aa r X i v : . [ qu a n t - ph ] D ec Foundations of Physics manuscript No. (will be inserted by the editor)
K.B. Wharton
A novel interpretation of theKlein-Gordon equation
Received: date / Accepted: date
Abstract
The covariant Klein-Gordon equation requires twice the bound-ary conditions of the Schr¨odinger equation and does not have an acceptedsingle-particle interpretation. Instead of interpreting its solution as a proba-bility wave determined by an initial boundary condition, this paper considersthe possibility that the solutions are determined by both an initial and a finalboundary condition. By constructing an invariant joint probability distribu-tion from the size of the solution space, it is shown that the usual mea-surement probabilities can nearly be recovered in the non-relativistic limit,provided that neither boundary constrains the energy to a precision near ¯ h/t (where t is the time duration between the boundary conditions). Otherwise,deviations from standard quantum mechanics are predicted. Keywords
Relativistic Quantum Mechanics · Classical Fields · QuantumFoundations
PACS · Because the Klein-Gordon equation (KGE) for a classical, scalar field wasthe conceptual precursor to Schr¨odinger’s “derivation” of the Schr¨odingerEquation (SE), there is a widely-held misconception that the SE must be the
Ken WhartonDepartment of Physics and AstronomySan Jos´e State University, San Jos´e, CA 95192-0106E-mail: [email protected] non-relativistic limit of the KGE. In fact, a second-order (in time) differen-tial equation like the KGE does not reduce to a single first-order differentialequation (like the SE) in any limit. One can reduce the KGE to a single SEonly by artificially discarding half of the solutions – the so-called “negative-energy” solutions that evolve like exp (+ iωt ). This may seem reasonable whenconsidered in the framework of standard, non-relativistic quantum mechanics(NRQM), but NRQM must be a limit of a generally covariant theory wheresuch a step is not natural. Indeed, this historical division between two arbi-trary halves of the solution has created severe problems for quantum gravity,for in curved space-time there is no well-defined way to separate the positive-and negative-energy terms [1].This paper motivates and analyzes a novel interpretation of the KGE(corresponding to a neutral, spinless particle) that promises to reproducethe predictions of standard NRQM without arbitrarily discarding any of thesolutions. Efforts to give the KGE a probabilistic interpretation analogousto the SE have historically led to two problems: apparent negative-energysolutions and no positive-definite probability four-current. Several recent pa-pers have addressed these issues [2][3][4][5], but there is not yet any acceptedresolution. Both of these problems arise from the presumption that any in-terpretation of the KGE must be directly analogous to the standard interpre-tation of the SE. However, such a direct analogy is inherently unlikely, dueto the mathematical disconnect between first-order and second-order differ-ential equations. Specifically, while the full solution to the SE requires onlyan initial boundary condition ψ ( t = 0), the full solution to the KGE requirestwo independent initial boundary conditions: both φ ( t = 0) and the first timederivative ˙ φ ( t = 0). Without some expanded measurement theory to explainhow to simultaneously impose independent boundary conditions on both φ and ˙ φ , one cannot even solve the KGE, let alone interpret the solutions.But if there was some way to impose additional initial boundary con-ditions (outside the scope of standard QM), then that extra informationshould allow for experimental predictions that surpass QM. A century ofexperimentation has not accomplished this feat, providing strong evidencethat one cannot physically impose the necessary initial conditions to solve See section 6 for a related discussion. the KGE. In other words, for a generic (neutral, spinless) particle, the maxi-mum amount of knowable information can be encoded by the instantaneousvalues of a single complex scalar field ψ – only half as much informationas the instantaneous values of the φ and ˙ φ fields needed to solve the KGEfor a complex scalar field. This “half-knowledge” situation is curiously simi-lar to the axiomatic foundation of Spekkens’s interesting toy model [6], butstill leaves the unanswered question: how can one calculate the solutions toa second-order differential equation like the KGE without sufficient initialboundary conditions?I propose that a natural resolution to this dilemma can be found by con-sidering time-symmetric approaches to NRQM; namely, those in which oneimposes two boundary conditions at two different times corresponding to apreparation and a measurement [7][8][9]. If only half of the required infor-mation to solve the KGE can be imposed as an initial boundary condition,then a subsequent measurement can still impose the remaining constraintsas a final boundary condition. This approach can also be extended to curvedspace-time by imposing a closed hypersurface boundary condition on clas-sical fields (similar to the closed boundaries imposed on quantum fields inrecent work by Oeckl [10][11]).To compare this approach to NRQM, one simply constrains the KGEby imposing two boundary conditions on two different instantaneous hyper-surfaces (corresponding to consecutive external interactions/measurements).The solution to the KGE can then be “retrodicted” in the space-time volumebetween the two measurements. Because the full solution cannot be deter-mined by the initial measurement alone, one is forced into a probabilisticdescription, weighting the solution space to determine the relative likelihoodof different pairs of measurements. After the final measurement, when theadditional boundary conditions become known, such a probabilistic interpre-tation is no longer necessary. The central result of this paper is to demonstratethe existence of a relativistically invariant weighting scheme that, in the non-relativistic (NR) limit, gives probabilities very similar (but not identical) toNRQM. Furthermore, the historical problems with the KGE are solved bythis approach: all possible solutions to the KGE represent positive energy (as is already the case for a classical scalar field), and the probabilities arealways positive definite.This paper is organized as follows: Section 2 solves the two-boundaryKlein-Gordon equation for generic time-even boundary conditions, circum-venting the infinite poles in the same manner as in quantum field theory. Sec-tion 3 then motivates an invariant joint probability distribution, and demon-strates that it is nearly identical to standard conditional probabilities in theNR limit (given other constraints that are derived in an appendix). Section 4addresses time-odd, multiple, and incomplete measurements. Section 5 exam-ines the constraints from the appendix, and finds that they are closely relatedto a time-energy uncertainty principle. Section 6 summarizes the postulatesused in this approach, details some of the next steps required to extend thisresearch program, and touches on other implications. In the absence of a potential, the Klein-Gordon equation on a complex scalarfield φ (and its solutions) can be written (cid:18) ∂ ∂t − c ∇ + m c ¯ h (cid:19) φ = 0 , (1) φ ( r , t ) = Z a ( k ) e i ( k · r − ωt ) + b ( k ) e