Projective Loop Quantum Gravity

II. Searching for1,2 Semi-Classical States1 Suzanne Lanéry and Thomas Thiemann 1 Institute for Quantum Gravity, Friedrich-Alexander University Erlangen-Nürnberg, Germany 2 Mathematics and Theoretical Physics Laboratory, François-Rabelais University of Tours, France

October 7, 2015

Abstract In [13] an extension of the Ashtekar-Lewandowski (AL) state space [2] of Loop Quantum Gravity was set up with the help a projective formalism introduced by Kijowski [12, 21, 15]. The motivation for this work was to achieve a more balanced treatment of the position and momentum variables (aka. holonomies and fluxes). Indeed, states in the AL Hilbert spaces describe discrete quantum excitations on top of a vacuum which is an eigenstate of the flux variables (a ‘no-geometry’ state): in such states, most holonomies are totally spread, making it difficult to approximate a smooth, classical 4-geometry. However, going beyond the AL sector does not fully resolve this difficulty: one uncovers a deeper issue hindering the constructionSU of states semi-classical with respect to a full set of observables. In the present article, we analyze this issue in the case of real-valued holonomies (we will briefly comment on the heuristic implications for other gauge groups, eg. (2) ). Specifically, we sho w that, in this case, there does not exist any state on the holonomy-flux algebra in which the variances of the holonomies and fluxes observables would all be finite, let alone small. It is important to note that this obstruction cannot be bypassed by further enlarging the quantum state space, for it arises from the structure of the algebra itself: as there are too many (uncountably many) non-vanishing commutators between the holonomy and flux operators, the corresponding Heisenberg inequalities force the quantum uncertainties to blow up uncontrollably. A way out would be to suitably restrict the algebra of observables. In a companion paper [16] we take the first steps in this direction by developing a general framework to perform such a restriction without giving up the universality and diffeomorphism invariance of the theory. arXiv:1510.01925v1 [gr-qc] 7 Oct 2015 Contents 1 Introduction 2

2 Obstructions to the constructionR of narrow states 2 2.1 Projective families of characteristic functions ...... 3 2.2 A no-go result in theG= case ...... 13

3 Outlook 26

A Appendix: Wigner characteristic functions 28 A.1 Weakly continuous representations of the Weyl algebra ...... 28 A.2 The Wigner-Weyl transform ...... 35

B References 1 49 1This Introduction work is set up in the context of a projective formalism for Quantum Field Theory, first introduced by Jerzy Kijowski in the late ‘70s [12] and further developed by Andrzej Okołów more recently [20, 21, 22]. Instead of describing quantum states as density matrices over a single big Hilbert space, one constructs them as projective families of density matrices over small ‘building block’ Hilbert spaces [15, subsection 2.1]. The projections are given as partial traces, so that each small Hilbert space can be physically interpreted as selecting out specific degrees of freedom.

The description, within this projective framework, of a theory of Abelian connections had been developed by Okołów in [21, section 5] and [22] , an important insight being to use ‘building blocks’ labeled by combinations of edges and surfaces (instead of edges only as in the case of the Ashtekar-Lewandowski (AL) Hilbert space). This construction was then generalized in [13] to an arbitrary gauge group : in particular, is neither assumed to be Abelian nor compact, because the factorization maps, which are the central objects of the formalism, can be expressed explicitly in terms of the groupGoperations [13, subsecG tion 3.1]. The motivation to set up such a projective state space for the holonomy-flux algebra was to obtain better semi-classical states. As argued in the introduction of [13] , the AL Hilbert space, being built out of a vacuum which is a momentum eigenstate (with vanishing fluxes), cannot accommodate states in which the quantum fluctuations would be balanced between position and momentum variables. However, now that this limitation could be overcome, we uncover a deeper issue, which has its root not in the restriction to a particular representation of the algebra of observables, but in the algebra itself.

As underlined in [13, subsection 3.2] the holonomy and flux variables do not come in independent, canonically conjugate pairs. In fact, any given flux observable has non trivial commutators with infinitely, uncountably many holonomies. As we will demonstrate below in the R case, the sheer amount of Heisenberg uncertainty relations arising from these non-vanishing commutators forces the quantum fluctuations to blow up uncontrollably as more and more degreesG= of freedom are taken into account: in the end, it is not even possible to keep the variance of all elementary variables finite, let alone small. Note that obstructions to the design of states with specific properties on the holonomy-flux algebra have been pointed out in earlier works [24] : in particular, the AL vacuum turns out to be the only diffeomorphism-invariant state [18] , which notably forbids the design of a diffeomorphism-invariant coherent state.

These issues can be traced back to the fact that holonomy-flux algebra along analytical (or semi- analytical) edges and surfaces is generated by uncountably many elementary observables, which motivates the construction we will present in a companion article [16] : we will explore a strategy to drastically reduce the algebra of observables, without denaturing the physical content of the theory. Such a reduction then allows the systematic construction of projective states, in particular semi-classical ones. As a proof of principle, this strategy can easily be implemented in the case of a (slightly simplified) one-dimensional version of the holonomy-flux algebra, while the generalization to higher dimensions, especially the physically relevant case, is currently a work in progress.

