Found Phys DOI 10.1007/s10701-014-9774-4

Theories of Variable Mass Particles and Low Energy Nuclear Phenomena Mark Davidson

Received: 8 May 2013 / Accepted: 17 January 2014 © Springer Science+Business Media New York 2014

Abstract Variable particle masses have sometimes been invoked to explain observed anomalies in low energy nuclear reactions (LENR). Such behavior has never been observed directly, and is not considered possible in theoretical nuclear physics. Nevertheless, there are covariant off-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 particles that arise in conventional Feynman diagrams in relativistic field theories. Effective Lagrangian models incorporating variable mass particle theories might be useful in describing anomalous nuclear reactions by combining mass shifts together with resonant tunneling and other effects. A detailed model for resonant fusion in a deuterium molecule with off-shell deuterons and electrons is presented as an example. Experimental means of observing such off-shell behavior directly, if it exists, is proposed and described. Brief explanations for elemental transmutation and formation of micro-craters are also given, and an alternative mechanism for the mass shift in the Widom–Larsen theory is presented. If variable mass theories were to find experimental support from LENR, then they would undoubtedly have important implications for the foundations of quantum mechanics, and practical applications may arise. Keywords

Hydrated palladium · Deuterated palladium · LENR · Fusion

JEL Classification

24.10.-i · 3.65.-w · 3.65.Nk · 3.70.+k

1 Introduction Many unexplained anomalies have been observed in heavily deuterated palladium, as well as other metal-hydrogen alloys [1,2] which suggest that nuclear reactions are M. Davidson (B) Spectel Research Corporation, Palo Alto, CA, USA e-mail: [email protected]

123

Found Phys

taking place in condensed matter. There is severe skepticism in the physics community about these claims. This literature is complex, and the subject cannot be mastered easily. It is this author’s opinion, after very extensive and long-term study of the literature, that the claims of experimental anomalies have a significant probability of being true. If pressed, he would roughly estimate a 75% probability that nuclear reactions are truly responsible for many of these anomalies. In this paper, we offer an explanation for these anomalies, assuming that they are true. The following experimental claims have been made by multiple experimenters: 1. In highly deuterated palladium, with loading factors (ratio of the number densities of d and pd) greater than 0.9, some electrolytic cells have produced heat in excess of what can be accounted for chemically. They also have produced Helium-4 in quantities which were approximately consistent with 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 to within a few thousand angstroms of the surface of the palladium. 2. Also in highly deuterated palladium, numerous elemental transmutations have been observed on the surface of the palladium after deuterium loading. 3. These reactions can sometimes be stimulated by injecting time-varying electromagnetic fields into the surface, or by laser illumination of the surface, or by sound waves, or by thermal cycling. We make an argument here that all anomalous phenomena which fall under the domain of LENR might be explained by variation of the rest masses of elementary particles in a condensed matter 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 that mass variation might occur for neutrons. Such behavior has never been directly observed, and it is considered impossible and ruled out by conventional physics. Nevertheless, there is a large body of theoretical work which suggests that it might be possible. The LENR anomalies may constitute indirect evidence of such mass deviations. Off-mass-shell covariant relativistic mechanics has a long and rather eminent history in physics. Early pioneering work was done by Fock [3] and Stueckelberg [4–6]. This was followed by applications of essentially the same theory by Feynman in his path integral formalism of quantum mechanics [7](see Appendix A), and later by elaborations and extensions of the theory by Horwitz and Piron [8], and by many others [9–22]. Other variable mass theories were proposed by Greenberger [20–23], 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 that they are required to describe any natural phenomena. If off-mass-shell quantum mechanics is needed in order to understand LENR results, then our understanding of quantum mechanics will be affected at a fundamental level. The standard on-shell wave equations would have to be considered as approximations to more general off-shell theories. There is a modern school of emergent quantum mechanics developing, looking for a deeper origin of the quantum laws [27–31]. Support from LENR for off-mass-shell dynamics would be of great relevance to this field. The standard relativistic wave equations (Klein-Gordon, Dirac, Proca, etc.) all

123

Found Phys

have problems with a localized position operator, or negative energies, or negative probabilities. Thus, modern physics regards quantum fields as preeminent over N particle wave mechanics. But a resurgence of the Fock–Stueckelberg theories could change this, and make N-particle wave mechanics prominent once again. We use units such that h¯ = c = 1, and a relativistic metric signature (−, +, +, +), unless otherwise noted. 2 The Well Known Difficulties 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 the penetration factor through the Coulomb barrier at room temperature [32] which gave reaction rates 50 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 unobserved affinity enhancement for 4 He in deuterated palladium. They and others concluded that either the experiments must be wrong, or that the effects were not due to fusion. Recent experiments on deuteron beam scattering off of deuterated metals of various kinds have exhibited much larger fusion cross sections than these calculations [36–41]. This enhancement has been attributed to unexpectedly large electron screening effects. The other main problem is to explain the extreme distortion of the branching ratios as compared with plasma fusion which has the well-measured ratios [42] d + d → t (1.01 MeV) + p(3.02 MeV), (Q = 4.04 MeV), 51 % (1) d + d →3 He(.82 MeV) + n(2.45 MeV), (Q = 3.27 MeV), 49 % d + d →4 He(0.076 MeV) + γ (23.77 MeV), (Q = 23.85 MeV), (3.8 ∗ 10 − 3) %

(2)

(3)

where Q is the kinetic energy released by the reaction in the center of mass Lorentz frame. Nagel has listed a number of phenomena as illustrative of the challenge to theory [43]. There are also extensive experimental claims of a variety of elemental transmutations taking place inside or on deuterated palladium and also in other metalhydrogen alloys [1,2]. The transmutation data are much harder to ignore than 4 He production which might be due to environmental contamination, and they represent strong evidence for nuclear reactions, although not necessarily fusion. Widom and Larsen [44] have presented an alternative to fusion models. A significant portion of the LENR research 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 conventional relativistic classical or quantum mechanics in a field-free region. The energy levels and therefore masses of electromagnetically bound systems can certainly be

123

Found Phys

changed by external fields. Examples are the Stark and Zeeman effects in atomic physics, where the mass of the atom is modified continuously according to E = mc2 depending on the strength of the field and the energy shift caused by it. A single charge monopole satisfies the minimal coupling prescription p μ = mv μ + q Aμ . This equation comes from the fact that electromagnetic stress energy of the electromagnetic field changes by the finite amount q Aμ due to the charge-field interaction as in [45–47], but it only has this very simple interpretation in the transverse gauge. So the rest mass is hard to define precisely. We can interpret this as meaning m is the mass contained in an infinitesimal volume around a point charge, and p μ pμ is the total mass of the system which includes the local rest mass plus the external field’s contribution. Charge is always conserved. Consider an N particle collision involving nuclear interactions. If the interaction is local,  point, then both  μso that the N particles meet at a space-time the total momentum pi and the electromagnetic contribution qi Aμ are independently conserved, ignoring variation of Aμ over the interaction volume. Consequently, μ the inertial momentum or m i Vi is also conserved, so that the presence of an external electromagnetic field does not affect the nuclear kinematics. The parameter m, the field-free inertial mass, changes discreetly in nuclear collisions in conventional theories, but never continuously. Of course, the concept of “effective mass” is frequently used in condensed matter to describe phenomenological mass parameters in transport equations in a solid. But this is not the fundamental rest mass of the particle which could affect 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 first papers by Fleischmann and Pons, noted that modified 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 fields (see 7), and then these “heavy” electrons can join with protons in a solid and form low energy neutrons through the weak interaction, which can then react with nuclei. The first suggestion that a variable mass covariant type of theory might be applicable to LENR was due to Evans [51,52] who applied it to electrons. In this paper, we consider mass changes for all types of particles. We contemplate up to about ±12 MeV for deuterons, and less for electrons.

4 Effective Lagrangians in Nuclear Physics and Off-Mass-Shell Propagators Feynman diagrams for relativistic field theories like the standard model of particle physics routinely involve off-mass-shell propagator lines. Such “particles” in the nomenclature of perturbation theory are called virtual. It is often stated that virtual particles are “not real”, but only mathematical constructions, unlike on-shell particles that appear in initial or final asymptotic scattering states and which are “real”. This point of view is reinforced by a fairly rigorous equivalence theorem in field theory which states that changing field variables will change off-shell Green’s functions while leaving the S-matrix invariant [53,54]. The relativistic single particle state in general has problems of localizability [55,56], and moreover quantum entanglement and wave

123

Found Phys

particle duality make the “reality” 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 to observe it. Thus all particles are in fact always “slightly” virtual, and perfectly on-shell particles are a mathematical abstraction. This forces us to conclude that most particles are a very little bit off the mass shell all the time. The fact that this deviation is empirically very small lets us approximate the true situation with theories in which the mass deviation is exactly zero. In the non-relativistic limit this leads to the Schrödinger equation with potential forces and fixed masses. This is the basis of almost all condensed matter physics, and so from the start, variable mass behavior is excluded from discussions about condensed matter. This is fine so long as there is no experimental evidence to the contrary. In Schrödinger wave mechanics perturbation theory the virtual particles become off the energy shell, but the mass is still equal to the rest mass. The experimental claims in the LENR area, if correct, point to the possibility that masses are in fact changing in some special condensed matter settings. We can choose one of two paths. Either reject the experimental data as impossible, or generalize condensed matter theory to allow for mass variation. We suggest the second 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 covariant footing [7](see Appendix A). He chose to introduce a second time variable, call it historical time τ . It is not the same as proper time in classical particle mechanics, although it plays a similar role which is to parametrize a curve in spacetime. It is useful to review his logic briefly. Consider the conventional Klein–Gordon equation for a spinless particle in the presence of an electromagnetic field (i∂ − A)μ (i∂ − A)μ  = −M 2 

(4)

M is the particle’s rest mass, and where A is the vector potential for an external electromagnetic field. There are well known problems with the probabilistic interpretation    of this equation because the conserved current is jμ = i ∂μ φ ∗ φ − φ∂μ φ ∗ and j0 takes on both positive and negative values. Moreover, a localizable Schrödinger type position operator cannot be defined for this equation [55,56]. Feynman wished to apply his path integration method of quantization to the KGE. In his words “... we try to represent the amplitude for a particle to get from one point to another as a sum over 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 trajectory in space-time by giving the four variables x μ (μ) as functions of some fifth parameter μ (rather than expressing x1 , x2 , x3 in terms of x4 )”. So Feynman replaces (4) with (we use x 0 = t, and τ instead of μ and we use spacelike metric, whereas Feynman used timelike) i

