OON THE ROLE OF QUANTUM EVENTS IN DOUBLE-SLITEXPERIMENTS
R. SCHUSTER
Abstract.
We formulate the Schr¨odinger equation as the equation of motionof a small external influence which serves as the initial boundary condition ofa physical system in classical laboratory space. The Hilbert space of possibleexternal influences for a given physical system is then equivalent to the Hilbertspace of a quantum system without spin. We discuss the double-slit experimentin the context of this approach and show that wave-particle dualism reducesto the choice of a basis in the geometrical construction of the Hilbert space ofsmall external influences. Introduction
The current development in mastering the creation of smallest objects rises anew era of tests, which challenges quantum theory on it’s most fundamental level[1]. The nature of these tests will move hypothetical Gedankenexperiments into thereach of experimental confirmation, and shine new light on the interpretation issues,which plague quantum mechanics from it’s very beginnings. As this recent progresstackles the issues at an interpretation level only, it needs to be accompanied by astructural rethinking of the theoretical fundamental concepts to gain substantiallynew insight.We approach quantum mechanics by using the mathematical formalism to vari-ational problems of Gelfand and Fomin [2], and investigate the time evolution ofan external influence, which serves as the initial boundary condition of a classicalphysical system given with free initial point.The necessary and sufficient condition for the external influence to serve as aboundary condition for the classical system in the entire time interval is given bythe Hamilton-Jacobi equation. We use the algebraic one-to-one correspondencebetween the elements of the algebra of transformations and the elements of thecorresponding covering group near the identity to derive the Schr¨odinger equationas the equation of motion of a field theory in classical configuration space for asmall external influence. The Hilbert space of quantum states in energy eigenstaterepresentation corresponds, then, to the set of possible solutions of the above fieldtheory, where only the set of basis vectors of this solution space is observable inclassical space due to the role of the Hamilton-Jacobi equation as the necessary andsufficient condition.The distinguished role of the energy eigenvalue representation enables us to applyour formalism to the experimental results of a double-slit experiment without mod-ification and explains the continuity of the interference pattern in a natural way.
Date : April 10, 2011.
Key words and phrases. wave-particle duality, non-locality, double-slit experiment, quantumsystem as disturbances. a r X i v : . [ qu a n t - ph ] A p r R. SCHUSTER
This is in contrast to standard quantum mechanics, where one needs to extendthe formalism by alternative methods, e.g. the positive operator valued measure(POVM) approach [3], [4], to formulate the observed results of the double-slit ex-periment. Our results allow us to show in particular that the wave-particle dualismobserved in the double-slit experiment reduces to the experimental realization ofthe choice of a basis in a degenerate Hilbert space constituted by the energy eigen-functions emanating from each slit.Another particular aspect in our approach is the close connection between thequantum object - the small external influence as a boundary condition - and theclassical laboratory configuration - the influenced classical physical system, whichemphasizes the geometry of the classical configuration space at the time of theappearance of the initial influence in the classical system. Our view shifts thequantum property, particle or wave, of a quantum object to the property of aquantum event, the event of appearance in classical laboratory space with config-uration space geometry given at this point in time. This explains the observedbehaviour of delayed choice experiments if we regard the contact of the quantumobject with the screen/counter as the quantum event of significance.From a formal perspective our derivation is closely related to Schr¨odinger’s firstnote [5]. But in contrast to Schr¨odinger, who abandoned this approach by lackof physical justification, we have an intuitive physical interpretation and mathe-matically rigorous derivation which gives new insight into the nature of quantummechanics. 2.
