aa r X i v : . [ m a t h - ph ] O c t On the mathematical origin of quantum space-time
G.Sardanashvily Department of Theoretical Physics, Moscow State University, 117234 Moscow, Russia
Abstract
An Euclidean topological space E is homeomorphic to the subset of δ -functions of thespace D ′ ( E ) of Schwartz distributions on E . Herewith, any smooth function of compact support on E is extended onto D ′ ( E ). One can think of these extensions as sui generis quantum deformations.In quantum models, one therefore should replace integration of functions over E with that over D ′ ( E ). A space-time in field theory, except noncommutative field theory, is traditionally de-scribed as a finite-dimensional smooth manifold, locally homeomorphic to an Euclideantopological space E = R n . The following fact (Proposition 1) enables us to think that aspace-time might be a wider space of Schwartz distributions on E .Let E = R n be an Euclidean topological space. Let D ( E ) be the nuclear space ofsmooth complex functions of compact support on E . Its topological dual D ′ ( E ) is thespace of Schwartz distributions on E , provided with the weak ∗ topology [1, 2]. Since D ( E )is reflexive and the strong topology on D ′ ( E ) is equivalent to the weak ∗ one, D ( E ) isthe a topological dual of D ′ ( E ). Therefore, any continuous form on D ′ ( E ) is completelydetermined by its restriction h φ, δ x i = Z φ ( x ′ ) δ ( x − x ′ ) d n x = φ ( x ) , x ∈ E, to the subset T δ ( E ) ⊂ D ′ ( E ) of δ -functions. Proposition 1.
The assignment s δ : E ∋ x → δ x ∈ D ′ ( E ) (1) is a homeomorphism of E onto the subset T δ ( E ) ⊂ D ′ ( E ) of δ -functions endowed with therelative topology (see Appendix for the proof ). As a consequence, T δ ( E ) is isomorphic to the topological vector space E with respect tothe operations δ x ⊕ δ x ′ = δ x + x ′ , λ ⊙ δ x = δ λx . Moreover, the injection E → T δ ( E ) ⊂ D ′ ( E )is smooth [3]. Therefore, we can identify E with a topological subspace E = T δ ( E ) of the E-mail: [email protected] φ of compact support on E = T δ ( E ) is extended to a continuous form e φ ( w ) = h φ, w i , w ∈ D ′ ( E ) , (2)on the space of Schwartz distributions D ′ ( E ). One can think of this extension as being aquantum deformation of φ as follows.The space D ( E ) is a dense subset of the Schwartz space S ( E ) of smooth complexfunctions of rapid decrease on E . Moreover, the injection D ( E ) → S ( E ) is continuous.The topological dual of S ( E ) is the space S ′ ( E ) of tempered distributions, which is asubset of the space D ′ ( E ) of Schwartz distributions. In QFT, one considers the Borchersalgebra A S = C ⊕ S ( E ) ⊕ S ( E ⊕ E ) ⊕ · · · ⊕ S ( k ⊕ E ) ⊕ · · · , (3)treated as a quantum algebra of scalar fields [4, 5]. Being provided with the inductive limittopology, the algebra A S (3) is an involutive nuclear barreled LF-algebra [6]. It follows thata linear form f on A S is continuous iff its restriction f k to each S ( k ⊕ E ) is well [1]. Thereforeany continuous positive form on A S is represented by a family of tempered distributions W k ∈ S ′ ( k ⊕ E ), k = 1 , . . . , such that f k ( φ ( x , . . . , x k )) = Z W k ( x , . . . , x k ) φ ( x , . . . , x k ) d n x · · · d n x k , φ ∈ S ( k ⊕ E ) . (4)For instance, the states of scalar quantum fields on the Minkowski space R are describedby the Wightman functions W k ⊂ S ′ ( R k ) [2].Any state of A S is also a state of its subalgebra A D = C ⊕ D ( E ) ⊕ D ( E ⊕ E ) ⊕ · · · ⊕ D ( k ⊕ E ) ⊕ · · · . This quantization can be treated as follows. Given a function φ ∈ D ( k ⊕ E ) on k ⊕ E , we haveits quantum deformation