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An Implementation of a Quantized Line Element in the

Context of the Klein-Gordon and Dirac Theories

D.L Bulathsinghala∗, K.A.I.L Wijewardena Gamalath†

Department of Physics, University of Colombo,

Colombo 3, Sri Lanka.

August 26, 2015

Abstract

In this paper, we examine an ansatz, where anti-commutation rules hold only as integrated averages over time

intervals, and not at every instant, giving rise to a time-discrete form solution to the Klein-Gordon equation. This

coarse-grained validation of the anti-commutation rules enables us to show that the relativistic energy-momentum

relation holds only over discrete time intervals (or at discrete time instants), ﬁtting well with the known time-energy

uncertainty relation. How this time-discrete scheme, when applied to 4-vector notation in relativity, could quantize

the line-element and thereby quantizing the physical attributes associated with time, space and matter is sketched. By

associating this time-discrete energy-momentum conservation, with the conservation of physical quantities of a particle

in relativistic motion, we identify particle-instants, separated by particle-intervals and thereby show that, more precisely

one determines a particle’s physical attributes associated with particle-instants, the less precisely its attributes associated

with particle-intervals can be known, and vice-versa, and how this scheme could potentially be extended to discuss the

Zeno’s arrow paradox within the classical limit is presented. As the solutions of the Dirac equation can be used to

construct solutions to the Klein-Gordon equation, we apply this temporal quantization rule to the Dirac equation and

ask how it interprets the solutions associated with the Dirac equation under such conditions. Finally, we introduce the

general relativistic effects to a line-element associated with a particle in relativistic motion and obtain a time quantized

line-element associated with gravity.

Keywords: Time quantization, Klein-Gordon equation, Dirac equation, Quantum gravity, Energy-momentum equa-

tion, Momentum-position uncertainty relation, Time-energy uncertainty relation, Continuity equation, Probability den-

sity, Probability current, Chronon, Zeno’s arrow paradox, Zeno’s Achilles and the tortoise paradox.

PACS: 04.60.-m, 03.70.+k, 03.50.-z

1 Introduction

While time is being treated as a continuous quantity in both quantum mechanics and relativity, many have suggested that a quantized

time model may help to produce a theory of quantum gravity. Chronon, a quantum of time, that is discrete and indivisible unit of time

was proposed by Lvi in 1927 [1]. In the early days of quantum theory, as a way to eliminate inﬁnities from quantum ﬁeld theory, the

idea of a quantum space time as a generalization of the usual space-time concept in which, some variables that ordinarily commute

are assumed not to commute, forming a different Lie algebra, was proposed by Heisenberg and Ivaneko. Using Lie algebra, Lorentz

invariant discrete space-time was then published by Snyder in 1947. Simplifying Snyder ’s Lie algebra, a quantum theory in which the

time is a quantum variable with a discrete spectrum, consistent with the special relativity was proposed by Yang [2] in the same year.

Margenau in 1950 suggested [3], that the chronon might be the time for light to travel the classical radius of an electron. A prominent

model was later introduced by Piero Caldirola in 1980 [4], giving a formula to obtain the value for the chronon, a quantization of the

evolution in a system along its world-line, which is much longer than the Planck-time, a universal quantization of time itself. Since

then, various theorists have done further research into time quantization or the chronon model. Sam Vaknin proposed that the time

is the result of the interaction of the chronons in the time ﬁeld, and the time asymmetry is generated as we observe them [5]. In order

to ﬁnd the time, that will be independent of one’s choice of coordinates, an upper limit of measurable time from an event, back to

the big-bang singularity was proposed by Suchard [6]. Tifft put forward the idea that the observed redshift quantization of galaxies

could possibly explain an underlying time quantization scheme [7]. A more recent developments on time quantized models are based

∗dinesh.b@phys.cmb.ac.lk

†imalie.gamalath@sci.cmb.ac.lk

1

on quasi-local invariant observables by Elze [8] and a perturbative approach to quantization by Barbero et.al [9]. Kitada [10] had

shown that the quantum mechanical clock is equivalent to the relativistic classical clock. A quantized simple cosmological model was

introduced by Halliwell [11], while Warner et.al [12] introduced the geometrodynamic quantization.

According to the Zeno’s arrow paradox, generally thought to have been devised by ancient Greek philosopher Zeno from Ela, if

everything is motionless at every instant, and time is entirely composed of many such instants, then motion is impossible. Further

investigation into this paradox reveals that if a moving arrow is motionless or at rest for every temporal instant, then by comparing it

with an arrow which is at rest for all time, one cannot distinguish between the two (arrows), at a given instant, which also suggests

that motion must be an illusion. Further, contrary to the view that treats the observable quantities arising from 4-vector notation in

relativity as conserved physical quantities at all instants, deﬁning a conserved momentum at all instants is problematic as the very

notion of the momentum requires a ﬁnite displacement over a ﬁnite time interval. Therefore, at a given instant, when conserved

physical quantities associated with a moving body and that of a body at rest are compared, there appears to be no concrete evidence

to suggest that one can be distinguished from the other. To resolve this problem, Bertrand Russell [13] offered what is known as “at-at

theory of motion”, where it agrees that there can be no motion during a duration-less instant, and states that all that is required for the

motion is that the arrow be at one point at one time and at another point another time. Peter Lynds [14] suggested a different solution,

that the whole notion of an instant is ﬂawed and argued that there really aren’t any instances, but intervals. The other notable efforts

to resolve this paradox came from the assumption that the space is comprised of ﬁnite and discrete units, but the reconciliation of that

view with the relativity, where space is being treated as a continuous quantity, was problematic.