Quantum System as External Influence
Let us consider a physical system with action(2.1) (cid:90) ba L ( t, q , . . . , q n , ˙ q , . . . , ˙ q n ) dt + g ( a, q ( a ) , . . . , q n ( a )) + g ( b, q ( b ) , . . . , q n ( b )) . and variable endpoints a and b . The action of this system differs from the familiaraction(2.2) (cid:90) ba L ( t, q , . . . , q n , ˙ q , . . . , ˙ q n ) dt by the additional terms g ( a, q ( a ) , . . . , q n ( a )) and g ( b, q ( b ) , . . . , q n ( b )), which arefunctions of the boundary points only. As known from elementary theory thesefunctions serve as the boundary conditions which single out a definite solution andrepresent the external influence on a physical system described by the Lagrangian Although this action is unusual in fundamental physics it is common in applied sciencein its generalized form given in optimal theory, see e.g. [6]. The task is to find an admissiblecontrol function u ( t ) which causes the system to follow an admissible trajectory x ( t ) subject tothe dynamical equation ˙ x = a ( x ( t ) , u ( t ) , t )and optimizes the action I ( u ) = ϕ ( x ( t f ) , t f , x ( t i ) , t i ) + (cid:90) t f t i L ( x ( t ) , u ( t ) , t ) dt. with boundary condition ϕ ( x ( t f ) , t f , x ( t i ) , t i ). N THE ROLE OF QUANTUM EVENTS IN DOUBLE-SLIT EXPERIMENTS 3 (2.3). As an example, suppose the above Lagrangian (2.3) represents an experimen-tal set up, which is kicked off and changes to its initial state at the free initial time a . Then, this Lagrangian describes the evolution of the classical physical systemover time t in the laboratory. For reasons which will be clear later in the text wewill call the system described by (2.3) the laboratory system in the following.The dynamics of the laboratory system is given by the Euler equations ∂L∂q i − ddt (cid:18) ∂L∂ ˙ q i (cid:19) = 0 , (2.3)and the additional boundary conditions ∂L∂ ˙ q i (cid:12)(cid:12)(cid:12)(cid:12) t = a − ∂g ( a, q ) ∂q i = 0(2.4)and ∂L∂ ˙ q i (cid:12)(cid:12)(cid:12)(cid:12) t = b − ∂g ( b, q ) ∂q i = 0 . (2.5)Instead of investigating the equation of motions of the laboratory system let usconcentrate on the initial boundary conditions and write the external influence g ( a, q ) = g ( t, q ) (cid:12)(cid:12)(cid:12)(cid:12) t = a (2.6)as a function of time evaluated at a and omit the subscript in the following. Notesimilar results can be observed for the boundary conditions at the terminal point,which we will ignore in the following by assuming that g ( b, q ( b ) , . . . , q n ( b )) = 0 atthe terminal point b .Condition (2.4) relates the external initial influence g ( a, q ) to the momentum ofthe laboratory system p i ( t, q, ˙ q ) (cid:12)(cid:12)(cid:12)(cid:12) t = a = ∂g ( t, q ) ∂q i (cid:12)(cid:12)(cid:12)(cid:12) t = a (2.7)at the initial point a , where we define the momentum as usual as the velocityderivative of the Lagrangian p i ( t, q, ˙ q ) = ∂L ( t, q, ˙ q ) ∂ ˙ q i . (2.8)Relations (2.7) and (2.8) allow us to rewrite the boundary conditions as functionsof configuration space variables˙ q i ( a ) = ϕ i ( q ) (cid:12)(cid:12)(cid:12)(cid:12) t = a ( i = 1 , . . . , n ) , (2.9)which can be thought to assign a direction to every point in the hyperplane t = a and allows us to define a family of boundary conditions as follows. The family ofboundary conditions ˙ q i ( t ) = ϕ i ( t, q ) ( i = 1 , . . . , n )(2.10)imposed for every t ∈ [ a, b ] represents a field of the functional (2.3) if In the following we follow closely the line of thoughts in [2], Chap. 6 and omit proofs whichcan be found in this book.
