Journal of Modern Physics, 2015, 6, 1586-1590
Published Online September 2015 in SciRes. http://www.scirp.org/journal/jmp
http://dx.doi.org/10.4236/jmp.2015.611160
How to cite this paper: Shapiro, V.E. (2015) The Energy Conservation Paradox of Quantum Physics. Journal of Modern
Physics, 6, 1586-1590. http://dx.doi.org/10.4236/jmp.2015.611160
The Energy Conservation Paradox
of Quantum Physics
V. E. Shapiro
Shapiro Physics, Vancouver, Canada
Received 25 August 2015; accepted 18 September 2015; published 21 September 2015
Copyright © 2015 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativ ecommon s.org/l icens es/by/4. 0/
Abstract
This work asserts that quantum theory runs into a fundamental conflict with the principles of
energy conservation inferred from the statistical evolution of interacting systems. The gist is the
energy of systems by the principles of Lagrangian mechanics leaves out of account their energy
associated with the phase flows of non-invariant phase volume. The quantum theory takes this
fact into account, but does that improperly. We show it by presenting insoluble inconsistencies
and a case study.
Keywords
Quantum Energy Transfer, Phase Volume Invariance, Detailed Balance, Vortex Energy
1. Introduction
The conserva tion of e nergy/ matter via tra nsformatio ns is in the hear t of phys ics, it s unifyi ng guid eline. I t relies
on the fact that the established consistent pattern formation of the world around us is basically relaxation to
recurrent states of rest and motion stable against ambient chaos. But with these attributes the notion of energy
assumed in line with the principles of analytical formalism going back to Euler and Lagrange leaves out of
account a complementary energy, we called it vortex energy, that makes an essential part of the whole of
systems energy. It exhibits in systems phase space by flows with broken phase volume invariance (Liouville
theor e m), ind ec ompo sab le int o i nvaria nt -volume flows. The notion of vortex energy was introduced into energy
percep tion in [1] [2].
An evident example of conserved vortex energy is the energy of rigid bodies rolling without sliding on a
plane. This and overwhelmingly diverse mechanics of phenomena inherit constraints on the motion not
reducible to independent variations of generalized coordinates of the systems. Such constraints called nonholo-
nomic by Hertz [3]-[5] are not integrable. They violate the variation principle of stationary action and the asso-
ciated energy decomposition formalism; in terms of kinetic rates they emerge as the limit opposite to detailed

V. E. Shapiro
1587
balance-utmost str ong fr ictio n causi ng the no nhol ono my of z ero -time relaxation. T he sa me no less app lies to the
world of syste ms evolution wi th d if fusion and rela xation of f inite r a te s beyond detailed b a la nce.
The vortex form of energy accounts for these factors. The process of breaking trends of phase volume
invariance can be viewed as due to the nonlinear cumulative effect of ambient chaos. Given that quantum
physics is postulated as inherent in trends beyond that of systems governed by a classical Hamiltonian, it is
imperative for this physics to be within that of vortex energy realm [2]. Our conclusion refutes the beliefs
banning the classical routes to the realms called quantum, unveils their vortex nature related to irreversible
processes. But what is more, and will be questioned below, is the steadfast adherence of quantum theory to the
principles of energ y conservation.
2. Problem Setting
The quantum theory assumes existing an ideal where the evolution of systems is described by a wave function
( )
,zt ψ
obeying Schroedinger equation
ˆ,iht H
ψψ ∂ ∂=
(1)
its Cauchy problem with suitable boundary conditions and governed by a quantum Hamiltonian
ˆ
H
, a
Hermitian operator having noncommuting terms bound to Planck’s co nstant h;
21i= −
, and the abso lute val ue
2
ψ
is taken for the densit y of statistical d istribution o f systems state s z at insta nt t. Wit h the obs ervab le va lue s
of
ˆ
H
defined in this q uantum ide a l in Dir ac’s notation as the averages
( )
ˆˆ ˆ,H Hz
ψψ =
(2)
and with all other observable system’s proper ties defi ned li ke i n (2) where t he e xplicitl y writte n
ˆ
H
is replaced
by Hermitian operators associated with observables, the eigen spectrum of
ˆ
H
governing the evolution of
ψ
is assigned the observable measure of energy.
While the evolution governed by a classic Hamiltonian, or a function of it within the concept of generalized
thermodynamic potential [6] and the wider concept in [1] [2], is reversible and preserves phase volume
invariance, it is clear that the distribution
2
ψ
may not preserve these features when
( )
,zt ψ
is continuously
diverged by noncommuting operations. But where is a niche for all this quantum ideal in the vortex energy
realm?