In quantum gravity, the quest of the research is to seek a description of gravity, or more precisely the principles governing gravity,

in accordance with the principles of quantum mechanics. As our current understanding of gravity is based on the relativity proposed

by Einstein, which is a formulation within the framework of classical physics, reconciliation requires that the classical view of space

and time being continuous requires some radical changes. However, difﬁculties arise when one attempts to introduce the usual

quantization techniques, as it becomes non-renormalizable and therefore cannot be used to make any meaningful predictions. As

a direct consequence, theorists have taken up more radical approaches to the problem of quantum gravity, with the most popular

among them being string theory and loop quantum gravity.

In this paper, the possibility of time quantization is considered as an intriguing question that may have relevance to quantum-

gravity, as well as foundations of quantum mechanics. One way to investigate such questions in a more limited way, rather than

within the framework of quantum ﬁeld theory, is to return back to the simpler relativistic quantum mechanics that predate ﬁeld

theory. Hence we return to the Klein-Gordon and Dirac equations and examine an ansatz, where anti-commutation rules hold only

as integrated averages over time intervals (or at discrete time instants), and not at every instant. This coarse-grained validation of

the anti-commutation rules, when applied to the Klein-Gordon equation, shows that the energy-momentum relation holds only over

discrete time intervals (or at discrete time instants), consistent with the time-energy uncertainty phenomenon in quantum mechanics.

The procedure to be followed in this paper is (a). to obtain a general ﬁrst order inner-product solution to the Klein-Gordon equation,

(b). introducing an ansatz, where the anti-commutation rules hold only as integrated averages over discrete time intervals (or at

discrete time instants), giving a temporally quantized solution to the Klein-Gordon equation, (c). use this rule to show that the

observable quantities associated with 4-vector notation in relativity are only conserved within this time-discrete scheme and thereby

quantizing the line-element, (d). by associating this time-discrete scheme of the energy-momentum relation, with the conservation

of physical quantities of a particle in motion, we identify particle-instants (energy-momentum relation holds at these discrete time

instants), separated by particle-intervals (energy-momentum relation holds over these discrete time intervals when averaged) and

discuss how this scheme can potentially resolve the Zeno’s Arrow paradox and the Achilles and the tortoise paradox within the

classical limit, (e). as the solutions of the Dirac equations are automatically solutions to the Klein-Gordon equation, explore how this

time-quantization scheme interprets the Dirac equation and its solutions and (f). show how this time-discrete scheme can be extended

to a line-element associated with gravity and discuss how the ansatz presented in this paper could potentially be further investigated

as a possible approach to quantum gravity.

2 A general ﬁrst order inner-product solution to the Klein-Gordon equa-

tion

In quantum mechanics the correspondence principle dictates that the momentum operator is associated with the spatial gradient,

and the energy operator with the time derivative. Since the Schrodinger equation [15] cannot be relativistically correct, as it does not

contain space and time derivatives of the same order, historically the ﬁrst attempt to construct a relativistic version of the Schrodinger

equation began by applying the familiar quantization rules to the relativistic energy-momentum invariant PµPµ:

PµPµ=E

c,pE

c,−p=E2

c2−p2= (mc)2(1)

where Pµis the four momentum, Eis the energy, pis the three dimensional momentum and mis the rest mass of the particle.

Applying the energy and momentum operators to Eq. 1, the Klein-Gordon equation is obtained [16], [17]:

−~2∂2

∂t2ψ+ (~c)2∇2ψ=mc22ψ(2)

∇2−1

c2

∂2

∂t2ψ=mc

~2ψ(3)

2

For any general plane-wave solution, we can show that the following set of conditions satisfy, for µ= 0,1,2,3where i,jare distinct

(i, j = 1,2,3):

h∂µψ, ∂µψi=−∂2

µψ, ψ(4)

h∂iψ, ∂jψi=−h∂i∂jψ , ψi(5)

In order to construct a ﬁrst-order inner-product solution that satisﬁes the Klein-Gordon equation, ﬁrst let us take the inner-product

of Eq. 3 with its corresponding plane-wave ψ, where ∂0=1

c

∂

∂t :

∇2−∂2

0ψ, ψ=mc

~2ψ, ψ(6)

and use Dirac’s heuristic procedure that leads to a more conventional scalar-product solution associated with ﬁrst-order operators

given by:

∇2−∂2

0ψ= (Ai∂i+ iD∂0) (Ai∂i+ iD∂0)ψ(7)

satisfying:

A2

i=D2= 1 (8)

and satisfying the anti-commutation relations:

{Ai, Aj}=AiAj+AjAi= 0 (9)

{D, Ai}=DAi+AiD= 0 (10)

where iand jare distinct (i, j = 1,2,3). Thus, Eq. 6 reads:

h(Ai∂i+ iD∂0) (Ai∂i+ iD∂0)ψ , ψi=mc

~2hψ, ψi(11)

to which, by applying the identities in Eqs. 4 and 5, we can construct a general ﬁrst-order inner-product solution that satisﬁes the

Klein-Gordon equation, where Ai= iγiand D =γ0correspond to Dirac matrices.

Diγi∂iψ+ iγ0∂0ψ, iγi†∂iψ+ iγ0†∂0ψE=mc

~2hψ, ψi(12)

As the cross-terms associated with anti-commutation rules for gamma-matrices vanish for every temporal instant:

γi, γj=γiγj+γjγi= 0 (13)

γ0, γi=γ0γi+γiγ0= 0 (14)

Eq. 12 satisﬁes the Klein-Gordon equation at all instants.

