Various representations of the quantity Newton called inertial mass

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by  J. L. Fry

Various representations of the quantity Newton called inertial mass
J.L. Fry and Z.E. Musielak
Department of Physics, The University of Texas at Arlington, Arlington, TX 76019, USA
(Received: date / Accepted: date)
Newtonintroducedtheconcept ofmassin hisPrincipiaandgaveanintuitiveexplanation forwhat
it meant. Centuries have passed and physicists as well as philosophers still argue overits meaning.
Three types of mass are generally identified: inertial mass, active gravitational mass and passive
gravitational mass. In addition to the question of what role mass plays in dynamical equations
2 and why,the origin of theparticular amount of matter associated with an elementary particle as a
1
consequence of fundamental fields has long been a topic of research and discussion. In this paper,
0
various representations of inertial mass are discussed within the framework of fundamental (either
2
Galilean or Poincar´e invariant) dynamical equations of waves and point particles. It is shown that

n thederivedequationshavemass-likeandmassparametersforwavesandpointparticles,respectively,
a and that the physical meaning of these parameters sheds a new light on the fundamental problem
J of thenature of inertial mass.

3
1
I. INTRODUCTION written for Physics Today by Wilczek7.

] There are also other definitions of mass such as that
h The concept of mass was originally introduced by originally introduced by Hertz8 and some based on ax-
p
- Newton1 who wrote in his Principia: ’The quantity of iomatized mechanics9. In his definition of mass, Hertz
n matter is the measure of the same, arising from its den- refers to a number of indestructible and unchangeable
e sityandbulkconjointly’. AccordingtoJammer2,amajor particles at a given point of space and at a given time,
g
. step in interpretation of Newton’s concept of mass was and defines mass by weight. A rather different approach
s made by Euler in his Mechanica3, where he suggested is presented in axiomatized mechanics, where Newto-
c
i that mass should be defined as the constant ratio be- nian inertial mass can only be determined in Galilean
s tween a constant force and the acceleration caused by reference frames in which the motion of the fixed (very
y
h this force. Euler’sdefinitionofmasshadbeenwidely ac- distant) stars must be a disjoint motion2,9. A different
p cepted in the nineteenth century, however, later in that axiomatization of mechanics proposed by Schmidt10 in-
[ century, Newton’s concept of force had become strongly tendedto introduce universalconceptofmass. However,

criticized and as a result new definitions of mass inde- the approachwasbasedonthe existence ofLagrangians,
1
v pendent of Newton’s second law were proposed. which requires solving the so-called Helmholtz inverse
6 The first step was made by Saint-Venant4 in 1845, problem11,12.
9 when he used the principle of conservation of linear mo- Differentconceptsofmasshavealsobeenconsideredby
8 mentum to express the ratio of masses of two bodies in Pendse13,Carnap14,Kamlah15,ZanchiniandBarletta16,
3 termsoftheirvelocityincrementsafteranimpact. Then, and others. A comprehensive review of different con-
1. Mach5 in 1867 introduced another definition of mass cepts of mass can be found in Jammer’s two books2,17,
0 thatwasbasedontwointeractingparticles,whichother- where the second book which was written more recently
2 wise werenotaffectedbyother particlesinthe Universe. alsoincludes ideas of massdeveloped inmodern physics.
1
Mach’s basic idea was to define mass in terms ratios of The book has one chapter devoted to relativistic mass
:
v accelerations caused by the particle’s interactions. How- andanotherdealingwiththemass-energyrelation. Both
i ever, this implied the existence of forces whose nature conceptshavebeenrecentlydiscussedbyOkun18 andRe
X
was not specified. It was also pointed out that Mach’s Fiorentin19, who givea new re-interpretationof the con-
r
a assumption of only two particles interacting with each cept of mass and the relativistic mass-energy relation.
