#  $E=mc^{2}$ versus Symmetry for Lorentz Covariant Physics

Otto C. W. Konga,b and Hock King Tinga aDepartment of Physics and Center for
High Energy and High Field Physics, National Central University, Chung-li,
Taiwan 32054
bQuantum Universe Center, Korea Institute for Advanced Study, Seoul, 02455
Republic of Korea

###### Abstract

The famous equation $E=mc^{2}$ is a version of particle mass being essentially
the magnitude of the (energy-)momentum four-vector in the setting of
‘relativistic’ dynamics, which can be seen as dictated by the Poincaré
symmetry adopted as the relativity symmetry. However, as Einstein himself
suggested, the naive notion of momentum as mass times velocity may not be
right. The Hamiltonian formulation perspective gives exactly such a setting
which in the case of motion of a charged particle under an electromagnetic
field actually has the right, canonical, momentum four-vector with an evolving
magnitude. The important simple result seems to have missed proper
appreciation. In relation to that, we present clear arguments against taking
the Poincaré symmetry as the fundamental symmetry behind ‘relativistic’
quantum dynamics, and discuss the proper symmetry theoretical formulation and
the necessary picture of the covariant Hamiltonian dynamics with an evolution
parameter that is, in general, not a particle proper time. In fact, it is
obvious that the action of any position operator of a quantum state violates
the on-shell mass condition. The phenomenologically quite successful quantum
field theories are ‘second quantized’ versions of ‘relativistic’ quantum
mechanics. We present a way for some reconciliation of that with our symmetry
picture and discuss implications.
Keywords : On-shell Mass Condition, Relativistic Dynamics, Casimir Invariants,
(Quantum) Relativity Symmetry, Lorentz Covariant Quantum Mechanics, Quantum
Field Theory

††preprint:  NCU-HEP-k096 Jan 2023 rev. Mar 2023 ed. Apr 2023

##  I Introduction

Poincaré symmetry has been generally taken as the relativity symmetry for
Lorentz covariant physics, the so-called ‘relativistic’ theories, quantum and
classical. The related equation $E=mc^{2}$ may be the most popular physics
equation, though the general understanding of the simple equation has been
loaded with many confusions O ; H , as would be expected. A key question we
want to address here though, is if the ‘correct’ picture about it, from the
‘standard’ ‘relativistic’ theories, is indeed correct. The article launches a
serious challenge to the question, at least for the case of quantum and
classical particle dynamics, mostly from the symmetry theoretical formulation
and the closely related Hamiltonian dynamics points of view. Implications for
quantum field theory would also be discussed.

The ‘standard’ ‘relativistic’ theories are supposed to be obtainable, symmetry
theoretically, from representations of the Poincaré symmetry. The irreducible
representations of a symmetry, for the description of elementary systems
(indecomposable into separated parts so long as the symmetry is concerned),
are classified by its Casimir invariants. Note that we use the word
representation in this article only in the straight symmetry theoretical
sense, and talk about representations of a symmetry instead of a group or a
Lie algebra, as a representation of a Lie algebra actually defines a
representation of the Lie group, the universal enveloping algebra, and the
group $C^{*}$-algebra as well as its various extensions all of which play some
role in the physical theory. The Poincaré symmetry has as generators the six
generators $J_{\mu\nu}$ of Lorentz symmetry as (spacetime) rotations and four
generators $P_{\mu}$ as (spacetime) translations. $P_{\mu}P^{\mu}$ gives a
Casimir invariant. For a particle, classical or quantum, the result says the
magnitude square of the momentum four-vector, representation of $P^{\mu}$ as
observables, has the value of $(-)m^{2}c^{2}$. That is called the on-shell
mass condition. The reading of it in the particle rest frame gives $E=mc^{2}$.
However, we are going to present arguments against seeing Poincaré symmetry as
the right symmetry for the theories. The correct symmetry picture gives a
different insight into the matter.

It is interesting to note that though it was Einstein who first gave the
$E=mc^{2}$ result and taught us that mass is a form of energy, he was not
exactly in support of the unconditional on-shell mass condition, here as
$p_{\mu}p^{\mu}=-m^{2}c^{2}$, for a (classical) particle H . In a 1935 paper E
, Einstein actually raised the concern that his ‘momentum’ four-vector as
$m\frac{dx^{\mu}}{d\tau}$, $\tau$ being the particle proper time, may not be
truly important conserved dynamical quantity in relation to the usual notion
of conservation of momentum. From the symmetry theoretical formulation and the
point of view of Hamiltonian dynamics, the latter is the canonical momentum as
the representation of the $P^{\mu}$ generators of the symmetry. That is to
say, Einstein thought that if the true energy-momentum four-vector $p^{\mu}$
is given by $m\frac{dx^{\mu}}{d\tau}$ and if the equation
$p_{\mu}p^{\mu}=-m^{2}c^{2}$ is satisfied in a general situation was an open
question.

Our focus is more on the quantum theories. It is well known that the usual
theorists of ‘relativistic’ quantum mechanics for particles of various spins,
as presented in the textbooks, are of little practical applications. At least
part of the reasons is related to the issues we are after here. That is a
stunning contrast to the case of the ‘nonrelativistic’ theory, which should be
like its approximation. We have a nice way to look at Newtonian mechanics as
an approximation to ‘nonrelativistic’ quantum mechanics from a symmetry
theoretical point of view, based on a scheme of symmetry contraction 070 .
Essentially, the contraction trivializes the Heisenberg commutation relation
in the representation picture reducing the quantum to the classical. That also
has deformation quantization fitting in as the deformation of the observable
algebra, say as the representation of the universal enveloping algebra, as a
result of the deformation of the basic symmetry Lie algebra. The deformation
is the exact inverse of the contraction. The Poincaré symmetry does not allow
any parallel. We have rather presented a new theory of Lorentz covariant
quantum mechanics 087  that fit in well with all that, having contractions
that give the ‘nonrelativistic’ or classical approximations correctly. That is
based on the representation of the $H_{\\!\scriptscriptstyle R}(1,3)$ symmetry
we discussed below. This article helps to clarify why the latter symmetry,
instead of the Poincaré symmetry, is the right one to be considered.

Most physicists are not much concerned about the unpleasant situation of
‘relativistic’ quantum mechanics because the field theory version seems to be
working perfectly well, giving all the success in the description of results
from experimental high energy physics. Yet, quantum field theories are
supposed to be only the ‘second quantized’ version of ‘relativistic’ quantum
mechanics. If the latter has to be revised with the symmetry picture, one has
to worry if the same applies to quantum field theories. And at least from the
theoretical point of view, not having a successful framework to look at
‘nonrelativistic’ quantum mechanics as an approximation of a ‘relativistic’
version of it, from the point of view of symmetry theoretical formulation, is
a shortcoming. Somewhat surprisingly, we see a room for the
$H_{\\!\scriptscriptstyle R}(1,3)$ symmetry to reconcile with most aspects of
the quantum field theories. However, that put the theories and their
corresponding quantum mechanical versions on independent footings in conflict
with the notion of the former being the ‘second quantized’ versions of the
latter. While more details may have to be studied carefully, the article also
presents some basic issues and implications about that.

The key motivation of the study is to work towards a single comprehensive
theoretical framework that incorporates all experimentally established
successful dynamical theories with some as approximations of others. Results
from the article, together with earlier efforts, score full success from the
level of ‘relativistic’ quantum particle dynamics down to all its
‘nonrelativistic’ as well as classical approximations. We illustrate further
that the framework accommodates quantum field theories for high energy theory
reasonably well. However, a good number of conceptual questions about the
relationship of the latter with the theories of particle dynamics remain to be
answered. Studies in this direction also provide an approach to pursue the
‘higher level’ theories beyond. Especially for that purpose we certainly need
to further improve our understanding of all the theoretical and conceptual
questions about the established theories.

