###### Abstract

A new formulation of relativistic quantum mechanics is presented and applied
to a free, massive, and spin zero elementary particle in the Minkowski
spacetime. The reformulation requires that time and space, as well as the
timelike and spacelike intervals, are treated equally, which makes the new
theory fully symmetric and consistent with the Special Theory of Relativity.
The theory correctly reproduces the classical action of a relativistic
particle in the path integral formalism, and allows for the introduction of a
new quantity called vector-mass, whose physical implications for nonlocality,
the uncertainty principle, and quantum vacuum are described and discussed.

Fully Symmetric Relativistic Quantum Mechanics and Its Physical Implications

B. D. Tran and Z. E. Musielak

##  1 Introduction

Relativistic Quantum Mechanics (RQM) primarily concerns free relativistic
fields [1,2] described by the Klein-Gordon [3,4], Dirac [5], Proca [6] and
Rarita-Schwinger [7] wave equations, whereas the fundamental interactions and
their unification are considered by the gauge invariant Quantum Field Theory
(QFT) [8,9]. The above equations of RQM are invariant with respect to all
transformations that form the Poincaré group $\mathcal{P}\ =\
SO(3,1)\otimes_{s}T(3+1)$, where $SO(3,1)$ is a non-invariant Lorentz group of
rotations and boosts and $T(3+1)$ an invariant subgroup of spacetime
translations, and this structure includes reversal of parity and time [10].
Bargmann and Wigner [11] used the representations of $\mathcal{P}$ [12] to
obtain the Bargmann-Wigner equations, which reduce to the RQM equations under
special conditions. Different methods of deriving the RQM equations were
proposed by using either symmetry properties [13] or the irreducible
representations (irreps) of $T(3+1)$ and the resulting eigenvalue equations
[14,15].

There were several attempts to generalize the RQM equations, specifically, the
Klein-Gordon [16-19], Dirac [20-25], or both Klein-Gordon and Dirac [26-28]
and other [29] equations. Physical motivations for these generalizations were
different and they range from including some supersymmetry effects to
unification of leptons and quarks and accounting for different masses of the
three generations of elementary particles. Recently, the Dirac equation was
generalized to include chiral symmetry and it was shown that the correct
identification of chiral bases may explain the small masses of neutrinos and
be used to predict a new massive particle of dark matter [30].

As required by Special Theory of Relativity (STR), time and space must be
treated equally, which means that all basic equations of RQM are symmetric in
their time and space derivatives. In STR, the timelike and spacelike intervals
are also equally allowed, even so RQM has been formulated using the timelike
intervals only [1,2]; known exceptions are some tachyonic versions of RQM and
QFT [31,32]. The main objective of this paper is to reformulate RQM by taking
into account both timelike and spacelike intervals and explore physical
implications of the developed theory. The presented theory is developed for a
free, massive, and spin zero elementary particle in the Minkowski spacetime.
We follow Wigner [12] and define an elementary (quantum) particle as an object
whose wavefunction transforms as one of the irreps of $\mathcal{P}$ [10].

The presence of the spacelike intervals in our theory modifies its linear
momentum that, in addition to its timelike eigenvalues, has also spacelike
eigenvalues. As a result, the STR energy-momentum relationship with its
scalar-mass is modified by an additional term with its vector-mass, which only
affects the spatial distribution of particle’s wavefunction. The physical
consequences of the presence of the vector-mass in the energy-momentum
relationship are far reaching, starting with the modifications to the mass
term in the Klein-Gordon equation, and then leading to nonlocality, some
modifications of the uncertainty principle, and a novel picture of vacuum.
Moreover, the developed RQM theory preserves causality, is self-consistent,
and is formulated using the path integral formalism, which correctly
reproduces the classical action of a relativistic particle; however, the
theory does not allow for tachyons [31,32]. The theory is not a hidden-
variable theory (e.g., [33]) and its physical interpretation is consistent
with the Copenhagen interpretation of Quantum Mechanics (QM) [34,35].

The paper is structured as follows: the generalized eigenvalue equations are
presented and discussed in Section 2; the basic postulates used to reformulate
RQM are described in Section 3; the generalized Klein-Gordon equation and its
solutions are given in Section 4; in Section 5, completeness of Hilbert space
and causality of the theory are demonstrated; in Section 6, the theory is
formulated by using the path integral approach; physical implications of the
theory are given in Section 7; and conclusions are in Section 8.

##  2 Generalized eigenvalue equations

The basis of our reformulation of RQM is a set of eigenvalue equations for the
four-momentum operator $\mathcal{K}_{\mu}=i\partial_{\mu}$ acting on a
relativistic quantum state $|\phi\rangle$ in the Minkowski spacetime, with the
metric tensor $\eta_{\mu\nu}$ of positive time signature. The eigenvalue
equations are given by

$\mathcal{K}_{\mu}|\phi\rangle=\left(P_{\mu}+V_{\mu}\right)|\phi\rangle\ ,$
(1)

where $P_{\mu}$ and $V_{\mu}$ are timelike and spacelike eigenvalues of
$\mathcal{K}_{\mu}$, respectively, and $P_{\mu}P^{\mu}=m^{2}$,
$V_{\mu}V^{\mu}=-\tilde{m}^{2}$ and $P_{\mu}V^{\mu}=0$, with $m$ being a
scalar mass and $\tilde{m}$ being a new quantity called here ”vector-mass”
(see Section 3 for discussion). The eigenfunction $\phi$ can be either scalar,
or vector, or any higher rank tensor, or spinor wavefunction in order to be
consistent with the irreps of $\mathcal{P}$ [12,10].

As shown previously [14,15], the eigenvalue equations for $\phi$ being a
scalar wavefunction represent the necessary conditions that this function
transforms as one of the irreps of $T(3+1)\in\mathcal{P}$; here equation (1)
is a generalization of the previous results, which are recovered when
$V_{\mu}=0$. A new RQM is formulated by using the above eigenvalue equations,
which means that time and space, as well as the timelike ($P_{\mu}$) and
spacelike ($V_{\mu}$) intervals, are treated equally as required by STR.