other was superficial. Despite these objections, Mach’s SincethemeasureofinertiainSpecialTheoryofRela-
definition of mass had gained some popularity and be- tivity(STR)isnotmassofaparticlebutitstotal(kinetic
come recognized2 as ”an acceptable operational defini- andrest)energy,Okun[18]arguesthatrelativisticmass,
tion of a theoretical construct”. which depends on particle’s velocity, cannot be used as
Euler’s and Mach’s definitions of mass are based on the measure of inertia. He points out that in the very
Netwon’ssecondandthirdlaws,respectively. Weyl6 also low velocity limit, the relativistic rest mass becomes the
proposed a definition of mass, which was based on the same as Newtonian mass and therefore STR and New-
conservationofmomentum. BothWeyl’sandMach’sdef- tonian mechanics are commensurate theories; see also
initions have much in common because the conservation Jammer’sdiscussioninhischapterdevotedtorelativistic
17 19
of momentum and the third law have the same physical mass . Now,ReFiorentin reachedsimilarconclusions,
content, namely, the former is the time-integrated result however, his approach was different as he used both the
ofthe latter. Amorerecentdiscussionofthese problems Minkowski metric and the principle of least action. His
can be found in a series of ’Reference Frame’ articles mainresultthatmassisanotherwayofmeasuringenergy
2
requiresthe explanationofthe nature ofthe rest-energy, concept of mass to begining physics students” published
for which the author refers to the Higgs mechanism. in American Journal of Physics28 more than 50 years
The basic idea of the Higgs mechanism is that space ago. Theauthor’smainpointisthattheconceptofmass
is permeated by a scalar field, which is called the Higgs shouldbeintroducedtobeginningstudentsbydiscussing
field, and that particles couple to this field to acquire variousphysicalphenomena,wheremassplaysanimpor-
someenergythatcanbeinterpretedasparticle’smass20. tant role, rather than by using formal definitions. Our
More massiveparticlescouple more stronglyto the field. paperpresentsamoremodernapproachtothisotherwise
Althoughthisis apromissingidea,the scientificcommu- old problem.
nity still awaits its experimental verification. Thepaperisorganizedasfollows. InSec. 2,webriefly
In our previous work21−25, we used the Principle of describe the method used to derive invariant dynamical
RelativityandthePrincipleofAnalyticitytoformallyde- equations in space-time with a given metric, and also
rivethefundamentalequationsofnon-relativisticandrel- present the obtained equations. In Sec. 3, we examine
ativistic mechanics of waves and particles. For the wave theroleofthemass-likeandmassparametersinthe fun-
mechanics, we considered free and spin-zero elementary damental theories of waves and point particles. In Sec.
particles described by scalar wave functions. We used 4, we determine the relationships between the mass-like
the extended Galileangroup26 and the Poincar´egroup27 andmassparametersofthetheories,andpresentvarious
to derive the respective Schr¨odinger21 andKlein-Gordon representations of inertial mass. The nature of mass is
equations24. We demonstrated that the Schr¨odinger discussedinSec. 5,andourconclusionsaregiveninSec.
equationistheonlyfundamental(Galileaninvariant)dy- 6.
namical equation in Galilean relativity22 and that the
second-order Klein-Gordon equation is the only funda-
mental (Poincar´e invariant) equation in space-time with
the Minkowski metric25. Moreover, we used the same II. FUNDAMENTAL EQUATIONS OF
GALILEAN AND MINKOWSKI SPACE-TIME
principlestoderiveNewton’sequationsofnon-relativistic
and relativistic point particle mechanics. In the derived
fundamental equations, we encountered mass-like and A. Basic procedure
mass parameters for waves and point particles, respec-
tively.
