between the two particles. The canonical four-momentum with minimal electromagnetic coupling is

$${\pi }_a^{\mu }=\frac{\hslash }{i}\frac{𝜕}{𝜕x_a^{\mu }}-\frac{e}{c}{A}_a^{\mu }, a=\mathrm{1,2}$$

(54)

Detailed examples with spin-0 and 2-component spin-1/2 particles are discussed in the literature [4,7,37] (and references therein).

4.5. Parametrized Relativistic Quantum Theory in Curved Spacetime

The formulation of PRQT in curved spacetime is developed and applied to free spinless particle motion in flat spacetime and curved spacetime [35]. The field equation in curved spacetime for a single spinless particle is

$$i\hslash \frac{𝜕\Psi }{𝜕s}=\frac{1}{2m}\left\{\left[\frac{\hslash }{i}\frac{𝜕}{𝜕x^{\mu }}-\frac{e}{c}{A}_{\mu }\right]\left[\frac{\hslash }{i}\frac{𝜕}{𝜕x_{\mu }}-\frac{e}{c}{A}^{\mu }\right]\Psi -i\hslash {\Gamma }_{\nu \mu }^{\mu }\left[\frac{\hslash }{i}\frac{𝜕}{𝜕x_{\nu }}-\frac{e}{c}{A}^{\nu }\right]\Psi \right\}+V \Psi$$

(55)

The metric is assumed to be independent of the invariant evolution parameter. The metric tensor for flat spacetime has nonzero elements

$$g_{00}=1=-g_{11}=-g_{22}=-g_{33}$$

(56)

Equations for flat spacetime are simplified because the contracted form ${\mathsf{\Gamma }}_{\nu \mu }^{\mu }$ of the Christoffel symbol of the second kind is zero for flat spacetime.

The field equation for a free scalar particle with mass $m$ moving in the gravitational field of a spherically symmetric mass $M$ can be modeled for the Schwarzschild metric

$$i\hslash \frac{𝜕{\psi }_{SM}}{𝜕s}=K_{SM}{\psi }_{SM}=\frac{1}{2m}\left\{\left[\frac{\hslash }{i}\frac{𝜕}{𝜕y^{\mu }}\right]\left[\frac{\hslash }{i}\frac{𝜕}{𝜕y_{\mu }}\right]-i\hslash {\mathsf{\Gamma }}_{\nu \mu }^{\mu }\left[\frac{\hslash }{i}\frac{𝜕}{𝜕y_{\nu }}\right]\right\}{\psi }_{SM}$$

(57)

Nonzero elements of the Schwarzschild metric are

$$g_{00}=-A(r), g_{11}=1/A(r),g_{22}=r^2,g_{33}=r^2{sin}^2\theta , A\left(r\right)=1-\frac{r_S}{r}$$

(58)

The term $r_S$ is the Schwarzschild radius

$$r_S=\frac{2M{G}_N}{c^2}$$

(59)

and $G_N$ is Newton’s gravitational constant. The differential equations for the Schwarzschild metric case were presented and solved in the large distance approximation. The solution to the radial equation has a $\frac{1}{r}$ dependence.

5. Conclusions

The formalism of Parametrized Relativistic Quantum Theory (PRQT) has been extended to multicomponent, many-body systems in curved spacetime. Probabilistic concepts with an invariant evolution parameter are introduced, and then used to derive a multicomponent, many-body formalism in curved spacetime. The extended PRQT formalism is a new approach for studying quantum gravity and comparing results to the approaches discussed in the Introduction. The metric tensor in the formalism presented here does not depend on the invariant evolution parameter.

A few applications are discussed to illustrate the range of applicability of the extended PRQT formalism. Applications in flat spacetime discuss the use of the formalism to illustrate the meaning of mass for a free particle, derivation of a quantum potential for a relativistic N-body system, and solution of a two-body bound state problem. A curved space application outlines the calculation of the motion of a free particle in a Schwarzschild metric. The extended PRQT formalism is a mathematical framework that can be further evaluated by applying it to widely recognized quantum gravitational phenomena such as Hawking radiation, quantum fluctuations in the early universe, and gravitational wave quantum effects.