Projective Limits of State Spaces III. Toy-Models
PProjective Limits of State SpacesIII. Toy-Models
Suzanne Lanéry , and Thomas Thiemann Institute for Quantum Gravity, Friedrich-Alexander University Erlangen-Nürnberg, Germany Mathematics and Theoretical Physics Laboratory, François-Rabelais University of Tours, France
November 11, 2014
Abstract
In this series of papers, we investigate the projective framework initiated by Jerzy Kijowski [7] and AndrzejOkołów [12, 13], which describes the states of a quantum theory as projective families of density matrices.A strategy to implement the dynamics in this formalism was presented in our first paper [10], which we nowtest in two simple toy-models. The first one is a very basic linear model, meant as an illustration of thegeneral procedure, and we will only discuss it at the classical level. In the second one, we reformulate theSchrödinger equation, treated as a classical field theory, within this projective framework, and proceed toits (non-relativistic) second quantization. We are then able to reproduce the physical content of the usualFock quantization.
Contents a r X i v : . [ g r- q c ] N ov Introduction
In [10, section 3], we introduced a strategy to deal with dynamical constraints in a projective limitof symplectic manifolds. After having convinced ourselves that a regularization of these constraintswill in general be necessary, since we cannot expect them to be adapted to the projective system,we adopted the perspective that a dynamical state can be identified with the family of successiveapproximations approaching an exact solution of the dynamics. On the one hand, this allowsus to put the dynamical state space into a projective form. On the other hand, it also provides asuitable ground for a notion of convergence, that will make it possible to define meaningful physicalobservables on this state space.However, applying this procedure demands that one sets up a regularization scheme fulfilling anumber of restrictive properties (summarized in [10, prop. 3.23]), which raises the question of itspracticability. Hence, we now want to discuss two simple examples, meant as ‘proofs of concept’that such schemes can indeed be designed.Note that the framework in [10, section 3] was purely classical. We have not yet undertakento formulate a general procedure regarding the resolution of dynamical constraints in projectivesystems of quantum state spaces [7, 13, 11]. Nevertheless, our second example will explore howanalogous ideas can be implemented at the quantum level, and will give us the opportunity todelineate an appropriate course and to underline possible difficulties.
This first example is arguably mostly artificial and does not pretend to have great physicalrelevance. Our motivation here is to illustrate the concepts introduced in [10, sections 2 and 3] inthe simplest possible setup. We consider an infinite dimensional Hilbert space H (which is nothingbut a linear Kähler manifold) and form its rendering by a projective structure of finite dimensionalHilbert spaces (to prevent any confusion: the Hilbert spaces in discussion here are the phase spacesof classical systems, there will be nothing quantum in the present section). This rendering is builtfrom an Hilbert basis of H by considering all the vector subspaces of H spanned by a finite numberof basis vectors and linking them by orthogonal projections (a more satisfactory rendering for H ,namely one that does not require the choice of a preferred basis, will be presented in section 3;however we do not want to use it here, since the constraints we will be looking at could be directlyformulated as an elementary reduction over a cofinal part of its label set, and it would thereforenot be appropriate as an example for the regularization procedure). Proposition 2.1
Let H , � · , · � be a complex Hilbert space and define: ∀� ∈ H , J � := � � ; ∀�, w ∈ H , Ω( �, w ) := 2 Im � ��, w� �. hen, H , Ω , J is a Kähler manifold. Proof
The real scalar product Re � · , · � equips H (seen as a real vector space) with a structureof real Hilbert space, therefore, any bounded real-valued real-linear form on H can be writtenas Re ��, · � = 2 Im � − � �, · � = Ω � − � �, · � for some � ∈ H . Hence, Ω is a strong symplecticstructure.Next, J is by construction a complex structure on H . We have ∀�, w ∈ H , Ω( � �, � w ) = Ω( �, w ),and � �→ Ω( �, � � ) = 2 Re ��, �� is positive definite.The integrability conditions for Ω and J are trivially satisfied since we actually have a Kähler vector space . � Proposition 2.2
Let H be a separable, infinite dimensional Hilbert space (equipped with the strongsymplectic structure Ω defined in prop. 2.1) and let ( � � ) �∈ N be an Hilbert basis of H . We define: L := {I ⊂ N | < I < ∞} equipped with the preorder defined by ⊂ ; ∀I ∈ L , H I := Vect {� � | � ∈ I} equipped with the induced symplectic structure Ω I (which isalso the natural symplectic structure on H I as a finite dimensional Hilbert space); ∀I ⊂ I � ∈ L , π I � →I := Π I | H I� → H I where Π I is the orthogonal projection on H I ; H N := H and ∀I ∈ L , π N →I := Π I | H → H I .Then, this defines a rendering [10, def. 2.6] of the symplectic manifold H by the projective systemof phase spaces ( L , H , π ) ↓ . We define σ ↓ : H → S ↓ ( L , H ,π ) as in [10, def. 2.6].Additionally, defining the dense vector subspace of H , D := Vect {� � | � ∈ N } (without com-pletion, ie. the space of finite linear combinations of the � � ), we have a bijective antilinear map ζ : D ∗ → S ↓ ( L , H ,π ) such that ζ − ◦ σ ↓ : H → D ∗ is the canonical identification of H with D � ⊂ D ∗ (where D ∗ is the algebraical dual of D and D � the topological one). Proof L is a directed set, since ∀I, I � ∈ L , I ∪ I � ∈ L and I, I � ⊂ I ∪ I � .Let I, I � ∈ L � { N } with I ⊂ I � . π I � →I is surjective by construction. Next, since H I is closed, wehave, for any bounded real-valued real-linear form υ on H I , a vector υ ∈ H I such that: ∀� ∈ H I , υ ( � ) = Ω I ( υ, � ) = Re � � υ, �� I .Hence, since Π I is the C -orthogonal projection on the complex vector subspace H I , it is also the R -orthogonal projection on the real vector subspace H I , and we have: ∀� ∈ H I � , υ ◦ π I � →I ( � ) = Re � � υ, Π I �� I = Re � � υ, �� I � = Ω I � ( υ, � ) ,and therefore π I � →I � υ ◦ π I � →I � = π I � →I ( υ ) = υ .Clearly for I ∈ L , we have π I→I = id H I and for I, I � , I �� ∈ L � { N } with I ⊂ I � ⊂ I �� , π I � →I ◦ π I �� →I � = π I �� →I .Lastly, we define: ζ : D ∗ → S ↓ ( L , H ,π ) υ → � υ| H I � I∈ L . here for all I ∈ L , ( · ) : H ∗I → H I is the canonical identification provided by the complex Hilbertspace structure on H I ( H I is finite dimensional, hence H ∗I = H �I ).The map ζ is well-defined, since ∀I ⊂ I � ∈ L , ∀� ∈ H I , � π I � →I � υ| H I� � , � � I = � υ| H I� , � � I � = υ ( � ) = � υ| H I , � � I , hence π I � →I � υ| H I� � = υ| H I .On the other hand, we define � ζ : S ↓ ( L , H ,π ) → D ∗ , by: ∀ ( � I ) I∈ L , ∀w ∈ D , � ζ �( � I ) I∈ L � ( w ) = �� I , w� I for any I ∈ L such that w ∈ H I .The map � ζ is well-defined since D = � I∈ L H I and if I, I � ∈ L are such that w ∈ H I ∩ H I � , then thereexists I �� ∈ L such that I, I � ⊂ I �� and: �� I , w� I = �π I �� →I ( � I �� ) , w� I = �� I �� , w� I �� = �� I � , w� I � .Now, we have � ζ ◦ ζ = id D ∗ , ζ ◦ � ζ = id S ↓ ( L , H ,π ) and ∀� ∈ H , ∀I ∈ L , ∀w ∈ H I ⊂ D , � ζ ◦ σ ↓ ( � ) ( w ) = �π N →I ( � ) , w� I = ��, w� H . � We now present the constraint surface of interest, as a real vector subspace of H admittinga description of a specific form (alternatively, we could characterize it by of a family of linearholomorphic second class constraints and a family of linear first class constraints). Additionally,we anticipate on the regularization of the constraints by providing a rendering (similar to the onewe adopted for H ) for the corresponding reduced phase space. Proposition 2.3
We consider the same objects as in prop. 2.2. Let � f j � j∈ N and ( � � ) �∈ N be two,mutually orthogonal, orthonormal families in H . We define: J := Vect C � f j �� j ∈ N � (equipped with the induced symplectic structure Ω J ) and K R :=Vect R {� � | � ∈ N } ; δ : J ⊕ K R → J by δ := Π J | J ⊕ K R → J where Π J is the orthogonal projection on J .Then ( J , J ⊕ K R , δ ) is a phase space reduction of H [10, def. A.1].Additionally, we define: ∀J ∈ L , J J := Vect C � f j �� j ∈ J � equipped with the induced symplectic structure Ω �J ; ∀K ∈ L , K K := Vect C {� � | � ∈ K } & K K , R := Vect R {� � | � ∈ K } ; ∀J ⊂ J � ∈ L , π �J � →J := Π �J | J J� → J J where Π �J is the orthogonal projection on J J ; J N := J and ∀J ∈ L , π � N →J := Π �J | J → J J .As in prop. 2.2, this provides a rendering of J by � L , J , π � � ↓ and we define σ �↓ : J → S ↓ ( L , J ,π � ) aswell as the bijective antilinear map ζ � : F ∗ → S ↓ ( L , J ,π � ) where F := Vect � f j �� j ∈ N � . Proof δ is a surjective linear map and for � ∈ J , we have δ − ��� = � + K R , hence δ − ��� is onnected. For �, w ∈ J ⊕ K R , we write � = � � + � �� and w = w � + w �� with � � , w � ∈ J and � �� , w �� ∈ K R . Then, we have:Ω( �, w ) = 2 Im ��, w� H = 2 Im �� � , w � � H + 2 Im �� �� , w �� � H (since J ⊥ K R )= 2 Im �� � , w � � J = Ω J ( δ ( � ) , δ ( w )) (since K R is the real vector subspace generated byan orthonormal family).Hence, ( J , J ⊕ K R , δ ) is a phase space reduction of H . � We are ready to turn to the core of the regularization procedure, namely formulating a set ofapproached implementations of the constraints (indexed by a label set E ), endowing E with anappropriate preorder, and linking together the approximate dynamics by supplying projecting mapsbetween their reduced phase spaces.Here we choose E to enumerate a large class of approximate solutions, ordered by comparinghow good they are at approximating the exact solution (the precise definition of E may at first seemto arise from nowhere but will become transparent when we will actually detail the correspondingapproximate constraint surfaces). This way of composing E will make the study the convergencemostly inexpensive: a large part of the work is actually done beforehand when checking that E with this preorder is really a directed set.It also has the advantage of partially getting rid of the arbitrariness inherent of working with anapproximating scheme. The philosophy is that an explicit, concretely implemented, approximatingscheme will correspond to a specific cofinal part of E , but that we have the option of consideringall such particular schemes at the same time, by arranging them into a (huge) set E , provided wecarefully tailor its preorder to our purpose.Besides, note that being quite broad in recruiting suitable approximate theories is, up to a certainextent, forced upon us by the fact that we are dealing with an unphysical and not further specifiedsystem, since, in a more realistic example, we could probably, from the physics of the system, inferguiding principles to be more selective.On the other hand, we could fear that such a loose label set E will leave us with a dispropor-tionately complicated projective structure for the dynamical theory. But, in fact, this dynamicalstructure (on EL ) gets spontaneously quotiented down to the projective structure we had alreadyintroduced above for the dynamical state space. The idea is that we can transparently match twopartial dynamical theories as soon as they have a common ancestor out of which they are carved inthe same way (recall this mechanism was presented at the end of [10, subsection 2.2], and expressedprecisely in [10, props. 2.8 and 2.9]). Definition 2.4
We consider the same objects as in prop. 2.3 and we define E as the set of allsextuples � I, I � , J, K , φ, � � such that: I ⊂ I � ∈ L & J, K ∈ L ; φ : J J ⊕ K K → H I � is a linear application and φ| J J ⊕ K K → Im φ is a unitary map; � > ∀� ∈ J J , �� − φ ( � ) � � � ��� ; Π I �φ � K K , R �� = Π I � K K , R � . n E we define a preorder � by � I , I � , J , K , φ , � � � � I , I � , J , K , φ , � � iff: I ⊂ I , I � ⊂ I � , J ⊂ J & K ⊂ K ; � � � . Proposition 2.5
We consider the same objects as in def. 2.4. Let
I ∈ L and � >
0. Let
J, K ∈ L such that:dim Π I � J J ⊕ K K � = dim ( J J ⊕ K K ).Then, there exist I � ∈ L and a linear application φ : J J ⊕ K K → H I � such that � I, I � , J, K , φ, � � ∈ E . Lemma 2.6
Let H be a Hilbert space and let F , G be two finite dimensional vector subspaces of H , such that dim Π G �F � = dim F , where Π G denotes the orthogonal projection on G .Then, there exists a unique linear application φ F→G : F → G satisfying: φ F→G | F→ Im φ F→G is a unitary map; � S F �µ S F ( � ) �� − φ F→G ( � ) � is minimal, where S F is the unit sphere of F equipped with themeasure induced by the euclidean structure of F .For � ∈ F , �� − φ F→G ( � ) � � F ��� sup �∈F��� =1 �� − Π G ( � ) � Proof
Existence and uniqueness.
