The Fermionic Projector, Entanglement, and the Collapse of the Wave Function
aa r X i v : . [ qu a n t - ph ] J u l THE FERMIONIC PROJECTOR, ENTANGLEMENT,AND THE COLLAPSE OF THE WAVE FUNCTION
FELIX FINSTERNOVEMBER 2010
Abstract.
After a brief introduction to the fermionic projector approach, we re-view how entanglement and second quantized bosonic and fermionic fields can bedescribed in this framework. The constructions are discussed with regard to decoher-ence phenomena and the measurement problem. We propose a mechanism leading tothe collapse of the wave function in the quantum mechanical measurement process.
Contents
1. An Introduction to the Fermionic Projector Approach 11.1. The Idea of Spontaneous Structure Formation 11.2. An Action Principle for Fermion Systems in Discrete Space-Time 31.3. Effects of Spontaneous Structure Formation 41.4. The Correspondence to Minkowski Space 51.5. A Formulation of Quantum Field Theory 62. Microscopic Mixing of Decoherent Subsystems 72.1. Description of Entangled Second Quantized Fermionic Fields 72.2. Description of Second Quantized Bosonic Fields 103. Physical Interpretation 113.1. The Superposition Principle 113.2. The Measurement Problem and Decoherence 123.3. The Wave-Particle Duality 153.4. The Collapse of the Wave Function 15References 171.
An Introduction to the Fermionic Projector Approach
In this section we give a brief introduction to the framework of the fermionic projec-tor, outlining a few ideas, methods and results (for details see [4, 8] and the referencesin the review papers [9, 11]).1.1.
The Idea of Spontaneous Structure Formation.
Due to the ultraviolet di-vergences of quantum field theory and the difficulties in quantizing gravity, it is gener-ally believed that our notions of space and time should be modified on a microscopicscale, usually thought of as the Planck scale. However, there is no agreement on whatthe right framework for physics on the Planck scale should be. A major difficulty is
Supported in part by the Deutsche Forschungsgemeinschaft. that if one modifies the microscopic structure of Minkowski space or of a Lorentzianmanifold in a naive way, the gauge invariance and the diffeomorphism invariance ofthe resulting physical theory are not preserved. For example, a simple ultravioletregularization is to replace Minkowski space by a four-dimensional lattice. If one nowworks instead of partial derivatives with difference quotients, then the gauge invarianceis lost. A better method is to work with the integrals of the gauge potentials alongthe links of the lattice, which restores gauge invariance and leads to the frameworkof lattice gauge theories. However, lattice gauge theory is not compatible with theequivalence principle, because after performing a diffeomorphism, the lattice pointsare no longer in a regular configuration. Taking a somewhat different point of view,the problem of a lattice is that it uses structures (like the nearest neighbor relation orthe lattice spacing) which refer to an ambient space-time, although we wanted to avoidthis background space-time for the formulation of the physical model. More generally,one would like to have a framework which is background independent and respects thefundamental physical principles.The fermionic projector approach provides a framework which meets the above re-quirements. We begin with a finite set of space-time points M = { , . . . , m } , withoutpresupposing any relations between the space-time points. On this finite set of points,we then define an “ensemble of wave functions” ψ , . . . , ψ f . We introduce an action S which to every configuration of wave functions associates a positive number. By min-imizing S , the wave functions tend to get into a configuration where the action issmaller. This process can be understood as a self-organization of the wave functions,driven by our action principle. In the resulting minimizing configuration, the wavefunctions are typically delocalized in the sense that they are non-zero on many space-time points. Thus by considering the correlations of the wave functions at differentspace-time points, one gets relations between these points. With this mechanism, re-ferred to as spontaneous structure formation , additional structures and objects arise inspace-time. The idea is that all the structures needed for the formulation of physics,like the causal structure, gauge potentials and fields, the Lorentzian metric, etc., aregenerated in this way. Finally, evaluating the Euler-Lagrange equations correspond-ing to our action principle in terms of all these structures should give the physicalequations.Before we can make this idea mathematically precise, we need to specify in whichspace the wave functions ψ i should take their value. To this end, we associate to everyspace-time point x ∈ M a complex vector space S x , referred to as the spin space .We endow this vector space with an indefinite inner product ≺ . | . ≻ x , the spin scalarproduct . Then a wave functions at x takes values in the corresponding spin space, ψ i : x ψ i ( x ) ∈ S x . In physical applications, a wave function should correspond to a four-component Diracwave function with the usual inner product ≺ ψ | φ ≻ x = ψ ( x ) φ ( x ), where ψ = ψ † γ isthe adjoint spinor (this will be made precise in Section 1.4 below). With this in mind,we assume that S x is four-dimensional, and that the spin scalar product is indefiniteof signature (2 , H = M x ∈ M S x , <ψ | φ> := X x ∈ M ≺ ψ ( x ) | φ ( x ) ≻ x , HE FERMIONIC PROJECTOR, ENTANGLEMENT AND COLLAPSE 3 we obtain an inner product space (
H, <. | .> ) of dimension 4 m and signature (2 m, m ).We can then describe the evaluation of a wave function at a space-time point x by aprojection operator E x , ψ ( x ) = E x ψ with E x : H → S x : ψ ψ ( x ) . Moreover, it is convenient to describe the ensemble of wave functions ψ , . . . , ψ f ∈ H by a projector P on the subspace of H spanned by these wave functions. This motivatesthe framework of fermion systems in discrete space-time which we now introduce.1.2. An Action Principle for Fermion Systems in Discrete Space-Time.
