Fermionic Fock Spaces and Quantum States for Causal Fermion Systems
aa r X i v : . [ m a t h - ph ] J a n FERMIONIC FOCK SPACES AND QUANTUM STATESFOR CAUSAL FERMION SYSTEMS
FELIX FINSTER AND NIKY KAMRANJANUARY 2021
Abstract.
It is shown for causal fermion systems describing Minkowski-type space-times that an interacting causal fermion system at time t gives rise to a distinguishedstate on the algebra generated by fermionic and bosonic field operators. Positivityof the state is proven and representations are constructed. Contents
1. Introduction 22. Preliminaries 42.1. Causal Fermion Systems and the Causal Action Principle 42.2. Spacetime and Physical Wave Functions 52.3. Connection to the Setting of Causal Variational Principles 62.4. The Euler-Lagrange Equations and Jet Spaces 82.5. The Euler-Lagrange Equations for the Physical Wave Functions 92.6. Surface Layer Integrals for Jets 122.7. The Linearized Field Equations and Bosonic Green’s Operators 122.8. Minkowski-Type Spacetimes 132.9. Inner Solutions, Arranging Jets without Scalar Components 142.10. The Extended Hilbert Space and Fermionic Green’s Operators 142.11. A Conserved Nonlinear Surface Layer Integral 152.12. Perturbative Description 163. The Partition Function 174. Interacting Quantum Fields in a Surface Layer 194.1. Field Operators in the Vacuum 194.2. Bosonic Variations of the Nonlinear Surface Layer Integral 204.3. Construction of the Fermionic Insertions 204.4. The State Induced by the Causal Fermion System at Fixed Time 224.5. Constructing Representations of the Field Algebra 234.6. Realizing the Insertions as Functional Derivatives 264.7. The Algebra of Observables and the Quantum Møller Map 295. Outlook: The Dynamics of the Quantum Fields 30References 31 Introduction
The theory of causal fermion systems is a recent approach to fundamental physics(see the basics in Section 2, the reviews [13, 15, 21], the textbook [12] or the website [1]).In this approach, spacetime and all objects therein are described by a measure ρ ona set F of linear operator of a Hilbert space ( H , h . | . i H ). The physical equations areformulated via the so-called causal action principle , a nonlinear variational principlewhere an action S is minimized under variations of the measure ρ .It is an important task to rewrite the dynamics as described by the causal actionprinciple in a form comparable to the usual description of physics. The connectionto an interaction of a quantized Dirac field via classical bosonic fields is obtained inthe so-called continuum limit as worked out in detail in [12]. The remaining task is towork out the connection to an interaction via bosonic quantum fields. A first step inthis direction is the work [11], where a Fock space dynamics was derived starting fromthe classical field equations obtained in the continuum limit and making additionalassumptions which were physically motivated but not justified mathematically fromfirst principles. In order to work out the quantum field theory limit of causal fermionsystems in a more fundamental and mathematically more convincing manner, oneshould not start from the classical field equations but work instead directly with theEuler-Lagrange (EL) equations corresponding to the causal action principle and thecorresponding conservation laws. The the first step in the resulting research programis the paper [18], where complex structures and related Fock space constructions werecarried out for the bosonic degrees of freedom. In the present paper, we include thefermionic degrees of freedom. Our main result is to show that the causal fermion systemgives rise at any given time to a canonical quantum state which can be represented onthe fermionic and bosonic Fock spaces built up of spinorial wave functions and solutionsof the linearized field equations. Our methods are based on the recent paper [19]where the dynamics of these spinorial wave functions is studied. The dynamics of theresulting Fock state will be analyzed in detail in the forthcoming paper [16]. Finally,the research program will be summarized in the survey paper [17].Our starting point consists of two causal fermion systems ( H , F , ρ ) (describing thevacuum) and ( ˜ H , ˜ F , ˜ ρ ) (describing the interacting system). We assume that bothmeasures ρ and ˜ ρ are critical points of the causal action. Our general strategy forconstructing the interacting quantum state is to “compare” these two systems at afixed time and to try to describe the interacting system in terms of linearized fields andwave functions in the vacuum spacetime. This “comparison” can also be understoodas a “measurement” performed in the interacting spacetime using objects from thevacuum spacetime as “measurement devices.” As we shall see, following this strategywill naturally give rise to Fock states having bosonic and fermionic components.As a preliminary step, we need to specify what we mean by “fixed time.” In thevacuum spacetime M := supp ρ we assume a global time function T (for example thetime of an observer in Minkowski space). In order to obtain a global time functionalso in the interacting spacetime ˜ M := supp ˜ ρ , we need to identify the two spacetimesby a mapping F : M → ˜ M .
Clearly, this identification is not unique. This arbitrariness was studied in detail in [18],and we can treat it here in the same way.
ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 3
There is another arbitrariness not considered in [18] which is tied to the fermionicdegrees of freedom: For a “comparison” of the two systems one needs to identify theHilbert spaces H and ˜ H . This can be done by choosing a unitary mapping V : H → ˜ H ,but this mapping is determined only up to a unitary transformation. We thus havethe freedom to transform V according to V → V U with U ∈ U( H ) . (1.1)This freedom has to be taken into account when “comparing” the two systems. It playsa crucial role in our analysis and will be responsible for the occurrence of quantumstates and effects like quantum entanglement. More specifically, the analysis in [18]revealed that, for the bosonic degrees of freedom, the Fock norm can be associated tothe exponential of a nonlinear surface layer integral γ t (˜ ρ, ρ ) (for basics and the notationwe refer to Section 2). In order to incorporate the fermionic degrees of freedom, weagain use this association, but treat the arbitrariness (1.1) by integrating over U . Thisleads us to introducing thepartition function Z = ˆ G exp (cid:16) β γ t (cid:0) ˜ ρ, U ρ (cid:1)(cid:17) dµ G ( U ) , (1.2)where G is a subgroup of the unitary group with Haar measure µ G , β is a real parameter,and U ρ is the unitarily transformed measure defined by ( U ρ )(Ω) = ρ ( U − Ω U ) (fordetails see Section 3). The name “partition function” is inspired by the similarity tostatistical physics. The formula (1.2) also resembles the path integral formulation ofquantum field theory (see for example [35, 28]). However, this similarity is only on aformal level, because in (1.2) we do not integrate over field configurations, but overthe unitary transformations arising in the identification of Hilbert spaces (1.1).Again in analogy to the procedure in the path integral formulation of quantum fieldtheory, we introduce the n -point functions defining the quantum state ω t by insertions,i.e. symbolically ω t ( · · · ) := 1 Z ˆ G ( · · · ) exp (cid:16) β γ t (cid:0) ˜ ρ, U ρ (cid:1)(cid:17) dµ G ( U ) , (1.3)where the dots on the left side stand for field operators, whereas the dots on the rightare suitable surface layer integrals (for details see Section 4). The insertions can beunderstood as “measurements.” The fact that we integrate over U at the very endcorresponds to the fact that the observer must choose U before performing a multi-particle measurement. The U -dependence of the insertions gives rise to correlationsbetween them. These correlations are crucial for getting an entangled state.It is a major objective of this paper to work out the detailed form of the insertionsand to prove that (1.3) has the positivity properties required for a quantum state.Once this has been shown, one can also construct representations of the algebra andrewrite the state ω t as the expectation value of a density operator σ t on a Fock space F ,i.e. ω t ( · · · ) = Tr F (cid:0) σ t · · · (cid:1) . Our analysis of the insertions explains in particular why the fermionic multi-particlewave functions are totally anti-symmetric. In this way, the Pauli Exclusion Principleis established for causal fermion systems on the level of totally anti-symmetric wavefunctions and anti-commuting field operators.The paper is organized as follows. Section 2 provides the necessary preliminaries oncausal fermion systems and the causal action principle. We also recall the definition
F. FINSTER AND N. KAMRAN of various surface layer integrals and summarize the main results from [19] on thedynamics of spinorial wave functions. In Section 3 the arbitrariness of the identificationof Hilbert spaces (1.1) is explained and built into the nonlinear surface layer integral,leading us to the definition of the partition function (1.2). In Section 4 the quantumstate at time t is introduced, its positivity properties are proven and representations areconstructed. In Section 5 we conclude the paper by a brief discussion of the dynamicsof the quantum state. The detailed analysis of this dynamics will be the objective ofthe follow-up paper [16]. 2. Preliminaries
Causal Fermion Systems and the Causal Action Principle.
We now recallthe basic setup and introduce the main objects needed later on.
Definition 2.1. (causal fermion systems)
Given a separable complex Hilbert space H with scalar product h . | . i H and a parameter n ∈ N (the “spin dimension” ), we let F ⊂ L( H ) be the set of all selfadjoint operators on H of finite rank, which (counting mul-tiplicities) have at most n positive and at most n negative eigenvalues. On F weare given a positive measure ρ (defined on a σ -algebra of subsets of F ), the so-called universal measure . We refer to ( H , F , ρ ) as a causal fermion system .A causal fermion system describes a spacetime together with all structures and ob-jects therein. In order to single out the physically admissible causal fermion systems,one must formulate physical equations. To this end, we impose that the universalmeasure should be a minimizer of the causal action principle, which we now intro-duce. For any x, y ∈ F , the product xy is an operator of rank at most 2 n . However,in general it is no longer a selfadjoint operator because ( xy ) ∗ = yx , and this is dif-ferent from xy unless x and y commute. As a consequence, the eigenvalues of theoperator xy are in general complex. We denote these eigenvalues counting algebraicmultiplicities by λ xy , . . . , λ xy n ∈ C (more specifically, denoting the rank of xy by k ≤ n ,we choose λ xy , . . . , λ xyk as all the non-zero eigenvalues and set λ xyk +1 , . . . , λ xy n = 0). Weintroduce the Lagrangian and the causal action by Lagrangian: L ( x, y ) = 14 n n X i,j =1 (cid:16)(cid:12)(cid:12) λ xyi (cid:12)(cid:12) − (cid:12)(cid:12) λ xyj (cid:12)(cid:12)(cid:17) (2.1) causal action: S ( ρ ) = ¨ F × F L ( x, y ) dρ ( x ) dρ ( y ) . (2.2)The causal action principle is to minimize S by varying the measure ρ under thefollowing constraints: volume constraint: ρ ( F ) = const (2.3) trace constraint: ˆ F tr( x ) dρ ( x ) = const (2.4) boundedness constraint: ¨ F × F | xy | dρ ( x ) dρ ( y ) ≤ C , (2.5)
ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 5 where C is a given parameter, tr denotes the trace of a linear operator on H , and theabsolute value of xy is the so-called spectral weight, | xy | := n X j =1 (cid:12)(cid:12) λ xyj (cid:12)(cid:12) . (2.6)This variational principle is mathematically well-posed if H is finite-dimensional. Forthe existence theory and the analysis of general properties of minimizing measureswe refer to [9, 10, 3]. In the existence theory one varies in the class of regular Borelmeasures (with respect to the topology on L( H ) induced by the operator norm), andthe minimizing measure is again in this class. With this in mind, here we alwaysassume that ρ is a regular Borel measure . Spacetime and Physical Wave Functions.