i ( k · r + ωt ) dk . (2)Here a ( k ) and b ( k ) are complex functions, and ω is assumed to be a positivefunction of k ; ω ( k ) = q c k + m c / ¯ h .Solutions of the form (2) have exactly twice the independent parametersas the solutions to the Schr¨odinger Equation (for which the b ( k ) terms are allidentically zero). The analysis from the previous section can equally well beapplied to this doubled parameter space: If an initial preparation constraintcan only specify parameters equivalent to a ( k ) (a single complex function),then specifying the two independent complex functions in the KGE solutionsmust require twice the constraints. Placing a second constraint at the nextmeasurement is independently motivated by CPT-symmetry [9], but thissymmetry implies both boundaries should be mathematically equivalent –not a strict initial boundary followed by some discontinuous “projection” atthe final boundary. There have been several prior efforts to impose two-time boundary con-ditions in NRQM without doubling the parameter space of the Schr¨odingerequation [12][13]; these efforts have revealed that the equations generally be-come overconstrained. In light of this difficulty, other two-boundary effortshave used a doubled-parameter space [7,8,14], where the ordinary quantumwavefunction ψ is constrained by an initial boundary and some other wave-function is constrained by a final boundary. In the present work, no artificialdivision is made between any two halves of the full Klein-Gordon field. (Re-call, it is this arbitrary splitting into so-called positive-energy and negative-energy solutions that causes trouble in curved space-time.) Instead, this ap-proach uses only one field, φ , all of which is constrained by both an initialand a final boundary. (Using this same approach, Rovelli has calculated theaction of a classical scalar field [15], but no direct use of this result has beenmade except for a noted similarity to the quantum field propagator.)In order to impose boundary conditions, it is convenient to use conven-tional measurement theory from NRQM, where a measurement constrains φ to be an eigenfunction of some operator at a particular time (on a space-likehypersurface s ). This permits a boundary to be imposed on, say, φ ( r , t = 0),but only in the reference frame somehow defined by the measurement it-self. (This will eventually have to be generalized, as discussed in section6.) Another complication is that the momentum operator in physical space P → − i ¯ h ∇ has been constructed for the SE, not the KGE. Applying it tosolutions of the form (2) leads to unphysical results, because the b ( k ) termsphysically propagate in the opposite direction as determined by the eigenval-ues of P . This is a general problem for any time-odd operator applied to theKGE. A natural solution will be presented in section 4; for now the analysisis simply restricted to time-even measurement operators such as X and P .Suppose the eigenfunction of the initial time-even measurement operatoron the hypersurface t = 0 is f ( r ), and the eigenfunction of the final time-evenmeasurement operator on the hypersurface t = t is g ( r ). (These eigenfunc-tions are determined via measurements in precisely the same manner as instandard NRQM.) Fourier-expanding (and ignoring the 2 π ’s, which will beautomatically incorporated into a later normalization) leads to the general boundary conditions φ ( r ,
0) = f ( r ) = Z F ( k ) e i k · r dk , (3) φ ( r , t ) = g ( r ) = Z G ( k ) e i k · r dk . (4)Comparing these equations to (2), the uniqueness of the Fourier transformimplies a ( k ) + b ( k ) = F ( k ) , (5) a ( k ) e − iωt + b ( k ) e iωt = G ( k ) . (6)This is the correct number of equations to solve for the coefficients a ( k )and b ( k ), but there is a problem for the particular values of k at which ω ( k n ) t = nπ . At these discrete values, solving (5) and (6) yield expressionsfor a ( k ) and b ( k ) with infinite poles (for arbitrary boundaries F and G ).But quantum field theory is no stranger to infinite poles – the propagatorfor the KGE contains a term ( E − p c − m c ) − which blows up on themass shell. The standard prescription for solving this problem is to make theassignment m → m − iǫ . After performing the integral, one takes the limit ǫ →
0. Because this is known to give acceptable results, it seems reasonable touse precisely the same method here. Implementing this change in m changesthe solutions to the KGE; φ ( r , t ) = Z ∞−∞ a ( k ) e i ( k · r − ωt ) e − ǫt + b ( k ) e i ( k · r + ωt ) e ǫt dk . (7)where, as usual, all positive constants are absorbed into ǫ itself. (A 1 /ω termhas also been absorbed into ǫ , which is approximately constant in a NRapproximation.) One concern raised by the form of (7) is that it appears todiverge as t → ±∞ . But recall that we are imposing both an initial and afinal boundary condition, so between these boundaries φ remains finite (aswe eventually will take ǫ → a ( k ) + b ( k ) = F ( k ) , (8) a ( k ) e − iωt e − ǫt + b ( k ) e iωt e ǫt = G ( k ) . (9) These are now exactly solvable for any two generic boundary conditions,yielding a ( k ) = F ( k ) e iωt e ǫt − G ( k ) e iωt e ǫt − e − iωt e − ǫt , (10) b ( k ) = G ( k ) − F ( k ) e − iωt e − ǫt e iωt e ǫt − e − iωt e − ǫt . (11)Notice that the addition of the small quantity ǫ now prevents the poles at ω ( k n ) t = nπ from diverging.Before proceeding, it is worth noting that this technique has already man-aged the feat of finding a continuous field φ ( r , t ) everywhere between any pairof time-even measurements. No “projection” is needed to get from one exactmeasurement to the next; one simply uses (10) and (11) to solve (7) exactly.Still, this cannot be done until after the second boundary becomes known,so this is a retrodiction, not a prediction. Also, some of the coefficients stilldiverge as ǫ →
0, making it difficult to ascribe any “reality” to φ . Variousphysical motivations for a non-zero ǫ will be discussed in section 6. For now,one can take the already-common “agnostic” viewpoint, treating φ as simplya mathematical tool for calculating measurement probabilities. In the nextsection, it is shown that this can be accomplished even if ǫ → Given a two-time-boundary framework, the use of conditional probabilityneeds to be revisited. In standard NRQM, some initial measurement ( Q F , F )at time t =0 determines the state/wavefunction ψ ( t = 0). (In this notation, Q F is the experimenter’s choice of measurement, and F is the particularoutcome.) Evolving ψ to the time t of the next measurement ( Q G , G ), onetraditionally uses the conditional probability P ( G | ψ ( t ) , Q G ). But this time-directed process simply does not work for the KGE because ( Q F , F ) is notsufficient to determine the solution φ . Furthermore, this paper speculates that( Q G , G ) determines φ ( t ), so one could hardly have a conditional probabilityof G in turn determined by φ ( t ).In this framework, one needs a time-symmetric implementation of proba-bility, closer in spirit to scattering amplitudes in quantum field theory. Such a symmetry can be naturally achieved by treating joint probability distri-butions (JPDs) as the fundamental quantities from which conditional prob-abilities can be derived. This approach assigns a joint probability to everypossible initial/final measurement outcome pair F i , G j and the time betweenthem ∆t = t . If there was some JPD given by J ( F i , G j , ∆t ) associated witheach possible pair of outcomes (for a given pair of measurement operators Q F , Q G ) one could generate any particular conditional probability P ( G | F , ∆t )via the standard normalization procedure: J ( F , G , ∆t ) / P j J ( F , G j , ∆t ).For operators with continuous eigenvalues, the sum becomes an integral overall eigenstates of the chosen operator Q G . Standard zero-potential probabil-ities in NRQM can be recovered from the JPD J ( F, G, t ) = (cid:12)(cid:12)(cid:12)(cid:12)Z F ( k ) G ∗ ( k ) e − iωt dk (cid:12)(cid:12)(cid:12)(cid:12) . (12)For any given F and t , the conditional probability of measuring anyfunction G is given by this same expression, P ( G | F, t ) = J ( F, G, t ), if oneconstrains the magnitude of F and G such that R | F | dk = R | G | dk = 1.