∂ϕ(x, τ ) 1 = (i∂ − A)μ (i∂ − A)μ ϕ(x, τ ), where x = {x 0 , x 1 , x 2 , x 3 } ∂τ 2

(5)

It is very similar to the time dependent Schrödinger equation, but with the “historical time” τ replacing the usual time variable t, and with the four coordinates of space-time

123

Found Phys

x μ replacing the usual three coordinates of space. Feynman points out that if Aμ does not depend on τ , then separable solutions exist so that ϕ(x, τ ) = (x)ex p(i 21 M 2 τ ) and ψ is the solution to the usual KGE. Because of the similarity to Schrödinger’s equation, (5) has a postive definite probability, and a localizable Schrödinger position operator. Moreover, it is amenable to solution using Feynman’s path integral formulation as he shows in [7]. In fact young Feynman had studied this idea more extensively than is commonly realized , but was discouraged by negative reactions to it [57]. Equations of this type were first studied by Fock [3] and Stueckelberg [4–6]. The path integral solution includes all paths in space-time connecting two space-time points and parametrized by τ , with no restrictions on the path, so that off-mass-shell paths are included in the path integral solutions. This reflects that fact that Feynman diagrams contain virtual particles which are not on the mass shell. This theory was extended to multi-particle systems by Horwitz and Piron [8] by postulating that a single τ variable acts as a historical time for all of the particles simultaneously. This greatly simplifies the mathematics for both the quantum and classical many particle relativistic equations, but at the expense of some new interpretative challenges. For example, the wave function describes an extended probability cloud not just in the spatial coordinates, but also in time. Let us refer to these types of theories as Fock–Stueckelberg or FS or historical-time theories. We note that, when restricted to a single particle, the mass of the particle is a constant of the motion for any arbitrary electromagnetic field described by the vector potential Aμ , as it is in ordinary classical relativistic mechanics, but the mass in Horwitz–Piron theory for two or more particles is not an invariant [8]. These type models can deviate in the mass, even in restricted non-relativistic approximation where the velocities of the particles are much less than c, but their mass can still change due to interactions. In the theory of photonic crystals, it has been found both theoretically and experimentally that the photon can acquire an effective mass [58–62]. This effect might enhance the mass deviation of charged particles which could interact with these massive photons. The nuclear active environment for LENR is known to occur near the surface of palladium, and in areas where the lattice has been deformed. The surface morphology and chemistry are also known to be critical factors. The possibility that off-mass-shell photons are a necessary condition for a nuclear active environment is worth considering. Because of the size of the strong coupling constant, the standard model is too complicated at low energies, and so effective Lagrangian approximations are routinely employed in nuclear physics. Many of these methods predate the standard model. They often exploit the low values for mass of the u and d quarks which when taken to zero lead to chiral-isospin symmetry which can then justify utilizing group theoretical methods to derive general forms for interacting potentials [63–65]. One would think that FS type of wave equations would have been considered in the context of effective Lagrangians for nuclear physics, but they haven’t been. Fanchi [10] gives a historical explanation for the general lack of interest. The current literature contains treatment of any spin in a path integral approach [15], along with significant other literature on relativistic wave equations for spin 0 and spin 1/2 systems [9,10]. Also there is an elegant examination of off-shell quantum electrodynamics [66] for spinless charged

123

Found Phys

particles. Most of theoretical nuclear physics calculations are based on the Schrödinger equation with effective nuclear and electromagnetic potential functions which do not allow any off-mass-shell behavior, even though the relativistic field theory perturbation does. Up until LENR effects were discovered, there was no need for such behavior in nuclear physics. It is this author’s opinion that there is such a need now since conventional approaches have failed by more than 50 orders of magnitude [32] to explain what is being seen in LENR experiments, and it seems that off-mass-shell behaviour of one type or another can explain all of the experimental anomalies. In the modern theory of multi-dimensional tunneling for many-body systems there are many unexpected phenomenon. In Coleman’s classic study [67], a quasi-stable vacuum state’s tunneling decay 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 that tunneling to the true ground state through nuclear reactions happens at such a slow rate that it can 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 these possibilities, there is certainly no rigorous theorem that says the experimental claims of LENR cannot occur. There may exist effective off-mass-shell Lagrangians for such systems which facilitate calculations of multi-dimensional tunneling, and which give the correct S-matrix when all particles are separated spatially in the initial and final state. So in this paper we propose that perhaps in a condensed matter setting, the electromagnetic interaction of charged particles can be described by an effective Lagrangian of the off-shell FS type, or some other variable mass type of theory. We acknowledge that the effective Lagrangian is not unique because of the equivalence theorem [53,54]. It is the author’s hope that such an effective Lagrangian could be derived from or at least reconciled with the standard model of elementary particles or perhaps string theory. This is not an easy task, but if direct experimental proof that off-shell behavior of the type proposed here is discovered, then these experiments should provide a critical guide on how to proceed.

5 Horwitz–Piron Theory We start with a brief review of the first of the modern variations of the Fock– Stueckelberg off-shell theory by Horwitz and Piron [8,9] who presented a solution to the two particle problem of relativistic classical and quantum physics by proposing that a single historical time could simultaneously parametrize two space-time paths for different particles. This is easily extended to any number of particles. They postulated a Hamiltonian K , conjugate to the historical time τ . They chose K as follows (written in the most general form)

K =

 n  

 

1 Vi j xi − x j

( pi − e A(xi ))μ ( pi − e A(xi ))μ + 2Mi i=1

(6)

i< j

123

Found Phys

where Aμ is an external electromagnetic four vector potential, and V is an inter-particle potential energy. The “dot” notation denotes τ derivatives . Hamilton’s equations of motion become ∂K ∂K μ μ x˙i = = ( pi − e A(xi )) /Mi , p˙ i = − (7) ∂ piμ ∂ xiμ This leads to the Lorentz force equation



 ∂ Vi j ( xi − x j ) μ μ μ μν Mi x¨i = eF x˙iν − , wher e Mi x˙i = pi − e Aμ ∂ xiμ

(8)

j=i

The mass of a particle is given by μ

m i2 = −Mi 2 x˙iμ x˙i = − ( pi − e A(xi ))μ ( pi − e A(xi ))μ

(9)

The particle’s mass m i is not necessarily equal to Mi , and is not even necessarily a constant of the motion if V is not zero. It is straightforward to quantize this system because it is mathematically very similar to the non-relativistic Newtonian particle system, where the usual time variable for each particle is a function of the historical time just as the three spatial coordinates for each particle are. In a nutshell, this 5D theory let’s you have both a relativistic time variable t along with a Newtonian absolute time variable τ , with both managing to coexist, or at least that is the assumption. The quantum wave equation is (ˆ denotes an operator) i

∂ ∂ μ ψ(x, τ ) = Kˆ ψ(x, τ ), pˆ i = −i ∂τ ∂ xiμ

(10)

6 Pre-Maxwell 5D Theory of Electromagnetic Interaction of Massive Charged Particles Off-shell electrodynamics was generalized to a five dimensional gauge invariant theory by Saad et al. [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 gauge transformations of the form ψ(x, τ ) −→ eie0 (x,τ ) ψ(x, τ )

(11)

 ∂ 1 i + e0 a5 (x, τ ) ψ(x, τ ) = ( p − e0 a(x, τ ))μ ( p − e0 a(x, τ ))μ ψ(x, τ ) ∂τ 2M (12) This leads to a non-trivial generalization of electromagnetism. The parameter e0 has units of length and is proportional to the electric charge e e=

123

e0 λ

(13)

Found Phys

where the constant parameter λ is new. The fields transform as a μ (x, τ ) → a μ (x, τ ) + ∂ μ (x, τ )

a5 (x, τ ) → a5 (x, τ ) + ∂τ (x, τ ) (14)

Non-vanishing values of a5 leads to mass variation. The Schrödinger Eq. (12) leads to a five dimensional conserved current ∂μ j μ + ∂τ j 5 = 0

j 5 = ρ = |ψ(x, τ )|2

jμ =

(15)

    −i ∗  μ ψ ∂ − ie0 a μ ψ − ψ ∂ μ + ie0 a μ ψ ∗ 2M (16)

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

+∞

J (x) =

μ

j (x, τ )dτ

+∞ A (x) = a μ (x, τ )dτ μ

−∞

(17)

−∞

where Aμ and J μ are the usual Maxwell potential fields and 4-current respectively. Saad et al [72] suggested an action which had higher 5-dimensional spacetime symmetry so that the Lorentz group O(3,1) is a subgroup. This requires either O(4,1) or O(3,2) symmetry. Both are considered in the literature, and to handle this, a 5D metric tensor is introduced g αβ = diag(−1, 1, 1, 1, σ ), where σ = +1 for O(4,1) and σ = −1 for O(3,2). The covariant action formula is (μ and ν range over the four space-time indexes, whereas α and β range over the five indexes including τ )  S=

  ∂ 1 ∗ + e0 a5 (x, τ ) ψ − ψ ( p − e0 a(x, τ ))μ d 4 xdτ ψ ∗ i ∂τ 2M  λ (18) ( p − e0 a(x, τ ))μ ψ(x, τ ) − f αβ f αβ 4

where f αβ = ∂α aβ − ∂β aα

(19)

This theory has been studied extensively by Horwitz in particular along with a number of co-authors [12–14,16,66,72–76]. Path integral quantization has been analyzed [15] as has more canonical second quantization approaches to off-shell quantum electrodynamics for spinless charged particles [OSQED][66]. A theory of relativistic wave functions generalizable to arbitrary spin and based on Wigner’s induced representations of the homogeneous Lorentz group has been described by Horwitz in [77] and references therein. This theory produces relativistic wave fuctions which satisfy all

123

Found Phys

the desired requirements by introducing an arbitrary timelike vector n μ for massive particles, and which is then used to define the induced representation and Wigner’s little group for the particle. Various FS spin 1/2 wave equations have been proposed [9,10,15,51,52,77–81]. They have not yet been included in the OSQED. The deuteron has spin 1 and positive parity J π = 1+ . Although it is composite particle of six quarks, it is more commonly treated as a composite of a proton and a neutron with an effective Lagrangian. For low energy interactions, it can be treated as an elementary particle. The conventional spin 1 wave equation is the Proca equation [82,83], although it is not the only possibility [84]   ∂μ ∂ μ − M 2 V α = 0