Let ρ be a minimizing measure. Spacetime is defined as the support of this measure, M := supp ρ . Thus the spacetime points are selfadjoint linear operators on H . On M we considerthe topology induced by F (generated by the operator norm on L( H )). Moreover,the universal measure ρ | M restricted to M can be regarded as a volume measure onspacetime. This makes spacetime into a topological measure space .The operators in M contain a lot of information which, if interpreted correctly, givesrise to spacetime structures like causal and metric structures, spinors and interactingfields (for details see [12, Chapter 1]). He we restrict attention to those structuresneeded in what follows. We begin with the following notion of causality: Definition 2.2. (causal structure)
For any x, y ∈ F , the product xy is an operator ofrank at most 2 n . We denote its non-trivial eigenvalues (counting algebraic multiplic-ities) by λ xy , . . . , λ xy n . The points x and y are called spacelike separated if all the λ xyj have the same absolute value. They are said to be timelike separated if the λ xyj are allreal and do not all have the same absolute value. In all other cases (i.e. if the λ xyj arenot all real and do not all have the same absolute value), the points x and y are saidto be lightlike separated.Restricting the causal structure of F to M , we get causal relations in spacetime.Next, for every x ∈ F we define the spin space S x by S x = x ( H ); it is a subspaceof H of dimension at most 2 n . It is endowed with the spin inner product ≺ . | . ≻ x definedby ≺ u | v ≻ x = −h u | xv i H (for all u, v ∈ S x ) . A wave function ψ is defined as a function which to every x ∈ M associates a vectorof the corresponding spin space, ψ : M → H with ψ ( x ) ∈ S x M for all x ∈ M .
In order to introduce a notion of continuity of a wave function, we need to comparethe wave function at different spacetime points. Noting that the natural norm on thespin space ( S x , ≺ . | . ≻ x ) is given by (cid:12)(cid:12) ψ ( x ) (cid:12)(cid:12) x := (cid:10) ψ ( x ) (cid:12)(cid:12) | x | ψ ( x ) (cid:11) H = (cid:13)(cid:13)(cid:13)p | x | ψ ( x ) (cid:13)(cid:13)(cid:13) H F. FINSTER AND N. KAMRAN (where | x | is the absolute value of the symmetric operator x on H , and p | x | is thesquare root thereof), we say that the wave function ψ is continuous at x if for every ε > δ > (cid:13)(cid:13)p | y | ψ ( y ) − p | x | ψ ( x ) (cid:13)(cid:13) H < ε for all y ∈ M with k y − x k ≤ δ . (2.7)Likewise, ψ is said to be continuous on M if it is continuous at every x ∈ M . Wedenote the set of continuous wave functions by C ( M, SM ).It is an important observation that every vector u ∈ H of the Hilbert space givesrise to a unique wave function. To obtain this wave function, denoted by ψ u , we simplyproject the vector u to the corresponding spin spaces, ψ u : M → H , ψ u ( x ) = π x u ∈ S x M .
We refer to ψ u as the physical wave function of u ∈ H . A direct computation showsthat the physical wave functions are continuous (in the sense (2.7)). Associating toevery vector u ∈ H the corresponding physical wave function gives rise to the waveevaluation operator Ψ : H → C ( M, SM ) , u ψ u . Every x ∈ M can be written as (for the derivation see [12, Lemma 1.1.3]) x = − Ψ( x ) ∗ Ψ( x ) . (2.8)In words, every spacetime point operator is the local correlation operator of the waveevaluation operator at this point. This formula is very useful when varying the system,as will be explained in Section 2.5 below.2.3. Connection to the Setting of Causal Variational Principles.
For the anal-ysis of the causal action principle it is most convenient to get into the simpler settingof causal variational principles. In this setting, F is a (possibly non-compact) smoothmanifold of dimension m ≥ ρ a positive Borel measure on F (the universalmeasure ). Moreover, we are given a non-negative function L : F × F → R +0 (the Lagrangian ) with the following properties:(i) L is symmetric: L ( x, y ) = L ( y, x ) for all x, y ∈ F .(ii) L is lower semi-continuous, i.e. for all sequences x n → x and y n ′ → y , L ( x, y ) ≤ lim inf n,n ′ →∞ L ( x n , y n ′ ) . The causal variational principle is to minimize the action S ( ρ ) = ˆ F dρ ( x ) ˆ F dρ ( y ) L ( x, y ) (2.9)under variations of the measure ρ , keeping the total volume ρ ( F ) fixed ( volume con-straint ). If the total volume ρ ( F ) is finite, one minimizes (2.9) over all regular Borelmeasures with the same total volume. If the total volume ρ ( F ) is infinite, however, it isnot obvious how to implement the volume constraint, making it necessary to proceedas follows. We need the following additional assumptions:(iii) The measure ρ is locally finite (meaning that any x ∈ F has an open neighbor-hood U with ρ ( U ) < ∞ ).(iv) The function L ( x, . ) is ρ -integrable for all x ∈ F , giving a lower semi-continuousand bounded function on F . ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 7
Given a regular Borel measure ρ on F , we then vary over all regular Borel measures ˜ ρ with (cid:12)(cid:12) ˜ ρ − ρ (cid:12)(cid:12) ( F ) < ∞ and (cid:0) ˜ ρ − ρ (cid:1) ( F ) = 0(where | . | denotes the total variation of a measure). These variations of the causalaction are well-defined. The existence theory for minimizers is developed in [24].There are several ways to get from the causal action principle to causal variationalprinciples, as we now explain in detail. If the Hilbert space H is finite-dimensional and the total volume ρ ( F ) is finite, one can proceed as follows: As a consequence ofthe trace constraint (2.4), for any minimizing measure ρ the local trace is constant inspacetime, i.e. there is a real constant c = 0 such that (see [3, Theorem 1.3] or [12,Proposition 1.4.1]) tr x = c for all x ∈ M .
Restricting attention to operators with fixed trace, the trace constraint (2.4) is equiv-alent to the volume constraint (2.3) and may be disregarded. The boundedness con-straint, on the other hand, can be treated with a Lagrange multiplier. More precisely,in [3, Theorem 1.3] it is shown that for every minimizing measure ρ , there is a Lagrangemultiplier κ > ρ is a critical point of the causal action with the Lagrangianreplaced by L κ ( x, y ) := L ( x, y ) + κ | xy | , (2.10)leaving out the boundedness constraint. Having treated the constraints, the differenceto causal variational principles is that in the setting of causal fermion systems, the setof operators F ⊂ L( H ) does not have the structure of a manifold. In order to give thisset a manifold structure, we assume that a given minimizing measure ρ is regular in thesense that all operators in its support have exactly n positive and exactly n negativeeigenvalues. This leads us to introduce the set F reg as the set of all operators F on H with the following properties:(i) F is selfadjoint, has finite rank and (counting multiplicities) has exactly n positiveand n negative eigenvalues.(ii) The trace is constant, i.e tr( F ) = c > . (2.11)The set F reg has a smooth manifold structure (see the concept of a flag manifold in [30]or the detailed construction in [20, Section 3]). In this way, the causal action principlebecomes an example of a causal variational principle.This finite-dimensional setting has the drawback that the total volume ρ ( F ) ofspacetime is finite, which is not suitable for describing asymptotically flat spacetimesor spacetimes of infinite lifetime like Minkowski space. Therefore, it is importantto also consider the infinite-dimensional setting where dim H = ∞ and consequentlyalso ρ ( F ) = ∞ (see [12, Exercise 1.3]). In this case, the set F reg has the structure ofan infinite-dimensional Banach manifold (for details see [25]). Here we shall not enterthe subtleties of infinite-dimensional analysis. Instead, we get by with the followingsimple method: Given a minimizing measure ρ , we choose F reg as a finite-dimensionalmanifold which contains M := supp ρ . We then restrict attention to variations of ρ inthe class of regular Borel measures on F reg . In this way, we again get into the settingof causal variational principles. We refer to this method by saying that we restrictattention to locally compact variations . Keeping in mind that the dimension of F reg can be chosen arbitrarily large, this method seems a sensible technical simplification.In situations when it is important to work in infinite dimensions (for example for F. FINSTER AND N. KAMRAN getting the connection to the renormalization program in quantum field theory), itmay be necessary to analyze the limit when the dimension of F reg tends to infinity, oralternatively it may be suitable to work in the infinite-dimensional setting as developedin [25]. However, this is not a concern of the present paper, where we try to keep themathematical setup as simple as possible.For ease of notation, in what follows we will omit the superscript “reg.” Thus F stands for a smooth (in general non-compact) manifold which contains the support M of a given minimizing measure ρ .2.4. The Euler-Lagrange Equations and Jet Spaces.