Welet (
H, <. | .> ) be a finite-dimensional complex vector space endowed with an indefiniteinner product (thus < . | . > is non-degenerate, but not positive definite). Next, welet M = { , . . . , m } be a finite set. To every point x ∈ M we associate a projector E x (a projector in H is defined just as in Hilbert spaces as a linear operator whichis idempotent and symmetric). We assume that these projectors are orthogonal andcomplete in the sense that E x E y = δ xy E x and X x ∈ M E x = 11 . (1.1)Furthermore, we assume that the images E x ( H ) ⊂ H of these projectors are non-dege-nerate subspaces of H , which all have the same signature (2 , x ∈ M arecalled discrete space-time points , and the corresponding projectors E x are the space-time projectors . The structure ( H, <. | .> , ( E x ) x ∈ M ) is called discrete space-time . Theparticles of our system are described by one more projector P in H , the so-called fermionic projector , which has the property that its image P ( H ) is a negative definitesubspace of H . The resulting system ( H, <. | .> , ( E x ) x ∈ M , P ) is referred to as a fermionsystem in discrete space-time .Let us briefly discuss these definitions and introduce a convenient notation. Thevectors in the image of P have the interpretation as the occupied fermionic states ofour system, and thus the rank of P gives the number of particles f := dim P ( H ). Thespace-time projectors E x can be used to project vectors of H to the subspace E x ( H ) ⊂ H . Using a more graphic notion, we also refer to this projection as the localization at the space-time point x . For the localization of a vector ψ ∈ H we use the shortnotation ψ ( x ) := E x ψ (1.2)and refer to ψ ( x ) as the corresponding wave function . The wave functions at x spanthe vector space E x ( H ), referred to as the spin space at x and denoted by S x . It isendowed with the inner product < . | E x .> of signature (2 , ≺ . | . ≻ and refer to as the spin scalar product . Using the relations (1.1), we can thenwrite <ψ | φ> = X x ∈ M ≺ ψ ( x ) | φ ( x ) ≻ . (1.3)The localization of the fermionic projector is denoted by P ( x, y ) := E x P E y . Thisoperator maps S y ⊂ H to S x , and we usually regard it as a mapping between thesesubspaces, P ( x, y ) = E x P E y : S y → S x . F. FINSTER
Again using (1.1), we can write the wave function corresponding to
P ψ as follows,(
P ψ )( x ) = E x P ψ = X y ∈ M E x P E y ψ = X y ∈ M ( E x P E y ) ( E y ψ ) , and thus ( P ψ )( x ) = X y ∈ M P ( x, y ) ψ ( y ) . (1.4)This relation resembles the representation of an operator with an integral kernel, andtherefore we call P ( x, y ) the discrete kernel of the fermionic projector.In order to introduce our action principle, for any x, y ∈ M we define the closedchain A xy by A xy = P ( x, y ) P ( y, x ) : S x → S x . (1.5)We denote the eigenvalues of A xy counted with algebraic multiplicities by λ xy , . . . , λ xy .We define the Lagrangian L by L [ A xy ] = X i,j =1 (cid:16) | λ xyi | − | λ xyj | (cid:17) (1.6)(for simplicity we here leave out the constraints and consider only the so-called criticalcase of the auxiliary Lagrangian). Our action principle is tominimize the action S [ P ] = 18 X x,y ∈ M L [ A xy ] , (1.7)where we consider variations of the fermionic projector, keeping the number of particlesas well as discrete space-time fixed. For a more general mathematical exposition of thisvariational principle and the existence theory we refer to [7]. For a discussion of theunderlying physical principles see [5, Section 2]. We finally remark that generalizingthe above setting leads to the framework of causal fermion systems (see [13] and thereferences therein).1.3. Effects of Spontaneous Structure Formation.
We now discuss two effectsof spontaneous structure formation: the spontaneous breaking of the permutationsymmetry and the generation of a discrete causal structure.We first point out that in the above definitions, we did not distinguish an orderingof the space-time points; all our notions are symmetric under permutations of thepoints of M . This suggests that a minimizing fermionic projector should also bepermutation symmetric. Surprisingly, this is not the case. Indeed, in [6] it is shownunder general assumptions that that the fermionic projector cannot be permutationsymmetric. This shows rigorously that the effect of spontaneous structure formationreally occurs, giving rise to a non-trivial symmetry group acting on the discrete space-time points. Before stating the main result, we need to specify what we mean by asymmetry of our fermion system. The group of all permutations of the space-timepoints is the symmetric group, denoted by S m . Definition 1.1.
A subgroup
O ⊂ S m is called outer symmetry group of the fermionsystem in discrete space-time if for every σ ∈ O there is a unitary transformation U such that U P U − = P and U E x U − = E σ ( x ) for all x ∈ M . (1.8)
HE FERMIONIC PROJECTOR, ENTANGLEMENT AND COLLAPSE 5
Theorem 1.2. (spontaneous breaking of the permutation symmetry)
Supposethat ( H, <. | .> , ( E x ) x ∈ M , P ) is a fermion system in discrete space-time. Assume thatthe number of space-time points and the number of particles f are in the range m > and < f < m − . Then the fermion system cannot have the outer symmetry group O = S m . In particular, this theorem applies in the physically interesting case of many particlesand even more space-time points.More detailed information on the space-time relations generated by the spontaneousstructure formation is obtained by analyzing the eigenvalues of the closed chain. Notethat in an indefinite inner product space, the eigenvalues of a self-adjoint operator A need not be real, but alternatively they can form complex conjugate pairs (see [16]or [7, Section 3]). This fact can be used to introduce a notion of causality. Definition 1.3. (causal structure)
Two space-time points x, y ∈ M are called time-like separated if the spectrum of A xy is real. The points are spacelike separated if thespectrum of A xy forms two complex conjugate pairs having the same absolute value. Inall other cases, the two points are lightlike separated. This notion is compatible with our action principle in the following sense. Supposethat two space-time points x and y are spacelike separated. Then the eigenvalues λ xyi all have the same absolute value, and thus the Lagrangian (1.6) vanishes. A shortcalculation shows that the first variations of the action also vanish, so that A xy doesnot enter the corresponding Euler-Lagrange equations. This can be seen in analogyto the usual notion of causality that points with spacelike separation cannot influenceeach other.We point out that by analyzing the kernel of the fermionic projector in more detail,one can even distinguish a direction of time, define a parallel transport of spinors andintroduce curvature, thus giving a proposal for a “Lorentzian quantum geometry.” Theinterested reader is referred to [12] and the survey article [13].1.4. The Correspondence to Minkowski Space.