In this paper, we consider only scalar wavefunctions and follow Wigner [10,12]
to identify an elementary particle with one of the irreps of $\mathcal{P}$. In
other words, the scalar wavefunction $\phi$ represents an elementary particle
if, and only if, the wavefunction satisfies the above generalized eigenvalue
equations.

The eigenvalue equations given by Eq. (1) are expressed in terms of a general
coordinate basis that is not necessarily orthonormal. Therefore, we formulate
our theory in the local orthonormal basis, called tetrads, which are shown to
be of great advantage in the development of our new RQM. The tetrads are
defined as local coordinate transformations from a global curvilinear
coordinate system to a locally flat orthogonal coordinate system that is
considered as the local inertial frame of an observer in the Minkowski
spacetime.

We used Greek indices for global coordinate components and Latin indices to
denote the components of local tetrad coordinates with
$\eta_{ab}=g_{\mu\nu}U_{a}^{\mu}U_{b}^{\nu}$ and $g_{\mu\nu}$ being the metric
tensor of the global coordinate system. The tetrads are defined as
$U^{a}_{\mu}=\frac{\partial\xi^{a}}{\partial x^{\mu}}$, where $\xi^{a}$ is the
local coordinate function. Then, the invariant spacetime element
$\mathrm{d}s^{2}$ of the Minkowski metric and the eignevalue equations in the
new local tetrad basis can be written as

$\mathrm{d}s^{2}=g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=\eta_{ab}\mathrm{d}\xi^{a}\mathrm{d}\xi^{b}\
,$ (2)

and

$\mathcal{K}_{a}|\phi\rangle=\left(P_{a}+V_{a}\right)|\phi\rangle=K_{a}|\phi\rangle\
.$ (3)

We express the contravariant components of the four-vectors $P$ and $V$ in the
local basis as $P=\\{E,\boldsymbol{P}\\}$ and $V=\\{U,\boldsymbol{V}\\}$ that
locally satisfy the following conditions:
$P_{a}P^{a}=E^{2}-\boldsymbol{P}\cdot\boldsymbol{P}=m^{2}$,
$V_{a}V^{a}=U^{2}-\boldsymbol{V}\cdot\boldsymbol{V}=-\tilde{m}^{2}$ and
$P_{a}V^{a}=0$, where $m$ and $\tilde{m}$ are the so-called scalar and vector
mass, respectively. It must be noted that both $m$ and $\tilde{m}$ are
Lorentz-invariant constants, and that $U=\boldsymbol{P}\cdot\boldsymbol{V}/E$.

Let $\boldsymbol{\beta}=\boldsymbol{P}/E$ be the relativistic velocity of an
elementary particle, which allows writing $P=\\{E,\boldsymbol{\beta}E\\}$,
$V=\\{\boldsymbol{\beta}\cdot\boldsymbol{V},\boldsymbol{V}\\}$, and
$K=P+V=\\{E+\boldsymbol{\beta}\cdot\boldsymbol{V},\boldsymbol{V}+\boldsymbol{\beta}E\\}$.
We refer to the vector $K$ as the four-mass vector and use it to define the
eigenstate wavefunction of a relativistic quantum particle. The four-vector
$P$ provides the energy-momentum relationship of STR, which is used in
standard RQM. However, the presence of the four-vector $V$ modifies this
relationship to become

$\left(E^{2}-|\boldsymbol{P}|^{2}\right)+\left(U^{2}-|\boldsymbol{V}|^{2}\right)=m^{2}-\tilde{m}^{2}=M^{2}\
,$ (4)

which is the energy-momentum relationship for a massive and spin zero quantum
particle used in the reformulated RQM. This energy-momentum relationship
reduces to that of STR, which is the basis for standard RQM, when $V=0$.

It must be also noted that $M^{2}$ is not characterized as the mass of the
particle, as in this reformulated RQM, a particle of definite mass and
position in space must be in a superposition of different vector-mass
$\tilde{m}$, and that the scalar-mass $m$ represents is responsible for the
particle’s world-line to be confined within the light cone. On the other hand,
the vector-mass $\tilde{m}$ does not play any role in the motion of the
particle, even if its squared magnitude exceeds that of the scalar-mass,
because of its spacelike origin. Therefore, the theory developed in this paper
only allows for particles with positive $m^{2}$ term, which rules out tachyons
and tachyonic fields [31,32] as unphysical within the framework of this
theory.

The origin of the four-vector $V$ is the generalized eigenvalue equations
given by Eq. (1), and the above results show that the vector-mass $\tilde{m}$
is associated with the presence of $V$. In our reformulated RQM, $m$ is
responsible for the time evolution of the wavefunction $\phi$, like in
standard RQM, while $\tilde{m}$ affects only the spatial distribution of the
wavefunction $\phi$, which makes our reformulated RQM to be a fully symmetric
theory that accounts for both the timelike and spacelike intervals. Let us now
describe the basic postulates of our new theory.

##  3 Basic postulates of new theory

Our reformulations of RQM is based on the following postulates:

  1. 1.

The canonical commutation relation of the conjugate operators
$(\xi^{a},\mathcal{K}_{b})$ is given as

$\left[\xi^{a},\mathcal{K}_{b}\right]=-i\delta^{a}_{b}\ .$ (5)

  2. 2.

The particle’s relativistic three-velocity,
$\boldsymbol{\beta}=\boldsymbol{P}/E$, is the boost parameter of a proper
Lorentz transformation

$\Lambda_{\boldsymbol{\beta}}=\begin{pmatrix}\gamma&\gamma\boldsymbol{\beta}^{\top}\\\
\gamma\boldsymbol{\beta}&\gamma\mathrm{I}+\frac{\gamma^{2}}{1+\gamma}\mathrm{B}^{2}\end{pmatrix}\
,$ (6)

where $\boldsymbol{\beta}^{\top}$ is a row vector, the symbol $\top$ denotes
matrix transposition, the $3\times 3$ identity matrix is represented by
$\mathrm{I}$, $\gamma=\left(1-|\boldsymbol{\beta}|^{2}\right)^{-1/2}$ is the
Lorentz factor with $|\boldsymbol{\beta}|<1$, and