In the next section, we present a simple and direct illustration of the
invalidity of the on-shell mass condition so long as the true momentum, the
canonical momentum is concerned and contrast it with the validity for the
‘momentum’ $m\frac{dx^{\mu}}{d\tau}$ as a defining property of little
dynamical interest. Note that the result is an exact contradiction of having
the Poincaré symmetry as the fundamental symmetry behind it because a Casimir
invariant of the latter is violated. That is followed by a section on a sketch
of the correct, $H_{\\!\scriptscriptstyle R}(1,3)$, symmetry formulation.
There are mathematically minor but important modifications compared to the
previous presentation 087 , in line with the parallel for the
‘nonrelativistic’ case 095 . Details as available in Ref.087  would not be
repeated here. Discussions on how a true Poincaré symmetry formulation fails
and how some of the $H_{\\!\scriptscriptstyle R}(1,3)$ symmetry features have
actually been incorporated into the usual textbook presentations of the
subject matter can be found. Sec.IV then looks into the basic features of the
Hamiltonian dynamics. The phenomenologically quite successful quantum field
theories are ‘second quantized’ versions of ‘relativistic’ quantum mechanics,
hence have related symmetry issues. Interestingly enough, there is a way to
have quite a reconciliation of that through the case of zero Newtonian mass.
We present that in Sec.V. Some of the implications are among the concluding
remarks presented in the last section.

##  II Generic Canonical Momentum versus $E=mc^{2}$

A practical setting to look at ‘relativistic’ particle dynamics is the
electrodynamics of a particle of charge $e$. It is interesting that even the
simple ‘nonrelativistic’ Hamiltonian of
$H=\frac{1}{2m}\left(p_{i}-\frac{e}{c}A_{i}\right)\\!\\!\left(p^{i}-\frac{e}{c}A^{i}\right)$
actually gives the $m\frac{dx^{i}}{dt}=p^{i}-\frac{e}{c}A^{i}$, i.e. the
canonical momentum as different from mass times velocity. The latter is a
‘momentum’ that has then a magnitude constant in time, while the same
certainly does not hold for the true momentum vector $p^{i}$. From the
perspective of Hamiltonian dynamics, when $p\neq mv$, $mv$ is not a quantity
of particular interest. We advocate against even using the word momentum for
it. We may not then have the Newtonian form of the equation of motion as
$F=ma$, but maintains force being the rate of change of (canonical) momentum
and that has been generally accepted as correct. The exact parallel, which we
prefer to write in the quantum sense, is given by

$\displaystyle\hat{H}_{s}=\frac{1}{2m}\left(\hat{P}_{\mu}-\frac{e}{c}\hat{A}_{\mu}\right)\\!\left(\hat{P}^{\mu}-\frac{e}{c}\hat{A}^{\mu}\right)\;,$
(1)

is natural to be accepted, in line with the notion of $\hat{A}_{\mu}$ as a
connection, to be a Lorentz covariant Hamiltonian for the ‘relativistic’ case.
We put an $s$ there to denote the evolution parameter, which can be seen here
as the particle proper time $\tau$. The Hamiltonian gives

$\displaystyle\frac{d\hat{X}^{\mu}}{ds}=\frac{1}{m}\left(\hat{P}^{\mu}-\frac{e}{c}\hat{A}^{\mu}\right)\equiv\frac{1}{m}\hat{\pi}^{\mu}\;,\qquad\frac{d}{ds}\left(\hat{P}^{\mu}-\frac{e}{c}\hat{A}^{\mu}\right)=0\;.$
(2)

Note that $\hat{P}^{\mu}$ being the observable in the Heisenberg commutation
relation, as essentially the Poisson bracket between canonical variables 081 ,
is again the canonical momentum and it is not $m\frac{d\hat{X}^{\mu}}{d\tau}$,
even for $s=\tau$. The classical case is obtained simply by turning the
quantum observables into classical ones. The equations give the correct
Lorentz force picture. Note that $\frac{d(\pi_{\mu}\pi^{\mu})}{ds}=0$, which
is in line with $\pi_{\mu}\pi^{\mu}=-m^{2}c^{2}$. The latter is, however, a
necessary consequence of the definition of $\tau$ ($=s$) as the time
coordinate in the particle rest frame. Hence, it has really no dynamical
content. $\frac{d(p_{\mu}p^{\mu})}{ds}$ cannot be zero, and the true momentum
four-vector has an evolving magnitude. Note that the covariant $H_{s}$ gives
exactly the same equations of motion as the more conventional
$H_{t}=c\pi^{\scriptscriptstyle 0}+eA^{\scriptscriptstyle 0}$ with
$\pi^{\scriptscriptstyle 0}=\sqrt{\pi_{i}\pi^{i}+m^{2}c^{2}}$ on the
$(x^{i},p^{i})$ phase space.

The mass in $E=mc^{2}$ is a notion of ‘rest mass’ O . For the problem at hand,
$mc^{2}$ is the constant timelike magnitude of the four-vector $c\pi^{\mu}$
and the value of its ‘energy’ or time component at the particle rest frame.
Yet, the charged particle is under an accelerating (Lorentz) force. The
particle rest frame is hence not even an inertial frame. The quantity should
not have any particular importance, and the same goes for the proper time
$\tau$. There is actually no good reason then to think about the latter as the
right evolution parameter for the covariant Hamiltonian formulation. That is
the reason for us to use $s$ instead of $\tau$ to begin with. It is amazing
that the simple and clear result of not having a meaningful on-shell mass
condition for the charged particle seems not to have received the proper
attention and most physicists still see the condition as straightly valid at
least so long as particle dynamics is concerned.

##  III The Correct Symmetry Theoretical Formulation

The above result runs in direct contradiction with having Poincaré symmetry as
the ‘relativistic’ symmetry. On the other hand, our earlier presentation of a
serious group theoretical formulation of fully Lorentz covariant quantum
mechanics 087  was based on a different, bigger, symmetry. It is the
$H_{\\!\scriptscriptstyle R}(1,3)$, as given by the Lie algebra

$\displaystyle[J^{\prime}_{\mu\nu},J^{\prime}_{\rho\sigma}]=i\hbar\,c\left(\eta_{\nu\sigma}J^{\prime}_{\mu\rho}+\eta_{\mu\rho}J^{\prime}_{\nu\sigma}-\eta_{\mu\sigma}J^{\prime}_{\nu\rho}-\eta_{\nu\rho}J^{\prime}_{\mu\sigma}\right)\;,$
$\displaystyle[J^{\prime}_{\mu\nu},Y_{\rho}]=i\hbar\,c\left(\eta_{\mu\rho}Y_{\nu}-\eta_{\nu\rho}Y_{\mu}\right)\;,$
$\displaystyle[J^{\prime}_{\mu\nu},E_{\rho}]=i\hbar\,c\left(\eta_{\mu\rho}E_{\nu}-\eta_{\nu\rho}E_{\mu}\right)\;,$
$\displaystyle[Y_{\mu},E_{\nu}]=i\hbar\,c\,\eta_{\mu\nu}\,M\;,$ (3)

with the Minkowski four-vector indices going from $0$ to $3$, where
$\eta_{\mu\nu}=\mbox{diag}\\{-1,1,1,1\\}$. That is actually more in line with
the earlier works of Refs.Z ; J . The presentation here has been slightly
modified from that of Ref.087 , along the exact parallel of that of the
‘nonrelativistic’ case in Ref.095 . The key point is to see the central charge
generator $M$ as essentially a Casimir element, eigenvalues of which
constitutes a Casimir invariant as a label of irreducible representations. The
latter is the Newtonian mass $m$. A generator in a Lie algebra is not usually
considered a Casimir element. However, the central generator completely
satisfies the requirement that an eigenspace of its operator representation
would be invariant under any other operators of the symmetry representation,
hence giving an irreducible representation with its eigenvalue as one of the
basic characteristics. Having the Newtonian mass obtained as a basic
characteristic of a (quantum) particle from the formulation is a very
appealing feature. In the consideration of a composite system, say of two
particles, the formulation here further avoids the otherwise inescapable
conclusion of having additive position observables. An irreducible
representation 087  would have $\hat{X}_{\mu}\equiv\frac{1}{m}\hat{Y}_{\mu}$
and $\hat{P}_{\mu}\equiv\frac{1}{c}\hat{E}_{\mu}$, satisfying the commutation
relation $[\hat{X}_{\mu},\hat{P}_{\nu}]=i\hbar\eta_{\mu\nu}\hat{I}$. A
composite system should correspond to the product representation of the
symmetry with all abstract generators $G$ of the Lie algebra represented by
$\hat{G}=\hat{G}_{a}\otimes\hat{I}+\hat{I}\otimes\hat{G}_{b}$ where
$\hat{G}_{a}$ and $\hat{G}_{b}$ are the operator representations of $G$ in the
irreducible representations for particle $a$ and particle $b$, respectively.
Hence, the symmetry theoretical formulation gives additive $\hat{M}$,
$\hat{P}_{\mu}$, and $\hat{Y}_{\mu}$ for the corresponding observables as
physical quantities. Yet, instead of additive $\hat{X}_{\mu}$, it dictates the
notion of center of mass from

