aa r X i v : . [ phy s i c s . g e n - ph ] D ec Pilot wave model without configuration or Fock spaces
Roman SverdlovRaman Research Institute,C.V. Raman Avenue, Sadashivanagar, Bangalore –560080, IndiaDecember 6, 2010
Abstract
The goal of this article is to come up with interpretation of quantum phenomena that isboth local and deterministic. This is done by the means of envoking two different metrics, g o and g s . These two metrics give very different ”speeds of light”: c o and c s , respectively. The g o and c o are, respectively, ”ordinary” metric and speed of light that we are used to. On theother hand, c s is superluminal. In this paper I propose a model in which newly introducedsignals, which are subject to g s , are responsible for key quantum phenomena.
1. Introduction
As was discussed in [1] and [2], one of the biggest problems in interpretation of quantummechanics is the need to envoke a configuration space. After all, if we only had one particle,we could have interpretted ψ as merely a classical field obeying Schrodinger’s equation.Granted, the behavior of ψ would have been ”odd”. In particular, for some strange reasons ψ would happen to ”collapse” into a δ function, the definition of ”measurement” that causesa collapse would be unknown, and the location of collapse would be ”probabilistic”. However,as odd as the theory would be, we would still know what ψ is , even though we might notknow why it behaves the way it does.Things become very different once we have more than one particle. In ordinary space, wecan think of ψ as the ”amount of stuff” in a given location, whatever that ”stuff” might referto. In a configuration space this is no longer the case. Since we no longer have an intuitivepicture of ψ , we are forced to call it ”probability amplitude”. This creates a paradox in itself:how can something related to ”probability” be complex? Obviously, the way of removingthat paradox is to go back to viewing ψ as ”physical” field. But this, again, brings back aquestion: if ψ is physical, how can it be defined in a ”configuration space”?1n this paper I propose to remove the concept of configuration space in favor of ordinaryone. First of all, I will postulate a large set of point particles in the ordinary space whoselocation is precisely defined in a classical sense. I will then postulate that some groups ofthem are ”strongly correlated”. Each ”strongly correlated” group of particles in the ordinaryspace corresponds to a single particle in a ”configuration space”. Naturally, the numbers ofparticles in each of the ”correlated groups” might differ. This naturally gives us a union ofdifferent configuration spaces corresponding to all possible particle numbers. This, of course,is a Fock space .I will then define ψ as an internal oscillation of each individual particle in the ordinary space. Due to the strong correlation between the groups of particles, each two particleswithin the same group have nearly the same value of ψ at any given time.This part, of course, violates relativity. However, I propose to have two differet metrics:an ordinary one, g o (or, in component form, g o ; µν ), and ”superluminal” one, g s (or, incomponent form, g s ; µν ). The speed of light ( c s ) corresponding to g s is much larger thanthe speed of light ( c ) corresponding to g . In fact, a particle moving with the speed c s would circle the universe multiple times within unnoticeably short time period.Nevertheless, its c s is still finite. This allows us to impose ”superluminal” correlationswithout sacrificing locality. Of course, the price for this is that the correlations are no longerexact. Thus, different particles within the same ”group” might have slightly different valuesof ψ . Thus, the exact value of ψ corresponding to the ”group” of particles is not defined; butthat is okay since the latter is non local and, therefore, not physical. However, due to thefact that c s is very large, the fact that the values of ψ within the particles of the same groupare nearly the same. This allows us to propose a model whose predictions are consistentwith quantum field theory as we know it.I will then go one step further and incoprporate Pilot Wave model proposed by D¨urr etal ([3]) into my theory. The key concept behind the Pilot Wave models is that, on top ofwave function, there is a ”beable” which corresponds to ”classical” reality. Different PilotWave models propose different things to be ”beables”: it might be position of the particles,or their momenta, or a classical field. The common denominator of most of these modelsis that the probability amplitudes evolve according to ordinary quantum mechanical laws,while ”beables” evolve strictly in response to the influence of the former (according to somedynamics known as ”guidence equation”).Accordoing to the proposal due to D¨urr et al ([3]) the ”probability amplitude” is viewedfrom the point of view of quantum field theory as opposed to quantum mechanics. Likewise,the ”beable” is a quantum state in a Fock space. Thus, at any given point in time, we havenon-zero probability amplitudes for all states and despite that only one state is actuallyrealized. From time to time the ”jumps” occur between different states. If, at the time t ,the ”realized” state | e i , then the probability that, at the time t + dt , a realized state will be | e ′ i is σ ( | e i , | e ′ i ) dt where σ ( | e i , | e ′ i ) = ( h ψ | e ih e | H | e ′ ih e ′ | ψ i ) † |h ψ | e i| (1)2here x † is equal to x if x ≥ ψ of each particle, I claim that the ”choice” of a ”beable” state correspondsto the internal oscillation B . Again, B is ”correlated” so that particles belonging to the samegroup have very similar values of B . At the same time, B has extra restriction: only one ofthe correlated groups has ”large” value of B . All other groups have B close to 0. Thus, B allows us to ”highlite” the ”realized” configuration.Finally, we will ”add” gravity to this picture. In order to avoid the problems related toattempts to quantize gravity, we will instead propose that gravity is a classical field. We areable to do this because the above picture ”converts” non-gravitational quantum field theoryinto classical framework. Thus, adding gravity is just a matter of ”coupling” two ”classical”theories. Now, since only the end-product of the work is ”classical”, it is important thatgravity is added only after all of the non-gravitational fields are ”converted” into classicalones based on my description.The goal of the paper is to come up with quantitative dynamics that accomplishes theabove goals.
2. Desired final outcomes
In light of the fact that the theory is quite complicated, it is important to summarize theend-results that I am trying to accomplish, and then in the Chapter 3, we will actually workthrough the details of the theory.
As was emphasized in introduction, one of the key components of the theory is that there aredifferent groups of particles that ”correlate”. This, naturally, leads one to ask: what makesparticles ”join” into ”groups”? According to the proposal, each particle has an integer-valued ”charge”. This ”charge” is not to be confused with the one in electrodynamics: infact the version of a ”charge” proposed here has integer values 1 through N , where N is verylarge. Furthermore, all particles posess this version of a charge, regardless of their electricproperties or anything else.Now, this ”charge” allows the particles to emit and receive certain frequencies. Thus,the particles with the same ”charge” are, effectively, ”listening” to the same radio station,and this is what makes them correlated. Now, we will assume that the particles are distin-guisheable and each particle has number. Thus, q k is a ”charge” of particle number k . Aswas said in the introduction, ψ corresponds to the internal oscillation of each particle. The ψ -osciallation of particle number k will be denoted by ψ k . Therefore, the ”correlation” of3 -values can be defined as a statement q a = q b ⇒ ψ a ( t ) ≈ ψ b ( t ) (2)where the approximation sign is due to finite value of ”superluminal” velocity c s . In theabove equation, t represents ”quasi-global” time: in light of the fact that c s is superluminalbut finite, we have a very good approximation to a global theory. It is also important tonotice that, as t changes, the values of ψ a and ψ b ”change together. That is, they are onlyapproximately equal to each other if taken at approximately the same time t .From the above picture we see that each configuration of particles corresponds to a givenvalue of charge. Since the value of q ranges from 1 to N , we only have N points in Fock space.In other words, Fock space is discrete. At the same time, however, our original spacetime iscontinuous. Thus, we propose a discrete Fock space in a background of a continuum ordinaryspace. Furthermore, when we add gravity into the picture, we will see thatNow, apart from defining ψ , we would like to also define transition amplitudes, H ,between different states. Essentially, we would like to look at the probability amplitudes oftransitions between different ”local” parts of each state and ”multiply” them to get overallprobability of transition. However, in light of the fact that the things we are ”multiplying”are complex-valued, we have similar problem as we have with ψ .We would like to use the same trick as we did with ψ . Now, if H is correlated betweenparticles of the same ”charge”, then each value of H will be associated with one configuration.But, in light of the fact that H represents a transition between two statees, it should beassociated with two configurations. In order to do this, we will introduce a different kind ofcharge, q ′ , that also varies from 1 to N .Thus, if a given particle, k , has charges q k = s and q ′ k = s , then H k represents aprobability amplitude of transition from the state | s i to the state | s i . Thus, in order for H to be ”consistent”, we need the following correlations:( q a = q b ) ∧ ( q ′ a = q ′ b ) ⇒ H a ≈ H b (3)Now, as we will see from the following sections, the mechanism of ”correlation” will involveemission and absorbtion of signals with various frequencies. Now, in light of the fact thatoscillation involves changing the sign, it would be meaningless to say that a given signal hascommunicated ψ = − ψ = +1. The only thing a signal can communicate isthe amplitude of oscillation, which is a positive quantity.Therefore, in order to be able to communicate complex valued quantities, we need tointroduce four separate real and positive fields ψ , ψ − , ψ i and ψ − i , and do the same for H .Then we can, respectively, define ψ and H as ψ = ψ − ψ − + iψ i − iψ − i ; H = H − H − + iH i − iH − i (4)The above definition leads to a redundancy, since ψ and H remain unchanged under thetransformation ψ c → ψ c + k ; ψ − c → ψ − c + k ; H c → H c + l ; H − c → H − c + l (5)4owever, since our goal in this paper is to be as ”literal” as possible, I claim that the aboveeight fields are all physical, and the two ”pictures” in the above transformation are not thesame. But, it just happends that our dynamics is dependent on ”overall” H and ψ . Asa result, it happened that the two pictures are dynamically indistinguisheable even thoughthey are not the same.In order to have a more compact notation, let us now introduce two classes of sets ofparticles: S k = { a | q a = k } ; T k = { a | q ′ a = k } (6)If we can accept a level of approximation due to finiteness of ”superluminal” c s , we canassign a ”common value” to the particles of the same set. Thus, h ψ | s k i is approximated by ψ ( S k ) and h s k | H | s l i is approximated by H ( S k ∩ T l ). Thus, our desired dynamics is d ( ψ ( t, S k )) dt ≈ X l iH ( S k , S l ) ψ ( x l ) (7)Let us now make the above equation more precise. Suppose we have two sets of particles: a n and b kl , where k , l , and n go from 1 to N . Suppose we choose them in such a waythat q a n = n , q b kl = k and q ′ b kl = l . Then, according to the above equation, the followingapproximation should hold: dψ a k dt ≈ X k,l iH bkl ψ a l (8)The above approximation is independent of the actual choice of a n and b kl that meet theabove criteria. This independence is due to the ”correlations” that we have introduced.Let us now rewrite a dynamics in terms of real components. In light of the fact thatboth ψ and H has four components, our equations would be enormously complicated. So, inorder to simplify them, we will notice a sum over parameter c that only takes four values: 1, − i and − i . This can be described by a single restriction, c = 1. Thus, in this language, ψ = X c =1 cψ c ; H = X c =1 cH c (9)In this notation it can be verified that iHψ is given by X l iH ( S k , S l ) ψ ( S l ) = N X l =1 X c = c =1 c H c ( S k , S l ) ψ c /c ( S l , t ) (10)And, therefore, the evolution equation becomes dψ c ( S k , t ) dt ≈ N X l =1 X c =1 H c ( S k , S l ) ψ − ic /c ( S l , t ) (11)If we again take the selection of particles a n and b kl described earlier, then the above can berewritten as dψ c ( a k ) dt ≈ N X l =1 X c =1 H c ( b kl ) ψ − ic /c ( a l , t ) (12)5ne can notice in the above equation that H is taken at a single particle, not two particles.This is due to the fact that each particle has two different charges. Thus, the combinationof the two charges of one particle tells us the information about the two states involved intransition: the ”in” state, given by q k and the ”out” state, given by q ′ k .Once again, there are many different choices of particles that can be made with the same q and q ′ . The above approximation simulteneously holds for all such choices. This is the keyplace where we take advantage of the ”correlations”. The actual dynamics that describeshow these correlations arise will be given in detail in section 3.1. However, that section canbe understood without reading the rest of chapter 2. As was stated in the Introduction, after ”converting” the probability amplitudes into ourframework (sec 2.1 and 3.1) we intend to do the same with Pilot Wave model (sec 2.2 and3.2). We have also stated that we would like to choose the Pilot Wave model proposed byD¨urr ([3]) as our basis. The reason for this is that, as we have seen, our Fock space only hasfinitely many states. Thus, we would like to have a discrete Pilot Wave model in order todescribe it. This makes D¨urr et al the best possible choice.However, for the sake of convenience of a reader, we would like to first summarisethe proposal of D¨urr (sec 2.2.1) before showing how we ”convert” this proposal into ourframework (sec 2.2.2). Therefore, a reader who is already familiar with D¨urr proposal in”standard” case can move straight to sec 2.2.2. However, sec 2.2.2 is important and can not be skipped if one wishes to understand the proposed ”local” development of Pilot Wavemodel in sec 3.2, since sec 2.2.2 outlines the key ideas of my own contribution to the theory.Nevertheless, a reader that is only interested in probability amplitudes as opposed toPilot Wave models, can skip chapter 2 altogether and only read sec 3.1. It must be said,however, that Pilot Wave model is important in order to understand gravity part of sec 3.3,since gravity will be ”coupled” to a Pilot Wave model. Sec 3.1 is non-gravitational and canbe understood without chapter 2.