A minimizer of a causalvariational principle satisfies the following
Euler-Lagrange (EL) equations : For a suit-able value of the parameter s >
0, the lower semi-continuous function ℓ : F → R +0 defined by ℓ ( x ) := ˆ M L ( x, y ) dρ ( y ) − s is minimal and vanishes on spacetime M := supp ρ , ℓ | M ≡ inf F ℓ = 0 . (2.12)The parameter s can be understood as the Lagrange parameter corresponding to thevolume constraint. For the derivation and further details we refer to [23, Section 2].The EL equations (2.12) are nonlocal in the sense that they make a statement onthe function ℓ even for points x ∈ F which are far away from spacetime M . It turns outthat for the applications we have in mind, it is preferable to evaluate the EL equationsonly locally in a neighborhood of M . This leads to the weak EL equations introducedin [23, Section 4]. Here we give a slightly less general version of these equations whichis sufficient for our purposes. In order to explain how the weak EL equations comeabout, we begin with the simplified situation that the function ℓ is smooth. In thiscase, the minimality of ℓ implies that the derivative of ℓ vanishes on M , i.e. ℓ | M ≡ Dℓ | M ≡ Dℓ ( p ) : T p F → R is the derivative). In order to combine these two equations ina compact form, it is convenient to consider a pair u := ( a, u ) consisting of a real-valuedfunction a on M and a vector field u on T F along M , and to denote the combinationof multiplication of directional derivative by ∇ u ℓ ( x ) := a ( x ) ℓ ( x ) + (cid:0) D u ℓ (cid:1) ( x ) . (2.14)Finally, compactly supported jets are denoted by a subscript zero, like for example J test ρ, := { u ∈ J test ρ | u has compact support } . Then the equations (2.13) imply that ∇ u ℓ ( x ) vanishes for all x ∈ M . The pair u =( a, u ) is referred to as a jet .In the general lower-continuous setting, one must be careful because the directionalderivative D u ℓ in (2.14) need not exist. Our method for dealing with this problem isto restrict attention to vector fields for which the directional derivative is well-defined.Moreover, we must specify the regularity assumptions on a and u . To begin with, wealways assume that a and u are smooth in the sense that they have a smooth extensionto the manifold F (for more details see [18, Section 2.2]). Thus the jet u should be anelement of the jet space J ρ := (cid:8) u = ( a, u ) with a ∈ C ∞ ( M, R ) and u ∈ Γ( M, T F ) (cid:9) , ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 9 where C ∞ ( M, R ) and Γ( M, T F ) denote the space of real-valued functions and vectorfields on M , respectively, which admit a smooth extension to F .Clearly, the fact that a jet u is smooth does not imply that the functions ℓ or L are differentiable in the direction of u . This must be ensured by additional conditionswhich are satisfied by suitable subspaces of J ρ which we now introduce. First, welet Γ diff ρ be those vector fields for which the directional derivative of the function ℓ exists, Γ diff ρ = (cid:8) u ∈ C ∞ ( M, T F ) (cid:12)(cid:12) D u ℓ ( x ) exists for all x ∈ M (cid:9) . This gives rise to the jet space J diff ρ := C ∞ ( M, R ) ⊕ Γ diff ρ ⊂ J ρ . For the jets in J diff ρ , the combination of multiplication and directional derivative in (2.14)is well-defined. We choose a linear subspace J test ρ ⊂ J diff ρ with the property that itsscalar and vector components are both vector spaces, J test ρ = C test ( M, R ) ⊕ Γ test ρ ⊆ J diff ρ , and the scalar component is nowhere trivial in the sense thatfor all x ∈ M there is a ∈ C test ( M, R ) with a ( x ) = 0 . (2.15)Then the weak EL equations read (for details cf. [23, (eq. (4.10)]) ∇ u ℓ | M = 0 for all u ∈ J test ρ . (2.16)Before going on, we point out that the weak EL equations (2.16) do not hold only forminimizers, but also for critical points of the causal action. With this in mind, allmethods and results of this paper do not apply only to minimizers, but more generallyto critical points of the causal variational principle. For brevity, we also refer to ameasure with satisfies the weak EL equations (2.16) as a critical measure .Here and throughout this paper, we use the following conventions for partial deriva-tives and jet derivatives: ◮ Partial and jet derivatives with an index i ∈ { , } only act on the respectivevariable of the function L . This implies, for example, that the derivatives com-mute, ∇ , v ∇ , u L ( x, y ) = ∇ , u ∇ , v L ( x, y ) . ◮ The partial or jet derivatives which do not carry an index act as partial deriva-tives on the corresponding argument of the Lagrangian. This implies, for exam-ple, that ∇ u ˆ F ∇ , v L ( x, y ) dρ ( y ) = ˆ F ∇ , u ∇ , v L ( x, y ) dρ ( y ) . We point out that, with these conventions, jets are never differentiated .2.5.
The Euler-Lagrange Equations for the Physical Wave Functions.
Forcausal fermion systems, the Euler-Lagrange Equations can be expressed in terms ofthe physical wave functions, as we now recall. These equations were first derived in [12, § expressing the spacetime point operator as a local correlation operator. Varying thewave evaluation operator gives a vector field u on F along M , u ( x ) = − δ Ψ( x ) ∗ Ψ( x ) − Ψ( x ) ∗ δ Ψ( x ) . (2.17)In order to make mathematical sense of this formula in agreement with the conceptof restricting attention to locally compact variations, we choose a finite-dimensionalsubspace H f ⊂ H , i.e. f f := dim H f < ∞ and impose the following assumptions on δ Ψ (similar variations were first consideredin [14, Section 7]):(a) The variation is trivial on the orthogonal complement of H f , δ Ψ | ( H f ) ⊥ = 0 . (b) The variations of all physical wave functions are continuous and compactly sup-ported, i.e. δ Ψ : H → C ( M, SM ) . We choose Γ f ρ, as a space of vector fields of the form (2.17). For convenience, weidentify the vector field with the first variation δ Ψ and write δ Ψ ∈ Γ f ρ, (this rep-resentation of u in terms of δ Ψ may not be unique, but this is of no relevance forwhat follows). Choosing trivial scalar components, we obtain a corresponding spaceof jets J f ρ, , referred to as the fermionic jets . We always assume that the fermionic jetsare admissible for testing, i.e. J f ρ, := { } ⊕ Γ f ρ, ⊂ J test ρ . Moreover, in analogy to the condition (2.15) for the scalar components of the test jets,we assume that the variation can have arbitrary values at any spacetime point, i.e.for all x ∈ M, χ ∈ S x and φ ∈ H f there is δ Ψ ∈ Γ f ρ, with δ Ψ( x ) φ = χ . (2.18)For the computation of the variation of the Lagrangian, one can make use of thefact that for any p × q -matrix A and any q × p -matrix B , the matrix products AB and BA have the same non-zero eigenvalues, with the same algebraic multiplicities.As a consequence, applying again (2.8), xy = Ψ( x ) ∗ (cid:0) Ψ( x ) Ψ( y ) ∗ Ψ( y ) (cid:1) ≃ (cid:0) Ψ( x ) Ψ( y ) ∗ Ψ( y ) (cid:1) Ψ( x ) ∗ , (2.19)where ≃ means that the operators have the same non-trivial eigenvalues with corre-sponding algebraic multiplicities. Introducing the kernel of the fermionic projector P ( x, y ) by P ( x, y ) := Ψ( x ) Ψ( y ) ∗ : S y → S x , we can write (2.19) as xy ≃ P ( x, y ) P ( y, x ) : S x → S x . In this way, the eigenvalues of the operator product xy as needed for the computationof the Lagrangian (2.1) and the spectral weight (2.6) are recovered as the eigenvaluesof a 2 n × n -matrix. Since P ( y, x ) = P ( x, y ) ∗ , the Lagrangian L κ ( x, y ) in (2.10) canbe expressed in terms of P ( x, y ). Consequently, the first variation of the Lagrangiancan be expressed in terms of the first variation of this kernel. Being real-valued andreal-linear in δP ( x, y ), it can be written as δ L κ ( x, y ) = 2 Re Tr S x (cid:0) Q ( x, y ) δP ( x, y ) ∗ (cid:1) (2.20) ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 11 with a kernel Q ( x, y ) which is again symmetric (with respect to the spin inner product),i.e. Q ( x, y ) : S y → S x and Q ( x, y ) ∗ = Q ( y, x )(more details on this method and many computations can be found in [12, Sections 1.4and 2.6 as well as Chapters 3-5]). Expressing the variation of P ( x, y ) in terms of δ Ψ,the variations of the Lagrangian can be written as D , u L κ ( x, y ) = 2 Re tr (cid:0) δ Ψ( x ) ∗ Q ( x, y ) Ψ( y ) (cid:1) D , u L κ ( x, y ) = 2 Re tr (cid:0) Ψ( x ) ∗ Q ( x, y ) δ Ψ( y ) (cid:1) (where tr denotes the trace of a finite-rank operator on H ). Likewise, the variationof ℓ becomes D u ℓ ( x ) = 2 Re ˆ M tr (cid:0) δ Ψ( x ) ∗ Q ( x, y ) Ψ( y ) (cid:1) dρ ( y ) . The weak EL equations (2.16) imply that this expression vanishes for any u ∈ Γ f ρ, .Using that the variation can be arbitrary at every spacetime point (see (2.18)), onemay be tempted to conclude that ˆ M Q ( x, y ) Ψ( y ) φ dρ ( y ) = 0 for all x ∈ M and φ ∈ H f . However, we must take into account that the local trace must be preserved in thevariation (2.11). This can be arranged by rescaling the operator x in the variation (fordetails see [14, Section 6.2]) or, equivalently, by treating it with a Lagrange multiplierterm (see [12, § EL equation for the physical wave functions ˆ M Q ( x, y ) Ψ f ( y ) dρ ( y ) = r Ψ f ( x ) for all x ∈ M , (2.21)where r ∈ R is the Lagrange parameter of the trace constraint, and Ψ f := Ψ | H f denotesthe restriction of the wave evaluation operator to the finite-dimensional subspace H f .Let us briefly discuss the structure of the obtained EL equation (2.21). Being a linearequation for every physical wave function, it has similarity with the Dirac equation.The interaction is taken into account because the kernel Q ( x, y ) also depends on theensemble of wave functions. As a major difference to the Dirac equation, the ELequation (2.21) only describes the occupied states of the system. More concretely, inthe example of the Dirac sea vacuum (see for example [34, 26]), the physical wavefunctions correspond to the negative-energy solutions of the Dirac equation. But thesolutions of positive energy are not described by (2.21). It is the main goal of thepresent paper to extend (2.21) in such a way that it includes the solutions of positiveenergy.We finally comment on the significance of the subspace H f ⊂ H . Choosing afinite-dimensional subspace is a technical simplification, made in agreement with themethod of restricting attention to locally compact variations discussed in Section 2.3.The strategy is to choose H f large enough to capture all the relevant physical effects.Our physical picture is that H f should contain all physical wave functions whose en-ergies are much smaller than the Planck energy and which are therefore accessibleto measurements as describing particle or anti-particle states. If necessary, one couldanalyze the limit where the dimension f f of H f tends to infinity. In what follows, weleave H f unspecified as being any finite-dimensional subspace of H . Surface Layer Integrals for Jets.
Surface layer integrals were first introducedin [22] as double integrals of the general form ˆ Ω (cid:18) ˆ M \ Ω ( · · · ) L κ ( x, y ) dρ ( y ) (cid:19) dρ ( x ) , (2.22)where ( · · · ) stands for a suitable differential operator formed of jets. A surfacelayer integral generalizes the concept of a surface integral over ∂ Ω to the setting ofcausal fermion systems. The connection can be understood most easily in the casewhen L κ ( x, y ) vanishes unless x and y are close together. In this case, we only get acontribution to (2.22) if both x and y are close to the boundary of Ω. A more detailedexplanation of the idea of a surface layer integrals is given in [22, Section 2.3].We now recall those surface layer integrals for jets which will be of relevance in thispaper. Definition 2.3.
We define the following surface layer integrals, γ Ω ρ : J ρ, sc → R (conserved one-form) γ Ω ρ ( v ) = ˆ Ω dρ ( x ) ˆ M \ Ω dρ ( y ) (cid:0) ∇ , v − ∇ , v (cid:1) L ( x, y ) (2.23) σ Ω ρ : J ρ, sc × J ρ, sc → R (symplectic form) σ Ω ρ ( u , v ) = ˆ Ω dρ ( x ) ˆ M \ Ω dρ ( y ) (cid:0) ∇ , u ∇ , v − ∇ , u ∇ , v (cid:1) L ( x, y ) (2.24)( ., . ) Ω ρ : J ρ, sc × J ρ, sc → R (surface layer inner product) ( u , v ) Ω ρ = ˆ Ω dρ ( x ) ˆ M \ Ω dρ ( y ) (cid:0) ∇ , u ∇ , v − ∇ , u ∇ , v (cid:1) L ( x, y ) . (2.25)Here J ρ, sc denotes the jets in J vary ρ with spatially compact support (for details see [5,Section 5.3]), where J vary ρ is a suitably chosen subspace of J test ρ (for details see [5,Section 3.2]).2.7. The Linearized Field Equations and Bosonic Green’s Operators.