(Note these constraints are simply chosen for familiarity, and should not beinterpreted as a separate normalization; changing the “1” to any constantwould have no effect, due to the normalization in the standard J → P proce-dure described in the previous paragraph.) Of course, the above expressionfor J is not relativistically invariant. The challenge for any relativistic ex-tension of quantum mechanics is to devise an invariant JPD that is equal to J in the appropriate limit.Note that JPDs are not probability densities, so the conditional proba-bilities derived from them need not satisfy a continuity equation. Indeed, totalk about the probabilities of a measurement at a time between t = 0 and t = t is a contradiction in terms, because the next measurement happens at t = t by definition . A similar conclusion was reached in Oeckl’s recent workon “general boundary” quantum field theory [10].In relativistic quantum mechanics, the charge density ρ ( r , t ) cannot rep-resent probability because it can be negative. Still, because of the usefulnessof this density in relativistic quantum mechanics, this is a reasonable placeto begin searching for an appropriate JPD. For the Klein-Gordon equation, the charge density is given by ρ ( r , t ) = i ¯ h mc (cid:18) φ ∗ ∂φ∂t − φ ∂φ ∗ ∂t (cid:19) , (13)and is real, although not invariant (it is one component of a four-current). Itis not possible to make ρ invariant without some reference unit four-vectorthat defines the time direction.Fortunately, the boundary conditions define two hypersurfaces, s and s . These surfaces have associated inward-pointing normal four-vectors η and η (pointing forward in time from the initial boundary and backward intime from the final boundary). In the limit comparable to NRQM, s and s are the hyperplanes t = 0 and t = t , so it is plausible that η is the neededreference unit vector, but of course it is only defined on the hypersurfaces.Because of this constraint, ρ is not well-defined in the volume between theboundaries – but one can still integrate ρ along the hypersurfaces themselves.So, generalizing to any closed hypersurface defined by the two boundaryconditions (e.g. the two infinite planes t = 0 and t = t ), the hypersurfaceintegral W = ¯ hmc I s ,s Im ( φη µ ∂ µ φ ∗ ) ds (14)is a scalar that is plausibly related to probability, motivated both by in-variance and known results from relativistic quantum mechanics. (In thisnotation, η µ is the inward-pointing four-vector unit normal to the integra-tion surface and ∂ µ is the four-gradient with ∂ = ∂/∂ ( ct ). Summation overthe index µ is implied.)In the special case that the closed hypersurface is defined by the planes t = 0 and t = t , W can be evaluated using (7). (Expand both φ and φ ∗ asintegrals in k and k ′ , and then the spatial integral yields 2 πδ ( k − k ′ ), leavingonly a single integral in k .) Dropping the overall constants, the simplifiedresult is W = Z ω | a | (cid:0) − e − ǫt (cid:1) − ω | b | (cid:0) − e ǫt (cid:1) − Im (cid:2) ǫab ∗ (cid:0) − e − iωt (cid:1)(cid:3) dk . (15) Next, a ( k ) and b ( k ) can be written in terms of the boundary conditions F ( k )and G ( k ) using (10) and (11). To lowest surviving orders in ǫ , this yields W = Z ωǫt (cid:2) | F | + | G | (cid:3) − ωǫt Re [ F G ∗ ] cos ( ωt ) + ǫsin ( ωt )( ... ) ǫ t + sin ( ωt ) dk . (16)This does not vanish as ǫ → ǫ → ǫt ǫ t + sin ( ωt ) = ∞ X n =0 π δ ( ωt − nπ ) . (17)The sin ( ωt ) term in the numerator of (16) was not fully shown because itis odd over the delta function, and vanishes; the cos ( ωt ) term is either 1 or − n . Expanding dk in spherical k -coordinates ( k = | k | ), onecan use the relationship k dk = ω dω to reduce the radial integral to a sum,leaving only a 2-D integral; W = Z ∞ X n = n − n ω n k n t (cid:0) | F n | + | G n | − Re [ F n G ∗ n ] (cid:1) sinθ k dθ k dφ k . (18)In this notation F n = F ( k n ), G n = G ( k n ), and k n = | k n | , where k n is thesolution to ω n ( k n ) = nπ/t , and n is the smallest value of n for which thereis such a solution.I now propose that an invariant joint probability distribution can be con-structed from W according to J = ( W max − W min ) . (19)Here W max is the maximum value W can attain when varying any uncon-strained parameters; similarly W min is the minimum value. Such a rangeparameter is common in probability theory: one way to interpret (19) is thatany pair of boundary conditions permits a two-dimensional space of solutions,with the allowed range in each dimension determined by allowed values of W . Picking a solution at random, particular pairs of boundaries that havea larger solution space would be more likely than other pairs with a smallersolution space.Still, W has no range at all if the parameters F , G , and t are all exactlyspecified. But regardless of the precision of the measurements, standard mea-surement theory always allows for one free parameter: the unknown relative phase between the two eigenfunctions f ( r ) and g ( r ). This relative overallphase, θ , does not affect the first two terms in (18), so for complete measure-ments these terms will not contribute to any calculation of J using (19). J must match the standard JPD in the NR limit (12), but J is given by a3-D integral over k-space, not a 2-D integral and a discrete sum. Still, the sumin (18) approximates an integral if neither F ( k ) nor G ( k ) change very rapidlywhen the magnitude | k | changes by k n +1 − k n ; the precise conditions arederived in the appendix and discussed in section 5. Given these constraints,the appendix demonstrates that the third term in (18) can be excellentlyapproximated as Z ∞ X n = n − n k n t Re [ F n G ∗ n ] sinθ k dθ k dφ k ≈ ¯ h m Z Re [ F ( k ) G ∗ ( k )] cos ( ωt ) dk . (20)To examine whether or not J ≈ J , one must now use the fact thatboth f ( r ) and g ( r ) were originally eigenfunctions of time-even operators(time-odd operators are considered in the next section). The eigenfunctionsof non-degenerate time-even operators can be constrained to be entirely realif their relative phase exp ( iθ ) is added explicitly [16]. (For the degeneratecase, there is no relative phase information, so one is still free to choosereal combinations of the real eigenfunctions.) So, forcing f ( r ) and g ( r ) tobe real by adding the relative overall phase exp ( iθ ), the approximation (20)simplifies the JPD (19) into the expression J = (cid:18)Z Re [ F ( k ) G ∗ ( k ) e iθ ] cos ( ωt ) dk (cid:19) max . (21)This assumes that one is in the NR limit such that the ω n in (18) is roughlyconstant