R. SCHUSTER • there exists a function g ( t, q ) such that p i ( t, q, ϕ ( q )) (cid:12)(cid:12)(cid:12)(cid:12) t = a = ∂g ( t, q ) ∂q i (cid:12)(cid:12)(cid:12)(cid:12) t = a . (2.11) That is, the external influence represented by g ( t, q ) is in contact with thelaboratory system imposing the initial momentum p i ( a, q ( a ) , ˙ q ( a )) at time a . • every extremal, i.e. solution of the Euler equations, satisfying the boundaryconditions ˙ q i ( t ) = ϕ (1) i ( q ) (cid:12)(cid:12)(cid:12)(cid:12) t = t (2.12) also satisfies the boundary conditions˙ q i ( t ) = ϕ (2) i ( q ) (cid:12)(cid:12)(cid:12)(cid:12) t = t (2.13) at the different point in time t and vice versa, i.e. the boundary conditionsare traceable in time. Boundary conditions of this type are called consistent.Thus, the influence function g ( t, q ) acts as a kind of potential in configurationspace for the family of boundary conditions. Note the potential g ( t, q ) is given atspecific points in time where the configuration space points should be regarded asa set of points at this given time and not parameterized by t as (2.11) is a relationfor equal times. Since boundary conditions describe the external influence of anunknown source to a physical system, which can be completely unrelated at differentpoints in time, this means physically spoken, that we restrict our investigation to theclass of external influences which are historically traceable and have a configurationspace representation of fields which are derivable from a potential.Then, the natural question arises which condition must fulfill the external influ-ence function g ( t, q ) to keep the ability to kick off the laboratory system describedby (2.3) at an arbitrary point t ∈ [ a, b ] in time. Gelfand and Fomin, [2] p. 146,showed that the necessary and sufficient condition, called consistency condition, isthe Hamilton-Jacobi equation ∂g ( t, q ) ∂t + H (cid:18) t, q, ∂g ( t, q ) ∂q , . . . , ∂g ( t, q ) ∂q n (cid:19) = 0 , (2.14)with Hamilton function H ( t, q, p ). Thus, the set of external influence functions { g i } , which constitute the solutions of the Hamilton-Jacobi equation (2.14) in localcoordinates, are the generators of the canonical transformation in the laboratorysystem and, therefore, the external influence function g ∈ { g i } is an element of aLie algebra.Suppose we have an external influence g which causes a very small transformationin laboratory space. Then, we can use the isomorphism between elements in theneighbourhood of 0 of a Lie algebra g and the Lie subgroup G of elements connectedto the identity established via the inverse exponential map g = ln ψ (2.15) Note the following derivation based on this equation is of a local character as the Jacobian forthe transformation to the generalized momentum leading to (2.14) is only locally valid as pointedout in footnote 2 of [2], p68. That is, we will have an explicitly local derivation of quantummechanics in contrast to the standard approach to quantum mechanics.
N THE ROLE OF QUANTUM EVENTS IN DOUBLE-SLIT EXPERIMENTS 5 and do the ansatz g ( t, q ) = k ln ψ ( t, q )(2.16)in local coordinates. Physically spoken, this means we want to investigate the objectitself represented as an element of the group of objects and are not interested inthe information space generated by the external influence given by the tangentialspace, which is isomorphic to the Lie algebra. Then (2.14) reads kψ ( t, q ) ∂ψ ( t, q ) ∂t + H (cid:18) q, kψ ( t, q ) ∂ψ ( t, q ) ∂q (cid:19) = 0 , (2.17)where the constant k is assumed to be small to guarantee the smallness of g ( t, q )over entire configuration space and time. In fact we know from experiment thatnature has chosen a very small value k ∝ (cid:126) (2.18)with Planck’s constant (cid:126) .For conservative systems H = E relation (2.17) decouples into kψ ( t, q ) ∂ψ ( t, q ) ∂t + E = 0(2.19)and − E + H (cid:18) q, kψ ( t, q ) ∂ψ ( t, q ) ∂q (cid:19) = 0 . (2.20)The first equation (2.19) reads k ∂ψ∂t = Eψ, (2.21)where we suppress the arguments in the following, and represents the time evolutionof the external influence, which is in contact with laboratory space with energy E .The second condition (2.20) can be reformulated in terms of the kinetic andpotential energy of the Hamiltonian H = T + V in laboratory space k (cid:18) ∂ψ∂q (cid:19) + V ψ − Eψ = 0 . (2.22)To describe the dynamics of the external influence in laboratory space we inter-pret the