We now describe how to get aconnection between discrete space-time and Minkowski space. The simplest methodfor getting a correspondence to relativistic quantum mechanics in Minkowski spaceis to replace the discrete space-time points M by the space-time continuum R andthe sums over M by space-time integrals. For a vector ψ ∈ H , the correspondinglocalization E x ψ should be a 4-component Dirac wave function, and the spin scalarproduct ≺ ψ | φ ≻ on E x ( H ) should correspond to the usual Lorentz invariant scalarproduct on Dirac spinors ψφ with ψ = ψ † γ the adjoint spinor. In view of (1.4), thediscrete kernel should go over to the integral kernel of an operator P on the Diracwave functions, ( P ψ )( x ) = Z M P ( x, y ) ψ ( y ) d y . The image of P should be spanned by the occupied fermionic states. We take Dirac’sconcept literally that in the vacuum all negative-energy states are occupied by fermionsforming the so-called Dirac sea . This leads us to describe the vacuum by the integralover the lower mass shell P sea ( x, y ) = Z d k (2 π ) ( k j γ j + m ) δ ( k − m ) Θ( − k ) e − ik ( x − y ) . (1.9) F. FINSTER (Here Θ is the Heaviside function, and in the terms k and k ( x − y ) we use theMinkowski inner product, using the signature convention (+ − −− )). Computingthe Fourier integral, one sees that P ( x, y ) is a smooth function, except on the lightcone { ( y − x ) = 0 } , where it has poles and singularities.Let us compare Definition 1.3 with the usual notion of causality in Minkowski space.Even without computing the Fourier integral (1.9), it is clear from Lorentz symmetrythat for every x and y for which the Fourier integral exists, P ( x, y ) can be written as P ( x, y ) = α ( y − x ) j γ j + β
11 (1.10)with two complex coefficients α and β . Taking the conjugate, we see that P ( y, x ) = α ( y − x ) j γ j + β . As a consequence, A xy = P ( x, y ) P ( y, x ) = a ( y − x ) j γ j + b
11 (1.11)with real parameters a and b given by a = αβ + βα , b = | α | ( y − x ) + | β | . (1.12)Applying the formula ( A xy − b = a ( y − x ) , one can easily compute the roots ofthe characteristic polynomial of A xy , λ = λ = b + p a ( y − x ) , λ = λ = b − p a ( y − x ) . (1.13)If the vector ( y − x ) is timelike, we see from the inequality ( y − x ) > λ j are all real. If conversely the vector ( y − x ) is spacelike, the term ( y − x ) < λ j form a complex conjugate pair. This shows thatfor Dirac spinors in Minkowski space, Definition 1.3 is indeed consistent with the usualnotion of causality. Moreover, from the fermionic projector (1.9) one can deduce theMinkowski metric as well as all all the familiar objects of Dirac theory. For example,the non-negative quantity ≺ ψ | γ ψ ≻ has the interpretation as the probability densityof the Dirac particle. Polarizing and integrating over space yields the scalar product( ψ | φ ) = Z t =const ≺ ψ ( t, ~x ) | γ φ ( t, ~x ) ≻ d~x . (1.14)For solutions of the Dirac equation, current conservation implies that this scalar prod-uct is time independent. In this way, we can describe Dirac theory in Minkowski spacein the setting of fermionic projector.A more difficult question is if and in which sense the distribution (1.9) is a minimizerof our action principle (1.7). We refer the interested reader to the survey article [9]and the references therein.1.5. A Formulation of Quantum Field Theory.
The above correspondence tovacuum Dirac sea structures can also be used to analyze our action principle for in-teracting systems in the so-called continuum limit . We now outline a few ideas andconstructions (for details see [4, Chapters 4], [8] and the survey article [11]). First,it is helpful to observe that the vacuum fermionic projector (1.9) is a solution of theDirac equation ( iγ j ∂ j − m ) P sea ( x, y ) = 0. To introduce the interaction, we replacethe free Dirac operator by a more general Dirac operator, for example involving gaugepotentials or a gravitational field. Thus, restricting attention to the simplest case ofan electromagnetic potential A , we demand that (cid:0) iγ j ( ∂ j − ieA j ) − m (cid:1) P ( x, y ) = 0 . (1.15) HE FERMIONIC PROJECTOR, ENTANGLEMENT AND COLLAPSE 7
Moreover, we introduce particles and anti-particles by occupying (suitably normalized)positive-energy states and removing states of the sea, P ( x, y ) = P sea ( x, y ) − π n f X k =1 | ψ k ( x ) ≻≺ ψ k ( y ) | + 12 π n a X l =1 | φ l ( x ) ≻≺ φ l ( y ) | . (1.16)Using the so-called causal perturbation expansion and light-cone expansion, the fermio-nic projector can be introduced uniquely from (1.15) and (1.16).It is important that our setting so far does not involve the field equations; in particu-lar, the electromagnetic potential in the Dirac equation (1.15) does not need to satisfythe Maxwell equations. Instead, the field equations should be derived from our actionprinciple (1.7). Indeed, analyzing the corresponding Euler-Lagrange equations, onefinds that they are satisfied only if the potentials in the Dirac equation satisfy certainconstraints. Some of these constraints are partial differential equations involving thepotentials as well as the wave functions of the particles and anti-particles in (1.16).In [8], such field equations are analyzed in detail for a system involving an axial field.In order to keep the setting as simple as possible, we here consider the analogous fieldequation for the electromagnetic field ∂ jk A k − (cid:3) A j = e n f X k =1 ≺ ψ k | γ j ψ k ≻ − e n a X l =1 ≺ φ l | γ j φ l ≻ . (1.17)With (1.15) and (1.17), the interaction as described by the action principle (1.7) re-duces in the continuum limit to the coupled Dirac-Maxwell equations . The many-fermion state is again described by the fermionic projector, which is built up of one-particle wave functions . The electromagnetic field merely is a classical bosonic field .Nevertheless, regarding (1.15) and (1.17) as a nonlinear hyperbolic system of partialdifferential equations and treating it perturbatively, one obtains all the Feynman dia-grams which do not involve fermion loops. Taking into account that by exciting seastates we can describe pair creation and annihilation processes, we also get all dia-grams involving fermion loops. In this way, we obtain agreement with perturbativequantum field theory (for details see [8, § Microscopic Mixing of Decoherent Subsystems
In the recent paper [10], a method was developed for describing entanglement andsecond quantized fermionic and bosonic fields in the framework of the fermionic pro-jector. We now outline a few constructions and results from this paper.2.1.