$\mathrm{B}^{2}=\boldsymbol{\beta}\boldsymbol{\beta}^{\top}-|\boldsymbol{\beta}|^{2}\mathrm{I}$
(7)

with

$\mathrm{B}=\begin{pmatrix}0&\beta_{z}&-\beta_{y}\\\ -\beta_{z}&0&\beta_{x}\\\
\beta_{y}&-\beta_{x}&0\end{pmatrix}\ .$ (8)

From this, we have $E=\gamma m$ as the usual definition for the energy of a
particle of scalar-mass $m$, and the new energy contribution related to
vector-mass $\boldsymbol{k}$ is defined as
$\boldsymbol{V}=\mathrm{L}\boldsymbol{k}$, where $\mathrm{L}$ is given by

$\mathrm{L}=\gamma\mathrm{I}+\frac{\gamma^{2}}{1+\gamma}\mathrm{B}^{2}=\mathrm{I}+\frac{\gamma^{2}}{1+\gamma}\boldsymbol{\beta}\boldsymbol{\beta}^{\top}\
.$ (9)

Thus, the four-mass vector in the rest frame, i.e.
$\boldsymbol{\beta}=\boldsymbol{0}$, is simply $K=k=\\{m,\boldsymbol{k}\\}$.

  3. 3.

Let $\mathcal{U}_{\boldsymbol{\beta}}$ be an operator identified as a unitary
representation of the Lorentz boost that acts on the state vector
$|k\rangle=|m,\boldsymbol{k\rangle}$ in a Hilbert space

$|m,\boldsymbol{k};\boldsymbol{\beta}\rangle=\mathcal{U}_{\boldsymbol{\beta}}|m,\boldsymbol{k}\rangle\
.$ (10)

Here, the states $|m,\boldsymbol{k}\rangle$ are the so-called inertial states
describing a particle at rest, with $m$ and $\boldsymbol{k}$ forming the four-
mass vector $K$, and the velocity $\boldsymbol{\beta}=\boldsymbol{0}$. A
particle can be boosted to a moving frame with
$\boldsymbol{\beta}\neq\boldsymbol{0}$, so the state of a moving particle is
labeled as $|m,\boldsymbol{k};\boldsymbol{\beta}\rangle$. The dual states of
the inertial states are called the spacetime states and they are represented
by the vector $|\xi\rangle$, or $|\tau,\boldsymbol{\xi}\rangle$, in a rest-
frame of an observer.

The states of all $m$ and $\boldsymbol{k}$ form a complete basis of the
relativistic quantum Hilbert space of a particle at rest, and the vector
$|\xi\rangle$, or $|\tau,\boldsymbol{\xi}\rangle$, form also a complete basis
for the spacetime location of the particle (see Section 5). The fact that
neither $m$ nor $\boldsymbol{k}$ are quantities with set up values (see
Section 6) makes the presented RQM to be a significantly different theory than
standard RQM, and its physical implications are far reaching (see Section 7).

  4. 4.

The inertial state $|m,\boldsymbol{k}\rangle$ is an eigenvector of the
operators $\mathcal{K}_{a}$ such that

$\mathcal{K}_{a}|m,\boldsymbol{k}\rangle=k_{a}|m,\boldsymbol{k}\rangle\ ,$
(11)

with the eigenvalues $k=\\{m,\boldsymbol{k}\\}$. It follows that
$k_{a}k^{a}=m^{2}-|\boldsymbol{k}|^{2}=m^{2}-\tilde{m}^{2}=M^{2}$. It must be
also noted that the vector-mass is completely independent from the scalar-
mass.

  5. 5.

The wavefunction of a quantum particle at rest, with the scalar-mass $m$ and
vector-mass $\boldsymbol{k}$, expressed in the spacetime basis is given by

$\langle\tau,\boldsymbol{\xi}|m,\boldsymbol{k}\rangle=\frac{e^{-i\left(m\tau-\boldsymbol{k}\cdot\boldsymbol{\xi}\right)}}{(2\pi)^{2}}\
,$ (12)

where $\tau=\xi^{0}$, and the Lorentz-transformed wavefunction of a particle
moving with a well-defined velocity $\boldsymbol{\beta}$ is given by

$\langle\tau,\boldsymbol{\xi}|m,\boldsymbol{k};\boldsymbol{\beta}\rangle=\frac{e^{-i\tau(E+\boldsymbol{\beta}\cdot\boldsymbol{V})}e^{i\boldsymbol{\xi}\cdot(\boldsymbol{V}+\boldsymbol{\beta}E)}}{(2\pi)^{2}}\
.$ (13)

The state in Eq. (12) describes a particle at rest, nevertheless, the
wavefunction seems to be propagating in the direction of $\boldsymbol{k}$.
However, a more careful look shows that it is not propagating, but instead it
is the winding of the complex wavefunction. Thus, there is nothing actually
moving though space and the only way to assign motion to the particle is to
change the motion of the inertial frame of the observer relative to the frame
of the particle through a Lorentz boost, as given in Eq. (13).

  6. 6.

The timelike differential operator $Q_{t}=t^{a}\mathcal{K}_{a}$, where $t^{a}$
is a timelike vector, is the generator of translation in time of the
relativistic wavefunction. However, the spacelike operator
$Q_{s}=s^{a}\mathcal{K}_{a}$, where $s^{a}$ is spacelike and is orthogonal to
$t^{a}$, is the translation generator in space. The symmetries generated by
these generators correspond to conservation laws, in which the energy and
momentum conservation laws are unified into one, which is the conservation of
scalar-mass energy generated by the timelike operator. For the spacelike
operator, the theory suggests that there is also another conservation law that
pertains to the vector-mass energy, and its physical implications are
discussed in Section 7.

The above postulates form the basis of our new RQM, which is developed here
for a massive and spin-zero elementary particle uniquely described by the
scalar wavefunction $\phi$ for which we now obtain its wave equation.

