aa r X i v : . [ qu a n t - ph ] F e b Geometric and algebraic approaches to quantumtheory
A. SchwarzDepartment of MathematicsUniversity of CaliforniaDavis, CA 95616, USA,schwarz @math.ucdavis.edu
Abstract
We show how to formulate quantum theory taking as a starting point theset of states (geometric approach). We give the equations of motion and theformulas for probabilities of physical quantities in this approach. A heuristicproof of decoherence in our setting is used to justify the formulas for prob-abilities. The geometric approach can be used to formulate quantum theoryin terms of Jordan algebras, generalizing the algebraic approach to quantumtheory. The scattering theory can be formulated in geometric approach.
Let us start with some very general considerations.Almost all physical theories are based on the notion of state at the moment t. The set of states will be denoted C .We can consider mixture of states: taking states ω i with probabilities p i we obtainthe mixed state denoted P p i ω i . Similarly if we have a family of states ω λ labeledby elements of a set Λ and a probability distribution on Λ ( a positive measure µ onΛ obeying µ (Λ) = 1) we can talk about the mixed state R Λ ω ( λ ) dµ. We assume that the set of states C contains also mixed states. Then the set ofstates is a convex set. We assume that it is a subset of a topological linear space L . The extreme points of C are called pure states.Instead of the set C we can work with the corresponding cone C (the set ofpoints of the form αx where α is a positive real number, x ∈ C ). The elements1f this cone are called non-normalized states. Two elements x, y ∈ C determinethe same normalized state if they are proportional : y = Cx where C is a positivenumber.An observable specifies a linear functional a on L . This requirement agrees withthe definition of mixed state: if ω = P p i ω i then a ( ω ) is an expectation value of a ( ω i ).(The non-negative numbers p i obeying P p i = 1 are considered as probabilities.)We consider deterministic theories. This means that the state in the moment t = 0 (or in any other moment t ) determines the state in arbitrary moment t .Let us denote by σ ( t ) the operator transforming the state in the moment t = 0into the state in the moment t (the evolution operator). The evolution operatorsconstitute a one-parameter family σ ( t ) of invertible maps σ ( t ) : C → C , thatcan be extended to linear maps of L . In other words the operators σ ( t ) belong tothe group U of automorphisms of C (to the group of linear bicontinuous maps of L inducing invertible maps of C onto itself). In some cases one should impose anadditional condition σ ( t ) ∈ V where V is a subgroup of U . The evolution operator satisfies the equation dσdt = H ( t ) σ ( t ) (1)(equation of motion). Here H ( t ) ∈ Lie ( V ) is an element of the tangent space to thegroup V at the unit element (the ”Hamiltonian”). The equation (1) can be regarded as a definition of H ( t ) . However, usually we goin opposite direction: the physical system we consider is specified by the operator H ( t ) (by the equation of motion) and our goal is to calculate the evolution operatorsolving the equation of motion.ExamplesClassical mechanicsThe cone C consists of positive measures µ on symplectic manifold M , we assumethat µ ( M ) < ∞ The set C consists of probability distributions (normalized positive measures, µ ( M ) = 1).Observables are functions on symplectic manifold V is the group of symplectomorphisms Lie ( V ) is the algebra of Hamiltonian vector fields Knowing the topology in L we can define in various ways the topology in V ⊂ U . This allows usto define the Lie algebra of the group V as the tangent space at the unit element. ( We disregardthe subtleties related to the fact that the group V is in general infinite-dimensional.) dρdt = { H, ρ } where ρ stands forthe density of the measure and {· , ·} denotes the Poisson bracket.Quantum mechanicsThe set C consists of density matrices ( positive trace class operators in realor complex Hilbert space H having unit trace: T rK = 1). Omitting the condition