left side of this equation as the Lagrangian L ( ψ, ∂ q ψ ) = k (cid:18) ∂ψ∂q (cid:19) + V ψ − Eψ (2.23) R. SCHUSTER of a field theory in ψ and its space derivative ∂ q ψ = ∂ψ/∂q , which is equivalent tothe variational problem L ( ψ, ∂ q ψ ) = k (cid:18) ∂ψ∂q (cid:19) + V ψ (2.24)with constraint (cid:90) | ψ | dq = 1(2.25)where the integration is over entire configuration space and the energy constant E plays the role of the Lagrange multiplier.The equation of motion can be easily read from (2.23) and results into thestationary equation H (cid:18) q, (cid:126) i ∂∂q (cid:19) ψ = Eψ (2.26)with k = (cid:126) /i and Hamiltonian H in operator notation. Insertion of the aboveequation into (2.19) leads to the time-dependent Schr¨odinger equation i (cid:126) ∂ψ∂t = Hψ (2.27)for the external influence ψ and we can identify the Schr¨odinger equations as the dy-namics of a very small external influence fulfilling the consistency condition (2.14),which is necessary and sufficient to keep the external influence in contact withlaboratory space.Moreover, it is well known, see e.g. [7], that the set of solutions of the above vari-ational problem form an orthonormal basis of a Hilbert space of square integrablefunctions over laboratory space M , which is subject to the experimental setting byvirtue of the consistency condition (2.14). Thus, the external influences connectedto laboratory space do not only fulfil the Schr¨odinger equations but also constitutethe orthonormal basis of a Hilbert space which is isomorphic to the space of statesof scalar quantum theory in energy eigenvalue representation with normalizationcondition (cid:90) | ψ | dq = 1 . (2.28)Let us call an arbitrary element ψ ∈ H of this space a possible external in-fluence. This possible external influence is in general a linear combination of thebasis elements, which we consider as actual external influences by virtue of theirrole in laboratory space expressed in (2.2) and the consistency condition (2.14). Remember in the derivation of the consistency condition (2.14), which is our point of de-parture, the connection between the external influence function and the momentum in laboratoryspace is given by the equal time relation p i ( t, q, ˙ q ) (cid:12)(cid:12)(cid:12)(cid:12) t = a = ∂g ( t, q ) ∂q i (cid:12)(cid:12)(cid:12)(cid:12) t = a . That is, the function g ( t, q ), and ψ ( t, q ), is considered for an arbitrary but fixed point in timein laboratory space and the points q should be considered as the set of points constituting theconfiguration space at this point in time. As the set of solutions of the laboratory space dynamicsof the field ψ ( q ) will result in the Hilbert space of states, this view is similar to the standardapproach in quantum mechanics where the Hilbert space of states is derived for a fixed point intime and the time-dependent Schr¨odinger equation is postulated additionally to describe the timeevolution of the system. N THE ROLE OF QUANTUM EVENTS IN DOUBLE-SLIT EXPERIMENTS 7
The possible external influences also fulfil the Schr¨odinger equation by linearityof (2.27) and from this end we are in formal alignment with standard quantummechanics. The difference, however, is that the actual external influence ψ mustfulfil the consistency condition (2.14) to be observable in laboratory space, whichhas an important consequence. We illustrate this shortly for the example of anon-degenerate Hilbert space. Suppose we have a general normed possible externalinfluence given as a superposition of eigenstates and subject to the normalization(2.28). We assume further that this external influence is observable in laboratoryspace. Then, this external influence fulfils the consistency condition (2.14) and itsbehaviour in laboratory space is given by the equation of motion of the variationalproblem (2.24) with constraint (2.25). However, the solutions of this problem serveas the basis of the Hilbert space which contradicts our assumption at the beginningof this derivation. Therefore, a superposition of external influences is not observablein laboratory space, which gives a mathematical explanation of the experimentalfact that only energy eigenstates are observable in measurements. Thus, the mea-surement problem for non-degenerate Hilbert spaces reduces to a mathematicalconsequence in our derivation.Let us highlight some key points in our derivation which are important for theinterpretation of the double-slit