##  4 Generalized Klein-Gordon equation and its solutions

The eigenvalue equations (see Eq. 11), in general, satisfy the following
second-order Klein-Gordon (KG) eigenvalue equation

$\mathcal{K}_{a}\mathcal{K}^{a}\phi\left(\tau,\boldsymbol{\xi}\right)=\left(m^{2}-|\boldsymbol{k}|^{2}\right)\phi\left(\tau,\boldsymbol{\xi}\right)=M^{2}\phi\left(\tau,\boldsymbol{\xi}\right)\
,$ (14)

where $\phi=\langle\tau,\boldsymbol{\xi}|\phi\rangle$ and $M^{2}$, whose value
ranges from $-\infty$ to $\infty$, is the eigenvalue of the operator
$\mathcal{K}_{a}\mathcal{K}^{a}$. The general solutions to Eq. 14 for a
particular $M^{2}$ is given as

$\displaystyle\phi\left(\tau,\boldsymbol{\xi}\right)$ $\displaystyle=$
$\displaystyle Ae^{-iK_{a}\xi^{a}}+Be^{-i\bar{K}_{a}\xi^{a}}$ (15)
$\displaystyle=$
$\displaystyle\left[Ae^{-iE(\tau-\boldsymbol{\beta}\cdot\boldsymbol{\xi})}+Be^{iE(\tau-\boldsymbol{\beta}\cdot\boldsymbol{\xi})}\right]e^{i\boldsymbol{V}\cdot(\boldsymbol{\xi}-\boldsymbol{\beta}\tau)}\
,$

where $K=P+V$ for the positive energy solution, and $\bar{K}=-P+V$ for the
negative energy solution, which corresponds to $m\to-m$. The coefficients $A$
and $B$ are constants of integration. Note that $M^{2}$ is not particle’s mass
despite its place in the KG equation. The main reason is that the particle’s
mass must be in a superposition of different vector-mass $\boldsymbol{k}$,
which does not play any role in the motion of the particle because of its
spacelike origin.

It must be noted that Eq. 14 is not a dynamical equation like the other wave
equations; in fact, it is the extended form of the Helmholtz equation with the
Minkowski metric, where the eigenvalue $M^{2}$ cannot be replaced by an
operator without introducing new structures to the particle or spacetime, such
as internal symmetries or extra dimensions, respectively. Therefore,
accounting for spins and different generations of particles is still required,
which is beyond the scope of this paper, as the main focus herein is to
develop an alternative theory of RQM and present its physical implications. In
the following, we discuss the relation between the completeness of Hilbert
space and the causal structure of this new RQM in detail.

##  5 Completeness of Hilbert space and causality

After defining the basic postulates of the new RQM and presenting the
resulting generalized Klein-Gordon eigenvalue equation, the completness of
Hilbert space and causality must be considered. This requires the
generalization of the concept of the inner product of the new state vectors in
Hilbert space, which is defined as

$\langle\tau^{\prime},\boldsymbol{\xi}^{\prime}|\tau,\boldsymbol{\xi}\rangle=\delta(\tau^{\prime}-\tau)\delta^{3}\left(\boldsymbol{\xi}^{\prime}-\boldsymbol{\xi}\right)\
,$ (16)

and

$\langle
m^{\prime},\boldsymbol{k}^{\prime}|m,\boldsymbol{k}\rangle=\delta(m^{\prime}-m)\delta^{3}\left(\boldsymbol{k}^{\prime}-\boldsymbol{k}\right)\
.$ (17)

It must be noted again that a spacetime state $|\tau,\boldsymbol{\xi}\rangle$
is completely defined throughout spacetime, which describes a particle that
exists exactly at the point $\boldsymbol{\xi}$ in space and $\tau$ in time of
a local inertial observer. For such state, an observer would see the particle
suddenly come in and out of existence, with undefined mass, energy, and
momentum. Conversely, an inertial quantum state
$|m,\boldsymbol{k};\boldsymbol{\beta}\rangle$ represents a particle of well-
defined scalar-mass, vector-mass, and velocity, but it does not have a well-
defined spacetime position.

Using Eq. (13), we write the projection of a inertial state onto a spacetime
state as

$\langle\tau,\boldsymbol{\xi}|m,\boldsymbol{k};\boldsymbol{\beta}\rangle=\frac{e^{-iE\left(\tau-\boldsymbol{\beta}\cdot\boldsymbol{\xi}\right)}e^{i\boldsymbol{V}\cdot\left(\boldsymbol{\xi}-\boldsymbol{\beta}\tau\right)}}{(2\pi)^{2}}=\frac{e^{-i\tau(E+\boldsymbol{\beta}\cdot\boldsymbol{V})}e^{i\boldsymbol{\xi}\cdot(\boldsymbol{V}+\boldsymbol{\beta}E)}}{(2\pi)^{2}}\
,$ (18)

which is a state of definite $K$ in flat spacetime, with
$\boldsymbol{\beta}=\boldsymbol{P}/E$ and $\tau=\xi^{0}$. Since the identity
operator $\mathcal{I}$ in the spacetime basis is given by

$\mathcal{I}=\int\mathrm{d}^{4}\xi\
|\tau,\boldsymbol{\xi}\rangle\langle\tau,\boldsymbol{\xi}|\ ,$ (19)

we use Eq. (18) to find

$\displaystyle\langle
m^{\prime},\boldsymbol{k}^{\prime};\boldsymbol{\beta}|\mathcal{I}|m,\boldsymbol{k};\boldsymbol{\beta}\rangle$
$\displaystyle=$ $\displaystyle\int\frac{\mathrm{d}^{4}\xi}{(2\pi)^{4}}\
e^{i\tau(\Delta
E+\boldsymbol{\beta}\cdot\Delta\boldsymbol{V})}e^{-i\boldsymbol{\xi}\cdot(\Delta\boldsymbol{V}+\boldsymbol{\beta}\Delta
E)}$ (20) $\displaystyle=$ $\displaystyle\delta(\Delta
E+\boldsymbol{\beta}\cdot\Delta\boldsymbol{V})\delta^{3}(\Delta\boldsymbol{V}+\boldsymbol{\beta}\Delta
E)\ ,$

where $\Delta K=K^{\prime}-K$. Note that for all allowed values of
$|\boldsymbol{\beta}|<1$, Eq. (20) is equivalent to Eq. (17), and that the
identity operator $\mathcal{I}$ is complete and the boost operator
$\mathcal{U}_{\boldsymbol{\beta}}$ (see Eq. 10) is unitary. Equivalently, the
identity operator can be expressed in the inertial basis as

$\mathcal{I}=\int\mathrm{d}m\
\mathrm{d}^{3}\boldsymbol{k}|m,\boldsymbol{k}\rangle\langle m,\boldsymbol{k}|\
,$ (21)

which is to show that the inertial states are also complete.