$\displaystyle\hat{X}_{\mu}\equiv\frac{1}{m}\hat{Y}_{\mu}=\frac{1}{m}(m_{\scriptscriptstyle
a}\hat{X}_{\mu a}\otimes\hat{I}+\hat{I}\otimes m_{\scriptscriptstyle
b}\hat{X}_{\mu b})\;,$ (4)

with $m=m_{\scriptscriptstyle a}+m_{\scriptscriptstyle b}$. The feature has
been more or less appreciated in the Galilean symmetry picture of the
‘nonrelativistic’ theory, though more in the language of $U(1)$ central
extension. The Heisenberg-Weyl symmetry perspective, however, the incorrect
picture of having the position observables as directly representing symmetry
generators is usually taken (see Ref.095 ). The modification is actually
complementary to the key notion under focus here. Poincaré symmetry gives
(Einstein rest) mass-square and spin as Casimir invariants. The former
dictates the non-negotiable on-shell condition for the four-vector
$\hat{P}_{\mu}$ of any representation. The $H_{\\!\scriptscriptstyle R}(1,3)$
symmetry, other than giving the real consistent picture of a formulation of
Lorentz covariant quantum mechanics, avoids running into a contradiction with
the above result in electrodynamics, yet gives also a notion of particle mass.
The latter has nothing to do with any on-shell condition, but agrees with the
same from the ‘nonrelativistic’ theory; both are to be inherited in the
classical theories as approximations to the quantum ones.

We give a summary of the key representation theoretical results in the
appendix, focusing especially on parts beyond a simple adaption of what has
been given in Ref.087 . Here, we first address the matching of
$H_{\\!\scriptscriptstyle R}(1,3)$ versus the Poincaré symmetry, which can be
seen as its subgroup generated by $J^{\prime}_{\mu\nu}$ and ${P}_{\mu}$.

The theory of quantum mechanics from $H_{\\!\scriptscriptstyle R}(1,3)$ can
easily be appreciated as a natural extension of the ‘nonrelativistic’ one,
from the analog $H_{\\!\scriptscriptstyle R}(3)$ 070 ; 095 , with three-vector
notions extended to (Minkowski) four-vector ones. Note that our
$H_{\\!\scriptscriptstyle R}(3)$ symmetry is mathematically exactly the $U(1)$
central extension of the that of the ‘standard’ Galilean one with the time
translation taken out. A Schrödinger wavefunction picture would have state
vectors given by functions $\phi(x^{\mu})$ with $\hat{X}_{\mu}$ and
$\hat{P}_{\mu}$ acting as $x_{\mu}$ and $-i\hbar{\partial}_{\mu}$. The usual
presentations of ‘relativistic’ quantum mechanics of course share the same.
But it is assumed, rather than in any sense obtained from a symmetry
theoretical construction. A serious symmetry theoretical construction gives
not only a vector space of states as a representation space, but also the
matching representations of all elements of the Lie algebra, Lie group,
universal enveloping algebra, or extensions of the group $C^{*}$-algebra as
operators on the vector space 087 ; 070 . The operators should include all
(quantum) observables. The position operators certainly cannot be obtained
from the Poincaré symmetry. Assuming the usual quantum mechanical
$\hat{X}_{\mu}$ and $\hat{P}_{\mu}$ is really looking at the space of
functions $\phi(x^{\mu})$ as a representation space of the $H(1,3)$
Heisenberg-Weyl symmetry (of ${Y}_{\mu}$, ${P}_{\mu}$, and $M$) or the spin-
zero representation of the full $H_{\\!\scriptscriptstyle R}(1,3)$. We simply
cannot have the position operators from the Poincaré symmetry. Notice that
while we have the standard picture of orbital angular momentum
$\hat{L}_{\mu\nu}\equiv\hat{X}_{\mu}\hat{P}_{\nu}-\hat{P}_{\mu}\hat{X}_{\nu}$,
which is $\hat{J}_{\mu\nu}$ for the spin-zero representation, from the
$H_{\\!\scriptscriptstyle R}(1,3)$ symmetry, there really seems to be no way
to obtained $\hat{X}_{\mu}$ from some operator representation of
$\hat{P}_{\mu}$ and $\hat{J}_{\mu\nu}$ without first putting the
$\hat{X}_{\mu}$ in $\hat{L}_{\mu\nu}$.

One may follow closely the supposed symmetry theoretical construction from the
Poincaré symmetry, as for example given in a common textbook by Tung T .
Starting with a (unitary) representation of the subgroup of translations,
generated by $\\{P_{\mu}\\}$, specifically a Hilbert/vector space spanned by
the eigenvectors $\\{\left|p^{\mu}\right\rangle\\}$, for a particle of rest
mass $m_{\\!\scriptscriptstyle E}>0$, an irreducible representation of spin-
zero is claimed under the condition $-p_{\mu}p^{\mu}=m_{\\!\scriptscriptstyle
E}^{2}c^{2}$ as dictated by the Casimir invariant. Here, and in the subsequent
discussions, we use $m_{\\!\scriptscriptstyle E}$ for the Einstein rest mass
to distinguish it from the conceptually different notion of our Newtonian
mass. Naively, the infinite-dimensional Hilbert space as the functional space
of $\phi(x^{\mu})\equiv\left\langle x^{\mu}|\phi\right\rangle$ is identified
as the representation space and such a Schrödinger wavefunction is seen as
related to the momentum space wavefunction
$\tilde{\phi}(p^{\mu})\equiv\left\langle p^{\mu}|\phi\right\rangle$, through a
Fourier transform from $\left\langle
x^{\mu}|p^{\mu}\right\rangle=e^{\frac{ip_{\mu}x^{\mu}}{\hbar}}$. The truth is
that many of the standard results for the so-called position and momentum
‘representations’ of ‘nonrelativistic’ quantum mechanics, as from rigorous
representation theory of the $H_{\\!\scriptscriptstyle R}(3)$, or simply
$H(3)$, symmetry have been implemented in a $1+3$ version. That is, of course,
effectively using the $H_{\\!\scriptscriptstyle R}(1,3)$ symmetry. Otherwise,
as the Poincaré symmetry has no origin for the operators $\hat{X}_{\mu}$, one
can have $\hat{P}_{\mu}$ as $p_{\mu}$ but no $\hat{X}_{\mu}$ as
$i\hbar\frac{\partial}{\partial p^{\mu}}$, and no notion of
$\left|x^{\mu}\right\rangle$ as states. In fact, even the momentum space
wavefunction $\tilde{\phi}(p^{\mu})$, or the Hilbert space spanned by
$\\{\left|p^{\mu}\right\rangle\\}$, cannot be justified from the Poincaré
symmetry. An irreducible representation of the subgroup of translations has to
be one-dimensional as the subgroup, unlike the $H(1,3)$, is an Abelian one.
Each eigenvector $\\{\left|p^{\mu}\right\rangle\\}$ spans one independent
irreducible representation. Even when the Lorentz symmetry part is
supplemented, at the best one can expect $\hat{J}_{\mu\nu}$ to rotate the
$p^{\mu}$ four-vector keeping $-p_{\mu}p^{\mu}=m_{\\!\scriptscriptstyle
E}^{2}c^{2}$ constant (as an irreducible representation of
$H_{\\!\scriptscriptstyle R}(1,3)$ induced by those of its $H(1,3)$ subgroup).
That gives only a Hilbert space spanned by $\\{\left|p^{i}\right\rangle\\}$,
or $\\{\left|\vec{p}\right\rangle\\}$ as independent basis vectors, hence only
$\tilde{\phi}(\vec{p})$. Then, it looks like one gets the same Hilbert space
as that of ‘nonrelativistic’ quantum mechanics, except that $\hat{X}_{i}$ we
want still has to be imported beyond the Poincaré symmetry. Using a
$\tilde{\phi}(p^{\mu})$ notation for such a $\tilde{\phi}(\vec{p})$, assuming
$-p_{\mu}p^{\mu}=m_{\\!\scriptscriptstyle E}^{2}c^{2}$ is misleading. Writing
$\phi(x^{\mu})$ is much worse, because it is completely unclear how the
$x^{\scriptscriptstyle 0}$ value can be obtained from the three independent
$x^{i}$ values from the three-dimensional Fourier transform. Note that the
mathematics of the latter cannot automatically be taken to give physical
meaning to the $x^{i}$ variables either. And in the actual physical
applications, one ’naturally’ thinks about having the full $x^{\mu}$ in
$\phi(x^{\mu})$ as a free Minkowski four-vector, which certainly cannot be
obtained from the construction, though Minkowski spacetime can indeed be
obtained from the Poincaré symmetry as a coset space representation. Taking
the theory as one of, representation of, the $H_{\\!\scriptscriptstyle
R}(1,3)$ symmetry obviously makes better sense. As none of the $\hat{X}_{\mu}$
commutes with the $\hat{P}_{\mu}\hat{P}^{\mu}$, an $\hat{X}_{i}$ action on the
quantum state of a particle would take the state beyond an eigenspace of
$\hat{P}_{\mu}\hat{P}^{\mu}$ hence violating the on-shell mass condition and
pulling the different eigenspaces from different irreducible representations
of the Poincaré symmetry (as a subgroup) into a single (induced) irreducible
representation of $H_{\\!\scriptscriptstyle R}(1,3)$. A Hamiltonian evolution
with an $\hat{X}_{\mu}$-dependent Hamiltonian is just an example of that. That
is really saying, a notion of a position observable cannot make sense for a
particle with the Einstein rest mass as a fixed characteristic as the
corresponding operator cannot be defined within its vector space of states.
Without the position observables and the Heisenberg commutation relation
between them and the $\hat{P}_{i}$, one has hardly a particle theory, nor a
quantum one.