and can be pulled out of the integral (overall constants are irrelevantto the unnormalized JPD). Given the complete freedom of the unmeasurablequantity θ , (21) implies W max = − W min , so the range W max − W min is just2 W max .Additional simplifications result from f ( r ) and g ( r ) being real; both Im [ F ( k ) G ∗ ( k )] cos ( ωt ) and Im [ F ( k ) G ∗ ( k )] sin ( ωt ) are odd functions in k ,so their integrals cancel out of both J (12) and J (21). Extracting the k-independent portion ω = mc / ¯ h from ω ( k ) = ω + ω ( k ) further simplifies these expressions to J = (cid:18)Z Re [ F G ∗ ] cos ( ω t ) dk (cid:19) + (cid:18)Z Re [ F G ∗ ] sin ( ω t ) dk (cid:19) , (22) J = (cid:18)Z Re [ F G ∗ ] cosθ [ cos ( ω t ) cos ( ω t ) − sin ( ω t ) sin ( ω t )] dk (cid:19) max . (23)While similar, these two expressions are unfortunately not equivalent if t is constrained to be a precise value. But is practically impossible to constrain t to an accuracy less than ω − ( ∼ − seconds for an electron, and evenless for more massive particles). No current experiments can determine themeasurement time to a sub-attosecond accuracy, so W must be maximizedover the completely unknown angle ω t as well as over θ . The latter maxi-mization is trivial, as one simply forces cos θ = 1 in (23). And with completefreedom of the angle ω t , the maximum value of ( A sinω t ± B cosω t ) is just A + B . Using this fact, one can compare (22) and (23) to confirmthat J ≈ J , to within the accuracy of the previous approximations.Note that this procedure only works if the precise value of t becomespart of the solution parameter space, and is not merely an unknown randomquantity. In other words, if t is constrained to be some particular value –even if that value is completely unknown – then J is no longer necessarilyequal to J .From a foundational perspective, using J is preferable to J , because J isrelativistically invariant, unlike (12). But theoretical preference is no longerthe only issue; even given realistic uncertainties in t , the above approxima-tions ensure that J and J are not exactly the same, opening the door toan experimental differentiation between this approach and NRQM. J and J begin to diverge in the relativistic regime, and also when the approximation(20) begins to fail as discussed in section 5.The other regime in which new effects would be expected is if the valueof t could be experimentally constrained to a value much less than ω − .Although such capabilities currently seem far out of reach, it is worth con-sidering a future experiment that might constrain ω t to be approximatelya multiple of π . In that case, the sin ( ω t ) term in (23) can be ignored, andone finds that J reduces to only the first term in (22). This term is the square the real portion of the standard NRQM transition amplitude, and of coursethere are many deviations from NRQM that would occur if the imaginarypart of the amplitude was ignored. The previous sections have assumed two consecutive measurements of time-even quantities, such as position, P , or L (angular momentum). Measure-ments of time-odd quantities leads to difficulties, because (as noted in Section2) the b ( k ) terms in (2) physically propagate in the opposite direction as de-termined by the eigenvalues of P . Another problem, as noted in Section 1, isthat there are two independent (but equally important) mathematical objectson which initial values can be specified: both φ and the first time derivative˙ φ . Initial boundaries have historically only been imposed on the field itselfbecause of the reliance on first-order differential equations, but when usingthe KGE one should consider the possibility of constraining the value of ˙ φ as well as φ .Both of the issues in the previous paragraph are linked to a simple fact: themeasurement of a time-even quantity must yield the same value under time-reversal of the entire system, while the measurement of a time-odd quantity(like momentum) must change sign under time-reversal. In spinless NRQM,time-reversal is accomplished via a complex conjugation, which is why (inthe position basis) time-even operators are real and time-odd operators areimaginary. But complex conjugation does not time-reverse the KGE, only theSE, so using “ i ’s” to distinguish time-even and time-odd operators must nowfail. What does get a sign-change upon time reversal is ˙ φ , the very objectthat standard measurement theory does not address. It is therefore temptingto replace the “ i ” in time-odd measurement operators with − ∂/∂t . Whilethe operator units are now wrong, this is easily fixed by multiplying thesetime-odd operators by the natural unit of time ω − = ¯ h/ ( mc ). This sectionwill demonstrate that such a replacement solves the momentum eigenvalueproblem while also allowing measurements to constrain ˙ φ instead of φ . Under the substitution i → − ω − ∂/∂t in all time-odd operators, thecorresponding eigenvalue equation Q φ = qφ then changes to become i Q ˙ φ = ω qφ (24)for any standard time-odd operator Q (and eigenvalue q corresponding toeigenfunction φ ).In the limit where one expects the ordinary NRQM operators to be valid,the frequency ω ( k ) at the relevant values of k are all approximately thesame; ω ≈ ω = mc / ¯ h . Using this approximation, the time-derivative of (2)is simply ˙ φ = ∂φ∂t ≈ − iω Z a ( k ) e i ( k · r − ωt ) − b ( k ) e i ( k · r + ωt ) dk . (25)Inserting this into (24), one finds that the ordinary eigenvalue equation Q φ = qφ is recovered when b ( k ) = 0 (the standard NRQM condition). On the otherhand, if a ( k ) = 0, the equation now has the opposite eigenvalue, − q . This isexactly the sign-reversal needed to solve the momentum eigenvalue problemnoted above; if Q is − i ¯ h ∇ , and yields a measurement eigenvalue ¯ h k at t = 0,then (24) implies a boundary condition a ( k )+ b ( − k ) = δ ( k − k ). Regardlessof the relative phase or weight between a ( k ) and b ( − k ), the combination a ( k ) exp ( i k · r − iωt ) + b ( − k ) exp ( − i k · r + iωt ) physically propagates ina direction aligned with k . (This would not have been the case if one wasusing the ordinary eigenvalue equation; the b ( k ) term has a phase velocityopposite that of the b ( − k ) term.)In general, (24) implies that the measured real eigenfunction (in k-space) F ( k ) of a time-odd operator from standard NRQM should be imposed on thecombination a ( k ) exp ( − iωt ) + b ( − k ) exp ( iωt ). The section 3 conclusion that J ≈ J therefore remains valid for two consecutive time-odd measurements;one need only change b ( k ) → b ( − k ) starting at (8).It remains to show that a time-odd measurement followed by a time-even measurement (or vice-versa) also gives results consistent with NRQM.Suppose a time-odd measurement at t = 0 yields a k-space measurement with(real) eigenfunction F ( k ), and a time-even measurement at t = t yields ameasurement with (real) eigenfunction g ( r ), both within an arbitrary overall phase. Then the new boundary conditions can be written in terms of F ( k )and G ( k ) (the latter is calculated from g ( r ) according to (4)) as a ( k ) + b ( − k ) = F ( k ) , (26) a ( k ) e − iωt e − ǫt + b ( k ) e iωt e ǫt = G ( k ) . (27)These equations have a solution different from (10) and (11). Substantial butstraightforward algebra yields a ( k ) = F ( k ) e iωt e ǫt + F ( − k ) − G ( k ) e − iωt − ǫt − G ( − k ) e iωt + ǫt e iωt +2 ǫt − e − iωt − ǫt , (28) b ( k ) = G ( k ) e iωt + ǫt + G ( − k ) e − iωt − ǫt − F ( k ) − F ( − k ) e − iωt − ǫt e iωt +2 ǫt − e − iωt − ǫt . (29)Inserting these new values of a ( k ) and b ( k ) into (15) yields a different valueof W than given in (16). The only relevant terms are those with both F and G , because all of the terms like G ( k ) G ∗ ( − k ) are independent of therelative overall phase θ and do not contribute to variation in W . Putting inthe relative phase explicitly ( F G ∗ → exp ( iθ ) F G ∗ ) one can simplify W usingthe fact that for the real function in k-space F ∗ ( k ) = F ( k ) and for the realfunction in position space G ∗ ( k ) = G ( − k ). To largest surviving orders of ǫ ,this yields W = Z − ωǫt Re (cid:2) F ( k ) G ∗ ( k ) (cid:0) e iωt + iθ + e − iωt + iθ + 2 e − iωt − iθ (cid:1)(cid:3) ǫ t + sin (2 ωt ) dk . (30)As ǫ →