T rK = 1 we obtain the cone C . A self-adjoint operator ˆ A ( an observable) specifies a linear functional by theformula K → T r ˆ AK V = U is isomorphic to the group of invertible isometries. (They are calledorthogonal operators if the Hilbert space is real and unitary operators if the Hilbertspace is complex.) An operator ˆ V ∈ U acts on C and on C by the formula K → ˆ V K ˆ V ∗ .The tangent space of this group at the unit element can be regarded as the Liealgebra Lie U ; it consists of bounded operators obeying ˆ A + ˆ A ∗ = 0 (skew-adjointoperators) . The equation of motion can be written in the form dKdt = A ( t ) K = − ˆ A ( t ) K + K ˆ A ( t ) . where ˆ A ( t ) is a family of skew-adjoint operators in H .In complex Hilbert space we can write the equation of motion as follows: dKdt = H ( t ) K = i ( ˆ H ( t ) K − K ˆ H ( t ))where ˆ H ( t ) is a family of self-adjoint operators ( here ˆ H ( t ) = i ˆ A ( t )).Quantum theory in algebraic approachThe starting point is a unital associative algebra A with involution ∗ Physicists always work in complex Hilbert space imposing the condition of reality on the statesif necessary. We prefer to work in real Hilbert space. In complex Hilbert space instead of skew-adjoint operator ˆ A we can work with self-adjointoperator ˆ H = i ˆ A The group U can be considered as Banach Lie group; the tangent space to it can be defined as theset of tangent vectors to curves (to one-parameter families of operators) that are differentiable withrespect to the norm topology. Notice, however, that the families of evolution operators appearingin physics usually do not satisfy this condition (but they can be approximated by differentiablefamilies). For a self-adjoint operator ˆ H in complex Hilbert space the operator ˆ A = − i ˆ H can beconsidered as a tangent vector to the one-parameter group of unitary operators e ˆ At = e − i ˆ Ht . Thisfamily is differentiable in norm topology only if the operator ˆ H is bounded. C is defined as the space of positive normalized linear functionalson A (the functionals obeying ω ( A ∗ A ) ≥ , ω (1) = 1). Omitting the normalizationcondition ω (1) = 1 we obtain the definition of the cone of states C . We can define acone A + of non-negative elements of A as a convex envelope of elements of the form A ∗ A ; the cone C of states is dual to this cone. V = U denotes the group of involution preserving automorphisms of A (they actnaturally on states)To relate the algebraic approach with the Hilbert space formulation we use theGNS (Gelfand-Naimark-Segal) construction: for every state ω there exists pre Hilbertspace H , representation A → ˆ A of A in H and a cyclic vector θ ∈ H such that ω ( A ) = h ˆ Aθ, θ i We say that vector θ is cyclic if every vector x ∈ H can be represented in the form x = ˆ Aθ where A ∈ A . Notice that instead of pre Hilbert space H one can work withits completion, Hilbert space H , then θ is cyclic in weaker sense: the vectors ˆ Aθ aredense in Hilbert space.In the present paper we describe the approach to quantum theory where the pri-mary notion is the convex set of states (geometric approach). We discuss its relationsto algebraic approach and to textbook quantum mechanics. We show that a gener-alization of decoherence is correct in geometric approach and use this statement toderive the formulas for probabilities from the first principles. We analyze the relationto Jordan algebras. Finally we discuss the notions of particle and quasiparticle inalgebraic and geometric approaches. We define particles as elementary excitations ofground state and quasiparticles as elementary excitations of any translation-invariantstationary state. We relegate the scattering theory to follow up papers [2],[3], [4].These papers are at least formally independent from the present paper. The paper[2] is devoted to the scattering theory in algebraic approach; it generalizes the resultsof [7] (see also the Chapter 13 of [11]. The main goal of this paper is to provide aconvenient way to compare the constructions of [3] with standard constructions. Inthe paper [3] we develop the scattering theory in geometric approach ( with weakeraxioms than in present paper). Notice that the conventional scattering matrix can-not be defined in this approach, but there exists a very natural definition of inclusivescattering matrix (see [6],[7], [11] ). The paper [4] is devoted to scattering theory The set of self-adjoint elements (observables) is not closed under multiplication, but it is closedwith respect to the operation a ◦ b = ( ab + ba ). This remark led to the notion of Jordan algebra( axiomatization of this operation). Jordan algebras can be regarded as the natural framework ofalgebraic approach. This statement is prompted by the Alfsen-Shultz theorem: Cones of states oftwo C ∗ -algebras are isomorphic iff corresponding Jordan algebras are isomorphic.