##  IV A Look at the Covariant Hamiltonian Dynamics

A covariant Hamiltonian formulation of classical and quantum dynamics has been
taken by some physicists starting probably from the works of Fock and
Strückelberg in the 40’s. For a general picture of the lines of work, and
sources of further references, books TS ; Ho ; F  may be consulted. Most
authors, however, has struggled to maintain the on-shell mass condition for
the momentum four-vector in some way. On the other hand, a no-interaction
theorem has been derived from imposing the straight on-shell mass condition
for the particle involved ni . The latter is unavoidable when each particle is
seen as from an irreducible representation of the Poincaré symmetry. Despite
that, quantum field theories supposed to be obtained from the symmetry talk
about ‘virtual’ states which violate the condition. From the perspective here,
all such theories should be re-examined from an $H_{\\!\scriptscriptstyle
R}(1,3)$ symmetry point of view.

A truly covariant Hamiltonian formulation should have free four-vector
canonical variables, the quantum $\hat{X}^{\mu}$ and $\hat{P}^{\mu}$ and the
${x}^{\mu}$ and ${p}^{\mu}$ in the classical approximation. A typical single-
particle Hamiltonian would have the form
$\hat{H}_{s}=\frac{\hat{P}_{\mu}\hat{P}^{\mu}}{2m}+V(\hat{X}^{\mu})$ which
dictates Hamiltonian flows in the phase space each as a one-parameter group of
transformations with an evolution parameter $s$. The latter is mathematically
the exact analog of the Newtonian time in the ‘nonrelativistic’ case so long
as the Hamiltonian formulation is concerned 095 , though it is not clear if it
is some sort of physical time at all. One has
$\frac{d}{ds}=-\frac{1}{i\hbar}[\hat{H}_{s},\cdot]$, the Hamiltonian vector
field in terms of the Poisson bracket as in the classical case 081 . (In fact,
$\hat{X}^{\mu}$ and $\hat{P}^{\mu}$ can be seen as noncommutative coordinates
of the exact phase space of the quantum states in the usual Schrödinger
picture 081 ; 087 ; 078 ). For the free particle case, with
${\hat{P}_{\mu}\hat{P}^{\mu}}=-m_{\\!\scriptscriptstyle E}^{2}c^{2}$ here as
the fixed initial value, we have $\frac{d\hat{P}^{\mu}}{ds}=0$ and
$\frac{d\hat{X}^{\mu}}{ds}=\frac{1}{m}\hat{P}^{\mu}$, at least easy to
interpret for initial momentum eigenstates or in the classical limit. Note
that the operator $d\hat{X}^{\mu}$ does not commute with $\hat{X}^{\mu}$, from
the perspective of the noncommutative differential geometry 078  or otherwise.
We have a constant $m_{\\!\scriptscriptstyle E}$ actually satisfying
$\frac{d\tau}{ds}=\frac{m_{\\!\scriptscriptstyle E}}{m}$. Identifying the
Einstein rest mass $m_{\\!\scriptscriptstyle E}$ with the Newtonian mass $m$
gives $s$ as $\tau$, in agreement with the usual textbook descriptions (more
of the classical case). Note that the $s$-independent Schrödinger equation
(for a spin-zero free particle) gives exactly the form of the Klein-Gordon
equation as
$\hbar^{2}\partial_{\mu}\partial^{\mu}\phi(x^{\mu})={m_{\\!\scriptscriptstyle
E}^{2}c^{2}}\phi(x^{\mu})$.

With the $H_{\\!\scriptscriptstyle R}(1,3)$ symmetry, as well as its classical
approximation, $P_{\mu}$ generate translations in $x^{\mu}$ and $Y_{\mu}$
generate translations in $p^{\mu}$. Such translational symmetries can be taken
as coordinate transformations on the phase spaces. As such, they are really
reference frame transformations. In fact, from the perspective of quantum
reference frame transformations qrf ; 093 , or that of our phase space as
quantum Minkowski spacetime 087 ; 078 , it is especially clear that the notion
of reference frames in a dynamical theory should really be one for the phase
space. Of course, the Lorentz transformations are also valid reference frame
transformations. With that understanding, one can see that taking any value of
$m_{\\!\scriptscriptstyle E}$ to the Newtonian mass $m$ can be achieved with a
reference frame transformation. Let us focus on the classical case as it is
more straightforward to interpret. For example, in a particle rest frame, a
translation in $p^{\scriptscriptstyle 0}$, or the energy value, is exactly one
in the $m_{\\!\scriptscriptstyle E}$ value. We are familiar with an arbitrary
absolute value of energy at least in Newtonian physics. It is easy to
appreciate that we can measure in practice only energy exchanges but never an
absolute energy value. In fact, without being able to do that, there is hardly
any way to check the exact validity of the on-shell mass condition
experimentally. Here, the symmetry picture suggests seeing setting a reference
to define the absolute energy value as a choice of frame of reference for the
Lorentz covariant phase space.