0, a periodic delta function arises according to (17), but withtwice as many poles as before: δ ( ωt − nπ/ exp (3 iωt ) = exp ( − iωt ), but the value of this quantity picks up a factor of i at eachconsecutive pole, making both the Re ( F G ∗ ) terms and the Im ( F G ∗ ) termsimportant. Applying the same approximation (20) to both of these terms,one finds that in the NR limit W ≈ cosθ Z Re [ F ( k ) G ∗ ( k )] cos ( ωt ) − Im [ F ( k ) G ∗ ( k )] sin ( ωt ) dk . (31)Using the same procedure as before (pulling out the k-independent part, ω ( k ) = ω + ω ( k ), and varying both the unknown angles θ and ω t ), onefinds J ≈ (cid:18) Re Z F ( k ) G ∗ ( k ) e − iω t dk (cid:19) + (cid:18) Im Z F ( k ) G ∗ ( k ) e − iω t dk (cid:19) . (32) Again, this is equal to the standard JPD J – this time without droppingany terms in J due to real eigenfunctions. Therefore, the analysis from theprevious section continues to hold for time-odd measurements followed bytime-even measurements (and vice-versa, due to the overall time-symmetryof this approach).For more than two consecutive measurements, this problem can appearto be over-constrained. For example, consider a time-even measurement Q at t = t , a time-even measurement Q at t = t , and a third time-evenmeasurement Q at t = t . While the outcomes of Q and Q fully determinethe solution from t ≤ t ≤ t , the outcomes of Q and Q independentlydetermine the solution from t ≤ t ≤ t . Although this implies a discontinuityin the solution to the KGE at t = t , note that it is not a discontinuity in φ itself, which is specified by the time-even measurement Q . Instead, thediscontinutity must occur at the unconstrained ˙ φ ( t = t ).The sort of discontinuity described in the previous paragraph is neithersurprising nor unphysical. If φ is supposed to describe a quantum system,one can only “measure” such a system by interacting with it. Because theinteraction is outside the scope of the KGE, one can hardly hope to finda continuous solution without taking this interaction into account. Further-more, the above discontinuity is precisely where the interaction occurs, at t = t . If no external system is present at the t = t hypersurface, then noinformation is available on this boundary, and the only external constraintson the system from t ≤ t ≤ t are the measurements at t = t and t = t . Inother words, one effectively “joins” together two adjacent regions of space-time into a single region – precisely as Hardy has recently argued must occurin any eventual theory of quantum gravity [17].Incomplete measurements, however, pose a mathematical challenge forthis two-boundary approach. A measurement that does not constrain a com-plete initial eigenfunction f ( r ) has additional “free” parameters which mustbe varied when calculating W max and W min . Possibly the biggest challengeis to determine how to implement commuting incomplete measurements, es-pecially when one measurement is time-even and the other is time-odd (e.g. L and L z ). The key will be to make sure that any discontinuity at thesecond measurement does not erase the information imposed at the prior (commuting) measurement. Due to the additional mathematical complexity,this paper can only conjecture that commuting incomplete measurementsin this formalism will give results (nearly) equivalent to NRQM. The mostpromising path forward seems to be to ignore the time duration betweentwo consecutive commuting measurements, and impose them as simultane-ous (but incomplete) constraints on both φ and ˙ φ .While much work is required to extend this approach to cover the fullrange of measurement theory from NRQM, such effort may be misguidedwhen it comes to a possible extension to quantum gravity. That is because therestriction of operators to spacelike hypersurfaces will have to be revisited ina general relativistic framework, possibly requiring a new theory of quantummeasurement without the use of operators at all (as in [18]). The importanceof imposing boundary conditions on arbitrary hypersurfaces will be discussedin section 6. When examining the limits in which the previous analysis fails, one findsthat some of these limits would manifest themselves as a time-energy uncer-tainty relation. Before addressing this result, it should be stressed that theusual position-momentum uncertainty relations are unchanged. As in stan-dard NRQM, a very precise measurement of φ in position space leads to alarge spread of φ in k-space due to the Fourier transform (3). There is onedifference; unlike standard NRQM, after a pair of complete measurementsthis approach can now reconstruct φ between the two measurements, leavingno uncertainty at all (except for the free angles θ and ω t ). Still, this istrue classically as well; one can “beat” Heisenberg’s uncertainty principle inhindsight, by measuring the position very accurately at two successive times,and then reconstructing the intermediate velocity. So knowing a more com-plete solution to φ after the fact does not contradict the usual uncertaintyprinciple, which only concerns what can be known at any given time.In NRQM, the time-energy uncertainty principle has never been put onan even footing with the position-momentum uncertainty principle [19], al-though in a relativistic theory they must of course be intimately related. The issue is not whether one can measure an arbitrarily precise energy at an arbi-trarily precise external clock time – in principle, one can do that in both thisapproach as well as in the standard formulation of quantum mechanics [20].Here, the question becomes whether or not precise energy measurements arereproducible over short periods of time.Without using the approximation (20), the exact invariant JPD (19) fortwo (time-even) energy measurements can be written J = Z ∞−∞ ∞ X n = n ωRe [ F ( k ) G ∗ ( k )] cos ( ωt ) δ ( ωt − nπ ) d k ! max . (33)From (33) it is clear that if both measurements are very precise energy mea-surements, then they must be nearly the same energy (with the same k ), orelse the product F G ∗ would be very small, leading to a small probability.But because of the delta function, there is now another constraint; the pre-cisely measured value of k must match up with a particular value k n (where k n solves ω n ( k n ) = nπ/t ). This can only happen if t is not exactly fixed,but is a parameter that has some freedom to maximize W , as discussed atthe end of section 3. This constraint on possible measurement times wouldnever be directly noticeable from a single measurement because of the smallchanges