Theories of variable mass particles and low energy nuclear phenomena

Theories of variable mass particles and low energy nuclear

phenomena

PREPRINT, May 4, 2013

Mark DavidsonSpectel Research Corp., Palo Alto, USA

E-mail: [email protected]

Abstract. Variable particle masses have sometimes been invoked to explain observed anomaliesin low energy nuclear reactions (LENR). Such behavior has never been observed directly, and isnot considered possible in theoretical nuclear physics. Nevertheless, there are covariant o-mass-shell theories of relativistic particle dynamics, based on works by Fock, Stueckelberg, Feynman,Greenberger, Horwitz, and others. We review some of these and we also consider virtual particlesthat arise in conventional Feynman diagrams in relativistic eld theories. Eective Lagrangianmodels incorporating variable mass particle theories might be useful in describing anomalousnuclear reactions by combining mass shifts together with resonant tunneling and other eects. Adetailed model for resonant fusion in a deuterium molecule with o-shell deuterons and electronsis presented as an example. Experimental means of observing such o-shell behavior directly, if itexists, is proposed and described. Brief explanations for elemental transmutation and formation ofmicro-craters are also given, and an alternative mechanism for the mass shift in the Widom-Larsentheory is presented. If variable mass theories were to nd experimental support from LENR, thenthey would undoubtedly have important implications for the foundations of quantum mechanics,and practical applications may arise.

Keywords: hydrated palladium, deuterated palladium, LENR, fusion

PACS numbers: 24.10.-i, 3.65.-w, 3.65.Nk, 3.70.+k

AMS classication scheme numbers:

Submitted to:

2 Theories of variable mass ...

1. Introduction

Many unexplained anomalies have been observed in heavily deuterated palladium, as well as othermetal-hydrogen alloys [1, 2] which suggest that nuclear reactions are taking place in condensedmatter. There is severe skepticism in the physics community about these claims. This literature iscomplex, and the subject cannot be mastered easily. It is this author's opinion, after very extensiveand long-term study of the literature, that the claims of experimental anomalies have a signicantprobability of being true. If pressed, he would roughly estimate a 75% probability that nuclearreactions are truly responsible for many of these anomalies. In this paper, we oer an explanationfor these anomalies, assuming that they are true. The following experimental claims have beenmade by multiple experimenters:

1. In highly deuterated palladium, with loading factors (ratio of the number densities of d andpd) greater than 0.9, some electrolytic cells have produced heat in excess of what can be accountedfor chemically. They also have produced Helium-4 in quantities which were approximately consistentwith the excess heat if the reaction were nuclear fusion with two deuterons forming an alpha particle,and with the entire energy release going into heat. This is a surface phenomenon restricted towithina few thousand angstroms of the surface of the palladium.

2. Also in highly deuterated palladium, numerous elemental transmutations have been observedon the surface of the palladium after deuterium loading.

3. These reactions can sometimes be stimulated by injecting time-varying electromagneticelds into the surface, or by laser illumination of the surface, or by sound waves, or by thermalcycling.

We make an argument here that all anomalous phenomena which fall under the domain ofLENR might be explained by variation of the rest masses of elementary particles in a condensedmatter environment. We shall consider deviation of rest masses for all charged particles: electrons,protons, deuterons, α particles, and even lattice and impurity nuclei. It's even concievable thatmass variation might occur for neutrons. Such behavior has never been directly observed, and itis considered impossible and ruled out by conventional physics. Nevertheless, there is a large bodyof theoretical work which suggests that it might be possible. The LENR anomalies may constituteindirect evidence of such mass deviations.

O-mass-shell covariant relativistic mechanics has a long and rather eminent history inphysics. Early pioneering work was done by Fock [3] and Stueckelberg [46]. This was followedby applications of essentially the same theory by Feynman in his path integral formalism ofquantum mechanics [7](see Appendix A), and later by elaborations and extensions of the theory byHorwitz and Piron [8], and by many others [922]. Other variable mass theories were proposed byGreenberger [2023], and Corben [24, 25]. Thorough reviews have been given by Fanchi [10, 26].Despite the theoretical interest in these theories, there is no conclusive experimental evidence thatthey are required to describe any natural phenomena.

If o-mass-shell quantum mechanics is needed in order to understand LENR results, then ourunderstanding of quantum mechanics will be aected at a fundamental level. The standard on-shellwave equations would have to be considered as approximations to more general o-shell theories.There is a modern school of emergent quantum mechanics developing, looking for a deeper originof the quantum laws [2731]. Support from LENR for o-mass-shell dynamics would be of greatrelevance to this eld. The standard relativistic wave equations (Klein-Gordon, Dirac, Proca, etc.)all have problems with a localized position operator, or negative energies, or negative probabilities.Thus, modern physics regards quantum elds as preeminent over N particle wave mechanics. But a

Theories of variable mass ... 3

resurgence of the Fock-Stueckelberg theories could change this, and make N-particle wave mechanicsprominent once again.

We use units such that ~ = c = 1, and a relativistic metric signature (-,+,+,+), unless otherwisenoted.

2. The well known diculties of explaining d-d fusion in a solid-state setting

Considering the possibility of d+d fusion in a d2 molecule, Koonin and Nauenberg calculated thepenetration factor through the Coulomb barrier at room temperature [32] which gave reaction rates50 orders of magnitude smaller than the claimed experimental results of Fleischmann and Pons [33].Leggett and Baym [34, 35] came to similar negative conclusions based on necessary but unobservedanity enhancement for 4He in deuterated palladium. They and others concluded that either theexperiments must be wrong, or that the eects were not due to fusion. Recent experiments ondeuteron beam scattering o of deuterated metals of various kinds have exhibited much largerfusion cross sections than these calculations [3641]. This enhancement has been attributed tounexpectedly large electron screening eects. The other main problem is to explain the extremedistortion of the branching ratios as compared with plasma fusion which has the well-measuredratios [42]

d + d → t(1.01MeV ) + p(3.02MeV ), (Q = 4.04MeV ), 51% (1)

d + d →3 He(.82MeV ) + n(2.45MeV ), (Q = 3.27MeV ), 49% (2)

d + d →4 He(0.076MeV ) + γ(23.77MeV ), (Q = 23.85MeV ), (3.8 ∗ 10− 3)%(3)

where Q is the kinetic energy released by the reaction in the center of mass Lorentz frame. Nagelhas listed a number of phenomena as illustrative of the challenge to theory [43]. There are alsoextensive experimental claims of a variety of elemental transmutations taking place inside or ondeuterated palladium and also in other metal-hydrogen alloys [1, 2]. The transmutation data aremuch harder to ignore than 4He production which might be due to environmental contamination,and they represent strong evidence for nuclear reactions, although not necessarily fusion. Widomand Larsen [44] have presented an alternative to fusion models. A signicant portion of the LENRresearch community do not believe that fusion is taking place.

3. Variable rest masses for charged particles can probably account for LENR

Rest masses for elementary particles like electrons and quarks cannot vary in conventionalrelativistic classical or quantum mechanics in a eld-free region. The energy levels and thereforemasses of electromagnetically bound systems can certainly be changed by external elds. Examplesare the Stark and Zeeman eects in atomic physics, where the mass of the atom is modiedcontinuously according to E = mc2 depending on the strength of the eld and the energy shiftcaused by it. A single charge monopole satises the minimal coupling prescription pµ = mvµ+qAµ.This equation comes from the fact that electromagnetic stress energy of the electromagnetic eldchanges by the nite amount qAµ due to the charge-eld interaction as in [4547], but it onlyhas this very simple interpretation in the transverse gauge. So the rest mass is hard to deneprecisely. We can interpret this as meaning m is the mass contained in an innitesimal volumearound a point charge, and pµpµ is the total mass of the system which includes the local rest massplus the external eld's contribution. Charge is always conserved. Consider an N particle collision


involving nuclear interactions. If the interaction is local, so that the N particles meet at a space-time point, then both the total momentum

∑pµi and the electromagnetic contribution

∑qiA

µ areindependently conserved, ignoring variation of Aµ over the interaction volume. Consequently, theinertial momentum or

∑miV

µi is also conserved, so that the presence of an external electromagnetic

eld does not aect the nuclear kinematics. The parameter m, the eld-free inertial mass, changesdiscreetly in nuclear collisions in conventional theories, but never continuously. Of course, theconcept of eective mass is frequently used in condensed matter to describe phenomenologicalmass parameters in transport equations in a solid. But this is not the fundamental rest mass of theparticle which could aect the kinematics of nuclear reactions. So when we talk about rest mass,we are talking about the parameter m. In conventional theory, this does not change.

A number of authors, shortly after the rst papers by Fleischmann and Pons, noted thatmodied mass values could lead to enhanced quantum tunneling rates in d-d fusion [1, 32, 48, 49]One well-studied related example of this is muon catalyzed fusion [32, 50]. Widom and Larsen [44]argued that the electron mass may increase due to electromagnetic interaction with local elds (see7), and then these heavy electrons can join with protons in a solid and form low energy neutronsthrough the weak interaction, which can then react with nuclei.

The rst suggestion that a variable mass covariant type of theory might be applicable to LENRwas due to Evans [51, 52] who applied it to electrons. In this paper, we consider mass changes forall types of particles. We contemplate up to about ±12 MeV for deuterons, and less for electrons.

4. Eective Lagrangians in nuclear physics and o-mass-shell propagators

Feynman diagrams for relativistic eld theories like the standard model of particle physics routinelyinvolve o-mass-shell propagator lines. Such particles in the nomenclature of perturbation theoryare called virtual. It is often stated that virtual particles are not real, but only mathematicalconstructions, unlike on-shell particles that appear in initial or nal asymptotic scattering statesand which are real. This point of view is reinforced by a fairly rigorous equivalence theorem ineld theory which states that changing eld variables will change o-shell Green's functions whileleaving the S-matrix invariant [53, 54]. The relativistic single particle state in general has problemsof localizability [55, 56], and moreover quantum entanglement and wave particle duality make thereality of even on-shell particles highly dubious from a fundamental ontological point of view.In addition, no particle is ever truly isolated in nature, for if it were then we wouldn't be able toobserve it. Thus all particles are in fact always slightly virtual, and perfectly on-shell particles area mathematical abstraction. This forces us to conclude that most particles are a very little bit othe mass shell all the time. The fact that this deviation is empirically very small lets us approximatethe true situation with theories in which the mass deviation is exactly zero. In the non-relativisticlimit this leads to the Schrödinger equation with potential forces and xed masses. This is the basisof almost all condensed matter physics, and so from the start, variable mass behavior is excludedfrom discussions about condensed matter. This is ne so long as there is no experimental evidenceto the contrary. In Schrödinger wave mechanics perturbation theory the virtual particles becomeo the energy shell, but the mass is still equal to the rest mass. The experimental claims in theLENR area, if correct, point to the possibility that masses are in fact changing in some specialcondensed matter settings. We can choose one of two paths. Either reject the experimental dataas impossible, or generalize condensed matter theory to allow for mass variation. We suggest thesecond path here for consideration.