The key idea behind Pilot Wave model is that we have two separate substances: a particleand a wave. The most traditional verision of that model is the one proposed by DeBroglieand Bohm. According to that model, a wave evolves according to Schrodinger’s equation,while a particle evolves according to guidence equation , d~xdt = 1 m ~ ∇ Im ln ψ, (13)The position of a particle is known at all times, and it is called position beable . It is easy to seethat, if the above equation is obeyed, then the ”classical” probability current ρ~v coincides6ith the ”probability current” of Schrodinger’s equation. As a result, if the ”classical”probability density ρ coincides with quantum mechanical, | ψ | , it will continue to do so. Inother words, ρ = | ψ | is an ”equilibrium point” of a theory.Of course there is a suddlety here: in light of time reversal, if a system can not ”leave”equilibrium, it can not ”enter” equilibrium either. However, we know from classical prob-ability theory that time reversal no longer holds on ”coarse grained” level. Thus, we canreasonably expect ρ ≈ | ψ | on some ”larger” scale.The precise definition of the statements that I have just made is still an unansweredquestion. But, for the purposes of this paper, it would suffice to say that this question isrelated to classical probability theory. Thus, it is still true that the model adresses ”quantummechanical” part of the problem: the probability we have to deal with is real, not complex;the particle moves along the deterministic trajectory and the probability is simply a measureof our ”ignorance” due to not knowing exact initial conditions.Now, in case of quantum field theory, things are not nearly as simple. Since the numberof particles change, we can no longer think of a beable as a ”position of particles”. In light ofthis, several proposals have been made in recent years in comming up with new ”beables” forquantum field theory, such as beables due to Struyve and Westman ([4]), Dirac see beablesdue to Colin ([5]), particle beables in extra dimension ([6]), and so forth. According to D¨urret al, however, the ”beable” is simply a highlighted quantum state. Thus, a ”wave function”in quantum mechanics generalizes to a set of probability amplitudes in quantum field theory,while a ”particle” in quantum mechanics generalizes to a state that is ”realised” at a giventime.Now, the essential goal is the same as it was in non-relativistic case: introduce thedynamics for a beable that would reproduce desired probability | ψ | . Now, since we havediscrete states, the only thing available to us is a ”jump”. Since the jump is discrete, it canno longer be described deterministically. Therefore, the dynamics involves probabilities ofjumps (but these probabilities are ”classical” and, therefore, are both positive and real).Thus, the desired statement of the dynamics is that if a state | e i is realized at the time t , then there is a probability σ ( | e i , | e ′ i ) that the state | e ′ i will be realised at the time t + dt .Now, if we denote by ρ the probability that a state | e i is realised at the time t , then ρ obeysthe following equation: dρ ( | e i ) dt = X e ′ ( σ ( | e i , | e ′ i ) ρ ( | e ′ i ) − σ ( | e ′ i , | e i ) ρ | e i ) (14)Therefore, we have to compare this equation with the one for d | ψ | /dt , and come up withthe definition of σ that would make the two match. It is easy to see that the latter can beexpressed as ddt ( ψ ∗ ψ ) = 2 Re (cid:16) ψ ∗ dψdt (cid:17) (15)Now, if we express the above in the form of quantum states, and insert the ”identity matrix”,7e get dψdt = i < e | H | Ψ > = i X e ′ < e | H | e ′ >< e ′ | Ψ > . (16)Finally, we have to multiply the above by ψ ∗ which is given by ψ ∗ = < Ψ | e > (17)This gives us ddt ( ψ ∗ ψ ) = 2 Im X e ′ ( h Ψ | e ih e | H | e ′ ih e ′ | Ψ i ) (18)Let us now compare this to our equation for ρ , which is dρ ( | e i ) dt = X e ′ ( σ ( | e i , | e ′ i ) ρ ( | e ′ i ) − σ ( | e ′ i , | e i ) ρ | e i (19)From our intuition, we know that if σ ( | e i , | e ′ i ) is large, then | e i and | e ′ i should be very similarto each other. Thus, we can replace ρ ( | e ′ i ) with ρ ( | e i ) to get the following approximation: dρ ( | e i ) dt ≈ ρ ( | e i ) X e ′ ( σ ( | e i , | e ′ i − σ ( | e ′ i , | e i )) (20)By comparing this to the result that we got from evolution equation, we see that our desiredresult is σ ( | e i , | e ′ i ) − σ ( | e ′ i , | e i ) ≈ Im h Ψ | e ih e | H | e ′ ih e ′ | Ψ i ρ ( | e i ) (21)Now, we recall that, when we did the Bohm’s case in the beginning of this subsection, wedid not prove that ρ = | ψ | . Instead we have proven that if ρ = | ψ | happens to hold at agiven point in time, then it will continue to do so. Now, in the present case, we would liketo be able to mimic the above argument. Therefore, we assume that ρ = | ψ | at a time t .In light of this, we can substitute ρ ( | e i ) = | ψ ( | e i ) | = h e | Ψ ih Ψ | e i (22)in a denominator, which gives us σ ( | e i , | e ′ i ) − σ ( | e ′ i , | e i ) ≈ Im h Ψ | e ih e | H | e ′ ih e ′ | Ψ ih e | Ψ ih Ψ | e i (23)Now we have a freedom of defining σ in different ways. We would like to utilize that freedomin order to make sure that σ is positive. It is easy to see that the right hand side reversessigns under the rearrangement of | e i and | e ′ i . Therefore, we can simulteneously keep σ positive and satisfy the above equation if we identify σ with (RHS) † , where x † is equal to x for x ≥ σ ( e ′ , e ) ≈ Im ( h Ψ | e ih e | H | e ′ ih e ′ | Ψ i ) † h e ′ | Ψ ih Ψ | e ′ i (24)8here we have used the fact that denominator is positive in order to pull it out of † . Now,since we have no other clues as to how turn the above approximation into exact one, wewill simply postulate the above equation as an exact dynamics. After all, ρ = | ψ | isa consequence of Pilot Wave model, and not hte other way around. Thus, we can alwayspostulate that the latter is exact, at the expense of assuming that the former is approximate.This brings us to the final proposal made by D¨urr et al: σ ( e ′ , e ) = 2 Im ( h Ψ | e ih e | H | e ′ ih e ′ | Ψ i ) † h e ′ | Ψ ih Ψ | e ′ i (25) We would like to modify the above model in two ways:A. We would like to turn a stochastic model into a deterministic oneB. We would like to get rid of references to Fock space in that model and ”simulate” itin our ordinary space by means of local, but superluminal, signals propagating with speed c s . We will leave the actual proposal of such model to section 3.2, but for now we will justoutline some of the key elements. In reality, these are simply something that we would like to obtain as an end result of some very complicated model that will be discussed in 3.2.First of all, we have to define what it means for a ”configuration” to be ”selected” interms of ordinary space. We have already seen in 2.1 how we can express ψ in ordinaryspace, despite the fact that ψ was originally meant to refer to ”configurations”. We will nowuse the same principle. In order to make the two problems similar, we will assume that, ontop of ψ , there is another function, B on a Fock space. That function is equal to zero forall states except for the ”highlighted” one. Then, during the ”transition” from | e i to | e ′ i , B ( | e i ) changes from large value to zero, while B ( | e ′ i ) does the opposite.Now, in light of the fact that one of our goals is determinism, we would like to be ableto write a differential equation. This means that we can not have discrete jumps. Therefore, B has some ”in between” values. During the ”transition” period it is quite possible that B is non-zero for both of the states. But, we would still like to make sure that most of the time B is cloase to zero for all states except one.Furthermore, in much the same way as the ”common” value of ψ is only an approxi-mation, what we have just said about B is approximation as well. In other words even faraway from the transition region , B is not exactly zero for non-selected states; it is simplyvery close to zero. Thus, we can formulate what we have said so far as follows:At a usual time t (that is, far away from ”transition”) the following conditions are met:1) If q a = q b , then B a ( t ) ≈ B b ( t )2) There is a unique q ( t ) ∈ N such that q a is large.The variable q ( t ) undergoes discrete jumps. The above approximations do not hold in9he vicinity of these jumps. Therefore, the exact time of the jump can not be defined. Thereis a continuous deterministic process that we interpret as the above ”jumps”. The desired deterministic theory implies that the probability of these jumps is consistent with D¨urr etal ([3]) and is given by σ ( e ′ , e ) = 2 Im ( h Ψ | e ih e | H | e ′ ih e ′ | Ψ i ) † h e ′ | Ψ ih Ψ | e ′ i (26)Another thing that is important to adress is creation and annihilation of particles. Suppose,for example, we have a process where an electron and positron annihilate into two photons.According to our proposal, there are four particles at all times (for convenience, we willnumber electron and photon with 1 and 2 and we will number the two photons with 3 and4). Now, it happens that q = q = Q and q = q = Q . At time t , all particles withcharge Q across the universe were highlighted. Thus, B k ( t ) = A whenever q k = Q , andit is 0 otherwise. On the other hand, at time t , the particles with charge Q were no longerhighlighted and, instead, particles with charge Q became highlighted. Thus, B l ( t ) = A whenever q l = Q and it is 0 otherwise.As a result of this ”global” phenomenon, electron and positron were ”highlighted” at t while the two photons were highlighted at t . Since we only see the particles that are”highlighted”, it appears to us that electron and proton have annihilated. In reality, however,they continue to exist, just in ”invisible” form. Likewise, it appears to us that the two photonswere ”created”. In reality, they always existed; they were simply ”invisible”. Thus, we canformulate it by defining a ”matter density” as ρ k ( x ) = X k Z dτ k B k ( τ k ) δ ( x − x k ) (27) WARNING: This section might be controversial. But the non-gravitational part of the paper(that is, sections 3.1 and 3.2) does NOT depend on it. Therefore, one of the options is toskip section 2.3 and go directly to 3.1 and 3.2. However, 3.3 picks up gravity again, and,therefore, might likewise be skipped
In order to avoid the difficulties that come from quatizing gravity, I propose to view it asa strictly ”classical” field. Now, while other things are also ”classical” thanks to Pilot Wavemodel, gravity is even more so. When it comes to other fields, we still had a probabilityampitude h ψ | s i . We simply re-interpretted it in classical terms. For gravity, however, we donot have ψ to begin with. Instead, we use ψ for non-gravitational fields and then after weobtain the behavior of beables of these fields, we look at how these fields produce gravity.More precisely, we would like to say that gravity is a solution of an exact Einstein’sequation, where the source term is given through the coupling of particles with B -field.10n order to avoid black hole singularities, δ -functions might be replaced with some finiteapproximation, ˜ δ . Thus, we would like to say f ( R µν ( x )) = X k Z dτ B k ( τ k )˜ δ ( x − x k ) (28)On the left hand sinde we wrote f ( R µν ) instead of standard Einstein’s equation for a reason.According to Bianchi identity, the violation of energy momentum conservation implies lack ofexact solutions. However, the variation of B -field changes the energy-momentum associatedwith particles.This issue is not unique to my specific Pilot Wave model. Even in the classical Bohmianmodel for fixed number of particles, they might accelerate through ”guidence equation”.Then, in more advanced models, such as D¨urr’s, their numbers are not fixed which makesthe situation even worse: during the ”jumps” explained in section 2.2, new partices can becreated ”out of nowhere”.There are deeper reasons as to why we can’t restore energy conservation. First of all,according to Pilot Wave model the influence is one way from wave to particle. Energyconservation, on the other hand, demands two-way interaction. Furthermore, even if PilotWave model did conserve energy-momentum, we would have to count all of the ”sources”of energy-momentum, which would include wave. However, due to decoherence, the wavesplits into ”parallel universes”. Thus, by counting the wave as one of the sources, we predictgravitational interaction between ”parallel universes” which, obviuosly, is very bad.It might be hoped, however, that once we do make sure that the interaction betweenthe particle and a wave is two-way, the particle will ”kill off” the unwanted ”parallel uni-verses”; then we will be able to count the energy of the wave without any fear of unwantedinteractions. In fact, one example of a theory where a particle ”kills off” parallel universeswas presented in [6]. However, that particular model is highly non-local and, therefore, doesnot restore energy conservation that we are looking for.As far as the current paper is concerned we will simply admit the violation of energyconservaton as given, and modify Einstein’s equation in such a way that it has solutionsfor non-conserved sources. This is done by introducing an extra term that breakes gaugesinvariance. However, we can not simply impose gauge fixing term. If we did, then theBianchi identity would imply that the divergence of the difference of T µν and gauge fixingis zero: ∇ ν ( T µν − ( GF ) µν ) = ∇ ν G µν = 0 (29)This, however, does not imply that GF = 0. If we assume that, due to quatnum field theory, T µν is conserved, the most information we will get is that gauge fixing term is conservedas well. But, obviously, conservation does not set it to zero. If we try to impose initialconditions on it, then, over a very large period of time (such as, for example, the age ofthe universe), the ”slight” non-conservation of T µν would ”build up”, so that the ”initialconditions” will no longer be relevent. And, unlike gauge theories, we do not have luxury ofdenying the small non-conservation; after all, that is what we are trying to adress to beginwith. 