4n the framework of Jordan algebras; it relies on the definitions given in [3], but ismostly independent of [2],[3].We do not perform any calculations in this paper. It is natural to ask whetherthe geometric approach is convenient for calculations. The answer to this questionis positive. The inspiration for geometric approach came in part from the formalismof L-functionals [5],[8], [6], [11]. In this formalism one works with the space of statesover Weyl or Clifford algebra. One can construct diagram techniques of perturbationtheory in this formalism; they are equivalent to Keldysh techniques (see for example[9], [10]) that were applied to many problems of statistical physics. These diagramsare useful also in calculation of inclusive scattering matrix.
We start with a bounded convex set C ⊂ L and a subgroup V of the automorphismgroup U of C . Here L is a normed space over R and automorphisms of C are by definitioninvertible linear bicontinuous operators in L mapping C onto itself. (More generallywe can assume that L is a linear topological space.)Together with C we consider the cone C spanned by C (the set of all points λx where λ ≥ x ∈ C . ) The set of all endomorphisms of C (the set of all linearcontinuous operators in L mapping C into itself) will be denoted by End C . This setis a semigroup with respect of composition of operators; it is also closed with respectto addition and with respect to multiplication by a positive scalar (is a semiring).We fix a subset W of End C that is also closed with respect to these operations andwith respect to the action of elements of V . (Notice that the automorphisms of C act naturally on C specifying elements of End C . )In examples W consists of quasilocal endomorphisms (quasilocal observables).The Lie algebra of the group V can be regarded as the Lie algebra of global observ-ables.Working in the framework of the algebraic quantum theory we define the dataof geometric approach taking as L the algebra A considered as a linear space orthe space A ∗ of linear functionals on A . The cone in A is spanned by A ∗ A where A ∈ A , the cone in A ∗ is dual to this cone. Automorphisms of algebra specifyautomorphisms of cones; we take V as the group of these automorphisms. The map A → B ∗ AB where B ∈ A specifies an endomorphism of the cone in A , the dual mapis an endomorphism of the dual cone. We define W as a semiring generated by thesemaps. 5very element B ∈ A specifies two operators on the space A ∗ : for a linear func-tional ω on A we define ( Bω )( A ) = ω ( AB ) , ( ˜ Bω )( A ) = ω ( B ∗ A ) . The endomorphismof C described above can be written as ˜ BB.
We assume that one can obtain the evolution operator σ ( t ) acting in C from aninfinitesimal automorphism A (”Hamiltonian”) using the equation of motion : dσdt = Aσ ( t ) (2)(we say that A is an infinitesimal automorphism if this equation has a solution σ ( t ) ∈ U .)We will consider also a more general case when A in (2) depends on t. We assume that there exists a basis of the complexification of L consisting ofeigenvectors of A ( we say in this case that A has discrete spectrum). Let us denoteby ( ψ j ) such a basis: Aψ j = ǫ j ψ j It follows from the boundedness of C that ǫ j ∈ i R . Example: textbook quantum mechanics. As we noticed in this case C consistsof density matrices and the equation of motion in the case of time- independentHamiltonian has the form dKdt = AK = i ( ˆ HK − K ˆ H ) . The space L consists of all self-adjoint operators belonging to trace class and thecomplexification of this space consists of all operators belonging to trace class. Thecone C consists of positive trace class operators.Let us assume that ˆ H has discrete spectrum. Then A also has discrete spectrumand eigenvalues of A are differences of eigenvalues of ˆ H ( up to a factor of i .)Let us suppose that A ( g ) is a continuous family of infinitesimal automorphismssuch that A (0) = A . Then for a right choice of the basis ( ψ j ) and for | g | < δ j we canconstruct vectors ( ψ j ( g )) that depend continuously on g in such a way that A ( g ) ψ j ( g ) = ǫ j ( g ) ψ j ( g )where ψ j (0) = ψ j . (We assume that all non-zero eigenvalues are at most finitelydegenerate) We say ψ j is a robust zero mode of A if ǫ j ( g ) = 0 . In other