In developing his path integral quantization method for the Klein-Gordon equation (KGE),


Feynman was faced with a dilemma. How to put spatial and time coordinates on a covariantfooting [7](see Appendix A). He chose to introduce a second time variable, call it historical timeτ . It is not the same as proper time in classical particle mechanics, although it plays a similar rolewhich is to parametrize a curve in space-time. It is useful to review his logic briey. Consider theconventional Klein-Gordon equation for a spinless particle in the presence of an electromagneticeld

(i∂ −A)µ

(i∂ −A)µ Ψ = −M2Ψ (4)

M is the particle's rest mass, and where A is the vector potential for an external electromagneticeld. There are well known problems with the probabilistic interpretation of this equation becausethe conserved current is jµ = i [(∂µφ

∗)φ− φ∂µφ∗] and j0 takes on both positive and negative values.Moreover, a localizable Schrödinger type position operator cannot be dened for this equation[55, 56]. Feynman wished to apply his path integration method of quantization to the KGE. In hiswords ... we try to represent the amplitude for a particle to get from one point to another as a sumover all trajectories of an amplitude exp(iS) where S is the classical action for a given trajectory.To maintain the relativistic invariance in evidence the idea suggests itself of describing a trajectoryin space-time by giving the four variables xµ(µ) as functions of some fth parameter µ (rather thanexpressing x1, x2, x3in terms of x4). So Feynman replaces (4) with (we use x0 = t, and τ insteadof µ and we use spacelike metric, whereas Feynman used timelike)

i∂ϕ(x, τ)

∂τ=

1

2(i∂ −A)µ (i∂ −A)

µϕ(x, τ), where x = x0, x1, x2, x3 (5)

It is very similar to the time dependent Schrödinger equation, but with the historical time τreplacing the usual time variable t, and with the four coordinates of space-time xµ replacing theusual three coordinates of space. Feynman points out that if Aµ does not depend on τ , thenseparable solutions exist so that ϕ(x, τ) = Ψ(x)exp(i 1

2M2τ) and ψ is the solution to the usual

KGE. Because of the similarity to Schrödinger's equation, (5) has a postive denite probability, anda localizable Schrödinger position operator. Moreover, it is amenable to solution using Feynman'spath integral formulation as he shows in [7]. In fact young Feynman had studied this idea moreextensively than is commonly realized , but was discouraged by negative reactions to it [57].Equations of this type were rst studied by Fock [3] and Stueckelberg [46]. The path integralsolution includes all paths in space-time connecting two space-time points and parametrized byτ , with no restrictions on the path, so that o-mass-shell paths are included in the path integralsolutions. This reects that fact that Feynman diagrams contain virtual particles which are not onthe mass shell. This theory was extended to multi-particle systems by Horwitz and Piron [8] bypostulating that a single τ variable acts as a historical time for all of the particles simultaneously.This greatly simplies the mathematics for both the quantum and classical many particle relativisticequations, but at the expense of some new interpretative challenges. For example, the wave functiondescribes an extended probability cloud not just in the spatial coordinates, but also in time. Letus refer to these types of theories as Fock-Stueckelberg or FS or hisotorical-time theories.

We note that, when restricted to a single particle, the mass of the particle is a constant ofthe motion for any arbitrary electromagnetic eld described by the vector potential Aµ, as it isin ordinary classical relativistic mechanics, but the mass in Horwitz-Piron theory for two or moreparticles is not an invariant [8].

These type models can deviate in the mass, even in restricted non-relativistic approximationwhere the velocities of the particles are much less than c, but their mass can still change due tointeractions.


In the theory of photonic crystals, it has been found both theoretically and experimentally thatthe photon can acquire an eective mass [5862]. This eect might enhance the mass deviation ofcharged particles which could interact with these massive photons. The nuclear active environmentfor LENR is known to occur near the surface of palladium, and in areas where the lattice hasbeen deformed. The surface morphology and chemistry are also known to be critical factors. Thepossibility that o-mass-shell photons are a necessary condition for a nuclear active environment isworth considering.

Because of the size of the strong coupling constant, the standard model is too complicated atlow energies, and so eective Lagrangian approximations are routinely employed in nuclear physics.Many of these methods predate the standard model. They often exploit the low values for mass ofthe u and d quarks which when taken to zero lead to chiral-isospin symmetry which can then justifyutilizing group theoretical methods to derive general forms for interacting potentials [6365]. Onewould think that FS type of wave equations would have been considered in the context of eectiveLagrangians for nuclear physics, but they haven't been. Fanchi [10] gives a historical explanation forthe general lack of interest. The current literature contains treatment of any spin in a path integralapproach [15], along with signicant other literature on relativistic wave equations for spin 0 andspin 1/2 systems [9, 10]. Also there is an elegant examination of o-shell quantum electrodynamics[66] for spinless charged particles. Most of theoretical nuclear physics calculations are based on theSchrödinger equation with eective nuclear and electromagnetic potential functions which do notallow any o-mass-shell behavior, even though the relativistic eld theory perturbation does. Upuntil LENR eects were discovered, there was no need for such behavior in nuclear physics. It isthis author's opinion that there is such a need now since conventional approaches have failed bymore than 50 orders of magnitude [32] to explain what is being seen in LENR experiments, andit seems that o-mass-shell behaviour of one type or another can explain all of the experimentalanomalies.

In the modern theory of multi-dimensional tunneling for many-body systems there are manyunexpected phenomenon. In Coleman's classic study [67], a quasi-stable vacuum state's tunnelingdecay into a lower state is described. Condensed matter systems are in fact quasistable systems.The true ground state - after allowing for all nuclear reactions - is at a much lower energy level,but there are large Coulomb barriers which prevent decay. Of course, it is always assumed thattunneling to the true ground state through nuclear reactions happens at such a slow rate that itcan be ignored. LENR phenomenon suggests otherwise. Mass deviation, chaos [68], resonances[69, 70], and driving forces [71] are all mechanisms for enhanced tunneling rates. Given all thesepossibilities, there is certainly no rigorous theorem that says the experimental claims of LENRcannot occur. There may exist eective o-mass-shell Lagrangians for such systems which facilitatecalculations of multi-dimensional tunneling, and which give the correct S-matrix when all particlesare separated spatially in the initial and nal state.

So in this paper we propose that perhaps in a condensed matter setting, the electromagneticinteraction of charged particles can be described by an eective Lagrangian of the o-shell FS type,or some other variable mass type of theory. We acknowledge that the eective Lagrangian is notunique because of the equivalence theorem [53, 54]. It is the author's hope that such an eectiveLagrangian could be derived from or at least reconciled with the standard model of elementaryparticles or perhaps string theory. This is not an easy task, but if direct experimental proof thato-shell behavior of the type proposed here is discovered, then these experiments should provide acritical guide on how to proceed.


5. Horwitz-Piron theory

We start with a brief review of the rst of the modern variations of the Fock-Stueckelberg o-shelltheory by Horwitz and Piron [8, 9] who presented a solution to the two particle problem of relativisticclassical and quantum physics by proposing that a single historical time could simultaneouslyparametrize two space-time paths for dierent particles. This is easily extended to any numberof particles. They postulated a Hamiltonian K, conjugate to the historical time τ . They chose Kas follows (written in the most general form)

K =

n∑i=1

[1

2Mi(pi − eA(xi))

µ(pi − eA(xi))µ

]+∑i<j

Vij(|xi − xj |) (6)

where Aµ is an external electromagnetic four vector potential, and V is an inter-particle potentialenergy. The dot notation denotes τ derivatives . Hamilton's equations of motion become

xµi =∂K

∂piµ= (pi − eA(xi)) /Mi, pµi = − ∂K

∂xiµ(7)

This leads to the Lorentz force equation

Mixµi = eFµν xiν −

∑j 6=i

∂Vij(|xi − xj |)∂xiµ

, where Mixµi = pµi − eA

µ (8)

The mass of a particle is given by

m2i = −M 2

i xiµxµi = − (pi − eA(xi))

µ(pi − eA(xi))µ (9)

The particle's mass mi is not necessarily equal to Mi, and is not even necessarily a constant of themotion if V is not zero.

It is straightforward to quantize this system because it is mathematically very similar to thenon-relativistic Newtonian particle system, where the usual time variable for each particle is afunction of the historical time just as the three spatial coordinates for each particle are. In anutshell, this 5D theory let's you have both a relativistic time variable t along with a Newtonianabsolute time variable τ , with both managing to coexist, or at least that is the assumption. Thequantum wave equation is ( denotes an operator)

i∂

∂τψ(x, τ) = Kψ(x, τ), pµi = −i ∂

∂xiµ(10)

6. Pre-Maxwell 5D theory of electromagnetic interaction of massive charged particles

O-shell electrodynamics was generalized to a ve dimensional gauge invariant theory by Saad,Horwitz, Arshansky, and Land [14, 66, 72]. Here we follow the notation in [66]. In this theory,τ -dependent gauge invariance was assumed so that the wave equation is invariant under local gaugetransformations of the form

ψ(x, τ) −→ eie0Λ(x,τ)ψ(x, τ) (11)(i∂

∂τ+ e0a5(x, τ)

)ψ(x, τ) =

1

2M(p− e0a(x, τ))

µ(p− e0a(x, τ))µ ψ(x, τ) (12)


This leads to a non-trivial generalization of electromagnetism. The parameter e0 has units of lengthand is proportional to the electric charge e

e =e0

λ(13)

where the constant parameter λ is new. The elds transform as

aµ(x, τ)→ aµ(x, τ) + ∂µΛ(x, τ) a5(x, τ)→ a5(x, τ) + ∂τΛ(x, τ) (14)

Non-vanishing values of a5 leads to mass variation. The Schrödinger equation (12) leads to a vedimensional conserved current

∂µjµ + ∂τ j

5 = 0 (15)

j5 = ρ = |ψ(x, τ)|2 jµ =−i2Mψ∗ (∂µ − ie0a

µ)ψ − ψ (∂µ + ie0aµ)ψ∗ (16)

The usual Maxwell theory is recovered by integrating over τ , a process termed concatenation.