We shall go into that of given b y the statistical d istr ibution function,
( )
,zt ρ
, smooth in the same phase space
as
2
ψ
and obe yi ng a ge neral continuity equati on
( ) [ ]
ˆ,,tdiv vHI
ρ ρρ ∂ ∂=−=+
(3)
its Cauchy problem under the same natural boundary conditions. Similarly to the evolution of
2
ψ
, the terms in
(3) and the solution to it are assumed existing, preserve
0 ρ ≥
and normalization
d1 ρ Γ=
∫
over the same
phase space
Γ
of system variables
{ }
,
ii
z xp =
, a set of n pairs of generalized coordinates x and momenta p.
The term
ˆ
v ρ
in (3) is the flux density of phase fluid, a 2n-vector functional of
ρ
non-anticipating and
nonlocal in z with
ˆ
v
an integro-differential operator of phase fluid velocity.
( )
,HH zt=
with
[ ]
,
a Poisson
bracket
[ ] ( )( )( )( )
,
ii ii
abax byaybx  =∂∂∂∂ −∂∂∂∂

∑
(4)
(summing is over all i’s of n) is a Hamiltonian governing an arbitrary chosen ideal of phase fluid flow in
Γ
,
ideal in the sense that
ˆ
v
reduces then to a vortex-free field of instant velocities
( ) [ ]
,,v ztzH=
, for then
[ ]
,0divz H≡
. The functional
[ ]
II ρ =
embodies the constraints beyond this ideal, including terms of
irreversible drift and diffusion. Continuous systems and relativism can be treated with n taken unlimited and t
the time of a single observer.
3. The Domain of Vortex Energy
Since the Poisson bracket expressions
[ ]
,ab
and phase volumes are invariant under canonical transformations,

V. E. Shapiro
1588
so the terms
t ρ ∂∂
and I of (3) are. But the evolution of
ρ
by Equation (3) and these terms are in effect
invariant only in the case of the ideal where I comes down to a Poisson bracket
[]
, H ρ
′
with
H′
a func tion of
z, t, thus making the sum
HHH
′
= +
a generalized potential (a dressed Hamiltonian) governing the system.
Obviously it is not the patches beyond this ideal that are rare in reality, it is the patches of ideal. Beyond this
ideal, the terms
[ ]
,H ρ
′
, I and
t ρ ∂∂
are hardly invariant and cause persistent flows already in the very
stati ona r y c o ndi tions of energ y co nse r va ti o n, wher e Equa tion (2) is a utonomo us ( see [2]). Such au t onomy offer s
a clearer view of the vortex energy realm versus the quantum postulates by (1), (2). Below we shall concentrate
on this case.
The realm of the classical ideal in point is then the domain of stationary vortex-free phase fluid flows of
density
()( )
()
Hzz ρρ =
with
H0 ρ ∂∂≤
and H bound from below for stability. This necessitates rigid
constraints on I: the irreversible drift and diffusion must then be in detailed balance-tend to zero in self-
compensation for each of all n degrees of system freedom. As diffusion means sources, the irreversible drift is to
be sinks, dissipatio n, cause relaxation to the state o f ideal. In conditions of e nergy conservation be yond detailed
balance, the irreversible drift and diffusion inevitably ca use persistent (not entrai ned in stationary chaos on the
average) phase fluid flows.
An e xotic exa mple of s uch i mb alanc e is t he s yst ems with no n-holonomic constraints that do no work on each
of n degrees of systems freedom; it can be viewed as a limit of zeroth diffusio n and utmost stron g frictio n, i.e.,
zero-time r ela xa t io n ca u si ng t he non -holonomy. Both this limit and a general imbalance between the irreversible
dri ft and d iffu sio n of fi nite rat es, whic h do work o n the syst e m, but in a wa y of ze ro th tot al work o n the syste m,
are within the realm of vortex energy. This encompasses all conditions of conserved vortex energy realm with
its persistent flows of phase f luid within autonomic Equation (3).
Let us now juxtapose the outlined realm of vortex energy with that of density distribution
2
ψ
for the
quantum-ideal proble m statem e nt.
4. Fundamental Contradictions
The quantum theory is indissolubly related to the uncertainty principle, hence, its domain is to be beyond the
purely dynamical non-holonomic systems of zeroth diffusion. The quantum phenomena to be observed as
recurrent against ambient chaos suppose conditions of relaxation, stability and ergodicity, and all that, if this
were the case, would be only in the vortex energy realm beyond the ideal nonholonomy. But there is no niche
for the systems having discrete energy spectrum of states in the vortex energy realm of diffusion and relaxation
of finite rates: finite rates inevitably cause spectra of finite line-widths.
So, the inherent hallmarks that together constitute the notion of quantum ideal have no niche in whatever
autonomic conditions in question. It means that the quantum ideal construction fails as a rigorously defined
bridge to energy conservation observations, appears without footing beyond the classic ideal. These things are
verities when treating the states of quantum systems as a mixture of pure states each given a statistical weight in
the sense used in classics, i.e., reso rting to a semi-cla ssical appr oach to the irreversible kinetics and the concept
of entropy.