Next, we introduce two new operators, $\mathcal{T}_{\boldsymbol{\beta}}$ and
$\mathcal{S}$, whose definitions are motivated by their application in the
next section for the construction of the path integral formulation of the new
RQM. The operators are defined as

$\mathcal{T}_{\boldsymbol{\beta}}(m)=\int\mathrm{d}^{3}\boldsymbol{k}\
|m,\boldsymbol{k};\boldsymbol{\beta}\rangle\langle
m,\boldsymbol{k};\boldsymbol{\beta}|=\int\frac{\mathrm{d}^{3}\boldsymbol{V}}{\gamma}\
|m,\boldsymbol{k};\boldsymbol{\beta}\rangle\langle
m,\boldsymbol{k};\boldsymbol{\beta}|\ ,$ (22)

and

$\mathcal{S}(\tau)=\int\mathrm{d}^{3}\boldsymbol{\xi}\
|\tau,\boldsymbol{\xi}\rangle\langle\tau,\boldsymbol{\xi}|\ .$ (23)

To understand the properties of these operators, we investigate the matrix
elements in their respective dual basis. First, we study the properties of
$\mathcal{T}_{\boldsymbol{\beta}}$ with its matrix element given by

$\displaystyle\langle\tau^{\prime},\boldsymbol{\xi}^{\prime}|\mathcal{T}_{\boldsymbol{\beta}}|\tau,\boldsymbol{\xi}\rangle$
$\displaystyle=$
$\displaystyle\frac{e^{-iE\left(\Delta\tau-\boldsymbol{\beta}\cdot\Delta\boldsymbol{\xi}\right)}}{2\pi\gamma}\int\frac{\mathrm{d}^{3}\boldsymbol{V}}{(2\pi)^{3}}\
e^{i\boldsymbol{V}\cdot\left(\Delta\boldsymbol{\xi}-\boldsymbol{\beta}\Delta\tau\right)}$
(24) $\displaystyle=$
$\displaystyle\frac{e^{-iE\left(\Delta\tau-\boldsymbol{\beta}\cdot\Delta\boldsymbol{\xi}\right)}}{2\pi\gamma}\
\delta^{3}\left(\Delta\boldsymbol{\xi}-\boldsymbol{\beta}\Delta\tau\right)\ .$

Note that we did not integrate over the scalar-mass $m$. This operator is
defined for a particle of definite scalar-mass. The immediate interpretation
of this result is that this is exactly the causal path of the particle
traveling between the two spacetime states, where
$\Delta\xi=\xi^{\prime}-\xi$. For a particle of definite mass $m$, the result
shows that the particle must be stationary in space when its velocity
$\boldsymbol{\beta}=0$, while the complex phase evolves over proper time of
the particle, i.e.

$\langle\tau^{\prime},\boldsymbol{\xi}^{\prime}|\mathcal{T}_{\boldsymbol{\beta}=\boldsymbol{0}}|\tau,\boldsymbol{\xi}\rangle=\frac{e^{-im\Delta\tau}}{2\pi}\
\delta^{3}\left(\Delta\boldsymbol{\xi}\right)\ .$ (25)

Alternatively, when $\boldsymbol{\beta}\neq 0$, the result (24) is nonzero if,
and only if, the states are causally separated, i.e. the path traced out by
the particle must stay within the light cone. The complex phase difference
between the final and initial states is given by the complex exponential
prefactor. Remarkably, this is simply the quantum path of a free point
particle and the matrix element is just the phase evolution along the path of
propagation. This operator is the propagation operator for a particle of mass
$m$ moving from $|\tau,\boldsymbol{\xi}\rangle$ to
$|\tau^{\prime},\boldsymbol{\xi}^{\prime}\rangle$; however, it is different
from time evolution operator in QM because the time dependence is already
contained in the spacetime states $|\tau,\boldsymbol{\xi}\rangle$, while the
operator $\mathcal{T}_{\boldsymbol{\beta}}$ simply connects the two spacetime
states by imparting the particle with a particular mass $m$ moving at the
velocity $\boldsymbol{\beta}$.

In fact, the particle itself can be associated with the operator
$\mathcal{T}_{\boldsymbol{\beta}}$. However, as we recall the general solution
to the KG eigenvalue equation in Eq. (15), we see that it is a linear
combination of both positive and negative energy terms; therefore, we must
also form a linear combination of the operator
$\mathcal{T}_{\boldsymbol{\beta}}$ with its negative energy counter part,
namely $m\to-m$, to describe a particle of mass $m>0$.

Let us refer to the linear combination

$\mathcal{P}_{\boldsymbol{\beta}}(m)=a^{+}_{\boldsymbol{\beta}}\mathcal{T}_{\boldsymbol{\beta}}(m)+b^{-}_{\boldsymbol{\beta}}\mathcal{T}_{\boldsymbol{\beta}}(-m)\
,$ (26)

as the ’particle operator’ that creates a particle or destroys an antiparticle
of mass $m$, and its corresponding mass conjugation

$\mathcal{P}^{\star}_{\boldsymbol{\beta}}(m)=a^{-}_{\boldsymbol{\beta}}\mathcal{T}_{\boldsymbol{\beta}}(-m)+b^{+}_{\boldsymbol{\beta}}\mathcal{T}_{\boldsymbol{\beta}}(m)\
,$ (27)

as the ’antiparticle operator’ that creates an antiparticle or destroys a
particle of mass $m$.