11n order to avoid the possibly-large unwanted impact from gauge-fixing term, we haveto make sure that our correction term comes with a very small coefficient, ǫ . Now, we knowthat if we set ǫ to 0 we will have no solution. This means that if ǫ is small, we should expectsome kind of singularity to arise. However, we also know that solution does exist for theconserved source. Therefore, the source of singularity is precisely non-conservation of energy.This means that as long as energy non-conservation is smaller than ǫ , there is no singularity.Now, the good news is that the conservation of energy momentum can be predictedfrom quantum field theory, independently of gravity. Thus, if Pilot Wave model is a goodapproximation to quantum field theory, it would not produce significant violation to energymomentum anyway. The bad news, however, is that, as long as particles are viewed asbeables, in light of their point size their impact on energy momentum is dramatic; whichautomatically means that their violation of energy momentum conservation is dramatic aswell. This means that it is important to ”average out” the energy-momentum of particleson a larger scale: ˜ T µν ( x ) ≈ Z n δ ( x ) d x ′ T µν ( x ′ ) (30)where n δ stands for a δ -neighborhood which is ”small” on classical scale but ”large” onquantum mechanical. In section 3.3 we will introduce a ”local” mechanism by which the”quasi-local” averaging works. This averaging, by the way, is also important in order toavoid ”black hole” singularities created by point particles; in fact, when we replaced δ withits finite version ˜ δ , we have assumed that we have some means of producing ˜ δ , throughaveraging.However, since the ”true” Lorentzian neighborhood is a vicinity of light cone and, there-fore, not compact, we have to break Lorentz covariance in order to define n δ . While we arenot explicilty doing the above integration, the ”local” mechanism that we introduce happensto violate relativity. The above issue might be a ”deeper reason” for this. Nevertheless, forthe purposes of this paper, we are okay with violation relativity, as long as we continue topreserve ”locality”. For that reason, I still included that mechanism.Once the ”classical” theory of producing gravity is under control, then our problemreduces to coupling two classical theories. It is easy to make an intuitive argument that itmight well have a solution. After all, at a hypersurface t = const, we already know g µν (bothspacelike and timelike components). Thus, we can substitute that ”fixed” g µν into a PilotWave model to see how our system will evolve between time t and t + dt . Then, we can usethis information as a ”source” of gravity which, through the modified Einstein’s equationwill give us g µν at t + dt . Then we will repeat that procedure between t + dt and t + 2 dt ,and so forth.There are, however, two more problems. First of all, if we are ”true” to general relativity,then the particles will have to move along geodesic. This conflicts with our usual notions.When we discretize ordinary space, the ”lattice” we use is stationary. Since our model is adiscretization of Fock space, and our ”groups of particles” correspond to ”lattice points” ona Fock space , we have to make sure they are also stationary, by analogy. In other words,states, themselves, would move as opposed to different motionless states being highlighted12t differnt times.This, however, does not dismiss the theory out of hand. After all, by uncertainty prin-ciple, if the ”position” of particle is precisely defined, the momentum is infinitely undertain.Therefore, setting momentum equal to zero is just as ”wrong” as setting a particle to movewith any other velocity. Thus, since the former ”works” the latter might ”work” as well. Infact, in some respect, it might even be ”better” to define state in terms of concrete particlesmoving along geodesics (as ”messy” as that picture might end up being) than to stick to atraditional view that in curved spacetime there is no such thing as ”state”.But, at the same time, it this would destroy flat space quantum field theory, since nowthe particles are allowed to move in that setting, too. So the key question is whether or notthe picture of ”stationary lattice” (and in our case we have a ”lattice” in Fock space) can bereplaced with the one involving ”moving lattice”. Analysis of this issue is beyond the scopeof this paper and is up to the verifiation in future research.If it turns out that the particles should not move, we still have a way of salvaging ourtheory. Throughout the rest of the paper we will explicitly use two different metrics, g o and g s , which, respectively, produce the speeds of signals c o and c s . Now, c o is the ”ordinary”speed of a signal that we are used to; on the other hand, c s is ”superluminal”. We account allof the seemingly-nonlocal processes in quantum mechanics to the signals propagating withspeed c s .Now, when we write down geodesic equation for the motion of a particle, we have to useeither g o or g s in a definition of Christoffel’s symbols. Now, if we use g s and if we assumethat the latter happened to be flat (after all, the only ”gravity” that we have observed isthe one involving g o ), then the particles will automatically go along straight line. Then, wecan always postulate they are stationary without any conflict with geodesic equation. Inthis picture we can explain the appearance of covariance with respect to g o in terms of a”coincidence” that Hamiltonian happened to couple everything to g o in identical way, whichmade g o look like geometry while, in reality, it is merely a field on a background of g s -basedgeometry.Another approach to get the same result, is to introduce a third metric g l (here l refersto lattice) for which speed of light, c l is much slower than c o . In this case the lattice pointsmight ”think” they are moving fast, when in fact, they move very slow ompared to anyof the velocities we are concerned about, which is why they can be safely approximated asstationary.Another thing to be weary of when it comes to g o is the fact that particles might move”backward” in global coordinate t , such is the case with a black hole. This might lead toconflict with Pilot Wave models that evolve ”forward” in time. It is possible, however, toattempt to avoid this issue by rewriting everything in terms of t . Thus, the geodesic equationis d x ka dt = Γ k + Γ k l dx la dt + Γ kij dx ia dt dx ja dt − dx ka dt (cid:16) Γ k + Γ k l dx la dt + Γ kij dx ia dt dx ja dt (cid:17) (31)While this might look like cheating, this is, in fact, consistent with the spirit of interpretationof quantum mechanics where we assume absolute time and view the appearance of relativity13s ”coincidence” in Hamiltonian. Now, the above equation is postulated if we choose theoption of sticking with g o . On the other hand, if we stick with g s we would violate relativityin a lot more obvious way, described in previous paragraph.Now, as far as this paper is concerned, I intend to leave a lot of freedom for futureresearch to decide between g o and g s in different aspects of the theory. In these cases I willuse other letters in place of o or s . As far as this paper is concerned, all of these choicesseem to ”work” but they have very different ideological implications. So I would like to bepluralistic and allow all of them.
3. The proposed model3.1. Quantum field theory in flat space
Note: in sections 3.1.1 – 3.1.3 we will assume that H is already known, and will be using itin order to obtain the dynamics for ψ . Then, in section 3.1.4 – 3.1.6, we will describe how H has been ”produced”. In this model we imagine that our universe is compact. Furthermore, the particles emitwaves that propagate with ”superluminal” speed c s . The latter happens to be so large, thatthe waves can circle the universe multiple times with a negligeable time period. Particlescan also ”absorb” waves. But, the process of ”absorbtion” is very unusual: when a particle”absorbs” a wave, it only has an impact on a particle but not on a wave! In other words,the wave continues to propagate with the same intensity after so-called ”absorbtion”. Thisallows this wave to be ”absorbed” multiple times by several particles.Now, each particle has emission and absorbtion frequencies that, respectively, are equalto its ”charges” q ′ and q . Thus, if a particle number k was ”triggered” to emit a wave,it would emit the one with frequency q ′ . On the other hand, if the wave happened to bepropagating from some other source, and it happened to reach a given particle, it will only be absorbed if its frequency happened to be q .The difference between emission and absorbtion frequencies accounts for the communi-cation between different ”configurations”. Suppose we have two particles, 1 and 2. It is easyto see that if q ′ = q , then the particle number 1 will be able to emit a signal that will bereceived by a particle number 2. But, since absorbtion has no effect on a wave that signalwill also be absorbed by a particle 3, satisfying q = q ′ . In other words, it will be collectively absorbed by all of the particles belonging to the same ”group” as particle number 2.By the same token, all of the signals that have ever been emitted are absorbed ”together”by all the particles in the group 2. Now, since the value of ψ is produced through theabsorbtion of signals, it is only logical that all of the particles in group 2 have approximately14he same value of ψ . The same, of course, is true for all other groups of particles. Now, inlight of time delay due to finite value of c s , the equality is not exact.In order for the approximation to be good, the time delay related to signal circlingthe universe should be negligeably small (that is, smaller than the time it takes for ”local”processes to occur that we use as ”watch”). Of course, if the universe was infinite, the timedelay would have been infite too, regardless of how large c s might be. That is why we areassuming that the universe is finite, which allows us to claim that the signal moving withvelocity c s can circle the universe within very short time. Furthermore, we assume that ouruniverse is compact in order to avoid various problems that might arise ”on the boundary”.Now, in light of compactness of the universe, the signals ”come back” multiple times.Since they are ”massless”, the signals emitted a million years ago have just as strong effectas the ones emitted just now! Of course, we do not want signals emitted millions of yearsago to cause change in ψ . This can be avoided by modeling the ”absorbtion” of signals as adriven harmonic oscillator with slight damping. The ”receptor” of a signal is the ”oscillating”quantity, while the signal itself is ”driving force”.Thus, ψ and not its time derivative is a function of the signals that are ”out there”.Whenever new signal is ”added”, this amounts to increase in the value of ψ c for one ofthe c ∈ { , − , i, − i } (and, therefore, changes the value of ψ ) among the configuration ofparticles whose charges coincide with its frequency. If, on the other hand, a signal would”die out” that would amount to ψ ”reversing back” to its earlier stage. Since this does nothappen, we do not want them to die out, which is why that is good that signals emittedmillions of years ago have the same intensity as the ones emitted recently.Now, on a first glance one might worry that this leads to a prediction of steady increase of ψ over time, while in reality it both increases and decreases. This can be explained interms of the ”four fields” ( ψ , ψ − , ψ i and ψ − i ). While it is true that the affected field in creases, the total ψ can go both ways. After all, if the in creased field is ψ − this wouldresult in de crease in ψ .Therefore, over time, the increases of ψ and ψ − ”cancel out” which leads to ψ averagingto the same thing. Nevertheless, it is true that the individual values of ψ and ψ − can serveas a ”clock” of the age of the universe. Thus, when the universe was ”young”, we couldhave had ψ = 5 − ψ = 10000 − ψ , we do not have any physical access to that extrainformation. The qualitative model presented in 3.1.1. has one gap. Namely, if the signals are beingemitted continuously, there might be interference. We would like interference terms to cancelout by making sure that emission events are ”random”. But, since we would like to have adeterministic theory, we do not want ”true” randmoness. Instead, we need to come up witha ”random generator” that is ran by deterministic