In sim-ple terms, the linearized field equations describe variations of the universal measurewhich preserve the EL equations. More precisely, we consider variations where wemultiply ρ by a non-negative function and take the push-forward with respect to amapping from M to F . Thus we consider families of measures (˜ ρ τ ) τ ∈ ( − δ,δ ) of the form˜ ρ τ = ( F τ ) ∗ (cid:0) f τ ρ (cid:1) , (2.26)where the f τ and F τ are smooth, f ∈ C ∞ (cid:0) M, R + (cid:1) and F ∈ C ∞ (cid:0) M, F (cid:1) , depend smoothly on the parameter τ and have the properties f ( x ) = 1 and F ( x ) = x for all x ∈ M (moreover, the star denotes the push-forward measure, which is definedfor a subset Ω ⊂ F by (( F τ ) ∗ µ )(Ω) = µ ( F − τ (Ω)); see for example [4, Section 3.6]). Weassume that the measures (˜ ρ τ ) τ ∈ ( − δ,δ ) satisfy the EL equations (2.12) hold for all τ ,i.e. ˜ ℓ τ | M τ ≡ inf F ℓ τ = 0 with ˜ ℓ τ ( x ) := ˆ F L κ ( x, y ) d ˜ ρ τ ( y ) − s . ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 13
Using the definition of the push-forward measure, we can rewrite the ˜ ρ -integral as a ρ -integral. Moreover, it is convenient to rewrite this equation as an equation on M and to multiply it by f τ ( x ). We thus obtain the equivalent equation ℓ τ | M ≡ inf F ℓ τ = 0with ℓ τ ( x ) := ˆ F f τ ( x ) L κ (cid:0) F τ ( x ) , F τ ( y ) (cid:1) f τ ( y ) d ˜ ρ τ ( y ) − f τ ( x ) s . In analogy to (2.16) we write the corresponding weak EL equations as ∇ u ℓ τ | M = 0 for all u ∈ J test ρ (for details on why the jet space does not depend on τ we refer to [14, Section 4.1]).Since this equation holds by assumption for all τ , we can differentiate it with respectto τ . Denoting the infinitesimal generator of the variation by v , i.e. v ( x ) := ddτ (cid:0) f τ ( x ) , F τ ( x ) (cid:1)(cid:12)(cid:12)(cid:12) τ =0 , we obtain the linearized field equations h u , ∆ v i ( x ) := ∇ u (cid:18) ˆ M (cid:0) ∇ , v + ∇ , v (cid:1) L κ ( x, y ) dρ ( y ) − ∇ v s (cid:19) , which are to be satisfied for all u ∈ J test ρ and all x ∈ M (for details see [14, Section 3.3]).We denote the vector space of all solutions of the linearized field equations by J lin ρ .In [5, Section 5] advanced and retarded Green’s operators are constructed as map-pings S ∨ , S ∧ : J ∗ → L ( M, dρ ) , where J ∗ is a space of compactly supported jets, whereas L ( M, dρ ) are locally square-integrable jets with spatially compact support. They are inverses of the linearized fieldoperator in the sense that ∆ S ∨ , ∧ = 11. The difference of these Green’s operators is theso-called causal fundamental solution G ; it maps to linearized solutions, G := S ∧ − S ∨ : J ∗ → J lin sc . (2.27)It satisfies for all u , v ∈ J ∗ the relation σ ( G u , G v ) = h u , G v i L ( M ) . (2.28)2.8. Minkowski-Type Spacetimes.
Let ( H , F , ρ ) be a causal fermion system, whichmay be thought of as describing the vacuum or the interacting physical system. Weassume that ρ is a critical point of the causal action principle. Moreover, we assumethat the corresponding spacetime M := supp ρ is diffeomorphic to a four-dimensionalspacetime with trivial topology, i.e. M ≃ M := R . Moreover, we assume that, using this identification, the universal measure is absolutelycontinuous with respect to the Lebesgue measure with a smooth weight function, i.e. dρ = h ( x ) d x with h ∈ C ∞ ( M , R + ) . (2.29)We also assume that h is bounded from above and below, i.e. there should be a con-stant C > C ≤ h ( x ) ≤ C for all x ∈ M . We also denote the coordinate x as time function T , T : M → R , ( t, x ) t . (2.30)For any t ∈ R , we let Ω t be the past of t ,Ω t := { x ∈ M | T ( x ) ≤ t } . Inner Solutions, Arranging Jets without Scalar Components.
We nowbriefly recall the definition of inner solutions as introduced in [18, Section 3].
Definition 2.4. An inner solution is a jet v of the form v = (div v, v ) with v ∈ Γ( M, T M ) , where the divergence is taken with respect to the measure in (2.29) , div v := 1 h ∂ j (cid:0) h v j (cid:1) . Under suitable regularity and decay assumptions, an inner solution solves the lin-earized field equations (for details see [18, Section 3.1]). We denote these inner solutionsby J in ρ . In [18, Proposition 3.6] it is shown that inner solutions can be used for testing.With this in mind, we always assume that J in ρ ⊂ J test ρ . Inner solutions can be regarded as infinitesimal generators of transformations of M which leave the measure ρ unchanged. Therefore, inner solutions do not change thecausal fermion system, but merely describe symmetry transformations of the universalmeasure. With this in mind, we can modify solutions of the linearized field equations byadding inner solutions. For conceptual clarity, it is preferable that the diffeomorphismgenerated by an inner solution does not change the global time function T in (2.30).Therefore, we consider vector fields which are tangential to the hypersurfaces N t := T − ( t ). As shown in [27, Lemma 2.7], the divergence of such a vector field can bearranged to be any given function a ∈ C ∞ ( M, R ). Thus, by adding the correspondinginner solutions we can achieve that all linearized solutions have no scalar components.Therefore, in what follows we may restrict attention to linearized solutions v ∈ J lin with vanishing scalar component. We also write these jets as v = (0 , v ) with v ∈ Γ lin ρ . Likewise, the causal fundamental solution maps to linearized solutions without scalarcomponents. Thus (2.27) is modified to G := S ∧ − S ∨ : J ∗ → Γ lin sc . The Extended Hilbert Space and Fermionic Green’s Operators.
In [19]the so-called extended Hilbert space ( H f ρ , h . | . i tρ ) was constructed. The dynamics in H f ρ is described by the equation Q dyn ψ = 0 , where Q dyn is an integral operator with continuous integral kernel Q dyn ( x, y ) : S y → S x . ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 15
The scalar product h . | . i tρ at time t has the form h ψ | φ i tρ = − i (cid:18) ˆ Ω t dρ ( x ) ˆ M \ Ω t dρ ( y ) − ˆ M \ Ω t dρ ( x ) ˆ Ω t dρ ( y ) (cid:19) × ≺ ψ ( x ) | Q dyn ( x, y ) φ ( y ) ≻ x . (2.31)Moreover, the fermionic causal fundamental solution is introduced as the mapping k := i (cid:0) s ∨ − s ∧ (cid:1) : W ∗ fpc ( M, SM ) → H fsc ( M, SM ) , (2.32)where s ∨ and s ∧ are the advanced and retarded Green’s operators, respectively, whichmap from a space of wave functions W ∗ fpc ( M, SM ) supported in finite time strips to aspace H fsc ( M, SM ) of spatially compact wave functions (for details see [19, Section 6]).The following relation holds for all η, η ′ ∈ W , h kη | kη ′ i = <η | k η ′ > , (2.33)where the last sesquilinear form is the Krein inner product (for details see [12, § <η | η ′ > := ˆ M ≺ η ( x ) | η ′ ( x ) ≻ x dρ ( x ) . (2.34)2.11. A Conserved Nonlinear Surface Layer Integral.
In [18, Section 4] a non-linear surface layer integral γ t ( ρ, ˜ ρ ) was introduced which gave a way of comparing themeasure ˜ ρ describing the interacting system with the vacuum measure ρ . The startingpoint is given by two critical measures ρ and ˜ ρ on F , which describe the vacuum andthe interacting system, respectively. Similar as in (2.26) we assume that the interactingmeasure can be written as ˜ ρ = F ∗ ( f ρ ) (2.35)with smooth functions f ∈ C ∞ ( M, R + ) and F ∈ C ∞ ( M, F (cid:1) . We assume that themapping F is injective and closed, implying that˜ M := supp ˜ ρ = F ( M ) , and that the mapping F : M → ˜ M is invertible. We denote the inverse of this mappingby Φ : ˜ M → M . Choosing a foliation ( N t ) t ∈ R of M , we obtain a correspondingfoliation ( ˜ N t ) t ∈ R of ˜ M given by ˜ N t = F ( N t ). Likewise, past sets Ω t ⊂ M correspondto past sets ˜Ω t ⊂ ˜ M . Then the nonlinear surface layer integral at time t is defined by(see [18, Definition 4.1]) γ t (˜ ρ, ρ ) = ˆ ˜Ω t d ˜ ρ ( x ) ˆ M \ Ω t dρ ( y ) L ( x, y ) − ˆ Ω t dρ ( x ) ˆ ˜ M \ ˜Ω t d ˜ ρ ( y ) L ( x, y ) . (2.36)Using the definition of the push-forward measure, this can be written alternatively as γ t (˜ ρ, ρ ) = ˆ Ω t dρ ( x ) ˆ M \ Ω t dρ ( y ) (cid:16) f ( x ) L (cid:0) F ( x ) , y (cid:1) − L (cid:0) x, F ( y ) (cid:1) f ( y ) (cid:17) . (2.37)We now recall the conservation law for the nonlinear surface layer integral. Aconservation law holds if the above surface layer integral vanishes when Ω t is replacedby any compact set Ω. In order to analyze when this is the case, we introduce a measure ν on M and a measure ˜ ν on ˜ M := F ( M ) (the so-called correlation measures )by dν ( x ) := (cid:18) ˆ ˜ M L ( x, y ) d ˜ ρ ( y ) (cid:19) dρ ( x ) d ˜ ν ( x ) := (cid:18) ˆ M L ( x, y ) dρ ( y ) (cid:19) d ˜ ρ ( x ) . Then (2.37) can be rewritten as γ t (˜ ρ, ρ ) = ˆ Ω d (cid:0) Φ ∗ ˜ ρ (cid:1) ( x ) ˆ M \ Ω dρ ( y ) L (cid:0) F ( x ) , y (cid:1) − ˆ Ω dρ ( x ) ˆ M \ Ω d (cid:0) Φ ∗ ˜ ρ (cid:1) ( y ) L (cid:0) x, F ( y ) (cid:1) = ˆ Ω d (cid:0) Φ ∗ ˜ ρ (cid:1) ( x ) ˆ M dρ ( y ) L (cid:0) F ( x ) , y (cid:1) − ˆ Ω dρ ( x ) ˆ ˜ M d ˜ ρ ( y ) L ( x, y )= ˆ Ω d (cid:0) Φ ∗ ˜ ν (cid:1) ( x ) − ˆ Ω dν ( x ) = (cid:0) Φ ∗ ˜ ν − ν (cid:1) (Ω) . We thus obtain the following result:
Proposition 2.5.