Jµ(x) =

ˆ +∞

−∞jµ(x, τ)dτ Aµ(x) =

ˆ +∞

−∞aµ(x, τ)dτ (17)

where Aµ and Jµ are the usual Maxwell potential elds and 4-current respectively. Saad et al [72]suggested an action which had higher 5-dimensional spacetime symmetry so that the Lorentz groupO(3,1) is a subgroup. This requires either O(4,1) or O(3,2) symmetry. Both are considered in theliterature, and to handle this, a 5D metric tensor is introduced gαβ = diag(−1, 1, 1, 1, σ), whereσ = +1 for O(4,1) and σ = −1 for O(3,2).

The covariant action formula is (µand ν range over the four space-time indexes, whereas α andβ range over the ve indexes including τ)

S =´d4xdτ

ψ∗(i ∂∂τ + e0a5(x, τ)

)ψ

− 12Mψ∗ (p− e0a(x, τ))

µ(p− e0a(x, τ))µ ψ(x, τ)− λ

4 fαβfαβ (18)

where

fαβ = ∂αaβ − ∂βaα (19)

This theory has been studied extensively by Horwitz in particular along with a number of co-authors[1214, 16, 66, 7276]. Path integral quantization has been analyzed [15] as has more canonicalsecond quantization approaches to o-shell quantum electrodynamics for spinless charged particles[OSQED][66].

Various FS spin 1/2 wave equations have been proposed [9, 10, 15, 51, 52, 7780]. They havenot yet been included in the OSQED.

The deuteron has spin 1 and positive parity Jπ = 1+. Although it is composite particleof six quarks, it is more commonly treated as a composite of a proton and a neutron with aneective Lagrangian. For low energy interactions, it can be treated as an elementary particle.The conventional spin 1 wave equation is the Proca equation [81, 82], although it is not the onlypossibility [83] [

∂µ∂µ −M2

]V α = 0 (20)


where V is a massive vector eld which transforms under Lorentz transformations Λµν as V (x)µ →V ′(x′)µ = Λ ν

µVν(x′). The appropriate Lagrangian is

Lp = −1

4WµυW

µυ − 1

2M2VµV

µ, Wµν = ∂µVν − ∂νVµ = ∂[µVν] (21)

which leads the Euler-Lagrange equations

∂µWµν = M2V ν (22)

from which it follows that ∂µV µ = 0 which leads to (20). The deuteron has a magnetic momentand an electric quadrupole moment [83], and these can be included in the Proca equation as wellwhen there are electromagnetic elds present.

We wish to generalize the Proca equation by adding a historical time τ . It does not appearthat this has been done in the extant literature, however path integral formalism for any spin hasbeen proposed in [15]. Since each component of V satises the Klein-Gordon equation, it is naturalto write the Stueckelberg version of the Proca equation as

i∂

∂τUα(x, τ) = − 1

2M[∂µ∂

µ]Uα = 0 (23)

We take the Lagrangian density to be

L = Uα∗i∂

∂τUα −

1

2MU∗pµpµUα −

1

4ZµυZ

µυ, pµ = −i∂µ (24)

where here Zµν = ∂µUν − ∂νUµ.The electromagnetic minimal interaction can be added by minimal coupling, although the deuterondoes have an anomalous magnetic moment as well as an electric quadrapole moment which caninuence the dynamics in strong elds [83, 84].

Land and Horwitz [66] have applied perturbative quantum eld theory methods to o-shellpre-Maxwell electromagnetism in interaction with spinless charged particles. They have developedFeynman rules and applied them to various scattering processes. This is a 5D theory, and thereare two possibilities for the 5D symmetry group that contains the Lorentz group as a subset. TheLorentz group O(3,1) is a subgroup of either O(4,1) or O(3,2), and these two possibilities lead totwo dierent theories with dierent physical properties. The results for particle-particle scatteringshow that in general the masses will change for particles in a scattering process. They presentdetailed cross-section calculations for both Møller and Compton scattering, and both calculationsshow mass changes after scattering. This is an extremely interesting paper as it provides ampleo-shell behavior. The mass of a particle in this theory is a function of the past history of theparticle.

The self interaction of a classical charged particle in pre-Maxwell theory has also been studied.The solutions are more complicated than the Lorentz-Dirac equation [16, 85, 86]. The runawaysolutions are replaced by chaotic nonlinear equations which include variation of the classicalparticle's mass with time.

One major problem with the o-shell theories is how or why particles tend to get back on themass shell if they have moved o of it through various interactions. Some unknown restorativemechanism must be at work, as has been acknowledged by advocates in this eld. Some clues havebeen found. In [16] a self-interaction theory for a charged particle was developed, and it was foundthat for many (but not all) initial conditions, the mass increases for a while and then decreasesback to the on-shell value. In [87] it was found that the mass distribution of certain distributions


of particles and anti particles can become sharply peaked in mass. A modication for the photonLagrangian in [88] was proposed, but similar modications might be required for massive particlesas well. It was pointed out that the classical behavior of charged particle scattering in pre-Maxwelltheory is unphysical without this modication, although the quantum versions of the theory didnot seem to have these problems [88]. On a more fundamental level, the kinetic term pupµ/2M isnot positive denite. Thus the Hamiltonian K is not bounded from below in Horwitz-Piron theory.Extra constraints, which are not derived from the basic theory must be imposed in order to avoidproblems. For example, in classical statistical mechanics of many body systems, various authors[18, 89] imposed the constraint on the system that in the non-relativistic limit all masses approachthe mass shell. This assumption leads to a narrow distribution of masses about their usual restmasses with a spread in values of width on the order of kBT . This assumption runs counter tothe behavior that we are proposing here, which is a signicant deviation from rest masses in anon-relativistic condensed matter setting. This constraint appears to have been imposed on thetheory because there was no evidence for o-mass-shell behavior in condensed matter. But LENRmight be evidence. Therefore, we assume here that this constraint can be relaxed in these statisticaltheories. Perhaps these problems could be solved by making the replacement in the HamiltonianK

pµpµ2M

⇒ pµpµ2M

+ η(pµpµ +M2

)2(25)

where η is a constant which might be dierent for dierent particles, and it might depend on thecondensed matter environment. This is an obvious generalization of [88] to massive particles. Alarge value for η would weigh against o-shell deviation, and a small value for η would allow largermass deviations. This replacement renders the Hamiltonian bounded from below for a suitable signof η, and may not aect the on-shell behavior very much. It adds a cost to going o shell and wouldact to inhibit the particle from moving o the mass shell during a collision for example. The ideathen would be that in normal matter η is large, but in a nuclear active environment, where LENRoccur, η would be small.

There are no rigorous bounds on how far o-shell the massive particles can wander in thevarious FS theories for realistic condensed matter systems, especially in non-equilibrium situations.

7. Some comments on Widom-Larsen theory

In the Widom-Larsen theory [44], it is argued that protons and heavy electrons can react to form aneutron and a neutrino e+ p→ n+ νe. They used the following formula for the mass shift inducedby electromagnetic elds

βme ≡Me

Me=

[1 +

(e

Mec2

)2

AµAµ

]1/2

(26)

where Me is the shifted mass value for the electron and Me the electron rest mass in isolation. Inorder to have enough energy to produce a neutron, it is required that βme

> 2.531. The sourcefor (26) is given as [90] (section 40, eq. 40.15), which contains a rigorous solution to the Diracequation in a monochromatic plane electromagnetic wave. The plane-wave in [90] is written in theLorentz gauge ∂µAµ=0, but since there are no sources for it, it is possible by appropriate choiceof gauge to arrange that in addition A0 = 0, which is assumed in [90], and consequently it followsthat ∇ · A = 0, and so that the eld is purely transverse. An estimated upper possible value of


βme = 20.6 on a Palladium-deuterium alloy surface is presented in [44], but this value has beenquestioned [91].

We can understand (26) by considering the change in the classical electromagnetic momentumvector that happens when a charged particle is moved into an electromagnetic eld in quasistaticapproximation. The contribution from the interaction with the eld is simply given by (up to aconstant that depends on units) [4547]

pµfield = qAµT (x(τ)) (27)

This formula is only correct in the transverse gauge ∇ · AT = 0. This xes the gauge uniquelyprovided AT vanishes at x = ∞, otherwise we could add a term ∇ϕ if 4ϕ = 0. This can bethought of as the justication for the minimal coupling formula which follows by adding pµfield tothe intrinsic 4-momentum of the particle expressed in terms of it's proper velocity

pµ = Mvµ + qAµT (28)

The Lagrangian for a charged particle in an electromagnetic eld is gauge invariant, and so if allone wants to do is solve for the equation of motion, one can drop the transverse gauge conditionfor practical purposes, and study a system with gauge symmetry, and this is routinely done byphysicists. Taking a time average of the instantaneous mass pµpµ and assuming the cross termscan be dropped results in (26).

There is a problem with the Widom-Larsen argument. Notice that the time averaged correctionto the mass is proportional to the charge squared, but the correction to the instantaneous 4-momentum is proportional to the charge q. The problem comes from the fact that the proton andthe electron have opposite charge, and this causes their mass shifts to be anti-correlated. Considerthe total instantaneous 4-momentum of their initial state

PµTot = pµe + pµp = (Mevµe − eA

µT (xe(τ)) + (Mpv

µp + eAµT (xp(τ)) (29)

Since the weak interaction which would produce a neutron is extremely local in space and time(because it is mediated by the massive W boson), and since the external eld is changing very littleover the short range and time of the weak interaction, the use of the instantaneous 4 momentumvalues seems more appropriate than the time-averages. So we see that when the positions of theelectron and proton are close enough to react weakly, the two electromagnetic mass terms cancelyielding simply

PµTot = Mevµe +Mpv

µp (30)

The center of momentum energy is then exactly the same as if there were no electromagnetic eldpresent. This is just an example of what was mentioned earlier in Section 3, that an externalelectromagnetic eld cannot change nuclear kinematics. Although this argument is classical, itdoesn't seem that appealing to quantization could change this conclusion. This sheds some doubtin this author's mind regarding the validity of the Widom-Larsen argument.