In a (classical) situation with a nontrivial interaction, with a nonzero
potential $V(x^{\mu})$ or under an electromagnetic field, the
$m_{\\!\scriptscriptstyle E}$ or $p_{\mu}p^{\mu}$ cannot be kept constant. It
goes without saying that any fixed choice of reference frame can only put
$m_{\\!\scriptscriptstyle E}$ as $m$ for one value of $s$, maybe only for one
of the particles involved. An interesting analysis of two interacting
particles has been presented in Ref.PR . The Hamiltonian is given as

$\displaystyle\frac{p_{a\mu}p_{a}^{\mu}}{2m_{a}}+\frac{p_{b\mu}p_{b}^{\mu}}{2m_{b}}+V(|x_{a}^{\mu}-x_{b}^{\mu}|)=\frac{p_{\mu}p^{\mu}}{2m}+\frac{q_{\mu}q^{\mu}}{2\mu}+V(r_{\mu}r^{\mu})\;,$
(5)

where we have $p^{\mu}=p_{a}^{\mu}+p_{b}^{\mu}$ as the conserved total
momentum, canonical variable conjugated to the center of mass position
$x^{\mu}=\frac{m_{a}x_{a}^{\mu}+m_{b}x_{b}^{\mu}}{m}$, $m=m_{a}+m_{b}$, and
$r^{\mu}$-$q^{\mu}$ being the other pair of canonical variables, for the
relative motion degree of freedom, given by $r^{\mu}=x_{a}^{\mu}-x_{b}^{\mu}$,
and $q^{\mu}=\frac{m_{a}p_{b}^{\mu}-m_{b}p_{a}^{\mu}}{m}$, with
$\mu=\frac{m_{a}m_{b}}{m}$ being the reduced mass. Actually, for the two
particles with an initial spacetime separation that is spacelike, i.e. with
$r_{\mu}r^{\mu}>0$, we can use a Lorentz transformation and a momentum, or
rather energy, translation symmetry to choose a frame of reference putting the
initial values of $r^{\scriptscriptstyle 0}$ and $q^{\scriptscriptstyle 0}$ to
zero, so long as the two particles do not have exactly the same mass.
Otherwise, as the authors noted, when the condition $j_{\mu\nu}j^{\mu\nu}>0$,
with $j^{\mu\nu}\equiv r^{\mu}q^{\nu}-r^{\nu}q^{\mu}$, is satisfied one can
achieve the same with simply a Lorentz boost. Hamilton’s equations of motion
then say $r^{\scriptscriptstyle 0}$ and $q^{\scriptscriptstyle 0}$ stay zero,
hence the nontrivial equations of motion, for the relative degree of freedom,
are really just

$\displaystyle
q^{i}=\mu\frac{dr^{i}}{ds}\;,\qquad\frac{dq_{i}}{ds}=-\frac{\partial
V(r_{i}r^{i})}{\partial r^{i}}\;,$ (6)

and $s$ the evolution parameter for the Hamiltonian. We can have then
$x_{a}^{\scriptscriptstyle 0}=x_{b}^{\scriptscriptstyle
0}=x^{\scriptscriptstyle 0}=\frac{p^{\scriptscriptstyle 0}}{m}s$, where
$p^{\scriptscriptstyle 0}c$ is the conserved total energy for the system,
giving $s$ as a rescaled value of the common coordinate time for the
particles. The dynamics is then essentially the same as the Newtonian theory.
Such frames may not be compatible with $p_{\mu}p^{\mu}=-m^{2}c^{2}$ though.
Note that what we used to think about as ‘relativistic’ dynamics is
essentially only about spacelike initial spacetime separation. More careful
analyses of other practical dynamical situations with the new symmetry
perspective taken into consideration are certainly of interest.

The basic picture of covariant Hamiltonian formulation, to the extent of the
actual Hamiltonians discussed, we have made no new contribution. That has all
been available in the literature. For the general validity of the theories,
classical or quantum, beyond the few cases discussed explicitly here, it
suffices to say that there is no problem at all for the formulation to at
least incorporate the successes of the usual formulation. At the classical
level, the textbook by Johns Jh , for example, discussed in some detail how
the usual Hamiltonian formulation can be cast in the ’equivalent’ alternative
of the covariant formulation in the ’extended’ phase space. In the quantum
case, the $s$-independent Schrödinger equations of motion can be seen as
exactly the usual equations. As discussed above, the $\phi(x^{\mu})$
wavefunction though can only truly be justified from our symmetry perspectives
presented. The current paper focuses on the symmetry theoretical and
background conceptual issues rather than the practical ones of applications.
Otherwise, the dynamical theory certainly can describe physics beyond what can
be done in the framework of the usual formulation. Some works in this
direction are already available in the literature, though in our opinion, not
enough attention has been paid on clarifying the physics pictures better. The
current article aims exactly at providing a better conceptual background for
such a task.

##  V Quantum Field Theories from $H_{\\!\scriptscriptstyle R}(1,3)$ Symmetry

Quantum field theories are formulated based on the perspective of ‘second
quantization’, promoting the notion of a wavefunction as the description of a
quantum multi-particle state to an operator that annihilates it, and the
Hermitian conjugate of the wavefunction to an operator that creates the state
from the vacuum as the ground state. So, a quantum field theory for a quantum
field is the ‘quantization’ of a ‘first quantized’, or ‘quantum particle’,
theory. From the point of view of symmetry theoretical formulation, the
‘single-particle’ state wavefunction should correspond to an irreducible
representation of the symmetry and the vector space of states, usually to be
taken as a Hilbert space, is the representation space. It is exactly in this
sense that textbooks on quantum field theories talk about the quantum fields
as irreducible representations of the Poincaré symmetry. The ‘multi-particle’
states for a quantum field at a fixed value of the ‘number of particles’
corresponds to an invariant part, with the proper exchange symmetry, of a
product representation. However, besides our critique about the ‘first
quantized’ ‘relativistic’ particle theory, for gauge fields such as the
electromagnetic field, the straight logic of ‘second quantization’ is at odd
with the classical picture of fields. There is certainly no two-step
quantization for the latter. Whether the ‘first quantized’ electromagnetic
field $A_{\mu}(x^{\nu})$ is a classical field or a quantum wavefunction of the
photon as a ‘particle’, or if some quantum wavefunctions can be seen as
classical fields, seems never been properly addressed. ‘Massless’ classical
particles are certainly not admissible in ‘nonrelativistic’ theories. It is
not clear either if there can be sensible theories about them without invoking
quantum notions even in the ‘relativistic’ setting.

In general, a quantum field theory cannot be seen as corresponding to a
representation of the background (relativity) symmetry. The creation and
annihilation operators, unlike the single- or multi-particle operators, cannot
be obtained in any way from a representation picture of that symmetry. For a
symmetry picture, they, or at least Hermitian linear combinations of them,
would be like generators of extra symmetries which extend the background
(relativity) symmetry of the corresponding ‘first-quantized’ theory to a
bigger symmetry. The quantum field theory would an irreducible representation
of the extended symmetry. Without a picture of the latter, we actually do not
have full symmetry theoretical formulation for a quantum field theory, not to
say a high energy theory involving a list of quantum fields with nontrivial
interaction among them. Though the ‘particle’ terminology is still in common
use in quantum field theories, ‘multi-particle’ states certainly have very
different physics beyond particle dynamics. At least from the symmetry
theoretical formulation point of view, the Newtonian notion of a particle as
an elementary constituent of the physical theory and a point mass cannot be
applied. Moreover, at least in principle, the state space of a quantum field
theory contains states which are nontrivial linear combinations of ‘multi-
particle’ states of a different ‘number of particles’. Note that direct
experimental results from high energy physics are essentially all obtained at
the classical limit.

While it may be interesting to construct a new quantum field theory from the
new theories of ‘relativistic’ quantum mechanics discussed, we do not intend
to do that here. The existing quantum field theories seem to have very
successful applications in high energy physics of the Standard Model. We want
to see if we can have some reconciliation of that with the new relativity
symmetry picture of $H_{\\!\scriptscriptstyle R}(1,3)$. With the above
clarification of the, actually quite limited, relation of a practical quantum
field theory in high energy physics and the background symmetry, we are ready
to illustrate how that can work. To focus more on the key features, working in
the simpler setting of the real scalar field serves the purpose best.