Let f = dim F . From dim Π G �F � = f , Π G induces a bijective map F → Π G �F � , hence � Π G ( · ) , Π G ( · ) � G defines a positive definite sesquilinear map on F . Therefore,there exists an orthonormal basis ( � � ) �∈{ ,���,f} such that: ∀�, j ∈ { , � � � , f } , �Π G ( � � ) , Π G ( � j )� = λ � δ �j with λ � > φ be a linear application F → G such that φ| F→ Im φ is a unitary map. We define B �j ∈ C for �, j ∈ { , � � � , f } and w � ∈ G ∩ (Π G �F � ) ⊥ for � ∈ { , � � � , f } by: ∀� ∈ { , � � � , f } , φ ( � � ) = 1 √λ � Π G ( � � ) + � j B �j λ j Π G ( � j ) + w � .From � φ ( � � ) , φ ( � j )� G = δ �j , we have: ∀�, j ∈ { , � � � , f } , B ∗�j + B j� + � � B ∗�� B j� + � w � , w j � = 0 . (2.6. )With these notations, we have:� S F �µ S F ( � ) �� − φ ( � ) � = � S F �µ S F ( � ) � Π G ( � ) − φ ( � ) � + �� − Π G ( � ) � = � �,j �� S C f �µ S C f ( � ) � ∗� � j � �Π G ( � � ) − φ ( � � ) , Π G ( � j ) − φ ( � j )� + � S F �µ S F ( � ) �� − Π G ( � ) � � � Vol ( S C f ) �1 + λ � − λ � Re (1 + B �� )� + � S F �µ S F ( � ) �� − Π G ( � ) � = � � Vol ( S C f ) �1 + λ � − λ � + � λ � � � |B �� | + � λ � �w � � �+� S F �µ S F ( � ) �� − Π G ( � ) � (using eq. (2.6. )).Hence, this expression is minimal if and only if ∀�, j ∈ { , � � � , f } , B �j = 0 and ∀� ∈ { , � � � , f } , w � =0. Therefore, we define φ F→G by: ∀� ∈ { , � � � , f } , φ F→G ( � � ) = 1 √λ � Π G ( � � ) . Bound on �� − φ
F→G ( � ) �. Let � = f � j =1 � j � j ∈ F . We have: �� − φ F→G ( � ) � � f � j =1 �� � j �� �� � j − φ F→G ( � j )�� � f ��� sup j �� � j − φ F→G ( � j )�� .Then, for j ∈ { , � � � , f } , �� � j − Π G ( � j )�� + λ j = 1 implies:���1 − � λ j ��� = �� � j − Π G ( � j )�� �� � j − Π G ( � j )��1 + � λ j � �� � j − Π G ( � j )�� ,therefore:�� � j − φ F→G ( � j )�� � �� � j − Π G ( � j )�� + ��Π G ( � j ) − φ F→G ( � j )��= �� � j − Π G ( � j )�� + ���� λ j − � � j − Π G ( � j )�� .Hence, �� − φ F→G ( � ) � � f ��� sup j �� � j − Π G ( � j )�� � f ��� sup �∈F��� =1 �� − Π G ( � ) � . � Proof of prop. 2.5
Since ( � � ) �∈ N is an orthonormal basis of H and J J has finite dimension, we canfind I � ∈ L such that: sup �∈ J J ��� =1 �� � − Π I � ( � )�� � � J ;and I � such that: I � ∩ I = ∅ & dim Π I � � J J � + dim ( J J ∩ H I ) = dim J J ; dim H I � � dim Π I � � J J � + dim K K .Let I � := I ∪ I � ∪ I � and I � := I � \ I . We have dim Π I � � J J � + dim ( J J ∩ H I ) = dim J J and K K ⊥ J J ,hence for all � ∈ K , there exists � �� ∈ Π I � � J J � such that: ∀j ∈ J, �Π I ( f j ) , Π I ( � � )� I + �Π I � ( f j ) , � �� � I � = 0 . his holds because, for all families of coefficients � α j � j∈J such that � j α j �Π I � ( f j ) , · � I � = 0, we have� j α j∗ f j ∈ H I (from point 2.5. above), and therefore � j α j �Π I ( f j ) , Π I ( � � )� I = �� j α j∗ f j , � � � H =0 . So, for all � ∈ K , ��Π I ( f j ) , Π I ( � � )� I � j∈J is in the image of ��Π I � ( f j ) , · � I � � j∈J .Next, using dim H I � � dim Π I � � J J � + dim K K , there exists a family of vectors � ��� ∈ H I � ∩ �Π I � � J J � � ⊥ for all � ∈ K such that: ∀�, � ∈ K , � Π I ( � � ) , Π I ( � � ) � I + �� �� , � �� � I � + �� ��� , � ��� � I � = 0 .Now, we define φ : J J ⊕ K K → H I � by: ∀j ∈ J, φ ( f j ) := φ J J → H I� ( f j ) (where φ J J → H I� is defined as in lemma 2.6),and ∀� ∈ K , φ ( � � ) := Π I ( � � ) + � �� + � ��� � Π I ( � � ) + � �� + � ��� � .From the proof of lemma 2.6, φ J J → H I� � J J � = Π I � � J J � , hence, for all � ∈ K , we have, by constructionof � �� and � ��� , φ ( � � ) ⊥ φ � J J � . Also by construction of � ��� , we have, for all �, � ∈ K , �φ ( � � ) , φ ( � � ) � = δ �� . Therefore φ induces an Hilbert space isomorphism J J ⊕ K K → Im φ .Finally, we can check that defs. 2.4. and 2.4. are fulfilled. � Proposition 2.7
With the notations of def. 2.4, E , � is a directed set. Proof
Let � I , I � , J , K , φ , � � ∈ E and � I , I � , J , K , φ , � � ∈ E . We define � I = I ∪I , J = J ∪J , K = K ∪ K and � = min( � , � ) >
0. Then, since ( � � ) �∈ N is an orthonormal basis of H , we canfind I ∈ L such that � I ⊂ I and dim Π I � J J ⊕ K K � = dim ( J J ⊕ K K ).From prop. 2.5, there exist � I � ∈ L and � φ : J J ⊕ K K → H � I � such that � I, � I � , J, K , � φ, � � ∈ E . Wedefine I � = I � ∪ I � ∪ � I � and φ : J J ⊕ K K → H I � by: ∀� ∈ J J ⊕ K K , φ ( � ) := � φ ( � ) .Then, � I, I � , J, K , φ, � � ∈ E and � I , I � , J , K , φ , � � , � I , I � , J , K , φ , � � � � I, I � , J, K , φ, � � . � Proposition 2.8
We consider the same objects as in def. 2.4. Let ε = � I, I � , J, K , φ, � � ∈ E . Wedefine: L ε := {I �� ∈ L | I � ⊂ I �� } ; ∀I �� ∈ L ε � { N } , J εI �� := J J , equipped with the induced symplectic structure Ω �J ; ∀I �� ∈ L ε � { N } , H εI �� := φ � J J ⊕ K K , R � ⊂ H I � ⊂ H I �� ; ∀I �� ∈ L ε � { N } , δ εI �� := � Π �J | J J ⊕ K K, R → J J � ◦ � φ| J J ⊕ K K, R →φ � J J ⊕ K K, R � � − where Π �J is the orthogonalprojection on J J ; ∀I �� , I �� ∈ L ε � { N } / I �� ⊂ I �� , π ε→εI �� →I �� := id J J . hen, L ε is cofinal in L and �( J εI �� ) I �� ∈ L ε �{ N } , ( H εI �� ) I �� ∈ L ε �{ N } , � π ε→εI �� →I �� � I �� ⊂I �� , ( δ εI �� ) I �� ∈ L ε �{ N } � is anelementary reduction [10, def. 3.7] of ( L ε � { N } , H , π ) ↓ . Proof
For �
I ∈ L , � I ∪ I � ∈ L ε , hence L ε is cofinal in L , so in particular it is directed (and so is L ε � { N } since it has a greatest element).Then, it is clear from the definitions that ( L ε � { N } , J ε , π ε→ε ) is a projective system of phasespaces.Replicating the proof of prop. 2.3, we can show that � J J , J J ⊕ K K , R , Π �J | J J ⊕ K K, R → J J � is a phasespace reduction of J J ⊕ K K . But since φ| J J ⊕ K K → Im φ is unitary, ( J J , H εI �� , δ εI �� ) is a phase spacereduction of Im φ , hence of H I �� , for all I �� ∈ L ε � { N } .Let I �� , I �� ∈ L ε � { N } such that I �� ⊂ I �� . We have π I �� →I �� � H εI �� � = Π I �� � H εI �� � = H εI �� since H εI �� = H εI �� ⊂ H I �� . Lastly, for � ∈ H εI �� , y ∈ J εI �� = J J , we have:� ∃ � ∈ H εI �� / δ εI �� ( � ) = y & π I �� →I �� ( � ) = � � ⇔ � ∃ � ∈ H εI �� / δ εI �� ( � ) = y & � = � � ⇔ � δ εI �� ( � ) = y � ,therefore def. [10, 3.7. ] is fulfilled. � Proposition 2.9
We consider the same objects as in prop. 2.8. We define: � E := E � { N } (we extend the preorder by ∀ε ∈ E , ε ≺ N ), ∀ε ∈ E , � L ε := L ε � { N } , and� L N := { N } ∀ε = � I, I � , J, K , φ, � � ∈ E , ∀I �� ∈ � L ε , � ( ε, I �� ) := J , and � ( N , N ) := N ; J NN := J N = J , H NN := J ⊕ K R and δ NN := δ ; ∀ ( ε , I �� ) , ( ε , I �� ) ∈ � E � L / ε � ε & I �� ⊂ I �� , π ε →ε I �� →I �� := π �� ( ε ,I �� ) →� ( ε ,I �� ) ( π �J � →J for J, J � ∈ L �{ N } with J ⊂ J � has been defined in defs. 2.3. and 2.3. ).Then, �� E , �� L ε � ε∈ � E , ( J εI �� ) ( ε,I �� ) ∈ � E � L , ( H εI �� ) ( ε,I �� ) ∈ � E � L , � π ε →ε I �� →I �� � ( ε ,I �� ) � ( ε ,I �� ) , ( δ εI �� ) ( ε,I �� ) ∈ � E � L � is a regularizedreduction [10, def. 3.16] of ( L � { N } , H , π ) ↓ .Additionaly, we have a bijective map κ : S ↓ (� E � L , J ,π ) → S ↓ ( L �{ N }, J ,π � ) . Proof � E is a directed set (for it has a greatest element) and �� L ε � ε∈ � E is a family of decreasingcofinal parts of L � { N } .Next, we have ∀ ( ε, I �� ) ∈ � E � L , J εI �� = J � ( ε,I �� ) hence for ( ε , I �� ) � ( ε , I �� ) ∈ � E � L , π ε →ε I �� →I �� is well-definedas a surjective map J ε I �� → J ε I �� and it is compatible with the symplectic structures.Moreover, for ε = ε ∈ E , this definition coincides with the map π ε→εI �� →I �� that has been introducedin prop. 2.8. Hence, for all ε ∈ E , we have from prop. 2.8 that ( J ε , H ε , π ε→ε , δ ε ) is an elementary eduction of �� L ε , H , π � ↓ .And � J N , H N , π N → N , δ N � is an elementary reduction of �� L N , H , π � ↓ since L N � has only oneelement and � J NN , H NN , δ NN � = ( J , J ⊕ K R , δ ) is a phase space reduction of H N = H (prop. 2.3).Lastly, using � : � E � L → L � { N } ( � satisfies that � �� E � L � is cofinal in L � { N } since it contains N , which is a greatest element in L �{ N } ), we have by [10, prop. 2.9] that �� E � L , J , π � ↓ is a projectivesystem of phase spaces, thus �� E , �� L ε � ε∈ � E , ( J εI �� ) ( ε,I �� ) ∈ � E � L , ( H εI �� ) ( ε,I �� ) ∈ � E � L , � π ε →ε I �� →I �� � ( ε ,I �� ) � ( ε ,I �� ) , ( δ εI �� ) ( ε,I �� ) ∈ � E � L �is a regularized reduction of ( L � { N } , H , π ) ↓ . And, in addition, there exists a bijective map κ : S ↓ (� E � L , J ,π ) → S ↓ ( L �{ N }, J ,π � ) . � Lastly, we can investigate the convergence and check that we are indeed in the optimal situationdiscussed at the end of [10, subsection 3.2] (more precisely in [10, prop. 3.23]). As announced above,the key ingredient for the convergence is the auxiliary result from prop. 2.5, that we proved in theprocess of establishing the directedness of E . Theorem 2.10
We consider the same objects as in prop. 2.9. Let ψ ∈ J = J NN . For ε ∈ E , wedefine: ψ ε := ( δ ε N ) − � π N →ε N → N ( ψ )� ⊂ H ε N ⊂ H N = H ; Ψ ε := � σ ↓ ( ψ ε ) , where � σ ↓ : P ( H ) → � S ↓ ( L , H ,π ) is defined as in [10, prop. 3.23].Then, the net �Ψ ε � ε∈ E converges in � S ↓ ( L , H ,π ) to � σ ↓ � δ − �ψ� � [10, def. 3.21]. Proof