(20)

where V is a massive vector field which transforms under Lorentz transformations μν as V (x)μ → V  (x  )μ = μν Vν (x  ). The appropriate Lagrangian is 1 1 Lp = − Wμυ W μυ − M 2 Vμ V μ , Wμν = ∂μ Vν − ∂ν Vμ = ∂[μ Vν] 4 2

(21)

which leads the Euler–Lagrange equations ∂μ W μν = M 2 V ν

(22)

from which it follows that ∂μ V μ = 0 which leads to (20). The deuteron has a magnetic moment and an electric quadrupole moment [84], and these can be included in the Proca equation as well when there are electromagnetic fields present. We wish to generalize the Proca equation by adding a historical time τ . We can use the induced representation method [77] to define a suitable wave function. A path integral formalism for any spin has also been proposed in [15]. Since each component of V μ satisfies the Klein–Gordon equation, it is natural to write the Stueckelberg version of the Proca equation as i

 1  ∂ α U (x, τ ) = − ∂μ ∂ μ U α = 0 ∂τ 2M

(23)

We take the Lagrangian density to be L = U α∗ i

∂ 1 ∗ μ 1 Uα − U p pμ Uα − Z μυ Z μυ , pμ = −i∂μ ∂τ 2M 4

(24)

where here Z μν = ∂μ Uν − ∂ν Uμ . We choose the arbitrary timelike vector [77] to can be taken in this be n μ = [1, 0, 0, 0] in some frame, and then the wave function 

− →

2 4 − → frame to be simply the 3-vector U with the normalization U d x=1. The wave function in other Lorentz frames is then determined by the transformation formulae of the induced representation theory [77] for a spin 1 particle. The electromagnetic minimal interaction can be added by minimal coupling, although the deuteron does

123

Found Phys

have an anomalous magnetic moment as well as an electric quadrapole moment which can influence the dynamics in strong fields [84,85]. Land and Horwitz [66] have applied perturbative quantum field theory methods to off-shell pre-Maxwell electromagnetism in interaction with spinless charged particles. They have developed Feynman rules and applied them to various scattering processes. This is a 5D theory, and there are two possibilities for the 5D symmetry group that contains the Lorentz group as a subset. The Lorentz group O(3,1) is a subgroup of either O(4,1) or O(3,2), and these two possibilities lead to two different theories with different physical properties. The results for particle-particle scattering show that in general the masses will change for particles in a scattering process. They present detailed cross-section calculations for both Møller and Compton scattering, and both calculations show mass changes after scattering. This is an extremely interesting paper as it provides ample off-shell behavior. The mass of a particle in this theory is a function of the past history of the particle. 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 [16,86,87]. The runaway solutions are replaced by chaotic nonlinear equations which include variation of the classical particle’s mass with time. One major problem with the off-shell theories is how or why particles tend to get back on the mass shell if they have moved off of it through various interactions. Some unknown restorative mechanism must be at work, as has been acknowledged by advocates in this field. Some clues have been found. In [16] a self-interaction theory for a charged particle was developed, and it was found that for many (but not all) initial conditions, the mass increases for a while and then decreases back to the on-shell value. In [88] it was found that the mass distribution of certain distributions of particles and anti particles can become sharply peaked in mass. A modification for the photon Lagrangian in [89] was proposed, but similar modifications might be required for massive particles as well. It was pointed out that the classical behavior of charged particle scattering in pre-Maxwell theory is unphysical without this modification, although the quantum versions of the theory did not seem to have these problems [89]. On a more fundamental level, the kinetic term p u pμ /2M is not positive definite. 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 avoid problems. For example, in classical statistical mechanics of many body systems, various authors [18,90] imposed the constraint on the system that in the non-relativistic limit all masses approach the mass shell. This assumption leads to a narrow distribution of masses about their usual rest masses with a spread in values of width on the order of k B T . This assumption runs counter to the behavior that we are proposing here, which is a significant deviation from rest masses in a non-relativistic condensed matter setting. This constraint appears to have been imposed on the theory because there was no evidence for off-mass-shell behavior in condensed matter. But LENR might be evidence. Therefore, we assume here that this constraint can be relaxed in these statistical theories. Perhaps these problems could be solved by making the replacement in the Hamiltonian K

123

Found Phys

 2 p μ pμ p μ pμ ⇒ + η p μ pμ + M 2 2M 2M

(25)

where η is a constant which might be different for different particles, and it might depend on the condensed matter environment. This is an obvious generalization of [89] to massive particles. A large value for η would weigh against off-shell deviation, and a small value for η would allow larger mass deviations. This replacement renders the Hamiltonian bounded from below for a suitable sign of η, and may not affect the on-shell behavior very much. It adds a cost to going off shell and would act to inhibit the particle from moving off the mass shell during a collision for example. The idea then would be that in normal matter η is large, but in a nuclear active environment, where LENR occur, η would be small. There are no rigorous bounds on how far off-shell the massive particles can wander in the various FS theories for realistic condensed matter systems, especially in nonequilibrium 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 a neutron and a neutrino e + p → n + νe . They used the following formula for the mass shift induced by electromagnetic fields βm e

 1/2 2  e M e μ ≡ = 1+ A Aμ Me Me c 2

(26)

e is the shifted mass value for the electron and Me the electron rest mass where M in isolation. In order to have enough energy to produce a neutron, it is required that βm e > 2.531. The source for (26) is given as [91] (Sect. 40, Eq. 40.15), which contains a rigorous solution to the Dirac equation in a monochromatic plane electromagnetic wave. The plane-wave in [91] is written in the Lorentz gauge ∂μ Aμ =0, but since there are no sources for it, it is possible by appropriate choice of gauge to arrange that in addition A0 = 0, which is assumed in [91], and consequently it follows that ∇ · A = 0, and so that the field is purely transverse. An estimated upper possible value of βm e = 20.6 on a Palladium-deuterium alloy surface is presented in [44], but this value has been questioned [92]. We can understand (26) by considering the change in the classical electromagnetic momentum vector that happens when a charged particle is moved into an electromagnetic field in quasistatic approximation. The contribution from the interaction with the field is simply given by (up to a constant that depends on units) [45–47] μ

μ

p f ield = q A T (x(τ ))

(27)

This formula is only correct in the transverse gauge ∇ · AT = 0. This fixes the gauge uniquely provided AT vanishes at x = ∞, otherwise we could add a term ∇ϕ if

ϕ = 0. This can be thought of as the justification for the minimal coupling formula

123

Found Phys μ

which follows by adding p f ield to the intrinsic 4-momentum of the particle expressed in terms of it’s proper velocity μ

p μ = Mv μ + q A T

(28)

The Lagrangian for a charged particle in an electromagnetic field is gauge invariant, and so if all one wants to do is solve for the equation of motion, one can drop the transverse gauge condition for practical purposes, and study a system with gauge symmetry, and this is routinely done by physicists. Taking a time average of the instantaneous “mass” p μ pμ and assuming the cross terms can be dropped results in (26). There is a problem with the Widom–Larsen argument. Notice that the time averaged correction to 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 and the electron have opposite charge, and this causes their 4momentum shifts to be anti-correlated. Consider the total instantaneous 4-momentum of their initial state μ

μ

μ

PT ot = peμ + p μp = (Me veμ − e A T (xe (τ )) + (M p v μp + e A T (x p (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 field is changing very little over the short range and time of the weak interaction, the use of the instantaneous 4 momentum values seems more appropriate than the time-averages. So we see that when the positions of the electron and proton are close enough to react weakly, the two electromagnetic mass terms cancel yielding simply μ

PT ot = Me veμ + M p v μp

(30)

The center of momentum energy is then exactly the same as if there were no electromagnetic field present. This is just an example of what was mentioned earlier in Sect. 3, that an external electromagnetic field cannot change nuclear kinematics. Although this argument is classical, it doesn’t seem that appealing to quantization could change this conclusion. This sheds some doubt in this author’s mind regarding the validity of the Widom–Larsen argument. However, in the FS type off-shell theories, and especially in Horwitz–Piron or preMaxwell 5D theories, the local mass of structureless point particles can change, and the mass shift can be positive or negative, even in the absence of any local field. Moreover, there is no bound known on how far the mass can shift off shell. So, these theories might justify the Widom–Larsen theory even given the above result, and moreover the objections raised in [92] may not apply in this case. So in our view the Widom–Larsen processes are possibly part of the picture, but not necessarily the whole story. They may compete with other processes for dominance in different reactions. All the effects we consider are due to mass shifts, but the shifts can be positive or negative and can apply to any particle. Aside from providing an alternative basis for the Widom–Larsen effect, we offer no opinions pro or con about the rest of their theory. We do consider other reactions that are enabled by mass-shift effects however.