in t ( < − sec) needed to set k = k n .But even with this freedom in the precise value of t , deviations fromstandard quantum mechanics will still occur if one cannot “ignore” the deltafunction in (33), as implied by (20). And (20) is only correct in certainlimits. The first limit derived in the appendix is (35), which constrains therate of change of both F ( k ) and G ( k ) for small values of k. (Specifically, for k ≤ k , where k ≡ (2 πm/ ¯ ht ) / .) For an example of such a constraint,consider a gaussian initial measurement F ( k ) = exp [ − k / (2 σ )]. From (35),one finds that this corresponds to σ > k , equivalent to a spread in the initialenergy greater than h/t . As this constraint is approached, (33) predictsthat new physics will begin to emerge. The form of this new physics will bea failure of the ordinary probability distributions, forcing sequential energymeasurements give slightly different results.Curiously, this “new physics” is almost what is expected to happen if thereis a time-energy uncertainty relation that is analogous to position-momentum uncertainty. The difference is that here the time t is not an uncertainty ina measurement time, but instead the duration between measurements. Still,in practice this might be the same thing, as it’s not possible to make twomeasurements (in a well-defined order) separated by a duration smaller thanthe amount of time it takes to make the measurements.The other constraint derived in the appendix also concerns the rate ofchange of F ( k ) and G ( k ); see equation (44). For values of k ≈ k , this isa similar constraint to the previously discussed condition (35). For largervalues of k , the condition becomes even less stringent, even for a precise non-zero energy measurement. Of course, values of k ≪ k can easily violate thiscondition, but so long as the constraint (35) continues to hold, this will justbe a very small perturbation on the approximation (20), because the integralon the right side scales like k , and (35) forces the important part of theintegral to extend out to at least k = k .The conclusion is that there are many pairs of boundary conditions forwhich the approximation (20) does not hold good, leading to a divergence of J from J , but these pairs of boundaries also approach the time-energy uncer-tainty limit that is known to be experimentally difficult. Still, it is promisingthat these results yield a time-energy uncertainty relation consistent withknown experimental limitations. Although the results in this paper have not yet been generalized to the pointwhere this is a full alternative interpretation of quantum mechanics, the indi-cations are promising that this approach might put quantum mechanics on afooting consistent with general relativity while also addressing many open in-terpretational questions. More immediately, the central result of an invariantscalar from which one can calculate positive definite outcome probabilitiesfor single-particle solutions to the Klein-Gordon equation is itself sufficientmotivation to further consider this novel framework.Three of the four postulates used in the previous sections appear com-patible with a general relativistic framework. These are as follows:
1) The correct wave equation governing generic (spinless, neutral) par-ticles is the Klein-Gordon equation on a complex scalar field φ . (This isassumed to be a classical field, not the operator-valued field of quantum fieldtheory.)Although this is a postulate, it still requires some justification. As notedin the introduction, the KGE and SE are not equivalent in the NR limit.This can be seen in both the amount of necessary Cauchy data (solving theKGE requires twice the initial data as the SE) as well as in the doubledsize of the KGE solution space (see Section 2). From this perspective, theKGE on a real scalar field would be more comparable to the SE, but sucha field has no natural U(1) symmetry. While it is true that the solutions tothe complex KGE in the NR limit can always be written as the sum of asolution to the SE and an independent solution to the complex conjugate ofthe SE, there are an infinite number of other ways to split the KGE solutionsinto the sum of two arbitrary sub-components. Assigning physical meaningto one of these particular sub-components under one particular splitting istherefore difficult to justify. For one, it raises the question of why one canignore the other term; both components have positive energy, as determinedby the zero-potential Hamiltonian H = P / (2 m ) (as well as by the energydensity T of classical scalar fields). Ignoring one term also raises severeproblems in curved spacetime where there is no covariant splitting of thefull solution into two analogous sub-components [1]. Therefore it seems morenatural to interpret the entire KGE solution as a single entity; that is thegoal of this paper.The next two postulates are:2) The solution φ is constrained by a boundary condition on a closedhypersurface s (with unit normal four-vector η µ ), imposed by external mea-surements. Any infinities are dealt with by giving the mass a small imaginaryvalue ǫ , and then taking the limit ǫ → J = ( ∆W ) , where ∆W is the allowed range of the real invariantscalar W = H s Im ( φη µ ∂ µ φ ∗ ) ds consistent with the boundary constraints.The usual conditional probabilities can be generated from J via normal-ization. In order to compare the above postulates to a preparation-measurement sequence in standard NRQM, one takes the hypersurface to be the pair ofhyperplanes t = 0 and t = t , presumably connected at some sufficiently largedistance where the integrand in W goes to zero. Imposing boundaries on suchhypersurfaces requires a fourth and final postulate;4) The measured eigenfunction of a time-even operator (from standardNRQM) is imposed on φ along an instantaneous hypersurface. Time-oddoperators are first adjusted by replacing the “ i ” with − ω − ∂/∂t .But this use of eigenfunctions and instantaneous surfaces is the only re-maining tool from NRQM, and this provides a strong motivation to developa measurement theory that could apply to any hypersurface. Postulate needs all of the KGE solutions to respond to the potential in the same man-ner. General relativity provides a way to add such a scalar potential: via ametric, g µν , in the weak-field limit. This does not work for NRQM, becausethe Schr¨odinger equation is not covariant, but it should work for the KGE,which can be written g µν ∇ µ ∇ ν φ = m φ (with covariant derivatives). Usingthis curved-space KGE to introduce potentials may be mathematically awk-ward, but if successful it would allow for a much easier extension to quantumgravity.For multiple particles, this approach could potentially allow a simplifi-cation that is not possible in NRQM: keeping the parameter space φ ( r , t )fixed for any number of (identical) particles. For two particles in standardNRQM, one expands the dimensionality of the Hilbert space, equivalent toa wavefunction ψ ( r , r , t ). This expansion is required because of the roleof probability in NRQM: the wavefunction ψ traditionally encodes all con-ditional probabilities of all possible outcomes, and the number of possibleoutcomes increases dramatically with the number of particles (because dif-ferent types of measurements can be made on each particle). Encoding all ofthese possibilities requires a wavefunction in a large configuration space.But the picture of probability described in section 3 is quite different. In-stead of encoding all possible outcomes, the field φ is a particular solution tothe pair of preparation/measurements that actually happen – not all possiblemeasurements. To find the outcome probability for any given experimentalset-up, one need only compare the permitted range of W over the outcomespermitted by that particular set-up. Instead of the wasteful increase in theparameter space of ψ to deal with potential measurements that never actuallyhappen, φ only “bothers” to encode the result of the actual next measure-ment. This framework therefore opens up the possibility that each type ofparticle might be represented by a field in physical space, and configurationspace would only be a tool to summarize knowledge of this physical field.