Here, we have promoted the numerical coefficients to creation and annihilation
operators that act on Fock space of occupancy number basis, $|n\rangle$, and
defined the mass conjugation of the operator
$\mathcal{T}_{\boldsymbol{\beta}}$ as

$\mathcal{T}^{\star}_{\boldsymbol{\beta}}(m)=\mathcal{T}_{\boldsymbol{\beta}}(-m)\
,$ (28)

and

$(a^{+}_{\boldsymbol{\beta}})^{\star}=a^{-}_{\boldsymbol{\beta}}\ .$ (29)

By promoting the coefficients to operators, the spacetime states must also be
promoted to the spacetime Fock’s states
$|\tau,\boldsymbol{\xi};n\rangle=|\tau,\boldsymbol{\xi}\rangle\otimes|n\rangle$,
such that $|\tau,\boldsymbol{\xi};0\rangle$ represents the vacuum state and
$|\tau,\boldsymbol{\xi}\rangle$ represents single particle state. Thus, we
have

$\langle\tau^{\prime},\boldsymbol{\xi}^{\prime}|\mathcal{P}_{\boldsymbol{\beta}}|\tau,\boldsymbol{\xi};0\rangle=\langle\tau^{\prime},\boldsymbol{\xi}^{\prime}|\mathcal{T}_{\boldsymbol{\beta}}|\tau,\boldsymbol{\xi}\rangle\
.$ (30)

Finally, we have the matrix element of the operator $\mathcal{S}$ expressed as
follows

$\langle
m^{\prime},\boldsymbol{k}^{\prime};\boldsymbol{\beta}|\mathcal{S}|m,\boldsymbol{k};\boldsymbol{\beta}\rangle=\frac{e^{-i\tau(\Delta
E+\boldsymbol{\beta}\cdot\Delta\boldsymbol{V})}}{2\pi}\delta^{3}\left(\Delta\boldsymbol{V}+\boldsymbol{\beta}\Delta
E\right)\ ,$ (31)

which is the spatial inner product of two inertial states at time $\tau$. The
$\mathcal{S}$ operator specifies a local inertial frame, in which the inner
product can be calculated, or the spatial projection of one particle state
onto another. As shown by Eq. (31), the projection is given for two states of
different four-mass vectors with the same velocity. This may be interpreted as
the transition amplitude of a particle changing its mass as it propagates. In
another case, we could project states of the same masses but different
velocities. The result would be the transition amplitude of the particle
changing its velocity. In either case, the particle is undergoing an
interaction, where the former could describe the particle decay process and
the latter for a particle experiencing a force. However, in this paper we
consider only free particles.

In the next section, the pair $\mathcal{S}$ and
$\mathcal{T}_{\boldsymbol{\beta}}$ will be used to recover the relativistic
classical Lagrangian of a free particle by following the Feynman path integral
formulation of QM. Our goal is to obtain a consistent relativistic path
integral and, ultimately, confirm the validity of our reformulated theory of
RQM.

##  6 Path integral formulation

We now present the path integral formulation of the developed RQM by deriving
the transition amplitude for a particle with a definite scalar-mass $m$. The
integration is performed over all possible $|\boldsymbol{\beta}|<1$, or
equivalently, $|\gamma\boldsymbol{\beta}|<\infty$, since a quantum particle
can take any path and the velocity integration is bounded because the particle
is massive. Then, using the operator $\mathcal{T}_{\boldsymbol{\beta}}$ from
previous section (see Eq. 22), the general transition operator $\mathcal{T}$
is given by

$\mathcal{T}(m)=\int_{|\gamma\boldsymbol{\beta}|<\infty}\frac{\mathrm{d}^{3}(\gamma\boldsymbol{\beta})}{\gamma}\
\mathcal{T}_{\boldsymbol{\beta}}(m)=\int_{|\boldsymbol{\beta}|<1}\mathrm{d}^{3}\boldsymbol{\beta}\
\gamma^{2}\mathcal{T}_{\boldsymbol{\beta}}(m)\ .$ (32)

The presence of the factor $\gamma$ is to keep the integration Lorentz-
invariant. Using the particle operator to create a particle of mass $m$ out of
the vacuum as shown in Eq. (30), we integrate Eq. (24) to obtain the
relativistic quantum transition amplitude

$\displaystyle\langle\tau^{\prime},\boldsymbol{\xi}^{\prime}|\mathcal{T}|\tau,\boldsymbol{\xi}\rangle$
$\displaystyle=$
$\displaystyle\int_{|\boldsymbol{\beta}|<1}\mathrm{d}^{3}\boldsymbol{\beta}\
\frac{e^{-iE\left(\Delta\tau-\boldsymbol{\beta}\cdot\Delta\boldsymbol{\xi}\right)}}{2\pi\gamma^{-1}}\
\delta^{3}\left(\Delta\boldsymbol{\xi}-\boldsymbol{\beta}\Delta\tau\right)$
(33) $\displaystyle=$
$\displaystyle\frac{e^{-im\Delta\bar{\tau}}}{2\pi\left|\Delta\bar{\tau}\right|}\
\Theta\left(1-\left|\frac{\Delta\boldsymbol{\xi}}{\Delta\tau}\right|^{2}\right)\
,$

where

$\Delta\bar{\tau}=\Delta\tau\sqrt{1-\left|\frac{\Delta\boldsymbol{\xi}}{\Delta\tau}\right|^{2}}\
,$ (34)

and the function $\Theta(x)$ is the Heaviside step function, whose presence is
necessary to prevent the particle from propagating faster than the speed of
light.

The Feynman path integral formalism describes the procedure of calculating the
propagator matrix element $G(\xi_{F};\xi_{I})$, a.k.a Green’s function, of a
free particle moving from $\xi$ to $\xi^{\prime}$. This is done by integrating
the Feynman propagation phase factor over all possible timelike paths that the
particle could take. To obtain this operator, firstly we subdivide spacetime
phase space into $N$ slices of spacelike hypersurfaces $\Sigma_{n}$ and the
corresponding dual surfaces $\Omega_{n}$, where
$\boldsymbol{\xi}_{n}\in\Sigma_{n}$ and $\boldsymbol{k}_{n}\in\Omega_{n}$.
Each surface $\Sigma_{n}$ is spatial surface that represents a time slice in
spacetime where the particle could be found at time $\tau_{n}$. The
integration over this surface is represented by the operator
$\mathcal{S}_{\tau_{n}}$.