3 A temporally quantized form of the Klein-Gordon equation

A general ﬁrst-order inner-product solution that satisﬁes the Klein-Gordon equation, whose cross terms associated with anti-commutation

rules vanish for every temporal instant was obtained in the previous section. On the other hand, if we could introduce some sort of

a rule that would make at least one or a few of these cross terms to vanish when integrated over a periodic temporal interval, then

the solution that satisﬁes the 2nd order relativistic wave equation is said to be time-discrete or temporally quantized. Thus, let us

introduce an ansatz: a particle in relativistic motion is associated with a pair of complementary plane waves, such that its temporal

derivative operator is associated with a plane wave ψand its spatial derivative operators are associated with its complementary plane

wave ψC, where ψC=ψ∗:

−1

c2

∂2

∂t2ψ+∇2ψC=mc

~2ψ(15)

to which by applying the general ﬁrst-order inner-product solution we derived in previous section, we obtain:

Diγi∂iψC+ iγ0∂0ψ, iγi†∂iψC+ iγ0†∂0ψE=mc

~2hψ, ψi(16)

Expanding the inner-product:

γi2h∂iψC, ∂iψCi+γ02h∂0ψ, ∂0ψi+1

2γiγj+γjγi(h∂iψC, ∂jψCi+h∂jψC, ∂iψCi) +

Diγ0∂0ψ, iγi†∂iψCE+Diγi∂iψC,iγ0†∂0ψE=mc

~2hψ, ψi(17)

and applying γ02=−γi2= 1 and the anti-commuting conditions in Eqs. 13 and 14, and the identity in Eq. 4, we obtain:

3

∇2ψC, ψC−∂2

0ψ, ψ+Diγ0∂0ψ , iγi†∂iψCE+Diγi∂iψC,iγ0†∂0ψE=mc

~2hψ, ψi(18)

As ψand ψCare complementary plane waves, the anti-commutation rule between the terms iγ0∂0ψand iγi∂iψCgives us:

Diγ0∂0ψ, iγi†∂iψCE+Diγi∂iψC,iγ0†∂0ψE=γ0γiω

ck(hψ, ψCi−hψC, ψi)(19)

By assigning (a) ψ= e+i(ωt−k.r)and ψC=ψ∗or (b) ψ= e−i(ωt−k.r)and ψC=ψ∗, we arrive at:

Diγ0∂0ψ, iγi†∂iψCE+Diγi∂iψC,iγ0†∂0ψE=γ0γiω

ck(2i) sin (2Ω) (20)

where:

Ω = ωt −k.r(21)

However, the anti-commutation reported in Eq. 20 does not hold for every temporal instant, as:

γ0γiω

ck(2i) sin (2Ω) 6= 0 (22)

for every Ω(or t) and it can be shown to hold only (a). at discrete time instants corresponding to Ω = nπ

2where nis a pos-

itive integer, and (b). when integrated over a periodic temporal interval corresponding to an angular interval of πradians, where

Rπsin (2Ω) dΩ=0. Further, in 4-vector notation, when one integrates the associated terms, one must be careful in selecting a suitable

differential. In this case, we select an invariant differential, so that it can be applied across all the terms associated with Eq. 18. By

selecting dΩ = d(KµXµ)as the invariant differential, where:

Ω = KµXµ=ωt −k.r=ω0τ(23)

and Kµis the 4-wave vector and Xµthe 4-position vector, we can show that the anti-commutation between iγ0∂0ψand iγi∂iψC

now holds, when integrated over a periodic interval π:

ZπDiγ0∂0ψ, iγi†∂iψCE+Diγi∂iψC,iγ0†∂0ψEdΩ = γ0γiω

ck(2i) Zπ

sin (2Ω) dΩ=0 (24)

Thus, by taking the integrated average of all the terms reported in Eq. 18 with respect to invariant differential dΩover a periodic

interval π:

1

πZπ∇2ψC, ψCdΩ−1

πZπ∂2

0ψ, ψdΩ+

1

πZπDiγ0∂0ψ, iγi†∂iψCE+Diγi∂iψC,iγ0†∂0ψEdΩ = 1

πZπmc

~2hψ, ψidΩ(25)

and applying the periodic condition reported in Eq. 24, we arrive at:

∇2ψC, ψC−∂2

0ψ, ψ=mc

~2hψ, ψi(26)

Selecting ψCand ψ, such that ∇2ψC=−k2ψCand ∂2

0ψ=−E2

~2c2ψand observing the conditions hψC, ψCi= 1 and hψ , ψi= 1:

−k2+E2

~2c2=mc

~2(27)

we arrive at the relativistic energy-momentum relation:

E2−p2c2= (mc2)2(28)

As the angular periodic interval πcorresponds to a temporal-interval associated with particle’s relativistic energy 4t=π

ω, a

time quantized solution to the Klein-Gordon equation, associated with two complementary waves is obtained, which satisﬁes the

energy-momentum relation (a). when one takes the integrated average over discrete time intervals corresponding to an angle π

radians, given by:

1

πZπDiγi∂iψC+ iγ0∂0ψ, iγi†∂iψC+ iγ0†∂0ψEdΩ = 1

πZπmc

~2hψ, ψidΩ(29)

and (b). at angular instants corresponding to Ω = nπ

2.