Description of Entangled Second Quantized Fermionic Fields.
Recallthat in the setting of Section 1.2 we described the fermion system by a projector P on an f -dimensional, negative definite subspace of H . It is now useful to choose anorthonormal basis ψ , . . . , ψ f of P ( H ) (meaning that < ψ i | ψ j > = − δ ij ). Then thediscrete kernel of the fermionic projector can be written in bra/ket-notation as P ( x, y ) = − f X j =1 | ψ j ( x ) ≻≺ ψ j ( y ) | . (2.1)In order to get a connection to the Fock space formalism, we take the wedge productof the one-particle wave functions,Ψ := ψ ∧ · · · ∧ ψ f , (2.2) F. FINSTER ∆ t M M ~xt Figure 1.
Example of a microscopic mixing of two space-time regions.to obtain a vector in the fermionic Fock space. This shows that the fermions can indeedbe described by a second-quantized field. However, as Ψ is a Hartree-Fock state, it isimpossible to describe entanglement (see for example [10, Example 3.1]).This shortcoming is removed by introducing the concept of microscopic mixing . It isbased on the assumption is that P has a non-trivial microstructure. This assumptionseems natural in view of setting in discrete space-time (see Section 1.2), where space-time is not smooth on the microscopic scale. “Averaging” this microstructure overmacroscopic regions of space-time gives rise to an effective kernel P ( x, y ) of a moregeneral form, making it possible to describe entanglement.To explain the construction, suppose that we want to describe the fermionic stateΨ = Ψ (1) + Ψ (2) , being a superposition of two states, each of which is an f -particleHartree-Fock state of the formΨ ( a ) := φ ( a )1 ∧ · · · ∧ φ ( a ) f ( a = 1 ,
2) (2.3)(the generalization to an arbitrary finite number of subsystems is straightforward).We now decompose Minkowski space M into two disjoint subsets M = M ∪ M with M ∩ M = ∅ , (2.4)which should be fine-grained in the sense that every macroscopic region of space-timeintersects both subsets (as indicated in Figure 1). We introduce one-particle wavefunctions ψ j which on M a coincide with φ ( a ) j , ψ j = ψ (1) j + ψ (2) j with ψ ( a ) j = χ M a φ ( a ) j (2.5)(here χ M a denotes the characteristic function defined by χ M a ( x ) = 1 if x ∈ M a and χ M a ( x ) = 0 otherwise). The form of the resulting fermionic projector (2.1) dependson whether its arguments x and y are in M or M , P ( x, y ) = − f X j =1 | ψ ( a ) j ( x ) ≻≺ ψ ( b ) j ( y ) | if x ∈ M a and y ∈ M b . If both arguments are in the same subsystem, then the fermionic projector is com-posed of the wave functions of this subsystem. But if the arguments are in differentsubsystems, then the fermionic projector involves both wave functions φ (1) j and φ (2) j ,thus giving correlations between the two subsystems. The idea is to remove all thesecorrelation terms by bringing the wave functions of the subsystems “out of phase.” Intechnical terms, we transform the wave functions of the second subsystem by a unitary HE FERMIONIC PROJECTOR, ENTANGLEMENT AND COLLAPSE 9 matrix U , ψ (1) j → ψ (1) j , ψ (2) j → ˜ ψ (2) j := f X k =1 U jk ψ (2) k with U ∈ SU( f ) . (2.6)This transformation does not change the fermionic projector if its two arguments arein the same subsystem, because in the case x, y ∈ M , the unitarity of U yields that P ( x, y ) → − f X j =1 | ˜ ψ (2) j ( x ) ≻≺ ˜ ψ (2) j ( y ) | = − f X j,k =1 ( U U † ) jk | ψ (2) j ( x ) ≻≺ ψ (2) k ( y ) | = − f X j | ψ (2) j ( x ) ≻≺ ψ (2) j ( y ) | = P ( x, y ) . However, if the two arguments of the fermionic projector are in different space-timeregions, the operator U does not drop out. For example, if x ∈ M and y ∈ M , P ( x, y ) → − f X j =1 | ˜ ψ (2) j ( x ) ≻≺ ψ (1) j ( y ) | = − f X j,k =1 U jk | ψ (2) j ( x ) ≻≺ ψ (1) k ( y ) | . (2.7)In the special case when U is a diagonal matrix whose entries are phase factors, U = diag( e iϕ , . . . , e iϕ f ) with f X j =1 ϕ j = 0 mod 2 π , the summations in (2.7) reduce to one sum involving the phase factors, P ( x, y ) → − f X j =1 e iϕ j | ψ (2) j ( x ) ≻≺ ψ (1) j ( y ) | . If the angles ϕ j are chosen stochastically, the phases of the summands are random. Asa consequence, there will be cancellations in the sum. At this point, it is importantto remember that the number f of fermions of our system is very large, because wealso count the states of the Dirac sea (see Section 1.5). As a consequence, P ( x, y ) willbe very small. More generally, we find that if U is a random matrix, P ( x, y ) involes ascale factor f − if x and y lie in different subsystems.Next we need to homogenize the wave functions by “taking averages” over macro-scopic regions of space-time. Our concept is that this homogenization should take placein the quantum mechanical measurement process, because the measurement device isalso composed of wave functions which are “smeared out” on the microscopic scale. Inorder to implement this idea, we introduce the so-called measurement scalar product ,where we replace the spatial integral in (1.14) by integrals over a strip of width ∆ t inspace-time (see Figure 1 and the more detailed treatment in [10, Section 4.2 and 4.3]).As a consequence of the homogenization process, the system is described effectively bythe sum of the fermionic projectors corresponding to the two subsystems.In order to clarify the above construction in the Fock space formalism, we nextconsider the many-particle wave function (2.2). Using (2.5), we obtainΨ = ψ ∧ · · · ∧ ψ f = ( ψ (1)1 + ψ (2)1 ) ∧ · · · ∧ ( ψ f + ψ f ) . (2.8) Multiplying out, we obtain many summands. Two of these summand correspond to theHartree-Fock states of the two subsystems (2.3), but we obtain many other contribu-tions. These additional contributions are again removed by the transformation (2.6).Namely, this transformation leaves the many-particle wave functions of the subsystemsremain unchanged, becauseΨ (2) → ˜Ψ (2) = ˜ ψ (2)1 ∧ · · · ∧ ˜ ψ (2) f = det U ψ (2)1 ∧ · · · ∧ ψ (2) f = Ψ (2) . (2.9)But all the summands obtained by multiplying out (2.8) which