For ε = � I, I � , J, K , φ, � � ∈ E , we have, by putting all definitions together: ψ ε = φ � Π �J ( ψ ) + K K , R � , where Π �J is the orthogonal projection on J J ,hence, for all I � ∈ L :[Ψ ε ] I � = π N →I � �ψ ε � = Π I � � φ � Π �J ( ψ ) + K K , R � � ⊂ H I � .Let I � ∈ L and let U be an open set in H I � such that U ∩ Π I � �ψ + K R � � = ∅ . Let ψ � ∈U ∩ Π I � �ψ + K R � and let � > ∀ψ �� ∈ H I � , �ψ �� − ψ � � � � ( �ψ� + 1) ⇒ ψ �� ∈ U .Next, there exits χ ∈ K R such that ψ � = Π I � �ψ + χ� . We choose J , K ∈ L and χ � ∈ K K , R such that �� ψ − Π �J ( ψ )�� � � ( �ψ� + 1) and �χ − χ � � � � ( �ψ� + 1). And we can find I ∈ L with I ⊃ I � such that dim Π I � J J ⊕ K K � = dim ( J J + K K ). So, from prop. 2.5, there exist I � ∈ L and φ : J J ⊕ K K → H I � such that ε := � I , I � , J , K , φ , � � ∈ E .Let ε = � I , I � , J , K , φ , � � ∈ E with ε � ε . Then, we have:��Π I � ( ψ ) − Π I � ◦ φ ◦ Π �J ( ψ )�� � �� ψ − φ ◦ Π �J ( ψ )�� � �� ψ − Π �J ( ψ )�� + ��Π �J ( ψ ) − φ ◦ Π �J ( ψ )�� �� ψ − Π �J ( ψ )�� + � ��Π �J ( ψ )�� � � ( �ψ� + 1) .Moreover, we have χ � ∈ K K , R ⊂ K K , R , so there exists χ �� ∈ K K , R such that Π I ◦ φ ( χ �� ) =Π I ( χ � ) and we have:��Π I � ( χ ) − Π I � ◦ φ � χ �� ��� � ��Π I ( χ ) − Π I ◦ φ � χ �� ��� (since I � ⊂ I ⊂ I ) � � Π I ( χ ) − Π I ( χ � ) � � � ( �ψ� + 1) .Therefore, ψ �� := Π I � ◦ φ �Π �J ( ψ ) + χ �� � ∈ U , but since ψ �� ∈ [Ψ ε ] I � , we have ∀ε � ε , [Ψ ε ] I � ∩U � = ∅ .Let K be a compact set in H I � such that K ∩ Π I � �ψ + K R � = ∅ . Hence, there exists � > ∀ψ �� ∈ H I � , � ∃ψ � ∈ Π I � �ψ + K R � / �ψ � − ψ �� � � � ( �ψ� + 1) � ⇒ � ψ �� /∈ K � .As above, we can, using prop. 2.5, construct ε := � I , I � , J , K , φ , � � ∈ E with I � ⊂ I and �� ψ − Π �J ( ψ )�� � � ( �ψ� + 1). Let ε = � I , I � , J , K , φ , � � ∈ E with ε � ε and let ψ �� ∈ [Ψ ε ] I � . Then, there exists χ �� ∈ K K , R such that ψ �� = Π I � ◦ φ �Π �J ( ψ ) + χ �� � and there exists χ � ∈ K K , R ⊂ K R such that Π I ◦ φ ( χ �� ) = Π I ( χ � ) . Moreover, we have again:��Π I � ( ψ ) − Π I � ◦ φ ◦ Π �J ( ψ )�� � � ( �ψ� + 1) .We define ψ � = Π I � � ψ + χ � � = Π I � ( ψ ) + Π I � ◦ φ � χ �� � (since I � ⊂ I ⊂ I ) and we have �ψ �� − ψ � � � � ( �ψ� + 1), hence ψ �� /∈ K , and therefore ∀ε � ε , [Ψ ε ] I � ∩ K = ∅ .So, the net �[Ψ ε ] I � � ε∈ E converges in P ( H I � ) to Π I � �ψ + K R � = π N →I � � δ − �ψ� �, thus, the net(Ψ ε ) ε∈ E converges in � S ↓ ( L , H ,π ) to � σ ↓ � δ − �ψ� �. � In this section we want to apply the formalism outlined in [10, sections 2 and 3] and [11, section 2]to the second quantization of the Schrödinger equation. In other words, we will consider the one-particle quantum mechanics defined on an Hilbert space H as a classical field theory (looking atthe wave function as a classical field, whose evolution is described by a linear partial differentialequation, namely the Schrödinger equation), and we will discuss how this field theory can bequantized. The standard way of doing this leads to the bosonic Fock space build on H [5, sectionI.3.4]. Here we want to compare this trusted path with the strategy inspired by [7, 13]: first, lookfor a rendering of the classical field theory by a collection of finite dimensional partial theories,then come up with a regularizing procedure to implement the dynamics, and, last but not least,take advantage of this classical insight to build a quantization of the theory, thus obtaining a rojective system of quantum state spaces. In particular, we want to use this example to illustratehow the classical regularization of the dynamics lays the stage for a corresponding procedure atthe quantum level. In [10, section 3], we only considered dynamics specified by constraints, whereas here we havea theory originally formulated with a ‘true’ Hamiltonian. However, this is quickly fixed, sincethere exists a routine trick (discussed in [15, section 1.8] and similar to the more general procedurepresented in [9]), that can be physically interpreted as introducing an artificial time parametrization,and allows to transform any theory on H with an non-vanishing Hamiltonian into a theory on H × R with an Hamiltonian constraint (the R part holds the time coordinate and its conjugatemomentum, aka. the energy variable).Note that there is a technical subtlety arising when we try to write the theory on an infinitedimensional symplectic manifold in the naive setup of [10, def. A.1], and we are forced to requirethe one-particle quantum Hamiltonian to be a bounded operator (we cannot simply restrict theconstraint surface so that it is included in an appropriate dense subspace, for it would then costthe reduced phase space its strong symplectic structure, by spoiling the needed non-degeneracyproperty). However, we will be able to lift this restriction without great efforts when switching tothe projective state space formalism. Proposition 3.1
Let H be a separable, infinite dimensional Hilbert space and H be a boundedself-adjoint operator on H . We equip M ��� := H × R with the strong symplectic structure: ∀ ( φ , � , � ) , ( φ , � , � ) ∈ M ��� , Ω ��� ( φ , � , � ; φ , � , � ) := 2 Im � �φ , φ � � + ( � � − � � ).We define: M ����� := { ( ψ, �, E ) ∈ M ��� | E = �ψ, Hψ�} ; M DY� := H with symplectic structure Ω DY� := 2 Im � · , · � ; δ : M ����� → M DY� ( ψ, �, E ) �→ exp ( � � H ) ψ .Then, ( M DY� , M ����� , δ ) is a phase space reduction of M ��� [10, def. A.1]. Proof
From prop. 2.1, Ω ��� , resp. Ω
DY� , defines a strong symplectic structure on M ��� , resp. M DY� .The map δ is surjective and, for ψ � ∈ M DY� , δ − �ψ � � = { (exp ( −� � H ) ψ � , �, E � ) | � ∈ R } , where E � := �ψ � , H ψ � � . So δ − �ψ � � is in particular connected.Let ( ψ, �, E ) ∈ M ����� . We have: T ( ψ,�,E ) ( M ����� ) = { ( φ, �, �φ, Hψ� ) | φ ∈ H , � ∈ R } ,and: ( ψ,�,E ) δ : T ( ψ,�,E ) ( M ����� ) → T � �H� ψ ( M DY� )( φ, �, �φ, Hψ� ) �→ � �H� φ + �� � �H� Hψ .Hence, T ( ψ,�,E ) δ is surjective and, for ( φ , � , �φ , Hψ� ) , ( φ , � , �φ , Hψ� ) ∈ T ( ψ,�,E ) ( M ����� ),we have:Ω ��� ( φ , � , �φ , Hψ� ; φ , � , �φ , Hψ� ) == 2 Im �φ , φ � + 2 � Im �φ , �Hψ� − � Im �φ , �Hψ� + 2 � � Im �� Hψ, � Hψ� = 2 Im � T ( ψ,�,E ) δ � φ , � , �φ , Hψ� � , T ( ψ,�,E ) δ � φ , � , �φ , Hψ� �� ,therefore Ω ��� | T ( ψ,�,E ) ( M ����� ) = [ δ ∗ Ω DY� ] ( ψ,�,E ) . � On H viewed as a phase space, we can define some remarkable observables (this defines thealgebra that we will latter endeavor to quantize): of interest are for us the scalar product with avector � ∈ H (that will give rise in the quantum theory to the corresponding creation and annihi-lation operators) and the expectation value of an operator on H . Additionally the Heisenberg (ie.time-dependent) operators of the first-quantized theory can be seen in a natural way as dynamicalobservables associated (in the sense of [10, def. A.2]) to particular kinematical observables (up to atechnical artefact: we restrict the support of the considered observables to spheres in H because wehad defined the map ( · ) DY� translating a kinematical observable into its dynamical version only forbounded observables; note that, alternatively, we could just weaken this requirement, for it wouldbe enough to only demand the kinematical observables to be bounded on orbits of the dynamics).
Proposition 3.2
We consider the same objects as in prop. 3.1. Let � ∈ H . On H we can definethe observables:a � : H → C ψ �→ ��, ψ� and a ∗� : H → C ψ �→ �ψ, �� .We have, for all �, f ∈ H : { a � , a f } H = 0 , { a ∗� , a ∗f } H = 0 , and { a � , a ∗f } H = � ��, f � .Let A be a bounded self-adjoint operator on H . We define on H the observable �A� by: ∀ψ ∈ H , �A� ( ψ ) := �ψ, A ψ� .We have, for all A, B bounded self-adjoint operators on H and � ∈ H :� �A� , �B� � H = � � [ A, B ] H � , �a � , �A� � H = � a A� , and �a ∗� , �A� � H = −� a ∗A� .Lastly, for A a bounded self-adjoint operator on H , N > � � ∈ R , we can define on M ��� the observable: �A, N, � � � : M ��� → R ( ψ, �, E ) �→ � �A� ( ψ ) if �ψ� = N & � = � � ψ � ∈ M DY� , �A, N, � � � DY� ( ψ � ) := sup ( ψ,�,E ) ∈δ − �ψ � � �A, N, � � � ( ψ, �, E )= �� � �� � H A � −�� � H � ( ψ � ) if �ψ � � = N ) Proof
In order to compute the Poisson brackets between observables of the type a � and a ∗� , wehave to be careful not to mix up the complex structure on H with the complex structure coming froma � and a ∗� being C -valued. Therefore, we will write J φ for the scalar multiplication of φ by � (in H seen as a C -vector space) and � φ for the vector ( � ⊗ R φ ) in C ⊗ R H ≈ T C ( H ) (for H seen asa real manifold). Extending Im � · , · � and Re � · , · � by C -bilinearity on T C ( H ) (because we want { · , · } H to be C -bilinear), we then have:Im �φ � , J φ� = − Im �J φ � , φ� = Re �φ � , φ� & Im �φ � , � φ� = Im � � φ � , φ� = � Im �φ � , φ� .With this we can compute the Hamiltonian vector fields at ψ ∈ H of a � and a ∗� , for � ∈ H :[ � a � ] ψ ( φ ) = ��, φ� = 2 Im � − J � � , φ � & [ � a ∗� ] ψ ( φ ) = �φ, �� = 2 Im � − J � − � � , φ � .Hence, for �, f ∈ H : { a � , a f } H ,ψ = 2 Im �X a f ,ψ , X a � ,ψ � = 2 Im � − J f f , − J � � { a ∗� , a ∗f } H ,ψ = 2 Im � X a ∗f ,ψ , X a ∗� ,ψ � = 2 Im � − J f − � f , − J � − � � { a � , a ∗f } H ,ψ = 2 Im � X a ∗f ,ψ , X a � ,ψ � = 2 Im � − J f − � f , − J � � � (Re �f , �� − � Im �f , �� ) = � ��, f � .Similarly, we have for any A bounded self-adjoint operator on H and at every ψ ∈ H :[ � �A� ] ψ = 2 Re �A ψ, φ� = 2 Im �−J A ψ, φ� ,hence, for A, B bounded self-adjoint operators on H , � ∈ H , and ψ ∈ H :� �A� , �B� � H ,ψ = 2 Im � X �B�,ψ , X �A�,ψ � = 2 Im �−J B ψ, −J A ψ� = −� ( �B ψ, A ψ� − �A ψ, B ψ� ) = � � [ A, B ] H � ( ψ ) ,�a � , �A� � H ,ψ = 2 Im � X �A�,ψ , X a � ,ψ � = 2 Im � −J A ψ, − J � � � ( −� Im �ψ, A �� + Re �ψ, A �� ) = � a A� ( ψ ) ,and �a ∗� , �A� � H ,ψ = 2 Im � X �A�,ψ , X a ∗� ,ψ � = 2 Im � −J A ψ, − J � − � � −� ( � Im �ψ, A �� + Re �ψ, A �� ) = −� a ∗A� ( ψ ) .Lastly, eq. (3.2. ) comes from: ∀ψ � ∈ M DY� , δ − �ψ � � = { (exp ( −� � H ) ψ � , �, �H� ( ψ � )) | � ∈ R } . � The projective system we will use here differs significantly from the one we were using in theprevious section (prop. 2.2), for we do not rely any more on the choice of a particular basis to definea family of vector subspaces: instead, we simply take as label set the set of all finite dimensionalvector subspaces of H (this structure is of course more satisfactory from a physical point of view;as mentioned at the beginning of section 2, we could not use it in the previous example, for ouraim was to illustrate the regularizing strategy, while this larger label set contains a cofinal familyon which the linear constraints we were considering form an elementary reduction).Note that the space of states of this projective system can be naturally identified with the algebraicdual on H , in such a way that the injection of H into the projective state space (in the sense of arendering, as introduced in [10, def. 2.6]) corresponds to the identification with its topological dual. Proposition 3.3
Let H be a separable, infinite dimensional Hilbert space. We define L as the setof all finite dimensional vector subspaces of H and we equip it with the preorder ⊂ . We define: ∀ I ∈ L � { H } , M ��� I := I × R , equipped with the symplectic structure Ω ��� , I induced from M ��� , Ω ��� ; ∀ I , I � ∈ L � { H } , with I ⊂ I � , π ��� I � → I := Π I | I � → I × id R where Π I is the orthogonal projectionon I Then, this defines a rendering [10, def. 2.6] of M ��� by the projective system of phase spaces( L , M ��� , π ��� ) ↓ . We define σ ��� ↓ : M ��� → S ↓ ( L , M ��� ,π ��� ) as in [10, def. 2.6].Additionally, we have a bijective map ζ ��� : H ∗ × R → S ↓ ( L , M ��� ,π ��� ) such that ζ ��� ,− ◦ σ ��� ↓ : M ��� → H ∗ × R corresponds to the canonical identification of H with H � ⊂ H ∗ . Proof