laws.15he specific way of ”triggering” the emission is not important, as long as it is random. Inthis paper, we will propose that the emission is triggered when a particle ”collides” with so-called ”a-particles”. Now, since particles are point-like, there is no ”true” collision. Instead,we have to claim that a-particle has a ”density” field, ρ a . This field is very short acting.Thus, it forms a ”cloud” that ”looks” like a particle but really has size. In other words,we can think of ρ a as a ”density” of ”non-point” particle that pointlike particle ”formed”around itself.Now, the ”relativistic” version of that density is a current, j αa . Just like ρ a is really a field rather than density, in the same way j αa is a vector field. The source of that vector fieldis a true δ -function corresponding to point particle. We have to be careful though: sinceour universe is compact, j a might ”accumulate” in the universe (similarly to ψ ) which wouldruin our purpose of having it large only in a vicinity of a particle. We adress this issue byadding a ”mass term” to the dynamics of j a .Thus, if we have N a a -particles, and each of these particles has a trajectory a k ( τ j ), thenthe resulting equation of motion for j αa is ∇ βj ∇ j ; β j αa + m j a g j ; µν j µa j νa = N a X k =1 Z dτ k δ ( x − a k ( τ )) (32)where j in g j refer to the choice of a metric. As we said earlier we have two different metrics: g o for ”ordinary” signals and g s for superluminal ones. In our case, since the field is shortlived it is not important which we choose to use. Thus, I am leaving it up to the philosophyof a reader and/or future research by simply using g j and leaving it an open option for futureto use either o or s in place of g .As far as the j in ∇ j , this means that we define it in terms of g j -based Christoffel’ssymbols. It is also understood that the indeces in ∇ s are ”raised” and ”lowered” by g s ,while the indeces in ∇ o continue to be raised and lowered through g o : ∇ αs f = g αβs ∇ s ; β f ; ∇ αo f = g αβo ∇ o ; β f (33)and, finally, dτ k ; j is defined by using g j : dτ k ; j = r g j ; βγ dr β dτ dr γ dτ (34)We are now ready to talk about the actual emission of a signal once our particle of interestis being ”triggered” by r -particle. As we have said in sec 2.1, we break our complex field ψ into four positive real ones: ψ = ψ − ψ − + iψ i − iψ − i = X c =1 cψ c (35)where it is understood that c = 1 under the sum is equivalent to c ∈ { , − , i, − i } . There-fore, we need four different means of emitting each of these four fields. We have seen from16ec 2.1 that the desired equation for ψ is dψ c ( a k ) dt ≈ N X l =1 X c =1 H c ( b kl ) ψ − ic /c ( a l , t ) (36)Therefore, we would like this to factor into our ”emittor”. At the same time we would alsolike to include the interaction of a particle with j a . Thus, we would like to couple the velocityof particle to j a and then couple the product to both ψ and H . So our proposed coupling is g e ; αβ j αa dx β dτ e X d =1 H d ( k, τ e ) ψ − id/c ( k, τ e ) (37)However, we would like to make sure that emission frequency coincides with the charge q ′ k . Thus, we have to say that the collision with a -particle first creates oscillatory processwith frequency q ′ and then this process causes the emission of µ -field. Since the value of q ′ is specific to the particle, this has to be internal oscillations of the particle itself, and,therefore, should not be confused with oscillations of µ ψ c .I define these internal oscillations of particle by e ψ c . These have to be triggered byinteraction with j a . This means that once a -particle has passed, they have to ”die out” verysoon. For this reason, I model e ψ c as a damped harmonic oscillator, driven by j a : d e ψ c ; k dτ e = − q ′ k e ψ c ; k − λ e de ψ c ; k dτ e + g e ; αβ j αa dx β dτ e X d =1 H d ( k, τ e ) ψ − id/c ( k, τ e ) (38)We are now ready to define a dynamics of µ -field with its ”source” being e -field. While with e -field we were undecided which g to use (hence we used g e ), for µ we know we should usesuperluminal one, g s , since we would like µ to circle the universe multiple times in order toassure ”consistency” of ψ . Thus for the sake of generality, we use two different g -s, and writethe dynamics of µ -field as follows: ∇ αs ∇ s ; α µ ψ c = N X k =1 Z dτ k ; s e ψ c ( τ k ; s ) δ ( ~x − ~x k ( τ s )) (39)It should be noticed that we did not put mass term for µ . This time, for the reasons explainedin sec 3.1.1, we want µ to circle universe multiple times (which is what we were trying to avoid in case of j by putting the mass term). Therefore, unlike the situation with j , wemake µ massless. In sec 3.1.2 we have discussed the mechanism of emission of signal µ ψ c . We denoted it µ ψ c instead of ψ c for a reason. As was described in sec 3.1.1., when the signal is omitted it fillsthe whole space. But, at the same time, it only gets absorbed by some particles, namelythe ones whose charge coincides with its frequency. What defines ψ c ( | s i ) is the internal17scillations of that specific group of particles. Thus, in order to get ψ c we have to define amechanism of absorbing µ ψ c .Now, as we said in sec 3.1.1., the absorbtion of signal is a one way process: it hasinfluence on the particle but it does not diminish the intensity of a signal. Therefore, wecan view this process as a driven harmonic oscillator, with natural frequency q : dr ψ c ; k ( τ s ) dτ r = − q k r ψ c ; k ( τ r ) − λ r dr ψ c ; k ( τ r ) dτ r + µ ψ c ( x k ( τ k ; r )) (40)where we have used τ r (”r” standing for ”reception”) instead of τ s because, while it is crucialthat the ”transmission” of µ is superluminal, it is not important when it comes to ”reception”(just like it was not important for ”emission” either).In the above expression we have introduced the damping λ r in order to avoid singularity.After all, in light of the fact that possible charges are discrete, the ”received” frequency is exactly equal to the charge (as opposed to ”approximately”), which is why we would havehad singularity if we didn’t introduce damping. However, λ is assumed to be very small, soits effect on anything else is negligeable.Now, as one can see, r ψ c ; k oscillates between positive and negative values. That is whywe did not identify it with ψ c , since the latter is strictly positive. Therefore, we need to findsome mechanism of ”extracting” the amplitude from the oscillations. The latter, of course,is positive, as desired. We do that by postulating the following equation: ψ c ; k = r q k r ψ c ; k + (cid:16) dr ψ c ; k dτ r (cid:17) (41)where, again, we used a special τ r corresponding to g r for ”reception” since we are not surewhich g to use. This last equation is, in fact, the final definition of ψ c . It is easy to see thatthe extracted quantity is strictly positive. That is the ultimate reason why we decided touse four different fields to define complex valued function. In the previous three subsections we have shown in detail the mehanism for evolution of ψ .However, we know from quantum field theory that the derivative of ψ is a function of H .Therefore, if we did our job correctly, we must have used H at some point . In fact, we have!The factor of H have shown up in the equation for the dynamics of e ψ c .However, it is not correct to assume that we know H . After all, as we have seen fromsection 2.1, in our reinterpretation H is not a Hamiltonian but rather an internal degree offreedom of a particle. In fact, when we used H in the equation for e ψ c , we explicitly saidthat H is a function of k , where k is a number of a particle. This, again, reinforces the factthat H is an internal degree of freedom of a particle.The reason why H is related to Hamiltonian is very similar to a reason why ψ is relatedto probability amplitude. There is a global correlation between the values of H among the18articles with the same q and q ′ . Thus, the ”common” value of H among these particles”encodes” a transition probability between the group of particles with ”first” charge q andthe group of particles with ”first” charge q ′ .The key point is that the common value of H corresonds to the transition amplitudegiven by quantum mechanical Hamiltonian. This statement is non-trivial and can not besimply postulated. We have to come up with dynamics of H that would show that this is,in fact the case. After all, we had to introduce dynamics of ψ in order to show that the”desired” evolution was ”true” (even though in standard quantum field theory the evolutionequation is simply postulated, just like H is).The good news, however, is that the mechanism of ”generation” of H is very similar tothe one of ”generation” of ψ , which is evident from the similarities between the statementsof both questions. Therefore, wherever possible, we will borrow the evolution equations for ψ with appropriate modifications.We know that in standard quantum field theory H represents the transition between two global states; that is, each ”state” describes the entire universe. At the same time, the ”re-sultant” probability amplitude H is ”produced” by ”local” probability amplitudes, h . Thisfeature is similar to classical probability theory where a probability of ”complicated” eventis a product of probabilities of ”simpler” sub-events. This time, however, the probabilitiesare ”complex” which means that we need to find a mechanism for this feature to arise.In order to understand this point, consider the following example. Suppose first stateconsists of electron and positron near the point A and a W -boson near the point B . Onthe other hand, the second state consists of two photons near the point A , and an electron-neutrino pair near the point B . Thus, in order to find the probability amplitude of transitionbetween the two states, we have to first ask ourselves what is the probability of annihilationof electron-positron pair near point A into two photons. We then have to ask ourselves whatis the probability of a decay of W -boson near point B , and then multiply those.Now, in light of the fact that our configuration space is descrete (which is due to finitenumber of possible ”charges” particles might have), the electrons are ”displaced” by finitedistance from the two photons they are supposedly decaying into. The same is true forthe W boson, electron and neutrino. Thus, from the ”continuum” quantum field theory,probability of either process should be strictly zero. We, therefore, assume that we are given”effective” quantum field theory according to which the probability is non-zero as long asdistances are sufficiently small. Formulating such effective theory is beyond the scope of thispaper. Here, we will just assume that we know what it is, and our only goal is to ”convert” itfrom position-based Fock space formulation into the ordinary space framework we propose.Now, the distances between photons and electron might differ from the ones betweenelectron, neutrino and W-boson. As a result, the probabilities of these processes might bedifferent as well. Suppose that electron, positron and two photons are so close to each otherthat the annihilation is ”very likely”. At the same time, the W-boson, neutrino and electronare displaced from each other sufficiently far that the decay of W -boson is ”unlikely”. Then,due to the latter, the product of the two probability amplitudes might be small. Thus, the global transition will not occur. But since we are ”not allowed” to do local transitions, the19ecay of the electron and positron will not happen either.Now, in terms of h and H the picture is the following: the h -variable does not have globalcorrelation. Thus, h for electron, positron and photon is large, while h for W -boson, electronand neutrino is small. But, at the same time, H does have global correlation. Therefore, H for both sets of particles has to be the same, and it happened to be ”small” in both cases.But, at the same time, the global correlation of H is produced through local process; justlike with ψ , the appearance of non-locality is due to the speed c s of the relevent signals.This is similar to voting for president. Suppose a person a is in favor of candidate A ,while a person b is in favor of candidate B . Suppose candidate a got elected. In this case, aperson b continues to favor candidate B but, at the same time, he obeys candidate A . Thecandidate that each person favors is h , while the candidate each person obeys is H . Thus,the fact that people a and b continue to favor different candidates indicates that h does nothave global correlation. However, the fact that they obey the same person indicates that H does, in fact have one.Now, strictly speaking both h and H are local. After all, person b ”learned” about theresults of election through radio waves, which propagate ”locally”. Besides, person b decided as an individual to obey the elected candidate A, despite the fact that he does not favorhim. After all, if every single individual choses to disobey the elected president, then therewould be no means of enforcing his ”election”. And, the decision of individuals to obey theelected president is local , although this ”local” quantity is subject to ”global correlation”.Finally, the globally-correlated quantity H (that is, a decision to obey a given person) isproduced through non-correlated quantity