(Such speculation agrees with the conclusion of a recent paper by Montina[21].) The outcome of such a research path is far from certain, and would com-plicate development of a generalized measurement theory (the amplitude ofthe field must then be related to the number of particles present) but thisalso would bring quantum theory more in line with general relativity. Finally, spin and charge will need to be added to this generic-particleframework. Incorporating spin should be straightforward, as it is known toinvolve expanding the scalar field φ into a multi-component spinor. For exam-ple, the above methodology could be extended to neutral spin-1/2 particlessimply by using the second-order van der Waerden equation instead of theKGE. Adding charge is less obvious, and might even seem problematic, asthe standard interpretation of the KGE treats the b ( k ) terms evolving like exp ( iωt ) as antimatter. Meanwhile, the above approach requires both the a ( k ) and b ( k ) terms to describe a single species, consistent with standard rel-ativistic QM for spinless, neutral particles. But doubling the dimensionalityof the scalar field φ would provide parameters for both matter and antimatterwhile still retaining the above two-boundary formalism. So, for example, theelectron/positron field would then require a four-component (Dirac) spinor;one doubling of the scalar field to introduce antimatter, and another dou-bling to permit spin-1/2. But instead of simply using the first-order Diracequation, this framework requires a second-order differential equation. Forthe electron/positron field, the most likely candidate would be the square ofthe Dirac equation. In the absence of electromagnetic coupling, this simplyreduces to the KGE on a Dirac spinor, making the connection to the aboveresults quite apparent.This approach also has implications for quantum foundations. One on-going disagreement concerns whether or not the solution to the Schr¨odingerequation ψ corresponds to some observer-indepedent “reality”, or is instead aconstruct concerning our knowledge of a system. This approach offers a mid-dle ground; ψ would be a construct, while the underlying system would bebest described by some classical field φ . Without knowing the future bound-ary condition, the best approximation one can make concerning φ would bea guess that looks something like ψ , but after the fact, one can reconstructwhat actually happened between measurements. Non-classical behavior ofthose fields, such as violations of Bell’s inequality, might now be explainedby virtue of the parameters t , a ( k ), and b ( k ) all being hidden variables thatare not “local” as defined by Bell [22] (because they depend on future events).Such a “realistic” view of φ is more plausible if ǫ does not exactly equalzero, because otherwise φ has discrete infinite coefficients at k = k n . But there is some hope for a physical motivation of a non-zero epsilon; invokinga small but non-zero imaginary mass component has a history that goesback to Dirac [23] and has been revived several times since [24][25]. Anotherpromising avenue to a real φ can be found by writing the Klein-Gordonequation in curved space, where the Christoffel symbols enter the equationin a similar manner as ǫ [26]. Given that no quantum mechanics experimenthas been performed in an exactly flat space, it would be possible for a slightlycurved space-time metric to be the source of a non-zero ǫ .It may be considered a disadvantage to this approach that φ depends onfuture events. The most serious concern is the possibility of constructing acausal paradox. If a system φ is constrained between two temporal bound-aries, then φ contains some information about the future. At first glance,it seems logical that an experimenter could 1) measure φ before the futureboundary, 2) learn about future events, and 3) change those events to some-thing different. This would create an irresolvable paradox, sufficient reasonto end this line of inquiry. But given the assumed equivalence between mea-surements and boundary conditions, step 1) is automatically forbidden; themeasurement of φ is the physical constraint corresponding to the futureboundary condition. To measure φ before the imposed constraint would be acontradiction in terms; in this framework there is no such thing as extract-ing information without also imposing that same information as a boundarycondition.Indeed, there is a perspective that makes this retrocausal aspect an ad-vantage, not a disadvantage [27]. It is obvious that quantum mechanics iscounter-intuitive, but it must be counter-intuitive for a reason – some hu-man intuition that fundamentally contradicts some physical principle. Oneexample of this would be the well-known conflict between our directed ex-perience of time and the more symmetric treatment of time in fundamentalphysics. If the counter-intuitive aspects of quantum mechanics could be ex-plained via classical fields symmetrically constrained by both past and futureevents, then it would be a mistake to reject such a solution based solely onour time-asymmetric intuitions.Regardless, the ultimate test for any theoretical proposal is experiment.Perhaps the strongest reason to pursue this line of research is that it leads to results which contradict standard quantum mechanics (albeit only inexperimentally-difficult regimes). Hopefully an incorporation of potentialsand a relativistically correct measurement theory will lead to practical ex-periments which can confirm or deny the validity of this general approach. APPENDIX