However, in the FS type o-shell theories, and especially in Horwitz-Piron or pre-Maxwell5D theories, the local mass of structureless point particles can change, and the mass shift can bepositive or negative, even in the absence of any local eld. Moreover, there is no bound known onhow far the mass can shift o shell. So, these theories might justify the Widom-Larsen theory evengiven the above result, and moreover the objections raised in [91] may not apply in this case. Soin our view the Widom-Larsen processes are possibly part of the picture, but not necessarily thewhole story. They may compete with other processes for dominance in dierent reactions. All the


eects we consider are due to mass shifts, but the shifts can be positive or negative and can applyto any particle. Aside from providing an alternative basis for the Widom-Larsen eect, we oer noopinions pro or con about the rest of their theory. We do consider other reactions that are enabledby mass-shift eects however.

8. The deuterated palladium system and a possible explanation of d-d fusion:

mass-tuned quantum tunneling

Consider two neighboring deuterons in a palladium lattice. We assume the masses of the deuteronsand possibly the nearby electrons too are moving slowly o the mass shell due to interaction withthe condensed matter system according to an o-shell FS type theory, as in (1). We further assumethat the nal state masses in the fusion process are the usual rest masses of the particles, thatspecial conditions inside the solid are required for this process to occur, and that these are roughlyequivalent to the conditions required for anomalous LENR eects to occur. Finally, we assume thatafter a period of time, the system returns to normal and all mass values return to their standardvalues, except for those that have experienced a nuclear reaction. We propose that an active d+dpair reduces its mass slowly until it is approximately equal to the mass of the 4He (alpha particle),about a 0.63% reduction, or 11.9 MeV per deuteron. This has never been observed in nature. It is aradical assumption which could be possible if an o-mass-shell eective lagrangian were describinga small volume of the lattic. Koonin and Nauenberg [32] modeled the electron screening eectby assuming that the d+d system acts similar to a d2 molecule. They showed that the fusionrate was far too low (by over 50 orders of magnitude) in this case to explain the fusion claims ofFleischmann and Pons. They also calculated what the electron mass would have to be in orderfor the tunneling rate in d2 to explain the Fleischmann-Pons results for excess heat. They found amass of 10me was required. There is beam-scattering evidence of enhanced screening in deuteratedpalladium [3640, 9296]. This enhancement has thus far been attributed to the higher density ofelectrons surrounding the deuterons without resorting to heavy electrons. But, it may also be thatheavy electrons are playing a role. Resonant tunneling would occur if the sum of the two deuteronsequaled the mass of Helium-4, regardless of the type of screening. In this case, any photon producedwould have a low energy. The increase in the electron mass enhances tunneling, and the decreasein the deuterium mass allows resonant tunneling directly into an alpha particle and the suppressionof neutrons and tritium. Figure 1 illustrates these ideas qualitatively.

There is a complication to this basic idea, and that is the existence of a 0+ resonance of4He which occurs at 20.210 MeV above the mass of 4He [97]. This excitation decays almostexclusively into t+p with a half-life of 1.3 × 10−21s. Only the s-wave component of the resonancewill contribute to the fusion rate, but the proportionality factor is unknown. Assuming a signicants-wave contribution we should see enhanced tritium production from this resonance as the two-deuteron total energy passes through it with the following kinematics when the two deuterons haveexactly the peak energy.

4He(20.21MeV )→ t(0.099MeV ) + p(0.2968MeV )Q = 0.396MeV, Γ = 0.5MeV

(31)

Because the Breit-Wigner width is .5 MeV, the proton and tritium energies would actually besmeared out. This helps explain the observed enhanced tritium production relative to neutronproduction in experiments with deuterated palladium. The stopping range of 0.3 MeV protons inwater is 4.27µ [98], and thus they would not be observed directly in electrolysis experiments.


0 .0

1 .0

d/pd

loading

0

3 .27

Q(3He+n)

(MeV)

0

4 .04

Q(t+p)

(MeV)

0

23 .85

Q(4He+ )

(MeV)

time

dd

(s-1

)

(a)

(b)

(c)

(d)

(e)

(f)

(g)

me

2md

2md

me

max me

m mass of 4He

1020

10-40

10-60

resonance 2md m=weak resonance at

2md m=

+20.21MeV

end of neutron

generation

end of tritum

generationpossible enhanced

tritium at weak resonance

Energy of produced goes to zero

Electron mass increasing

reducing the Coulomb barrier

due to screening.

very strong, but very narrow peak in the

fusion rate due to resonance

Figure 1. Qualitative time evolution of d+d mass variation to resonant tunneling and fusion (a)shows a plausible but ctional loading process; (b) shows presumed decrease in deuteron mass;(c) shows presumed increase in electron mass; (d) shows the resulting reduction in the energyrelease from d + d → 3He + n; (e) the energy release from d + d →3 H+ t; (f) the energy releasefrom d + d →4 He + γ; (g) The non-resonant fusion rate of Koonin and Nauenberg with the verysharp resonant peak superimposed.

We see in Fig. 1 that the main fusion is occurring at the instant that m2d = mα, but this is notwhen most of the heat is added to the solid, because the Q value for fusion is essentially zero thenas the masses of the two deuterons sum to very nearly the mass of an alpha particle at resonance.The energy has been given up to the condensed matter prior to fusion due to the continuouslyvarying masses, and transients continue until all masses eventually return to their on-shell values .Energy is conserved by the following formula

dEextdt

=d(ndmd)

dt+d(neme)

dt+d(npdmpd)

dt+∑i

d(nimi)

dt+dK

dt+dEe&mdt

(32)

where K is the kinetic energy density of the system, and nd, ne, npd, ni are the number densitiesof deuterons, electrons, palladium nuclei, and any other particles. These number densities are notconserved because of nuclear reactions. The mass terms md, etc. are local mean values for themasses in the solid at a given location, and they are presumed to be varying with time. The term


dEext

dt is the net energy density rate of change due to conventional transport mechanisms such asradiation, thermal conduction, etc, and Ee&m denotes the electromagnetic energy density. In thenon-relativistic limit, Ee&m can be replaced by a pairwise Coulomb interaction plus the energydensity of radiation. Thus we have energy conservation at every step of the way. We see fromFigure 1 on page 13 that neutrons can be produced only at the very beginning of the time intervalshown so long as Q(3He + n) > 0. Tritium can be produced so long as Q(3H + p) > 0. We don'texpect all of the particles of a given type at a given location to necessarily have the same mass. Sowe dene a probability density ρd, ρe, ρpd, etc. with functional form ρ(x, t,m), where ρdm is theprobability that the mass m lies between m and m+ dm. We restrict the sign of the particle massto be positive, and so we write the normalization condition asˆ ∞

0

ρ(x, t,m)dm = 1 (33)

and m(x, t) =´∞

0mρ(x, t,m)dm. The masses in Figure 1 on page 13 are these mean values

for masses. At most locations, the critical condition m2d(x, t) ≈ Mα will not ever be achieved,and therefore signicant fusion will never occur there. In order for many fusion reactions tooccur, the critical condition must be satised at a number of locations. Fusion depends ontwo tuning parameters - the mean deuteron mass, and the mean electron mass. Correlationsin mass may well occur between particles of the same or dierent species. Therefore, in generalρ(x, t,m1,m2) 6= ρ(x, t,m1)ρ(x, t,m2).

As a simplifying approximation to this situation, let us assume that at a given location themass values are sharply peaked about a mean value that depends on time. It is dicult to sayanything precise about the solid, so let us rather study the d2 molecule, as was done in [32]. Thisfour-body system allows us to estimate the eect of resonant tunneling. For simplicity let thedeuteron masses both be the same, and similarly for the two electrons in the d2 molecule. We treatthe masses as slowly varying in time so that a quasistatic approximation can be used.

The non-relativistic Hamiltonian for d2, is

H =p2d1

2md+

p2d2

2md+

p2e1

2me+

p2e2

2me+ V (xd1,xd2,xe1,xe2) (34)

where we treat the d as spinless, and where V is a sum of six two-body terms. One can obtainthis from a Horwitz-Piron or other FS theory provided the masses of the deuterons and electronshave been modied by prior and ongoing interaction with the solid and are changing slowly.

V (xd1,xd2,xe1,xe2) = Vdd (|xd1 − xd2|)+∑2i,j=1 Vde (|xdi − xej |) + Vee (|xe1 − xe2|) + Vmasses

(35)

Vmasses = 2md + 2me (36)

Vmasses is slowly varying with time, and so it must be included in the energy. We take Vee andVde to be pure Coulomb potentials

Vee(r) = −Vde(r) =e2

r(37)

but we tak Vdd to be a modied Coulomb potential which includes nuclear forces of connementfor r less than an eective nuclear force range an.

Vdd(r) =

e2/r, r > an ≈ 10 fmUdd(r), r ≤ an

(38)


where Udd includes the short-distance attractive nuclear force between the deuterons, as well aselectromagnetic forces. Even if Udd were known accurately, solving these equations exactly iscomplicated and requires numerical techniques. We ignore the time variation of the masses insolving for the wave functions. The standard treatment for the hydrogen molecule uses the adiabaticapproximation and the clamped nuclei computation [99]. The calculation begins by solving for theground state of the clamped nuclei Hamiltonian H0:

H0 =p2e1

2me+

p2e2

2me+Vdd (|xd1 − xd2|) +

2∑i,j=1

Vde (|xdi − xej |) +Vee (|xe1 − xe2|)(39)

The nuclei are slowly moving compared to the electrons, and their positions are held xed(clamped) in this rst step. In the center of mass system, let R = xd1−xd2, R = |R| , and xe1 , xe2

are the electron coordinates. The eigenfunctions for the electronic states are solved rst with Rheld xed

H0ψj(x,R) = E0j (R)ψj(x,R) (40)

Expanding the full wave function in terms of these by ψ(x,R) =∑χj(R)ψj(x,R) we must

solve the full Schrödinger equation Hψ(x,R) = Eψ(x,R). This can be written in the form of anequation for the nuclei in an eective potential after making an adiabatic approximation [99][

1

2µ4R + Uj(R)− E

]χj(R) = 0 (41)

where µ is the reduced mass of the two o-shell deuterons, and where the eective potential is givenby

Uj(R) = E0j (R) + Cj(R) (42)

The rst term in this equation is the Born-Oppenheimer approximation. Several correctionsmaking up Cn are included in [99] to improve accuracy. Koonin and Nauenberg [32] used theseresults in their estimate of fusion rates. Here we shall only consider the ground state, and theeective potential Veff (R) = U0(R) which will depend on the electron masses. The barrierpenetration factor obtained from the WKB approximation is

B = exp

[−2

ˆ aN

rtp

(2µ(Veff (r)− E))1/2dr

](43)

This factor is real-valued and very sensitive to the electron mass through Veff . It also dependson the deuteron mass, but for the 0.6% change we are contemplating here, it can be treated asindependent of deuteron mass to a rst approximation.