involve factors ψ (1) j as well as ψ (2) k contain matrix elements of U . These matrix elements again becomesmall if f gets large. As a consequence, the system is described effectively by the sumof the Hartree-Fock states of the subsystems. The homogenization process can againbe described by the measurement scalar product. It is most convenient to introduceon the many-particle wave functions the scalar product induced by the measurementprocess, giving rise to the effective fermionic Fock space F eff .The above consideration can be understood using the notion of decoherence. Ifthe one-particle wave functions ψ (1) j and ψ (2) j are coherent or “in phase”, then thefermionic projector P ( x, y ) has the usual form, no matter whether x and y are in thesame subsystem or not. If however the wave functions in the subregions are decoherentor “out of phase”, then the fermionic projector P ( x, y ) becomes very small if x and y are in different subregion. We refer to this effect as the decoherence between space-timeregions . It should be carefully distinguished from the decoherence of the many-particlewave function (see for example [17]). Namely, as we saw in (2.9), in our case the many-particle wave functions remain unchanged. Thus they remain coherent, no decoherencein the sense of [17] appears. But the one-particle wave functions are decoherent (2.6),implying that correlations between the two subsystems become small.We finally remark that the mixing of subsystems can be generalized to the so-called holographic mixing , where the subsystems need no longer be localized in disjoint regionsof space-time. We refer the interested reader to [10, Section 5.3].2.2. Description of Second Quantized Bosonic Fields.
As outlined in Section 1.5,in the continuum limit the interaction is described by the Dirac equation coupled toclassical field equations. If our system involves a microscopic mixing of several sub-systems, we can take the continuum limit of each subsystem separately. Labeling thesubsystems by an index a = 1 , . . . , L and again considering for simplicity the elec-tromagnetic interaction, the interaction of the a th subsystem is described in analogyto (1.15) and (1.17) by the coupled Dirac-Maxwell equations (cid:16) iγ j ( ∂ j − ieA ( a ) j ) − m (cid:17) P ( a ) ( x, y ) = 0 ∂ kj A ( a ) k − (cid:3) A ( a ) j = e n f X k =1 ≺ ψ ( a ) k | γ j ψ ( a ) k ≻ − e n a X l =1 ≺ φ ( a ) l | γ j φ ( a ) l ≻ . (2.10)In particular, the subsystems have an independent dynamics as described by theseequations. To clarify the physical picture, we again point out that the underlying space-time is not smooth on the microscopic scale, but it is decomposed into subsystems M a which are microscopically mixed. If x and y are in different subsystems, the fermionicprojector P ( x, y ) is very small due to decoherence, implying that the causal relationsof Definition 1.3 no longer agree with the causal structure in Minkowski space. In other HE FERMIONIC PROJECTOR, ENTANGLEMENT AND COLLAPSE 11 words, the usual causal relations are not valid between points in different subsystems.This means that the subsystems should be regarded as having a simultaneous andindependent existence in our space-time (as indicated in Figure 1 by the two “space-time sheets”).The bosonic field in (2.10) is purely classical. The key observation for passing froma classical to a second-quantized bosonic field is that the dynamics of a free second-quantized bosonic field can be described equivalently by a superposition of solutionsof the classical field equations (for details see [10, Section 5.1]). Thus to describea free second-quantized bosonic field, it suffices to consider the microscopic mixingof subsystems (2.10) (each with a classical bosonic field A ( a ) satisfying the classicalfield equations), and to associate to every subsystem a complex number φ ( a ). Theresulting function φ ( a ) can be interpreted as a wave function on the space of classicalfield configurations. For clarity, we first construct φ ( a ) in the case when no fermions arepresent. In this case, we take the wedge product of all the states ψ ( a ) j which form thefermionic projector of the subsystem a to obtain a many-particle wave function Ψ ( a ) .We then compare this wave function with the corresponding wave function ˇΨ of areference system (i.e. of a Dirac sea configuration in the presence of the externalfield A ( a ) ). These two wave functions coincide up to a proportionality factor, and thuswe can define φ ( a ) by Ψ ( a ) = φ ( a ) ˇΨ with φ ( a ) ∈ C . (2.11)The last construction immediately generalizes to the situation when fermionic parti-cles and/or anti-particles are present. In this case, one decomposes the many-particlestate into a component involving the particles and anti-particle states in (1.16) andthe sea sates,Ψ ( a ) = (cid:16) ψ ( a )1 ∧ · · · ∧ ψ ( a ) n f ∧ φ ( a )1 ∧ · · · ∧ φ ( a ) n a (cid:17) ∧ h ψ n +1 ∧ · · · ∧ ψ f i . (2.12)Then after homogenizing, the particles and anti-particles give rise to a vector of thefermionic Fock space F eff , whereas the sea states can be used to define in analogyto (2.11) the second-quantized bosonic wave function φ .We finally point that in the above constructions, we worked with free bosonic fields.To get the connection to interacting quantum fields, we recall that in the continuumlimit (see Section 1.5 above), one gets all the Feynman diagrams. We thus obtainagreement also with the quantitative predictions of perturbative quantum field theory.Nevertheless, the fermionic projector approach is not equivalent to a local interactingquantum field theory (for example in the canonical formulation on a Fock space).Clarifying the connections and differences of the two frameworks is work in progress.3. Physical Interpretation
We now discuss the previous constructions and results from a conceptual point ofview. For clarity, we try to explain the physical picture in simple examples, withoutaiming for mathematical rigor nor maximal generality.3.1.