Representation theory of the $H_{\\!\scriptscriptstyle R}(1,3)$ has a special,
kind of degenerated, case. That is the case of zero Newtonian mass (as the
Casimir invariant from generator $M$). The case is tricky. For the key part of
$H(1,3)$, it is usually given as representation for the Lie group T . But at
the Lie algebra level, it is like trivializing the Lie brackets, similar to
the contraction to the classical limit or removing the relevance of the
central extension. Moreover, as the position observables are to be introduced
on the representation space as $\frac{1}{m}\hat{Y_{\mu}}$, they cannot be
properly defined. Yet, when we are looking at fields instead of particles,
that is not a problem. While the notion of a field has a background picture of
spacetime, it should not be expected to have a specific position in it. For
each value of $m\neq 0$, we have an irreducible representation. For $m=0$, we
have actually one-dimensional irreducible representations for the $H(1,3)$
group that has a group element
$e^{\frac{i}{\hbar}(\tilde{v}^{\mu}Y_{\mu}-\tilde{x}^{\mu}P_{\mu}+\tilde{u}M)}$
represented by the operator
$e^{\frac{i}{\hbar}(\tilde{v}^{\mu}y_{\mu}-\tilde{x}^{\mu}p_{\mu})}$, for real
four-vector parameters $\tilde{v}^{\mu}$ and $\tilde{x}^{\mu}$ T . It is easy
to appreciate. When $\hat{M}$ is effectively zero, all operators representing
elements of the Lie algebra commute. They can be taken as real numbers acting,
as scalar multiplications, in a one-dimensional vector space to make the
representations unitary. Consider such representation spaces as the span of a
single $\hat{P}_{\mu}$ eigenstate, the action of a $\hat{Y_{\mu}}$ as
multiplication of a real number $y_{\mu}$ need not be much of a concern for
moving towards the ‘second quantization’ framework. Extending an irreducible
representation of $H(1,3)$ to a spin-zero representation of the
$H_{\\!\scriptscriptstyle R}(1,3)$, different representation spaces of
$\left|p_{\mu}\right\rangle$ related by Lorentz transformations are pulled
together as a single irreducible representation. The resulting representation
space is the space of $\left|p_{\mu}\right\rangle$ with constant Lorentz
invariant $-m_{\\!\scriptscriptstyle E}^{2}c^{2}=p_{\mu}p^{\mu}$, or rather
the space of $\left|\vec{p}\right\rangle$. It is exactly like what we
discussed above for the Poincaré symmetry case. In fact,
$\hat{P}_{\mu}\hat{P}^{\mu}$ now commutating with all operators representing
elements of the $H_{\\!\scriptscriptstyle R}(1,3)$ Lie algebra serves as an
Casimir operator for the representation.

The $m=0$ irreducible representations of $H_{\\!\scriptscriptstyle R}(1,3)$
seem to have little structure even in comparison to a picture of classical
mechanics. It is hard to think about the systems described as quantum. Yet,
that may be enough for implementing the so-called ‘second quantization’, which
may proceed as follows. Take a $\left|\vec{p}\right\rangle$ state as the
‘single-particle’ state. Introduce the creation and annihilation operators
$\hat{a}_{\vec{p}}^{\dagger}$ and $\hat{a}_{\vec{p}}$ and the vacuum state
$\left|0\right\rangle$ with

$\displaystyle\hat{a}_{\vec{p}}\left|0\right\rangle=0\;,\qquad\left|\vec{p}_{1}\dots\vec{p}_{n}\right\rangle=\sqrt{2E_{\vec{p}_{1}}\dots
2E_{\vec{p}_{n}}}\hat{a}_{\vec{p}_{1}}^{\dagger}\dots\hat{a}_{\vec{p}_{n}}^{\dagger}\dots\left|0\right\rangle\;,$
(7)

where $E_{\vec{p}_{n}}\equiv\sqrt{|\vec{p}|^{2}+m_{E}^{2}c^{2}}(c)$, together
with the Lorentz invariant inner product
$\left\langle\vec{p}_{1}|\vec{p}_{2}\right\rangle={2E_{\vec{p}_{1}}}(2\pi)^{3}\delta^{3}(\vec{p}_{1}-\vec{p}_{2})$.
Of course, the usual commutation relations among those creation and
annihilation operators are also assumed. What one has then is essentially a
picture of quantum field theory, as for example presented in Maggiore M . In
fact, it is exactly this momentum eigenstate picture that is mostly used in
applications like Feynman diagram calculations. One can even define the field
operator
$\phi(x)\equiv\int\\!\\!\frac{d^{3}p}{(2\pi)^{3}\sqrt{2E_{\vec{p}}}}(e^{-ipx}\hat{a}_{\vec{p}}+e^{ipx}\hat{a}_{\vec{p}}^{\dagger})$,
as a linear combination. The basic mathematical framework of the usual quantum
field theory is thus obtained, though under a quite different line of
thinking.

We have seen above that the so-called ‘second quantization’ offers a procedure
to construct the ‘quantum field theories’ that requires very little
background. A usual picture of some ‘first quantized’ quantum particle
theories may be by-passed from the perspective of a symmetry theoretical
formulation. It can work from the $m=0$ irreducible representations of
$H_{\\!\scriptscriptstyle R}(1,3)$ which in themselves hardly give an
interesting dynamical setting, unlikely the $m\neq 0$ cases we discussed in
the previous section.

##  VI Concluding Remarks

It should be clear that in all considerations of ‘relativistic’ quantum
mechanics, one always has the position and momentum observables satisfying the
Heisenberg commutation relation incorporated, though probably only the spatial
part may be taken explicitly. That is, of course, what we need in the theories
but not what a first principle Poincaré symmetry theoretical formulation can
offer. We are really dealing with representations of $H_{\\!\scriptscriptstyle
R}(1,3)$ but in the name of its subgroup of Poincaré symmetry. The magnitude
(square) of the momentum four-vector changes under the action of a position
operator and hence cannot be a constant of the theory. Physicists keep
thinking the latter is the right symmetry, we think, mostly for two main
reasons. First of all, the symmetry seems to be the natural symmetry of
Minkowski spacetime and seems to be fine for the classical mechanics though
one cannot quite have a covariant phase space (see Ref.086 ). Secondly, it is
mostly from the on-shell mass condition, though really that of free particles,
that we appreciate the famous equation $E=mc^{2}$ giving the great
understanding that mass is a form of energy. Poincaré symmetry dictates the
condition. $H_{\\!\scriptscriptstyle R}(1,3)$ actually does not admit that.
And naturally, most physicists do not check through the straight logic of a
rigorous symmetry theoretical formulation, yet easily accept a claim that it
works. From our presentations, it should be clear that it is the
$H_{\\!\scriptscriptstyle R}(1,3)$ that really is the right symmetry. We have
discussed the perspective that it is the symmetry for the particle phase space
that gives a full formulation of a theory of particle dynamics that should be
taken as the true relativity symmetry of the theories 095 . A picture of the
space(time) as some may naively conceive, other than the picture of the
configuration (sub)space of single-particle phase space is not physical and of
no interest to physics so long as particle dynamics is concerned. While
logically speaking, the notion of spacetime in a field theory may be
different, how one can justify a physical notion of spacetime based completely
on the mathematical formulation of a field theory in itself looks, in our
opinion, quite problematic. We have presented earlier a consistent theory of
Lorentz covariant quantum mechanics from the symmetry theoretical formulation
based on $H_{\\!\scriptscriptstyle R}(1,3)$ 087 . At the proper symmetry
contraction limits, the full dynamical theory gives the ‘nonrelativistic’ and
the classical theories as approximations. We further highlighted in this
article that the on-shell mass condition actually does not hold even in the
simple case of the dynamics of a charged particle under a background
electromagnetic field, and is violated by the action of any position operator.
The case for $H_{\\!\scriptscriptstyle R}(1,3)$ against the Poincaré symmetry
should be seen as firmly established.