4 Quantization of the Line Element

In the theory of relativity, a four-vector Aµ(µ= 0,1,2,3), is a four dimensional vector in Minkowski space-time deﬁned as a quantity

which transforms under Lorentz transformations in the same way as coordinates of a point (x0, x1, x2, x3), giving rise to a scalar

product which is invariant under Lorentz transformation. By starting from the temporally quantized solution to the Klein-Gordon

equation, for a particle in relativistic motion given in Eq. 29, and multiplying out by ~2

m2which is invariant:

4

~2

m21

πZπDiγi∂iψC+ iγ0∂0ψ, iγi†∂iψC+ iγ0†∂0ψEdΩ = ~2

m21

πZπmc

~2hψ, ψidΩ(30)

~2

m2−k2+E2

~2c2=~2

m2mc

~2(31)

we obtain:

−γ2u2+ (γc)2=c2(32)

to which by applying dτ =dt/γ:

−dr2+c2dt2=c2dτ 2⇒dxµdxµ=c2dt2−dr2(33)

we obtain the square of the line-element associated with a particle in relativistic motion, which is time-discrete:

ds2=c2dt2−dr2(34)

which means the relation reported in Eq. 34 only holds (a) at discrete time instants and (b) over discrete time intervals, when

averaged. Using this time-discrete line-element, a time-discrete differential 4-position vector can be obtained:

dxµ=cdt dx dy dz (35)

whose space-time differential elements, namely time and space are time-discrete or time-quantized, over an interval 4t=π

ω,

where ω=γmc2/~. By integrating the terms in Eq. 35, the classical 4-position vector is obtained, whose components are also

time-discrete.

Xµ=ct x y z (36)

Similarly, multiplying out Eq. 29 by ~2, one can obtain the conventional 4-momentum vector whose components, namely energy

and momentum, are also time-discrete over 4t:

Pµ=E

cpxpypz(37)

Hence, the above work suggests that (a) the line-element associated with a particle in relativistic motion is time-discrete, (b) time,

space and matter components associated with a particle in relativistic motion are time-discrete and thereby (c) the other observable

quantities associated with time, space and matter components in 4-vector notation are conserved only over discrete time-intervals (or

at discrete time instants) in accordance with the principles of quantum mechanics, and (d) the time-quantum 4tassociated with this

quantization-scheme depends on the relativistic particle’s matter-wave angular frequency ω. Further, the periodic angular interval π,

which corresponds to a time-quantum 4tover which the conservation of the observable quantities hold, can be regarded as a chronon

which can be expressed in terms of a particle’s total-energy:

4t=π

ω=h

21

γmc2(38)

Taking a limiting case where the maximum energy a particle can possess is bound by the planck-energy EP, we now can derive

an absolute smallest time-quanta or chronon, for a particle associated with planck-energy:

(4t)min =1

2

h

EP

=1

2tP(39)

where tPrepresents the planck-time:

tP= 5.39106 ×10−44s(40)

4.1 Zeno’s arrow paradox

Many centuries ago, a Greek philosopher named Zeno from Ela proposed that, a ﬂying arrow is at rest, at each point of time, during

its ﬂight and comparing this ﬂying arrow with an arrow at rest, at each point of time (or instants), one cannot distinguish them apart,

thus he argued that motion must be an illusion.

In order to resolve this paradox, Peter Lynds [14] suggested that there really aren’t any instants but intervals. However, as the

very boundary, where the intervals are combined together, gives rise to an instant, the validity of his claim that there are no instants,

would collapse. The other notable effort came from Bertrand Russell [13], with his “at-at theory of motion”, in which he agrees that

there can be no motion during a duration-less instant, and states that all that is required for the motion is that the arrow be at one point

at one time and at another point another time. However, a closer investigation into his claim suggests that, if one can deﬁne inﬁnitely

many instants, in-between two given ﬁnite instants, then motion becomes impossible, as the arrow, now needs to be at rest at all such

in-between instants (inﬁnitely many), and the very notion of motion then becomes problematic, i.e. while the arrow is in motion,

during however small interval, one can ﬁnd yet another sub set of instants during the said ﬂight interval, where at those instants

the arrow must be at rest. Thus, it implies that, unless there is an absolute minimum temporal separation between two consecutive

instants (where there are no further in-between instants or sub-divisions), the “at-at theory” would collapse.

5

Let us now re-visit, the work presented in the previous chapters, where, by investigating a new ansatz (a particle in a relativistic

motion is associated with a pair of complementary waves), we showed that the energy-momentum relation holds over discrete time

intervals (or at discrete time instants), and thereby the observable physical quantities associated with 4-vector notation, are also

conserved over discrete time intervals (or at discrete time instants). Further, if one were to associate the conservation of the energy-

momentum relation of a particle in relativistic motion (or the conservation of its physical quantities associated with 4-vector notation),

with how we deﬁne the particle itself, then the time-discrete scheme presented in this paper suggest that, we can only talk about a

particle (a). at discrete time instants and (b). over discrete time intervals, and that it is not proper to state that the particle exists or can

be deﬁned (based on the conservation of its physical quantities) for all time instants, rather we have particle-instants (conservation

holds at these discrete time instants), separated by particle-intervals (conservation holds over these intervals).

In light of this interpretation, that the conservation of observable physical quantities (associated with 4-vector notation) are only

held over discrete time-intervals (when averaged) or at discrete time instants, let us re-examine the Zeno’s arrow paradox stated

above. Contrary to Lynds’ claim that there aren’t really any instants but intervals, the time-discrete scheme discussed above suggests

that, it is the conservation of the observable physical quantities which are time-discrete, and not the time itself. Further, this scheme

prohibits us from deﬁning a set of conserved physical quantities attributed to an arrow, or more precisely to a particle, at all instants,

but rather for discrete time-intervals and at discrete time-instants, thus the conservation of the physical quantities that deﬁnes the

arrow (or the particle) then becomes discrete, enabling us to resolve the paradox within the notion of continuous time, consistent with

classical mechanics. Additionally, the time instants where the conservations hold, can be identiﬁed as duration-less instants or points,

where the arrow (or the particle) is said to be at rest, whereas during the intervals in between such instants (where the conservation

holds when averaged), the arrow can be regarded as in state of motion.