b φ = φ + f k ( φ ) ⊂ C ∞ ( k ⊕ E ) . (5)Let k ⊕ E be identified to the subspace T δ ( k ⊕ E ) ⊂ D ′ ( k ⊕ E ) of δ -functions on k ⊕ E . Thenthe quantum deformation b φ (5) of φ comes from the extension of φ onto D ′ ( k ⊕ E ) by theformula b φ ( z ) = φ ( z + W k ) , z + W k ∈ S ′ ( k ⊕ E ) ⊂ D ′ ( k ⊕ E ) . Generalizing this construction, let us consider a continuous injection s : k ⊕ E ∋ z → s z ∈ D ′ ( k ⊕ E )2nd a continuous function s φ : k ⊕ E ∋ z → s z ( φ ) ∈ C . for any φ ∈ D ( k ⊕ E ). For instance, the map s δ (1) where s δ,φ = φ is of this type. Given afunction φ ∈ D ( k ⊕ E ), we agree to call b φ = φ + s φ , b φ ( z ) = φ ( z ) + s z ( φ ) = φ ( z + s z ) (6)the quantum deformation of φ and to treat it as a function on the quantum space b E =( s δ + s )( E ) ⊂ D ′ ( E ).For instance, let φ ( x, y ) ∈ D ( E ⊕ E ) be a symmetric function on E ⊕ E . Then itsquantum deformation (6) obeys the commutation relation b φ ( x, y ) − b φ ( y, x ) = h φ, s x,y − s y,x i . Let E be coordinated by ( x λ ), and let us consider a function x x on E , though it isnot of compact support. Let us choose a map s such that all distributions s x , x ∈ E , are ofcompact support. Its quantum deformation is d x x = x x + s x ( x x ). It is readily observedthat d x x − d x x = 0, i.e., coordinates on a quantum space commute with each other, incontrast to a space in noncommutative field theory.Bearing in mind quantum deformations b φ (2) of functions φ on E , one should replaceintegration of functions over E with that over D ′ ( E ). Here, we summarize the relevantmaterial on integration over the space of Schwartz distributions D ′ ( E ). I. Due to the homeomorphism (1), the space T δ ( E ) is provided with the measure d n x ,invariant with respect to translations δ x → δ x + a . II.
The space M ( E, C ) of measures on E is the topological dual of the space K ( E, C ) ofcontinuous functions of compact support on E endowed with the inductive limit topology(see Appendix). The space M ( E, C ) is provided with the weak ∗ topology. It is homeomor-phic to a subspace of D ′ ( E ) provided with the relative topology. It follows that, for anymeasure ν on E , there exists an element w ν ∈ D ′ ( E ) and the Dirac measure ε ν of supportat w ν such that, for each φ ∈ D ( E ), we have Z E φν ( x ) = h φ, w ν i = Z D ′ ( E ) h φ, w i ε ν ( w ) . Let T x ⊂ D ′ ( E ) denote a subspace of point measures λδ x , λ ∈ C , on E = T δ ( E ). It is aBanach space with respect to the norm || λδ x || = | λ | . Let us consider the direct product T ( E ) = Y x ∈ E T x . (7)3y analogy with the notion of a Hilbert integral [7], we define the Banach space integral( T ( E ) , L ( E ) , d n x ) where L ( E ) is a set of fields b ϕ : E ∋ x → ϕ x δ x ∈ T ( E )such that: • the range of L ( E ) is a vector subspace of the direct product T ( E ) (7); • there is a countable set { ϕ i } of elements of L ( E ) such that, for any x ∈ E , the set { ϕ ix } is total in T x ; • the function x → || ϕ x || = | ϕ x | is d n x -integrable for any ϕ ∈ L ( E ).Let L ( E ) = L ( E, d n x ) be the space of complex square d n x -integrable functions on E .Clearly, D ( E ) ⊂ L ( E ), and there is an injection L ( E ) → M ( E, C ) ⊂ D ′ ( E ) such that ϕ ( φ ) = Z φ ( x ′ ) ϕ x δ ( x ′ − x ) d n xd n x ′ . Therefore, let Z φ x δ x d n x denote the image of ϕ in D ′ ( E ). Then any d n x -equivalent measure ν = c d n x (where c ∈ L ( E, d n x ) is strictly positive almost everywhere on E ) defines the correspondingelement w ν = Z c ( x ) δ x d n x of D ′ ( E ). For instance, if ν = d n x , we have ϕ x = 1 and w ν = Z δ x d n x. III.