The surface layer integral (2.37) vanishes for every compact Ω ⊂ M if and only if ν = Φ ∗ ˜ ν . (2.38)Exactly as explained in [18, Appendix A], the existence of a diffeomorphism Φ followsfrom a general result in [29].Having fixed the foliations by N t = ∂ Ω t and ˜ N t = F ( N t ) , (2.39)there remains the freedom to perform diffeomorphisms on each leaf ˜ N t , i.e. to performthe transformation ( t, x ) ∈ ˜ N t → ( t, φ t ( x )) ∈ ˜ N t for a family of diffeomorphisms φ t on N t , which for simplicity we assume to be smoothin t and x . By choosing these diffeomorphisms appropriately, one can change the weightof the measure ρ arbitrarily (see [29] and the constructions in [18, Appendix A]). Inthis way, we can arrange that˜ ρ = F ∗ ρ with F ∈ C ∞ ( M, F (cid:1) . (2.40)2.12. Perturbative Description.
Following the procedure in [14] and [18], the per-turbation map is defined as a mapping which to a linearized solution associates acorresponding nonlinear solution, i.e. P ρ : U ⊂ ( J test ρ ∩ J lin ρ ) → B , (2.41)where B denotes the set of all critical measures. In the applications, one usuallyassumes that the perturbation map has a power series expansion of the form P ρ ( λ w ) = ∞ X p =1 λ p P ( p ) ρ (cid:0) w , . . . , w | {z } p arguments (cid:1) . Note that the operator P ( p ) ρ maps to jets which are in general not linearized solutions, P ( p ) ρ : (cid:0) J lin ρ ∩ J test ρ (cid:1) p → J ∞ ρ ∩ J test ρ . ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 17
From the perspective of differential geometry, the perturbation map defines distin-guished charts. A specific choice of chart is obtained by working with the retardedGreen’s operator, i.e. P ( p ) ρ ( w , . . . , w ) = S ∧ E ( p ) ( w , . . . , w ) , where E ( p ) denotes the inhomogeneity which contains the error of order ( p −
1) inperturbation theory which arises to the order p (for details see [14]; for the existenceof retarded Green’s operators see [5]).As explained in Section 2.9, by adding suitable inner solutions one can arrange thatthe jet w has no scalar component. Similarly, as explained at the end of Section 2.11, bymodifying F by diffeomorphisms on the leaves N t , one can arrange that the function f in the ansatz (2.26) is identically equal to one. In the perturbative description, thismeans that ˜ ρ = P ρ ( w ) , where both the jet w and the image of P have no scalar components, i.e. P ρ ( λw ) = ∞ X p =1 λ p P ( p ) ρ (cid:0) w, . . . , w | {z } p arguments (cid:1) and P ( p ) ρ : (cid:0) Γ lin ρ ∩ Γ test ρ (cid:1) p → Γ ∞ ρ ∩ Γ test ρ . In this way, we have arranged that all the jets which enter the following analysis havevanishing scalar components.3.
The Partition Function
As outlined in Section 2.11, the nonlinear surface layer integral γ t (˜ ρ, ρ ) was definedby (2.36) in the setting of causal variational principles. When working with causalfermion systems, however, there is the complication that the vacuum system and theinteracting system are described by two causal fermion systems ( H , F , ρ ) and ( ˜ H , ˜ F , ˜ ρ )which are defined on two different Hilbert spaces H and ˜ H . Therefore, in order to makesense of the nonlinear surface layer integral, we need to identify the Hilbert spaces H and ˜ H by a unitary transformation denoted by V , V : H → ˜ H unitary . Then operators in ˜ F can be identified with operators in F by the unitary transforma-tion, F = V − ˜ F V . (3.1)An important point to keep in mind is that this identification is not canonical, but itleaves the freedom to transform the operator V according to V → V U with U ∈ L( H ) unitary . (3.2)For ease in notation, in what follows we always identify H and ˜ H via V , makingit possible to always work in the Hilbert space H . Then the non-uniqueness of theidentification still shows up in the unitary transformation of the vacuum measure ρ → U ρ , where U ρ is defined by( U ρ )(Ω) := ρ (cid:0) U − Ω U (cid:1) for Ω ⊂ F . (3.3) Our strategy for dealing with this arbitrariness is to “symmetrize” the nonlinear sur-face layer integral by integrating over the unitary transformations. In order to makethe construction mathematically well-defined, we only integrate over a compact sub-group G of the (possibly infinite-dimensional) unitary group U( H ). In the applications,this subgroup must be chosen sufficiently large. Moreover, in view of the physical ap-plications in mind, it seems sensible to assume that the unitary transformations are ofrelevance only for the low-energy states, i.e. the wave functions describing particles andanti-particles. With this in mind, we choose a finite-dimensional subspace H f ⊂ H andconsider the freedom to transform ˜ ρ with elements of a Lie subgroup G of the unitarygroup on H f , G ⊂ U( H f ) . This freedom must be taken into account systematically in the subsequent construc-tions.
Definition 3.1.
The symmetrized nonlinear surface layer integral is defined by γ t (cid:0) ˜ ρ, ρ (cid:1) = G γ t (cid:0) ˜ ρ, U ρ (cid:1) dµ G ( U ) , where dµ G denotes the normalized Haar measure on G , and U ρ is the unitarily trans-formed measure (3.3) . Writing the symmetrized nonlinear surface layer integral as γ t (cid:0) ˜ ρ, ρ (cid:1) = γ t (cid:0) ˜ ρ, ρ symm (cid:1) , where ρ symm is the symmetrized measure defined in analogy to (3.3) by ρ symm (Ω) = G ρ (cid:0) U − Ω U (cid:1) dµ G ( U ) , we can again apply Proposition 2.38 with ν replaced by its symmetrization ν symm .The existence of a diffeomorphism Φ again follows from the general result in [29] (fordetails see again [18, Appendix A]). In what follows, we always identify M and ˜ M viathis diffeomorphism. Then (2.38) can be written as (cid:18) ˆ ˜ M L (cid:0) U x U − , y ) d ˜ ρ ( y ) (cid:19) dρ ( x ) = (cid:18) G dµ G ( U ) ˆ M dρ ( y ) L (cid:0) x, U y U − (cid:1)(cid:19) d ˜ ρ ( x ) . As noted in [18], the Fock norm is to be identified with the exponential of γ t ( ρ, ˜ ρ ).In order to extend this connection to the fermionic Fock space, it is important tointegrate over G after exponentiating. This motivates the following definition. Definition 3.2.
The partition function is defined by Z t (cid:0) β, ˜ ρ (cid:1) = G exp (cid:16) β γ t (cid:0) ˜ ρ, U ρ (cid:1)(cid:17) dµ G ( U ) with the unitarily transformed measure U ρ as in (3.3) . We point out that Z t will in general not be conserved. Moreover, Z t as well as thenonlinear surface layer integral depend crucially on the choice of the foliations (2.39).We take the view that in the vacuum spacetime M , the foliation ( N t ) t ∈ R is givencanonically by the time function of an observer in an inertial frame in Minkowskispace. The choice of the foliation ( ˜ N t ) t ∈ R of the interacting spacetime, however, is lessobvious. As worked out in [7], this foliation can be obtained by a variational principlein which Z t is minimized under variations of ˜ N t keeping γ t fixed. This minimization ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 19 process is also shown in [7] to give a connection to a notion of entropy and clarifiesthe significance of the parameter β and of the freedom in choosing the group G .4. Interacting Quantum Fields in a Surface Layer
Field Operators in the Vacuum.
We now introduce creation and annihilationoperators operators in the spacetime M describing the vacuum (the connection to thealgebra of observables and Møller operators will be explained in Section 4.7). For thebosonic fields, this involves the choice of a complex structure. In the setting of causalfermion systems, a canonical complex structure J is induced on Γ lin ρ, sc by the surfacelayer integrals ( ., . ) tρ and σ tρ (see (2.25) and (2.24) for Ω = Ω t ; for ease in notation weshall use the time parameter as upper index). The construction, which is carried outin detail in [18, Section 6.3], is summarized as follows. We assume that the surfacelayer integral ( ., . ) tρ defines a scalar product on Γ lin ρ, sc . Dividing out the null space andforming the completion, we obtain a real Hilbert space denoted by ( h R , ( ., . ) tρ ). For theconstruction of J , one assumes that σ tρ is bounded relative to the scalar product ( ., . ) tρ .Then we can represent σ tρ as σ tρ ( u, v ) = ( u, T v ) tρ , where T is a uniquely determined bounded operator on the Hilbert space h R . Assumingthat T is invertible, we set J := − ( − T ) − T . Next, we complexify the Hilbert space h R and denote its complexification by h C . Onthis complexification, the operator J has eigenvalues i and − i . The correspondingeigenspaces are referred to as the holomorphic and anti-holomorphic subspaces , respec-tively. We write the decomposition into holomorphic and anti-holomorphic componentsas v = v hol + v ah . We also complexify the symplectic form to a sesquilinear form on h C (i.e. anti-linearin its first and linear in its second argument). On the holomorphic jets we introducea scalar product ( . | . ) tρ by( . | . ) tρ := σ tρ ( . , J . ) : Γ hol ρ × Γ hol ρ → C . Taking the completion gives a Hilbert space, which we denote by ( h , ( . | . ) tρ ). This scalarproduct has the useful property thatIm( u | v ) tρ = Im σ tρ ( u, J v ) = Re σ tρ ( u, v ) . (4.1)The bosonic creation and annihilation operators are denoted by a † ( z ) and a ( z ) with z ∈ h (the overline in a ( z ) simply clarifies that this operator is anti-linear). They satisfy thecanonical commutation relations (cid:2) a ( z ) , a † ( z ′ ) (cid:3) = ( z | z ′ ) tρ , (4.2)and all other operators commute, (cid:2) a ( z ) , a ( z ′ ) (cid:3) = 0 = (cid:2) a † ( z ) , a † ( z ′ ) (cid:3) . The fermionic creation and annihilation operators denoted byΨ † ( φ ) and Ψ( φ ) for φ ∈ H f ρ, sc . satisfy the canonical anti-commutation relations (cid:8) Ψ( φ ) , Ψ † ( φ ′ ) (cid:9) = h φ | φ ′ i tρ , (4.3)and all other operators anti-commute, (cid:8) Ψ( φ ) , Ψ( φ ′ ) (cid:9) = 0 = (cid:8) Ψ † ( φ ) , Ψ † ( φ ′ ) (cid:9) . Bosonic Variations of the Nonlinear Surface Layer Integral.