The proof works in the same way as the proof of prop. 2.2. � We are ready to go on to the formulation of an approximating scheme for the dynamics. Theapproximation here will take place in two different directions. First, we introduce a deformationof the constraint surface, controlled by a small parameter � > , to replace the non-compact orbitsof the exact dynamics (going in time from −∞ to + ∞ ) by compact orbits (running only througha finite time interval): the rough idea is that instead of having a ‘free particle’ in the energy-timevariable, we put an harmonic oscillator, thus preventing the time variable to grow for ever. Thiswill be more comfortable when switching to the quantum theory: having compact orbits is closely elated to having well-normalized states solving the quantum constraints (heuristically, quantumsolutions of the constraints have much in common with classical statistical states, supported by theconstraint surface and constant on the gauge orbits, and these will only exist as properly normalizedprobability measures if the orbits are compact).The other aspect of the approximation is what will allow us to build, for the approximated dy-namics, a corresponding elementary reduction on a cofinal part of the projective system introducedpreviously. For this, we truncate the exact Hamiltonian H of the first-quantized theory as Π J H Π J where J is a finite vector subspace of H , such that H is bounded on J (from now on, we canindeed relax the requirement we had above, and we allow H to be an unbounded, densely defined,operator on H ). In other words, we project the Hamiltonian flow on the symplectic submanifold J × R of H × R . Moreover, we include in the approximated dynamics additional second classconstraints, forcing the wave function ψ to belong to J (by definition of the truncated Hamiltonianthese additional constraints are preserved by the truncated evolution): the point is that it does notmake sense to keep the degrees of freedom orthogonal to the subspace J since with the truncatedHamiltonian we would not evolve them at all and they would soon lie very far away from theircorrect values (note that the degrees of freedom along J are not evolved exactly either, but at leastthey are evolved approximately; the error comes from neglecting the backreaction of the degrees offreedoms along J ⊥ , due to the cross-terms of the exact Hamiltonian H between J and J ⊥ ).The side effect of these additional second class constraints is to make the approximated reducedphase space finite dimensional (aka. M DY� ,ε∞ , using the notations of [10, prop. 3.24]): this is notneeded for the construction (in general only the ‘partial’ reduced phase space M DY� ,εη , arising fromthe constraint surface projected on M ��� η for some η ∈ L ε , is expected to be finite dimensional), butit will simplify the structure of the dynamical projective system. Definition 3.4
We consider the same objects as in prop. 3.3. Let H be a densely defined (possiblyunbounded) self-adjoint operator on H . We define E as the set of all pairs ( J , � ) such that: J ∈ L and ∀ψ ∈ J , �Hψ� < ∞ ; � > E we will use the preorder: ( J , � ) � ( J � , � � ) ⇔ � J ⊂ J � & � � � � � . Proposition 3.5 E , � is a directed preordered set. Proof
Let ( J , � ) , ( J , � ) ∈ E . Then, we have J + J ∈ L , ∀ψ ∈ J + J , �Hψ� < ∞ andmin ( � , � ) >
0. Hence, ( J + J , min ( � , � )) ∈ E and ( J , � ) , ( J , � ) � ( J + J , min ( � , � )) . � Proposition 3.6
We consider the same objects as in def. 3.4. Let ε = ( J , � ) ∈ E . We define: L ε := { I ∈ L | J ⊂ I } ; ∀ I ∈ L ε �{ H } , M DY� ,ε I := J equipped with the symplectic structure Ω DY� , J induced from M DY� , Ω DY� ; ∀ I ∈ L ε � { H } , M ����� ,ε I := �( ψ, �, E ) ∈ M ��� J ⊂ M ��� I ��� ( E − �ψ, Hψ� ) + � � = � �; ∀ I ∈ L ε � { H } , ∀ ( ψ, �, E ) ∈ M ����� ,ε I , δ ε I ( ψ, �, E ) := exp ( � � Π J H ) ψ ∈ J ; . ∀ I , I � ∈ L ε � { H } , with I ⊂ I � , π DY� ,ε→ε I � → I := id J .Then, L ε is cofinal in L and �� M DY� ,ε I � I ∈ L ε �{ H } , � M ����� ,ε I � I ∈ L ε �{ H } , � π DY� ,ε→ε I � → I � I ⊂ I � , ( δ ε I ) I ∈ L ε �{ H } � isan elementary reduction [10, def. 3.7] of ( L ε � { H } , M ��� , π ��� ) ↓ . Proof
For I ∈ L , I + J ∈ L ε , hence L ε is cofinal in L , so in particular it is directed (and so is L ε � { H } since it has a greatest element).Then, it is clear from the definitions that ( L ε � { H } , M DY� ,ε , π DY� ,ε→ε ) is a projective system ofphase spaces.Now, we define: M ε := �( ψ, �, E ) ∈ M ��� J ⊂ M ��� ��� ( E − �ψ, Hψ� ) + � � = � � ; ∀ ( ψ, �, E ) ∈ M ε , δ ε ( ψ, �, E ) := exp ( � � Π J H ) ψ ∈ J ;and we want to show that ( J , M ε , δ ε ) is a phase space reduction of M ��� .Π J H| J → J is a bounded (by definition of E ), self-adjoint operator on J . Therefore, the map δ ε is surjective and, for ψ � ∈ J , δ ε,− �ψ � � = �� � − �� sin θ Π J H ψ � , � sin θ, E � + � cos θ � ��� θ ∈ [0 , π [�,where E � := �ψ � , H ψ � � . So δ ε,− �ψ � � is in particular connected.Let ( ψ, �, E ) ∈ M ε . T ( ψ,�,E ) ( M ε ) is given by:�( φ, �, � ) ∈ M ��� J ��� φ ∈ J & � √ − � � ( � − �φ, Hψ� ) + 2 � � � = 0� , (3.6. )and we have: T ( ψ,�,E ) δ ε : T ( ψ,�,E ) ( M ε ) → J ( φ, �, � ) �→ � �� � � Π J H Π J H ψ + � � � Π J H φ .Hence, T ( ψ,�,E ) δ ε is surjective and, for ( φ , � , � ) , ( φ , � , � ) ∈ T ( ψ,�,E ) ( M ε ), we have:Ω ��� ( φ , � , � φ , � , � ) == 2 Im �φ , φ � + � �2 Re �φ , Hψ� − � �√ − � � � � −� �2 Re �φ , Hψ� − � �√ − � � � �(using eq. (3.6. ))= 2 Im �T ( ψ,�,E ) δ ε ( φ , � , � ) , T ( ψ,�,E ) δ ε ( φ , � , � ) � (like in the proof of prop. 3.1),therefore Ω ��� | T ( ψ,�,E ) ( M ε ) = [ δ ε,∗ Ω DY� , J ] ( ψ,�,E ) .Thus, for all I ∈ L ε � { H } , � M DY� ,ε I , M ����� ,ε I , δ ε I � = ( J , M ε , δ ε ) is a phase space reduction of M ��� , hence of M ��� I .Let I , I � ∈ L ε � { H } , with I ⊂ I � . We have: π ��� I � → I � M ����� ,ε I � � = �(Π I ψ, �, E ) ��� ψ ∈ J & ( E − �ψ, Hψ� ) + � � = � � = M ����� ,ε I ,since J ⊂ I . Lastly, for ( ψ, �, E ) ∈ M ����� ,ε I , ψ �� ∈ M DY� ,ε I � = J , we get:� ∃ ( ψ � , � � , E � ) ∈ M ����� ,ε I � / δ ε I � ( ψ � , � � , E � ) = ψ �� & π ��� I � → I ( ψ � , � � , E � ) = ( ψ, �, E )� ⇔ � ∃ ( ψ � , � � , E � ) ∈ M ����� ,ε I / δ ε I ( ψ � , � � , E � ) = ψ �� & ( ψ � , � � , E � ) = ( ψ, �, E )� ⇔ � δ ε I ( ψ, �, E ) = ψ �� = π DY� ,ε→ε I � → I ( ψ �� )� ,therefore def. [10, 3.7. ] is fulfilled. � Now, as we did in the previous section (prop. 2.3), we introduce a more concise dynamicalprojective system, that we will be able to identify with the one on the label set EL using the facilitydeveloped in [10, prop. 2.8]. This dynamical projective system could be thought of as a rendering[10, def. 2.6] of the dense domain D of the operator H , except for the fact that D is actually not a strong symplectic manifold (unless H is bounded, in which case D = H ). For the same reason,the assertion in prop. 3.8 could not be put in the form of [10, prop. 3.24], since we are lacking aphase space reduction at the level of the infinite dimensional manifold M ��� = H × R when H is unbounded. Instead, we collect in prop. 3.9 a set of properties imitating the framework of [10,prop. 3.24], and we will formulate the convergence on this substitute ground.It is worth mentioning that here, as in the previous example, we are able to directly give aprojective system rendering the space of dynamical states, and more generally being able to finda regularizing scheme in the sense of [10, subsection 3.2] implies that one can construct such adynamical projective structure. This perhaps requires a few comments. At first it sounds as ifimplementing and solving the constraints requires to already know completely the structure of thedynamical theory. However, one should keep in mind that solving the dynamics and obtaining thedynamical theory is not simply constructing the space of physical states: the more crucial part is toconstruct the dynamical observables, not simply as a space of functions on the reduced phase space,but as a family of non functionally independent elementary observables, each of which should belinked to a physical meaning (aka. an experimental protocol).This point is transparently illustrated by the toy model we are studying in the present section.The submanifold � = 0 is obviously a gauge fixing surface of the theory we are considering, andthis is what allows us to obtain immediately a description of the reduced phase space. But, clearly,having realized this property of the dynamics does not mean we have solved the theory: if we wantto know how a given system will evolve we need to define dynamical observables associated tokinematical ones with support on other constant time surfaces. Indeed, the dynamical observablesassociated with time � = 0 are the only ones that can be directly defined on the reduced phase spacedefined through the aforementioned gauge fixing. And, although they provide a parametrization ofthe dynamical state space, they do not allow us to compute predictions for any arbitrary experiment,since, as underlined many times in the discussion of the handling of constraints [10, section 3],the predictive content of the theory is encoded in the functional relations among an overcompleteset of dynamical observables, arising from functionally independent kinematical observables.Note that in any theory admitting some obvious gauge fixing (which needs not to be singled outnor preferred in any sense: in the example at hand, selecting � = 0 rather than any other timesurface is an arbitrary choice), we can use this gauge fixing surface as a starting point to design anapproximating scheme: it provides an explicit description of the reduced phase space, and we canuse it as a pivot to define projections between the successive approximated dynamical theories (forwe can relate approximated orbits depending on their intersection with the gauge fixing surface, aswe indeed do in the present example). In particular, this suggests that such approximating schemescould be obtained without many difficulties within the so called ‘deparametrization’ framework [4]. roposition 3.7 Under the same hypotheses as in def. 3.4, we define: L H := { J ∈ L | ∀ψ ∈ J , �Hψ� < ∞} with the preorder defined by ⊂ ; ∀ J ∈ L H , M DY� J := J , equipped with the symplectic structure Ω DY� , J induced from M DY� , Ω DY� ; D := {ψ ∈ H | �Hψ� < ∞} ( D is the dense domain of the self-adjoint possibly unboundedoperator H ) and M DY� D := D ; ∀ J , J � ∈ L H , with J ⊂ J � , π DY� J � → J := Π J | J � → J where Π J is the orthogonal projection on J ; ∀ J ∈ L H , π DY� D → J := Π J | D → J .Then, ( L H , M DY� , π
DY� ) ↓ is a projective system of phase spaces and we can define (in analogy to[10, def. 2.6]) a map σ DY� ↓ as: σ DY� ↓ : M DY� D → S ↓ ( L H , M DY� ,π DY� ) ψ �→ � π DY� D → J ( ψ )� J ∈ L H .Additionally, we have a bijective antilinear map ζ DY� : D ∗ → S ↓ ( L H , M DY� ,π DY� ) such that ζ DY� ,− ◦ σ DY� ↓ : M DY� D → D ∗ is the restriction to D of the canonical identification of H with D � ⊂ D ∗ . Proof