The Superposition Principle.
In the standard formulation of quantum physics,it is a general principle that superpositions of quantum states can be formed (for agood exposition see [17, Section 2.1]). The superposition principle holds for the one-particle wave functions in quantum mechanics just as well as for the quantum statedescribing the whole physical system. In the framework of the fermionic projector, however, where the whole system is described by a projector in an indefinite innerproduct space, the validity of the superposition principle is not obvious. We nowexplain this point in detail.In the framework of the fermionic projector, the superposition principle arises ondifferent levels. On the fundamental level of discrete space-time, the wave functions arevectors in the indefinite inner product space ( H , <. | .> ), so that superpositions of one-particle wave functions can be formed. In the continuum limit, where the interaction isdescribed by the Dirac equation (1.15) coupled to a classical field (1.17), we thus obtainthe superposition principle for the Dirac wave functions. Moreover, since the Maxwellequations are linear, the superposition principle also holds for classical electromagneticwaves.For many-particle states, however, the superposition principle does not hold on thefundamental level. In particular, it is impossible to form the naive superposition of twophysical systems, simply because the linear combination of two projectors in generalis no longer a projector. But the superposition principle does hold for the effectivefermionic many-particle wave function obtained by decomposing the system into de-coherent subsystems and homogenizing on the microscopic scale (see Section 2.1). Forthe free second-quantized electromagnetic field as described in Section 2.2, the su-perposition principle corresponds to taking linear combinations of the complex-valuedwave function φ defined on the classical field configurations. Since the function φ is constructed out of the fermionic many-particle wave function (see (2.11)), linearcombinations are again justified exactly as for the fermionic many-particle.We conclude that in the framework of the fermionic projector, the superpositionprinciple again holds. It is possible to form linear combinations of macroscopic systems(like a dead and a living cat). Nevertheless, the framework of the fermionic projectordiffers from standard quantum theory in that for many-particle wave functions, thesuperposition principle does not hold on the fundamental level, but it is merely aconsequence of the microscopic mixing of decoherent subsystems. This means that thesuperposition principle is overruled in situations when our action principle in discretespace-time (1.7) needs to analyzed beyond the continuum limit. We will come back tothis point in Section 3.4 in the context of collapse phenomena.3.2. The Measurement Problem and Decoherence.
One of the most contro-versial and difficult points in the understanding of quantum physics is the so-calledmeasurement problem. In simple terms, it can be understood from the dilemma thaton one side, the dynamics of a quantum system is described by a linear evolution equa-tion in a Hilbert space (for example the Schr¨odinger equation), and considering themeasurement apparatus as part of the system, one would expect that this linear anddeterministic quantum evolution alone should give a complete description of physics.But on the other side, the Copenhagen interpretation requires an external observer,who by making a measurement triggers a “collapse” or “reduction” of the wave func-tion to an eigenstate of the observable. It is not obvious how the external observer canbe described within the linear quantum evolution. Also, the statistical interpretationof the expectation value in the measurement process does not seem to correspond tothe deterministic nature of the quantum evolution.The measurement problem has been studied extensively in the literature, and manydifferent solutions have been proposed (see for example [20, 3, 17, 2, 1, 19]). Here we
HE FERMIONIC PROJECTOR, ENTANGLEMENT AND COLLAPSE 13 ψ ↑ ψ ↓ ψ Figure 2.