experiment. First, our approach to quantum me-chanics emphasizes the energy eigenvalue representation of the Hilbert space. Anexternal influence is always associated to a specific value of the constant E , which isconstrained by the experimental setting in laboratory space. Thus, we can assumethat for an external influence in empty laboratory space the energy constant E canhave any positive value and our Hilbert space L ( R ) is the space of square inte-grable function in R . This fact solves not only the problem of the introduction ofan infinite dimensional Hilbert space in the standard Hilbert space approach to theexperiment, see e.g. [3]. But, as we will see below, gives also a natural frameworkfor the positive operator valued measure (POVM) approach, since by Naimark’stheorem a POVM is a measure given on a subset of a standard Hilbert space [3],[4].Another key point in our derivation is that the influence is explicitly externalto the physical system which experiences/detects this external influence. This isreflected in the very beginning of our derivation by the role of the external influencefunction g in (2.2) as the boundary condition which kicks off the system in labo-ratory space. The transition to the wave function ψ as the Lie group element in(2.16) is the change of the description about the obtainable information to the de-scription of the object itself. Thus, in our derivation a quantum object representedby the wave function ψ is explicitly external to the physical system in laboratoryspace represented by the variational problem (2.2). The Schr¨odinger equation andHilbert space description describe the necessary and sufficient conditions in lab-oratory time and space to be detectable by the experimental setting representedby (2.2). Therefore, from the perspective of laboratory space our derivation isexplicitly non-realistic as the external influence does not exist in this space untilit kicked off the physical system. Also, our description of dynamics in laboratoryspace is explicitly non-local which can be seen by the employment of the Lagrangian(2.23) of the field theory. This Lagrangian uses the unparameterized space variables q and the ”equation of motion” needs to be interpreted in time-unparameterizedform. Therefore, the stationary Schr¨odinger equation (2.26) is explicitly non-local R. SCHUSTER in terms of space and time locality as the space coordinates do not depend on thetime variable in this formulation. Thus, we are in alignment with modern experi-ments which give strong evidence that quantum theory can not be interpreted as alocal and realist theory but needs to be build as a non-local and non-realist theory[8], [9].However, we need to emphasize that our versions of non-realism and non-localityare more restricted than the general usage of these terminologies since this formal-ism is explicitly local as pointed out in footnote 2 of [2], p68. Thus, the separationof object space - the space of small external influences - and laboratory space - thespace of the experimental setting - must be regard in the context of local spacesonly. Therefore, in our case, the notions non-realistic and, in particular, non-localhave the paradoxical meaning that they are applied to the local spaces of a fieldtheory only, and reflect our missing knowledge about the structure of a global man-ifold similar to the ignorance of the global metric and global space-time structurein general relativity, which forces us to reject the picture of an absolute space. Infact most of the interpretation problems of quantum mechanics are connected tothe tacit assumption of an absolute global space, which we give up in favor to alocal description of quantum mechanics.Moreover, the wave function is the representation of the initial disturbing objectin our approach and, therefore, we should regard a wave function as the repre-sentation of a quantum event instead of attaching this function permanently to aquantum system. This quantum event is closely related to the experimental settingwhich determines the possible outcomes perceivable in the laboratory mathemat-ically expressed in equation (2.2). An advantage of this view is that the objects,e.g. electrons, can change their quantum and classical roles arbitrarily as the eventof appearance in laboratory space explicitly determines the observability and thevisibility of observables, which removes this interpretation problem from the begin-ning. 3.