h (that is, voting). In the same way, in quantumfield theory the ”correlated” quantity H that represents global probability of transition isproduced through non-correlated quantities h that represent ”local” probabilities of transi-tion. Therefore, in order to formulate the mechanism of generation of H we have to do thefollowing steps:1) Formulate the mechanism of generation of h . This is, in itself, non-trivial since,as mentioned earlier, due to discreteness of Fock space the ”nearby” particles are slightlydisplaced from each other. Since our intended theory is trully local as opposed to quasi-local,we need some ”local” mechanism of ”gathering” quasi-local information into the same pointin order to produce h .2) Find a superluminal, but local , mechanism by which the correlated quantity H isgenerated once h is produced at all points. We would now like to come up with a concrete model of generating the h . As we haveestablished in the previous section, while H is global, h is quasi-local. However, h is nottruly local either. This is due to the fact that the number of possible charges is very large butfinite. Therfore, Fock space is discrete. Thus, the ”transition” between two ”local” elementsof Fock space is quasi-local, which means that h has to be quasi-local as well.20ince we would like to come up with truly local theory, we need to come up with a”local” mechanism for quasi-local h to arise. As we have indicated previously, both H and h are incoded in one particle rather than two, because each one particle has two charges.Thus, both H k and h k has something to do with transition from | q k i to | q ′ k i .Thus, both H k and h k are the functions of some information regarding the particles l satisfying q l = q ′ k . H k is a function of all such particles, throughout the universe, while h k is only a function of ”nearby” ones. Now, in order for the mechanism of production of h k tobe truly local, we need to introduce signals that would encorce it.We have already described such signals in previous section when we talked about µ j and µ ψ c . Therefore, we would like to come up with yet another µ , namely µ ρ ′ . While the theoryis similar, there are a couple of changes that need to be made. First of all, we have to take H -dependence out of the equation for the ”emitter”. Thus, our new equation is d e ρ ′ ; k dτ e = − q ′ k e ψ ′ ; k − λ e de ρ ′ ; k dτ e + g e ; αβ j αa dx β dτ e (42)Secondly, in order to make sure that our theory is quasi-local as opposed to global, we haveto make sure that µ ρ ′ dies out over a very short distance (as opposed to µ ψ c that ”circles theuniverse” multiple times). Thus, we have to introduce a mass term, where it is understoodthat the ”mass” is very large. Thus, our equation for µ ρ ′ is ∇ αs ∇ s ; α µ ρ ′ + m µ ρ ′ µ ρ ′ = N X k =1 Z dτ k ; s e ψ c ( τ k ; s ) δ ( ~x − ~x k ( τ s )) (43)The equation for ”reception” r ρ ′ is the same as before. As before, there is an importanttrick: the ”absorbtio mechanism” is based on q k while the ”emission mechanism” is basedon q ′ k . This is what allows the desired ”communication” between two different states whichwould produce h (which is a function of both of them). Anyway, after we change the lettersto suit our current purposes, the equation for r ρ ′ becomes d r ρ ′ ; k dτ r = − q k r ρ ′ ; k ( τ r ) − λ r dr ρ ′ ; k dτ r + µ ρ ′ ( x k ( τ k ; r )) (44)And, finally, as before, in order to get ρ ′ we extract amplitude from the oscillation of r ρ ′ ,which gives us ρ ′ = r q r ρ ′ + (cid:16) dr ρ ′ dτ r (cid:17) (45)Thus, while the value of ρ ′ is inside the ”receiver” particle, it represents the total density ofthe ”emittors”. In other words, we can picture that each ”point” particle produced a ”finitesize” particle around itself. Thus, a ”point” receiver-particle ”overlapses” with ”finite sized”emittor particles. ρ ′ tells us about the ”density” of emmitor particles at the location of a”point” receiver-particle.Now, the information about the behavior of ρ should give a given point particle theinformation about the distances to all the other particles. Thus, if a given particle ”knows”21ur version of quasi-local field theory, it can ”use” the local behavior of ρ in order to calcu-late quasi local information that it will then ”substitute” into the equation for h and thus”calculate” the value of h it is supposed to have.There is a bit of a difficulty, however. If a point particle is allowed to use only thevalues of ρ itself as well as its first derivatives, then this information can be ”mimiked” bymany different configurations of particles. Thus, it is possible for 10 photons to ”mimic” thetwo-photons in terms of the local densities at key points; as a result, electron and positronwill annihilate into 10 photons since they will ”think” they are annihilating into 2.In order to avoid this difficulty, we have to come up with some mechanism by which theparticle ”watches” the behavior of ρ ”over time” and analyses it to get the actual distances.After all, in light of the fact that the emission of ρ is triggered by the interaction with a -particles, and the latter keeps changing, it is nearly impossible for two different configurationsto ”mimic” each other’s behavior over time , despite the fact that it is possible for them tomimic each other at an instant .The problem, though, is that we do not want non-locality in time, just like we do notwant one in space. Therefore, we need to come up with some local mechanism by whicha particle ”remembers” what went on over time. Our goal is for that mechanism to besomething of the form d h c dt = f (cid:16) ρ ′ , dρ ′ dτ h , h c , dh c dτ h (cid:17) (46)The fact that the right hand side has h -dependence implies ”non-linearlity” which mightprovide a key for the ”memory” to occur. However, we have not found the specific way ofwriting down the function f on right hand side. The definition of f is one, and only, gap inthe theory. But this is a very important gap: until we know what f is, we can not hope toexplicitly reproduce any of the results of quantum field theory.Nevertheless, the same can be said about ”standard” quantum field theory as well: untilwe know H we can’t dream to reproduce the evolution of states. The difference between whatwe do here and the standard case is that in case of the latter H is viewed as ”fundamental”while in our case f is fundamental and H is derived from it. But, in either case, we have to”guess” what the ”fundamental” equation is, and, until we do, we can not make any kind ofpredictions.However, it is possible to discuss the interpretation of quantum field theory withoutknowing what the Hamiltonian actually is. In the same way, in this paper we can discussthe interpretational aspects of our theory without knowing what f is. Therefore, the endproduct of our work is that we will be able able to say ”the interpretation is clear, and nowget to work”. Actually finding f is left for the future research. As we have explained in section 3.1.5, there are two quantities, h and H . The former only hasquasi-local correlation, while the latter has global one. As we promised before, now that we22ave obtained h we will proceed in finding the value of H through it. We have to slow down,however, and decide whether we want to take a ”sum” or a ”product”. On the one hand,we know from quantum field theory that total Hamiltonian is an integral of Hamiltoniandensity; this suggests ”sum”. On the other hand, based on what we were saying so far, theHamiltonian represents probability. This implies ”product”.Let us, therefore, do a simple example. We have four identical particles at locations x , x , x ′ and x ′ , all of spin 0. We assume that x ′ ≈ x and x ′ ≈ x , and we also assumethat q = q and q ′ = q ′ . Thus, we are interested in transition probability from { x , x } to { x ′ , x ′ } . Now, if we denote the creation and annihilation operators on position space by φ † and φ , respectively, then we can write down Hamiltonian as H = h + h = k ( φ † ( x ′ ) φ ( x ) + φ † ( x ) φ ( x ′ )) + k ( φ † ( x ′ ) φ ( x ) + φ † ( x ) φ ( x ′ )) (47)where the difference between coefficients k and k is due to the difference between thedistances within corresponding pairs of particles (the above expression can be easilly obtainedby ”discretization” of ∂ µ φ † ∂ µ φ ). Now, it is easy to see that, up to the normalization factors, h x x | H | x ′ x ′ i = 0 ; h x x | H | x ′ x ′ i = k k (48)Thus, the ”sum” comes from the first equation, while the ”product” comes from the lastone. However, product appears only through H term. If we do take this term seriously, itwould give us problems if we consider situation with more than one kind of particle, whichwould lead to interference between different ”channels” of going from initial to final state.That term, however, can be avoided if we realize that we are only talking about proba-bility amplitudes within a small time interval δt . In fact, as we established earlier, due to thelarge but finite amount of ”charges” our configuraiton space is discrete. This implies that the”transition time” should have small but finite value. That value should be approximated,at least to the order of magnitude, based on the total amount of ”states” in fock space (or,equivalently, the number of available integer values of ”charges”). Since the latter is verylarge, δt is very small, which means that H term gives neglegeable contribution.In order for h x x | H | x ′ x ′ i = 0 not to cause a problem, we would like to have a process { x , x } → { x ′ , x } → { x ′ , x ′ } . This, however, is ”not allowed”: the first and the secondstate, respectively, imply that particles at locations x and x ′ have the same ”charge” as theparticle at x ; this, trivially, imply that x has the same charge as x ′ which means that wecan not have a state containing x that does not contain x ′ or visa versa.In order to solve this problem, we introduce another particle at x ′′ which is ”very close”to x . In other words, the distance between x and x ′′ is much smaller than the one between x and x ′ . Thus, the transition probabilities between the two are of the order of 1 despite the smallness of δt , and we can think of them as essentially the same point. Therefore, theprocess { x , x } → { x ′ , x ′′ } → { x ′ , x ′ } will have probability of the order of δt of each step,which is what we want.At first it might look troubling because the product of two transitions, each havingprobability of the order of δt , will have ”total probability” of the order of ( δt ) , which is the23xact thing we said earlier we were trying to avoid. This, however, stops being the issueonce we consider finite time intervals, say, t ≤ t ≤ t . Then the number of choices of one transition is approximately ( t − t ) /δt , which the one of two transitions is ( t − t ) / (2( δt ) .Thus, a probability that single trial will produce ”single transition” is of the order of δt ,and the number of choices of ”single trial” is of the order of 1 /δt . At the same time, theprobability that ”double trial” will produce ”double transition” is of the order of ( δt ) andthe number of different possible double trials is of the order of 1 / ( δt ) . Thus, in both casesthe product is finite.Let us now go back to the original question: is H a sum, or a product, or what?From what we have just seen, we do not care about H term. Therefore, we might as wellmake it our goal to forbit a single-step process { , } → { ′ , ′ } althogether, and only allow { , } → { ′ , ′′ } where 2 ′′ is ”very close” to 2 (see earlier discussion). Now, the way we can”forbid” things is through ”products”: the consequence of postulating a product is that H ( { , } → { ′ , ′′ } ) = h (1 → ′ ) h (2 → ′′ ) = δt × δt (49)and H ( { , } → { ′ , ′ } ) = h (1 → ′ ) h (2 → ′ ) = δt × δt = ( δt ) (50)and, since whatever is of the order of ( δt ) is inconsequential, this will, effectively, ”forbid” { , } → { ′ , ′ } . It is important to notice that ( δt ) term might come with ”wrong” coeffi-cients; in particular, this might be the case if we have many different ”channels” to go frominitial to final state, which the above equation does not take into account. That is why avery important part of the argument was our ”indifference” regarding anything of the orderof ( δt ) . This argument implies that our goal is to simply make sure that whatever is of theorder of ( δt ) continues to be of that order, while the coefficients don’t matter as long as this is the case.However, our goal is to come up with local mechanism for every non-local ingredient inour model. From the example with ψ , we have seen that the local mechanism is additive . Toour luck, we know that the logarithm of a product is a sum of logarithms. Thus, we wouldlike to use logarithms in our theory. But, unfortunately, we have already used H , instead of ln H in the previous sections. So, for the purposes of notation, we will say that˜ H = ln H (51)Again, for the purposes of notation, we will further say that the ”integral” of h produces ˜ H , not H . Thus, the ”local probability” is e h . Therefore,˜ H = X h k ; H = exp ( ˜ H ) (52)Now, if h ≈ ln δt at two different places, we would have ˜ H ≈ ln δt , which would lead to H = exp ˜ H being of the order of ( δt ) . Therefore, we would like h to be ln δt at only one place, while being finite everywhere else. The ”one” place where h is very large correspondsto x → x ′ , while all the other places correspond to x → x ′′ .Now, from the above equation we see a peculiar pattern: whenever h is ”large”, it is always approximately equal to ln δt , without any extra coefficient (unless, of course, there24s a generic one)! All the ”physics” is hidden in ”much smaller” ln k term which, uponexponentiation, becomes a coefficient that determins transition probability. This is evenmore peculiar than the constant value of ln δt , since our intuition will probably tell us todisregard ln k as ”random fluctuations”. However, as they say, ”if there is a will there is away”. Since we have not defined h , we certainly have a freedom to do so in such a way thatwe avoid ”random fluctuations” and make the ”small term” important.Our freedom of defining h is logically parallel to the freedom of defining Hamiltonianin quantum field theory. We do not need to know Hamiltonians of any fields in orderto understand the concept of ordinary QFT. The specific Hamiltonians are merely being”pluged into” that overall ”framework”. In the same way, the point of this paper is todevelop a ”framework” for the new, superluminally-local, QFT, and, later, we will ”plug” h into this ”framework”. The key difference between what we are doing here and ordinaryQFT is the presence of non-trivial transition from h to H in our case, which is what makes h , as opposed to H , an item to be ”plugged into” our framework.Now, in case of ordinary QFT they were using compactibility with an experiment as abasis of defining Hamiltonian. In our case, in light of the complexity of the theory, we wereunable so far to make a proposal for h based on an experimental data. We do, however, havesome ”clues” of how h should ”look like”. The above expression with ”large” logarithmicterm and ”small” k -term is one such clue. We do, therefore, know that when h will beproduced, it will, in some approximation, take the above form.This, of course, is a very vague statement. If h has ln -term only at some point , thencontinuity demands that there should really be a region where h looks like this. And, thereshould be a ”transition region” where h is, say, 1 / ln δt . That ”smaller” value of h shouldnot ”mess things up” since the situation is identical in ”transition regions” around all other”large regions”; thus, upon ”averaging out” all coefficients are affected in the same way. But,of course, this argument is very vague. The more precise picture would only emerge whenwe will give quantitative definition of h .As we have said in the paragraph before last, this situation is not too different fromordinary QFT in its embriotic form. We do not have enough evidence to provide equationfor h . We do, however, have some clues about its properties. And, the idea that the”intermediate” vales of h in ”transition regions” average out to the same thing is one suchproperty. In the previous section we have seen that H is, indeed, the ”product”, but ˜ H = ln H is thesum. From what we have done with ψ we see that sum, and not a product is what we havea ”mechanism” of generating. Therefore, in this section we will ”copy” what we have donefor ψ in order to obtain ˜ H and then, at the very end, we will set up H = exp ˜ H .As we have seen in the previous section, ˜ H is obtained by summing h (which the localprobability is given by e h ). We have already done one important part of the work: we25brought” quasi-local information into ”one point” through ”large-mass” signals. So, now h is defined pointwise , and our task reduces to ”summing” the ”pointwise” information. Thismeans that what we do here is even closer copy to what was done for ψ . The ”emitor”equation is the same, up to change of letters: d e ˜ H c ; k dτ e = − ω ( q k , q ′ k ) e ˜ H c ; k − λ e de ˜ H c ; k dτ e + g ˜ H ; αβ j αa h c ; k ( τ ˜ H ) dx β dτ ˜ H (53)However, the ”propagation” equation has to be somewhat different. In case of ψ , due to thefact that µ ψ is massless, it has infinite life, which causes ψ to ”accumulate” and, therefore,change in time. In case of ˜ H we do not want it to change in time. This means that we wouldlike to supply µ ˜ H with some pass in order to prevent it from living ”for too long”. At thesame time, however, we would like H to be globally-correlated.Thus, µ ˜ H has to be very small (much smaller than, say, m µ j or m µ h ). While µ j and µ h can only travel a short distance, we would like µ ˜ H to be ”light” enough to circle theuniverse several times. At the same time, we would also like it to be ”heavy” enough inorder for its lifetime to be very small (and, as said before, the two requirenments do notcontradict each other since the ”superluminal” speed c s is very large). On any event, forappropriately-chosen m µ ˜ H , the dynamics for µ ˜ H is the following: ∇ αs ∇ s ; α µ ˜ H c + m µ ˜ H = N X k =1 Z dτ k ; s e ˜ H c ; k ( τ k ; s ) δ ( ~x − ~x k ( τ s )) (54)The equation for the ”reception” of the signal is carbon copy of the ones we found earlier, d r ˜ H c ; k dτ r = − q k r ˜ H c ; k ( τ r ) − λ r dr ˜ H c ; k dτ r + µ ˜ H c ( x k ( τ k ; r )) (55)which then, by our usual trick, gives us a final answer for H c :˜ H c = s q k r H c + (cid:16) dr ˜ H c dτ r (cid:17) (56)Now that we have produced ˜ H , we convert it into H through H = exp ˜ H (57)As before, our ultimate answer should be in component form, for which we have a lot offreedom. As far as the expression of ˜ H in terms of h , there is one natural choice:˜ H a = X h k ; a , a ∈ { , − , i, − i } (58)As far as H = exp ˜ H , there is no one best choice. In this paper we will randomly decide toset H = − H − and H i = − H − i , which gives H = − H − = 12 e ˜ H − ˜ H − cos ( ˜ H i − ˜ H − i ) ; H i = − H − i = 12 e ˜ H − ˜ H − sin ( ˜ H i − ˜ H − i ) (59)26 .2 Beables (flat case)3.2.1 Encoding the information about σ We would now like to come up with local mechanism of generating the Pilot Wave modeldescribed in section 2.2. That is, we would like to be able to ”highlight” different states with B (as described in sec 2.2) and then make ”jumps” which ”translate” into us ”changing”the state we highlight. As was explained in sec 2.2, the probability that such a jump occurswithin a time interval dt is given by σ ( e, e ′ ) = ( h ψ | e ih e | H | e ′ ih e ′ | ψ i ) † |h ψ | e i| (60)where x † is equal to x when x ≥ local definition of σ . Wehave two challenges here. First of all, σ is a function of configurations, and, secondly, it is afunction of two different configurations. Both of these challenges are similar to the ones wehad to face when we were dealing with Hamiltonian. So we will use some of the tricks wehave learned there.We already know how to deal with the first challenge. In fact, we have already defined h ψ | e i at a point of our interest. Our problem is to define h ψ | e ′ i at the same point. Since | e ′ i is different from | e i , they can not possibly refer to the same collection of points. Sincethe collections of points are defined based on the same charge, q , it is clear that they do notoverlap. Thus, with original prescription, | e ′ i can not possibly be read at the same point.In other words, ψ k does not give us the information we want.What comes to our rescue, however, is a charge q ′ . Thus, we introduce a different internal degree of freedom, ψ ′ k . The trick is that the ”emitor” and ”messenger” for ψ ′ is thesame as for ψ : it is e ψ c and µ ψ c as opposed to e ψ ′ c or µ ψ ′ c . At the same time, however, themechanism of reception is based on q ′ , not q : dr ψ ′ c ; k ( τ s ) dτ s = − q ′ k r ψ ′ c ; k ( τ s ) − λ r dr ψ ′ c ; k ( τ s ) dτ s + µ ψ ′ c ( x k ( τ s )) (61)Thus, due to the fact that µψ c is the same, ψ ′ ”listens” to the same thing as one of the ψ -s.But, due to the fact that r ψ ′ c is different the ”one of the ψ -s” we have just mentioned is not h e | ψ i . Instead, it is h e ′ | ψ i which gives us the informatoin we are missing. Both ψ and ψ ′ aretaken at the same point (namely, at a particle number k ), as desired: ψ k = h ψ | s q i ; ψ ′ k = h ψ | s q ′ i (62)The other ingredient, namely the H -matrix, has been already ”converted” into a local quan-tity in the previous. For our current purposes, we can just write it symbolially as h s q | H | S q i = H k , q k = q , q ′ k = q (63)27hus, we obtain σ ( S q , S q ) = σ k ( τ ) (64)where σ k ( τ ) = Im ( ψ k ( τ ) ψ ′ k ( τ ) H k ( τ )) † | ψ k ( τ ) | . (65)This, in fact, ”encodes” the probability of transition between two states within a single particle number k . As with everything else, we have a large choice of k . The approximationwill hold for all particles with the same q and q ′ due to global correlations. In the previous section we were able to ”locally” define a ”transition probability” σ . Aswe have extensively discussed in section 2.2, σ is ”classical” probability. That is, while”quantum mechanical” probability H is complex, the ”classical” σ is both real and positive.However, we are not done yet. At this point, σ is only a desired probability; not the actualone. In order for this to be actual probability in a classical sense, we have to come upwith a deterministic process and find out that if we don’t know the initial conditions, theprobability happens to be σ .In this section we will attempt to come up with that process. We propose that the”transitions” between two states are triggered by the ”collision” between one of the particlesof interst, and an a -particle. Since both are point particles, there is no actual collision.Instead, there is an overlap with j a . Therefore, we would like to impose some ”threshold” M : if g γδ j γa dx δ /dτ is greater than M then a signal will be sent that triggers a transitionbetween the states | q k i and | q ′ k i .We would like to make sure that the probability of exceeding M is, in fact, proportionalto σ . According to the above definition of collision, that probability is proportional to the”volume” of the ”cloud” around a -particle that is defined by j -field exceeding the abovethreshold. We know that the j is produced through Laplace’s equation, ∇ βj ∇ j ; β j αa + m j a g j ; µν j µa j νa = N a X k =1 Z dτ k δ ( x − a k ( τ )) (66)and the trajectory of a -particle is given by geodesic equation, d a χ dτ a = Γ χa ; ηλ da η dτ da λ dτ (67)This might and might not lead to spherical distribution, depending on our choices of g . Inparticular, if we set g j = g s and g a = g o then this would, in fact, be spherical distribution.In other cases, we would have to count Lorentz contraction, which might complicate thecalculation of probabilities due to the fact that we are ”not allowed” to assume Boltzmanndistribution, unless we know mechanism by which it is produced.28egardless of the details of the above, however, it is clear from dimensional analysis that σ is proportional to 1 /r . Of course, due to mass, there is also e − mr factor. But we can simplyassume that the threshold is so high that in order for the information to be ”relevent”, r hasto be very small, which results in e − mr ≈
1. Now, dropping the unknown coefficient, we seethat the probability of the collision is proportional to r .Thus, in spherical case, we would like the ”threshold radius” to be proportional to σ / .Now since j is inverseley proportional to r and M is linearly proportional to j , it means that M should be proportional to σ − / . Thus, the ”emission” criteria is g s ; γδ j γa dx δ dτ s ( τ emission ) > p emit σ k ( τ emit ) (68)Let us now provide a mechanism by which the ”collision” triggers a ”transition”. As we saidbefore, the ”highlighted” particles are defined by the internal degree of freedom B . Therefore,if a ”transition” is to happen, we are to make sure that B of one group of particles changesfrom something large to zero, while B of another group of particles does the opposite. This,of course, requires a dynamics on B .We propose the following mechanism. There are two other internal degrees of freedom, A k and C k . The degree of freedom A k is responsible for ”starting” B k , while the degree offreedom C k is responsible for ”stoping” B k . Thus, the dynamics on B k is defined as follows: ∂ B K ∂τ k = A k ( τ k ) − B k ( τ k ) C k ( τ k ) (69)Now, if we want to make a ”transition” from a group of particles with charge q to the groupof particles with charge q ′ , we have to simulteneously trigger A among the particles withcharge q and, at the same time, trigger B among the particles with charge q ′ . This canbe done by a similar trick that we used in defining ψ ′ in the previous section. Namely, wewill use the same emittor and messenger fields, e AC and µ AC for two different receivers r A and r C . If these two receivers are being ”tuned” to different frequencies, then, as we havepreviousy discuss in detail, this would do a trick.Now, the key element in the argument is that the common ”emittor” has to have fre-quency q . In order to impose the above threshold, that ”emittor” has to be triggered by astep function. Thus, its dynamics is given by d e k ; AB dτ k = − q k e k ; AB − λ AB de k ; AB dτ k + step (cid:16) g s ; γδ j γa dx δ dτ s − p emit σ k ( τ ) (cid:17) (70)The dynamics of messenger is defined based on emittor by our usual formula. This time wewould like the ”messenger” to be massive since we don’t want a transition to ”continue” tooccur beyond