Then, we connect the different hypersurfaces using the summation over all
world lines that the particle could take. This is done by including the
transition operators $\mathcal{T}_{n}$ of a fixed scalar-mass in between
$\mathcal{S}_{\tau_{n}}$. Lastly, we take the continuum limit as
$N\rightarrow\infty$ and $\Delta\tau_{n}\rightarrow 0$. This construction
gives us the following the time-ordered products

$\displaystyle G_{N}(\xi_{F},\xi_{I})$ $\displaystyle=$
$\displaystyle\langle\xi_{F}|\mathcal{U}_{N}|\xi_{I}\rangle$ (35)
$\displaystyle=$
$\displaystyle\langle\xi_{N}|\mathcal{T}_{N}C_{N-1}\mathcal{S}_{\tau_{N-1}}\mathcal{T}_{{N-1}}\cdots
C_{n}\mathcal{S}_{\tau_{n}}\mathcal{T}_{n}\cdots
C_{1}\mathcal{S}_{\tau_{1}}\mathcal{T}_{1}|\xi_{0}\rangle$ $\displaystyle=$
$\displaystyle\left[\prod_{n=1}^{N-1}\int_{\Sigma_{n}}\mathrm{d}^{3}\boldsymbol{\xi}_{n}C_{n}\right]\left[\prod_{j=1}^{N}\langle\xi_{j}|\mathcal{T}_{j}|\xi_{j-1}\rangle\right]\
,$

where $\tau_{n}>\tau_{n-1}$ and $C_{n}$ is some normalization constant to be
determined. The matrix elements of $\mathcal{T}_{j}$ are given by Eq. (33).
Thus, we have

$\langle\xi_{j}|\mathcal{T}_{j}|\xi_{j-1}\rangle=\frac{e^{-im\Delta\bar{\tau}_{j}}}{2\pi\Delta\bar{\tau}_{j}}\Theta\left(1-\left|\frac{\Delta\boldsymbol{\xi}_{j}}{\Delta\tau_{j}}\right|^{2}\right)\
,$ (36)

which leads to

$G_{N}(\xi_{F},\xi_{I})=\left[\prod_{n=1}^{N-1}\int\mathrm{d}^{3}\boldsymbol{\xi}_{n}C_{n}\right]\prod_{j=1}^{N}\left(\frac{e^{-im\Delta\bar{\tau}_{j}}}{2\pi\Delta\bar{\tau}_{j}}\right)\
,$ (37)

where $\Delta\xi_{j}=\xi_{j}-\xi_{j-1}$. Also, we have omitted the Heaviside
functions for legibility and

$\Delta\bar{\tau}_{j}=\Delta\tau_{j}\sqrt{1-\left|\frac{\Delta\boldsymbol{\xi}_{j}}{\Delta\tau_{j}}\right|^{2}}\
.$ (38)

With the endpoints fixed and the velocity bounded by the speed of light, we
take the limit $N\to\infty$ and obtain

$G(\xi_{F},\xi_{I})=\int\mathrm{D}\boldsymbol{\xi}(\bar{\tau})\ Ce^{iS}\ ,$
(39)

where the integral is appropriately normalized by $C$. Then, the path-integral
measure in the continuum limit is

$\mathrm{D}\boldsymbol{\xi}(\bar{\tau})=\lim_{N\to\infty}\prod_{n=1}^{N-1}\mathrm{d}^{3}\boldsymbol{\xi}_{n}\
.$ (40)

The above results demonstrate that the classical action of a relativistic free
particle is correctly recovered from the path-integral formulation, with the
action given by

$S=-m\int_{\bar{\tau}_{I}}^{\bar{\tau}_{F}}\mathrm{d}\bar{\tau}\ ,$ (41)

where $\bar{\tau}$ is the proper time of a relativistic particle. We may
finally define the evolution operator $\mathcal{U}$ that acts on states in
spacetime basis as

$\mathcal{U}(\tau)=\lim_{N\to\infty}\mathcal{T}_{N}\prod_{n=1}^{N-1}C_{n}\mathcal{S}_{\tau_{n}}\mathcal{T}_{n}\
.$ (42)

Now, developing an analytical method to evaluate the relativistic path
integral given in Eq. (39) is out the scope of this paper. Nonetheless, for a
free particle this problem is equivalent to showing that it reduces to the
result (see Eq. 33) for the operator $\mathcal{T}$.

Table 1: Differences between the standard and new RQM Concepts | Standard RQM | New RQM
---|---|---
Four momentum | Timelike | Timelike and spacelike
Invariant mass | Scalar | Scalar and vector
Particle’s velocity | Implicit | Explicit
Conjugate variables | Energy - time | Scalar-mass - proper time
Conjugate variables | Momentum - position | Vector-mass - proper distance
Conserved quantities | Timelike components | Timelike and spacelike components
Path integral | All trajectories | Trajectories inside the light cone

##  7 Physical implications

The main differences between the standard RQM and the reformulated RQM
presented in this paper are summarized in Table 1. Let us now briefly comment
on the concepts given in Table 1. As a consequence of symmetry between the
timelike and spacelike intervals, the four momentum is no longer confined to
the light cone, and that the theory requires both the scalar-mass and vector-
mass; although only scalar-mass is observable, the four-mass vector labels the
mass state of the particle. The (linear) velocity is a parameter of the
Lorentz boost. The reformulation makes explicit distinction between the
vector-mass and the velocity vector. The quantum nature of the particle is
encoded in the superposition of different vector-masses and velocities. The
proper time and distance are defined relative to an inertial observer and the
reformulation redefines the Heisenberg uncertainty principles with the new
conjugate variable pairs. Furthermore, the generalized KG equation obtained in
this paper is not a wave equation but an eigenvalue equation. Unlike the
square of the scalar-mass in the conventional theory being a constant, $M^{2}$
is an eigenvalue. On the one hand, the conservation of timelike component
incorporates both the conservation of momentum and energy; on the other hand,
the conservation of spacelike component could be related to the conservation
of quantum information and responsible for the nonlocal quantum effects. Most
importantly, the reformulated theory reproduces the well-known relativistic
Lagrangian of a classical particle for the path integral formulation.