We prove that L H is directed like in the proof of prop. 3.5.For J ∈ L H , we have that π DY� D → J is surjective (since J ⊂ D ) and, for J ⊂ J � ∈ L H , π DY� J � → J ◦ π DY� D → J � = π DY� D → J (but speaking of compatibility with symplectic structure does not make sense for π DY� D → J since D is not a strong symplectic manifold).The rest of the proof works as for prop. 2.2. � Proposition 3.8
We consider the objects introduced in props. 3.6 and 3.7. We define: ∀ε ∈ E , L �ε := L ε � { H } ; ∀ε = ( J , � ) ∈ E , ∀ I ∈ L �ε , � DY� ( ε, I ) := J ; ∀ ( ε , I ) , ( ε , I ) ∈ EL � / ( ε , I ) � ( ε , I ) , π DY� ,ε →ε I → I := π DY� � DY� ( ε , I ) →� DY� ( ε , I ) .Then, � E , � L �ε � ε∈ E , � M DY� ,ε I � ( ε, I ) ∈ EL � , � M ����� ,ε I � ( ε, I ) ∈ EL � , � π DY� ,ε →ε I → I � ( ε , I ) � ( ε , I ) , ( δ ε I ) ( ε, I ) ∈ EL � � is aregularized reduction of ( L � { H } , M ��� , π ��� ) ↓ and we have a bijective map κ DY� : S ↓ ( EL � , M DY� ,π DY� ) → S ↓ ( L H , M DY� ,π DY� ) . Proof E is a directed set (prop. 3.5) and � L �ε � ε∈ E is a family of decreasing cofinal parts of L � { H } .Next, we have ∀ ( ε, I ) ∈ EL � , M DY� ,ε I = M DY� � DY� ( ε, I ) hence for ( ε , I ) � ( ε , I ) ∈ EL � , π DY� ,ε →ε I → I is well-defined as a surjective map M DY� ,ε I → M DY� ,ε I and it is compatible with the symplecticstructures.Moreover, for ε = ε ∈ E , this definition coincides with the map π DY� ,ε→ε I → I that has been introducedin prop. 3.6. Hence, for all ε ∈ E , we have from prop. 3.6 that ( M DY� ,ε , M ����� ,ε , π DY� ,ε→ε , δ ε ) is anelementary reduction of � L �ε , M ��� , π ��� � ↓ .Lastly, using � DY� : EL � → L H (with � DY� � EL � � = L H ), we have by [10, prop. 2.8] that � EL � , M DY� , π
DY� � ↓ is a projective system of phase spaces, thus: E , � L �ε � ε∈ E , � M DY� ,ε I � ( ε, I ) ∈ EL � , � M ����� ,ε I � ( ε, I ) ∈ EL � , � π DY� ,ε →ε I → I � ( ε , I ) � ( ε , I ) , ( δ ε I ) ( ε, I ) ∈ EL � �is a regularized reduction of ( L � { H } , M ��� , π ��� ) ↓ . And, in addition, there exists a bijective map κ DY� : S ↓ ( EL � , M DY� ,π DY� ) → S ↓ ( L H , M DY� ,π DY� ) . � Proposition 3.9
We consider the same objects as in prop. 3.8 and we additionally define: M DY� , DH := M DY� D = D ; M ����� , DH := { ( ψ, �, E ) ∈ M ��� H | ψ ∈ D & E = �ψ, Hψ�} ; δ DH : M ����� , DH → M DY� , DH ( ψ, �, E ) �→ exp ( � � H ) ψ ; ∀ ( ε, I ) ∈ EL � , π DY� , D →ε H → I := π DY� D →� DY� ( ε, I ) .Then, we have: δ DH is surjective and, for all ψ � ∈ M DY� , DH , � δ DH � − �ψ � � is connected; for all ( ε, I ) ∈ EL � , π DY� , D →ε H → I is surjective and, for all ( ε , I ) � ( ε , I ) ∈ EL � , π DY� ,ε →ε I → I ◦π DY� , D →ε H → I = π DY� , D →ε H → I . Proof
Since H is self-adjoint, exp ( −� � H ) defines a unitary operator on H , and this operatorstabilizes D , for ∀ψ ∈ D , �H exp ( −� � H ) ψ� = � exp ( −� � H ) H ψ� = �H ψ� < ∞ . Hence, for ψ � ∈ M DY� , DH :� δ DH � − �ψ � � = { ( ψ, �, E ) ∈ M ��� H | � ∈ R , ψ = exp ( −� � H ) ψ � ∈ D & E = �ψ � , Hψ � � < ∞} .Next, the statements 3.9. follows from the proof of prop. 3.7 (for ∀ ( ε, I ) ∈ EL � , M DY� ,ε I = M DY� � DY� ( ε, I ) and M DY� , DH = M DY� D ). � We close the discussion of the classical part of this toy model by proving that we indeed havesuitable convergence at least for the dynamical states corresponding to vectors in D , and, moreprecisely, that the successive projective families of orbits approximating such a state on the kine-matical side correctly converge to the family arising from its associated orbit in the pseudo phasespace reduction of M ��� (that we introduced in prop. 3.9). Theorem 3.10
We consider the same objects as in prop. 3.9. Let ψ � ∈ D . For ε ∈ E , we define: ψ ε := ( δ ε H ) − � π DY� , D →ε H → H ( ψ � )� ⊂ M ����� ,ε H ⊂ M ��� H = M ��� ; Ψ ε := � σ ��� ↓ ( ψ ε ) , where � σ ��� ↓ : P ( M ��� ) → � S ↓ ( L , M ��� ,π ��� ) is defined as in [10, prop. 3.23].Then, the net (Ψ ε ) ε∈ E converges in � S ↓ ( L , M ��� ,π ��� ) to � σ ��� ↓ �� δ DH � − �ψ � � � [10, def. 3.21]. Proof
For ε = ( J , � ) ∈ E , we have, from the proof of prop. 3.6: ε = �� � − �� sin θ Π J H Π J ( ψ � ) , � sin θ, E J + � cos θ � ��� θ ∈ [0 , π [� ,where E J := � Π J ( ψ � ) , H Π J ( ψ � ) � . Hence, for all I ∈ L :[Ψ ε ] I = ��Π I � − �� sin θ Π J H Π J ( ψ � ) , � sin θ, E J + � cos θ � ��� θ ∈ [0 , π [� .And, from the proof of prop. 3.9:�� σ ��� ↓ �� δ DH � − �ψ � � �� I = ��Π I � −� � H ψ � , �, E D � �� � ∈ R � where E D := �ψ � , H ψ � � .Let I ∈ L and let U be an open set in M ��� I , such that U ∩ �� σ ��� ↓ �� δ DH � − �ψ � � �� I � = ∅ . Let � ∈ R such that:�Π I � −� � H ψ � , �, E D � ∈ U ,and let � > ∀ψ ∈ I , ∀E ∈ R , ��� ψ − Π I � −�� H ψ � �� � � ( �ψ � � + 1) & |E − E D | � � � ⇒ (( ψ, �, E ) ∈ U ) .Let � = |�| log �1 + � � > � = min � � , |�| � > H � := � � � H � (where � · � denotes the floor function).Then, we have [ H, H � ] = 0, �H − H � � � � and H � has discrete spectrum included in � Z .Hence, there exists N ∈ N such that:����� ψ � − + N � � = −N ψ � ,�� ����� � � � ∈ Z , ψ � ,�� is the projection of ψ � on the eigenspace of H � with eigenvalue � � (defining these eigenspace to be { } if � � is not in the spectrum of H � ).We define J = Vect � ψ � ,�� �� � ∈ {−N, � � � , N} � , J is finite dimensional, and ∀ψ ∈ J , �Hψ� � � �ψ� + N� �ψ� < ∞ , hence J ∈ L H . Moreover, J is stabilized by H � and �ψ � − Π J ψ � � � � . Next, we define J = J + Vect {ψ � } ∈ L H (because ψ � ∈ D ).Now, we consider ε = ( J , � ) ∈ E , with ( J , � ) � ( J , � ). We choose θ ∈ [0 , π [ such thatsin θ = � � ( |��| � � |�| �
1) and we define: ψ := Π I � − �� sin θ Π J H Π J ( ψ � ) .We have:�� ψ − Π I � −�� H ψ � �� � �� � −�� Π J H Π J ( ψ � ) − � −�� H ψ � �� � �� � −�� Π J ( H−H � ) − id J �� �� � −�� Π J H � Π J ( ψ � )�� + �� � −�� Π J H � Π J ( ψ � ) − � −�� H � ψ � �� ++ �� � −�� ( H−H � ) − id H �� �� � −�� H � ( ψ � )�� � � |�| �H−H � � − �ψ � � + �� � −�� Π J H � Π J ( ψ � ) − � −�� H � Π J ( ψ � )�� + �� � −�� Π J H � ( ψ � − Π J ( ψ � )) − � −�� H � ( ψ � − Π J ( ψ � ))�� � � |�| � − �ψ � � + �� � −�� H � Π J ( ψ � ) − � −�� H � Π J ( ψ � )�� + 2 �ψ � − Π J ( ψ � ) � � � �ψ � � + 2 � � � (1 + �ψ � � ) ,and: |E J + � cos θ − E D | � |�ψ � , ( H − Π J H Π J ) ψ � �| + � = � (since ψ � ∈ J ⊂ J ).Therefore, � ψ, � sin θ, E J + � cos θ � ∈ U , but since � ψ, � sin θ, E J + � cos θ � ∈ [ ψ ε ] I , we have ∀ε = ( J , � ) � ( J , � ) , [ ψ ε ] I ∩ U � = ∅ .Let K be a compact subset of M ��� I , such that K ∩ �� σ ��� ↓ �� δ DH � − �ψ � � �� I = ∅ . Hence, there exist T > � > ∀ψ ∈ I , ∀� ∈ R , ∀E ∈ R , ���� ψ − Π I � −�� H ψ � �� � � ( �ψ � � + 1) & |E − E D | � � � or |�| � T � ⇒ (( ψ, �, E ) /∈ K ) .Following the same path as above, we can define: � = T log �1 + � � > J ∈ L H such that: ∀ J ∈ L H / J ⊃ J , ∀� ∈ R , �� � −�� Π J H Π J ( ψ � ) − � −�� H ψ � �� � � |�| � − �ψ � � + � .Analogously, we define � = � and J = J + Vect {ψ � } ∈ L H .Now, we consider ε = ( J , � ) ∈ E , with ( J , � ) � ( J , � ) , and θ ∈ [0 , π [. If �� � sin θ �� < T , thenwe have, with � = � sin θ :���Π I � − �� sin θ Π J H Π J ( ψ � ) − Π I � −�� H ψ � ��� � � (1 + �ψ � � ) ,and: |E J + � cos θ − E D | � � .Therefore, ∀ε = ( J , � ) � ( J , � ) , [ ψ ε ] I ∩ K = ∅ .So, for every I ∈ L , the net �[Ψ ε ] I � ε∈ E converges in P ( M ��� I ) to �� σ ��� ↓ �� δ DH � − �ψ � � �� I , thus, thenet (Ψ ε ) ε∈ E converges in � S ↓ ( L , M ��� ,π ��� ) to � σ ��� ↓ �� δ DH � − �ψ � � � . � We now want to implement this construction at the quantum level, with the aim of using thissimple toy model to get a first hold on the implementation of constraints in projective systems ofquantum state spaces. o fix the notations, we begin by summarizing the main properties of (bosonic) Fock spaces [5,section I.3.4]. Definition 3.11
Let H be a separable Hilbert space. We define the Fock space � H by:� H := � �∈ N H ⊗�, sym where H ⊗�, sym is the symmetric vector subspace of H ⊗� .For ( � � ) �∈I an orthonormal basis of H ( I ⊂ N ), we define:Λ I := { ( � � ) �∈I | ∀� ∈ I, � � ∈ N & � � � � < ∞} ,indexing the orthonormal basis � | ( � � ) �∈I , ( � � ) �∈I � � ( � � ) �∈I ∈ Λ I of � H : ∀ ( � � ) �∈I ∈ Λ I , | ( � � ) �∈I , ( � � ) �∈I � := � Π �∈I ( � � !) N ! � � ,���, � N ∀�∈I, {� | � � = �} = � � |� � � ⊗ � � � ⊗ |� � N � ,where N = � �∈I � � .If � f j � j∈I is an other orthonormal basis of H , we have: ∀ ( � � ) �∈I , � m j � j∈I ∈ Λ I , �( � � ) �∈I , ( � � ) �∈I ��� � m j � j∈I , � f j � j∈I � == � � �,j ∈ N I×I ∀�, � � =� j � �,j ∀j, m j =� � � �,j � �∈I �� � � !� j � �,j ! � � j∈I �� m j !� � � �,j ! � � �,j � � � , f j � � �,j . (3.11. ) Definition 3.12
We consider the same objects as in def. 3.11. Let � ∈ H , N � � ∈ { , � � � , N} .We define the operators �a N,�� : H ⊗N → H ⊗N− and ��a N,�� � + : H ⊗N− → H ⊗N by: ∀φ , � � � , φ N ∈ H , �a N, �� φ (1)1 ⊗ � � � ⊗ φ ( N ) N := ��, φ � �√N φ (1)1 ⊗ � � � ⊗ φ ( � ) � ⊗ � � � ⊗ φ ( N− N ,and ∀φ , � � � , φ N− ∈ H , ��a N, �� � + φ (1)1 ⊗ � � � ⊗ φ ( N− N− := 1 √N φ (1)1 ⊗ � � � ⊗ � ( � ) ⊗ � � � ⊗ φ ( N ) N− .Then, on � H we can define (unbounded) operators �a � and �a + � , such that: ∀ψ ∈ H ⊗N, sym , �a � ψ = N � � =1 �a N, �� ψ ∈ H ⊗N− , sym ,and ∀ψ ∈ H ⊗N− , sym , �a + � ψ = N � � =1 ��a N, �� � + ψ ∈ H ⊗N, sym . et A be a bounded self-adjoint operator on H . We can define an (unbounded) operator � A on � H such that: ∀ψ ∈ H ⊗N, sym , � A ψ = N � � =1 id (1) H ⊗ � � � ⊗ A ( � ) ⊗ � � � ⊗ id ( N ) H ψ ∈ H ⊗N, sym .For ( � � ) �∈I is an orthonormal basis of H , we have:� A = � �,j∈I � � � , A � j � �a + � � �a � j .Lastly, let �, f ∈ H and let A, B be bounded self-adjoint operator on H . The commutatorsbetween the operators defined above are given by:[�a � , �a f ] = 0, ��a + � , �a + f � = 0, and ��a � , �a + f � = ��, f � id � H ,�� A, � B � = � [ A, B ] H , ��a � , � A � = �a A� , and ��a + � , � A � = − �a + A� . Before going on to the quantization using projective structures, we recall the more conventionalquantization of M DY� , Ω DY� (ie. a reduced phase space quantization for the theory we are considering)using Fock spaces techniques. The notable fact is that this direct quantization of the Schrödingerequation (considered as a classical field theory, aka. second quantization) can be identified withthe (bosonic) Fock space describing an arbitrary number of independent, indistinguishable quantumparticles of the corresponding first quantized theory [2]. This identification is not merely a naivematching of the Hilbert spaces: we can check that the quantized observables correspond in a naturalway to the observables built on the Fock space.