Our nextconstruction step is to associate the above creation and annihilation operators tosurface layer integrals. Using these surface layer integrals as insertions in the par-tition function, we shall obtain the n -point functions defining our quantum state (seeSection 4.4). We begin with the bosonic insertions (the fermionic insertions will beconstructed in the next Section 4.3.In preparation, we construct variations of the measure ˜ ρ describing the interactingspacetime. To this end, we simply linearize the perturbation map P (see (2.41) in Sec-tion 2.12) to get from a linearized solution in the vacuum to a corresponding linearizedsolutions in the interacting spacetime jet v in M to a jet ˜ v in M , D P ρ | w : J lin ρ, sc → J lin ˜ ρ, sc . Complexifying, we set ˜ z := D P ρ | w z and ˜ z := D P ρ | w z . Before going on, we remark that, in this way, we transfer holomorphic and anti-holomorphic jets in the vacuum to corresponding jets of the interacting measure. Thejets need not be holomorphic or anti-holomorphic with respect to the intrinsic complexstructure of the interacting system as induced by σ t ˜ ρ and ( . | . ) t ˜ ρ .Next, we use the above jets to vary the surface layer integral γ t (˜ ρ, U ρ ) for fixed U , D ˜ z γ t (˜ ρ, U ρ ) and D ˜ z γ t (˜ ρ, U ρ ) . For multi-particle measurements, we simply take products of such variations, i.e. ex-pressions of the form D ˜ z ′ γ t (˜ ρ, U ρ ) · · · D ˜ z ′ p γ t (˜ ρ, U ρ ) D ˜ z γ t (˜ ρ, U ρ ) · · · D ˜ z q γ t (˜ ρ, U ρ ) . One can think of these variations as describing measurements by fictitious observerswho have chosen the identification of the Hilbert spaces H and ˜ H given by U .4.3. Construction of the Fermionic Insertions.
For the fermionic insertions, giventhe unitary transformation U and time t , we want to perform a “measurement” in H f ρ which gives us the probability that a state described by a physical wave function ˜ ψ inthe spacetime with interaction is occupied. This makes it necessary to map the wavefunction ˜ ψ (˜ x ) to a corresponding vector ψ ( x ) in H f ρ in the vacuum Hilbert space. Onemethod would be to follow the procedure in [19, Section 4.4] and to take the orthogonalprojection of the vector ˜ ψ (˜ x ) ∈ S ˜ x to the spin space S x . Keeping in mind that thetwo spacetimes are defined on different Hilbert spaces which are unitarily identified,the transformations (3.1) with freedom (3.2) would lead us to define the wave functionin M by ψ ( x ) := π x U − ˜ ψ (˜ x ) with ˜ x = F ( x ) . (4.4)This construction has the disadvantage that it does not only depend on the measures ρ and ˜ ρ , but also on the choice of the mapping F used in for the identification of themeasures (see (2.35) or (2.40)). More precisely, (4.4) changes if F is transformed ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 21 according to F → F ◦ ϕ with a volume-preserving diffeomorphism ϕ : M → M whichleaves the the global time function T invariant. Since such transformations do notchange the causal fermion systems, it seems desirable that they should also not affectthe mapping from ˜ ψ to ψ . For this reason, we here prefer to set ψ = π ρ, ˜ ρ ˜ ψ with ψ ( x ) := 1˜ t ( x ) ˆ ˜ M π x U − ˜ ψ ( y ) | xy | d ˜ ρ ( y ) and (4.5)˜ t ( x ) := ˆ ˜ M | xy | d ˜ ρ ( y ) . (4.6)Here | xy | is again the spectral weight defined in (2.6). Comparing with (2.1), we findthat L ( x, y ) ≤ n X i (cid:12)(cid:12) λ xyi (cid:12)(cid:12) ≤ | xy | . As a consequence, ˜ ℓ ( x ) := ˆ ˜ M L κ ( x, y ) d ˜ ρ ( y ) − s ≤ (1 + κ )˜ t ( y ) − s . Therefore, if ˜ ρ is a minimizer, the EL equations (2.12) imply that the function ˜ t ( x ) isbounded from below by s / (1 + κ ) >
0. Therefore, dividing by ˜ t in (4.4) is unproblem-atic. The integral in (4.5) might diverge, however, in which case we simply set ψ ( x )to zero. We denote the space spanned by these wave functions by π tρ, ˜ ρ H f = span { ψ ( x ) = π x U − ˜ ψ (˜ x ) with ˜ ψ ∈ ˜Ψ( H f ) } H f ,tρ ⊂ H f ,tρ , (4.7)where for simplicity we assumed that the wave functions in the span are all in H f ,tρ (if this is not the case, one must project to H f ,tρ before taking the completion). Thissubspace can be regarded as the space of all occupied states for an observer at time t who has chosen the identification of the Hilbert spaces H and ˜ H given by U . Wedenote the orthogonal projection onto this subspace by π t U : H f ,tρ → π tρ, ˜ ρ H f . (4.8)We next consider a subspace I ⊂ H f ,tρ of dimension p . Diagonalizing the operator π t U on this subspace, we obtain an orthonormal basis ψ , . . . , ψ p of I with h ψ i | π t U ψ j i tρ = ν i δ ij . (4.9)We can interpret the vectors ψ , . . . , ψ p as the wave functions detected in simultane-ous one-particle measurements with corresponding probabilities ν , . . . , ν p . Hence theproduct ν · · · ν p can be interpreted as the probability of the p -particle measurement.Clearly, this product of eigenvalues coincides with the determinant of π t U on I , ν · · · ν p = det (cid:0) π I π t U | I (cid:1) , where π i : H f ρ → I ⊂ H f ρ is the orthogonal projection. This determinant can also beexpressed directly in terms of the sesquilinear form h . | π t U . i t ˜ ρ, U by working with totally anti-symmetrized wave functions, ν · · · ν p = 1 p ! X σ,σ ′ ∈ S p ( − sign( σ )+sign( σ ′ ) h ˜ f σ (1) | π t U ˜ f σ ′ (1) i tρ · · · h ˜ f σ ( p ) | π t U ˜ f σ ′ ( p ) i tρ . The State Induced by the Causal Fermion System at Fixed Time.
Wedenote the unital ∗ -algebra generated by the field operators of Section 4.1 by A . A state ω is a linear mapping from the algebra which is positive, i.e. ω : A → C with ω ( A ∗ A ) ≥ A ∈ A . In this section we will show that the interacting measure ˜ ρ gives rise to a distinguishedstate at time t . Using linearity and the canonical commutation and anti-commutationrelations (4.2) and (4.3), the state is determined by defining how it acts on a productof operators where all creation operators are on the left and all annihilation operatorsare on the right. We define this expectation value by inserting the variations of thenonlinear surface layer integral in Section 4.2 as well as the fermionic expectationvalues constructed in Section 4.3 into the integrand of the partition function (seeDefinition 3.2). Definition 4.1.
The state ω t at time t is defined by ω t (cid:16) a † ( z ′ ) · · · a † ( z ′ p ) Ψ † ( φ ′ ) · · · Ψ † ( φ ′ r ′ ) a ( z ) · · · a ( z q ) Ψ( φ ) · · · Ψ( φ r ) (cid:17) := 1 Z t (cid:0) β, ˜ ρ (cid:1) δ r ′ r p ! X σ,σ ′ ∈ S r ( − sign( σ )+sign( σ ′ ) × ˆ G e β γ t (˜ ρ, U ρ ) h ˜ φ σ (1) | π t U ˜ φ ′ σ ′ (1) i tρ · · · h ˜ φ σ ( r ) | π t U ˜ φ ′ σ ′ ( r ) i tρ × D ˜ z ′ γ t (˜ ρ, U ρ ) · · · D ˜ z ′ p γ t (˜ ρ, U ρ ) D ˜ z γ t (˜ ρ, U ρ ) · · · D ˜ z q γ t (˜ ρ, U ρ ) dµ G ( U ) . We point out that this state vanishes unless there are as many fermionic creation asannihilation operators. This means that our state is an eigenstate of the fermionicparticle number operator (where, as usual, we count anti-particles with a minus sign).We next work out under which assumptions the positivity properties of the stateare satisfied.
Theorem 4.2.
The state ω t is positive, i.e. ω t (cid:0) A ∗ A (cid:1) ≥ for all t ∈ R and A ∈ A . Proof.
A general state A can be written as a finite linear combination of operatorproducts of the form A = X p,p ′ ,q,q ′ X k c ( p, p ′ , q, q ′ , k ) a † ( z ′ α, ) · · · a † ( z ′ α,p ′ ) a ( z α, ) · · · a ( z α,p ) × Ψ † ( φ ′ α, ) · · · Ψ † ( φ ′ α,q ′ ) Ψ( φ α, ) · · · Ψ( φ α,q ) , where α := ( p, p ′ , q, q ′ , k ) is a multi-index. We form the product A ∗ A and multiplyout.Let us first treat the fermionic field operators. We get a nonzero contribution to theexpectation value only if there are as many creation as annihilation operators. More-over, we obtain pairings between a creation and an annihilation operator accordingto Ψ † ( φ ) Ψ( ψ ) gives h ψ | π t U φ i tρ and Ψ( ψ ) Ψ † ( φ ) gives h ψ | (11 − π t U ) φ i tρ . ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 23
There are pairings between operators within A and within A ∗ . For those pairings, wesimply replace the operator products by the corresponding expectation values. Thenit remains to consider pairings in which one field operator in A is combined with onefield operator in A ∗ .Next, in order to treat the bosonic field operators, we use the commutation relationssuch as to bring all the creation operators to the left and all the annihilation operatorsto the right. This gives rise to pairings according to a ( z ) a † ( z ′ ) gives ( z | z ′ ) tρ , where always one annihilation operator A ∗ is combined with one creation operatorin A . After having performed all these commutations, we end up with a productof bosonic field operators of the form as in Definition 4.1, where each creation andannihilation operator gives rise to an insertion D z γ t and D z γ t , respectively.After these transformations, the expectation value can be written as follows, ω t ( A ∗ A ) = ˆ G dµ G ( U ) ∞ X p,q,r =0 T i ...i p , j ··· j q k ··· k r ( U ) T l ...l p , m ··· m q n ··· n r ( U ) ( z k | z n ) tρ · · · ( z k r | z n r ) tρ × h ψ i | π t U ψ l i tρ · · · h ψ i p | π t U ψ l p i tρ h ψ j | (11 − π t U ) ψ m i tρ · · · h ψ j q | (11 − π t U ) ψ m q i tρ , where the functions T i ...i p , j ··· j q k ··· k r are symmetric in the lower “bosonic” indices andtotally anti-symmetric in the upper “fermionic” indices i , . . . , i p and j , . . . , j q . Theintegrand is non-negative because all the involved inner products are positive semi-definite. This concludes the proof. (cid:3) Constructing Representations of the Field Algebra.
The GNS Representation.
The state ω t can be represented abstractly usingthe GNS construction, as we now briefly recall (for details see for example [2, Sec-tion 1.6] or [31, Section 5.1.3]). To this end, on the unital ∗ -algebra A we introducethe sesquilinear form h A | A ′ i := ω t ( A ∗ A ′ ) : A × A → C . Using the positivity of the state ω t (see Theorem 4.2), this sesquilinear form is positivesemi-definite. Dividing out the null space and forming the completion, we obtain aHilbert space ( F , h . | . i ). Since the multiplication in the algebra gives a representationof A on itself, we also obtain a canonical representation of A on F . Moreover, bydefinition, the vector Φ representing the identity in A has the property that h Φ | A Φ i = ω t (11 ∗ A
11) = ω t ( A ) . Thus in the above representation, the state is realized as the expectation value of Φ.This is the abstract GNS representation.We point out that, unless the state is quasi-free, the GNS representation is not aFock representation. In order to remedy the situation, we now proceed by constructinga Fock representation on the free Fock space.