Similarly, Zeno’s Achilles and the tortoise paradox states, in order for Achilles to overtake the tortoise, he must reach inﬁnite

number of points, where tortoise has already been, thus he can only get ever so closer to the tortoise, but cannot overtake it. Thus, if

space is inﬁnitely divisible, according to this paradox, Achilles will have to perform inﬁnite number of steps, before overtaking the

tortoise. When investigated further, this paradox questions, whether a ﬁnite continuum can be sub divided to give inﬁnite number

of positions, therein. Through out the history, several solutions have been proposed to resolve this paradox, notable among them

are the works by Aristotle [18], Thomas Aquinas [19], Bertrand Russell [13] and Hans Reichenbach [20] to name a few. However, the

time-discrete scheme presented in this paper (that the particle’s deﬁnition is associated with the conservation of its physical quantities,

in a time-discrete manner), clearly suggests that, one cannot keep sub dividing the space, inﬁnitely many times, and expect Achilles

(or a particle) to reach each of those inﬁnitely many points at different instants consecutively, as the very deﬁnition of Achilles (or

the particle) now only exists or can be deﬁned (a). at discrete time instants and (b). over discrete time intervals, i.e. if we consider

any moment in between two consequative instants where the conservations hold, Achilles (or the particle) now will not obey the

conservations, but rather obeys them over discrete time intervals (as a whole), when one takes the integrated average of its physical

quantities. Therefore, it is not proper to state, that Achilles (or the particle) must traverse through all those inﬁnitely many possible

points consecutively (at different instants), before overtaking the tortoise, as the very existence or the conservations that deﬁnes

Achilles (or the particle) now has become discrete. That is, Achilles (or the particle) now cannot claim to exist at each and every

possible consecutive spatial points, one after the other. Thus, we are able to show that, a resolution to this centuries old paradox is

possible, within the notion of continuous time and space, consistent with classical mechanics, when one investigates the time-discrete

scheme presented in this paper.

4.2 Momentum-position uncertainty

One of the most important consequences of the wave-particle duality of nature was discovered by Werner Heisenberg in 1927, and is

called the uncertainty principle, which states: “The more precisely the position of some particle is determined, the less precisely its

momentum can be known, and vice-versa”.

In this paper, by investigating an ansatz (where a particle in relativistic motion is associated with a pair of complimentary waves),

we showed that the conservation of the energy-momentum relation holds at discrete time instants or over discrete time intervals (when

averaged). Then, by associating the conservation of a particle’s physical attributes with the conservation of its energy-momentum

relation, we showed that its conservations become discrete, and obtained a discrete set of particle-instants (conservations hold at

these instants) separated by particle-intervals (conservations holds over these intervals). Finally, we identiﬁed those instants as space-

time points, where the particle is said to be at rest, whereas we associated the particle’s motion with the particle-intervals. This also

suggests, that one cannot know, both the position and the momentum of a particle simultaneously, as it infers that, the particle’s

position can only be known precisely, at particle-instants, where the particle is presumed to be at instantaneous rest. However,

as there can be no motion, during this duration-less instant, the momentum of the particle at this instant becomes undetermined.

Similarly, during the particle-intervals, we cannot ﬁnd particle instants, thus one looses the precision of the particle’s position, while

its momentum is observed.

4.3 Time-energy uncertainty

The time-discrete scheme we examined above shows that the total energy associated with a particle in relativistic motion is conserved

only over discrete time-intervals, thus a measure of a particle’s energy content Eover its conserved discrete time-interval 4tyields a

classical product between them:

E4t=γmc2h

21

γmc2=⇒E4t=h

2(41)

6

In other words, the time-discrete conservation of energy give rise to a type of relativistic time-energy uncertainty. However,

based on the interpretation given for uncertainty principles in quantum mechanics, the two observables associated in an uncertainty

relation must arise from two operators which satisfy the canonical-commutation rule. As time is not considered a quantum mechanical

operator, there has been some confusion with regards to the time-energy uncertainty relation in quantum mechanics. Nevertheless,

in 1945, Mandelshtam and Tamm derived a non-relativistic time-energy uncertainty [21] for a quantum system in a stationary state,

taking into account the life-time of the state to change its expectation value appreciably.

The confusion related to time not being considered a quantum mechanical operator does not arise with the time-discrete scheme

we examined in this paper, as we relate the terms in the time-energy uncertainty relation to a particle’s total energy and a discrete

time-interval over which its energy is conserved. This suggests that the time-energy uncertainty relation, is not a statement related to

the observer-effect or the measurement-disturbance based on the precision of a measurement, but perhaps, a fundamental statement

arising from the time-discrete scheme we examined in this paper.

5 Time quantized spinors

5.1 Spinors associated with the classical Dirac equation for a particle in

relativistic motion

The classical Dirac equation Eψ −(αipic)ψ=βmc2ψ[22], can be written in the form of a 4-component wave-function Ψ:

E−mc21 0

0 1 −cσ.p

−cσ.p E+mc21 0

0 1

Ψ=0 (42)

by introducing one possible set of solutions given by:

αi=0σi

σi0;β=I20

0−I2(43)

where σis are Pauli matrices, given by:

σ1=0 1

1 0 ;σ2=0−i

i 0 ;σ3=1 0

0−1(44)

and the wave-function Ψis associated with a constant 4-spinor u, which is associated with 2 coupled 2-spinors uAand uB:

Ψ = uψ;u=uA

uB(45)

uA=u1

u2;uB=u3

u4(46)

Expanding Eq. 42 and considering its positive-energy solutions:

(cσ.p)uAψ=E+mc2uBψ(47)

where cσ.p is given by:

cσ.p =cpzpx−ipy

px+ ipy−pz(48)

we can obtain an expression for uB, in terms of uA:

uB=c

(E+mc2)pzpx−ipy

px+ ipy−pzuA(49)

Now, by making a choice for uAbetween uA=1

0or uA=0

1, we obtain the following set of positive energy solutions

for the wave-function, where the term N=√E+mc2is a normalized factor, normalizing Ψ†Ψ=2E.