The Stern-Gerlach experiment.shall not try to enter an exhaustive discussion or comparison of the different interpre-tations of quantum mechanics. We only explain how our concepts fit into the pictureand give one possible interpretation which corresponds to the personal preference ofthe author. But it is well possible that the framework of the fermionic projector canbe adapted to other interpretations as well.We begin by considering the
Stern-Gerlach experiment . Thus a beam of atomspasses through an inhomogeneous magnetic field. Decomposing the wave function ψ into the components with spin up and down, ψ = ψ ↑ + ψ ↓ , (3.1)these two components feel opposite magnetic forces. As a consequence, the beamsplits up into two beams, leading to two exposed dots on the photographic material(see Figure 2). If the intensity of the beam is so low that only one atom passes throughthe magnetic field, then either the upper or the lower dot will be exposed, both withprobability one half. It is impossible to predict whether the electron will fly up ordown; only probabilistic statements can be made.Let us try to describe the Stern-Gerlach experiment in the framework of the fermionicprojector. For simplicity, we replace the atom by an electron (disregarding the Lorentzforce due to the electron’s electric charge). Then at the beginning, the system is de-scribed by a classical external magnetic field and a Dirac wave function ψ ( t, ~x ) whichat time t = 0 has the form of a wave packet moving towards the magnetic field. Thissituation is modeled by the fermionic projector in the continuum limit (1.16) for oneparticle; the dynamics is described by the Dirac equation (1.15) in the given externalfield. Solving the Dirac equation, the wave function splits into two components (3.1),which are deflected upwards and downwards, respectively. Writing the contribution ofthe wave function to the fermionic projector (1.16) as − π | ψ ( x ) ≻≺ ψ ( y ) | = − π (cid:16) | ψ ↑ ( x ) ≻≺ ψ ↑ ( y ) | + | ψ ↓ ( x ) ≻≺ ψ ↓ ( y ) | (3.2)+ | ψ ↑ ( x ) ≻≺ ψ ↓ ( y ) | + | ψ ↓ ( x ) ≻≺ ψ ↑ ( y ) | (cid:17) , (3.3)one gets contributions of different type. Namely, the two summands in (3.2) arelocalized at the upper and lower electron beam, respectively. The two summandsin (3.3), however, are delocalized and give correlations between the two beams. Asobserved in [8, Chapter 10], the Euler-Lagrange equations in the continuum limitcannot be satisfied if general delocalized contributions to the fermionic projector arepresent. This means that there should be a mechanism which tries to avoid nonlocalcorrelations as in (3.3). A possible method for removing the nonlocal correlations isto divide the system into two decoherent subsystems (as shown in Figure 1, although at this stage they do not necessarily need to be microscopically mixed), in such a waythat ψ ↑ belongs to the first and ψ ↓ to the the second subsystem. Then the continuumlimit is to be taken separately in the two subsystems. The contribution of the wavefunction in the two subsystems simply is − π | ψ ↑ ( x ) ≻≺ ψ ↑ ( y ) | and − π | ψ ↓ ( x ) ≻≺ ψ ↓ ( y ) | , respectively. Thus the delocalized terms (3.3) no longer occur, so that the problemof solving the Euler-Lagrange equations observed in [8, Chapter 10] has disappeared.This consideration gives a possible mechanism for the generation of subsystems .Let us carefully discuss different notions of decoherence . First of all, the space-timepoints of the two subsystems should be decoherent, in the sense that the states ofthe fermionic projector are unitarily transformed in the second subsystem (2.6). Thisdecoherence has the effect that the fermionic projector P ( x, y ) becomes very smallif x and y are in different subsystems (see (2.7)), implying that the two subsystemshave an independent dynamics in the continuum limit. However, the many-particlewave function of the system is not affected by the decoherence between the space-timepoints (see (2.9)). Rewriting it similar to (2.12) as the wedge product of the one-particle wave function ψ with the sea states, one sees that the quantum mechanicalwave functions of the two subsystems are still coherent. In particular, if the twobeams interfered with each other (for example after redirecting them with additionalStern-Gerlach magnets), they could be superposed quantum-mechanically, giving riseto the usual interference effects of the double slit experiment. We also point out thatthe dynamics of each subsystem is still described by the Dirac equation (1.15) in theexternal magnetic field. Since the Dirac equation is linear, solving it for ψ is the sameas solving it separately for the two components ψ ↑ and ψ ↓ . Thus at this point, thedynamics is not affected by the decomposition into subsystems; we still have the lineardeterministic dynamics as described by the Dirac equation.As just explained, at this stage the generation of subsystems has no effect on thedynamics of the system. This suggests that it should not be observable whether thesubsystems have formed or not. This motivates us to demand that expectation valuestaken with respect to the measurement scalar product (as introduced in Section 2.1)should not be affected by the generation of subsystems. Keeping in mind that, af-ter a suitable homogenization process, the measurement scalar product coincides withthe integral (1.14), we can say alternatively that the process of generation of subsys-tems should preserve the probability densities . This condition also ensures that whenwe get the connection to the statistical description of the measurement process, theprobabilities are indeed given by the spatial integrals of the absolute square of thewave functions. Since the probability density is the zero component of the probabilitycurrent ≺ ψ | γ j | ψ ≻ , we can say equivalently that the generation of subsystems shouldrespect current conservation . This assumption seems reasonable, because current con-servation holds in each subsystem as a consequence of the Dirac equation, and wemerely extend this conservation law to the situation when the number of subsystemschanges.The dynamics becomes more complicated when the wave function approaches thescreen, because the interaction with the electrons of the photographic material canno longer be described by an external field. Instead, one must consider the coupledDirac-Maxwell equations (1.15) and (1.17) for a many-electron system. Since theinteraction is no longer linear, it now makes a difference that the two subsystems have HE FERMIONIC PROJECTOR, ENTANGLEMENT AND COLLAPSE 15 an independent dynamics. More specifically, in the first subsystem the wave functionin the upper beam interacts with the electrons near the upper impact point, whereasthe second subsystem describes the interaction of the lower beam. The whole systemis a superposition of these two systems, described mathematically by a vector in theFock space F eff . Exactly as explained in [17, Chapter 3], the different interactionwith the environment leads to a decoherence of the many-particle wave functions ofthe two subsystems. Thus now it is no longer possible to form quantum mechanicalsuperpositions of the wave functions of the two subsystems. The whole system behaveslike a statistical ensemble of the two subsystems. Following the resolution of themeasurement problem as proposed in [17], one should regard the human observer asbeing part of the system. Thus the observer is also decomposed into two observers, onein each subsystem. The two observers measure different outcomes of the experiment.Due to the decoherence of their wave functions, the two observers cannot communicatewith each other and do not even experience the existence of their counterparts. Fromthe point of view of the observer, the outcome of the experiment can only be describedstatistically: the electron moves either up or down, both with probability one half.Other experiments like the spin correlation experiment can be understood similarly.One only needs to keep in mind that if entanglement is present, then the subsystemsmust be microscopically mixed. The quantum state of each subsystem is not entangled.But homogenizing on the microscopic scale leads to an effective description of thesystem by an entangled state in the Fock space F eff .We finally remark that the mechanism for the generation of subsystems proposedabove could be made mathematically precise by analyzing the action principle (1.7)in the discrete setting, going beyond the approximation of the continuum limit. Oneshould keep in mind that on this level, our action principle violates causality . Thus itis conceivable that the formation of subsystems depends on later measurements or thatsubsystems tend to form eigenstates of the subsequent measurement device. However,such effects can hardly be verified or falsified in experiments, and therefore we will notconsider them here (for a discussion of a measurable effect of causality violation see [8,Section 8.2]).3.3. The Wave-Particle Duality.