words a zero mode ψ j is a robust zero mode of A if in any neighborhood of ψ j and sufficiently small g we can find a zero mode of A ( g ) in this neighborhood.6et us model the interaction with environment by random adiabatic Hamiltonian A ( g ( t )) . Then in the adiabatic approximation σ ( t ) ψ j = e ρ j ( t ) ψ j ( g ( t )) , where dρ j dt = ǫ j ( g ( t )) . ( In adiabatic approximation we can neglect the derivative ˙ g ( t ) . It follows that σ ( t ) obeys the equations of motion in this approximation.)Imposing some conditions on the random Hamiltonian A ( t ) one can prove thatin average the random phase factors e ρ j ( t ) vanish unless φ j is a robust zero mode.One can apply the above considerations to the case when A is an arbitrary in-finitesimal automorphism (not necessarily the Hamiltonian). If Ax = 0 ( i.e. x isa zero mode of A ) we say that x is a robust zero mode if for every infinitesimalautomorphism A ′ in a small neighborhood of A we can find a zero mode x ′ in a smallneighborhood of x. Let us define an observable as a pair (
A, a ) where A is an infinitesimal auto-morphism and a is an A -invariant linear functional on the space of states (it has aphysical meaning of the expectation value of the observable). For example for energy( H, h ) the infinitesimal automorphism H is the Hamiltonian and h is the expectationvalue of the energy. The state with minimal value of h is a ground state.Let us denote by ( KerA ) r the space of robust zero modes of A and by P ′ theprojection P ′ : L → ( KerA ) r sending all eigenvectors of A that are not robust zeromodes to zero. To calculate probabilities of A in the state x we should represent therobust zero mode P ′ x as a mixture of robust pure zero modes: P ′ x = P p k z k . Then p k is the probability to find the the value a ( z k ) measuring A . (We assume that thenumbers a ( z k ) are different. If this condition is not satisfied we should calculate theprobability to obtain the value α summing all p k with a ( z k ) = α. )In the textbook quantum mechanics we take A as a commutator with a self-adjoint operator ˆ A multiplied by i and define a ( K ) = T r ˆ AK . Notice that a ( K ) isnot necessarily finite; for example, in translation-invariant state the value of energyis in general infinite (but we can talk about the density of energy and about thedifference of energies). The situation in the geometric approach is similar.In ˆ A -representation the basis of eigenvectors of A consists of matrices having onlyone non-zero entry equal to 1 . Diagonal matrices of this kind are robust zero modes.The projection P ′ in ˆ A -representation sends every density matrix into its diagonalpart (decoherence, collapse of wave function).Define a projection P : L → L by the formula
P x = lim T →∞ T Z T dtσ A ( t ) x σ A ( t ) is the group of automorphisms generated by A. It is obvious that P C ⊂C . It is easy to check, that
P ψ j = 0 if ǫ j = 0 , P ψ j = ψ j if ǫ j = 0 . If all zero modes are robust then P = P ′ , hence we can calculate the probabilitiesusing P. Jordan algebra can be defined as a unital commutative algebra where the operators R x and R x ◦ x commute. ( Here R u is an operator of multiplication by u , i.e. R u ( v ) = u ◦ v. ) A subalgebra of Jordan algebra generated by one element is associative ( theJordan algebra is power- associative). This means that we can talk about powers x n ofan element x. For every unital associative algebra we can define a structure of Jordanalgebra introducing the operation x ◦ y = ( xy + yx ) . One says that a subalgebraof a Jordan algebra obtained this way is a special Jordan algebra; algebras that arenot special are called exceptional.If a unital associative algebra is equipped with an involution the set of self-adjointelements can be regarded as a Jordan algebra with respect to the operation x ◦ y. One can consider Jordan algebras over any field; for definiteness we considerJordan algebras over R . The formulation of quantum theory in terms of the set of states is closely relatedto the formulation in terms of Jordan algebras. For every Jordan algebra B we definea cone of positive elements B + as a convex set spanned by the elements of the form x where x ∈ B . We can consider also the dual cone consisting of linear