Representation on the Free Fock Space.
We now explain how the state ω t canbe represented on the Fock space of the free field theory.In preparation, we build up the Fock space for the vacuum measure ρ . In orderto get the connection to the usual vacuum state used in Dirac theory, we need toimplement the fermionic frequency splitting and the Dirac sea picture. To this end,we introduce the orthogonal projections π − : H f ρ → H f ⊂ H f ρ and π + = (11 − π i ) : H f ρ → ( H f ) ⊥ and set Ψ †± ( ψ ) := Ψ † ( π ± ψ ) and Ψ ± ( ψ ) := Ψ( π ± ψ ) . Obviously, these operators satisfy the anti-commutation relations (cid:8) Ψ s ( φ ) , Ψ † s ′ ( φ ′ ) (cid:9) = h φ | π s,s ′ φ ′ i tρ (4.10)(and all other fermionic operators anti-commute). Moreover,Ψ † = Ψ † + + Ψ †− and Ψ = Ψ + + Ψ − . The subscript ± can be understood as a generalization of the usual splitting of thesolution space into solutions of positive and negative frequencies, respectively.We introduce the vacuum | i ∈ F as the vector with the property0 = a ( z ) | i = Ψ + ( ψ ) | i = Ψ †− ( ψ ) | i for all z ∈ h and ψ ∈ H f ρ . (4.11)We now consider the vector space generated by A | i with A ∈ A . The scalar productbetween such vectors is determined by (4.11), the commutation relations (4.2) andthe anti-commutation relations (4.10). Taking the completion then gives the Fockspace ( F , h . | . i F ).Before going on, we make a few explanatory remarks. In view of (4.11), the Fockspace is built up by acting on the vacuum state by the operators a † ( z ) , Ψ † + ( ψ ) and Ψ − ( ψ ) . (4.12)These operators can be viewed as generating the Fock space by creating particles andanti-particles. The annihilation operator Ψ − of the vectors in H f is to be interpretedas the operator creating a hole in the Dirac sea. By renaming the creation and anni-hilation operators according to Ψ − ↔ Ψ †− , one gets to the usual notation in quantumfield theory (in order to avoid confusion, we shall not adopt this convention here).Since the bosonic and fermionic field operators commute, the Fock space is a tensorproduct of the subspaces generated by the bosonic and fermionic operators, F := F b ⊗ F f . The vectors in the bosonic subspace F b can be identified with the symmetric tensorproduct of h , i.e. F b = ∞ M n =0 F b ,n and F b ,n := ( h n ) s , where the index s stands for the total symmetrization, i.e. (cid:0) ψ ⊗ · · · ⊗ ψ n (cid:1) s := 1 n ! X σ ∈ S n ψ σ (1) ⊗ · · · ⊗ ψ σ ( n ) . Details on this way of constructing the bosonic Fock space can be found in [18, Sec-tion 7.1]. The vectors of the fermionic subspace F f , on the other hand, are products ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 25 of totally antisymmetric wave functions describing the particles and the anti-particles,i.e. vectors of the form (cid:0) ψ ∧ · · · ∧ ψ p (cid:1) ∧ (cid:0) φ ∧ · · · ∧ φ q (cid:1) with ψ i ∈ H f and φ i ∈ ( H f ) ⊥ , endowed with the scalar product induced by h . | . i tρ .Our task is to construct an operator σ t on F with the property that ω t ( A ) = tr F ( σ t A ) for all A ∈ A . (4.13)Bringing all the operators in (4.12) to the left with the commutation and anti-commu-tation relations, it clearly suffices to satisfy this relations (4.13) for all operators A ofthe form A = a † ( z ′ ) · · · a † ( z ′ r ′ ) Ψ † + ( ψ ′ + , ) · · · Ψ † + ( ψ ′ + ,p + ) Ψ − ( ψ − , ) · · · Ψ − ( ψ − ,p − ) × a ( z ) · · · a ( z r ) Ψ + ( ψ + , ) · · · Ψ + ( ψ + ,q + ) Ψ †− ( ψ ′− , ) · · · Ψ †− ( ψ ′− ,q − ) . (4.14)Regarding A as describing a measurement, the parameters r, r ′ , p ± and q ± tell us howmany particles and anti-particles are involved in the process. In order to clarify issuesof convergence, we first consider the case that the process involves only a finite numberof particles and anti-particles. Definition 4.3.
The state ω t involves a finite number of particles and anti-particles if there is N ∈ N such that ω t ( A ) = 0 for all A of the form (4.14) wheneverone of the parameters r, r ′ , p ± or q ± exceeds N . Theorem 4.4.
Assume that the states ω t involves a finite number of particles oranti-particles. Then there is a density operator σ t on F such that (4.13) holds.Proof. We first explain the idea for a purely bosonic state. The fermions will be treatedbelow with similar methods. Thus, disregarding the fermions, the operator A in (4.14)simplifies to A = a † ( z ′ ) · · · a † ( z ′ r ′ ) a ( z ) · · · a ( z r ) . (4.15)The assumption that σ t involves a finite number of particles and anti-particles meansthat for some R > r , ω t ( A ) = 0 if r > R . We refer to r as the number of particles involved in the measurement and to R as themaximal number of such particles. Our goal is to show that there is a linear operator σ t which involves at most R particles and represents ω t for measurements involving R particles, i.e. ω t ( A ) − tr F ( σ t A ) = 0 if r > R − . (4.16)Once this has been achieved, we can proceed inductively for decreasing values of R until R = 0. Adding the operators σ t of the finite number of induction steps gives thedesired operator σ t satisfying (4.13) for all A of the form (4.15).In order to construct an operator σ t which satisfies (4.16), we first choose an or-thonormal basis ( z i ) of h . Then the operators A j ,...,j R i ,...,i r ′ := a † ( z i ) · · · a † ( z i r ′ ) a ( z j ) · · · a ( z j R ) with indices i ≤ i ≤ · · · ≤ i r ′ and j ≤ j ≤ · · · ≤ j R form a basis of the opera-tors (4.15) with R = r . Setting σ t = X r ′ ω t (cid:0) A j ,...,j R i ,...,i r ′ (cid:1)(cid:18) r ′ Y ℓ =1 m ( i ℓ )!) m ( iℓ ) (cid:19)(cid:18) R Y ℓ =1 m ( j ℓ )!) m ( jℓ ) (cid:19) × | a † ( z j ) · · · a † ( z j R ) | ih | a ( z i ) · · · a ( z i r ′ ) | , where m ( i ℓ ) and m ( j ℓ ) denote the multiplicities of the corresponding indices, a directcomputation using the canonical commutation relations shows that (4.16) holds (notethat the powers 1 /m i ℓ and 1 /m j ℓ are needed in order to compensate for the fact thatthe corresponding factors appear several times). This concludes the proof for a purelybosonic state.The general case with fermions can be proved similarly by proceeding inductivelyfor decreasing values of r as well as of the fermionic particle numbers q + and q − .Moreover, the operator σ t must be complemented by fermionic operators of the form | Ψ † + ( ψ + ,i ) · · · Ψ † + ( ψ + ,i q + ) Ψ − ( ψ − ,j ) · · · Ψ − ( ψ − ,j q − ) | i× h | Ψ + ( ψ + ,k ) · · · Ψ + ( ψ + ,k p + ) Ψ †− ( ψ − ,l ) · · · Ψ †− ( ψ − ,l p − ) | , where the fermionic indices are strictly increasing due to the nilpotence of the fermioniccreation and annihilation operators. Consequently, there are no multiplicities for thefermionic states. After these obvious modifications, the above arguments again gothrough. (cid:3) In order to extend this result to a general state ω t , one can decompose the stateinto a series ω t = ∞ X N =1 ω tN , where each ω tN involves at most N particles or anti-particles (the ω tN could be con-structed for example by projection to subspaces of the Fock space). With the help ofTheorem 4.4 we can represent each ω tN by a linear operator σ tN . Taking their sum, σ t := ∞ X N =1 σ tN , (4.17)one obtains the desired representation of ω t by a density operator σ t . However, oneshould keep in mind that the sum in (4.17) can be understood only formally. Wecannot expect that this sum converges for any state ω t . This corresponds to the well-known problem in quantum field theory of inequivalent Fock representations (see forexample [32, 33] for this problem in the context of quantum fields in a classical externalfield). Here we shall not enter the analysis of such convergence issues. Nevertheless,the representation constructed in Theorem 4.4 seems useful at least in a perturbativetreatment.4.6. Realizing the Insertions as Functional Derivatives.