The reformulated RQM presented in this paper takes into account the spacelike
eigenvalues of the linear momentum, in addition to its timelike eigenvalues.
The timelike and spacelike eigenvalues are associated with the scalar-mass $m$
and vector-mass $\boldsymbol{k}$, respectively, with the latter being a new
physical concept. The theory demands that $m$ and vector-mass $\boldsymbol{k}$
are independent observable quantities. However, in practice only $m$ can be
observed since all observers and their experiments are confined to their local
light cone. On the other hand, $\boldsymbol{k}$ does not describe the
particle’s causal motion to an inertial observer, therefore, it cannot be
measured experimentally because of its spacelike nature outside of the light
cone. Despite this spacelike nature of $\boldsymbol{k}$, the developed theory
is not a hidden-variable theory and it preserves causality.

Moreover, the presence of vector-mass does not affect particle’s probability
amplitude because $\boldsymbol{k}$ is integrated out of the complex phase when
the operator $\mathcal{T}_{\boldsymbol{\beta}}$ is projected onto the
spacetime states. The result is that the complex phase that involves
$\boldsymbol{k}$ becomes a delta function representing the localization of the
particle in spacetime. The remaining complex phase contains the scalar-mass
and the proper time, which becomes the probability amplitude of the particle
for a particular path that it takes as it is shown by the path integral
formulation (see Section 6). Other important physical implications of the
presence of the vector-mass in the theory include nonlocality, a novel picture
of vacuum, and some modifications of the uncertainty principle, and they are
now discussed.

There is a large body of literature on the subject of nonlocality and
incompleteness in QM and the subsequent developments triggered by the original
Einstein, Podolsky and Rosen paper [36] and Bell’s work [37-39]. Bell showed
that hidden-variables theories must be nonlocal, in order to be in agreement
with the empirically verified predictions of QM, and that nonlocality of QM
cannot be attributed to incompletness [40]. The work on nonlocality in QM
initiated formulations of different nonlocal theories of QFT. In the earlier
work summarized in [41], the nonlocal theories were constructed by either
introducing a new universal length that allows for a slight violation of
locality on small distances, or defining a constant that characterizes the
domain of nonlocal interactions. In more recent work, nonlocal QFT for scalar
fields were constructed [42,43].

In the RQM theory presented in this paper, the scalar-mass is responsible for
the time evolution of the wavefunction, similar to standard RQM; however, the
vector-mass affects only the spatial distribution of the wavefunction, which
makes our theory a fully symmetric theory and, thus, different than those
local and nonlocal theories previously constructed [41-43]. We identify the
spacelike nature of the wavefunction spatial distribution with nonlocality in
our theory and suggest that due to the presence of vector-mass, the particle
may potentially interact nonlocally with another particle by overlapping the
spatial distributions of their wavefunction. Since our theory considers only
one free, massive and spin zero elementary particle, such nonlocally
interacting particles are out of the scope of this paper.

In this reformulated RQM, the scalar-mass is responsible for the the
particle’s world-line being within the light cone. Moreover, a particle of
definite mass and position in space must be in a superposition of different
vector-masses, which means that $\boldsymbol{k}$ does not play any role in the
motion of the particle. As a result, $M^{2}$ given by Eq. (4) is not the mass
of the particle and the theory allows only for particles with positive $m^{2}$
term, which prevents tachyons and tachyonic fields [31,32] from being defined
in this theory.

The vector-mass, as the spacelike component responsible for the nonlocal
behavior of quantum particles, is the true conjugate variable of position in
$\mathbb{R}^{3}$ space, thus, the canonical commutation relations given by Eq.
(5) can be used to write the uncertainty principle in the following form

$\prod_{i=1}^{3}\Delta\xi^{i}\Delta k^{i}>1\ .$ (43)

There is also uncertainty of the scalar-mass due to the presence of
$\boldsymbol{k}$ and this uncertainty is inversely proportional to particle’s
lifetime, then the resulting uncertainty relation becomes

$\Delta\tau\Delta m>1\ ,$ (44)

and it describes virtual particles that are present in this reformulated RQM;
note that according to our theory only particles with infinite lifetime are
stable and observable. The presence of virtual particles in the theory leads
to a novel picture of vacuum that is now briefly described.

From a point of view of RQM and QFT, vacuum is defined as the state of lowest
energy filled with virtual particles spontaneously coming out of the void and
disappearing in it (e.g., [44,45]), with the notion that known elementary
particles may form pairs with their antiparticles and become virtual on the
time and energy scales corresponding to these particles (e.g., [46-48]). The
theory presented in this paper and its uncertainty relation given by Eq. (44)
shows masses of virtual particles do not have to be limited by experimentally
established masses of known elementary particles, as virtual particles may
have any mass, including masses of all known particles, which are required by
this theory to be stable and detectable if, and only if, their lifetimes are
infinititely long.

##  8 Conclusions

Standard RQM that is based on the timelike intervals is reformulated by taking
into account both the timelike and spacelike intervals. The presence of the
spacelike intervals in the reformulated RQM modifies its linear momentum by
including its spacelike eigenvalues, in addition to timelike eigenvalues. The
theory is developed for a free, massive, and spin zero elementary particle in
the Minkowski spacetime, where the particle is called elementary if its
wavefunction transforms as one of the irreps of the Poincaré group. The STR
energy-momentum relationship of standard RQM with its scalar-mass is modified
by an additional term with its vector-mass, which only affects the spatial
distribution of particle’s wavefunction. The physical consequences of the
presence of the vector-mass in the energy-momentum relationship are far
reaching, starting with the modification fo the mass term in the Klein-Gordon
equation, and then leading to nonlocality, some modifications of the
uncertainty principle, and a novel picture of vacuum. Moreover, the developed
RQM theory preserves causality, is self-consistent, and is formulated using
the path integral formalism, which correctly reproduces the classical action
of a relativistic particle; however, the theory does not allow for tachyons
and tachyonic fields. The theory is not a hidden-variable theory; its physical
interpretation is consistent with the Copenhagen interpretation of QM, and
preserves causality.

Data Availability The data that supports the findings of this study is
available within the article.

Acknowledgments We are grateful to two anomynous referees for their comments
and suggestions that allowed us to improved the original version of this
paper. We also thank Lesley Vestal for reading our manuscript and commenting
on it.

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