functionalson B that are positive on B + . We can use one of these cones in geometric approachto quantum theory.If B is a linear topological space and algebraic operations are continuous we saythat B is a topological Jordan algebra. For such algebras we consider only continuousfunctionals and maps.We will consider J B -algebras B defined as Jordan algebras that can be equippedwith Banach norm obeying || x ◦ y || ≤ || x || · || y || , || x || = || x || , || x || ≤ || x + y || . ( The first condition means that J B -algebra is a Banach algebra. An associativeBanach algebra obeying || x || = || x || is called C ∗ -algebra. The set of self-adjointelements of C ∗ -algebra is a J B -algebra with respect to the operation x ◦ y = ( xy + yx ) . J B -algebras of this kind and their subalgebras are called J C -algebras.)8inite-dimensional
J B -algebras coincide with Euclidean Jordan algebras classi-fied by Jordan, von Neumann, Wigner. Almost all simple algebras of this type canbe realized as algebras of Hermitian n × n matrices with real, complex, quater-nionic or octonionic entries. (In octonionic case we should take n = 3; we obtain27-dimensional algebra called Albert algebra. The Albert algebra is exceptional.)One more series of simple Euclidean algebras consists of algebras with generators1 , e , ..., e n obeying relations e i ◦ e j = 0 for i = j , e i ◦ e i = 1 . We defined a positive cone B + in any Jordan algebra B as a convex set spannedby all squares. In the case of J B -algebra one can say that the positive cone consistsof squares. Equivalently a ∈ B + if the spectrum of R a consists of non-negative realnumbers.If a J B -algebra comes from C ∗ -algebra this definition coincides with the definitionof the positive cone in C ∗ -algebra (recall, that the positive cone in an associativealgebra with involution is spanned by the elements of the form A ∗ A ).The group V is defined as the group of automorphisms of the algebra B actingon the cone. Its Lie algebra consists of derivations.If the Jordan algebra B consists of self-adjoint elements of an algebra with in-volution A then every skew-adjoint element T of A specifies a derivation α T (as acommutator with T ). An even polynomial p ( T ) is a self-adjoint element of A com-muting with T , hence it is a zero mode of the derivation α T .This is a robust zeromode of the derivation: a derivation α T ′ where T ′ is close to T has a zero mode p ( T ′ )that is close to p ( T ) . There exists unique exceptional (not special) simple Jordan algebra. Any deriva-tion of it has three robust zero modes. To prove this fact we realize this algebra asthe algebra of 3 × SO (8)can be regarded as automorphisms of this algebra, elements of so (8) = LieSO (8)specify infinitesimal automorphisms having diagonal matrices as zero modes. Thesezero modes are robust. To prove this we notice that generic elements of so (8) haveonly these three zero modes. From the other side all infinitesimal automorphismscan be transformed into elements of so (8) by means of inner automorphisms of theautomorphism group. This means that the every infinitesimal automorphism has atleast three zero modes and generic infinitesimal automorphism has precisely threezero modes. It is easy to conclude from this fact that these zero modes are robust.Transformations Q a ( x ) = { a, x, a } where { a, x, b } = ( a ◦ x ) ◦ b + ( x ◦ b ) ◦ a − ( a ◦ b ) ◦ x B + . ) If a is invertible, then Q a is anautomorphism of the cone.Noticing that Q a (1) = a we obtain that the cone of J B -algebra is homogeneous(the automorphisms of the cone act transitively on the interior of the cone). Ifthe algebra is finite-dimensional then the cone is self-dual and all self-dual homo-geneous cones can be obtained this way. Therefore Jordan-von Neumann-Wignertheorem gives the classification of finite-dimensional self-dual homogeneous cones(round cones, cones of positive self-adjoint operators in real, complex and quater-nionic vector spaces and the exceptional 27-dimensional cone).All finite-dimensional homogeneous cones were described by E. Vinberg [13]It was conjectured that superstring is related to exceptional Jordan algebra (Foot-Joshi [12]). The group SO (1 ,
9) acts as a subgroup of the automorphisms of the coneof this algebra.