Proposition 3.13
We consider the objects introduced in prop. 3.2 and def. 3.12. We define the Fockquantization of M DY� as � M DY�Fock := � H . For A a bounded self-adjoint operator on H and � ∈ H wedefine the following quantizations for the observables on M DY� :� �A� Fock := � A , �(a � ) Fock := �a � , and �(a ∗� ) Fock := �a + � .Then, we have: ∀O, O � ∈ { a � | � ∈ H } ∪ { a ∗� | � ∈ H } ∪ {�A� | A bounded, self-adj on H } , � � O Fock , � O � Fock � = −� � � {O, O � } DY� � Fock . Proof
This can be directly checked by comparing prop. 3.2 with def. 3.12. � The key tool for constructing a projective system of quantum state spaces reproducing the classicalstructure from prop. 3.3 is the realization that the Fock space arising from a direct orthogonal sumof two Hilbert space can be naturally identified with the tensor product of the two correspondingFock spaces. This is in fact a special case of the well-known property of quantization, that translatesa Cartesian product of symplectic manifold into a tensor product of Hilbert spaces (for a direct sum s indeed a Cartesian product). Proposition 3.14
Let I be a separable Hilbert space. Let J be a vector subspace of I and J ⊥ the orthogonal complement of J in I . Let ( � � ) �∈J be an orthonormal basis of J and ( � � ) �∈I\J be anorthonormal basis of J ⊥ (with I ⊃ J ). Hence, ( � � ) �∈I is an orthonormal basis of I = J ⊕ J ⊥ .We consider the corresponding Fock spaces � I , � J & � J ⊥ (def. 3.11) and we define the linearapplication � φ I → J : � I → � J ⊥ ⊗ � J by its action on the orthonormal basis � | ( � � ) �∈I , ( � � ) �∈I � � ( � � ) �∈I ∈ Λ I of � I :� φ I → J | ( � � ) �∈I , ( � � ) �∈I � := ��( � � ) �∈I\J , ( � � ) �∈I\J � ⊗ | ( � � ) �∈J , ( � � ) �∈J � . (3.14. )Then, � φ I → J is an Hilbert space isomorphism. Moreover, � φ I → J does not depend on the choice ofthe bases ( � � ) �∈J and ( � � ) �∈I\J . Proof � φ I → J sends an orthonormal basis to an orthonormal basis, since the map:Λ I → Λ J\I × Λ J ( � � ) �∈I �→ ( � � ) �∈I\J , ( � � ) �∈J ,is bijective.Then, if ( f � ) �∈J is an other orthonormal basis of J and ( f � ) �∈I\J is an other orthonormal basis of J ⊥ ,we have, using eq. (3.11. ) for � m j � j ∈ I ∈ Λ I :� φ I → J ���� m j � j∈I , � f j � j∈I � == � � �,j ∈ N I×I ∀j∈I, m j =� � � �,j � �∈I ���� j∈I � �,j �!� � j∈I �� m j !� � �,j∈I � � � , f j � � �,j � �,j ! � φ I → J ����� j∈I � �,j � �∈I , ( � � ) �∈I � .Now, for �, j ∈ J × ( I \ J ) or (
I \ J ) × J , � � � , f j � = 0 since � J , J ⊥ � = 0. Therefore, the only non-vanishing terms in the sum above are such that � �,j = ( �,j ) ∈ ( I\J ) � �,j + ( �,j ) ∈J � �,j with � �,j ∈ N ( I\J ) × ( I\J ) and � �,j ∈ N J×J . Hence, using eq. (3.14. ):� φ I → J ���� m j � j∈I , � f j � j∈I � == � � �,j ∈ N ( I\J ) × ( I\J ) ∀j∈ ( I\J ) , m j =� � � �,j � � �,j ∈ N J×J ∀j∈J, m j =� � � �,j � �∈I\J ���� j∈I\J � �,j �!� � �∈J ���� j∈J � �,j �!� � j∈I\J �� m j !� ×× � j∈J �� m j !� � �,j∈I\J � � � , f j � � �,j � �,j ! � �,j∈J � � � , f j � � �,j � �,j ! ������ j∈I\J � �,j � �∈I\J , ( � � ) �∈I\J � ⊗ ����� j∈J � �,j � �∈J , ( � � ) �∈J �= ���� m j � j∈I\J , � f j � j∈I\J � ⊗ ���� m j � j∈J , � f j � j∈J � , here we used again eq. (3.11. ), both in � J ⊥ and in � J . � Proposition 3.15
We consider the objects introduced in props. 3.3 and 3.14. We define: ∀ I ∈ L , � M ��� I := � I ⊗ T where T := L ( R , �µ ) ( µ being the Lebesgue measure on R ); ∀ I ⊂ I � ∈ L , � M ��� I � → I := � I ⊥ ∩ I � (with the convention that � M ��� I � → I = C if I � = I ); ∀ I ⊂ I � ∈ L , � φ ��� I � → I := � φ I � → I ⊗ id T : � I � ⊗ T → � I ⊥ ∩ I � ⊗ �� I ⊗ T �; ∀ I ⊂ I � ⊂ I �� ∈ L , � φ ��� I �� → I � → I := � φ ( I ⊥ ∩ I �� ) → ( I ⊥ ∩ I � ) : � I ⊥ ∩ I �� → � I �⊥ ∩ I �� ⊗ � I ⊥ ∩ I � (note that� I ⊥ ∩ I � � ⊥ ∩ � I ⊥ ∩ I �� � = I �⊥ ∩ I �� since I ⊂ I � ).� L , � M ��� , � φ ��� � ⊗ is a projective system of quantum state spaces [11, def. 2.1]. Proof L is directed since for I , I � ∈ L , I + I � ∈ L . And, for I ⊂ I � ⊂ I �� ∈ L , � φ ��� I � → I and � φ ��� I �� → I � → I are Hilbert space isomorphisms.Let I ⊂ I � ⊂ I �� ∈ L . We choose an orthonormal basis ( � � ) �∈I of I , an orthonormal basis ( � � ) �∈I � \I of I � ∩ I ⊥ (with I � ⊃ I ) and an orthonormal basis ( � � ) �∈I �� \I � of I �� ∩ I �⊥ (with I �� ⊃ I � ). Since eq. (3.14. )is valid for any choice of orthonormal bases, we have for ( � � ) �∈I �� ∈ Λ I �� :�id � I �� ∩ I �⊥ ⊗ � φ I � → I � ◦ � φ I �� → I � | ( � � ) �∈I �� , ( � � ) �∈I �� � == ��( � � ) �∈I �� \I � , ( � � ) �∈I �� \I � � ⊗ ��( � � ) �∈I � \I , ( � � ) �∈I � \I � ⊗ | ( � � ) �∈I , ( � � ) �∈I � = �� φ ( I �� ∩ I ⊥ ) → ( I � ∩ I ⊥ ) ⊗ id � I � ◦ � φ I �� → I | ( � � ) �∈I �� , ( � � ) �∈I �� � . (3.15. )Hence, [11, eq. (2.1. )] is fulfilled:�id � M ��� I ��→ I � ⊗ � φ ��� I � → I � ◦ (� φ ��� I �� → I � ) = �� φ ��� I �� → I � → I ⊗ id � M ��� I � ◦ � φ ��� I �� → I . � Proposition 3.16
We consider the objects introduced in props. 3.7 and 3.14. We define: ∀ J ∈ L H , � M DY� J := � J ; ∀ J ⊂ J � ∈ L H , � M DY� J � → J := � J ⊥ ∩ J � & � φ DY� J � → J := � φ J � → J ; ∀ J ⊂ J � ⊂ J �� ∈ L H , � φ DY� J �� → J � → J := � φ ( J ⊥ ∩ J �� ) → ( J ⊥ ∩ J � ) .� L H , � M DY� , � φ DY� � ⊗ is a projective system of quantum state spaces.Let A be a bounded self-adjoint operator on H and suppose that (Ker A ) ⊥ ∈ L H . For J ∈ L H suchthat (Ker A ) ⊥ ⊂ J , we define � A J := � � A| J → J � (def. 3.12). For J , J � ∈ L H such that (Ker A ) ⊥ ⊂ J , J � ,we have:� A J ∼ � A J � (with the equivalence relation ∼ defined in [11, eq. (2.3. )] ), ence, we can define � A L H := �� A J � ∼ ∈ O ⊗ ( L H , � M DY� , � φ DY� ) [11, prop. 2.5]. Proof
We know from prop. 3.7 that L H is a directed set. Then, we can show that � L H , � M DY� , � φ DY� � ⊗ is a projective system of quantum state spaces exactly like in the proof of prop. 3.15.Let J ⊂ J �� ∈ L H , ( � � ) �∈J be an orthonormal basis of J and ( � � ) �∈J �� \J (with J �� ⊃ J ) an orthonormalbasis of J �� ∩ J ⊥ . For �, � ∈ J and ( � � ) �∈J �� ∈ Λ J �� , we have:� φ − J �� → J �id � J �� ∩ J ⊥ ⊗ �a J , + � � �a J � � � � φ J �� → J | ( � � ) �∈J �� , ( � � ) �∈J �� � == √� � � � � + 1 − δ �� | ( � � − δ �� + δ �� ) �∈J �� , ( � � ) �∈J �� � = �a J �� , + � � �a J �� � � | ( � � ) �∈J �� , ( � � ) �∈J �� � .Now, if (Ker A ) ⊥ ⊂ J , we have J �� ∩ J ⊥ ⊂ J ⊥ ⊂ Ker A , therefore: � � A| J �� → J �� � = � �,�∈J �� �� � , A � � � �a J �� , + � � �a J �� � � = � �,�∈J �� � , A � � � �a J �� , + � � �a J �� � � = � φ − J �� → J id � J �� ∩ J ⊥ ⊗ � �,�∈J �� � , A � � � �a J , + � � �a J � � � φ J �� → J = � φ − J �� → J �id � J �� ∩ J ⊥ ⊗ � � A| J → J �� � φ J �� → J . (3.16. )Hence, � A J �� = � φ DY� ,− J �� → J �id � M DY� J ��→ J ⊗ � A J � � φ DY� J �� → J .Finally, if J , J � ∈ L H are such that (Ker A ) ⊥ ⊂ J , J � , we can find J �� ∈ L H such that J , J � ⊂ J �� (because L H is directed), so � A J ∼ � A J � . � Using the general result derived in [11, theorem 2.9], we are able to embed the space of densitymatrices on the Fock space into the larger quantum state space constructed by projective techniques,and to precisely characterize the image of this embedding, by giving a condition for a projectivestate to be representable as a density matrix on � H . Proposition 3.17
We consider the same objects as in prop. 3.16. There exists an injective map� σ ↓ : S Fock → S ⊗ ( L H , � M DY� , � φ DY� ) (where S Fock is the space of (self-adjoint) positive semi-definite, traceclassoperators over � M DY�
Fock and S ⊗ ( L H , � M DY� , � φ DY� ) was defined in [11, def. 2.1]) satisfying, for any boundedself-adjoint operator A on H with (Ker A ) ⊥ ∈ L H , and any ρ ∈ S Fock :Tr � M DY�Fock � ρ I W � � �A� Fock �� = Tr �� σ ↓ ( ρ ) I W �� A L H �� , (3.17. ) here W is a measurable subset in the spectrum of � �A� Fock , and I W ( · ) denotes the correspondingspectral projectors.Moreover, we have:� σ ↓ � S Fock � = �( ρ J ) J ∈ L H ���� sup J ∈ L H inf J � ⊃ J Tr � M DY� J � � ρ J � �Π J � | J � = 1� ,where S Fock is the space of density matrices over � M DY�
Fock and: ∀N ∈ N , ∀ψ ∈ J �⊗N, sym , �Π J � | J ψ := (Π J ) ⊗N ψ ∈ J �⊗N, sym ,Π J being the orthogonal projection on J . Proof
For J ⊂ J � ∈ L H , we define ζ J � → J ∈ � M DY� J � → J as the vacuum state of � M DY� J � → J = � J � ∩ J ⊥ (ie. ζ J � → J = | ( O ) �∈I , ( � � ) �∈I � for any basis ( � � ) �∈I of J � ∩ J ⊥ ). The family of vectors ( ζ J � → J ) J ⊂ J � fulfillsthe hypotheses of [11, theorem 2.9].Next, for all J ∈ L H , we define an injection τ Fock ← J from � M DY� J = � J into � M DY�
Fock = � H by: τ Fock ← J = � φ − H → J ( ζ Fock → J ⊗ ( · )) ,where ζ Fock → J is the vacuum state of � J ⊥ . Using eq. (3.15. ), we can show that ∀ J ⊂ J � ∈ L H , τ Fock ← J � ◦ τ J � ← J = τ Fock ← J (where τ J � ← J is defined from ζ J � → J as in [11, theorem 2.9]).Now, we can choose an orthonormal basis ( � � ) �∈ N of H such that ∀� ∈ N , �H � � � < ∞ andconsider for N � J N := Vect {� � | � � N} ∈ L H . Using eq. (3.14. ) with the orthonormal basis( � � ) � � N of J N and the orthonormal basis ( � � ) �>N of J ⊥N , we get: τ Fock ← J N �� J N � = Vect � | ( � � ) �∈ N , ( � � ) �∈ N � �� ( � � ) �∈ N ∈ Λ N N � ,where Λ N N := { ( � � ) �∈ N ∈ Λ N | ∀� > N, � � = 0 } . Hence, from Λ N = � N � Λ N N , we have:� M DY�
Fock = � J ∈ L H Im τ Fock ← J .Therefore, we can identify � M DY�
Fock with the inductive limit � M DY� ζ introduced in [11, theorem 2.9], sowe have an injection � σ ↓ : S Fock → S ⊗ ( L H , � H DY� , � φ DY� ) , satisfying:� σ ↓ � S Fock � = �( ρ J ) J ∈ L H ���� sup J ∈ L H inf J � ⊃ J Tr � M DY� J � � ρ J � �Π J � | J � = 1� ,where: ∀ J ⊂ J � ∈ L H , �Π J � | J = � φ − J � → J ◦ � |ζ J � → J �� ζ J � → J | ⊗ id � J � ◦ � φ J � → J .Let J ⊂ J � ∈ L H and let ( � � ) �∈J , resp. ( � � ) �∈J � \J be an orthonormal basis of J , resp. J � ∩ J ⊥ . For( � � ) �∈ J � ∈ Λ J � , we have:�Π J � | J | ( � � ) �∈J � , ( � � ) �∈J � � = � | ( � � ) �∈J � , ( � � ) �∈J � � if ∀� ∈ J � \ J, � � = 00 otherwise Π ⊗ � �∈J� � � J | ( � � ) �∈J � , ( � � ) �∈J � � ,therefore ∀N ∈ N , ∀ψ ∈ J �⊗N, sym , �Π J � | J ψ = (Π J ) ⊗N ψ .Lastly, let A be a bounded self-adjoint operator on H such that J := (Ker A ) ⊥ ∈ L H . Eq. (3.17. ) isthen an application of [11, prop. 2.5], using the definition of � σ ↓ (given in the proof of [11, theorem 2.9])together with:� �A� Fock := � A = � φ − H → J �id � J ⊥ ⊗ � A J � � φ H → J ,which can be shown like in the proof of prop. 3.16 (eq. (3.16. ) ). � We can now implement and solve in the quantum theory the approximated constraints we hadon the classical side, and thus define a family of maps (indexed by the regularization parameter ε )from the dynamical projective system of quantum state spaces introduced above into the kinematicalone. Proposition 3.18
We consider the objects introduced in def. 3.4 and prop. 3.15. Let ε = ( J , � ) ∈ E and let I ∈ L ε . We define the map:� δ ε I : � J → � I ⊗ T ψ �→ �� φ − I → J ⊗ id T � � ζ I → J ⊗ exp � −� � � (Π J H Π J ) | J → J � ⊗ � T � ( ψ ⊗ δ � )� ,where ζ I → J is the vacuum state in � I ∩ J ⊥ , Π J is the orthogonal projection on J , � T is the positionoperator on T = L ( R , �µ ) and δ � ∈ T is defined by: ∀� ∈ R , δ � ( � ) = √�π / exp � − � � δ ε I �� J � = � ψ ∈ I ��� ��Π I | J ⊗ id T � ψ = ψ & � C ε ψ = ψ � ,with � C ε = 1 � �id � I ⊗ � E − � � (Π J H Π J ) | I → I � ⊗ id T � + � id � I ⊗ � T ,where �Π I | J is defined as in prop. 3.17 and � E is the operator � ∂ � on T . Moreover, � δ ε I ��� � J → � δ ε I � � J � is aunitary map. Proof
We define:� δ : � J → � J ⊗ T ψ �→ ψ ⊗ δ � , � δ : � J ⊗ T → � J ⊗ T ψ �→ exp � −� � H ε J ⊗ � T � ψ with H ε := Π J H Π J ,� δ : � J ⊗ T → � I ∩ J ⊥ ⊗ � J ⊗ T ψ �→ ζ I → J ⊗ ψ , � δ : � I ∩ J ⊥ ⊗ � J ⊗ T → � I ⊗ T ψ �→ �� φ − I → J ⊗ id T � ψ . e have � δ �� J � = � J ⊗ Vect {δ � } = � J ⊗ � ψ ∈ T ��� � C ψ = ψ �, where � C := 1 � � E + � � T , and� δ ��� � J → � δ � � J � is a unitary map.� δ is a unitary map, because � H ε J and � T are essentially self-adjoint ( ∀N ∈ N , � H ε J stabilize H ⊗N, sym and the restriction of � H ε J to H ⊗N, sym is a bounded self-adjoint operator, for so is H ε | J → J ,by definition of L H ). And we have:� δ ◦ � δ �� J � = � ψ ∈ � J ⊗ T ��� � C ψ = ψ � ,with:� C := � δ ◦ �id � J ⊗ � C � ◦ � δ − = exp � −� � � H ε J ⊗ � T , · �� �id � J ⊗ � C �= 1 � �id � J ⊗ � E − � H ε J ⊗ id T � + � id � J ⊗ � T .Next, we compute:� δ ◦ � δ ◦ � δ �� J � = � ζ I → J ⊗ ψ ∈ � I ∩ J ⊥ ⊗ � J ⊗ T ��� � C ψ = ψ �= � ψ ∈ � I ∩ J ⊥ ⊗ � J ⊗ T ��� id � I ∩ J ⊥ ⊗ � C ψ = ψ & � |ζ I → J �� ζ I → J | ⊗ id � J ⊗ T � ψ = ψ � ,and � δ ��� � J ⊗ T → � δ � � J ⊗ T � is a unitary map.Finally, � δ is unitary (from prop. 3.14) and:� δ ε I �� J � = � δ ◦ � δ ◦ � δ ◦ � δ �� J � = � ψ ∈ � I ⊗ T ��� � C ψ = ψ & � D ψ = ψ � ,with:� C := �� φ − I → J ⊗ id T � �id � I ∩ J ⊥ ⊗ � C � (� φ I → J ⊗ id T )= 1 � �id � I ⊗ � E − �� φ − I → J �id � I ∩ J ⊥ ⊗ � H ε J � � φ I → J � ⊗ id T � + � id � I ⊗ � T = 1 � �id � I ⊗ � E − � H ε I ⊗ id T � + � id � I ⊗ � T (using eq. (3.16. ) ),and: � D := �� φ − I → J ⊗ id T � � |ζ I → J �� ζ I → J | ⊗ id � J ⊗ T � (� φ I → J ⊗ id T )= �Π I | J ⊗ id T (as was shown in the proof of prop. 3.17). � Proposition 3.19
We consider the same objects as in prop. 3.18. For ε = ( J , � ) ∈ E and ρ J a self-adjoint) positive semi-definite, traceclass operator on � J , we define: ∀ I ∈ L ε , �Δ ε I ( ρ J ) := � δ ε I ρ J �� δ ε I � + .Then, ��Δ ε I ( ρ J )� I ∈ L ε ∈ S ⊗ ( L ε , � M ��� , � φ ��� ) .Hence, for ρ = ( ρ J ) J ∈ L H ∈ S ⊗ ( L H , � M DY� , � φ DY� ) (prop. 3.16), we can define:�Δ ε ( ρ ) = � σ − ���Δ ε I ( ρ J )� I ∈ L ε � ,where the map � σ : S ⊗ ( L , � M ��� , � φ ��� ) → S ⊗ ( L ε , � M ��� , � φ ��� ) is defined as in [11, prop. 2.6] (and is bijective, since L ε is cofinal in L ). Proof
We need to prove that ∀ I , I � ∈ L ε , with I ⊂ I � , Tr I � → I �Δ ε I � ( ρ J ) = �Δ ε I ( ρ J ). We have: ∀ψ ∈ � J , � φ ��� I � → I ◦ � δ ε I � ( ψ ) = ��� φ I � → I ◦ � φ − I � → J � ⊗ id T � � ζ I � → J ⊗ � −� � H ε J ⊗ � T ( ψ ⊗ δ � )�= ���id � I � ∩ I ⊥ ⊗ � φ − I → J � ◦ �� φ I � → I → J ⊗ id � J �� ⊗ id T � � ζ I � → J ⊗ � −� � H ε J ⊗ � T ( ψ ⊗ δ � )�= ��id � I � ∩ I ⊥ ⊗ � φ − I → J � ⊗ id T � � ζ I � → I ⊗ ζ I → J ⊗ � −� � H ε J ⊗ � T ( ψ ⊗ δ � )�= ζ I � → I ⊗ � δ ε I ( ψ ) ,hence � φ ��� I � → I ◦ �Δ ε I � ( ρ J ) ◦ � φ ��� ,− I � → I = |ζ I � → I �� ζ I � → I | ⊗ �Δ ε I ( ρ J ), therefore:Tr I � → I �Δ ε I � ( ρ J ) = Tr � I � ∩ I ⊥ |ζ I � → I �� ζ I � → I | ⊗ �Δ ε I ( ρ J ) = �Δ ε I ( ρ J ) . � As a preparation for the study of convergence, we define a subset � R of the space of statesover the quantum projective structure. The motivation is to implement a quantum version of theregularity condition that was ensuring convergence on the classical side: at the classical level wehave proved the convergence for normalized states, so in analogy we consider here states witha bounded expectation value for the total number of particles (which indeed corresponds to thequantization of the classical observable ψ �→ �ψ, ψ� ).Note that, as we show in the following result, the regular states (the elements of � R ) can be seenas states in the Fock space via the embedding of prop. 3.17. This is not really surprising, since weknow that the Fock space quantization is appropriate for a basic non-interacting field theory likethe Schrödinger equation. Proposition 3.20
We consider the same objects as in prop. 3.17 and we define:� R := � ρ ∈ S ⊗ ( L H , � M DY� , � φ DY� ) ���� sup J ∈ L H Tr � ρ �(Π J ) L H � < ∞ � , here Tr � ρ �(Π J ) L H � := � �∈ N � Tr � ρ I {�} � �(Π J ) L H �� and I {�} � �(Π J ) L H � denotes the spectral projec-tor as in [11, prop. 2.5].Then, � R ⊂ � σ ↓ � S Fock � .
Proof
Let ρ ∈ � R and N = sup J ∈ L H Tr � ρ �(Π J ) L H �. If ρ = 0, then ρ = � σ ↓ (0). Otherwise, Tr ρ = � > ρ = � � � ρ � with � ρ ∈ S ⊗ ( L H , � M DY� , � φ DY� ) . Let J , J � ∈ L H with J ⊂ J � . Let ( � � ) �∈J be an orthonormalbasis of J and ( � � ) �∈J � \J ( J � ⊃ J ) be an orthonormal basis of J � ∩ J ⊥ . For ( � � ) �∈J � , ( m � ) �∈J � ∈ Λ J � , wehave:�( � � ) �∈J � , ( � � ) �∈J � ��� � � Π J � ∩ J ⊥ | J � → J � � ��� ( m � ) �∈J � , ( � � ) �∈J � � = � �∈J � \J � � if ∀� ∈ J � , � � = m � � � ) �∈J � , ( � � ) �∈J � ��� �id � J � − �Π J � | J � ��� ( m � ) �∈J � , ( � � ) �∈J � � = 1 if ∀� ∈ J � , � � = m � & � �∈J � \J � � �
10 else ,therefore:Tr � J � � ρ J � �Π J � | J � � � − � �∈ N � Tr � J � � ρ J � I {�} � � � Π J � ∩ J ⊥ | J � → J � ��� =: � − Tr � J � � ρ J � � � Π J � ∩ J ⊥ | J � → J � �� .Now, Tr � J � � ρ J � � � Π J � ∩ J ⊥ | J � → J � �� = � �∈ N � Tr � J � � ρ J � I {�} � �(Π J � ) J � �� − � �∈ N � Tr � J � � ρ J � I {�} � �(Π J ) J � �� =Tr � ρ �(Π J � ) L H � − Tr � ρ �(Π J ) L H �, hence:inf J � ⊃ J Tr � J � � ρ J � �Π J � | J � � � − sup J � ⊃ J Tr � ρ �(Π J � ) L H � + Tr � ρ �(Π J ) L H � .Finally, �Tr � ρ �(Π J � ) L H �� J � ∈ L H is increasing, so sup J � ⊃ J Tr � ρ �(Π J � ) L H � = N and:sup J ∈ L H inf J � ⊃ J Tr � J � � ρ J � �Π J � | J � � � − N + N = � .On the other hand, ∀ J ⊂ J � , Tr � J � � ρ J � �Π J � | J � � � , thus, using prop. 3.17, � ρ ∈ � σ ↓ � S Fock � , and therefore ρ ∈ � σ ↓ � S Fock �. � Finally, we prove a convergence result at the quantum level. We define here two differentnotions of convergence, one stronger than the other, in both cases requiring convergence of theexpectation values for a certain class of observables. To assess how exactly the convergence shouldbe adjusted would require a closer study of which observables are really measured in practice, forthese constitute the class of kinematical observables that we want to be able to transport on thedynamical side. n addition, we need to introduce an � -dependent normalization parameter N that accountsfor the fact that states solving the exact dynamics cannot be correctly normalized (they describeprobability distributions invariant under a transformation running along the full time line from � = −∞ to � = + ∞ ) so that it only makes sense to consider partial probability, measuring theprobability of measuring the system in a certain state, knowing that the measurement takes placeat a certain time. So, as we lift the � -regularization (that was making the gauge orbits compact andthe solution of the quantum constraint normalizable), the probability of measuring the system in acertain time interval is dropping and needs to be accordingly compensated. Theorem 3.21
We consider the same objects as in props. 3.19 and 3.20. Let I ∈ L , A be a boundedoperator on � I and φ, φ � ∈ T . We additionally assume that φ, φ � have compact support. On � M ��� I ,we define the operator: R I A, φ, φ � := A ⊗ |φ �� φ � | ,and, for ε = ( J , � ) ∈ E and ρ ∈ � σ ↓ � S Fock �, we define: R I ,εA, φ, φ � ( ρ ) := 1 N ( �, φ, φ � ) Tr � M ��� I ��Δ ε I ( ρ ) R I A, φ, φ � � ,where N ( �, φ, φ � ) = Tr T |φ �� φ � | |δ � �� δ � | = �φ � , δ � � �δ � , φ� .Then, the net � R I ,εA, φ, φ � ( ρ )� ε∈ E converges.For �, f ∈ I , we also define on � M ��� I the operator: R I �, f, φ, φ � := �a I , + � �a I f ⊗ |φ �� φ � | ,and, for ε = ( J , � ) ∈ E and ρ ∈ � R , we define: R I ,ε�, f, φ, φ � ( ρ ) := 1 N ( �, φ, φ � ) Tr � M ��� I ��Δ ε I ( ρ ) R I �, f, φ, φ � � .Then, the net � R I ,ε�, f, φ, φ � ( ρ )� ε∈ E converges. Proof
Bounded operator & Fock state.