123

Found Phys

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 deuterons and possibly the nearby electrons too are moving slowly off the mass shell due to interaction with the condensed matter system according to an off-shell FS type theory, as in (1). We further assume that the final state masses in the fusion process are the usual rest masses of the particles, that special conditions inside the solid are required for this process to occur, and that these are roughly equivalent to the conditions required for anomalous LENR effects to occur. Finally, we assume that after a period of time, the system returns to normal and all mass values return to their standard values, except for those that have experienced a nuclear reaction. We propose that an active d + d pair reduces its mass slowly until it is approximately equal to the mass of the 4 He (alpha particle), about a 0.63 % reduction, or 11.9 MeV per deuteron. This has never been observed in nature. It is a radical assumption which could be possible if an off-mass-shell effective Lagrangian were describing a small volume of the lattic. Koonin and Nauenberg [32] modeled the electron screening effect by assuming that the d + d system acts similar to a d2 molecule. They showed that the fusion rate was far too low (by over 50 orders of magnitude) in this case to explain the fusion claims of Fleischmann and Pons. They also calculated what the electron mass would have to be in order for the tunneling rate in d2 to explain the Fleischmann—Pons results for excess heat. They found a mass of 10m e was required. There is beam-scattering evidence of enhanced screening in deuterated palladium [36–40,93–97]. This enhancement has thus far been attributed to the higher density of electrons surrounding the deuterons without resorting to heavy electrons. But, it may also be that heavy electrons are playing a role. Resonant tunneling would occur if the sum of the two deuterons equaled the mass of Helium-4, regardless of the type of screening. In this case, any photon produced would have a low energy. The increase in the electron mass enhances tunneling, and the decrease in the deuterium mass allows resonant tunneling directly into an alpha particle and the suppression of 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 of 4 He which occurs at 20.210 MeV above the mass of 4 He [98]. This excitation decays almost exclusively into t+p with a half-life of 1.3 × 10−21 s. Only the s-wave component of the resonance will contribute to the fusion rate, but the proportionality factor is unknown. Assuming a significant s-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 have exactly the peak energy. → t (0.099 MeV) + p(0.2968 MeV) Q = 0.396 MeV,  = 0.5MeV

4 H e(20.21MeV)

(31)

Because the Breit–Wigner width is .5 MeV, the proton and tritium energies would actually be smeared out. This helps explain the observed enhanced tritium production relative to neutron production in experiments with deuterated palladium. The stopping

123

Found Phys

(a)

(b) (c) (d) (e) (f)

(g)

^

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

range of 0.3 MeV protons in water is 4.27 μ [99], and thus they would not be observed directly in electrolysis experiments. We see in Fig. 1 that the main fusion is occurring at the instant that m 2d = m α , but this is not when most of the heat is added to the solid, because the Q value for fusion is essentially zero then as 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 continuously varying masses, and transients continue until all masses eventually return to their on-shell values . Energy is conserved by the following formula d(n d m d ) d(n e m e ) d(n pd m pd )  d(n i m i ) d K d E e&m d E ext = + + + + + dt dt dt dt dt dt dt i

(32) where K is the kinetic energy density of the system, and n d , n e , n pd , n i are the number densities of deuterons, electrons, palladium nuclei, and any other particles. These number densities are not conserved because of nuclear reactions. The mass terms m d , etc. are local mean values for the masses in the solid at a given location, and they are presumed to be varying with time. The term d Edtext is the net energy density

123

Found Phys

rate of change due to conventional transport mechanisms such as radiation, thermal conduction, etc, and E e&m denotes the electromagnetic energy density. In the nonrelativistic limit, E e&m can be replaced by a pairwise Coulomb interaction plus the energy density of radiation. Thus we have energy conservation at every step of the way. We see from Fig. 1 that neutrons can be produced only at the very beginning of the time interval shown so long as Q(3 H e + n) > 0. Tritium can be produced so long as Q(3 H + p) > 0. We don’t expect all of the particles of a given type at a given location to necessarily have the same mass. So we define a probability density ρd , ρe , ρ pd , etc. with functional form ρ(x, t, m), where ρdm is the probability that the mass m lies between m and m + dm. We restrict the sign of the particle mass to be positive, and so we write the normalization condition as ∞ ρ(x, t, m)dm = 1 (33) ∞

0

and m(x, t) = 0 mρ(x, t, m)dm. The masses in Fig. 1 are these mean values for masses. At most locations, the critical condition m 2d (x, t) ≈ Mα will not ever be achieved, and therefore significant fusion will never occur there. In order for many fusion reactions to occur, the critical condition must be satisfied at a number of locations. Fusion depends on two tuning parameters—the mean deuteron mass, and the mean electron mass. Correlations in mass may well occur between particles of the same or different species. Therefore, in general ρ(x, t, m 1 , m 2 ) = ρ(x, t, m 1 )ρ(x, t, m 2 ). As a simplifying approximation to this situation, let us assume that at a given location the mass values are sharply peaked about a mean value that depends on time. It is difficult to say anything precise about the solid, so let us rather study the d2 molecule, as was done in [32]. This four-body system allows us to estimate the effect of resonant tunneling. For simplicity let the deuteron masses both be the same, and similarly for the two electrons in the d2 molecule. We treat the masses as slowly varying in time so that a quasistatic approximation can be used. The non-relativistic Hamiltonian for d2 , is ˆ = H

p2 p2d1 p2 p2 + d2 + e1 + e2 + V (xd1 , xd2 , xe1 , xe2 ) 2m d 2m d 2m e 2m e

(34)

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

 

V (xd1 , xd2 , xe1 , xe2 ) = Vdd x d1 − xd2

  



2 + i, j=1 Vde xdi − xej + Vee xe1 − xe2 + Vmasses

(35)

Vmasses = 2m d + 2m e

(36)

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

123

Found Phys

Vee (r ) = −Vde (r ) =

e2 r

(37)

but we tak Vdd to be a modified Coulomb potential which includes nuclear forces of confinement for r less than an effective nuclear force range an .  Vdd (r ) =

e2 /r, r > an ≈ 10 fm Udd (r ), r ≤ an

(38)

where Udd includes the short-distance attractive nuclear force between the deuterons, as well as electromagnetic forces. Even if Udd were known accurately, solving these equations exactly is complicated and requires numerical techniques. We ignore the time variation of the masses in solving for the wave functions. The standard treatment for the hydrogen molecule uses the adiabatic approximation and the clamped nuclei computation [100]. The calculation begins by solving for the ground state of the clamped nuclei Hamiltonian H0 : ˆ0 = H

2 

 

p2e1 p2 + e2 + Vdd (|xd1 − xd2 |) + Vde xdi − xej

2m e 2m e i, j=1

+Vee (|xe1 − xe2 |)

(39)

The nuclei are slowly moving compared to the electrons, and their positions are held fixed (clamped) in this first 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 first with R held fixed ˆ 0 ψ j (x, R) = E 0j (R)ψ j (x, R) H

(40)

 Expanding the full wave function in terms of these by ψ(x, R) = χ j (R)ψ j (x, R) ˆ we must solve the full Schrödinger equation Hψ(x, R) = Eψ(x, R). This can be written in the form of an equation for the nuclei in an effective potential after making an adiabatic approximation [100] 

1

R + U j (R) − E χ j (R) = 0 2μ

(41)

where μ is the reduced mass of the two off-shell deuterons, and where the effective potential is given by U j (R) = E 0j (R) + C j (R)

(42)

The first term in this equation is the Born–Oppenheimer approximation. Several corrections making up Cn are included in [100] to improve accuracy. Koonin and Nauenberg [32] used these results in their estimate of fusion rates. Here we shall only

123

Found Phys

consider the ground state, and the effective potential Ve f f (R) = U0 (R) which will depend on the electron masses. The barrier penetration factor obtained from the WKB approximation is ⎡

⎤ a N ⎢ ⎥ B = ex p ⎣−2 (2μ(Ve f f (r ) − E))1/2 dr ⎦

(43)

rt p

This factor is real-valued and very sensitive to the electron mass through Ve f f . It also depends on the deuteron mass, but for the 0.6 % change we are contemplating here, it can be treated as independent of deuteron mass to a first approximation. We follow the basic approach of a two-level quantum system as given by Hagelstein when near to a resonance [69]. There are two weakly coupled eigenstates 1. ψa is the ground-state for the hydrogen-like molecule constructed from the two slightly off-shell deuterons and off-shell electrons. Ve f f is changed by the masses being off-shell. ψa is a solution to the modified off-mass-shell 2-body problem where the parameter an is taken to zero and the nuclear force is not included. 2. ψn is a nuclear bound state consisting of two off-shell deuterons bound together by the nuclear force and surrounded by a two-(off-shell) electron cloud around them in a ground state solution to a 4 He like atom. There are two resonant states to be considered at the masses of 4 He and it’s first excited resonance which is at 20.210 MeV. In the interest of simplicity, we shall now make a crude approximation and develop the two-level theory using only the stable state for 4 He. We also consider a third state ψ4 He which is an on-shell 4 He 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 that molecule. Near a resonance, the states ψa and ψn can tunnel back and forth reversibly through the screened Coulomb barrier that separates them. We start with the following formula from [32] to relate the nuclear fusion rate to the deuteron wave functions when electrons and deuterons are on-shell dd = Add |ψa (an )|2

(44)

where Add = 1.5 × 10−16 cm3 s−1 is the rate constant for dd fusion, and ψa (an ) is a normalized 3D wave function with units (1/L)3/2 . We only know Add for on-massshell deuterons. This same rate formula was also used for describing muon catalyzed fusion—a similar problem—in a seminal paper by Jackson [50]. It has problems in the present context. It should be applicable for any wave function ψ, but consider a wave function which vanishes at r = an , but is non-zero for r < an . The formula then wrongly predicts dd = 0. But the deuterons have a non-zero probability of being even closer together than an in this case, and consequently they should fuse with some non-zero probability. For a molecular eigenfunction which varies slowly with position for two deuterons, (44) works fine. The following generalization of (44) fixes this

123

Found Phys

problem and agrees with (44) for slowly varying s-wave radial wave functions  dd = Add

r
|ψ(r)|2 d 3r 4πan3

(45)

The radial s-wave function R(r ) is defined by ψ(r ) =

∞

R(r ) (4π )1/2 r

|R(r )|2 dr = 1

,

(46)

0

and it is easy to show that (45) is√consistent with (44), and that for slowly varying radial s-wave functions R(0) ≈ an dd /Add . The following off-mass-shell formula, based on accurate hydrogen molecular orbitals, is presented in [32] where m e is the off-shell electron mass and m e its usual value. Add dd (m e ) = 3 a0



μ MN

3 106.5−79

√ m e /m e

(47)

where a0 is the Bohr radius, μ is the reduced mass of the two deuterons, and M N is the nucleon mass. We assume that the rate constant Add does not depend on electron mass, but it could and probably does depend on the deuteron mass. We see then that the rate dd depends dramatically on 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 verified phenomenon from muon-catalyzed fusion [50]. Now we proceed with the resonant theory, following Hagelstein’s two level approach [69]. The two quantum states |ψa  and |ψn  denote two deuterons in a molecular bound state, and bound in a nuclear 4 He stable ground state respectively. ˆ 0 . We presume that the nuclear bound state has They are orthonormal eigenstates of H the same mass as 4 He, indepdendent of the mass of the deuterons. The state vector and Hamiltonian operator are |ψ(t) = ca (t) |ψa  + cn (t) |ψn 

(48)

  ˆ H(t) = Ea (t) |ψa  ψa | + En (t) |ψn  ψn | + β(t) |ψa  ψn | + β(t)∗ |ψn  ψa | (49) ˆ 0 = Ea |ψa  ψa | + En |ψn  ψn | H

(50)