In geometric approach particles and the scattering of particles can be defined if anabelian Lie group interpreted as a group of space-time translations acts on the coneof states C . We denote spatial translations by T x where x ∈ R d and time translationsby T τ . We assume also that translations act also on W and this action is compatiblewith the action on the states. We use the notation A ( τ, x ) for the translated operator A. In Lorentz-invariant theory the action of translations can be extended to theaction of Poincar´e group P . In geometric approach an excitation of translation-invariant stationary state ω can be defined as a state of the form W ω where W ∈ W . Alternatively one cansay that an excitation is a state σ obeying T x σ → Cω as x → ∞ (here T x standsfor spatial translation, C is a constant factor). The second definition is the mosttransparent one. One can say not very precisely that the excitation essentially differsfrom ω only in a bounded spatial domain.Let us establish the relation between two definitions in algebraic quantum fieldtheory. Recall that in this case W is the smallest semiring containing elements ofthe form ˜ BB where B ∈ A . Hence to prove that an excitation in the first sense is anexcitation in second sense one should check that the state σ ( A ) = ω ( B ∗ AB ) obeysthe conditions of the second definition. We assume that ω obeys the cluster property.This means, in particular, that ω ( B ∗ A ( τ, x ) B ) − ω ( B ∗ B ) ω ( A ( τ, x )) → x → ∞ . Using translation invariance of ω we obtain that in this limit T x σ → Cω with C = ω ( B ∗ B ) . Quasi-particles can be defined as elementary excitations of translation-invariantstationary state ω. Particles are defined as elementary excitations of ground state.To make these definitions precise we should explain the notion of elementaryexcitation. We start with the explanation in the algebraic approach to quantumtheory. In this approach the action of translations on states is induced by the actionof translations on the algebra A . The time and spatial translations are defined asinvolution-preserving automorphisms α ( τ, x ); we use the notation A ( τ, x ) = α ( τ, x ) A for A ∈ A . The GNS ( Gelfand-Naimark-Segal) construction gives a representation A → ˆ A of the algebra A in the pre Hilbert space H , a cyclic vector θ corresponding tothe state ω (i.e. obeying ω ( A ) = h ˆ Aθ, θ i ). The translations descend to the space H asunitary (or orthogonal) operators T τ , T x (this follows from our assumption that ω isa stationary translation-invariant state). Namely, we define define T τ ˆ Aθ as \ A ( τ, θ , T x ˆ Aθ as \ A (0 , x ) θ. Notice that \ A ( τ, x ) = T τ T x ˆ AT − x T − τ . The operators of energy andmomentum ˆ H, ˆ P are defined as infinitesimal translations (if we are working in realHilbert space, they act in its complexification.). We say that the states correspondingto the elements of H are excitations of ω. This definition agrees with the definitionin geometric approach: if Θ = Bθ and σ denotes the state corresponding to Θ then σ ( A ) = h A Θ , Θ i = h B ∗ ABθ, θ i = ω ( B ∗ AB ) = ( W ω )( A )where W = ˜ BB ∈ W .)In Lorentz-invariant theory the Poincar´e group P acts as a group of automor-phisms of the algebra A . This action induces an action of P on states and a unitary(or orthogonal ) representation of this group on the space H . An elementary excita-tion can be defined as an irreducible subrepresentation of this representation.Notice that we consider H as a pre Hilbert space; by definition a unitary represen-tation in pre Hilbert space is irreducible if it induces an irreducible representation