Let Q be an arbitrary nuclear space (e.g., D ( E ), S ( E )) and Q ′ its topological dual(e.g., D ′ ( E ), S ′ ( E )). A complex function Z ( q ) on Q is called positive-definite if Z (0) = 1and X i,j Z ( q i − q j ) λ i λ j ≥ q , . . . , q m of elements of Q and arbitrary complex numbers λ , . . . , λ m .In accordance with the well-known Bochner theorem for nuclear spaces [8, 9, 10], anycontinuous positive-definite function Z ( q ) on a nuclear space Q is the Fourier transform Z ( q ) = Z exp[ i h q, w i ] µ ( w ) (8)4f a positive measure µ of total mass 1 on the dual Q ′ of Q , and vice versa .Note that there is no translationally-invariant measure on Q ′ . Let a nuclear space Q beprovided with a separately continuous non-degenerate Hermitian form h . | . i . In the case of Q = D ( E ), we have h φ | φ ′ i = Z φφ ′ d n x. Let w q , q ∈ Q , be an element of Q ′ given by the condition h q ′ , w q i = h q ′ | q i for all q ′ ∈ Q .These elements form the image of the monomorphism Q → Q ′ determined by the Hermitianform h . | . i on Q . If a measure µ in (8) remains equivalent under translations Q ′ ∋ w w + w q ∈ Q ′ , ∀ w q ∈ Q ⊂ Q ′ , in Q ′ , it is called translationally quasi-invariant. However, it does not remains equivalentunder an arbitrary translation in Q ′ , unless Q is finite-dimensional.Gaussian measures exemplify translationally quasi-invariant measures on the dual Q ′ ofa nuclear space Q . The Fourier transform of a Gaussian measure reads Z ( q ) = exp (cid:20) − B ( q ) (cid:21) , where B ( q ) is a seminorm on Q ′ called the covariance form. Let µ K be a Gaussian measureon Q ′ whose Fourier transform Z K ( q ) = exp[ − B K ( q )]is characterized by the covariance form B K ( q ) = h K − q | K − q i , where K is a boundedinvertible operator in the Hilbert completion e Q of Q with respect to the Hermitian form h . | . i . The Gaussian measure µ K is translationally quasi-invariant. It is equivalent µ ifTr( − KK + ) < ∞ . For instance, the Gaussian measures µ and µ ′ possessing the Fourier transforms Z ( q ) = exp[ − λ h q | q i ] , Z ( q ) = exp[ − λ ′ h q | q i ] λ, λ ′ ∈ R , are not equivalent if λ = λ ′ .If the function R ∋ t → Z ( tq ) is analytic on R at t = 0 for all q ∈ Q , then one canshow that the function h q | u i on Q ′ (e.g., the extension e φ (2) of φ onto D ′ ( E )) is square µ -integrable for all q ∈ Q . Moreover, the correlation functions can be computed by theformula h q · · · q n i = i − n ∂∂α · · · ∂∂α n Z ( α i q i ) | α i =0 = Z h q , w i · · · h q n , w i µ ( w ) .
5n particular, an integral of the function e φ (2) over D ′ ( E ) reads Z e φµ ( w ) = Z h φ ′ w i µ ( w ) = i ∂∂α Z ( αφ ) . Appendix
Let K ( E, C ) be the space of continuous complex functions of compact support on E = R n . For each compact subset K ⊂ E , we have a seminorm p K ( φ ) = sup x ∈ K | φ ( x ) | on K ( E, C ). These seminorms provide K ( E, C ) with the topology of compact convergence.At the same time, K ( E, C ) is a Banach space with respect to the norm k f k = sup x ∈ E | φ ( x ) | . Its normed topology, called the topology of uniform convergence, is finer than the topologyof compact convergence. The space K ( E, C ) can also be equipped with another topology,which is especially relevant to integration theory. For each compact subset K ⊂ E , let K K ( E, C ) be the vector subspace of K ( E, C ) consisting of functions of support in K . Let U be the set of all absolutely convex absorbent subsets U of K ( E, C ) such that, for everycompact K , the set U ∩ K K ( E, C ) is a neighborhood of the origin in K K ( E, C ) underthe topology of uniform convergence on K . Then U is a base of neighborhoods for theinductive limit topology on K ( E, C ) [11]. This is the finest topology such that the injection K K ( E, C ) → K ( E, C ) is continuous for each K . The inductive limit topology is finer thanthe topology of uniform convergence and, consequently, the topology of compact converges.The space M ( E, C ) of complex measures on E is the topological dual of K ( E, C ), endowedwith the inductive limit topology. The space M ( E, C ) is provided with the weak ∗ topology,and K ( E, C ) is its topological dual. The following holds [10]. Lemma 2.
Let ε x denote the Dirac measure of support at a point x ∈ E . The assignment s ε : E ∋ x → ε x ∈ M ( E, C ) (9) is a homeomorphism of E onto the subset T ε ⊂ M ( E, C ) of Dirac measures endowed withthe relative topology. Of course, D ( E ) ⊂ K ( E, C ), but the standard topology of D ( E ) is finer than its relativetopology as a subset of K ( E, C ). Let D R ( E ) denote D ( E ) ⊂ K ( E, C ) provided with therelative topology, and let D ′ R ( E ) be its topological dual endowed with the weak ∗ topology.6hen M ( E, C ) is homeomorphic to a subspace of D ′ R ( E ) provided with the relative topology.At the same time, D ′ R ( E ) is a subspace of D ′ ( E ) endowed with the relative topology. Thus,we have the morphisms E s ε −→ M ( E, C ) −→ D ′ R ( E ) −→ D ′ ( E ) , whose composition leads to the homeomorphism x → ε x = δ x d n x → δ x (1). References [1] F.Trevers,
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