We are interested in describing a physical object (an
Themainobjectiveofthispaperistodemonstratethe
elementary particle or a classical point particle) by us-
relevance of the mass-like and mass parameters to the
ing dynamical equations, which depend upon space and
conceptsofinertialmassdiscussedabove,andtodescribe
time variables that are characterized by a given metric.
variousrepresentationsofinertialmasswithintheframe-
The dynamical equations of a given metric may be de-
work of the fundamental (either Galilean or Poincar´ein- rived by the procedure used in our earlier work23. Since
variant) theories of waves and point particles. This pa-
the procedure explains the appearance of mass-like and
per was stimulatedby the two booksonmass written by
mass parametersin the derived dynamicalequations, we
Jammer2,17, and specifically by his statement that can
now briefly describe it. The basic procedure of deriving
be found in the last chapter of the second book:
dynamical equations for free particles is as follows:
(i) Establish a class of observers who define a physical
”Ifitwerepossibletodefinethemassofabodyorpar-
law; for example those in isometric frames of reference.
ticle on its own in purely kinematicalterms and without
anyimplicit referencetoaunitofmass,suchadefinition (ii) Decide upon the type of theoretical description to
might be expected to throw some light on the nature of be employed; two examples are a point particle (clas-
mass. Suchadefinition, ifitexisted,wouldintegratedy- sical) description, and a wave description. The theory
namics into kinematics and eliminate the dimension M may introduce new quantities, which require an addi-
of mass in terms of length L and time T.” tional metric to interpret the dynamical equations, such
as the measure of the amplitude of a wave in wave theo-
It is now our purpose to show that we have already ries.
accomplished the task suggested by Jammer, and that
(iii)EmploythePrincipleofRelativity,whichstatesthat
ourresultsdoindeedshedanewlightonthefundamental
all observers must identify the same physical object and
problem of the nature of mass.
write down the same dynamical equations describing its
Moreover,wehopethatourpresentationoftheconcept
space-time evolution. This could equally well be taken
of mass will benefit scientists working in different fields
as the definition of a law instead of a principle. Clearly
of natural sciences and that it will be especially helpful
changingtheclassofobserverscouldchangetheformand
tobeginingstudents,whoarelikelytoencounterconcep-
apparent nature of the laws.
tual difficulties with mass in their introductory physics
courses. The fact that this indeed can be a serious edu- (iv) Employ the Principle of Analyticity, which requires
cationalproblemwasfirst(to thebestofourknowledge) that all things that can be measured must be described
recognizedby Jacksonin his article ”Presentationof the by analytic functions of the space-time variables.
3
B. Galilean and Poincar´e invariant equations Galilean invariant and we call it the wave mass21. The
origin of this parameter is the definition of an elemen-
Inourpreviouswork,weconsideredfreeandspin-zero taryparticle. Thewavevector~k andthefrequencyω are
elementary particles described by scalar wave functions. the eigen-labels by which its wave representation may
To derive Schr¨odinger and Klein-Gordon equations, we be labeled in free space, and the Galilean invariant ra-
used the extended Galilean group26 and the Poincar´e tio of these labels upon which all inertial observersmust
group27, respectively, and obtained agree21,23: M = k2/2ω. Now, M may be determined
independently of(Newtonian) massm,has units derived
i ∂ + 1 ∇2 ψ =0 , (1) from space and time only, and is listed for various ele-
(cid:20) ∂t 2M (cid:21) mentary particles in Table 2 of our paper23. It occurs
naturally and cannot be avoided in a Galilean wave de-
and
scription of an elementary particle.
ω2 Fromthedispersionrelationω/k2 =1/2M,wededuce
∂µ∂ + 0 φ=0 , (2)
(cid:20) µ c2(cid:21) that if a particle is caused to change its state to a new
value of k in a given frame of reference, then the change
where ψ and φ are scalar wave functions, M and ω0 are in ω is proportional to 1/M. The larger M, the smaller
the so-called wave mass21 and invariant frequency23,24, the change in the state label ω. Thus M measures the
respectively, and c is the speed of light. resistance to change in frequency of the state of a free
In addition to free and spin-zero elementary particles particle,apropertywerelatetotheinertiaoftheparticle.