inthe completion. An irreducible unitary representation in Hilbert space is isomorphicto the representation in the space L of square integrable functions , a representationin pre Hilbert space is isomorphic to the representation in a dense subspace of L . An irreducible unitary representation of Poincar´e group with positive energy isisomorphic to a representation of this group in the space of (multicomponent) func-tions depending on the momentum k ; the momentum operator ˆ P can representedas a multiplication by k and the translation T x is an operator of multiplication by e i xk . This fact prompts the definition of elementary excitation in general case: we11ssume that the representation of the group of spatial translations is the same as inLorentz-invariant situation.Let us consider an algebra over complex numbers A , a stationary translationinvariant state ω , a complex pre Hilbert space H and a vector θ ∈ H obtainedfrom ω by means of GNS construction. Then the elementary excitation can bedefined as a generalized multicomponent function Φ( k ) = (Φ ( k ) , · · · Φ m ( k )) suchthat ˆ P Φ( k ) = k Φ( k ), ˆ H Φ( k ) = E ( k )Φ( k ) where E ( k ) is a matrix function takingvalues in Hermitian matrices.We assume that Φ( k ) takes values in H and is delta- normalized: h Φ( k ) , Φ( k ′ ) i = δ ( k − k ′ ) . In other words, we have a linear operator φ → Φ( φ ) that assigns to every φ ∈ h a vector Φ( φ ) = R φ ( k )Φ( k ) d k in H . This operator should be an isometryobeying ˆ H Φ( φ ) = Φ( ˆ Eφ ) , ˆ P Φ( φ ) = Φ(ˆ k φ ) where ˆ E and ˆ k stand for multiplicationoperators by E ( k ) and k . Here we take as the space h the space of complex square-integrable functions on R d × I ( here I is a finite set consisting of m elements) or anydense linear subspace of this space that is invariant with respect to the operators ˆ E and ˆ k . (In other words, these functions depend on the momentum variable k ∈ R d and discrete parameter j ∈ I . For definiteness we assume that these functionsbelong to the space S of smooth functions such that all of their derivatives tend tozero faster than any power.) We can work also in coordinate representation assumingthat the momentum operator ˆ P = i ∇ is the infinitesimal spatial translation (spatialtranslations are represented as shifts with respect to the coordinate variable x ).We say that h is an ”elementary space” over C . If A is an algebra over real numbers with action of spatial and time translationswe can define the elementary excitations of translation-invariant stationary state ω in the following way.Let us fix the space h as a subspace of the space of real square-integrable func-tions on R d × I where I is a finite set. For definiteness we take h as the space S of smooth functions of x decreasing faster than any power . Then h is invariant withrespect to spatial translations T a : φ ( x , j ) → φ ( x − a , j ); we assume that it is invari-ant with respect to time translations (one-parameter group of orthogonal operatorscommuting with spatial translations). The time translations can be written in theform T τ = e − τ ˆ E where ˆ E is a skew-adjoint operator with translation-invariant ker-nel. (In other words the kernel of the operator ˆ E has the form E ab ( x − y ) where E ab ( x ) = − E ba ( − x ) , a, b ∈ I . ) We say that h is an ”elementary space” over R . Definition 1.
An elementary excitation of a stationary translation-invariant state ω is an isometric map Φ of h into the space H of the corresponding GNS -representationsuch that the translations in h agree with translations in H (i.e. T x Φ( φ ) = Φ( T x φ ) , T τ Φ( φ ) =Φ( T τ φ ).