Following the above arguments, one can also un-derstand the wave-particle duality in a way where the wave function is the basicphysical object, whereas the particle character is a consequence of the interactionas described by the action principle (1.7). To explain the idea, we return to theStern-Gerlach experiment of Figure 2. In the previous section, we justified that if oneelectron flies through the magnetic field, the atom will expose either the upper or thelower dot on the screen, much in contrast to the behavior of a classical wave, whichwould be observable at both dots of the screen at the same time. Repeating the argu-ments of the previous section on the scale of the atoms of the photographic material,we conclude that more and more subsystems will form, which become decoherent asexplained in [17, Chapter 3]. For an observer in one of the subsystems, the electronwill not expose the whole dot on the screen uniformly, but it will only excite one atomof the photographic material. As a consequence, the electron appears like a pointparticle. Again, the outcome of the measurement can only be described statistically.3.4.
The Collapse of the Wave Function.
The resolution of the measurementproblem in Section 3.2 is conceptually convincing and explains the experimental ob-servations. It goes back to Everett’s “relative state interpretation” and has found many different variations (see [20] and [2]). But no matter which interpretation one prefers,there always remains the counter-intuitive effect that all the possible outcomes of ex-periments are realized as components of the quantum state of the system. Thus whentime evolves, the quantum state disintegrates into more and more decoherent compo-nents, which should all describe a physical reality. This phenomenon, which is oftensubsumed under the catchy but oversimplified title “many-worlds interpretation,” isdifficult to imagine and hard to accept. Another criticism is that decoherence leadsto an effective description by a density operator, which however does not uniquelydetermine the Fock states of the decoherent components (for details see [14, Section 4]or [1, Section 6]). In order to avoid these problems, it has been proposed to introducea mechanism which leads to a “collapse” or “reduction” of the wave function. Differentmechanism for a collapse have been discussed, in particular models where the collapseoccurs at discrete time steps [15] or is a consequence of a stochastic process [18]. Thesemodels have in common that the superposition principle is overruled by a nonlinearcomponent in the dynamics. The nonlinearity is chosen to be so weak that it doesnot contradict the experimental evidence for a linear dynamics. In other words, thenonlinear term is so small that it cannot be detected experimentally. But nevertheless,this term can be arranged to prevent superpositions of macroscopically different wavefunctions.In the framework of the fermionic projector, the dynamics as described by the ac-tion principle (1.7) is nonlinear. This indeed provides a new collapse mechanism , as wenow explain. Suppose that our system is described by many decoherent subsystems.Since perfect decoherence seems impossible to arrange, the kernel of the fermionicprojector P ( x, y ) will in general not vanish identically if x and y are in two differentsubsystems. This gives rise to a contribution to the action (1.7) which, although beingsmall due to decoherence, is strictly positive. These contributions, which “mix” dif-ferent subsystems, grow quadratically with the number of subsystems, thus penalizinga very large number of subsystems. This shows that there is a mechanism which triesto reduce the number of subsystems, violating the superposition principle and the in-dependent dynamics of the subsystems. It seems very plausible that this mechanismleads to a collapse of the wave function (although a derivation from the action princi-ple (1.7) or a quantitative analysis has not yet been given). We also point out that, justas in other collapse theories, the collapse itself seems very difficult to observe. Namely,as systems whose quantum states are decoherent no longer interact with each other, itis impossible to decide whether there are other worlds beside the one observed by us,or whether the whole system has collapsed into the state which we observe. For thisreason, despite its importance for the interpretation of quantum theory, the issue of thecollapse of the wave function is often regarded as being speculative. But at least, theframework of fermion systems in discrete space-time outlined in Section 1.2 providesa well-defined mathematical setting for studying collapse phenomena. Analyzing thisframework in more detail might help to overcome the open problems discussed in [19],thus leading to a fully satisfying quantum theory.To summarize the physical interpretation, the framework of the fermionic projectorseems in agreement with the superposition principle and the decoherence phenomenawhich explain the appearance of our classical world as well as the wave-particle duality.Our description is more concrete than the usual Fock space formulation because thedecoherent components of the quantum state should all be realized in space-time bythe states of the fermionic projector. Moreover, we saw qualitatively that our action HE FERMIONIC PROJECTOR, ENTANGLEMENT AND COLLAPSE 17 principle (1.7) provides a mechanism for a collapse of the wave function and a reduc-tion of the number of decoherent components. In view of the fact that this collapseseems very difficult to observe in experiments, the remaining question of how many“different worlds” are realized in our space-time seems of more philosophical nature.The personal view of the author is that the fermionic projector should only realize onemacroscopic world, but at present this is mere speculation.
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Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, D-93040 Regensburg, Germany
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