Let ρ ∈ � σ ↓ � S Fock �. For ε = ( J , � ) ∈ E and I � ∈ L ε , wehave:�Δ ε I � ( ρ ) = � φ ��� ,− I � → J � |ζ I � → J �� ζ I � → J | ⊗ � � −� � H ε J ⊗ � T ( ρ J ⊗ |δ � �� δ � | ) � � � H ε J ⊗ � T �� � φ ��� I � → J = � φ ��� ,− I � → J � −� id � I �∩ J ⊥ ⊗ � H ε J ⊗ � T ( |ζ I � → J �� ζ I � → J | ⊗ ρ J ⊗ |δ � �� δ � | ) � � id � I �∩ J ⊥ ⊗ � H ε J ⊗ � T � φ ��� I � → J = � −� � H ε I � ⊗ � T � φ ��� ,− I � → J ( |ζ I � → J �� ζ I � → J | ⊗ ρ J ⊗ |δ � �� δ � | ) � φ ��� I � → J � � � H ε I � ⊗ � T (like in eq. (3.16. ) )= � −� � H ε I � ⊗ � T �� τ I � ← J ρ J τ + I � ← J � ⊗ |δ � �� δ � | � � � � H ε I � ⊗ � T ,where τ I � ← J = � φ − I � → J ( ζ I � → J ⊗ ( · )) . Hence, for I ⊂ I � : I ,εA, φ, φ � ( ρ ) = 1 N ( �, φ, φ � ) Tr � M ��� I � �Δ ε I ( ρ ) A ⊗ |φ �� φ � | � == Tr � M ��� I � ��Δ ε I � ( ρ ) �� φ − I � → I �id � I � ∩ I ⊥ ⊗ A � � φ I � → I � ⊗ |φ �� φ � | � N ( �, φ, φ � )= Tr � M ��� I � � � −� � H ε I � ⊗ � T �� τ I � ← J ρ J τ + I � ← J � ⊗ |δ � �� δ � | � � � � H ε I � ⊗ � T �� φ − I � → I �id � I � ∩ I ⊥ ⊗ A � � φ I � → I � ⊗ |φ �� φ � | � N ( �, φ, φ � )= � T−T �� �� � φ ( � ) φ �∗ ( � � ) δ � ( � � ) δ ∗� ( � ) Z ε I � ( �, � � )� T−T �� �� � φ ( � ) φ �∗ ( � � ) δ � ( � � ) δ ∗� ( � ) ,where T > φ and φ � is included in [ −T , T ], and Z ε is defined as: Z ε ( �, � � ) = Tr � I � � � −�� � � H ε I � � τ I � ← J ρ J τ + I � ← J � � �� � H ε I � �� φ − I � → I �id � I � ∩ I ⊥ ⊗ A � � φ I � → I ��= Tr � J ρ J � τ + I � ← J � �� � H ε I � �� φ − I � → I �id � I � ∩ I ⊥ ⊗ A � � φ I � → I � � −�� � � H ε I � τ I � ← J �= Tr � J ρ J � � �� � H ε J τ + I � ← J �� φ − I � → I �id � I � ∩ I ⊥ ⊗ A � � φ I � → I � τ I � ← J � −�� � � H ε J � ,for any I � ∈ L such that I , J ⊂ I � .Next, √π� δ � ( � � ) δ ∗� ( � ) converges uniformly to 1 for �, � � ∈ [ −T , T ], when � →
0. Therefore, weneed to show that the net � Z ε ( �, � � )� ε∈ E converges uniformly for �, � � ∈ [ −T , T ].Let ρ Fock ∈ S Fock such that ρ = � σ ↓ ( ρ Fock ) . Using the definition of � σ ↓ , we can show: Z ε ( �, � � ) = Tr � H ρ Fock � φ − H → J � � J ⊥ ⊗ � �� � H ε J τ + I � ← J �� φ − I � → I �id � I � ∩ I ⊥ ⊗ A � � φ I � → I � τ I � ← J � −�� � � H ε J � � φ H → J = Tr � H ρ Fock � �� � H ε � φ − H → J � � J ⊥ ⊗ τ + I � ← J �� φ − I � → I �id � I � ∩ I ⊥ ⊗ A � � φ I � → I � τ I � ← J � � φ H → J � −�� � � H ε (like in eq. (3.16. ) )= Tr � J �Tr H → J � −�� � � H ε ρ Fock � �� � H ε � � τ + I � ← J �� φ − I � → I �id � I � ∩ I ⊥ ⊗ A � � φ I � → I � τ I � ← J �(by definition of Tr H → J )And using twice [11, eq. (2.1. )] (for J , I ⊂ I � ⊂ H ), we have, for any ψ ∈ J :� φ H → I ◦ τ Fock ← J ( ψ ) = � φ H → I ◦ � φ − H → J ( ζ Fock → J ⊗ ψ )= � φ H → I ◦ � φ − H → J ◦ �� φ − J ⊥ → I � ∩ J ⊥ ⊗ J � ( ζ Fock → I � ⊗ ζ I � → J ⊗ ψ )= � φ H → I ◦ � φ − H → I � ( ζ Fock → I � ⊗ τ I � ← J ( ψ ))= �� φ − I ⊥ → I � ∩ I ⊥ ⊗ id I � ( ζ Fock → I � ⊗ � φ I � → I ◦ τ I � ← J ( ψ )) . ence, we get: Z ε ( �, � � ) = Tr � J �Tr H → J � −�� � � H ε ρ Fock � �� � H ε � � τ +Fock ← J �� φ − H → I �id � I ⊥ ⊗ A � � φ H → I � τ Fock ← J �= Tr � H � τ Fock ← J �Tr H → J � −�� � � H ε ρ Fock � �� � H ε � τ +Fock ← J � �� φ − H → I �id � I ⊥ ⊗ A � � φ H → I �But we have:��� φ − H → I �id � I ⊥ ⊗ A � � φ H → I �� = �A� < ∞ ,therefore, what remains to be shown is that the net:� τ Fock ← J �Tr H → J � −�� � � H ε ρ Fock � �� � H ε � τ +Fock ← J � ε =( J ,� ) ∈ E converges in trace norm (which was defined in [11, lemma 2.10]), uniformly for �, � � ∈ [ −T , T ] .Let � � >
0. We have:� H = � J ∈ L H ,N � � J N where � J N = � � � N � J ⊗�, sym ,because H = � J ∈ L H J . Hence, we can prove, using the spectral decomposition of the self-adjointtraceclass operator ρ Fock and the directed preorder on L H and N , that there exist J � ∈ L H and N � � �ρ Fock − ρ � Fock � � � � ρ � Fock := �Π J � ,N � ρ Fock �Π J � ,N � (with �Π J � ,N � the orthogonal projection on � J N � � ) and � · � denotesthe trace norm.Since � J N � � is finite dimensional, there exist vectors ψ α ∈ � J N � � , α ∈ { , � � � , K } (with K ∈ N ) suchthat: ρ � Fock = K � α =1 |ψ α �� ψ α | .We define: � := 1 N �
11 + |T | log 1 + � � K �1 + max α �ψ α � � > H � := � � � H � (as in the proof of theorem 3.10). Then, since H � has discrete spectrum and K , N � < ∞ , we can construct J ∈ L H , such that J is stabilized by H � and: ∀α � K , ��� ψ α − �Π J ,N � ψ α ��� � � � K (1 + max α �ψ α � ) .Thus, we get:��� ρ Fock − �Π J ,N � ρ � Fock �Π J ,N � ��� � � � K � α =1 ��� |ψ α �� ψ α | − ����Π J ,N � ψ α �� �Π J ,N � ψ α ������ � � K � α =1 ��� ψ α − �Π J ,N � ψ α ��� �ψ α � + ����Π J ,N � ψ α ��� ��� ψ α − �Π J ,N � ψ α ��� � � � J ∈ L H such that J � + J ⊂ J . For all �, � � ∈ [ −T , T ], we have:���� � −�� � � ( Π J H Π J ) ρ Fock � �� � ( Π J H Π J ) − � −�� � � H � �Π J ,N � ρ � Fock �Π J ,N � � �� � H � ���� �� � � � −�� � � ( Π J H Π J ) − � −�� � � ( Π J H � Π J )���� �ρ � Fock � + �ρ � Fock � ���� � �� � ( Π J H Π J ) − � �� � ( Π J H � Π J )����(since H � stabilizes J ⊂ J ) � � � � T N � � − �ρ � Fock � � � � � −�� � � H ρ Fock � �� � H − � −�� � � H � �Π J ,N � ρ � Fock �Π J ,N � � �� � H � ��� � � � H � stabilizes J ⊂ J , we also have: τ Fock ← J �Tr H → J � −�� � � H � �Π J ,N � ρ � Fock �Π J ,N � � �� � H � � τ +Fock ← J = � −�� � � H � �Π J ,N � ρ � Fock �Π J ,N � � �� � H � .Hence, for any ε = ( J , � ) ∈ E such that ( J � + J , � ε , and any �, � � ∈ [ −T , T ] , we have:��� τ Fock ← J �Tr H → J � −�� � � H ε ρ Fock � �� � H ε � τ +Fock ← J − � −�� � � H ρ Fock � �� � H ��� � � � ,which provides the desired convergence. Transition operator & regular state. Let ρ ∈ � R , I ∈ L and �, f ∈ I . Since � R ⊂ � σ ↓ � S Fock �(prop. 3.20), there exists ρ Fock ∈ S Fock such that ρ = � σ ↓ ( ρ Fock ). Like above, a sufficient condition forthe convergence of the net � R I ,ε�, f, φ, φ � ( ρ )� ε∈ E is uniform convergence for �, � � ∈ [ −T , T ] of the net:�Tr � H � τ Fock ← J �Tr H → J � −�� � � H ε ρ Fock � �� � H ε � τ +Fock ← J � T �,f � ε =( J ,� ) ∈ E ,where we define: T �,f := � φ − H → I �id � I ⊥ ⊗ �a I , + � �a I f � � φ H → I .Choosing an orthonormal basis ( � � ) �∈I of I and completing it into an orthonormal basis ( � � ) �∈ N of I � ( I ⊂ N ), we get: T �,f = � �,j∈I �� � , �� � f , � j � � φ − H → I �id � I ⊥ ⊗ �a I , + � � �a I � j � � φ H → I = � �,j∈I �� � , �� � f , � j � ��a Fock , + � � �a Fock � j � (like in the proof of prop. 3.16)= �a Fock , + � �a Fock f . ow, from the definition of the creation and annihilation operators, we have: T �,f = ∞ � � =0 �Π ( � ) �a Fock , + � �a Fock f �Π ( � ) ,where, for all � ∈ N , �Π ( � ) is the orthogonal projection on the subspace H ⊗�, sym of � H , and:����Π ( � ) �a Fock , + � �a Fock f �Π ( � ) ��� � � ��� �f � .On the other hand, we have, by definition of � R , sup J ∈ L H Tr � ρ �(Π J ) L H � =: N tot < ∞ , so: ∞ � � =0 � Tr � H ρ Fock �Π ( � ) = � �∈ N � Tr � H ρ Fock I {�} � � � id H � Fock �= � �∈ N � sup J ∈ L H Tr � H ρ Fock I {�} � � � Π J � Fock � (using [11, lemma 2.10])= � �∈ N � sup J ∈ L H Tr � J ρ J I {�} � �(Π J ) J � = N tot (from eq. (3.17. ) ).Let � � >
0. Then, there exists N � � �>N � � Tr � H ρ Fock �Π ( � ) � � � ε = ( J , � ) ∈ E :� �>N � � ���� τ Fock ← J �Tr H → J � −�� � � H ε ρ Fock � �� � H ε � τ +Fock ← J � �Π ( � ) ��� == � �>N � � ��� τ Fock ← J �Tr H → J � −�� � � H ε ρ Fock � �� � H ε � �Π ( � ) J τ +Fock ← J ��� (where for all � ∈ N , �Π ( � ) J is the orthogonal projection on the subspace J ⊗�, sym of � J )= � �>N � � ����Tr H → J � −�� � � H ε ρ Fock � �� � H ε � �Π ( � ) J ��� = � �>N � � ���Tr H → J � −�� � � H ε ρ Fock � �� � H ε � φ − H → J �id � J ⊥ ⊗ �Π ( � ) J � � φ H → J ��� � � �>N � � ��� � −�� � � H ε ρ Fock � �� � H ε � φ − H → J �id � J ⊥ ⊗ �Π ( � ) J � � φ H → J ��� � � �>N � � � � � � ��� � −�� � � H ε ρ Fock � �� � H ε � φ − H → J ��Π ( � � ) J ⊥ ⊗ �Π ( � ) J � � φ H → J ��� � � �>N � � � � � ( � + � � ) ��� � −�� � � H ε ρ Fock � �� � H ε � φ − H → J ��Π ( � � ) J ⊥ ⊗ �Π ( � ) J � � φ H → J ��� � �>N � � � � � ( � + � � ) ��� � −�� � � H ε ρ Fock � φ − H → J ��Π ( � � ) J ⊥ ⊗ �Π ( � ) J � � φ H → J � �� � H ε ��� (for � H ε stabilizes the subspaces �� J ⊥ � ⊗� � ⊗ J ⊗� � sym for all �, � � ) � � �>N � � � � � ( � + � � ) ��� ρ Fock � φ − H → J ��Π ( � � ) J ⊥ ⊗ �Π ( � ) J � � φ H → J ��� = � �>N � � � � � ( � + � � ) Tr � H ρ Fock � φ − H → J ��Π ( � � ) J ⊥ ⊗ �Π ( � ) J � � φ H → J � � � �� >N � � �� Tr � H ρ Fock �Π ( � �� ) � � � ε � ∈ E such that, for all ε = ( J , � ) � ε � :��� τ Fock ← J �Tr H → J � −�� � � H ε ρ Fock � �� � H ε � τ +Fock ← J − � −�� � � H ρ Fock � �� � H ��� � � � N � ,thus:���Tr � H � τ Fock ← J �Tr H → J � −�� � � H ε ρ Fock � �� � H ε � τ +Fock ← J � T �,f − Tr � H � −�� � � H ρ Fock � �� � H T �,f ��� � ���� τ Fock ← J �Tr H → J � −�� � � H ε ρ Fock � �� � H ε � τ +Fock ← J � − � −�� � � H ρ Fock � �� � H ��� ×× ����� � � � N � �Π ( � ) �a Fock , + � �a Fock f �Π ( � ) ����� + � �>N � ���� τ Fock ← J �Tr H → J � −�� � � H ε ρ Fock � �� � H ε � τ +Fock ← J � �Π ( � ) ��� ×× ����Π ( � ) �a Fock , + � �a Fock f �Π ( � ) ��� + � �>N � ��� � −�� � � H ρ Fock � �� � H �Π ( � ) ��� ����Π ( � ) �a Fock , + � �a Fock f �Π ( � ) ��� � � � N � N � ��� �f � + � � ��� �f � + � �>N � ��� � −�� � � H ρ Fock �Π ( � ) � �� � H ��� � ��� �f � (for � �� � H stabilizes the subspaces H ⊗�, sym for all � ) � � � ��� �f � ,which concludes the proof. � While it will be essential to play with more toy models (and especially with more sophisticatedones), in order to sharpen our still rather crude proposal for dealing with constraints, we have atleast ascertained that this program can be applied to the most simple quantum field theory, whereit satisfactorily reproduces established results. Indeed, we found that we can define a sensibleconvergence at the quantum level, on a subspace of states that can either be identified with the ock space or with a subset of it. This is reassuring, for we know that the Fock space is theright arena to describe interaction-free theory (since such a theory preserves the subspaces of fixedparticles number). It would be interesting to study whether more general quantum field theoriescan be translated in this language too.On the classical side, we would like to develop systematic recipes to generate the input neededfor the regularization. On the quantum side, we still have to provide a rigorous procedure, includ-ing rules for defining an effective and physically meaningful notion of convergence. As a generalguiding principle, we should strive to reflect the concrete experimental implementation of the ob-servables. In particular, when considering a theory of gravity, it might prove legitimate to definethe convergence in a way that completely ignores the gravitational degrees of freedom: indeed,geometry is only probed by matter, and never measured directly.Additionally, we might be able to gain a deeper understanding of the formalism considered hereby studying its relations to approaches that incorporate similar ingredients, like lattice quantumfield theory or other discretization techniques. This could help shed light on issues that are sharedwith these approaches, notably the problem of ‘universality’: in other words, the concern about howto ensure that the results we are getting are robust, and do not depend critically on some arbitrarychoices entering the definition of the regularization scheme. We have displayed in section 2 atrick to circumvent this pitfall: by assembling all reasonable approximations into a huge label set E , and ordering them by their respective quality, we can view a specific regularization prescriptionas simply selecting a cofinal subset in E . However, it is not clear whether this could still be donefor less trivial systems, because it could become difficult to arrange for E to be directed. Hence, wewill probably need to invent more subtle ways of ensuring universal properties. Acknowledgements
This work has been financially supported by the Université François Rabelais, Tours, France.This research project has been supported by funds to Emerging Field Project “Quantum Geometry”from the FAU Erlangen-Nuernberg within its Emerging Fields Initiative.
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