This appendix derives the limits in which the approximation (20) holds good.Recall k n are the solutions to the equation ω n ( k n ) = nπ/t , and n is thesmallest value of n for which there is a solution. It will also be useful to definethe quantity k n +1 / using the similar equation ω n ( k n +1 / ) = ( n + 1 / π/t .To compare to NRQM, one is interested in the limit where ω ≈ mc / ¯ h +¯ hk / (2 m ).Working backwards from the right side of (20), using spherical k-coordinates,the “radial” component of this integral can then be written as the discretesum of integrals Z ∞ Re [ F G ∗ ] cos ( ωt ) k dk = P ∞ n = n +1 R k n +1 / k n − / Re [ F G ∗ ] cos ( ωt ) k dk + R k / Re [ F G ∗ ] cos ( ωt ) k dk. (34)The final integral (where k n +1 / is written as k / for clarity) is differentfrom the others because the lower limit must be forced to zero, and the precisevalue of k n − / depends on the values of m and t – indeed, it may not evenexist. In the special case that k n − / = 0, then k n +1 / = (2 πm/ ¯ ht ) / ≡ k . This value, k , is a reasonable estimate for the value of k n , given that m and t are not known precisely.The last integral in (34) is just one of many similar integrals that scale like k , so this lowest-k integral it is not likely to be important unless F ( k ) G ∗ ( k )is only large for values of k < k . But such a narrow range is only possibleif either (1 /F ) dF/dk or (1 /G ) dG/dk evaluated at k ≤ k is on the order of k . This leads to the first constraints on the approximation (20), (cid:12)(cid:12)(cid:12)(cid:12) ∂F∂k (cid:12)(cid:12)(cid:12)(cid:12) < | F ( k ) | k , (cid:12)(cid:12)(cid:12)(cid:12) ∂G∂k (cid:12)(cid:12)(cid:12)(cid:12) < | G ( k ) | k f or k ≤ k . (35)This constraint is discussed above in section 5 on the uncertainty principle. Each integrals in the sum in (34) will be referred to as Y ( k n ); writing outthe k-dependence explicitly (in the NR limit) Y ( k n ) = Z k n +1 / k n − / Re [ F ( k ) G ∗ ( k )][ cos ( ω t ) cos ( α k ) − sin ( ω t ) sin ( α k )] k dk, (36)using the constants ω = mc / ¯ h and α = p ¯ ht / (2 m ).Taylor expanding the nth integrand around g ( k n ) = Re [ F ( k n ) G ∗ ( k n )],the zero-order term in the expansion can be integrated exactly: Y ( k n ) = g ( k n ) k α sin ( ω t + α k ) + O [ g ′ ( k n )] − g ( k n ) r π α [ cos ( ω t ) S ( αk ) + sin ( ω t ) C ( αk )] (cid:12)(cid:12)(cid:12)(cid:12) k n +1 / k n − / . (37)Here C ( x ) and S ( x ) are the Fresnel integrals, which for x > π can be excel-lently approximated by [28] C ( x ) = 12 + 1 √ πx sin ( x ) − √ πx cos ( x )+ O ( x − ) sin ( x )+ O ( x − ) cos ( x ) , (38) S ( x ) = 12 − √ πx cos ( x ) − √ πx sin ( x ) − O ( x − ) cos ( x )+ O ( x − ) sin ( x ) . (39)With x = αk , the constraint x > π corresponds to k > k , reinforcing theearlier discovery that this approximation fails when the first integral in (34)dominates the others, and therefore is equivalent to the earlier constraint(35). Plugging in the limits in (37), dramatic simplifications occur because sin ( ω n +1 / t ) = ( − n and cos ( ω n +1 / t ) = 0; Y ( k n ) = g ( k n )( − n " ( k n +1 / + k n − / )2 α + ( k − n +1 / + k − n − / )8 α + O ( α − k − ) + O ( g ′ ) . (40) Y ( k n ) ≈ g ( k n )( − n mk n ¯ ht + O ( α − k − ) + O ( g ′ ) . (41)The last step is an approximation, good to better than 1% even for the firstterm ( n = n + 1), and rapidly improving for the higher terms in the sum.Within a constant, the first term in (41) is exactly the left side of (20), withthe correct scaling of both k n and t . The next term in the Taylor expansion, g ′ ( k ) = d Re [ F ( k ) G ∗ ( k )] /dk , isunimportant if the variation with k is sufficiently slow. This integral is notmuch harder than the last, because the k cos ( k ) integral can be done byparts without requiring the Fresnel integrals. To highest surviving orders,this term becomes O ( g ′ ) = − g ′ ( k n ) ( − n α n k n +1 / + k n − / − k n h k n +1 / + k n − / + ( k − n +1 / + k − n − / ) / (4 α ) io , (42) Y ( k n ) ≈ g ( k n )( − n mk n ¯ ht − π − (cid:18) m ¯ ht (cid:19) − n g ′ ( k n ) k n + O ( α − k − ) . (43)Again, this last step is a quite accurate numerical approximation thatincreases in accuracy with n . Comparing the magnitude of these two terms,one finds that the approximation (20) fails unless the fourier expansion ofboth boundary conditions vary sufficently slowly over k such that (cid:12)(cid:12)(cid:12)(cid:12) ∂∂k [ F ( k ) G ∗ ( k )] (cid:12)(cid:12)(cid:12)(cid:12) k n ≪ | F ( k n ) G ∗ ( k n ) | k n k (44)This result is also is discussed above in section 5 on the uncertainty principle.The conditions on the higher order terms in the Taylor expansion are notworked out here, but they should also be taken into account. Acknowledgments
The author is eternally grateful to J. Finkelstein for careful and thought-ful criticism. Many additional improvements arose thanks to detailed analy-sis from E. Cavalcanti. Further thanks go to M. Derakhshani, P. Goyal, F.Kleefeld, D. Miller, R. Schafer, L.S. Schulman, R.W. Spekkens, W. Struyve,and W.R. Wharton.
References
1. DeWitt, B.S.: Quantum Theory of Gravity I: The Canonical Theory.
Phys. Rev. , 1113 (1967)2. Horton, G., Dewdney, C., and Nesteruk, A.: Time-like flows of energy momentumand particle trajectories for the Klein-Gordon equation.
J. Phys. A , 7337(2000)3. Mostafazadeh, A. and Zamani, F.: Quantum mechanics of KleinGordon fieldsI: Hilbert Space, localized states, and chiral symmetry. Annals Phys. , 2183(2006)84. Kleefeld, F.: On some meaningful inner product for real Klein-Gordon fieldswith positive semi-definite norm. arXiv:quant-ph/0606070 (2006)5. Nikoli´c, H.: Probability in Relativistic Quantum Mechanics and Foliation ofSpace-Time.
Int. J. Mod. Phys. A , 6243 (2007)6. Spekkens, R.W.: Evidence for the epistemic view of quantum states: A toy the-ory. Phys. Rev. A , 032110 (2007)7. Aharonov,Y. and Vaidman, L.: Complete description of a quantum system at agiven time. J. Phys. A , 2315 (1991); Aharonov,Y. and Vaidman, L.: The Two-State Vector Formalism: An Updated Review. In: Time in Quantum Mechanics ,J.G. Muga et al. eds. Springer, Berlin, (2002)8. Sutherland, R.: Causally Symmetric Bohm Model. arXiv:quant-ph/06010959. Wharton, K.B.: Time-Symmetric Quantum Mechanics.
Found. Phys. , 159(2007)10. Oeckl, R.: States on timelike hypersurfaces in quantum field theory. Phys. Lett.B , 172 (2005); arXiv:hep-th/050526711. Oeckl, R.: Probabilites in the general boundary formulation.
J. Phys. : Conf.Ser. , 12049 (2007); arXiv:hep-th/061207612. Gell-Mann, M. and Hartle, J.B.: Time Symmetry and Asymmetry in QuantumMechanics and Quantum Cosmology. In: Proceedings of the NATO Workshop onthe Physical Origins of Time Asymmetry , J.J. Halliwell, J. Perez-Mercader, andW. Zurek, eds. Cambridge Univ. Press, Cambridge (1994)13. Schulman, L.S.:
Time’s Arrows and Quantum Measurement.
Cambridge Uni-versity Press, Cambridge (1997)14. Cramer, J.G.: Generalized absorber theory and the Einstein-Podolsky-Rosenparadox.
Phys. Rev. D , 362 (1980); Cramer, J.G.: The transactional interpre-tation of quantum mechanics. Rev. Mod. Phys. , 647 (1986)15. Rovelli, C.: Quantum Gravity.
Cambridge University Press, Cambridge (2004)16. Sakurai, J.J.:
Modern Quantum Mechanics.
Addison-Wesley, Redwood City(1985)17. Hardy, L.: Towards quantum gravity: a framework for probabilistic theorieswith non-fixed causal structure.
J. Phys. A Found. Phys. Phys. Rev. , 1649 (1961)21. Montina, A.: Exponential complexity and ontological theories of quantum me-chanics.
Phys. Rev. A , 22104 (2008)22. Bell, J.S.: On the Problem of Hidden Variables in Quantum Mechanics. Rev.Mod. Phys. , 447 (1966)23. Dirac, P.A.M.: The electron wave equation in de-Sitter space. Ann. Math. ,657 (1935)24. Nakanishi, N.: Covariant Formulation of the Complex-Ghost Relativistic FieldTheory and the Lorentz Noninvariance of the S Matrix. Phys. Rev. D , 1968(1972)25. Kleefeld, F.:On Symmetries in (Anti)Causal (Non)Abelian Quantum Theories. Proc. Inst. Math. NAS Ukraine , 1367 (2004)26. Evans, J., Alsing, P.M., Giorgetti, S.,and Nandi, K.K.: Matter waves in a gravi-tational field: An index of refraction for massive particles in general relativity. Am.J. Phys. , 1103 (2001)27. Miller, D.J.: Quantum mechanics as a consistency condition on initial and finalboundary conditions. arXiv:quant-ph/0607169 (2006)28. Cody, W.J.: Chebyshev approximations for the Fresnel integrals. Math. Comp.22