Without loss of generality, we can take β(m e , m d ) to be real, since any phase of β can be absorbed into the relative phase difference between ψa and ψn . In the WKB approximation, β is proportional to the barrier penetration B (43), and thus will not depend on the deuteron mass very much. The Hamiltonian is Hermitian, and the

123

Found Phys

dynamics will be reversible, as a first approximation. Because masses are presumed changing, the parameters Ea and β will be slowly varying functions of time as in (1). We assume that at the initial time the coupling β is extremely small, and that the system is in the ground state of the d2 molecule with electrons and deuterons on the mass shell. Although tempting, we do not use the adiabatic approximation here because β could be so small that the relaxation tunneling time would be larger than the run-time of an experiment. The coupling β can grow greatly over time because the electron mass is assumed to increase. The fusion rate at any instant follows from (45), and is given by dd = Add

|cn (t)|2 4πan3

(51)

The Hamiltonian is presumed to be slowly varying with time due to the variation of the ˆ 0 along electron and deuteron masses. We include the rest masses of the deuterons in H ˆ with the binding energy, but we ignore time derivatives of H0 . We don’t need to add the electron masses because they would be the same in the two states |ψa  and |ψn . Assuming both deuterons have the same mass, we have Ea = 2m d (t) + mbea (t) and En = m 4 H e = m α , where mbe denotes molecular binding energy. The time evolution equation is then simply ˆ |ψ(t) = i ∂ |ψ(t) = i c˙a (t) |ψa  + i c˙n (t) |ψn  H(t) ∂t

(52)

We need a functional form for β in order to proceed. We can deduce it from [32] ˆ at t = 0 which we take as the start time for by calculating the true eigenstates of H deuterium loading. This amounts to finding the eigenvalues and eigenvectors of a 2×2 matrix    Ea (0) β(0) φa φa =E (53) β(0) En (0) φn φn with solutions E± = 

φa φn

! 1 Ea + En ± (Ea − En )2 + 4β 2 2 2

±

=!

1 1 + χ±2



1 , χ± = (E± − Ea )/β χ±

(54)

(55)

For on-mass-shell electrons, ε = β/ |Ea − En |  1 and we can approximate and find     φa β/(En − Ea ) φa 1 + O( 2 ) + O( 2 ), = = φn + φn − 1 −β/(En − Ea ) (56)

123

Found Phys

Calculating the fusion rate for the (−) eigenvector with φa ≈ 1, we find at t = 0 when the deuterons are on-shell dd = Add

|β/(En − Ea )|2 |β/(m α − 2m d )|2 = A dd 4πan3 4πan3

(57)

Solving for β, we find " β(m e ) =

dd (m e ) 4πan3 (m α − 2m d )2 , Add

(58)

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

d cn (t) = En cn (t) + βca (t) dt

(59)

i

d ca (t) = Ea ca (t) + βcn (t) dt

(60)

The solutions are [69,101], ignoring any time dependence in β and in Ea for a first approximation, and with initial conditions cn (0) = 0 |cn (t)| =

4β 2

2

(En − Ea )2 + 4β 2

=

 sin

2

!  (En − Ea )2 + 4β 2

t 2

(61)

(62)

For small t we find |cn (t)| ≈ βt, so that β is just the barrier transmission coefficient. At exact resonance, En = Ea , which can happen if the deuterons are off-shell and their sum equals the mass of 4 He, we obtain r es = 2β(m e )

(63)

The two energy eigenvalues (54) are not degenerate at resonance, the degeneracy having been split by the coupling β. The time to first maxima for |cn (t)|2 is given by Tmax = π/ . This is a critical parameter because we don’t want to have to wait around for a long time before a resonance has a chance to build up. A value of 1 day for Tmax requires an electron mass about 2.4 times the standard electron mass. At

123

Found Phys

resonance |cn (Tmax )| = 1.0, representing 100 % barrier penetration! Using (51) we find the following fusion rate at resonance dd =

Add |cn (t)|2 ≈ 1.19 × 1019 s −1 at resonance and at peak time 4πan3

(64)

In arriving at the numerical estimate, we have made a questionable assumption that Add does not depend on m d , 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 fusion rate originally claimed by Fleischmann and Pons requires an electron mass ten times heavier than normal for a non-resonant theory. This translates into a fusion rate of  F P = 10−9.1 s −1 from Table 2 in [32]. Thus, at resonance peak, we have a fusion rate here which is 28 orders of magnitude larger than claimed by Fleischmann and Pons, but which persists for only a very short time. To achieve this resonance value, the energy difference term must be zero to extremely high precision so that |En − Ea | < 2β(m e ). This precision would be unfeasible if it had to be experimentally controlled as has been already pointed out by Hagelstein [69]. We can relax this precision and still achieve a rate of  F P by the following dd ≈ 2

4β 2 Add = F P 3 4πan (En − Ea )2 + 4β 2

(65)

So the factor 4β 2 can be as small as 10−28 , or |En − Ea | ≤ 2β(m e ) × 1014 . (En −Ea ) But even this would require extreme precision. What saves this theory is that we have assumed that the deuteron mass is slowly and continuously varying with time as is illustrated in Fig. 1. So provided the asymptotic value of this mass is low enough, the combined masses of the two deuterons will at some time or another have to pass through the resonant value, and then the particle will fuse at that point with a probability of |cn (t)|2 . If this coincides with the broad maxima in time of (61) then the pair will fuse with near 100 % probability owing to the huge rate at resonance (64). The fact that resonance is required for significant fusion serves as a safeguard which prevents harmful radiation from being produced because at resonance there is no energy (or Q) left over to produce it. This is an enormous benefit to this form of nuclear energy if it can be verified. The energy given to the lattice by the fusion event is actually given up prior to the event as reflected in the reduced masses of the two deuterons which subsequently fuse into 4 He with zero Q. The d2 molecule has been treated as a closed system, but the slow mass variation would require continuous soft electromagnetic interaction with the lattice. As the deuteron mass decreases, the n +3 He phase space reaches zero before the t +3 H channel does. Moreover, as the electron mass is supposedly increasing during this time, the t +3 H channel benefits from a relatively lower Coulomb barrier, as well as the resonance 20.21 MeV (31). This explains why significantly more tritium is produced than neutrons. After all reactions have ceased, we expect all particles to return to their normal rest masses, and so the energy stored in the electrons’ higher masses would be returned to the solid then too. This relaxation process may take some time, and might result in apparent heat production after all driving factors such as

123

Found Phys

(a)

(b)

(c)

Fig. 2 Fusion rates versus electron mass. Deuterons are on-shell. a shows the non-resonant fusion rate of Koonin and Nauenberg; b shows the estimated β parameter as a function of electron mass; c shows the amount of time in days until the first maxima of the cyclic, two-level model resonant tunneling if the electron’s mass is held fixed

electrolysis have ceased. This could explain the so-called life-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 of two-level systems with dynamic coupling [101]. From a practical standpoint, Tmax must not be too long. This requires that the Coulomb barrier penetration factor β not be too small. Figure 2 shows a plot of dd , β, and Tmax . This sets a limit on how small the electron mass can be and still achieve the observed fusion rates. But the effect of screening in the actual palladium lattice should be included too, and this goes beyond our simple d2 model. It’s important to note that dd is the conventional fusion rate which does not include the effects of resonance as calculated in [32]. For understanding the effects of resonance, it is more imprortant to look at the Tmax parameter. Any deuteron pair which goes through resonance near to this time will fuse with nearly 100 % probability because of the extremely high fusion rate at resonance. Thus the energy release depends on how many deuteron pairs are passing through resonance near to Tmax at some time or other. If we randomize 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 excess energy effect. This could have a number of effects on the dynamics of this system as currently described, but the details remain to be explored. One speculation is that the deuteron

123

Found Phys

mass might acquire a small ripple in time, and this could cause the resonance value to be crossed more than once, which in some circumstances may enhance the fusion rate. Or maybe the β term can become time dependent due to this effect. 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, and because a theory for the off-mass-shell dependence of this parameter is not known. We can guess at some expected behavior of Add as the deuteron mass decreases and approaches resonance. First of all, as the phase space for d+d → n+3 He and d + d → t + p both go to zero, Add is expected to decrease by as much as 5 orders of magnitude. This is still a small effect on a logarithmic scale compared to the quantum tunneling factor. But, Add may be increased somewhat by the unstable resonance at 20.210 MeV. As the stable 4 He resonance is approached, we can expect a Breit–Wigner resonance form for the S-matrix of d + d → γ +4 He, and this resonance should more than offset the reduction in the photon phase space as resonance is approached. As a consequence, there should be an enhancement in the number of γ produced as the deuteron mass approaches resonance, and the γ energy approaches zero. Very near to the 4 He resonance, multiple photon emission events will probably become important as well. We defer a more detailed model for the functional variation of Add to a future publication. 9 Transmutation in Deuterated Palladium With varying particle masses, transmutations can conceivably in theory occur in a number of ways in a palladium-deuterium lattice. These include electron capture, beta and alpha decay of nuclei whose mass has increased, resonant fusion of deuterium with other nuclei including palladium, resonant fusion of alpha particles and other nuclei including palladium, and even fission of palladium and/or other impurity elements after a mass increase. Also, there is the possibility of neutron creation and subsequent capture as in the Widom–Larsen theory, leading to many possible reactions with no Coulomb barrier to be overcome. In short a world of possibilities exist. The fusion possibilities mentioned in this list have a much higher Coulomb barrier than the deuterium molecule. Nevertheless, if resonant tunneling occurs, there might be small numbers of such events occuring, as has been reported in [102]. We shall defer a detailed examination of this subject to a future publication. 10 Micro-craters Many experiments have revealed micro-crater damage to the Palladium surface after LENR activity has been observed [103–106]. It has been commonly thought that these craters were evidence of micro explosions. We offer a different explanation. If all (or a substantial fraction) of the electron masses were to increase in a small local region, the lattice spacing would be reduced approximately inversely proportional to their average mass, and this would cause a severe mechanical deformation of the surface of the palladium which could leave a crater caused by shrinkage. If all the electrons were to increase in mass by say a factor of 3 in a small volume on the surface, the volume

123

Found Phys

would decrease by a factor of 33 or 27, which might well leave a crater on the surface. This would explain why transmutations are often observed to have occurred in or near these craters, as these are locations where the mass variation would be expected to have been the largest. Later, after all reactions were over and the electrons returned to their original mass, the volume would re-expand and possibly look like excess material on the rim of a crater.