Particle dynamics from the $H_{\\!\scriptscriptstyle R}(1,3)$ and its
classical limit gives no constant Einstein rest mass in general. Though one
can choose a reference frame to put any initial value of it to the fixed
Newtonian mass $m$, it may not be maintained in the dynamical evolution. That
can clearly be established from the dynamics of a charged particle under a
Lorentz force. The key is to identify the correct momentum as the canonical
momentum. Mass as a form of energy is a great insight from Einstein, which we
do not contest. But there is no fixed value of the magnitude of the momentum
four-vector given to be the particle mass or otherwise.
$m\frac{dx^{\mu}}{d\tau}$ has constant magnitude by definition. But that has
no dynamic content. The actual evolution parameter $s$ from a general
covariant Hamiltonian formulation may have a nontrivial relation to any
particle proper time $\tau$, i.e. $\frac{d\tau}{ds}$ is not necessarily a
constant in a generic situation. The latter depends on the actual interaction
potential, which is not allowed from a Poincaré symmetry formulation with the
on-shell mass condition. Up to this point, we do not have a practical
situation where such a particle interaction picture applies. However, we do
have solid applications of the kind of particle interaction picture in
Newtonian mechanics, and the quantum analog. The corresponding particle
interaction picture from the $H_{\\!\scriptscriptstyle R}(1,3)$ formulation
seems to be the only Lorentz covariant quantum particle dynamics picture from
which the ‘nonrelativistic’ theories can be retrieved as approximations.

The dynamical ‘velocity’ $\frac{dx^{\mu}}{ds}$ is not the Einstein four-
velocity $\frac{dx^{\mu}}{d\tau}$. We want to emphasize that, from the
Hamiltonian formulation, a choice of Hamiltonian $H_{s}$ comes along with its
evolution parameter $s$ which is non-negotiable. In a dynamical setting
involving multiple particles and even fields, for example, we do not have to
guess at the relation between $s$ and the other physical time parameters such
as the various individual particle proper times. Nor do we have to ponder the
question if the latter can be correlated a priori. Find the correct $H_{s}$
and the rest follow from the mathematics. The correct $H_{s}$ can be found
from practical considerations or, on the theoretical side, from feasible
generalizations of successful ‘nonrelativistic’ theories and checked from the
validity of the results. Any mathematical $s$-evolution results may have to be
physically interpreted in terms of some time evolution picture, under a choice
of frame of reference (for the phase space).

A very practical ‘relativistic’ theory is the theory of electrodynamics.
However, a consistent formulation of the classical theory in the general
setting of more than one charged particle is quite a nontrivial matter R .
There is also a version of it without the electromagnetic field as a true
independent constituent FW . Otherwise, electrodynamics generally involved
interacting particles and the electromagnetic field. Our current theory of
quantum electrodynamics is one of interacting quantum electromagnetic (or
photon field) and quantum fields of ‘charged particles’ like the electron. We
illustrate above how quantum field theories may be obtained from applying some
form of ‘second quantization’ to $m=0$ irreducible representations of
$H_{\\!\scriptscriptstyle R}(1,3)$. Yet, the latter hardly looks like any
‘first quantized’ particle theory. Actually, from the perspective of the usual
formulation as given in textbooks, so long as a classical field theory is
concerned, there is really no two-step quantization. That picture of
‘quantization’ of the classical electromagnetic field and the ‘second
quantization’ leading to the matter fields like the electron field are
conceptually quite different. We do not really have a theoretically and
mathematically rigorous way to formulate our theory of quantum electrodynamics
as the quantization of (one of the) theory of classical electrodynamics, or
the latter as the classical approximation of the former. Other quantum fields
such as the quark fields and the nonabelian gauge fields have no practical
classical picture anyway. A comprehensive formulation framework for all the
fundamental theories from Newtonian mechanics to Standard Model high energy
theory is still a task to be accomplished, not to say with quantum gravity
included. Our current study is only a small effort in relation.

A comment on the two different notions of masses is in order. While it is
clear that we have given a nice symmetry theoretical notion for the Newtonian
mass $m$ in both ‘nonrelativistic’ 095  and ‘relativistic’ quantum setting, as
a very different notion from the Einstein rest mass $m_{\\!\scriptscriptstyle
E}$. For a quantum field theory as discussed, we have $m=0$ and an independent
$m_{\\!\scriptscriptstyle E}$ as characteristic properties. One may worry
about their compatibility. For this matter, it is interesting to note that all
the quantum fields other than the Higgs doublet in the Standard Model of high
energy physics are actually massless. They are chiral fermionic fields and
gauge fields nonzero masses for which are ruled out by the gauge symmetries.
The Higgs doublet has a negative mass-square. The masses in quantum field
theories are really energy, or renormalization, scale-dependent parameters.
The only possible constant value as basic characteristic properties of the
entities is the kind of symmetry-protected vanishing value. The negative mass-
square parameter for the Higgs doublet is never a notion of physical
characteristic. Besides, the doublet is widely believed to be a composite. All
physical masses come from interaction couplings to the doublet after
spontaneous electroweak symmetry breaking. So, apart from the general fact
that a more serious and mathematically rigorous understanding of the
relationship among the various theories, from the point of view of the
symmetry theoretical formulation or otherwise, still has to be studied, the
analysis presented in the article does not post new compatibility issues for
‘mass’.

We have mentioned above the admissible noncommutative geometric picture the
phase space for quantum mechanics with the position and momentum operators as
noncommutative coordinate observables 087 ; 078 . The perspective, when taken
seriously, certainly asks for different thinking about the existing field
theories. The Schrödinger wavefunction is really a set of complex number
coordinates for the phase space, for the description of a state in quantum
mechanics as a point inside. In our opinion, there has never been any good
reason to think about the real ‘position’ variables for the wavefunction as
coordinates for the physical space other than as some form of the classical
approximation of such. The position operators as observables describing the
particle position should rather be the natural choice. The idea of quantum
fields being fields of the classical, Minkowski, model of spacetime may be
much restricted to looking at the quantum theory from a far too classical
perspectives. We do not look into the construction of quantum field theory as
a ‘second quantized’ version of quantum mechanics from the
$H_{\\!\scriptscriptstyle R}(1,3)$ symmetry perspective and the noncommutative
geometric picture. That is because we see up to this point no good reason for
believing in the direction. After all, our main aim here is to look at the
proper symmetry theoretical formulation of ‘relativistic’ quantum physics and
its implications. ‘Second quantization’ in itself, as illustrated, is not much
of a symmetry theoretically based notion.

We have brought up many questions about consistent conceptual and theoretical
pictures of Lorentz covariant theories of quantum physics fully compatible
with symmetry theoretical perspectives. A lot more effects are certainly
needed to clarify that all. The existing quantum field theory is never short
of conceptual difficulties. Many have taken the practical approach of Feynman
not to worry about that. However, to push for further advance in fundamental
theoretical physics, it is important to take up the kind of question more
seriously.

## Appendix: Supplement to the symmetry theoretical formulation: particles
with nonzero spin.

Here, we sketch some more results from the $H_{\\!\scriptscriptstyle R}(1,3)$
symmetry theoretical formulation, along the line of 095 , including explicitly
irreducible representation for nontrivial spin cases.