We follow the basic approach of a two-level quantum system as given by Hagelstein when nearto a resonance [69]. There are two weakly coupled eigenstates

(i) ψa is the ground-state for the hydrogen-like molecule constructed from the two slightly o-shelldeuterons and o-shell electrons. Veff is changed by the masses being o-shell. ψa is a solutionto the modied o-mass-shell 2-body problem where the parameter an is taken to zero and thenuclear force is not included.


(ii) ψn is a nuclear bound state consisting of two o-shell deuterons bound together by the nuclearforce and surrounded by a two-(o-shell) electron cloud around them in a ground state solutionto a 4He like atom. There are two resonant states to be considered at the masses of 4He andit's rst excited resonance which is at 20.210 MeV.

In the interest of simplicity, we shall now make a crude approximation and develop the two-leveltheory using only the stable state for 4He. We also consider a third state ψ4He which is an on-shell4He atom with a nucleus that has irreversibly (we assume) changed from a two-deuteron state ψn

into an α particle. Once this transformation happens, the reversible dynamics are over for thatmolecule.

Near a resonance, the states ψa and ψn can tunnel back and forth reversibly through thescreened Coulomb barrier that separates them. We start with the following formula from [32] torelate the nuclear fusion rate to the deuteron wave functions when electrons and deuterons areon-shell

Λdd = Add |ψa(an)|2 (44)

where Add = 1.5 × 10−16cm3s−1 is the rate constant for dd fusion, and ψa(an) is a normalized3D wave function with units (1/L)3/2. We only know Add for on-mass-shell deuterons. This samerate formula was also used for describing muon catalyzed fusion - a similar problem - in a seminalpaper by Jackson [50]. It has problems in the present context. It should be applicable for any wavefunction ψ, but consider a wave function which vanishes at r = an, but is non-zero for r < an. Theformula then wrongly predicts Λdd = 0. But the deuterons have a non-zero probability of beingeven closer together than an in this case, and consequently they should fuse with some non-zeroprobability. For a molecular eigenfunction which varies slowly with position for two deuterons, (44)works ne. The following generalization of (44) xes this problem and agrees with (44) for slowlyvarying s-wave radial wave functions

Λdd = Add

´r<an

|ψ(r)|2 d3r

4πa3n

(45)

The radial s-wave function R(r) is dened by

ψ(r) =R(r)

(4π)1/2

r,

ˆ ∞0

|R(r)|2 dr = 1 (46)

and it is easy to show that (45) is consistent with (44), and that for slowly varying radial s-wavefunctions R(0) ≈ an

√Λdd/Add. The following o-mass-shell formula, based on accurate hydrogen

molecular orbitals, is presented in [32] where me is the o-shell electron mass and me its usualvalue.

Λdd(me) =Adda3

0

(µ

MN

)3

106.5−79√me/me (47)

where a0 is the Bohr radius, µ is the reduced mass of the two deuterons, and MN is the nucleonmass. We assume that the rate constant Add does not depend on electron mass, but it could andprobably does depend on the deuteron mass. We see then that the rate Λdd depends dramaticallyon the electron mass, and as the electron gets more massive the rate goes up dramatically as in(47), and illustrated in (2). This is a well-known and veried phenomenon from muon-catalyzedfusion [50].


Now we proceed with the resonant theory, following Hagelstein's two level approach [69]. Thetwo quantum states |ψa〉 and |ψn〉 denote two deuterons in a molecular bound state, and boundin a nuclear 4He stable ground state respectively. They are orthonormal eigenstates of H0. Wepresume that the nuclear bound state has the same mass as 4He, indepdendent of the mass of thedeuterons. The state vector and Hamiltonian operator are

|ψ(t)〉 = ca(t) |ψa〉+ cn(t) |ψn〉 (48)

H(t) = Ea(t) |ψa〉〈ψa|+ En(t) |ψn〉〈ψn|+ [β(t) |ψa〉〈ψn|+ β(t)∗ |ψn〉〈ψa|] (49)

H0 = Ea |ψa〉〈ψa|+ En |ψn〉〈ψn| (50)

Without loss of generality, we can take β(me,md) to be real, since any phase of β can beabsorbed into the relative phase dierence between ψa and ψn. In the WKB approximation, β isproportional to the barrier penetration B (43), and thus will not depend on the deuteron mass verymuch. The Hamiltonian is Hermitian, and the dynamics will be reversible, as a rst approximation.Because masses are presumed changing, the parameters Ea and β will be slowly varying functionsof time as in (1). We assume that at the initial time the coupling β is extremely small, and thatthe system is in the ground state of the d2 molecule with electrons and deuterons on the massshell. Although tempting, we do not use the adiabatic approximation here because β could be sosmall that the relaxation tunneling time would be larger than the run-time of an experiment. Thecoupling β can grow greatly over time because the electron mass is assumed to increase. The fusionrate at any instant follows from (45), and is given by

Λdd = Add|cn(t)|2

4πa3n

(51)

The Hamiltonian is presumed to be slowly varying with time due to the variation of the electronand deuteron masses. We include the rest masses of the deuterons in H0 along with the bindingenergy, but we ignore time derivatives of H0. We don't need to add the electron masses becausethey would be the same in the two states |ψa〉 and |ψn〉. Assuming both deuterons have the samemass, we have Ea = 2md(t) +mbea(t) and En = m4He = mα, where mbe denotes molecular bindingenergy. The time evolution equation is then simply

H(t) |ψ(t)〉 = i∂

∂t|ψ(t)〉 = ica(t) |ψa〉+ icn(t) |ψn〉 (52)

We need a functional form for β in order to proceed. We can deduce it from [32] by calculatingthe true eigenstates of H at t = 0 which we take as the start time for deuterium loading. Thisamounts to nding the eigenvalues and eigenvectors of a 2× 2 matrix[

Ea(0) β(0)β(0) En(0)

] [φaφn

]= E

[φaφn

](53)

with solutions

E± =Ea + En

2± 1

2

√(Ea − En)

2+ 4β2 (54)[

φaφn

]±

=1√

1 + χ2±

[1χ±

], χ± = (E± − Ea)/β (55)


For on-mass-shell electrons, ε = β/ |Ea − En| 1 and we can approximate and nd[φaφn

]+

=

[β/(En − Ea)

1

]+O(ε2),

[φaφn

]−

=

[1

−β/(En − Ea)

]+O(ε2) (56)

Calculating the fusion rate for the (-) eigenvector with φa ≈ 1, we nd at t = 0 when thedeuterons are on-shell

Λdd = Add|β/(En − Ea)|2

4πa3n

= Add|β/(mα − 2md)|2

4πa3n

(57)

Solving for β, we nd

β(me) =

√Λdd(me)

Add4πa3

n(mα − 2md)2, (58)

We can substitute (47) into (58). Note that md is the on-mass-shell deuteron mass and Add is themeasured on-shell value in (58). Having thus established β(me), we can use it for the range ofelectron masses. The equations for the evolution of the two-level system are then

id

dtcn(t) = Encn(t) + βca(t) (59)

id

dtca(t) = Eaca(t) + βcn(t) (60)

The solutions are [69, 100], ignoring any time dependence in β and in Ea for a rst approximation,and with initial conditions cn(0) = 0

|cn(t)|2 =4β2

(En − Ea)2

+ 4β2sin2(

Ωt

2) (61)

Ω =

√((En − Ea)

2+ 4β2

)(62)

For small t we nd |cn(t)| ≈ βt, so that β is just the barrier transmission coecient. At exactresonance, En = Ea , which can happen if the deuterons are o-shell and their sum equals the massof 4He, we obtain

Ωres = 2β(me) (63)

The two energy eigenvalues (54) are not degenerate at resonance, the degeneracy having beensplit by the coupling β. The time to rst maxima for |cn(t)|2 is given by Tmax = π/Ω. This is acritical parameter because we don't want to have to wait around for a long time before a resonancehas a chance to build up. A value of 1 day for Tmax requires an electron mass about 2.4 timesthe standard electron mass. At resonance |cn(Tmax)| = 1.0, representing 100% barrier penetration!Using (51) we nd the following fusion rate at resonance

Λdd =Add |cn(t)|2

4πa3n

≈ 1.19× 1019s−1 at resonance and at peak time (64)

In arriving at the numerical estimate, we have made a questionable assumption that Adddoes not depend on md, but as this is an enormously large fusion rate, Add could be smaller by


many orders of magnitude without changing the basic conclusion. According to [32], the fusionrate originally claimed by Fleischmann and Pons requires an electron mass ten times heavier thannormal for a non-resonant theory. This translates into a fusion rate of ΛFP = 10−9.1s−1 from table2 in [32]. Thus, at resonance peak, we have a fusion rate here which is 28 orders of magnitude largerthan claimed by Fleischmann and Pons, but which persists for only a very short time. To achievethis resonance value, the energy dierence term must be zero to extremely high precision so that|En − Ea | < 2β(me). This precision would be unfeasible if it had to be experimentally controlledas has been already pointed out by Hagelstein [69]. We can relax this precision and still achieve arate of ΛFP by the following

Λdd ≈Add4πa3

n

4β2

(En − Ea)2

+ 4β2= ΛFP (65)

So the factor 4β2

(En−Ea)2can be as small as 10−28, or |En − Ea | ≤ 2β(me) × 1014. But even

this would require extreme precision. What saves this theory is that we have assumed that thedeuteron mass is slowly and continuously varying with time as is illustrated in Figure 1 on page13. So provided the asymptotic value of this mass is low enough, the combined masses of thetwo deuterons will at some time or another have to pass through the resonant value, and thenthe particle will fuse at that point with a probability of |cn(t)|2. If this coincides with the broadmaxima in time of (61) then the pair will fuse with near 100% probability owing to the huge rateat resonance (64). The fact that resonance is required for signicant fusion serves as a safeguardwhich prevents harmful radiation from being produced because at resonance there is no energy (orQ) left over to produce it. This is an enormous benet to this form of nuclear energy if it can beveried. The energy given to the lattice by the fusion event is actually given up prior to the eventas reected in the reduced masses of the two deuterons which subsequently fuse into 4He with zeroQ. The d2 molecule has been treated as a closed system, but the slow mass variation would requirecontinuous soft electromagnetic interaction with the lattice. As the deuteron mass decreases, then+3He phase space reaches zero before the t+3H channel does. Moreover, as the electron mass issupposedly increasing during this time, the t+3H channel benets from a relatively lower Coulombbarrier, as well as the resonance 20.21 MeV (31). This explains why signicantly more tritium isproduced than neutrons. After all reactions have ceased, we expect all particles to return to theirnormal rest masses, and so the energy stored in the electrons' higher masses would be returned tothe solid then too. This relaxation process may take some time, and might result in apparent heatproduction after all driving factors such as electrolysis have ceased. This could explain the so-calledlife-after-death phenomenon which has been observed in LENR reactions [1].