12e formulated this definition for the case when A is an algebra over real numbers,but it can be applied also in the case when A is an algebra over C . Considering the elements of h as test functions we can say that elementaryexcitations are generalized functions Φ( x , j ) taking values in H (here j is a dis-crete index: j ∈ I ). (We define the generalized function by the formula Ψ( φ ) = P j R d x Φ( x , j ) φ ( x , j ) . )One can work in momentum representation. Then the test functions dependon the momentum variable k ∈ R d and discrete variable j ∈ I ; they obey thereality condition φ ∗ ( k , j ) = φ ( − k , j ) . The spatial translation T a can be understoodas multiplication by e i ak . The time translations act as multiplication by a matrixfunction e iτE ( k ) where k ∈ R d and E ( k ) is a Hermitian ( m × m ) -matrix. (Here m is the number of elements in I . ) Diagonalizing E ( k ) we can calculate the matrixfunction e iτE ( k ) ; it has the form e iτE ( k ) = X j a j ( k ) e iǫ j ( k ) τ (3)where ǫ j ( k ) are eigenvalues of E ( k ) . This formula allows us to analyze the asymptotic behavior of T τ as τ → ∞ incoordinate representation.Let us denote by U φ an open subset of R d containing all points having the form ∇ ǫ s ( k ) where k belongs to supp( φ ) = ∪ j supp φ j ) (to the union of supports of thefunctions φ ( k , j )). Lemma 2.
Let us assume that supp( φ ) is a compact subset of R . Then for large | τ | we have | ( T τ φ )( x , j ) | < C n (1 + | x | + τ ) − n where x τ / ∈ U φ , the initial data φ = φ ( x , j ) is the Fourier transform of φ ( k , j ) , and n is an arbitrary integer. The proof of this lemma can be given by means of the stationary phase method.We can express Lemma 2 saying that τ U φ is an essential support of ( T τ φ )( x , j )for large | τ | . Let us consider now quantum theories in geometric approach. To define elemen-tary excitations we need the action of spatial and time translations on the cone C . Definition 3.
In geometric approach we define an elementary excitation of translation-invariant stationary state ω as a quadratic map of h into the space of excitations of ω . This map should agree with the action of spatial and time translations.
13o relate this definition to the definition of elementary excitations in algebraicapproach we notice that starting with a map Φ : h → H specifying an elementaryexcitation we can construct a quadratic map σ sending φ ∈ h into a state σ φ definedby the formula σ φ ( A ) = h A Φ( φ ) , Φ( φ ) i . ( Here a state is a positive linear functionalon A where A is an algebra with involution over R .)Notice that in scattering theory we impose some additional conditions on excita-tions of ω in geometric approach.If we are starting with classical field theory in Hamiltonian or Lagrangian ap-proach then the classical vacuum can be regarded as stationary translation-invariantfield configuration with minimal energy density. We can consider ”excitations” oftranslation-invariant field as fields having finite energy or as fields that coincide withtranslation-invariant field at spatial infinity. ( Talking about the energy of an exci-tation we assume that that energy of translation-invariant field is equal to zero.) Allexcitations of classical vacuum should have non-negative energy.Quantizing classical field theory we expect that the ground state (physical vac-uum) is obtained from the classical vacuum and that the quadratic part of the actionfunctional in the neighborhood of classical vacuum governs the excitations of groundstate (quantum particles). The quantum particles corresponding to the quadraticpart of action functional are called elementary particles. However, it is possible thatthere exist other (composite) particles. Especially interesting particles correspondto solitons ( to finite energy solutions to the classical equations of motion having theform s ( x − v t )). Usually we have a family of solitons labelled by momentum p (inLorentz-invariant theories this is always the case). Then the set of fields s p ( x − a )is a symplectic submanifold of phase space that is invariant with respect to spatialand time translations. The restriction of the Hamiltonian to this manifold has theform H ( p , a ) = E ( p ) . Quantizing this manifold we obtain a quantum particle thatin zeroth order with respect to ~ corresponds to ”elementary space” over C with m = 1 and with infinitesimal time translation governed by the function E ( p ) . Gen-eralized solitons also correspond to symplectic submanifolds of phase space that areinvariant with respect to spatial and time translations; after quantization they leadto quantum particles described by ”elementary spaces” over C with m > . Finally a remark about the elementary excitations in the formulation in terms ofJordan algebras. We assume that time and spatial translations act as automorphismsof
J B -algebra; then they act also on the positive cone and on the dual cone.Let us fix a linear map ρ : h → B commuting with translations. Using thequadratic map Q : B →