Along with the incompa tibility set-forth abo ve, the followi ng basic inconsiste ncy takes place. Its crucial point
is the density distr ibu tion
2
ψ
of the quantum ideal, its pure states, is independent of t in autono mic co nditions.
The solutions to autonomic (3) in detailed balance are also independent of t as then
( )
( )
Hz ρρ =
, and it is
assumed so in the semi-classics modeling in question. However, the conditions of energy conservation of
autonomic Equation (3) beyond detailed balance is another story and makes a difference.
Thereat, it is not just that
0t ρ ∂ ∂≠
in (3) links to imbalance, it is the variations (correlations) between
densities of phase fl uid at different t’s is a matter of princip le, signifie s persistent flo ws of fluid as observable in
the
( )
,zt
space of systems in the stationary realm of vortex energy. So, while t-dependent phenomena in
stationary outer conditions are usually referred to non-equilibria, this is not the case of vortex energy realm,
where it is just the equilibriu m states of motio n with conse rved energy [2]. The persistent flows in point can be
fast and stable depending on conditions of self-sustained imbalance. It is largely admitted of a test.
The same should be expected from the quantum phenomena, including its ideal by (1) and (2) for
ˆ
H
independent of t. However, since
( )
,zt ψ
reduces then to a sum
()( )
,e
k
it k
zt z
λ
ψψ
−
=
∑
(5)

V. E. Shapiro
1589
over the basis of orthogonal eigen-functions
()
{}
k
z ψ
of
ˆ
H
with reals
{ }
k
λ
its eigen values, it yields
vanishing temporal variations of
2
ψ
in pure states. The null variance means that the energy of a pure state
cannot be associated with circulating vortex energy conserved in the system, never represents its energy level.
This is the point.
It cannot be otherwise, since the vortex energy as a function (generally a functional) dual to that of given by a
function of state s in heren t in flows of invaria nt p hase volum e is integral, ind eco mposab le i nto flows o f in variant
volume. To identify persistent currents with a pure quantum state, a ground state or some other state, is not
viable in this sense. It pertains to the symmetry in space-time and balance theorems of quantum theory going
back to Heitler, Coester, Watanabe [7]-[9]. Treating the pure states of quantum systems as statistically
independent, of random phases and this or that statistics of quantum energy spectrum levels treated as energy
levels, appears a theory also lacking steadfast adherence to the vortex energy realm of energy conservation.
5. Case Study
A telling exa mple is the much -used Manley-Rowe p o wer -fre quenc y r elati ons o f no nli near wave -mixi ng. Pro ved
first ad hoc for classical resonant systems of nonlinear capacitors and inductors [10], these relations are accepted
since then widely, see review [11], as intrinsic also in t he qu antum e nergy tran sfer a nd ste mming fro m its theor y,
e.g. [12] [13]. For a three-level quantum scheme the relations mean
12 1321 23
0 and0NN NN+= +=
(6)
where
ij ji
NN= −
is the number of quanta per second going from level i to level j, t he e nerg y in e ac h qua ntu m
is
hf
with f the frequency of the transition, so
12 12
N hf
is construed as the power in frequency
12
f
. The
general multi-level r elations are co mmonly reasoned analogously-as equ ivale nt o f t wo p ri ncip les, t hat of e ner gy
conservation and that of detailed balance.
However, the two principles are in contradiction, not in the least compatible beyond the classic limit, since
detailed balance is referred there to balance between quantum energy levels, hence, beyond the realm of energy
conception where each level is a measurable energy state. The mistaken view is detrimental to search for new
methods of wave mixing.
It has a direct bearing also on the popular concept of quasi-energy for the systems in high frequency fields.
The averaged impact of hf fields reduces then to that of an effective potential. Treated classically the concept
completely abstracts away of the vortex energy, which appears in the long run particularly incorrect for the
phenomena of laser cooling and trapping in optics and other hf fields, see [1] [2]. Treated by quantum theory or
semi-classically, it does not clear up the trouble while adds new ones.
A telling example is also the concept of quantum computers, since its quantum principle is untenable, in
complete dependence of the quantum ideal in question. There, as well as in many other things, forming
conclusions based on quantum principles needs correction. It includes the entropy concepts for reasoning of
stability of systems in vortex energy realm; the unfit of classical entropy concepts there was noted in [2].
6. Resume
The contradictions unveiled raise fundamental questions about the whole point of energy and matter—its
integral, cumulative bond with the ambient chaos. We addressed here one of its basic aspects, the quantum
energy transfer, and have shown that its theory appears in an insoluble conflict with the energy conservation
principles which the phenomena called quantum really display. Really they display the energy duality with
respect to phase volume invariance, while quantum postulates depart from its principles.
The alternative, keep putting the quantum postulates as a guiding idea above all, appears fraught with
depriving the physics based on energy of tangible ground based on work done by the systems. Proving the
contradictions, the term vortex energy was invoked for classification, but with its principles we have provided
the stuff for rethinking the conventional practice.
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V. E. Shapiro
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