This theory can be generalized to the case of time varying masses by utilizing the theory oftwo-level systems with dynamic coupling [100]. From a practical standpoint, Tmax must not be toolong. This requires that the Coulomb barrier penetration factor β not be too small. Figure 2 onpage 21 shows a plot of Λdd, β, and Tmax. This sets a limit on how small the electron mass can beand still achieve the observed fusion rates. But the eect of screening in the actual palladium latticeshould be included too, and this goes beyond our simple d2 model. It's important to note that Λdd isthe conventional fusion rate which does not include the eects of resonance as calculated in [32]. Forunderstanding the eects of resonance, it is more imprortant to look at the Tmax parameter. Anydeuteron pair which goes through resonance near to this time will fuse with nearly 100% probabilitybecause of the extremely high fusion rate at resonance. Thus the energy release depends on howmany deuteron pairs are passing through resonance near to Tmax at some time or other. If werandomize the resonance time, then on average the value of |cn|2 will be 1/2 as can be seen from


(61). It is known that external time dependent stimulation of the system can enhance the excessenergy eect. This could have a number of eects on the dynamics of this system as currentlydescribed, but the details remain to be explored. One speculation is that the deuteron mass mightacquire a small ripple in time, and this could cause the resonance value to be crossed more thanonce, which in some circumstances may enhance the fusion rate. Or maybe the β term can becometime dependent due to this eect.

In our calculations we have assumed that the rate constant for dd fusion Add was a constant,independent of the deuteron mass. This assumption was made in the interest of simplicity, andbecause a theory for the o-mass-shell dependence of this parameter is not known. We can guessat some expected behavior of Add as the deuteron mass decreases and approaches resonance. Firstof all, as the phase space for d+ d→ n+3 He and d+ d→ t+ p both go to zero, Add is expectedto decrease by as much as 5 orders of magnitude. This is still a small eect on a logarithmic scalecompared to the quantum tunneling factor. But, Add may be increased somewhat by the unstableresonance at 20.210 MeV. As the stable 4He resonance is approached, we can expect a Breit-Wignerresonance form for the S-matrix of d+d→ γ+4He, and this resonance should more than oset thereduction in the photon phase space as resonance is approached. As a consequence, there should bean enhancement in the number of γ produced as the deuteron mass approaches resonance, and theγ energy approaches zero. Very near to the 4He resonance, multiple photon emission events willprobably become important as well. We defer a more detailed model for the functional variation ofAdd to a future publication.

9. Transmutation in deuterated palladium

With varying particle masses, transmutations can conceivably in theory occur in a number ofways in a palladium-deuterium lattice. These include electron capture, beta and alpha decayof nuclei whose mass has increased, resonant fusion of deuterium with other nuclei includingpalladium, resonant fusion of alpha particles and other nuclei including palladium, and even ssionof palladium and/or other impurity elements after a mass increase. Also, there is the possibility ofneutron creation and subsequent capture as in the Widom-Larsen theory, leading to many possiblereactions with no Coulomb barrier to be overcome. In short a world of possibilities exist. Thefusion possibilities mentioned in this list have a much higher Coulomb barrier than the deuteriummolecule. Nevertheless, if resonant tunneling occurs, there might be small numbers of such eventsoccuring, as has been reported in [101]. We shall defer a detailed examination of this subject to afuture publication.

10. Micro-craters

Many experiments have revealed micro-crater damage to the Palladium surface after LENR activityhas been observed [102105]. It has been commonly thought that these craters were evidence ofmicro explosions. We oer a dierent explanation. If all (or a substantial fraction) of the electronmasses were to increase in a small local region, the lattice spacing would be reduced approximatelyinversely proportional to their average mass, and this would cause a severe mechanical deformationof the surface of the palladium which could leave a crater caused by shrinkage. If all the electronswere to increase in mass by say a factor of 3 in a small volume on the surface, the volume woulddecrease by a factor of 33 or 27, which might well leave a crater on the surface. This would explainwhy transmutations are often observed to have occurred in or near these craters, as these are


10-70

10-60

10-50

10-40

10-30

10-20

10-10

100

0 5 10 15 20

dd

(s-1

)

me (me)

10-20

10-15

10-10

10-5

100

105

1010

1015

0 5 10 15 20

Tmax

(days)

me=2.3971me

me (me)

10-20

10-15

10-10

10-5

100

105

1010

1015

0 5 10 15 20

(s-1

)

me (me)

(a) (b)

(c)

Figure 2. Fusion rates vs. electron mass. Deuterons are on-shell. (a) shows the non-resonantfusion rate of Koonin and Nauenberg; (b) shows the estimated β parameter as a function ofelectron mass; (c) shows the amount of time in days until the rst maxima of the cyclic, two-levelmodel resonant tunneling if the electron's mass is held xed.

locations where the mass variation would be expected to have been the largest. Later, after allreactions were over and the electrons returned to their original mass, the volume would re-expandand possibly look like excess material on the rim of a crater.

11. Predictions and an experimental test of o-mass-shell variation

Assuming that this theory is correct, and that masses of deuterons are decreasing slowly as theloading factor d/pd increases, then we can make a very simple experimental prediction that istestable. We predict that the Q values for the reactions in (1), (2), and (3) will be observed todecrease with time, as the loading progresses, as in Figure 1 on page 13, in those systems that exhibitexcess heat production. Perhaps observable eects will even be seen in systems where no excessheat is produced, as excess heat requires that the deuteron mass must decrease until it is resonantwith the 4He channel, but less mass change could still reduce the Q values for the reactions. As thephase space for all three channels will change as the Q values diminish, then the relative branchingratios for the 3 channels will change as well. The rst channel to zero out would be the 3He + n


channel. Then, once the mass of the deuterons decrease below the threshold for producing t + p,the only channel open would the 4He+ γ. Such a reduction in Q values for a fundamental nuclearreaction has never been observed before. It would be a clear and undeniable proof that the deuteronrest mass was changing in these settings. We also expect that the energy of the gamma rays will bereduced from the expected value of 23.77MeV continuously down to zero at the resonance point.These non-resonant fusion events would be governed by the conventional fusion rate formula Λdd,and thus would be generally much fewer in number than the resonant fusion events which produceonly 4He and heat.

Perhaps the simplest apparatus to look for reduced Q values in d+ d fusion would be the gaspermeation methods of Iwamura [106, 107]. Alternatively, beam experiments might be used withlow-energy deuterons incident on a palladium metal surface. In either case, precision detection ofenergetic charged particles, neutrons, and γ leaving the surface would be desired. The chargedparticles would include Tritons, Helium-3 nuclei, protons, and α particles. The loading factorincreases in both systems over time. The energies of the charged particles produced are functionsof the deuteron rest mass. If the deuteron rest masses are changing, then this would show upas a broadening of the energy spectrum for a given type of charged particle. Beam experimentshave been performed already by Huke, Czerski, et al. [3640, 92, 93]. Various charged particleanomalies have already been observed in those experiments and others [1]. Iwamura et al. [106]have measured radiation produced in experiments with deuterium diusing through pd foil. Theynd a broad spectrum of X-rays along with neutron detection, but no correlation between neutronsand X-rays, or with excess heat. These X-rays could indicate fusion events between two o-shelldeuterons, which are relatively close to resonance (64).

We leave the job of determining whether this phenomenon is occurring to experimentalists,and will not try and draw any conclusions from the various anomalous radiation events that havebeen reported in the literature here.

12. Conclusion

We want to emphasize that there is no direct experimental evidence yet that masses of electrons,nucleons, or nuclei can change signicantly in a condensed matter setting. What is being proposedhere is a radical departure from existing accepted nuclear and condensed matter theory, and deservesto be treated with a great deal of skepticism. Nevertheless, it is this author's opinion that Fock-Stueckelberg or other type of o-mass-shell theories are a possible explanation for such variationsand that all of the experiments in LENR can potentially be explained if they are occurring. Theexperimental claims that have been made about the occurrence of fusion, transmutation, ssion, andlack of signicant radiation in most experiments have been easily judged as absurd and preposterousby many physicists. Yet each year that passes sees more experimental papers claiming to validatethese phenomena in the international scientic literature. Any theory that could describe thisgrowing body of experimental evidence will likely seem equally absurd. The author freely admitsthat the theory he has proposed here is radical and hard to believe, but he sees no other way toexplain these experiments, and thus feels compelled to persist.

If experiments conrm that mass variation is occurring in deuterated palladium, then it is likelythat all of the eects of LENR can be explained by this when applied to various types of chargedparticles. Controlling the mass variation will consequently become the key engineering challengerequired for controlling low energy nuclear reactions. It is ironic that on the one hand in the case ofLENR, the acceptance of the experimental results has been impeded for over 20 years by the lack

REFERENCES 23

of a theoretical framework in which the results could be contemplated as even conceivable, whereasin the case of the o-mass-shell covariant relativistic dynamics, the acceptance of this theory intothe canon of physics has not happened for over 70 years because of the lack of any experimentalevidence for it. If o-mass-shell behavior is conrmed, then it will undoubtedly have profoundimplications for the foundations of quantum mechanics, as well as for other branches of physics andchemistry.

We have made clear and simple predictions that the Q values for the deuterium fusion reactionchannels will reduce with time in deuterated palladium experiments that produce excess heat, andat resonance the Q values will all be zero so that no radiation is produced for the bulk of the fusionevents which occur then. If this behavior is conrmed experimentally, then it will be proof thatrest masses are changing in a condensed matter setting.

Although we have concentrated on the deuterated palladium system here, it is clear that manyreactions can occur in other systems such as nickel and hydrogen if masses vary there too. Thesame can be said for combinations of deuterium and metals other than palladium.

The body of experimental evidence in this eld has now grown rather large, and it is quitecomplex. It takes a good deal of time and eort to gain a good understanding of it. In 1990, giventhe 50 orders of magnitude discrepancy between nuclear theory and the original Fleischmann-Ponsexperiments, the conclusion was reached that the experiments were wrong. Today this discrepancyremains, mitigated somewhat by enhanced electron screening arguments, but worsened by theabundant transmutation evidence now observed and requiring an explanation. The growing weightof evidence has been slowly tilting the verdict in favor of the experiments. Extrapolating this trend,it seems likely that scientists in the future at some point may come to believe that the nuclear theorycirca 1990 was incomplete, and that the experiments showing LENR anomalies were and are in largemeasure correct. The nal chapter in this epic story has clearly not yet been written.