Revisit of the Double-Slit Experiment
Although the double-slit experiment lies at the heart of quantum mechanics itis amazing how many difficulties one encounters if one tries to actually describethis experiment in terms of the standard approach to quantum mechanics. Thesedifficulties can be easily seen as the assumption that the two wave functions ψ and ψ originating from the two slits can be regarded as state vectors of a Hilbertspace constituting the basis of a two-dimensional Hilbert space only, which doesnot cover the continuity of measured values.Modern approaches use a positive operator valued measure (POVM) to describethe experiment in the language of quantum mechanics, see e.g. ([3], [4]), whichextends the standard measure of quantum mechanics to reflect the difference be-tween the number of measurement values and the dimension of the Hilbert spaceassociated to the quantum system. But every POVM is connected to a standardHilbert space measure in an extended Hilbert space by Naimark’s theorem ([3], [4]),and thus indicates this approach as the special case of the description of a quantumeffect in a subspace of a larger Hilbert space. Therefore, the usage of a POVM isan indicator of the lack of knowledge of the correct Hilbert space of the regardedquantum system. N THE ROLE OF QUANTUM EVENTS IN DOUBLE-SLIT EXPERIMENTS 9
An alternative approach ([3], p. 343) assumes that the wave functions ψ and ψ emanating from each slit should be regarded as solutions of the time-independentSchr¨odinger equation for the same energy E which again leads to the usage ofan operator valued measure for the position measurement in the Hilbert spaceapproach. This ansatz is very close to our description of quantum mechanics, whichemphasizes the role of the space of external influences with wave functions subjectto the stationary Schr¨odinger equation (2.26) for the set of energy eigenvalues { E i } .In general a common fact is that regardless of the used approach the geometricalconfiguration of the experiment is crucial to the construction of the Hilbert spaceand interpretation of the measurement. That is, in every case we deal with twowave functions ψ and ψ associated to the slits of the experimental setting.Let us use our approach to describe the double-slit experiment. Analogous tothe latter approach we regard the wave functions ψ ,j and ψ ,j originating at slit1 and 2 as external influences which kick off a physical system in laboratory spacegiven by (2.2) with energy E j .The equation of motion (2.26), i.e. the stationary Schr¨odinger equation, of theexternal influence in laboratory space is given by the Laplace equations Hψ i,j = ∂ ∂q ψ i,j = E j ψ i,j , (3.1)which have a continuous spectrum of eigenvalues E j for each value of the index i . Thus, the vectors ψ i,j build the basis of an infinite-dimensional Hilbert spaceavoiding the difficulty of a finite eigenvalue spectrum which occurs in the standardapproach formulation of the experiment.Each individual set { ψ i,j } , i = 1 , H i ofpossible external influences emanating from slit i , which is experimentally realizedby opening one slit only during the entire investigation time of the experiment. TheHilbert space of all external influences for both slits individually measured is thengiven as the sum of the Hilbert space of the individual slits H = H ⊕ H . (3.2)Furthermore, the common set S = { ψ i,j , i = 1 , , Hψ i,j = E j ψ i,j } of all functions ψ i,j also constitutes a basis of a Hilbert space H of external influence originatingfrom both slits, which is experimentally realized by the usage of a screen as themeasuring device. This Hilbert space H is twofold degenerate by virtue of thetwo wave functions ψ ,j and ψ ,j . It is a well-known fact that every degenerateHilbert space with eigenstates ψ i , i = 1 , . . . , m of common eigenvalue λ can berepresented either in a basis of the individual eigenvectors ψ i or in a basis givenby a superposition of these eigenvectors. While the individual basis vectors areagain equivalent to external influences originating from an individual slit, the basisvectors given as the superposition ψ j = (cid:88) i =1 α i ψ i,j (3.3)represent an external influence with energy E j given as the superposition of externalinfluences emanating from both slits. Therefore, the role of the wave functions ψ i,j as basis vectors in Hilbert space H ⊕ H or H and the measured values dependson the experimental set up as one would naturally expect. Individual Visibility Full VisibilityPartial VisibilitySource SourceSourceSlit 1 Slit 1
Slit 1
Slit 2 Slit 2Slit 2Measuring Device Measuring DeviceMeasuring DeviceMeasuring Device
Figure 1.