We start with an irreducible representation of the Heisenberg-Weyl subgroup
$H(1,3)$, based on (pseudo-)Hermitian operators
$\hat{X}_{\mu}\equiv\frac{1}{m}\hat{Y}_{\mu}$ and
$\hat{P}_{\mu}\equiv\frac{1}{c}\hat{E}_{\mu}$, with
$[\hat{X}_{\mu},\hat{P}_{\nu}]=i\hbar\eta_{\mu\nu}\hat{I}$ for $m$ being the
eigenvalue of the effectively Casimir operator $\hat{M}$. The representation
is essentially as the one presented in Ref.087 , with only necessary easy to
trace adjustments involving the identification of $M$ as the central charge
generator. Similar to the $H_{\\!\scriptscriptstyle R}(3)$ case, an
irreducible representation of $H_{\\!\scriptscriptstyle R}(1,3)$ is given by
as a direct product of one for the Heisenberg-Weyl group $H(1,3)$ and one for
an $SO(1,3)$, or its double cover $SL(2,C)$, generated by
$T^{\prime}_{\mu\nu}\equiv
MJ^{\prime}_{\mu\nu}-({Y}_{\mu}{E}_{\nu}-{Y}_{\nu}{E}_{\mu})$. We have
$\hat{S}_{\mu\nu}=\frac{1}{mc}\hat{T}^{\prime}_{\mu\nu}=\hat{J}_{\mu\nu}-\hat{L}_{\mu\nu}$,
where $\hat{J}_{\mu\nu}=\frac{1}{c}\hat{J}^{\prime}_{\mu\nu}$ and
$\hat{L}_{\mu\nu}\equiv\hat{X}_{\mu}\hat{P}_{\nu}-\hat{P}_{\mu}\hat{X}_{\nu}$.
The corresponding two Casimir operators,
$\frac{1}{2}\hat{S}_{\mu\nu}\hat{S}^{\mu\nu}$ and
$\frac{1}{4}\epsilon^{\mu\nu\rho\sigma}\hat{S}_{\mu\nu}\hat{S}_{\rho\sigma}$
have eigenvalues $2\hbar^{2}[s_{\\!\scriptscriptstyle
L}(s_{\\!\scriptscriptstyle L}+1)+s_{\\!\scriptscriptstyle
R}(s_{\\!\scriptscriptstyle R}+1)]$ and $2\hbar^{2}[s_{\\!\scriptscriptstyle
L}(s_{\\!\scriptscriptstyle L}+1)-s_{\\!\scriptscriptstyle
R}(s_{\\!\scriptscriptstyle R}+1)]$, respectively, for the labeling with the
left-handed and right-handed spins $\\{m,s_{\\!\scriptscriptstyle
L},s_{\\!\scriptscriptstyle R}\\}$ of dimension $(2s_{\\!\scriptscriptstyle
L}+1)(2s_{\\!\scriptscriptstyle R}+1)$. Explicitly, we have
$\hat{S}_{ij}^{\pm}=\frac{1}{2}(\hat{S}_{ij}\pm
i{\epsilon_{ij}}^{k}\hat{S}_{{\scriptscriptstyle 0}k})$, from
${T^{\prime}}_{ij}^{\pm}=\frac{1}{2}(T^{\prime}_{ij}\pm
i{\epsilon_{ij}}^{k}T^{\prime}_{{\scriptscriptstyle 0}k})$, (here
${\epsilon_{ij}}^{k}=\epsilon_{ijh}\delta^{hk}$, with $\epsilon_{ijh}$ being
the totally antisymmetric tensor), as generators of the two independent
$SO(3)$ under the isomorphism $SO(1,3)\simeq SO(3)\oplus SO(3)$ as complex Lie
algebra. The picture of irreducible representations for particles with
nontrivial spins having the same form of wavefunctions as a zero-spin
wavefunction $\phi(x^{\mu})$ for each spin component though familiar from
textbooks cannot be fully justified from Poincaré symmetry picture as the
group cannot be seen as a direct product. Without the generators $Y_{\mu}$ in
the Lie algebra, the $T^{\prime}_{\mu\nu}$ do not exist as elements in the
universal enveloping algebra.

To get to the approximation with Lorentz boosts replaced by Galilean ones, we
take the Lie algebra and the representation of interest to the $c\to\infty$
limit of $K_{i}=\frac{1}{c^{2}}J^{\prime}_{i{\scriptscriptstyle 0}}$,
$P_{i}=\frac{1}{c}E_{i}$, $U=\frac{-1}{c}Y_{\scriptscriptstyle 0}$, and
$J_{ij}=\frac{1}{c}J^{\prime}_{ij}$ with the renaming $H\equiv-
E_{\scriptscriptstyle 0}$. Mathematically, it is to be formulated as a
symmetry contraction 087 , taken from the Lie algebra to the group, the proper
extension of the universal enveloping algebra or group $C^{*}$-algebra and
their relevant representation as a single representation 070 ; 087  for the
full dynamical theory, including the symplectic structure. In this case, it is
easy to see that the generator set $\\{J_{ij},K_{i},P_{i},H\\}$ gives exactly
a Galilean group $G(3)$ without the central extension. The set
$\\{J_{ij},Y_{i},P_{i},M\\}$ gives an $H_{\\!\scriptscriptstyle R}(3)$ as the
symmetry for the Galilean quantum particles. We have

$\displaystyle[U,H]=-i\hbar
M\;,\quad\quad[K_{i},Y_{j}]=-i\hbar\delta_{ij}U\;,$ (8)

as the extra nontrivial Lie products among the full set. For the irreducible
representation $\\{m,s_{\\!\scriptscriptstyle L},s_{\\!\scriptscriptstyle
R}\\}$, we have $\hat{P}_{i}$ directly representing $P_{i}$,
$\hat{H}=-c\hat{P}_{0}$ which apparently goes to infinite limit, and
$\hat{T}=\frac{1}{m}\hat{U}=\frac{-1}{c}\hat{X}_{\scriptscriptstyle 0}$ which
goes to the vanishing limit. $\hat{X}_{i}$ and $\hat{S}_{ij}$ are unchanged.
$\hat{J}_{ij}=(\hat{X}_{i}\hat{P}_{j}-\hat{P}_{i}\hat{X}_{j})+\hat{S}_{ij}$ is
retrieved. $\hat{K}_{i}=\frac{1}{c}\hat{S}_{i{\scriptscriptstyle
0}}-\frac{1}{c^{2}}\hat{X}_{i}\hat{H}+\hat{P}_{i}\hat{T}$ which has the limit
as $\hat{P}_{i}\hat{T}$. If one put the parallel Lorentz boost to Galilean
boost contraction onto the Lie algebra of the $\hat{S}_{\mu\nu}$, one can
easily see that the left- and right-handed spin generators both become
essentially just $\hat{S}_{ij}$ (with a factor $\frac{1}{2}$, which makes no
practical difference). Of course the particle mass as the key Casimir
invariant maintains. Hence, one obtains $\\{m,s\\}$ of
$H_{\\!\scriptscriptstyle R}(3)$ from $\\{m,s_{\\!\scriptscriptstyle
L},s_{\\!\scriptscriptstyle R}\\}$.

For a composite system, parallel analysis as shown above for the
$H_{\\!\scriptscriptstyle R}(3)$ symmetric ‘nonrelativistic’ case obviously
goes through, with additive mass $m=m_{a}+m_{b}$, the notion of center of mass
and $\frac{1}{2}\hat{S}_{\mu\nu}\hat{S}^{\mu\nu}$, here together with and also
$\frac{1}{4}\epsilon^{\mu\nu\rho\sigma}\hat{S}_{\mu\nu}\hat{S}_{\rho\sigma}$,
as effective Casimir operators for the composite are obtained from the product
representation. $\hat{S}_{\mu\nu}$ for the contains the ‘orbital’ angular
momentum $\hat{R}_{\mu}\hat{Q}_{\nu}-\hat{Q}_{\mu}\hat{R}_{\nu}$, which is a
contribution to the internal or intrinsic angular momentum for the composite.
$\hat{R}_{\mu}$ and $\hat{Q}_{\mu}$ are position and momentum variables for
the relative motion degree of freedom, respectively. We have
$\hat{S}_{\mu\nu}=\hat{S}_{a\mu\nu}+\hat{S}_{b\mu\nu}+\hat{R}_{\mu}\hat{Q}_{\nu}-\hat{Q}_{\mu}\hat{R}_{\nu}$
and $\hat{J}_{\mu\nu}=\hat{X}_{\mu}\hat{P}_{\nu}-\hat{P}_{\mu}\hat{X}_{\nu}$
with $\hat{X}_{\mu}$ and $\hat{P}_{\mu}$ being the center of mass position and
momentum.

Acknowledgments:  The work is partially supported by research grants number
110-2112-M-008-016 and 111-2112-M-008-029 of the MOST of Taiwan. OK also has
the support of a Visiting Professorship at Korea Institute for Advanced Study
during the later phrase of the study.

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