the very short period of time after it has been ”triggered”. At the same time,we do want the signal to circle the universe since the transition is global one. As we saidbefore, due to very large value of c s it is possible for a signal to be short lived and circlethe universe. But, what it menas is that its mass should be a lot smaller than short-range29ignals ( µ j and µ h ) but at the same time much larger than long lived ones ( µ ψ ). Anyway,for appropriately chosen mass, the equation is g αβs ∇ s ; α ∇ s ; β µ AC + m µ AC µ AC = N X k =1 Z dτ s ; k e k ; AB ( τ k ) δ ( x − x k ( τ s ; k )) (71)Now, since we hae used q k instead of q ′ k in the equation for j , we are assuming that we wouldlike to make a transition from a configuration defined by q k to a configuration defined by q ′ k .Thus, if q l = q ′ k , then we would like to activate B l and de-activate B k . This can be done byactivating A l and C k . Since C k happened to have ”the same” index k , this means that theequation for the receptoin of C is based on ”the same” q : d r C ; k dτ s = − q ′ k r C ; k ( τ s ) − λ C dr C ; k dτ s + µ C ; k ( x k ( τ k ; s )) (72)Since the key of our argument is the difference between the mechanism of reception of C and A , we immediately know that the reception of A has to be based on q ′ : d r A ; k dτ s = − q k r A ; k ( τ s ) − λ A dr A ; k dτ s + µ A ; k ( x k ( τ k ; s )) (73)Finally, the actual values of A and C are given by usual means, A = r q A r + (cid:16) dA r dτ s (cid:17) ; C = r q ′ A r + (cid:16) dA r dτ s (cid:17) (74)And, as was previously stated, the above are substitutted into a dynamics for B , ∂ B K ∂τ k = A k ( τ k ) − B k ( τ k ) C k ( τ k ) (75)Now, as we explained in section 2.2, we identify ”reality” with a probability density˜ ρ k ( ~x, t ) = B ( ~x, t ) X k δ ( ~x − ~x k ) (76)Thus, the variation of B might lead to the perceived creation and annihilation of particles.The ”creation” and ”annihilation” of similar particles nearby might lead to perceived motion,and so forth. In other words, the above gives us a complete description of classical reality. We are now done talking about non-gravitational part of the theory and we are now readyto couple it to gravity. In section 2.3 we have introduced the basic concept of ”coupling”30lassical general relativity with Pilot Wave model. We have stated, however, that non-conservation of energy momentum is a major issue. Therefore, we will focus this chapter onadressing that single issue.As we said in section 2.3, we would like to find a modification to gravity in such away that it would be able to accomodate the violation of energy momentum conservation.However, we have pointed out that the fact that the Einstein’s equation in its originalform has no solution for non-conserved case, the modified Einstein’s equation has nearly-singular solution provided that its modification is ”small” (we have also explained why”large” modifications through gauge terms are not acceptabe as they lead to ”large”, albeitdevergence-less, violations of Einstein’s equation).We have pointed out, however, that in case of ordinary Einstein’s equation the conserved sources do have a solution. Thus, the only source of ”singularity” is precisely violation ofconservation of energy momentum. Thus, if we make sure that the violation of energymomentum conservation is much smaller than the ”small” correction to Einstein’s equation,this would not lead to singularity.However, we have observed that, in light of zero size of particles, any effect they haveon gravity, including the violation of conservation of energy momentum, is arbitrarily large.Thus, we need ”average out” energy momentum over sufficiently large volume in order tohave any hope that the violation of its conservation is, in fact, ”small”. Appart from that, ofcourse, they create unwanted black holes even if the energy momentum tensor was conserved.This is another reason for having to average it out.In light of this, in section 2.3 we have proposed to adress the issue in two major steps:a) Average out energy momentum tensor over a sufficiently large neighborhood, in orderto make sure that its non-conservation is very smallb) Propose modification of gravity that would have a solution for that very small viola-tion of conservationWe will, therefore, do step a in sec 3.3.2, and step b in sec 3.3.3 We will now attempt to propose a ”local” way of ”averaging” energy-momentum tensor.That is, we would like to take ”large-fluctuating” T αβ and then produce ”reasonable” ˜ T αβ .In the previous sections, we have introduced various ”messenger fields” µ for various otherreasons. Since the ”messengers” µ ρ happened to obey regular wave equation, we can easilydefine energy-momentum tensor based on them. THe latter, of course, will be singular at theemission point, but we will soon be describing our ways of ”converting” it into non-singulartensor.However, in light of the fact that gravity is supposed to be produced by ”beables”, onlythe ”highlighted” particles can emit our current version of ρ (which should be contrastedwith all previous ρ -s which are produced both by highlighted and non-highlighted particles).31hus, this time we have to multiply the ”emittor” by B .Furthermore, when we change from state to state, each new set of particles gravitates”in the same way”. In fact, we are able to ”observe” the gravity that was ”emitted” inthe past, even though back then a different state was highlighted. This means that theemission frequency has to be independent of q and q ′ , so we have to remove this part fromour equation. Thus, with the above two modifications, the ”emittor” equation is d e T ; k dτ e = − λ e de T ; k dτ e + g e ; αβ B k ( τ e ) j αa dx β dτ e (77)where we have retained the ”damping” term in order for the gravity-production event to belimitted in time. Since e ”takes care” of the above modifications, the equation of propagationof µ is similar to the one used previously, namely ∇ αs ∇ s ; α µ T + m µ T µ T = N X k =1 Z dτ k ; s e T ; k ( τ k ; s ) δ ( x − x k ( τ s )) (78)Now, since we would like to ”average” our energy-momentum tensor over ”large enough”region, m µ B has to be ”small enough” for this to happen. In particular, it must be smallerthan m µ ρ since the ”size” of a particle (which we identify as a size of a ”cloud” formed by ρ )is quantum-mechanical, while the size of ”smearing” of energy momentum tensor is classical.In fact, I would like the gravity of particles making up a large object to ”smear” so muchthat we see a ”continuous” source of gravity which, naturally, is conserved.At the same time, however, m µ B has to be large enough in order to make sure that theregion µ B fills up is small on a classical scale. Thus, the mass of µ B is unusual comparedto all the masses we took earlier. In the past, any messenger was either ”heavy” enough tobe constrained in a very small region or ”light” enough to circle the universe. This time,however, we would like an in-between version, where it is light enough to fill a macroscopicalregion as long as that region is ”small” on our scale.Now, we define the energy momentum associated with µ B by mimicking the way it isusually done for scalar fields: T αβ = ∂ α µ ρ ∂ β µ ρ − g o ; αβ g γδo ∂ γ µ ρ ∂ δ µ ρ (79)We have a problem now, since in the vicinity of a particle T αβ blows up. One thing we cando is to put an ”upper bound” on T αβ by replacing it with T ′ αβ defined as follows: T ′ αβ = M T αβ exp (cid:16) − X µν T µν (cid:17) (80)where we have violated relativity by assigning the same sign to all terms in the exponent.This violation of relativity logically relates to the fact that Lorentzian neighborhood hasinfinite volume and, therefore, relativity has to be violated in order to define a ”small”neighborhood. The reason for infinite volume of Lorentzian neighborhood has to do withdifference in sign of a metric, and this is precisely what we got rid of in the above equation.32owever, we now have a different problem. What if we want T αβ to be larger than the”upper bound” M in the above equation? One way out of this situation is to say that theupper bound is only imposed on how much one particle can contribute, since that is theultimate source of singularity. But, as long as we are summing contributions from different particles, we can have the value of T αβ as large as we want.This, however, raises a question: the physics is not ”smart”, so how can it ”know” where T αβ comes from? What comes to rescue is the emission process of the particles that is being”triggered” by various ”random parameters” (which, in our case, is collision with a -particle).If we consider the case of µ ρ field, due to its very large mass, it is highly unlikely that twodifferent particles will emit µ ρ so close in time that it would ”add”. Thus, we can safelyinterpret µ ρ as a field produced by one particle, and, therefore, impose our upper bound onit. Now, we still want the version of ˜ T αβ that is produced by several particles. We canobtain it by comming up with a dynamics that would slowly contribute to ˜ T αβ from theresources of T ′ αβ . Thus, the former corresponds to the ”integral” over some time interval. Atthe same time, that time interval can not be too large, in order to allow ˜ T αβ to change over sufficiently large periods of time. We, therefore, propose the following dynamics: ∂ ˜ T αβ ∂t = k T αβ − k ˜ T αβ (81)This dynamics violates relativity, as it singles out t as ”preferred” time direction. But, aswe said before, the violation of relativity is linked to the non-local nature of true Lorentzianneighborhood, which is defined by a light cone. If the above process went on all timelikedirections, it would, in particular, go along near-lightlike one. So by going with near-lightlikevelocity in + t/ + x direction, and then ”comming back” at + t/ − x direction, we would havearbitrary large effects arbitrary far in time. This means that if we did want to allow multipletime directions, we would be forced to impose a ”cut off”. That cut off would violaterelativity. So, since we have to violate relativity anyhow, we might as well just consider one time direction, for the sake of simplicity.Let us now go back to the details of the model. In order to understand how the ”ab-sorbtion” mechanism works, imagine that we have started out from ˜ T ( −∞ ) = 0. Then, at atime t , we have a δ -function ”pulse”, T αβ = k f αβ ( t ) δ ( t − t ). The evolution equation for˜ T tells us that ˜ T αβ ( t − ) = 0 and ˜ T αβ ( t +1 ) = k f αβ . Then, due to the ”decay” process, at thetime t its value will become ˜ T αβ ( t ) = k f αβ e − k ( t − t ) . Since the above picture is clearlylinear, if we have a continuous source f αβ , the equation becomes˜ T αβ ( t ) = k Z f αβ e − k ( t − t ) dt (82)Now, since we postulated that T αβ is a source of excitation of ˜ T αβ , we can safely replace f with T , which gives us ˜ T αβ ( t ) = k Z T αβ e − k ( t − t ) dt (83)33hus, if k is sufficiently small, this, in fact, becomes an integral over large enough timeinterval in order for different contributions to ˜ T to ”average out”. Now, as was said earlier,each individual contribution has an upper bound, but, due to the fact that they come atdifferent times, they ”add up” to something unbounded. As we said before, the ”upperbound” of individual portions is enforced by replacing T with T ′ :˜ T αβ ( t ) = k Z T ′ αβ e − k ( t − t ) dt (84)where, as we said before, T ′ αβ = M T αβ exp (cid:16) − X µν T µν (cid:17) (85)As was previously stated, we are forced to violate relativity by having the same sign inexponent, as a result of lightcone singularities that come with any Lorentz-covariant neigh-borhoods.However, we never claimed that our theory is relativistic; we only claimed that it is”local”. In fact, we have already violated relativity by having multiple speeds of light, whichcan be ”combined” to produce a single preferred frame. In neither case, the locality of atheory did not suffer. In case of c s we claimed the locality was preserved because c s is stillfinite, even though very large. In case of preferred time direction, we have even strongercase: we don’t even envoke ”very large” velocities either. Thus, we do not view either of theabove as failures of the theory.What might be the issue, however, is the possibility that integral of exponent has stochas-tic jumps. Regardless of how ”small” they might be, that would be a problem since it wouldrequire us to come up with ”another” averaging procedure, which would effectively meanbegging the question. At this point I do not have rigourous proof one way or the other, butI hope that such is not the case. We know from linear algebra that the existence of soutions and their uniqueness are equiv-alent: after all, both are linked to the determinant of the matrix being non-zero. In thecontext of differential equations, ”uniqueness” is equivalent to lack of gauge freedom. Thus,by adding a term that would break gauge invariance we will restore ”uniqueness” and, there-fore, ”existence”, which means that we will have solutions for non-conserved sources, asdesired.However, while breaking guage invariance might be ”necessery”, not every way of doingso is ”sufficient”. After all, it is possible that we break invariance that we are familiar within favor of some other one, which we won’t recognize. Thus, we