described by scalar wave functions, we also considered In the Minkowski metric, the Klein-Gordon equation
free classical point particles29 and derived both non- (see Eq. 2) contains a single parameter ω0, which is
relativistic and relativistic versions of Newton’s second Poincar´e invariant and we called it the invariant fre-
law of dynamics. The obtained equations canbe written quency in our previous paper24. The origin of this pa-
in the following form: rameter is the requirement of a Poincar´e invariant de-
scriptionofan irreducible representationofthe Poincar´e
dVi
m =0 , (3) group. While ~k and ω must also be eigen-labels of the
dt
irreduciblerepresentations(irreps)ofthePoincar´egroup
and in any inertial frame of reference, a Poincar´e invariant
label is the length of the eigen four-vector kµ, where
dUµ
0
M0 =0 , (4) k = ω/c. The invariant frequency may be determined
dτ
independently of wave mass and Newtonian mass but it
24
whereτ isthepropertime,Vi isthethree-velocityvector is related to them . Its units are a derived quantity,
with i = 1, 2 and 3, and Uµ is the four-velocity vector depending upon units of time only.
withµ=0,1,2and3. Inaddition, mrepresentsNewto- Valuesofinvariantfrequenciesforvariousparticlesare
nianinertialmassthatismeasuredinkgintheSIsystem listed in Table 2 of our paper24. The parameter ω0 is
of units, and M0 is a derived parameter whose units are a measure of the inertial properties of matter, occurs
chosen here to be the same as the wave mass M. naturally and cannot be avoided in a Poincar´e wave de-
Inderivingtheabovedynamicalequations,weencoun- scription of an elementary particle. In a given frame of
2 2 2
tered the need for the four parameters (M, ω0, m and reference the dispersion relation ω = ω0 +k allows us
M0) that describe the elementary particles and have the to deduce that the greater ω0, the smaller the change in
same value in all inertial frames of reference. Each of ω for a given change in k. Thus ω0 is a measure of a
these parameters is a manifestation of inertial mass of particle’sresistancetochangeinfrequencyω ofthestate
an elementary particle, so we call M and ω0 the mass- of the elementary particle in a given frame of reference,
likeparameters,andmandM0 the massparameters;we a property we relate to the inertia of the particle.
call M0 the mass parameter despite its units, which are The form of the free particle dynamical equations in
the same as M, because it represents mass of a point point particle theories is very different from that of the
particle. Examining the invariant dynamical equations wave equations. The parameters remaining after setting
for free particles, we can offer an interpretation for the the forces equal to zero on the RHS of Eqs (3) and (4)
meaning of each of the invariant constants describing a are m and M0, respectively. The parameter m in New-
free particle in the above four different dynamical equa- tonian mechanics is customarily assigned a new funda-
tions. mentalunit ofmeasure,the kg inthe SI systemof units,
whileM0 isaderivedparameterwhichwehavechosento
have the same units as wave mass M. As already shown
III. INVARIANT MASS-LIKE AND MASS byus24, M0 mayberelatedtomaswellastoM andω0.
PARAMETERS The invariant mass-like and mass parameters for the
waveandpointparticletheoriesgivenbyEqs(1)through
Inthe Galileanmetric,theSchr¨odingerequationgiven (4) are listed in Table 1, which also contains the corre-
by Eq. (1) contains one single parameter M, which is spondingdispersionrelations. Otherlocalparametersfor
4
IV. RELATIONSHIP BETWEEN PARAMETERS
Metric/Theory Invariant Dispersion OF THEORIES
parameter relation
Galilean/wave M k2 =2Mω A. Galilean metric
Minkowski/wave ω0 ω2−k2 =ω02 Consider a free particle moving in space-time with a
Galileanmetric andcharacterizedby k , ω andM in the
i
Galilean/particle m p2 =2ME wavedescriptionandbyp ,E andminthepointparticle
i
description. Let us assume that it is possible to arrange
Minkowski/particle M0 PµPµ =M02 an interaction with a field so that both wave and parti-
cle descriptions may be employedindetermining the pa-
rameters associated with the elementary particle. These
conditions are described in most derivations of Ehren-
fest’s theorem and we assume they can be achieved for
TABLE I: Invariant parameters and dispersion relations for
purposeofdiscussionhere. Usingthefreeparticleparam-
the wave and point particle theories in space-time with the
eters listed above, one observer determines the direction
Galilean and Poincar´e metrics.