Acknowledgements

The author acknowledges valuable correspondence with Lawrence Horwitz, and valuable discussionswith Peter Hagelstein, Michael McKubre, David Nagel, and Paul Marto. He also acknowledgesVladimir Kresin for extensive critical but good-spirited and helpful discussions. Any errors areentirely the author's, and he makes no claim of endorsement of this theory by anyone.

References

[1] Storms E 2007 Science of Low Energy Nuclear Reaction: A Comprehensive Compilation ofEvidence and Explanations about Cold Fusion (World Scientic Publishing Company) ISBN9812706208

[2] Srinivasan M, Miley G and Storms E 2011 Low-energy nuclear reactions: TransmutationsNuclear Energy Encyclopedia ed Krivit S B, Lehr J H and Kingery T B (John Wiley &Sons, Inc.) p 503539 ISBN 9781118043493 URL http://onlinelibrary.wiley.com/doi/

10.1002/9781118043493.ch43/summary

[3] Fock V 1937 Phys. Z. Sowjetunion 12 404425

[4] Stueckelberg E 1941 Helv. Phys. Acta 14 322323

[5] Stueckelberg E 1941 Helv. Phys. Acta 14 588594

24 REFERENCES

[6] Lacki J, Ruegg H and Wanders G 2008 E.C.G. Stueckelberg, An Unconventional Figure ofTwentieth Century Physics: Selected Scientic Papers with Commentaries (Springer) ISBN9783764388775

[7] Feynman R P 1950 Physical Review 80 440457 URL http://link.aps.org/doi/10.1103/

PhysRev.80.440

[8] Horwitz L P and Piron C 1973 Helv. Phys. Acta 46 316326 URL http://www.osti.gov/

energycitations/product.biblio.jsp?osti_id=4355335

[9] Reuse F 1979 Foundations of Physics 9 865882 ISSN 0015-9018, 1572-9516 URL http:

//link.springer.com/article/10.1007/BF00708697

[10] Fanchi J R 1993 Parametrized Relativistic Quantum Theory (Springer-Verlag GmbH) ISBN9780792323761

[11] Horwitz L P and Lavie Y 1982 Physical Review D 26 819838 URL http://link.aps.org/

doi/10.1103/PhysRevD.26.819

[12] Horwitz L and Shnerb N 1998 Foundations of Physics 28 15091519 ISSN 0015-9018 URLhttp://www.springerlink.com/content/rw725423p4l33516/abstract/

[13] Land M C and Horwitz L P 1991 Foundations of Physics Letters 4 6171 ISSN 0894-9875,1572-9524 URL http://link.springer.com/article/10.1007/BF00666417

[14] Land M C 1998 Foundations of Physics 28 14791487 ISSN 0015-9018, 1572-9516 URLhttp://link.springer.com/article/10.1023/A%3A1018813429428

[15] Seidewitz E 2009 Annals of Physics 324 309331 ISSN 0003-4916 URL http://www.

sciencedirect.com/science/article/pii/S0003491608001668

[16] Aharonovich I and Horwitz L P 2012 Journal of Mathematical Physics 53 03290203290229 ISSN 00222488 URL http://jmp.aip.org/resource/1/jmapaq/v53/i3/p032902_s1?

bypassSSO=1

[17] Burakovsky L and Horwitz L 1993 Physica A: Statistical Mechanics and its Applications201 666679 ISSN 0378-4371 URL http://www.sciencedirect.com/science/article/

pii/037843719390135Q

[18] Burakovsky L and Horwitz L P 1994 Journal of Physics A: Mathematical and General 2726232631 ISSN 0305-4470, 1361-6447 URL http://iopscience.iop.org/0305-4470/27/

8/003

[19] Burakovsky L and Horwitz L P 1996 ArXiv High Energy Physics - Theory e-prints 4106 URLhttp://adsabs.harvard.edu/abs/1996hep.th....4106B

[20] Greenberger D M 1970 Journal of Mathematical Physics 11 23292340 ISSN 00222488 URLhttp://jmp.aip.org/resource/1/jmapaq/v11/i8/p2329_s1?isAuthorized=no

[21] Greenberger D M 1970 Journal of Mathematical Physics 11 23412347 ISSN 00222488 URLhttp://jmp.aip.org/resource/1/jmapaq/v11/i8/p2341_s1?isAuthorized=no

[22] Greenberger D M 1974 Journal of Mathematical Physics 15 395 ISSN 00222488 URLhttp://link.aip.org/link/?JMP/15/395/1&Agg=doi

[23] Greenberger D M 1974 Journal of Mathematical Physics 15 406 ISSN 00222488 URLhttp://link.aip.org/link/?JMP/15/406/1&Agg=doi

[24] Corben H C 1962 Proceedings of the National Academy of Sciences of the United States ofAmerica 48 15591566 ISSN 0027-8424 PMID: 16590996

REFERENCES 25

[25] Corben H C 1962 Proceedings of the National Academy of Sciences of the United Statesof America 48 17461752 ISSN 0027-8424 PMID: 16591007 PMCID: PMC221034 URLhttp://www.ncbi.nlm.nih.gov/pmc/articles/PMC221034/

[26] Fanchi J R 1993 Foundations of Physics 23 487548 ISSN 0015-9018, 1572-9516 URLhttp://link.springer.com/article/10.1007/BF01883726

[27] Adler S L 2004 Quantum theory as an emergent phenomenon: the statistical mechanics ofmatrix models as the precursor of quantum eld theory (Cambridge University Press) ISBN9780521831949

[28] 't Hooft G 2007 arxiv.org 0707.4568 AIPConf.Proc.957:154-163,2007 URL http://arxiv.

org/abs/0707.4568

[29] 't Hooft G 2005 Determinism beneath quantum mechanics Quo Vadis Quantum Mechanics?The Frontiers Collection ed Elitzur A C, Dolev S and Kolenda N (Berlin/Heidelberg: Springer-Verlag) p 99111 ISBN 3-540-22188-3 URL http://www.springerlink.com/content/

t545r734304254q3/

[30] 't Hooft G 2009 arxiv.org 0908.3408 URL http://arxiv.org/abs/0908.3408

[31] Weinberg S 2011 arXiv:1109.6462 URL http://arxiv.org/abs/1109.6462

[32] Koonin S E and Nauenberg M 1989 Nature 339 690691 ISSN $footerJournalISSN URLhttp://www.nature.com/nature/journal/v339/n6227/abs/339690a0.html

[33] Fleischmann M, Pons S and Hawkins M 1989 Journal of Electroanalytical Chemistry 261

301308 URL http://www.ftp.nic.funet.fi/pub/doc/Fusion/fp.ps

[34] Leggett A J and Baym G 1989 Physical Review Letters 63 191194 URL http://link.aps.

org/doi/10.1103/PhysRevLett.63.191

[35] Leggett A J and Baym G 1989 Nature 340 4546 ISSN $footerJournalISSN URL http:

//www.nature.com/nature/journal/v340/n6228/abs/340045a0.html

[36] Czerski K, Huke A, Biller A, Heide P, Hoeft M and Ruprecht G 2001 Europhysics Letters(EPL) 54 449455 ISSN 0295-5075, 1286-4854 URL http://iopscience.iop.org/epl/

i2001-00265-7

[37] Czerski K, Huke A, Heide P and Ruprecht G 2004 EPL (Europhysics Letters) 68 363369URL http://adsabs.harvard.edu/abs/2004EL.....68..363C

[38] Czerski K, Huke A, Heide P and Ruprecht G 2006 Experimental and theoretical screeningenergies for the ²H(d, p)³H reaction in metallic environments The 2nd InternationalConference on Nuclear Physics in Astrophysics ed Fülöp Z, Gyürky G and SomorjaiE (Springer Berlin Heidelberg) pp 8388 ISBN 978-3-540-32843-8 URL http://www.

springerlink.com/content/k740562323673840/abstract/

[39] Czerski K, Huke A, Martin L, Targosz N, Blauth D, Górska A, Heide P and Winter H 2008Journal of Physics G: Nuclear and Particle Physics 35 014012 ISSN 0954-3899, 1361-6471URL http://iopscience.iop.org/0954-3899/35/1/014012

[40] Czerski K 2009 Enhanced electron screening and nuclear mechanism ofcold fusion ICCF-15 vol 15 (Rome, Italy: ENEA) pp 197202 URLhttp://www.enea.it/it/produzione-scientifica/edizioni-enea/2012/

proceedings-iccf-15-international-conference-on-condensed-matter-nuclear-science

[41] Tsyganov E N 2012 Physics of Atomic Nuclei 75 153159 ISSN 1063-7788, 1562-692X URLhttp://link.springer.com/article/10.1134/S1063778812010140

26 REFERENCES

[42] Atzeni S and Meyer-ter Vehn J 2004 The Physics of Inertial Fusion:BeamPlasma Interaction,Hydrodynamics, Hot Dense Matter: BeamPlasma Interaction, Hydrodynamics, Hot DenseMatter (Oxford University Press) ISBN 9780198562641

[43] Nagel D J 2009 Innite Energy 88 21

[44] Widom A and Larsen L 2006 The European Physical Journal C - Particles and Fields 46107111 ISSN 1434-6044, 1434-6052 URL http://link.springer.com/article/10.1140/

epjc/s2006-02479-8

[45] Konopinski E J 1978 American Journal of Physics 46 499 ISSN 00029505 URL http:

//link.aip.org/link/?AJP/46/499/1&Agg=doi

[46] Calkin M G 1966 American Journal of Physics 34 921 ISSN 00029505 URL http://link.

aip.org/link/?AJP/34/921/1&Agg=doi

[47] Aguirregabiria J M, Hernández A and Rivas M 2004 European Journal of Physics 25 555567ISSN 0143-0807, 1361-6404 URL http://iopscience.iop.org/0143-0807/25/4/010

[48] Szalewicz K, Morgan J D and Monkhorst H J 1989 Physical Review A 40 28242827 URLhttp://link.aps.org/doi/10.1103/PhysRevA.40.2824

[49] Rabinowitz M 1990 Modern Physics Letters B 4 233247 ISSN 0217-9849 URL http:

//adsabs.harvard.edu/abs/1990MPLB....4..233R

[50] Jackson J 1957 Physical Review 106 330339 ISSN 0031-899X, 1536-6065 URL http:

//adsabs.harvard.edu/abs/1957PhRv..106..330J

[51] Evans A B 2009 J. Condensed Matter Nucl. Sci 2 712

[52] Evans A B 1990 Foundations of Physics 20 309335 ISSN 0015-9018, 1572-9516 URLhttp://link.springer.com/article/10.1007/BF00731695

[53] Fearing H W and Scherer S 2000 Physical Review C 62 034003 URL http://link.aps.org/

doi/10.1103/PhysRevC.62.034003

[54] Tyutin I V 2002 Physics of Atomic Nuclei 65 194202 ISSN 1063-7788, 1562-692X URLhttp://link.springer.com/article/10.1134/1.1446571

[55] Newton T D and Wigner E P 1949 Reviews of Modern Physics 21 400406 URL http:

//link.aps.org/doi/10.1103/RevModPhys.21.400

[56] Wightman A S 1962 Reviews of Modern Physics 34 845872 URL http://link.aps.org/

doi/10.1103/RevModPhys.34.845

[57] Alvarez E T G and Gaioli F H 1998 Foundations of Physics 28 15291538 ISSN 0015-9018,1572-9516 URL http://link.springer.com/article/10.1023/A%3A1018882101146

[58] Gorelik V S 2010 Physica Scripta 2010 014046 ISSN 1402-4896

[59] Gorelik V S 2006 Acta Physica Hungarica A) Heavy Ion Physics 26 3746 ISSN 1589-9535,1786-3767 URL http://link.springer.com/article/10.1556/APH.26.2006.1-2.6

[60] Gorelik V S 2007 Quantum Electronics 37 409432 ISSN 1063-7818, 1468-4799 URL http:

//www.turpion.org/php/paper.phtml?journal_id=qe&paper_id=13478

[61] John S and Wang J 1990 Physical Review Letters 64 24182421 URL http://link.aps.org/

doi/10.1103/PhysRevLett.64.2418

[62] J-M André P J 2011 Physica Scripta 84 035708 ISSN 1402-4896

[63] Weinberg S 1990 Physics Letters B 251 288292 ISSN 03702693 URL http://65.54.113.

26/Publication/18408213/nuclear-forces-from-chiral-lagrangians

REFERENCES 27

[64] Weinberg S 1979 Physica A: Statistical Mechanics and its Applications 96 327340 ISSN 0378-4371 URL http://www.sciencedirect.com/science/article/pii/0378437179902231

[65] Epelbaum E 2010 arXiv:1001.3229 URL http://arxiv.org/abs/1001.3229

[66] Land M C and Horwitz L P 1996 arXiv:hep-th/9601021 URL http://arxiv.org/abs/

hep-th/9601021

[67] Coleman S 1977 Physical Review D 15 29292936 URL http://link.aps.org/doi/10.

1103/PhysRevD.15.2929

[68] Tomsovic S and Ullmo D 1994 Physical Review E 50 145162 URL http://link.aps.org/

doi/10.1103/PhysRevE.50.145

[69] Hagelstein P L 2005 Resonant tunneling and resonant excitation transfer Proceedings, ICCF-12 (Cambridge, MA, USA) pp 871886 URL http://adsabs.harvard.edu/abs/2005cmns.

conf..871H

[70] Li X Z 1999 Czechoslovak Journal of Physics 49 985992 ISSN 0011-4626, 1572-9486 URLhttp://link.springer.com/article/10.1023/A%3A1021485221050

[71] Grifoni M and Hänggi P 1998 Physics Reports 304 229354 ISSN 0370-1573 URL http:

//www.sciencedirect.com/science/article/pii/S0370157398000222

[72] Saad D, Horwitz L P and Arshansky R I 1989 Foundations of Physics 19 11251149

[73] Horwitz L P, Arshansky R I and Elitzur A C 1988 Foundations of Physics 18 11591193 ISSN0015-9018, 1572-9516 URL http://link.springer.com/article/10.1007/BF01889430

[74] Land M C and Horwitz L P 1991 Foundations of Physics 21 299310 ISSN 0015-9018, 1572-9516 URL http://link.springer.com/article/10.1007/BF01883636

[75] Land M C 1997 Foundations of Physics 27 1941 ISSN 0015-9018, 1572-9516 URL http:

//link.springer.com/article/10.1007/BF02550153

[76] Land M 2011 Journal of Physics: Conference Series 330 012015 ISSN 1742-6596 URLhttp://iopscience.iop.org/1742-6596/330/1/012015

[77] Horwitz L P and Arshansky R 1982 Journal of Physics A: Mathematical and General 15 L659L662 ISSN 0305-4470, 1361-6447 URL http://iopscience.iop.org/0305-4470/15/12/002

[78] Fanchi J 1981 Foundations of Physics 11 493498 ISSN 0015-9018 URL http://www.

springerlink.com/content/v3211wt465115256/abstract/

[79] Piron C and Reuse F 1978 Helv. Phys. Acta 51 146176

[80] Horwitz L P, Piron C and Reuse F 1975 Helv. Phys. Acta, v. 48, no. 4, pp. 546-548 48 URLhttp://www.osti.gov/energycitations/product.biblio.jsp?osti_id=4018859

[81] Proca A 1936 J. Phys. Rad. 7 347353

[82] Lee T D and Yang C N 1962 Physical Review 128 885898 URL http://link.aps.org/

doi/10.1103/PhysRev.128.885

[83] Ruck H M and Greiner W 1977 Journal of Physics G: Nuclear Physics 3 657680 ISSN0305-4616 URL http://iopscience.iop.org/0305-4616/3/5/013

[84] Kaplan D B, Savage M J and Wise M B 1999 Physical Review C 59 617629 URLhttp://link.aps.org/doi/10.1103/PhysRevC.59.617

[85] Horwitz L P, Katz N and Oron O 2004Discrete Dynamics in Nature and Society 2004 179204ISSN 1026-0226, 1607-887X URL http://www.hindawi.com/journals/ddns/2004/205916/

abs/

28 REFERENCES

[86] Roitgrund A and Horwitz L 2010 Discrete Dynamics in Nature and Society 2010 136 ISSN1026-0226, 1607-887X URL https://eudml.org/doc/230976

[87] Burakovsky L, Horwitz L P and Schieve W C 1996 Physical Review D 54 40294038 URLhttp://link.aps.org/doi/10.1103/PhysRevD.54.4029

[88] Land M 2006 arXiv:hep-th/0603074 Found.Phys.33:1157-1175,2003 URL http://arxiv.

org/abs/hep-th/0603074

[89] Horwitz L, Schieve W and Piron C 1981 Annals of Physics 137 306340 ISSN 0003-4916 URLhttp://www.sciencedirect.com/science/article/pii/0003491681901998

[90] Berestetskii V B, Pitaevskii L P and Lifshitz E M 1982 Quantum Electrodynamics, SecondEdition: Volume 4 2nd ed (Butterworth-Heinemann) ISBN 0750633719

[91] Hagelstein P L and Chaudhary I U 2008 Journal of Physics B: Atomic, Molecular andOptical Physics 41 125001 ISSN 0953-4075, 1361-6455 URL http://iopscience.iop.org/

0953-4075/41/12/125001

[92] Huke A, Czerski K, Heide P, Ruprecht G, Targosz N and ebrowski W 2008 Physical ReviewC 78 015803 URL http://link.aps.org/doi/10.1103/PhysRevC.78.015803

[93] Huke A, Czerski K and Heidea P 2003 Nuclear Physics A 719 C279C282 ISSN 0375-9474URL http://www.sciencedirect.com/science/article/pii/S0375947403009321

[94] Huke A, Czerski K and Heide P 2007 Nuclear Instruments and Methods in Physics ResearchSection B: Beam Interactions with Materials and Atoms 256 599618 ISSN 0168-583X URLhttp://www.sciencedirect.com/science/article/pii/S0168583X07001784

[95] Arensburg A and Horwitz L P 1992 Foundations of Physics 22 10251039 ISSN 0015-9018,1572-9516 URL http://link.springer.com/article/10.1007/BF00733394

[96] Rolfs C 2006 Nuclear Physics News 16 911

[97] Firestone R B, Baglin C M and Chu S Y F 1999 Table of isotopes (Wiley) ISBN 9780471356332

[98] Ziegler J F, Biersack J P and Ziegler M D 2008 SRIM - The Stopping and Range of Ions inMatter (SRIM Co.) ISBN 9780965420716

[99] Koªos W and Wolniewicz L 1964 The Journal of Chemical Physics 41 36633673 ISSN 00219606 URL http://jcp.aip.org/resource/1/jcpsa6/v41/i12/p3663_s1?

isAuthorized=no

[100] Hagelstein P L, Senturia S D and Orlando T P 2004 Introductory Applied Quantum andStatistical Mechanics (Wiley) ISBN 9780471202769

[101] Iwamura Y, Itoh T and Tsuruga S 2012 Increase of reaction products in deuterium permeationinduced transmutation ICCF-17 (Daejeon, South Korea) p 6

[102] Nagel D 2013 Journal of Condensed Matter Nuclear Science 10 114

[103] Szpak S, Mosier-Boss P A, Young C and Gordon F E 2005 Naturwissenschaften 92

394397 ISSN 0028-1042, 1432-1904 URL http://link.springer.com/article/10.1007/

s00114-005-0008-7

[104] Toriyabe Y, Mizuno T, Ohmori T and Aoki Y 2006 Elemental analysis of palladium electrodesafter Pd/Pd light water critical analysis Proceedings, ICCF-12 (Yokohama, Japan: WorldScientic Publishing Co. Pte. Ltd.) pp 253263 ISBN 9789812569011 URL http://adsabs.

harvard.edu/abs/2006cmns...12..253T

REFERENCES 29

[105] Zhang W S and Dash J 2007 Excess heat reproducibility and evidence of anomalous elementsafter electrolysis in Pd/D2+H2SO4 electrolytic cells Proceedings, ICCF-13 (Sochi, Russia) p202

[106] Iwamura Y 1998 Fusion science and technology 33 476492

[107] Iwamura Y, Itoh T, Sakano M, Yamazaki N, Kuribayashi S, Terada Y, Ishikawa T and Kasagi J2006 Observation of nuclear transmutation reactions induced by d2 gas permeation through pdcomplexes ICCF-11 vol 11 (Marseilles, France) pp 339350 URL http://adsabs.harvard.

edu/abs/2006cmns...11..339I

Theories of variable mass particles and low energy nuclear phenomena

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