need some other means ofshowing whether or not Einstein’s equation does, in fact, have solution. One way of doingthat is selecting a ”preferred” time axis t and moving all of the ∂ t terms on the left side ofthe equation, while leaving ∂ t , ∂ x , ∂ t ∂ x , and so forth, on the right.34f we think of t as the only ”dynamical variable” describing the evolution of the metricas well as all of its first and second derivatives apart from ∂ t ∂ t , then we get a familiar ”secondNewton’s law” problem, and the dependence on ∂ t and ∂ t ∂ x can be simply interpretted as”viscocity”. Thus, the existence or absence of the solutions can be easilly shown based onwhether or not the determinant of coefficient matrix is zero.It is important to point out that t -axis is not a geodesic. After all, suppose two timelikegeodesics, γ and γ intersect at two points, p µ and q µ , where q µ is ”to the future” of p µ . Now,if we know ”initial metric and its derivatives” at p µ , we can determine the ”final metric andits derivatives” at q µ by two different methods: we can either track the evolution of metricalong γ , or along γ . It is easy to see that if, in both cases, the evolution is completelydetermined by Einstein’s equation, we might as well be ”forced” to get different results,unless metric obeys some restrictions to begin with, which, of course, begs the question.On the other hand, if we think of t as a ”flat” coordinate, independent of a metric, thenwe are guaranteed that the two lines along t -axis would not intersect, which would removethe ambiguity from our result. While this contradicts the spirit of general relativity, the endresult is ”covariant”. After all any dynamical equation is either invertable or it isn’t. Theanswer to this question should be the same, regardless of the ”coordinate frame” we have”chosen” to answer this.Since Einstein’s equation for arbitrary metric is too complicated to analyze, we willconsider a linear approximation. Within this approximation we will first convince ourselvesthat the equation is not invertible, and then find a way to ”modify” it to make it such. Thelinear approximation to Einstein’s equation is R µν − Rg µν = − ∂ α ∂ α h µν + 12 ∂ µ ∂ α h αν + 12 ∂ ν ∂ α h αµ − η µν ∂ α ∂ β h αβ − ∂ µ ∂ ν h + 12 η µν ∂ α ∂ α h (86)We will now compute the above expression for all choices of ( µν ). However, we will leaveout all terms except for the second time derivatives. We will denote the expressions not containing second time derivatives by dots. Thus, we get R − Rg = · · · ; R − Rg = 12 ( ∂ ∂ h + ∂ ∂ h ) + · · · R − Rg = · · · ; R − Rg = − ∂ ∂ h + · · · (87)The rest of the components are obvious from the permutting indeces. Now, in order to finda dynamics, we have to solve the above equation for ∂ ∂ h µν . Thus, in order to see whetheror not the equation is solvable, we have to compute a determinant of corresponding 10 × v , · · · , v ) = ∂ ∂ ( h , h , h , h , h , h , h , h , h , h ) (88)In these coordinates, our equation takes the form A~v = ~w , where A = 12 diag (0 × , B × , × , − I × ) (89)35nd B = (90)As a result of 0 × and 0 × components of a diagonal, the determinant is zero. This confirmsour earlier prediction that Einstein’s equation, as it stands, is not convertible. However, dueto the fact that the determinant of B is 2, it is easy to see that the problem can be cured byreplacing A with A + ǫI × . After all, this would replace 0 × and 0 × with ǫI × and ǫI × ,both of which have non-zero determinants. At the same time, by continuity, B × + ǫI × will have determinant close to 2 and, therefore, also non-zero.Now, adding this extra term is equivalent to adding ǫ∂ ∂ h µν to our equations. Sincewe would like to have a covariant theory, we instead add a term ǫ∂ ∂ α ∂ α h µν . Furthermore,we notice that derivatives of η µν are 0 and, therefore, we can freely replace ǫ∂ α ∂ α h µν with ǫ∂ α ∂ α ( η µν + h µν ). Finally, in order to generalize our equation to arbitrary metric, we replace η µν + h µν with g µν , which gives us R µν − Rg µν + ǫ∂ α ∂ α g µν = T µν (91)It is important to notice that, in light of ∂ α being used in place of ∇ α , the ǫ -term in theabove equation is not covariant. In fact, in light of the fact that ∇ α g µν = 0, we couldn’t havepossibly made it covariant even if we wanted to. But, like was mentioned earlier, violation ofcovariance is one of the necessary steps of turning non-invertible equation into the invertibleone. This expression corresponds to a Lagrangian S = Z d x √− g (cid:16) R + kT + ǫ ∂ α g µν ∂ α g µν (cid:17) (92)where ǫ -term likewise breaks the general relativistic covariance.It is important to admit, however, that in the above analysis we were working exclusivelyin the linear approximation. For the general metric, it is quite possibe for determinant to”accidentally” become zero. In fact, if there are two timelike-separated points, p µ and q µ ,and the determinant has opposite sign at these two points, then any curve γ that connectsthem will have some point r γ at which the determinant is 0. If we now alter the curve γ , thenthe collection of such points will form a surface, and the determinant will be zero everywhere on that surface!The good part is that the surface is ”infinitely smaller” than anything three-dimensional.The bad news, however, is that in a small-but-finite vicinity of a surface, the second timederivatives of gravitational field will undergo unwanted large-but-finite variation. This willhave large unwanted impact throughout the three-dimensional region ”to the future” of thatsurface. Thus, in order to avoid this, we have to ”tame” the behavior of time derivativessomehow.As a ”quick fix”, we are going to single out ”preferred” t -direction since, for the purposesof avoiding singularity, it only necessary to tame ∂ t ∂ t -terms. When we first introduced36preferred frame” few paragraphs earlier, we have stated that the choice of that frame hasno impact on our conclusion of whether or not a differential equation is convertible. However,now that we are trying to do modification of an equation in our ”preferred frame”, this does lead to a prediction of different results depending on that frame; although, of course, thedifference will be of the order of magnitude of ”small correction” we attempt to introduceand, therefore, undetectable.Of course, for the future, it is important to explore more covariant ways of modifyinggravity. But, just for the purposes of this paper, we will settle on violating relativity; ourexcuse being that we have already done that anyway when we have introduced Pilot Wavemodel. As we said before, we can write the second time derivatives in a form of a singlevector ~v ∈ R , which allows us to write the dynamics as ~v = A − ~w , for the tensor A specifiedearlier. We will now replace it with ~v = (cid:16) tan − ( ξ det A − ) ξ det A − (cid:17) / A − ~w (93)where ξ is a very small number. In light of the smallness of δ , the overall coefficient is veryclose to 1 as long as the determinant of A − does not blow up. However, once it does, thecoefficient approaches zero. It is easy to see that the determinant of the ”modified” versionof A − approaches π/ δ , which is small but finite. Thus, if the thickness of a ”problematic”region is much smaller then ξ , it does not have any detectible effects.In fact, we can do ”even better” by removing 1 /
10 from the power of the above coefficient.In this case, near the singular region the second time derivatives will approach zero, which isjust the opposite of the issue we were trying to avoid! We have to be careful if we want to dothat, because this might potentially ”freeze” the singularity once it is reached. What comesto our rescue is that the first time derivatives continue to be non-zero. The space variationof the latter will eventually allow us to ”get out” of the singular region. But, of course, thisneeds to be investigated more closely.
4. Conclusion
In this paper we have proposed a model in which quantum phenomena are reproduced bymeans of superluminal, but finite, speed of signals. The difference between”superluminal”and ”trully infinite” is a key idea of the model. While we embrace the former, we reject thelatter. Thus, we get all the ”benefits” of quantum non-locality, without having to pay theprice of a very counter-intuitive notions.The key idea of this paper is that there is no such thing as ”configuration space”. Instead,the same-charge particles are so strongly correlated that, up to a very good approximation,they act like a single point in a configuration space. That correlation is due to them com-municating signals with each other, with emission and absorbtion frequencies correspondingto their charges.The letter ψ , which usually denotes probability amplitude, is now viewed as the internaldegree of excitation of particles. But, due to strong correlation, the dynamics forces ψ to be37early the same among particles with the same ”charge”. As a result, we wrongly interpret ψ as a parameter on configuration space or Fock space.According to proposed model, the particles do not get greated or annihilated. When wemake a ”switch” from one state to another, what happens is that some partiles stop beinghighlighted while others begin to be. They are ”highlighted” through another internal degreeof freedom, B , which is also subject to similar global correlations. The values of B defineobserved reality, as particles with nearly-zero B are ”invisible”.Finally, gravity is introduced as a ”classical” field that is coupled to the end product ofthe above process (namely B ). Until B is produced, the only role g µν plays is defining a fixed geometrical background for the non-gravitational processes. Then, after B is produced, g ”looks” at it and ”decides” how it wants to alter; its aterations are, of course, identified withthe ”gravitatioal effects” of B .However, the energy-momentum ”source” produced by B happened not to be conserved.Thus, there is no exact solution of Einstein’s equation with that source. Since we would liketo view g µν as classical, we would like it to be exact solution of some equation. Thus, wehave modified Einstein’s equation by adding to it small kinetic term. While we have notproven the existence of solution in a general case, we have at least made a good argument infavor of solution being available in most ”regular” cases. We then attempted some maneursof getting around the problems in more exotic situations.This model should be compared to an earlier attempt of comming up with superluminallocality, [7] (since I happened to be the author of that attempt as well, I would like totake a liberty and criticize my own work). In that other attempt, instead of introducingcorrelations between particles, I have introduced correlations between regions of space intowhich particles happened to fall. I believe that was a lot less elegant than the currentwork. After all, back then I had to introduce oscillations of points in space not occupied byparticles. This amouts to a lot of extra unnecessary information, which fails Ocam’s razor.Apart from this, the ”structure” of the ”regions” introduced in [7] was very complicated.Thus, it would be difficult to combine that model with gravity since the latter is likely to”disturb” a very complicated structure that was employed. While the current approachalso has some problems when it comes to combining it with gravity, these problems are allunavoidable, as compared to the extra complications introduced in the previous approachthat could have easilly been avoided.The gravitational part of this work should be compared with [8] (which was also authoredby me). In that paper we have also attempted to describe the production of gravity by non-conserved sources, for the same reasons as we did it here. However, again, the approach ofthat work was a lot less natural than what we are doing now. In [8] we came up with amodel where gravitational field performs ”trial and error” method as it attempts different”changes” that would bring it closer to being a solution of Einstein’s equation. Even thoughI successfully described trial-and-error through a set of differential equations, it still feels abit artificial. On the other hand, in the current paper it was done in a lot more natural way,through 38 eferences [1] D. D¨urr, E. Rosinger`ı “Further on Pilot Wave model” arXiv:quant-ph/0910.0344 [2] D. D¨urr, A. Valentini `ı “Beyond the quantum” arXiv:quant-ph/0910.0344 [3] D. D¨urr, S. Goldstein, R. Tumulka and N. Zangh`ı “Trajectories and particle creationand annihilation in quantum field theory” J. Phys. A: Math. Gen. (2003) 4143-4149,and arXiv:quant-ph/0208072 .[4] W. Struyve and H. Westman 2007: “A minimalist pilot-wave model for quantum elec-trodynamics” arXiv:0707.3487v2 [5] S. Colin and W. Struyve “A Dirac sea pilot-wave model for quantum field theory” arXiv:quant-ph/0701085 [6] R. Sverdlov, ”Pilot Wave model that includes creation and annihilation of particles”(2010) ( arXiv:1011.3039)[7] R. Sverdlov, "Completely local interpretation of quantum field theory"(2010) (arXiv:1004.2933)[8] R. Sverdlov, "Can gravity be added to Pilot Wave models?" (2010) (arXiv:1010.0580)arXiv:1011.3039)[7] R. Sverdlov, "Completely local interpretation of quantum field theory"(2010) (arXiv:1004.2933)[8] R. Sverdlov, "Can gravity be added to Pilot Wave models?" (2010) (arXiv:1010.0580)