Possible ray visibilities for different experimental settingsAgain, recall the crucial point in our derivation that the Schr¨odinger equationsdescribe the appearance of a small external influence in classical laboratory spaceat a definite point in time. That is, the geometry of the configuration space ofthe laboratory at this specific point in time determines the quantum event, i.e.the action of the influence, and needs to be taken into account. In the case ofthe double-slit experiment the significant event is located in the detector behindthe slits, which gives evidence of the occurrence of the quantum particle in thelaboratory. Since we also restricted ourselves to the family of external influences,which are traceable as boundary conditions in time given by (2.12) and (2.13), the’ray visibility’ at the point of contact with laboratory space plays a crucial role inour approach. Therefore, we have three possibilities of ray visibility which match tothe experimental settings of full visibility, partial visibility and individual visibilityof the slits as indicated in figure 1, which lead to the realizations of the Hilbertspaces given above.Furthermore, the normalization condition (2.28) for a state vector in a Hilbertspace also plays the role of the constraint (2.25) for the variational problem (2.24),and we interpret this constraint as the condition of certain occurrence of the exter-nal influence in laboratory space. Additionally, only solutions for a specific energyeigenvalue E j of the variational problem are observable in an individual measure-ment process by virtue of the consistency condition (2.14), and every detectableexternal influence must be an element of the energy eigenvalue basis. Therefore, N THE ROLE OF QUANTUM EVENTS IN DOUBLE-SLIT EXPERIMENTS 11 energy eigenvalue superpositions are not detectable in laboratory space as confirmedby all quantum experiments.Thus, in the case of the double-slit experiment with one slit closed for eachmeasurement process, the Hilbert space of possible external influences H ⊕ H isthe sum of the individual Hilbert space of external influences emanating from oneindividual slit, and the normalization conditions (cid:90) | ψ i | dq = 1 , i = 1 , i with given energy E j , where we suppress the energyindex j in the following equations. Thus, the integral (3.4) over entire laboratoryspace expresses the certain appearance of an external influence in laboratory spaceoriginating from the individual slits. The integral over the interval [ x, x + δx ] givesthen the probability p i ( x ) = (cid:90) x + δxx | ψ i ( q ) | dq (3.5)to measure the external influence emanating from slit i in this region and thetotal probability is the sum of the individual probabilities normed to 1, which isequivalent to the particle picture of standard quantum mechanics.Let us consider now the Hilbert space of possible external influences originatingfrom both slits. As mentioned above, this Hilbert space is degenerate and can berepresented in two different energy eigenvalue bases. One possibility is to representthe states in the basis of individual functions ψ and ψ with energy eigenvalue E j ,which corresponds to the scenario of partial visibility as shown in the figure above.Then, each basis element ψ i is again subject to the consistency condition for theobservability of a small external influence in laboratory space with normalizationcondition (2.28) (cid:90) | ψ i | dq = 1 , i = 1 , . (3.6)This leads again to the probability (3.5) of finding the particle in the interval [ x, x + δx ] with energy E j . But, analogous to the non-degenerate case a superposition state ψ = α ψ + α ψ is not observable in laboratory space in this basis. Therefore,this basis describes exclusively the particle picture of the theory.On the other hand, we can use the superposition state ψ = ( α ψ + α ψ )(3.7)as the basis of the energy eigenvalue representation of the Hilbert space, whichcorresponds to full visibility of both slits at the instance in time and space ofthe quantum event. This basis allows us to identify the superposition state ofthe external influences with the normalization condition (2.28) which leads to theprobability p i ( x ) = (cid:90) x + δxx | α ψ + α ψ | dq (3.8)of finding the particle in the interval [ x, x + δx ] with energy E j . Since the ψ i areeigenstates of the degenerate eigenvalue E j these are not orthogonal and result intointerference terms leading to the wave interpretation of the experiment. Now, the individual wave functions ψ i are superposition states and are not observable in thisbasis by virtue of the consistency condition. Thus, the complementary of wave andparticle behaviour is a result of the basis change in a degenerate Hilbert space andno additional assumptions need to be introduced to explain this experimental fact.4. Conclusion