Ψ1=pE+mc2

1

0

cpz

(E+mc2)

cpx+icpy

(E+mc2)

ψ; Ψ2=pE+mc2

0

1

cpx−icpy

(E+mc2)

−cpz

(E+mc2)

ψ(50)

Similarly by considering the negative-energy solutions in Eq. 42:

(cσ.p)uBψ=E−mc2uAψ(51)

and repeating the above procedure to ﬁnd an expression for uAin terms of uB, where uB=1

0or uB=0

1gives:

7

uA=c

(E−mc2)pzpx−ipy

px+ ipy−pzuB(52)

Using the conventional “Feynman-Stueckelberg” interpretation [23], where “A negative energy solution with E < 0represents

a negative energy particle, traveling backwards in time, or equivalently, a positive energy particle going forward in time”, we deﬁne

the anti-particle states by just ﬂipping the signs of Eand p, thus Eq. 52 reads:

uA=c

(−E−mc2)−pz−px+ ipy

−px−ipypzuB=⇒uA=c

(E+mc2)pzpx−ipy

px+ ipy−pzuB(53)

which gives us the following set of negative energy solutions, where the associated wave with ﬂipped Eand psigns becomes ψ∗:

Ψ3=pE+mc2

cpz

(E+mc2)

cpx+icpy

(E+mc2)

1

0

ψ∗; Ψ4=pE+mc2

cpx−icpy

(E+mc2)

−cpz

(E+mc2)

0

1

ψ∗(54)

This enabled Dirac to predict the existence of an anti-electron or the positron in 1931 [24]. The solutions in Eq. 50 are associated

with spin-up and spin-down positive-energy particles, whereas the solutions in Eq. 54 are associated with spin-up and spin-down

negative-energy particles.

5.2 Time quantized spinors for a particle in relativistic motion

First, let us consider the Dirac equation associated with complementary waves: E ψ −(αipic)ψC=βmc2ψ, which can be written in

terms of a 4 component spinor u=uA

uB:

E−βmc2uA

uBψ= (αipic)uA

uBψC(55)

By using αiand β, and Pauli matrices σiand re-arranging the terms in Eq. 55, we obtain:

E−mc21 0

0 1 uAψ−cσ.puBψC

−cσ.puAψCE+mc21 0

0 1 uBψ

= 0 (56)

whose positive-energy solutions are given by:

E+mc21 0

0 1 uBψ=cσ.puAψC(57)

and the negative-energy solutions are given by:

E−mc21 0

0 1 uAψ=cσ.puBψC(58)

Applying the “Feynman-Stueckelberg” interpretation to the negative-energy solutions in Eq. 58, where the signs of Eand pare

ﬂipped, ψ→ψCand ψC→ψ, we can construct the following relation:

−E−mc21 0

0 1 uAψC=cσ.(−p)uBψ=⇒E+mc21 0

0 1 uAψC=cσ.puBψ(59)

Thus, by combining the positive-energy solutions in Eq. 57and the negative-energy solutions in Eq. 59, we can construct:

E+mc21 0

0 1 −cσ.p

−cσ.p E+mc21 0

0 1

Ψ=0 (60)

where the spinors associated with the time-quantized solution now take the following form:

Ψ = uAψC

uBψ;uA=u1

u2;uB=u3

u4(61)

When Eq. 60, which is a time-quantized equation, is expanded and its positive-energy solutions are considered, we simply cannot

write down a relation similar to Eq. 47 satisfying for every temporal instant, given by:

E+mc2uBψ= (cσ.p)uAψC(62)

as time-quantized equations hold only when integrated over discrete periodic time intervals (or at discrete time instants). How-

ever, by making use of the following identities:

Z+π

2

−π

2

e−iΩdΩ = Z+π

2

−π

2

e+iΩdΩ(63)

8

Z+3π

2

+π

2

e−iΩdΩ = Z+3π

2

+π

2

e+iΩdΩ(64)

we now can construct a relationship of the form given in Eq. 47 which holds (a). when integrated over angular periods of π

radians:

ZπE+mc2uBψdΩ = Zπ

(cσ.p)uAψCdΩ =⇒E+mc2uB= (cσ.p)uA(65)

or (b). at discrete angular instants, satisfying ψ=ψC, where Ω = nπ:

E+mc2uBψ= (cσ.p)uAψC=⇒E+mc2uB= (cσ.p)uA(66)

We then end up with the following solution for uBin terms of uA,

uB=c

(E+mc2)pzpx−ipy

px+ ipy−pzuA(67)

By making a choice for uAbetween uA=1

0or uA=0

1, we can now obtain the following set of positive-energy

solutions, where the solutions satisfy the Eq. 55 when integrated over discrete intervals (or at discrete instants), where N=√E+mc2

is a normalized factor, normalizing Ψ†Ψ=2E.

Ψ1=N

1

0

cpz

(E+mc2)

cpx+icpy

(E+mc2)

; Ψ2=N

0

1

cpx−icpy

(E+mc2)

−cpz

(E+mc2)

(68)

Similarly, by repeating this procedure for the negative energy solutions reported in Eq. 58, we obtain the following set of negative-

energy solutions, which satisfy the Eq. 55 over discrete intervals (or at discrete instants).