oftravelofawaveandtheotherdeterminesthedirection
oftravelofwhatheassumestobeapointparticle. Since
itis infactthesameobjecttheirlocalvectorparameters
~k and p~ must be parallel,so their magnitudes differ only
by a real constant. Thus, we may write λk = p and
i i
Metric/Theory Scalars Vectors
obtain from the dispersion relations
Galilean/wave ω ki E =λ2ωM . (5)
m
Minkowski/wave ω ki
Hereλisanarbitraryrealconstant. Wenotethatfrom
Galilean/particle E p its definition the units of classical mass are arbitrary, i.
e. changing them changes the units of force and energy
Minkowski/particle P0 Pi butnottrajectoriescomputedfromNewton’ssecondlaw.
Ontheotherhandtheunitsofwavemassareestablished
from the choice of units of length and time. We may
choose to measure m and M in the same units so that
forthesameparticletheyareequal. ThentheunitsofE
TABLE II: Frame of reference dependent labels for the wave
arethesameastheunitsofωifweλ=1,adimensionless
and point particle theories in space-time with the Galilean
number.
and Poincar´e metrics.
Onthe otherhanditiscustomarytointerpretEq. (5)
by writing λM = m and E = λω using the experimen-
tally determined value of λ, which is of course known as
the Planck constant, ~. The wave equations given by
Eqs (1) and (2) were both derivedwithout any reference
these theories, the three-vectors ki, pi and Pi, and the to the Plank constant and contain only the parameters
scalars ω, E and P0 are given in Table 2, with pi =mvi M and ω0, both of which can be determined without
and Pµ =M0dxµ/dτ. The three-vectors and scalars are reference to the Planck constant. Since the usual clas-
alsoacceptable labels in a givenframe ofreference,how- sical mass introduces an unnecessary fundamental unit
ever,they differ in value fromone inertialframeofrefer- into physics, we prefer relating m to the wave mass M
ence to another. for the same particle. For elementary particles the wave
mass may be determined to almost two orders of mag-
SinceelementaryparticlesinNatureappeartobebest nitude less residual error than the residual errors in the
describedbywaveequations,whichhaveparameterswith Planck constant or classical mass. Thus the wave equa-
derived units, the description of inertial mass by an ad- tions without classicalmass andthe Planckconstantare
ditional fundamental measure, the kilogram, is possible more accurate and they should be used to describe ele-
but unnecessary. For elementary particles it is less accu- mentary particles23.
ratelyknownthanthe correspondingwavemass23,24 and
thusitshouldnotbethe measureofchoice. Thedynam-
ical equations for free classical point particles given by B. Minkowski metric
Eqs (3) and (4) have solutions independent of the mass
parametersmorM0; thetrajectoriesandworldlinesare The relativistic wave equation and the relativistic
the same for all values of these parameters. point particle equation are completely independent of
5
each other, but in an appropriate limit the relativistic invariant dynamical equations, and determine the corre-
wave may be interpreted as a point particle24,25. Using sponding mass-like and mass parameters.