11 Predictions and an Experimental Test of Off-Mass-Shell Variation Assuming that this theory is correct, and that masses of deuterons are decreasing slowly as the loading factor d/ pd increases, then we can make a very simple experimental prediction that is testable. We predict that the Q values for the reactions in (1), (2), and (3) will be observed to decrease with time, as the loading progresses, as in Fig. 1, in those systems that exhibit excess heat production. Perhaps observable effects will even be seen in systems where no excess heat is produced, as excess heat requires that the deuteron mass must decrease until it is resonant with the 4 He channel, but less mass change could still reduce the Q values for the reactions. As the phase space for all three channels will change as the Q values diminish, then the relative branching ratios for the 3 channels will change as well. The first channel to zero out would be the 3 H e + n channel. Then, once the mass of the deuterons decrease below the threshold for producing t + p, the only channel open would the 4 H e + γ . Such a reduction in Q values for a fundamental nuclear reaction has never been observed before. It would be a clear and undeniable proof that the deuteron rest mass was changing in these settings. We also expect that the energy of the gamma rays will be reduced from the expected value of 23.77 MeV continuously down to zero at the resonance point. These non-resonant fusion events would be governed by the conventional fusion rate formula dd , and thus would be generally much fewer in number than the resonant fusion events which produce only 4 He and heat. Perhaps the simplest apparatus to look for reduced Q values in d +d fusion would be the gas permeation methods of Iwamura [107,108]. Alternatively, beam experiments might be used with low-energy deuterons incident on a palladium metal surface. In either case, precision detection of energetic charged particles, neutrons, and γ leaving the surface would be desired. The charged particles would include Tritons, Helium-3 nuclei, protons, and α particles. The loading factor increases in both systems over time. The energies of the charged particles produced are functions of the deuteron rest mass. If the deuteron rest masses are changing, then this would show up as a broadening of the energy spectrum for a given type of charged particle. Beam experiments have been performed already by Huke, Czerski et al. [36–40,93,94]. Various charged particle anomalies have already been observed in those experiments and others [1]. Iwamura et al. [107] have measured radiation produced in experiments with deuterium diffusing through pd foil. They find a broad spectrum of X-rays along with neutron detection, but no correlation between neutrons and X-rays, or with excess heat. These X-rays could indicate fusion events between two off-shell deuterons, which are relatively close to resonance (64).

123

Found Phys

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 have been 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 significantly in a condensed matter setting. What is being proposed here is a radical departure from existing accepted nuclear and condensed matter theory, and deserves to be treated with a great deal of skepticism. Nevertheless, it is this author’s opinion that Fock–Stueckelberg or other type of off-mass-shell theories are a possible explanation for such variations and that all of the experiments in LENR can potentially be explained if they are occurring. The experimental claims that have been made about the occurrence of fusion, transmutation, fission, and lack of significant radiation in most experiments have been easily judged as absurd and preposterous by many physicists. Yet each year that passes sees more experimental papers claiming to validate these phenomena in the international scientific literature. Any theory that could describe this growing body of experimental evidence will likely seem equally absurd. The author freely admits that the theory he has proposed here is radical and hard to believe, but he sees no other way to explain these experiments, and thus feels compelled to persist. If experiments confirm that mass variation is occurring in deuterated palladium, then it is likely that all of the effects of LENR can be explained by this when applied to various types of charged particles. Controlling the mass variation will consequently become the key engineering challenge required for controlling low energy nuclear reactions. It is ironic that on the one hand in the case of LENR, the acceptance of the experimental results has been impeded for over 20 years by the lack of a theoretical framework in which the results could be contemplated as even conceivable, whereas in the case of the off-mass-shell covariant relativistic dynamics, the acceptance of this theory into the canon of physics has not happened for over 70 years because of the lack of any experimental evidence for it. If off-mass-shell behavior is confirmed, then it will undoubtedly have profound implications for the foundations of quantum mechanics, as well as for other branches of physics and chemistry. We have made clear and simple predictions that the Q values for the deuterium fusion reaction channels will reduce with time in deuterated palladium experiments that produce excess heat, and at resonance the Q values will all be zero so that no radiation is produced for the bulk of the fusion events which occur then. If this behavior is confirmed experimentally, then it will be proof that rest masses are changing in a condensed matter setting. Although we have concentrated on the deuterated palladium system here, it is clear that many reactions can occur in other systems such as nickel and hydrogen if masses vary there too. The same can be said for combinations of deuterium and metals other than palladium. The body of experimental evidence in this field has now grown rather large, and it is quite complex. It takes a good deal of time and effort to gain a good understanding

123

Found Phys

of it. In 1990, given the 50 orders of magnitude discrepancy between nuclear theory and the original Fleischmann—Pons experiments, the conclusion was reached that the experiments were wrong. Today this discrepancy remains, mitigated somewhat by enhanced electron screening arguments, but worsened by the abundant transmutation evidence now observed and requiring an explanation. The growing weight of 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 theory circa 1990 was incomplete, and that the experiments showing LENR anomalies were and are in large measure correct. The final chapter in this epic story has clearly not yet been written. Acknowledgments The author acknowledges valuable correspondence with Lawrence Horwitz, and valuable discussions with Peter Hagelstein, Michael McKubre, David Nagel, and Paul Marto. He also acknowledges Vladimir Kresin for extensive critical but good-spirited and helpful discussions. Any errors are entirely the author’s, and he makes no claim of endorsement of this theory by anyone.

References 1. Storms, E.: Science of Low Energy Nuclear Reaction: A Comprehensive Compilation of Evidence and Explanations about Cold Fusion. World Scientific Publishing Company (2007) 2. Srinivasan, M., Miley, G., Storms, E.: Low-energy nuclear reactions: transmutations. In: Krivit, S.B., Lehr, J.H., Kingery, T.B. (eds.) Nuclear Energy Encyclopedia, pp. 503–539. Wiley (2011). URL http://onlinelibrary.wiley.com/doi/10.1002/9781118043493.ch43/summary 3. Fock, V.: Die Eigenzeit in der klassischen und in der Quantenmechanik. Phys. Z. Sowjetunion 12, 404 (1937) 4. Stueckelberg, E.: La signification du temps propre en mécanique ondulatoire. Helv. Phys. Acta 14, 322 (1941) 5. Stueckelberg, E.: Remarque à propos de la création de paires de particules en théorie de la relativité. Helv. Phys. Acta 14, 588 (1941) 6. Lacki, J., Ruegg, H., Wanders, G.: E.C.G. Stueckelberg, An Unconventional Figure of Twentieth Century Physics: Selected Scientific Papers with Commentaries. Springer (2008) 7. Feynman, R.P.: Mathematical formulation of the quantum theory of electromagnetic interaction. Phys. Rev. 80(3), 440 (1950). doi:10.1103/PhysRev.80.440 8. Horwitz, L.P., Piron, C.: Relativistic dynamics. Helv. Phys. Acta 46(3), 316 (1973). URL http://www. osti.gov/energycitations/product.biblio.jsp?osti_id=4355335 9. Reuse, F.: On classical and quantum relativistic dynamics. Found. Phys. 9(11–12), 865 (1979). doi:10. 1007/BF00708697 10. Fanchi, J.R.: Parametrized Relativistic Quantum Theory. Springer GmbH (1993). 11. Horwitz, L.P., Lavie, Y.: Scattering theory in relativistic quantum mechanics. Phys. Rev. D 26(4), 819 (1982). doi:10.1103/PhysRevD.26.819 12. Horwitz, L., Shnerb, N.: Second quantization of the stueckelberg relativistic quantum theory and associated gauge fields. Found. Phys. 28(10), 1509 (1998). doi:10.1023/A:1018841000237. URL http://www.springerlink.com/content/rw725423p4l33516/abstract/ 13. Land, M.C., Horwitz, L.P.: The Lorentz force and energy-momentum for off-shell electromagnetism. Found. Phys. Lett. 4(1), 61 (1991). doi:10.1007/BF00666417 14. Land, M.C.: Pre-maxwell electrodynamics. Found. Phys. 28(9), 1479 (1998). doi:10.1023/A: 1018813429428 15. Seidewitz, E.: Spacetime path formalism for massive particles of any spin. Ann. Phys. 324(2), 309 (2009). doi:10.1016/j.aop.2008.10.007. URL http://www.sciencedirect.com/science/article/pii/ S0003491608001668 16. Aharonovich, I., Horwitz, L.P.: Radiation-reaction in classical off-shell electrodynamics. I. The above mass-shell case. J. Math. Phys. 53(3), 032902 (2012). doi:10.1063/1.3694276. URL http://jmp.aip. org/resource/1/jmapaq/v53/i3/p032902_s1?bypassSSO=1