Ψ3=N

cpz

(E+mc2)

cpx+icpy

(E+mc2)

1

0

; Ψ4=N

cpx−icpy

(E+mc2)

−cpz

(E+mc2)

0

1

(69)

6 Conservation of probability current

Starting from the classical Dirac equation Eψ −(αipic)ψ=βmc2ψ, where γ0=βand γk=γ0αk:

iγµ∂µ−mc

~Ψ=0 (70)

one can obtain the conservation of the Dirac current, where ∂µ=1

c

∂

∂t ,∇:

∂µ¯

ΨγµΨ= 0 (71)

and the 4-current jµ:

jµ=¯

ΨγµΨ= (cρ, j)(72)

where ρis the probability density and jis the probability current. The probability density ρ, which is positive deﬁnite is given by:

j0=¯

Ψγ0Ψ(73)

As ¯

Ψγ0=¯

Ψβ= Ψ†:

j0= Ψ†Ψ(74)

similarly as αk=γ0γµ:

¯

ΨγµΨ = ¯

Ψγ0γ0γµΨ=Ψ†αkΨ(75)

Thus the continuity equation associated with the classical Dirac equation reads:

∂µ¯

ΨγµΨ= 0 (76)

1

c∂tΨ†Ψ+

3

X

k=1

∂kΨ†αkΨ= 0 (77)

However, by considering one of the solutions reported in Eq. 68 that satisﬁes the time-quantized Dirac equation in Eq. 55, (a).

over discrete intervals and (b). at discrete instants:

9

Ψ1=N

1

0

cpz

(E+mc2)

cpx+icpy

(E+mc2)

(78)

let us calculate the probability density associated with Ψ1:

cρ =Ψ†

1Ψ1(79)

cρ =pE+mc21 0 cpz

(E+mc2)

cpx−icpy

(E+mc2)pE+mc2

1

0

cpz

(E+mc2)

cpx+icpy

(E+mc2)

(80)

cρ =E+mc2 1+0+cpz

E+mc22

+(cpx+ icpy) (cpx−icpy)

(E+mc2)2!(81)

cρ = 2E(82)

Similarly, by expanding the probability current term j1=Ψ†

1α1Ψ1:

Ψ†

1α1Ψ1=pE+mc21 0 cpz

(E+mc2)

cpx−icpy

(E+mc2)

0001

0010

0100

1000

pE+mc2

1

0

cpz

(E+mc2)

cpx+icpy

(E+mc2)

(83)

Ψ†

1α1Ψ1=E+mc2cpx−icpy

(E+mc2)+0+0+ cpx+ icpy

(E+mc2)(84)

yields to classical probability current arising from the classical Dirac equation:

j1=2cpx(85)

and, by expanding the probability current term j2=Ψ†

1α2Ψ1, we obtain:

j2=2cpy(86)

and, by expanding the probability current term j3=Ψ†

1α3Ψ1,we obtain:

j3=2cpz(87)

We then take the spatial derivatives of the probability current j, and obtain:

3

X

k=1

∂kΨ†

1αkΨ1=∂x(2cpk) + ∂y(2cpy) + ∂z(2cpz)=0 (88)

whereas by taking the temporal derivative of the probability density reported in Eq. 82, we obtain:

∂tΨ†

1Ψ1=∂t(2E)=0 (89)

Repeating the above procedure for the remaining time-quantized solutions Ψ2,Ψ3,Ψ4reported in Eqs. 68 and 69 would also give us

a similar set of results from which, the validity of the continuity equation can be shown to hold over discrete angular-intervals of π

radians, or at discrete instants corresponding to Ω = nπ, hence giving us a time-quantized continuity equation:

1

c∂tΨ†Ψ+

3

X

k=1

∂kΨ†αkΨ= 0 (90)

7 Time quantized line-element associated with gravity

In section 4, we showed that the 4-position vector associated in Minkowskian or ﬂat space-time for a particle with rest mass min

constant velocity motion, which is associated with a set of plane waves ψand ψC, where:

ψ= e−i(ωt−kr);ψC=ψ∗(91)

gives rise to square of the time-discrete or time-quantized line-element:

ds2=dxµdxµ=c2dt2−dr2(92)

which holds only over discrete time-intervals of 4tgiven by:

10

4t=π

ω=1−u2

c2

1

2π

mc2~(93)

where ω=γmc2/~and:

γ=1−u2

c2−1

2

(94)

To study how this time-discrete scheme could be extended to general relativity, let us introduce the general relativistic effect to

the line-element reported in Eq. 92:

ds2=gµν dx

0µdx

0ν(95)

and using the +− −− notation, where g00 =|g00|and gii =− |gii|:

ds2=g00c2dt

02− |gii|dr

02(96)

we obtain:

c2dτ2=g00c2dt

02− |gii|ut

02dt

02(97)

c2=g00c2dt02

dτ2− |gii|ut

02dt02

dτ2(98)

c2=g00γ

0t

02c2− |gii|γ

0t

02ut

02(99)

where, γ0t02is given by:

dτ2

dt02=g00 − |gii|

ut02

c2= 1

γ0t02!2

(100)

Multiplying out Eq. 99 by an invariant m2

~2:

m2c2

~2=g00 γ0t0mc22

~2c2− |gii|γ0t0mu t02

~2(101)

m2c2

~2=g00

E02

~2c2− |gii|k

02(102)

we ﬁnally arrive at:

m2c2

~2hψ, ψi=−g00 D∂2

0ψ

0

, ψ

0E+|gii|D∇2ψ

0

C, ψ

0

CE(103)

where:

ψ

0

= e−iω

0

t

0

−k

0

r

0;ψ

0

C=ψ

0∗(104)

ω

0

=E0

~=

γ0t0mc2

~(105)

∂2

0ψ

0

=−E02

~2c2ψ

0

(106)

∇2ψ

0

C=−k

02ψ

0

C(107)

and the Eq. 103 holds (a). when integrated over discrete-time intervals of 4t0which satisﬁes:

Z4t0ω

0t

0dt

0

=π(108)

Z4t0ω

0t

0dt

0

=Z4t0

γ0(t0)mc2

~dt

0

=π(109)

Z4t0ω

0t

0dt

0

=Z4t0

g00 − |gii|

ut02

c2

−1

2

mc2

~dt

0

=π(110)

and (b). at discrete time instants corresponding to:

11

ω

0t

0t

0

=nπ

2(111)

Therefore, the square of the line-element ds2=g00c2dt02− |gii |dr02, which is associated with general-relativistic effects can

be shown to hold within the time-discrete scheme we proposed in this paper, and gives rise to the following 2nd order relativistic

wave-equation with complementary waves:

m2c2

~2ψ=|gii| ∇2ψ

0

C−g00∂2

0ψ

0

(112)

to which by introducing the time-quantization scheme in Eq. 29:

1

πZπ

(mc)2hψ, ψidΩ = 1

πZπDp|gii|i~γi∂iψ

0

C+√g00i~γ0∂0ψ

0

,p|gii|i~γi†∂iψ

0

C+√g00i~γ0†∂0ψ

0EdΩ(113)

we obtain a ﬁrst order time-quantized solution that satisﬁes the relativistic wave-equation given in Eq. 112, with usual notation:

βmc2ψ=√g00 Eψ

0

−p|gii|(αipic)ψ

0

C(114)

Using the Schwarzschild metric, where g00 =1−2GM

c2rand gii =−1−2GM

c2r−1=−|gii |, the square of the line-element

reported in Eq. 96 becomes:

ds2=1−2GM

c2rc2dt

02−1−2GM

c2r−1

dr

02(115)

from which we can obtain a 2nd order relativistic quantized wave-equation with complementary waves:

m2c2

~2ψ=1−2GM

c2r−1

∇2ψ

0

C−1−2GM

c2r∂2

0ψ

0

(116)

whose 1st order solution is given by:

βmc2ψ=1−2GM

c2r1

2

Eψ

0

−1−2GM

c2r−1

2

(αipic)ψ

0

C(117)

When the general relativistic effects are ignored by setting g00 =|gii|= 1 and the time-quantization condition is removed by

setting ψ0=ψ0

C=ψ, the line-element reported in Eq. 115 reduces to the classical line-element:

ds2=c2dt2−dr2(118)

and the 2nd order relativistic quantized wave-equation reported in Eq. 116 reduces to the classical Klein-Gordon equation:

mc

~2ψ=−1

c2

∂2

∂t2ψ+∇2ψ(119)

whereas the 1st order solution reported in Eq. 117 reduces to the classical Dirac equation:

βmc2ψ=Eψ −(αipic)ψ(120)

8 Conclusion

We have studied an ansatz where the energy-momentum anti-commutation holds over discrete time intervals (or at discrete time

instants), and not at every instant, giving rise to a type of time quantization scheme which treats time as a continuous quantity and

shows that the observable quantities like energy, momentum, displacement etc., associated with the relativistic energy-momentum re-

lation are conserved only over discrete time-intervals, enabling us to obtain a time-discrete line-element. This suggests that if one were

to measure the energy, momentum, or other observable quantities associated with the 4-vector notation, for a duration which is not

exactly an integer multiple of the time-quanta (or chronon), associated with a given particle state, then there will be an un-avoidable

uncertainty associated with the measurement, which ﬁts well with the known time-energy uncertainty relation in quantum mechanics.

Additionally, contrary to Lynds’ claim that there aren’t really any instants, but rather intervals, the present work suggests that it is the

conservation of the observable quantities which are time-discrete, and not the time itself. In light of this interpretation, when Zeno’s

arrow paradox was revisited within the classical limit, we observed that one cannot deﬁne conserved physical quantities attributed to

an arrow, for any given instant, enabling us to potentially resolve the paradox within the notion of continuous time, consistent with

classical mechanics. By associating particle conservation with its time-discrete energy-momentum conservation, Zeno’s Achilles and

the tortoise paradox was discussed and a possible resolution was presented, within the classical context. When we extended this work

to momentum-position uncertainty relation, the work suggests that, more precisely one determines the physical attributes of a par-

ticle associated with particle-intervals, the less precisely its attributes associated with particle instants can be known, and vice-versa.

Therefore, perhaps we can identify this statement as a generalization of the uncertainty relation, in the context of relativistic quantum

mechanics and the wave-particle duality.

Further, this ansatz leads to the concept that the wave-function of a particle in relativistic motion is associated with a pair of

complementary waves, giving rise to a solution that satisﬁes the Klein-Gordon equation, (a). over discrete time intervals and (b). at

12

discrete time instants, in terms of the complimentary waves. This treatment, when applied to the Dirac equation, whose solutions

satisfy the Klein-Gordon equation, gave us a ﬁrst order time-quantized solution with complementary waves, associated with Dirac

matrices, and showed that the relativistic energy-momentum relation holds only over discrete time intervals (or at discrete time in-

stants). The solutions associated with the Dirac equation, when the temporal quantization rules are applied, were discussed along

with their corresponding classical solutions and the validity of the continuity equation was shown to hold within this ’time-discrete’

scheme. Finally, we extended this scheme to a line-element associated with general relativistic effects and obtained a temporally quan-

tized line-element that incorporates gravity. As the discreteness in quantum mechanics is combined with the line-element associated

with gravity, perhaps, the ansatz presented in this paper could potentially be further investigated as a possible approach to quantum

gravity.

Acknowledgements

The authors thank Prof. Chandre Dharma-wardana and Prof. J.A Gunawardena for some discussions and critical comments on the

manuscript.

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