theresultsofthesepapers,werelatekµandPµtoω0and By studying the relationships between Newton’s iner-
M0 by using kµkµ = ω02 and PµPµ = M02. Since kµ and tialmassandtheseparameters,weestablishedthatallin-
Pµ both provide the directionof motion alongthe world ertialobserversmustagreeuponthe valueofthe massin
line for the same particle under the proper experimental order to identify the same elementary particle. Dynam-
setupsothatboththeoriesarevalid,thetwofour-vectors ical behavior of free elementary particles is governed by
are parallel and can differ only by their lengths. Since the mass-like and mass parameters and by the way they
M0 has arbitrary units, its units can be chosen so that enter each invariant dynamical equation. Their presence
the lengths are the same: ω0 =M0. in the Galilean and Poincar´e invariant dynamical equa-
In general, we have ω0 = M0c2 and M0 has the same tions leads to properties that we identify physically with
units as wave mass M, a unit derived from L and T. Newton’s original concept of inertia.
However, if the units with c = 1 are used, then both ω0 InNewtonianmechanics,the propertyidentifiedasin-
and M0 may be expressed in units of 1/T. Because of ertia is commonly known as a resistance to a change in
this relationship between the invariantfrequency ω0 and velocity of a particle with mass m, which is called the
the rest mass M0 it is possible to remove the fundamen- inertialmass. A generalizationof this property,valid for
tal definition of relativistic mass and replace it with a all four fundamental theories considered above, is that
derived unit of mass as it was already done in Galilean the mass-like and mass parameters reflect the resistance
relativity(seetheprevioussubsection). Wenotethatthe of a particle to a change in its free particle state. The
Planck constant did not enter in this relationship. The principal effect of a larger mass-like (or mass) parame-
dynamical equation of point particles in the Minkowski teristomakeitmoredifficulttoincreasethe energy-like
metric may be expressed in terms of wave mass units. measures of the state of the system as the momentum-
The non-relativistic limits of Eqs (2) and (4), given likeparametersareincreaseduponapplicationofagiven
in some textbooks30,31, lead to additional connections force. This concept has been used to provide a working
between the mass-like and mass parameters of the two definition of a classical elementary particle29.
metrics. Thus with units c = 1, we obtain m = M0 and
M =ω0 whenmandM aremeasuredinunitsof1/T in-
stead of kilogram units. Combining all relationships be-
V. THE NATURE OF MASS
tweentheinvariantparameters,anelementaryparticlein
a Minkowskimetric may be described under appropriate
Massoccursnaturallyinourinvariantdynamicalequa-
conditions by any one of four dynamical equations with
tions as a result of type of metric, definition of physi-
all invariant mass-like parameters being the same:
cal law, definition of an elementary particle, assumption
of analyticity, and resulting differential equations. The
m=M =ω0 =M0 . (6)
central idea is that mass labels the irreps of the group
In this process the familiar concepts of mass, length of the metric, and that it also characterizes the nature
andtime,whichareconsideredfundamentalunits ofNa- of the state function during its transformation from one
ture, have been replaced by one fundamental unit for isometric frame of reference to another. Thus, in our
time, and mass and length units have been reduced to approach, mass is a natural consequence of the Galilean
derivedunits 1/T for mass and T for length. Thus there and Minkowski metrics.
isnoneedforacirculardefinitionofmassandtheunitsof Some understanding of the inertial properties of mass
space and time are properly connectedin the Minkowski can be gained from the work of Barut32,33, who demon-
metric. The unit of mass was eliminated by the con- stratedthatitispossibletotakewaveequationsformass-
nection to the wave equations and the unit of space was less particles and by separatingvariables find a localized
eliminatedbytheMinkowskimetric. Thewaveequations solutioncorrespondingtoarestframefrequencyω0. The
appear to haveeliminated the circulardefinition of mass equationsthenappeartohavepropertiesofawaveequa-
critized by Jammer2,17. tion with mass proportional to the invariant frequency
ω0. Based on the results presented in this paper, as well
as on Barut’s results, we conclude (Barut did not state
C. Various representations of Newton’s inertial so)thatlocalizationistheprocessbywhichinertialmass
mass appears. What causes the localization with observed el-
ementary particle frequencies is not fully understood for
Accordingtothe aboveresults,Netwon’sinertialmass allparticles,butinterestingaccountsofmosttheneutron
34
may be represented by different mass-like and mass andprotoninertialmasseshavebeengivenby Wilczek
parameters that arise in the fundamental (Galilean or in his ”What Matters for Matter” discussions presented
Poincar´e invariant) equations of waves and point parti- in Physics Today.