123

Found Phys 17. Burakovsky, L., Horwitz, L.: Equilibrium relativistic mass distribution. Phys. A 201(4), 666 (1993). doi:10.1016/0378-4371(93)90135-Q. URL http://www.sciencedirect.com/science/article/ pii/037843719390135Q 18. Burakovsky, L., Horwitz, L.P.: Galilean limit of equilibrium relativistic mass distribution. J. Phys. A 27(8), 2623 (1994). doi:10.1088/0305-4470/27/8/003. URL http://iopscience.iop.org/0305-4470/ 27/8/003 19. Burakovsky, L., Horwitz, L.P.: Mass-Proper Time Uncertainty Relation in a Manifestly Covariant Relativistic Statistical Mechanics. ArXiv High Energy Physics-Theory e-prints, p. 4106 (1996). URL http://adsabs.harvard.edu/abs/1996hepth4106B 20. Greenberger, D.M.: Theory of particles with variable Mass. I. Formalism. J. Math. Phys. 11(8), 2329 (1970). doi:10.1063/1.1665400. http://jmp.aip.org/resource/1/jmapaq/v11/i8/p2329_s1? isAuthorized=no 21. Greenberger, D.M.: Theory of particles with variable mass. II. Some physical consequences. J. Math. Phys. 11(8), 2341 (1970). doi:10.1063/1.1665401. URL http://jmp.aip.org/resource/1/jmapaq/v11/ i8/p2341_s1?isAuthorized=no 22. Greenberger, D.M.: Some useful properties of a theory of variable mass particles. J. Math. Phys. 15(4), 395 (1974). doi:10.1063/1.1666658. URL http://link.aip.org/link/?JMP/15/395/1&Agg=doi 23. Greenberger, D.M.: Wavepackets for particles of indefinite mass. J. Math. Phys. 15(4), 406 (1974). doi:10.1063/1.1666659. URL http://link.aip.org/link/?JMP/15/406/1&Agg=doi 24. Corben, H.C.: Relativistic quantum theory of particles with variable mass I. Proc. Natl. Acad. Sci. USA 48(9), 1559 (1962) 25. Corben, H.C.: Relativistic quantum theory of particles with variable mass, II. Proc. Natl. Acad. Sci. USA. 48(10), 1746 (1962). URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC221034/. PMID: 16591007 PMCID: PMC221034 26. Fanchi, J.R.: Review of invariant time formulations of relativistic quantum theories. Found. Phys. 23(3), 487 (1993). doi:10.1007/BF01883726 27. Adler, S.L.: Quantum theory as an emergent phenomenon: the statistical mechanics of matrix models as the precursor of quantum field theory. Quantum Theory as an Emergent Phenomenon: The Statistical Mechanics of Matrix Models as the Precursor of Quantum Field Theory. Cambridge University Press (2004) 28. ’t Hooft, G.: The Free-Will Postulate in Quantum Mechanics. arxiv.org 0707.4568 (2007). doi:10. 1063/1.2823751. URL http://arxiv.org/abs/0707.4568. AIPConf.Proc. 957:154-163 29. ’t Hooft, G.: Determinism beneath quantum mechanics. Quo Vadis quantum mechanics? In: Elitzur, A.C., Dolev, S., Kolenda, N. (eds.) The Frontiers Collection, pp. 99–111. Springer, Berlin (2005). URL http://www.springerlink.com/content/t545r734304254q3/ 30. ’t Hooft, G.: Entangled quantum states in a local deterministic theory. arxiv.org 0908.3408 (2009). URL http://arxiv.org/abs/0908.3408 31. Weinberg, S.: Collapse of the State Vector. arXiv.org 1109.6462 (2011). URL http://arxiv.org/abs/ 1109.6462 32. Koonin, S.E., Nauenberg, M.: Calculated fusion rates in isotopic hydrogen molecules. Nature 339(6227), 690 (1989). doi:10.1038/339690a0. URL http://www.nature.com/nature/journal/v339/ n6227/abs/339690a0.html 33. Fleischmann, M., Pons, S., Hawkins, M.: Electrochemically induced nuclear fusion of deuterium. J. Electroanal. Chem. 261(2), 301–308 (1989). URL http://www.ftp.nic.funet.fi/pub/doc/Fusion/fp.ps 34. Leggett, A.J., Baym, G.: Exact upper bound on barrier penetration probabilities in many-body systems: Application to “cold fusion”. Phys. Rev. Lett. 63(2), 191 (1989). doi:10.1103/PhysRevLett.63.191 35. Leggett, A.J., Baym, G.: Can solid-state effects enhance the cold-fusion rate?. Nature 340(6228), 45 (1989). doi:10.1038/340045a0. URL http://www.nature.com/nature/journal/v340/n6228/abs/ 340045a0.html 36. Czerski, K., Huke, A., Biller, A., Heide, P., Hoeft, M., Ruprecht, G.: Enhancement of the electron screening effect for d + d fusion reactions in metallic environments. Europhys. Lett. (EPL) 54(4), 449 (2001). doi:10.1209/epl/i2001-00265-7. URL http://iopscience.iop.org/epl/i2001-00265-7 37. Czerski, K., Huke, A., Heide, P., Ruprecht, G.: The 2 H(d, p)3 H reaction in metallic media at very low energies. Europhys. Lett. 68, 363 (2004). doi:10.1209/epl/i2004-10209-3. URL http://adsabs. harvard.edu/abs/2004EL68.363C 38. Czerski, K., Huke, A., Heide, P., Ruprecht, G.: Experimental and theoretical screening energies for the 2 H(d, p)3 H reaction in metallic environments. In: The 2nd International Conference on Nuclear

123

Found Phys

39.

40.

41. 42.

43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

61. 62.

Physics in Astrophysics, ed. by Z. Fülöp, G. Gyürky, E. Somorjai (Springer, Berlin Heidelberg, 2006), pp. 83–88. URL http://www.springerlink.com/content/k740562323673840/abstract/ Czerski, K., Huke, A., Martin, L., Targosz, N., Blauth, D., Górska, A., Heide, P., Winter, H.: Measurements of enhanced electron screening in d+d reactions under UHV conditions. Journal of Physics G: Nucl. Part. Phys. 35(1), 014012 (2008). doi:10.1088/0954-3899/35/1/014012. URL http://iopscience. iop.org/0954-3899/35/1/014012 Czerski, K.: Enhanced electron screening and nuclear mechanism of cold fusion. in ICCF-15, vol. 15, pp. 197–202. ENEA, Rome, Italy (2009). URL http://www.enea.it/it/produzione-scientifica/edizionienea/2012/proceedings-iccf-15-international-conference-on-condensed-matter-nuclear-science Tsyganov, E.N.: Cold nuclear fusion. Phys. At. Nucl. 75(2), 153 (2012). doi:10.1134/ S1063778812010140 Atzeni, S., Meyer-ter Vehn, J.: The Physics of Inertial Fusion:BeamPlasma Interaction, Hydrodynamics, Hot Dense Matter: BeamPlasma Interaction, Hydrodynamics, Hot Dense Matter. J. Meyer-ter Vehn, The Physics of Inertial Fusion:BeamPlasma Interaction, Hydrodynamics, Hot Dense Matter: BeamPlasma Interaction, Hydrodynamics, Hot Dense Matter. Oxford University Press (2004). Nagel, D.J.: Scientific overview of ICCF15. Infinite Energy 88, 21 (2009). URL http://www. infinite-energy.com/images/pdfs/nageliccf15.pdf Widom, A., Larsen, L.: Ultra low momentum neutron catalyzed nuclear reactions on metallic hydride surfaces. Eur. Phys. J. C 46(1), 107 (2006). doi:10.1140/epjc/s2006-02479-8 Konopinski, E.J.: What the electromagnetic vector potential describes. Am. J. Phys. 46(5), 499 (1978). doi:10.1119/1.11298. URL http://link.aip.org/link/?AJP/46/499/1&Agg=doi Calkin, M.G.: Linear momentum of quasistatic electromagnetic fields. Am. J. Phys. 34(10), 921 (1966). doi:10.1119/1.1972282. URL http://link.aip.org/link/?AJP/34/921/1&Agg=doi Aguirregabiria, J.M., Hernández, A., Rivas, M.: Linear momentum density in quasistatic electromagnetic systems. Eur. J. Phys. 25(4), 555 (2004). doi:10.1088/0143-0807/25/4/010 Szalewicz, K., Morgan, J.D., Monkhorst, H.J.: Fusion rates for hydrogen isotopic molecules of relevance for “cold fusion”. Phys. Rev A 40(5), 2824 (1989). doi:10.1103/PhysRevA.40.2824 Rabinowitz, M.: High temperature superconductivity and cold fusion. Modern Phys. Lett. B 4, 233 (1990). doi:10.1142/S0217984990000301. URL http://adsabs.harvard.edu/abs/1990MPLB4.233R Jackson, J.: Catalysis of nuclear reactions between hydrogen isotopes by μ-mesons. Phys. Rev. 106(2), 330 (1957). doi:10.1103/PhysRev.106.330. URL http://adsabs.harvard.edu/abs/1957PhRv.106.330J Evans, A.B.: 4-Space Dirac theory and LENR. J. Condens. Matter Nucl. Sci. 2, 7 (2009) Evans, A.B.: Four-space formulation of Dirac’s equation. Found. Phys. 20(3), 309 (1990). doi:10. 1007/BF00731695 Fearing, H.W., Scherer, S.: Field transformations and simple models illustrating the impossibility of measuring off-shell effects. Phys. Rev. C 62(3), 034003 (2000). doi:10.1103/PhysRevC.62.034003 Tyutin, I.V.: Once again on the equivalence theorem. Phys. At. Nucl. 65(1), 194 (2002). doi:10.1134/ 1.1446571 Newton, T.D., Wigner, E.P.: Localized states for elementary systems. Rev. Mod. Phys. 21(3), 400 (1949). doi:10.1103/RevModPhys.21.400 Wightman, A.S.: On the localizability of quantum mechanical systems. Rev. Mod. Phys. 34(4), 845 (1962). doi:10.1103/RevModPhys.34.845 Alvarez, E.T.G., Gaioli, F.H.: Feynman’s proper time approach to QED. Found Phys. 28(10), 1529 (1998). doi:10.1023/A:1018882101146 Gorelik, V.S.: Effective mass of photons and the existence of heavy photons in photonic crystals. Phys. Scr. 2010(T140), 014046 (2010). doi:10.1088/0031-8949/2010/T140/014046 Gorelik, V.S.: Bound and dark photonic states in globular photonic crystals. Acta Physica Hungarica A 26(1–2), 37 (2006). doi:10.1556/APH.26.2006.1-2.6 Gorelik, V.S.: Optics of globular photonic crystals. Quantum Electron. 37(5), 409 (2007). doi:10. 1070/QE2007v037n05ABEH013478. URL http://www.turpion.org/php/paper.phtml?journal_ id=qe&paper_id=13478 John, S., Wang, J.: Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms. Phys. Rev. Lett. 64(20), 2418 (1990). doi:10.1103/PhysRevLett.64.2418 André, P.J.J.-M.: Effective mass of photons in a one-dimensional photonic crystal. Phys. Scr. 84(3), 035708 (2011). doi:10.1088/0031-8949/84/03/035708