Our view of quantum mechanics does not only allow the derivation of the Schr¨o-dinger equation and the Hilbert space of states from physically intuitive assump-tions, but gives also an explanation of wave-particle duality. In contrast to standardquantum mechanics we do not need to incorporate additional concepts to explainthe complementarity of waves and particles, but interpret this fact as the result ofa basis change in a Hilbert space, which is closely tied to a quantum event of anexperimental setting in a laboratory.In addition, all considerations above apply to delayed choice experiments, asexperimentally realized in e.g. [10], since our approach singles out for the quan-tum event of significance the laboratory space configuration at the definite pointin time of appearance in the detector, which removes the paradox that a quantumobjects changes its wave or particle nature during its experimental lifetime. Theinterpretation problems in traditional approaches stem from the implicit assump-tion of historic consistence which is connected to the tacit picture that the waveor particle property is attached to an object ”flying” from the source to the detec-tor, which must therefore change its property during the ”flight”. In our case thetime traceability of the quantum object, the small external influence, is reflectedby the restriction to the family of consistent boundary conditions, which gives usthe possibility to identify the quantum object at an arbitrary point in time bymeasuring in classical laboratory space. Thus, we can picture the same propositionthat if we measured an electron at time t we can conclude it is the same elec-tron we would have measured at point t somewhere else in the experimental setup. The difference, however, is that the small influence, which causes the quantumevent, is explicitly external until its appearance in classical laboratory space andonly the geometry of the classical configuration space at this point in time plays acrucial role. Therefore, we associate the wave or particle property of an object to aquantum event which is located at a specific point in time and space in laboratoryspace, which removes the paradoxical assumption that the particle must change itsproperty over time.However, this comes at the cost of a radical modification of quantum mechanicswhich is more on a conceptual level than a formal one. First, a quantum eventis not an independent property of an object, but is always closely related to theexperimental setting in laboratory space since the Hamilton-Jacobi equation for theexternal influence explicitly requires the Hamiltonian of the experimental environ-ment. This is not a serious restriction as it enables us to show that the reductionof a state vector is equivalent to the transition of a possible external influence to anactual external influence which must be a basis element of the energy representa-tion of the corresponding Hilbert space. In fact it enables us to show superpositionstates can only be measured for states of a degenerate energy eigenvalue and cannotbe measured in an energy superposition state for different energies. N THE ROLE OF QUANTUM EVENTS IN DOUBLE-SLIT EXPERIMENTS 13
Second, the separation of the external influence from the laboratory system andthe view of the wave function as a field in coordinate space only incorporate non-reality and non-locality into our derivation. Moreover, the explicit local construc-tion of our approach and the separation between object space and laboratory spacealso removes the implicit assumption of absolute space given in the standard ap-proach and introduces the concepts of non-reality and non-locality in a relativecontext between local spaces. We believe that our explicit incorporation of non-locality and non-reality into a local theory can be a first step to a better physicalunderstanding of quantum mechanics as modern investigations and experimentsgive evidence that one has to give up the concepts of reality and locality in quan-tum mechanics [8],[9], [11], [12].The author wants to thank M. Harris for reading the manuscript.
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