It is a natural questionwhether, in analogy to the path integral formulation of quantum field theory, the aboveinsertions in the state ω t (see Definition 4.1) can be realized as variational derivatives ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 27 of the partition function, i.e. symbolically ω t ( · · · ) = 1 β k Z t (cid:0) β, ˜ ρ (cid:1) D · · · D | {z } k derivatives Z t (cid:0) β, ˜ ρ (cid:1) . The short answer is yes, up to rather subtle technical issues. We now explain theconnection in more detail.The bosonic insertions are variational derivatives of the exponent of the partitionfunction. Therefore, in order to obtain the desired product of insertions, one cantake corresponding variational derivatives of Z t , but one must make sure that eachderivative acts on the exponential. This could be arranged for example by restrictingattention to the highest order in β , i.e. symbolically ω t ( · · · ) = 1 β k Z t (cid:0) β, ˜ ρ (cid:1) D · · · D | {z } k derivatives Z t (cid:0) β, ˜ ρ (cid:1) (cid:16) O (cid:0) β − (cid:1)(cid:17) . (4.18)The disadvantage of this method is that one would have to control the error term byshowing that the higher derivatives of the nonlinear surface layer integral are negligible.This analysis has not yet been carried out. This is why we prefer to define the bosonicstate with insertions, each of which involves one functional derivative.For the fermionic insertions, the situation is more involved. In preparation, it isuseful to choose an orthonormal basis ( f ℓ ) of the subspace π tρ, ˜ ρ H f introduced in (4.7),making it possible to write the insertion (4.9) as h ψ i | π t U ψ j i tρ = X ℓ h ψ i | f ℓ i tρ h f ℓ | ψ j i tρ . (4.19)Here each scalar product is a surface layer integral of the form (2.31). Before we cancompare this expression with variational derivatives of the exponent of the partitionfunction, we need to introduce fermionic jet spaces. To this end, we can make use ofthe fact that jets can be described by first variations of the wave evaluation operator.Indeed, varying the formula F ( x ) = − Ψ( x ) ∗ Ψ( x ) , a vector field v ∈ T x F can be written as v ( x ) = − δ Ψ( x ) ∗ Ψ( x ) − Ψ( x ) ∗ δ Ψ( x ) . (4.20)A wave function ψ ∈ H f ρ by itself does not give rise to a variation of ˜Ψ. As an additionalinput we need a vector u ∈ H f , making it possible to vary the wave evaluation operatorby δ Ψ( x ) v := ψ ( x ) h u | v i H . The corresponding jet (4.20) takes the form v ( x ) = −| u i H ≺ ψ ( x ) | Ψ( x ) ≻ x − ≺ Ψ( x ) ∗ | ψ ( x ) ≻ x h u | . Taking linear combinations of these jets gives the vector space of fermionic jets denotedby Γ f ρ . On the fermionic jets one has a natural complex structure inherited from thecomplex structure of H f ρ given by J v ( x ) = i | u i H ≺ ψ ( x ) | Ψ( x ) ≻ x − i ≺ Ψ( x ) ∗ | ψ ( x ) ≻ x h u | . As a consequence, the corresponding holomorphic and anti-holomorphic componentsare given by v hol ( x ) = −≺ Ψ( x ) ∗ | ψ ( x ) ≻ x h u | v ah ( x ) = −| u i H ≺ ψ ( x ) | Ψ( x ) ≻ x . We shall always work with this complex structure. For ease in notation, we denotethese jets by v hol = − ψ h u | and v ah = −| u i ψ with ψ ∈ H f ρ , u ∈ H f . In view of (2.20) and (2.36), the jet derivative of the nonlinear surface layer integraltakes the form D ψ h u | γ t (˜ ρ, U ρ ) = D ψ h u | γ t ( U − ˜ ρ, ρ )= 2 (cid:18) ˆ ˜Ω t d ˜ ρ (˜ x ) ˆ M \ Ω t dρ ( y ) − ˆ ˜ M \ ˜Ω t d ˜ ρ (˜ x ) ˆ Ω t dρ ( y ) (cid:19) × ≺ ˜ ψ U − u (cid:0) U − ˜ x U (cid:1) | Q (cid:0) U − ˜ x U , y (cid:1) ψ ( y ) ≻ U − ˜ x U , where in the first step we used the unitary invariance of the Lagrangian. We write thisequation in a shorter form as D ψ h u | γ t (˜ ρ, U ρ ) = i h ˜ ψ U − u | ψ i t U with the sesquilinear form h . | . i t U − ˜ ρ,ρ : W U − ˜ ρ × W ρ → C , h ˜ ψ | ψ i t U − ˜ ρ,ρ := − i (cid:18) ˆ ˜Ω t d ˜ ρ (˜ x ) ˆ M \ Ω t dρ ( y ) − ˆ ˜ M \ ˜Ω t d ˜ ρ (˜ x ) ˆ Ω t dρ ( y ) (cid:19) × ≺ ˜ ψ (cid:0) U − ˜ x U (cid:1) | Q (cid:0) U − ˜ x U , y (cid:1) ψ ( y ) ≻ U − ˜ x U . (4.21)This sesquilinear form can be understood as the analog of the surface layer inte-gral (2.31), but where the two arguments are wave functions in different spacetimes.Moreover, there are subtle differences, which we shall discuss in detail below. Choosingan orthonormal basis ( e ℓ ) of H f , we obtain X ℓ D | e ℓ i ψ i γ t (˜ ρ, U ρ ) D ψ j h e ℓ | γ t (˜ ρ, U ρ ) = X ℓ h ψ i | ˜ ψ e ℓ i tρ, U − ˜ ρ h ˜ ψ e ℓ | ψ j i t U − ˜ ρ,ρ . (4.22)The left side of this equation can be realized as an insertion by taking second variationalderivatives of the partition function, if again one leaves out second derivatives of thesurface layer integral. Thus, in analogy to (4.18), we can write ˆ G (cid:18) X ℓ D | e ℓ i ψ i γ t (˜ ρ, U ρ ) D ψ j h e ℓ | γ t (˜ ρ, U ρ ) (cid:19) e βγ t (˜ ρ, U ρ ) d U = 1 β Z t (cid:0) β, ˜ ρ (cid:1) (cid:18) X ℓ D | e ℓ i ψ i D ψ j h e ℓ | Z t (cid:0) β, ˜ ρ (cid:1)(cid:19) (cid:16) O (cid:0) β − (cid:1)(cid:17) . Moreover, the right side of (4.22) looks very similar to the desired fermionic inser-tion (4.19). However, there are the following differences:
ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 29 (a) The surface layer integral (2.31) involves the “dynamical” kernel Q dyn as ob-tained by the constructions in [19]. The surface layer integral (4.21), however,involves the kernel Q which comes directly from the first variation of the La-grangian (2.20).(b) In the inner product (2.31) both wave functions are in the vacuum spacetime M .In (4.21), however, ψ is a wave function in M , whereas ˜ ψ is a wave function inthe interacting spacetime ˜ M .(c) A related difference is that the operator π t U in the insertion (4.19) (see also (4.8)and (4.7)) involves the transformation (4.5) of wave functions in ˜ M to wavefunctions in M .Thus, in simple terms, in (4.19) the wave functions from ˜ M are transformed to wavefunctions in M , making it possible to work with the scalar product h . | . i tρ on wave func-tions in the vacuum spacetime. In (4.22), on the other hand, the two wave functionsare paired by means of the kernel Q ( U − ˜ x U , y ), where one argument lies in the vacuumspacetime and the other in the interacting spacetime.The considerations and constructions in [19] show that working directly with thekernel Q is not sensible unless one restrict attention to physical wave functions (i.e.wave functions in the image of the wave evaluation operator). Likewise, it seemsthat (4.21) gives a physically meaningful pairing only if the kernel Q is modified similaras explained in [19]. For working out this modification of Q , it seems preferable tofirst transform the wave function ˜ ψ according to (4.5) to a wave function in M , so thatone can work purely in the vacuum spacetime. After this transformation, the surfacelayer integrals (2.31) and (4.21) have the same form, except for the different integralkernels, namely Q dyn in (2.31) and the modified kernel Q in (4.21). If the latter kernelis chosen in such a way that the corresponding mapping (4.5) becomes an isometry,then (2.31) and (4.21) agree. However, the possible modifications of Q have not yetbeen studied, and they might leave more freedom than the constructions in [19]. Thisis the reason why we here prefer to work with the insertion (4.19).4.7. The Algebra of Observables and the Quantum Møller Map.
We finallyput our construction into the context of algebraic quantum field theory by discussingthe algebras of observables and making the connection to Møller operators.For clarity, we first introduce the algebra of observables of the vacuum , denotedby A ρ . For the bosonic operators, as in [18, Section 7.2] to every u ∈ J ∗ we associatea corresponding symmetric operator ˆΦ( u ). The bosonic algebra of observables is thefree algebra generated by these operators satisfying the commutation relations (cid:2) ˆΦ( u ) , ˆΦ( v ) (cid:3) = i h u , G v i L ( M ) , where G is the bosonic Green’s operator (2.27). Making use of (2.28) and (4.1),the bosonic observables can be expressed in terms of the creation and annihilationoperators by (for details see [18, Section 7.2])ˆΦ( u ) = a ( G u ) + a † ( G u ) . Likewise, to every ψ ∈ W we associate the operators ˆΨ ∗ ( ψ ) and ˆΨ( ψ ). The fermionicalgebra of observables is the algebra generated by these operators which satisfies theanti-commutation relations { ˆΨ( φ ) , ˆΨ ∗ ( ψ ) (cid:9) = <φ | k ψ> , (4.23) where k is the fermionic Green’s operator (2.32) and <. | .> is the Krein inner prod-uct (2.34). All other fermionic operators anti-commute, (cid:8) ˆΨ( φ ) , ˆΨ( φ ′ ) (cid:9) = 0 = (cid:8) ˆΨ ∗ ( φ ) , ˆΨ ∗ ( φ ′ ) (cid:9) . The algebra of observables A ρ is defined as the unital ∗ -algebra generated freely by thebosonic and fermionic algebras (which means that all bosonic field operators commutewith all fermionic field operators). The fermionic observables are expressed in termsof the creation and annihilation operators simply byˆΨ ∗ ( ψ ) = Ψ † ( kψ ) and ˆΨ( ψ ) = Ψ( kψ ) . Indeed, making use of (2.33), for any φ, ψ ∈ W , (cid:8) Ψ( kφ ) , Ψ † ( kψ ) (cid:9) = h kφ | kψ i tρ = <φ | k ψ> , giving agreement with (4.23). The algebra A ρ corresponds precisely to the algebra ofobservables for quasi-free quantum field theories in algebraic quantum field theory.In order to get further connection to the algebraic formulation, it would be desir-able to also have an algebra of observables A ˜ ρ for the interacting spacetime. Thecommutation and anti-commutation relations would involve the interacting Green’soperators. At present, it is not clear how the algebra A ˜ ρ could be constructed. Partof the problem is that the interacting spacetime may involve observables which do notcorrespond to the degrees of freedom described by the field operators in M . A moredetailed discussion of this point can be found in [16]. In view of these difficulties, herewe are more modest and identify the operators in A ˜ ρ with operators in A ρ only atfixed time t . More precisely, in view of the time slice axiom (or, more computationally,by taking the Hamiltonian time evolution in the Heisenberg picture), the algebras areuniquely determined by all field operators in a time strip [ t , t + ∆ t ], where ∆ t > A ˜ ρ and A ρ inthis time strip, we obtain an algebra homomorphism ι t : A ˜ ρ → A ρ . (4.24)Identifying the algebras only at a fixed time has the advantage that the equal timecommutation and anti-commutation relations of the vacuum can also be used for theinteracting operators. In other words, we describe the interacting quantum fields attime t with the field operators of the vacuum. The state of the interacting system isdescribed at time t by the state ω t .A homomorphism (4.24) identifying the interacting algebra with the algebra in thevacuum appears in algebraic quantum field theory as the so-called quantum Møllermap . For details and the broader context we refer to the recent paper [6] and thereferences therein.5. Outlook: The Dynamics of the Quantum Fields
In the previous section, for a given time t we constructed a state ω t describing theinteraction as encoded in the measure ˜ ρ . By performing this construction for differenttimes, one can recover the dynamics of the system. However, it is a-priori not clearwhether this dynamics can be described by a time evolution operator acting on thestate L tt : ω t → ω t . ERMIONIC FOCK SPACES AND QUANTUM STATES FOR CAUSAL FERMION SYSTEMS 31
Going one step further, one would like to realize this time evolution by a unitarytransformation U tt on the Fock space, i.e. U tt : F → F unitary and σ t = U tt σ t ( U tt ) − . The question whether such time evolution operators exist and how they looks like willbe the objective the forthcoming paper [16].
Acknowledgments:
We would like to thank Claudio Dappiaggi, J¨urg Fr¨ohlich, MarcoOppio, Claudio Paganini, Moritz Reintjes and J¨urgen Tolksdorf for helpful discussions.We are grateful to the “Universit¨atsstiftung Hans Vielberth” for support. N.K.’s re-search was also supported by the NSERC grant RGPIN 105490-2018.
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Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, D-93040 Regensburg, Germany
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