123

Found Phys 63. Weinberg, S.: Nuclear forces from chiral lagrangians. Phys. Lett. B 251(2), 288 (1990). doi:10.1016/0370-2693(90)90938-3. URL http://65.54.113.26/Publication/18408213/ nuclear-forces-from-chiral-lagrangians 64. Weinberg, S.: Phenomenological lagrangians. Phys. A 96(1–2), 327 (1979). doi:10.1016/ 0378-4371(79)90223-1. URL http://www.sciencedirect.com/science/article/pii/0378437179902231 65. Epelbaum, E.: Nuclear forces from chiral effective field theory: a primer. arXiv:1001.3229 (2010). URL http://arxiv.org/abs/1001.3229 66. Land, M.C., Horwitz, L.P.: Off-Shell Quantum Electrodyn (1996). URL http://arxiv.org/abs/hepth/ 9601021 arXiv:hepth/9601021 67. Coleman, S.: Fate of the false vacuum: semiclassical theory. Phys. Rev. D 15(10), 2929 (1977). doi:10. 1103/PhysRevD.15.2929 68. Tomsovic, S., Ullmo, D.: Chaos-assisted tunneling. Phys. Rev. E 50(1), 145 (1994). doi:10.1103/ PhysRevE.50.145 69. Hagelstein, P.L.: Resonant tunneling and resonant excitation transfer. In: Proceedings, ICCF-12, pp. 871–886. Cambridge, MA, USA (2005). doi:10.1142/9789812701510_0079. URL http://adsabs. harvard.edu/abs/2005cmns.conf.871H 70. Li, X.Z.: Overcoming of the gamow tunneling insufficiencies by maximizing the damp-matching resonant tunneling. Czechoslovak J. Phys. 49(6), 985 (1999). doi:10.1023/A:1021485221050 71. Grifoni, M., Hänggi, P.: Driven quantum tunneling. Phys. Rep. 304(5–6), 229 (1998). doi:10.1016/S0370-1573(98)00022-2. URL http://www.sciencedirect.com/science/article/pii/ S0370157398000222 72. Saad, D., Horwitz, L.P., Arshansky, R.I.: Off-shell electromagnetism in manifestly covariant relativistic quantum mechanics. Found. Phys. 19(10), 1125–1149 (1989). doi:10.1007/BF00731876 73. Horwitz, L.P., Arshansky, R.I., Elitzur, A.C.: On the two aspects of time: The distinction and its implications. Found. Phys. 18(12), 1159 (1988). doi:10.1007/BF01889430 74. Land, M.C., Horwitz, L.P.: Green’s functions for off-shell electromagnetism and spacelike correlations. Found. Phys. 21(3), 299 (1991). doi:10.1007/BF01883636 75. Land, M.C.: Particles and events in classical off-shell electrodynamics. Found. Phys. 27(1), 19 (1997). doi:10.1007/BF02550153 76. Land, M.: Abraham-Lorentz-Dirac equation in 5D Stuekelberg electrodynamics. J. Phys. Conf. Ser. 330, 012015 (2011). doi:10.1088/1742-6596/330/1/012015. URL http://iopscience.iop.org/ 1742-6596/330/1/012015 77. Horwitz, L.: Spin, angular momentum and spin-statistics for a relativistic quantum many-body system. J. Phys. A 46(3), 035305 (2013). URL http://stacks.iop.org/1751-8121/46/i=3/a=035305 78. Horwitz, L.P., Arshansky, R.: On relativistic quantum theory for particles with spin 1/2. J. Phys. A 15(12), L659 (1982). doi:10.1088/0305-4470/15/12/002. URL http://iopscience.iop.org/0305-4470/ 15/12/002 79. Fanchi, J.: Resolution of the Klein paradox for spin-1/2 particles. Found. Phys. 11(5), 493 (1981). doi:10.1007/BF00727077. URL http://www.springerlink.com/content/v3211wt465115256/abstract/ 80. Piron, C., Reuse, F.: Relativistic dynamics for the spin 1/2 particle. Helv. Phys. Acta 51, 146–176 (1978) 81. Horwitz L.P., Piron, C., Reuse, F.: Relativistic dynamics for the Spin 1/2 particle. Helv. Phys. Acta 48(4), 546–548; 48(4) (1975). URL http://www.osti.gov/energycitations/product.biblio.jsp? osti_id=4018859 82. Proca, A.: J. Phys. Rad. 7, 347–353 (1936) 83. Lee, T.D., Yang, C.N.: Theory of charged vector memons interacting with the electromagnetic field. Phys. Rev. 128(2), 885 (1962). doi:10.1103/PhysRev.128.885 84. Ruck, H.M., Greiner, W.: A study of the electromagnetic interaction given by relativistic spin-1 wave equations in elastic scattering of polarized spin-1 nuclei or mesons. J. Phys. G 3(5), 657 (1977). doi:10.1088/0305-4616/3/5/013. URL http://iopscience.iop.org/0305-4616/3/5/013 85. Kaplan, D.B., Savage, M.J., Wise, M.B.: Perturbative calculation of the electromagnetic form factors of the deuteron. Phys. Rev. C 59(2), 617 (1999). doi:10.1103/PhysRevC.59.617 86. Horwitz, L.P., Katz, N., Oron, O.: Could the classical relativistic electron be a strange attractor? Discret. Dyn. Nat. Soc. 2004(1), 179 (2004). doi:10.1155/S1026022604401034. URL http://www. hindawi.com/journals/ddns/2004/205916/abs/

123

Found Phys 87. Roitgrund, A., Horwitz, L.: Simulation of the radiation reaction orbits of a classical relativistic charged particle with generalized off-shell Lorentz force. Discret. Dyn. Nat. Soc. 2010 1(2010). doi:10.1155/ 2010/602784. URL https://eudml.org/doc/230976 88. Burakovsky, L., Horwitz, L.P., Schieve, W.C.: New relativistic high-temperature Bose-Einstein condensation. Phys. Rev. D 54(6), 4029 (1996). doi:10.1103/PhysRevD.54.4029 89. Land, M.: Higher-Order Kinetic Term for Controlling Photon Mass in Off-Shell Electrodynamics (2006) 2003, doi:10.1023/A:1025670806787. URL http://arxiv.org/abs/hepth/0603074. Found. Phys. 33:1157-1175 arXiv:hepth/0603074 90. Horwitz, L., Schieve, W., Piron, C.: Gibbs ensembles in relativistic classical and quantum mechanics. Ann. Phys. 137(2), 306 (1981). doi:10.1016/0003-4916(81)90199-8. URL http://www.sciencedirect. com/science/article/pii/0003491681901998 91. Berestetskii, V.B., Pitaevskii, L.P., Lifshitz, E.M.: Quantum Electrodynamics, Second Edition: Volume 4, 2nd edn. Butterworth-Heinemann (1982). 92. Hagelstein, P.L., Chaudhary, I.U.: Electron mass shift in nonthermal systems. J. Phys. B 41(12), 125001 (2008). doi:10.1088/0953-4075/41/12/125001. URL http://iopscience.iop.org/0953-4075/ 41/12/125001 ˙ 93. Huke, A., Czerski, K., Heide, P., Ruprecht, G., Targosz, N., Zebrowski, W.: Enhancement of deuteronfusion reactions in metals and experimental implications. Phys. Rev. C 78(1), 015803 (2008). doi:10. 1103/PhysRevC.78.015803 94. Huke, A., Czerski, K., Heidea, P.: Experimental techniques for the investigation of the electron screening effect for d+d fusion reactions in metallic environments. Nucl. Phys. A 719(0), C279 (2003). doi:10.1016/S0375-9474(03)00932-1. URL http://www.sciencedirect.com/science/article/ pii/S0375947403009321 95. Huke, A., Czerski, K., Heide, P.: Measurement of the enhanced screening effect of the d + d reactions in metals. Nucl. Instrum. Methods Phys. Res. B 256(2), 599 (2007). doi:10.1016/j.nimb.2007.01.082. URL http://www.sciencedirect.com/science/article/pii/S0168583X07001784 96. Arensburg, A., Horwitz, L.P.: A first-order equation for spin in a manifestly relativistically covariant quantum theory. Found. Phys. 22(8), 1025 (1992). doi:10.1007/BF00733394 97. Rolfs, C.: Enhanced electron screening in metals: a plasma of the poor man. Nucl. Phys. News 16(2), 9 (2006) 98. Firestone, R.B., Baglin, C.M., Chu, S.Y.F.: Table of isotopes. Wiley (1999). 99. Ziegler, J.F., Biersack, J.P., Ziegler, M.D.: SRIM–The Stopping and Range of Ions in Matter. SRIM Co. (2008). 100. Kołos, W., Wolniewicz, L.: Accurate adiabatic treatment of the ground state of the hydrogen molecule. J. Chem. Phys. 41(12), 3663 (1964). doi:10.1063/1.1725796. URL http://jcp.aip.org/resource/ 1/jcpsa6/v41/i12/p3663_s1?isAuthorized=no 101. Hagelstein, P.L., Senturia, S.D., Orlando, T.P.: Introductory Applied Quantum and Statistical Mechanics. Wiley (2004). 102. Iwamura, Y., Itoh, T., Tsuruga, S.: Increase of reaction products in deuterium permeation induced transmutation. In: ICCF-17, p. 6. South Korea, Daejeon (2012). 103. Nagel, D.: Characteristics and energetics of craters in LENR experimental materials. J Condens. Matter Nucl. Sci. 10, 1 (2013) 104. Szpak, S., Mosier-Boss, P.A., Young, C., Gordon, F.E.: Evidence of nuclear reactions in the Pd lattice. Naturwissenschaften 92(8), 394 (2005). doi:10.1007/s00114-005-0008-7 105. Toriyabe, Y., Mizuno, T., Ohmori, T., Aoki, Y.: Elemental analysis of palladium electrodes after Pd/Pd light water critical analysis. In: Proceedings, ICCF-12. World Scientific Publishing Co., Pte. Ltd., Yokohama, Japan (2006), pp. 253–263. doi:10.1142/9789812772985_0025. URL http://adsabs. harvard.edu/abs/2006cmns12.253T 106. Zhang, W.S., Dash, J.: Excess heat reproducibility and evidence of anomalous elements after electrolysis in Pd/D2 + H2 SO4 electrolytic cells. In: Proceedings, ICCF-13, p. 202. Russia, Sochi (2007). 107. Iwamura, Y.: Detection of anomalous elements, X-Ray, and excess heat in D2-Pd system. Fusion Sci. Technol 33(4), 476 (1998) 108. Iwamura, Y., Itoh, T., Sakano, M., Yamazaki, N., Kuribayashi, S., Terada, Y., Ishikawa, T., Kasagi, J.: Observation of nuclear transmutation reactions induced by D2 gas permeation through pd complexes. In: ICCF-11, vol. 11, pp. 339–350. Marseilles, France (2006) doi:10.1142/9789812774354_0027. URL http://adsabs.harvard.edu/abs/2006cmns11.339I

123