cles. To obtain this important result, we assumed that We have accomplished the task suggested by Jammer
themostbasicelementsofourapproachwerethemetrics, (see Sec. 1) by defining the mass ofanelementary parti-
which we used to define elementary particles, derive the cle on its own without any specific reference to the unit
6
of mass ’kilogram’. This elimination of the dimension of to identify elementary particles. The particular way in
mass has allowedus to formulate the fundamental quan- which the inertial mass-like and mass parameters enter
tumtheoriesbasedontheSchr¨odingerandKlein-Gordon eachinvariantdynamicalequationgovernsits dynamical
equations without making any reference to the Planck behavior, leading to properties that we identify physi-
constant. We have also contributed to the challenging callywiththe conceptofinertia. Inertialmassisaframe
problemofthenatureofmassbyshowingthatthemass- ofreferenceindependentdescriptionoftheparticle,while
like and mass parameters are related to the concept of energy-likeandmomentum-likelabelsonthefreeparticle
inertial mass originally introduced by Newton, and that areframe ofreferencedependent. The lattertwo quanti-
amongtheseparameterstheinvariantfrequencyω0isthe ties are nonetheless very useful in the description of the
most fundamental one as the other parameters may be state of a particle relative to a given frame of reference.
derived from it. Why only selected values of this pa- It is our hope that our presentation of the concept
rameter occur in Nature must be determined from con- of mass given in this paper will be helpful to scientists
siderations other than the free particle dynamical equa- working in different fields of natural sciences and that it
tions considered here. Why ω0 takes the special values willespeciallybenefitbeginingstudents,whoarelikelyto
observed in Nature is not fully understood for elemen- encounterconceptualdifficultieswithmassintheirintro-
taryparticles,butthoughttoarisefromsomeunderlying ductoryphysicscourses. Themainmessageofthispaper
20
fields, like for instance the Higgs field . for the begining students is that the concept of mass oc-
curs in Nature naturally once the metric of space-time
in which we live is determined. Moreover, the mass of
VI. CONCLUSIONS an elementary particle can be defined on its own, with-
out any reference to the specific unit of mass ’kilogram’.
We have discussed the fundamental dynamical equa- This has important physical consequences as it allows
tions for waves and point particles in space-time with formulating the fundamental quantum theories without
both the Galilean and Minkowskimetrics. The obtained formallyintroducingthePlanckconstantbutusingwhat
equations are either Galilean or Poincar´e invariant and is called here a mass-like parameter. Hence, the theories
theydescribefreespin-zeroelementaryparticlesthatare ofphysicsmaybeformulatedbyusingeithertheclassical
represented by scalar wave functions, and free classical conceptofmasswithitsunitof’kilogram’andthePlanck
particles that are treated here as point particles. There constant, or by using only one the mass-like parameter.
arefourinvariantmass-likeandmassparametersinthese It must also be pointed out that the theories based on
equations, andwe have shownthat these parametersare the mass-like parameter can attain higher accuracy of
various representations of Newton’s inertial mass. Our performing computations23,24.
discussion of the relationships between the parameters
andtheir physicalmeaning shedsa new lightonthe fun- ACKNOWLEDGMENTS We thank Alex Weiss and
damental problem of the nature of inertial mass. Andy White for discussions and their comments on the
Fromtheperspectiveofthispaper,inertialmassisjust manuscript. Z.E.M. acknowledges the support of this
a parameter that all inertial observers must agree upon work by the Alexander von Humboldt Foundation.
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