�=3

2 2We Obstructions will, in subsection to2.1, the work co innstruction the context of a ofgeneral narrow linear projective states system L M : namely, a projective system such that all small phase spaces M are linear and across which the distinction between configuration and momentum variables can be defined consisten( tly, (ie., π all) η projections preserve this distinction). Note that this is precisely the setup of [21] and we recalled in [15, subsection 3.1] how to quantize such a classical projective limit into a projective system of quantum state space in the position representation.

In projective quantum state spaces of this kind, we can systematically study the construction of narrow states, namely states in which the measurement probabilities of all configuration and momentum variables have finite variance. In particular, we will spell out the conditions, arising from the Heisenberg uncertainty relations, for a certain assignment of variances to be realizable in a quantum state. Of special interest for the construction of semi-classical states are the Gaussian states, and, in fact, we will see that there exists, for any narrow state, a Gaussian state with the same variances.

In subsection 2.2, we will apply this study to the projective quantum state space set up in [13] , in the special case R (where it constitutes a linear projective system in the sense above, as will be shown in prop. 2.13). We will then be able to prove (prop. 2.14) that there are not any narrow states in this stateG= space: in other words, all states therein have infinite variance for at least some of the variables, and thus are not good candidates as semi-classical states. We will also formulate some weaker notions of semi-classicality that are excluded as well (props. 2.15 and 2.16).

2.1 Projective families of characteristic functions The central tool of the present subsection will be Wigner characteristic functions (see [29, 11] and the review in appendix A): any quantum state can be represented by a function defined on the dual of the classical phase space, and this alternative representation is strictly equivalent to its representation as a density matrix . This is the quantum version of characteristic distributions in classical statistical physics, in the sense that the moments of the probability distributions gov- erning quantum measurements can beρ recovered from the derivatives of the characteristic function (prop. A.8). However, the positivity requirement for quantum characteristic functions (eq. (A.6.1), ensuring the positivity of ) differs from its classical counterpart (that would ensure positivity of the corresponding probability distribution), as it encodes the non-commutation of quantum observables (see the discussion precedingρ prop. A.6).

Unsurprisingly, the characteristic functions of the partial density matrices composing a projec- tive state combine again into a projective family (the dual spaces M form naturally an inductive structure, with injections given by the pullback under the projections ∗ , so functions on them ar- η range themselves in a projective structure, see also the discussion at the� beginning of appendix A.2). η →η This is the advantage of working with Wigner characteristic functionsπ rather than Wigner quasi- probabilities, which would be in analogy to classical probability distributions (the computation of partial traces, or, at the classical level, of marginal probability distribution, is translated into a

3 simple restriction when working with characteristic functions).

L M L ↓ C P PropositionP 2.1C Let ( ,C , π ) be a projectiveC system of phase spaces such that:M C P 1. η η ∀η ∈ , there∗ exists two∗ finite-d imensional real vectorC spacesP , , an invertible linear map η η η η η η η η η Ξ : → (where denotes the dual of ) and a symplectomorphismL : → × , C P η η η with respect to the symplectic structure Ω defined on × by: � � � � � � η η η η η ∀(��, �),(�L, � )∈ × ,Ω (�, �;� , � ) := Ξ (� )(�)−Ξ (�)(� ); C C P � P � −1 � � � � � � 2. η →η η η →η η →η η η →η η →η η η ∀η η �∈ , π =L ◦(Q ×P )◦L withQ , resp.P , a linear map → , η η L C � L resp. → . � C η η We� choose, for any� η∈ , a Lebesgue measureµ on . For anyη η ∈ , we define: 3. η →η C ηC→η C := KerQ ; C C C � � � η →η η η →η η � � � � 4. φ : � → � ×� � � η →η η η →η η →η η →η C η →η , withR : → the projection on parallel � �→ R (� ),Q (� ) � −�1 ∗ ∗ � η η →η η � η to Ξ ◦ P ◦Ξ � � (where (·) denotes the dual map). L C L H Then, we can× complete these elements into a factorizing system of measured manifolds⊗ [15, def. 3.1] ( ,( , µ), φ ) , from which a projective system of quantum state spaces ( , ,Φ ) can be con- structed as described in [15, prop. 3.3]. C

Proof Since a finite-dimensional vector space is in particular an additive simp� ly-connected� Lie η →η η →η group, this is a special case of [15, theorem 3.2]. The explicit expressions for andφ are� obtained by adapting the ones from the proof of [15, theorem 3.2] to the slightly different notations we are using here. L Definition 2.2 We consider the same objects as in prop. 2.1. A projective family of characteristic L � η� C P C functions is a family W η∈ such that: ∗ ∗ L η N η η C P C 1. for anyη∈ ,W is a continuous function × → ; ∗ ∗ 1 1 N N η η 1 N 2. for anyη∈ , anyN∈ , any (� , � ), � � � ,(� , � )∈ × and anyz , � � � , z ∈ : N � �� � � ξ � � η � � � � 1 �,�=1 z z � W (� − � , � − � ) 0, (2.2. )

1 −1 −1 ��� 2 � �L� η � � � � η � �� whereξ := � Ξ (� ) − � Ξ (� ) ; � � ∗� ∗� W η η � η →η η →η� 3. for anyη η ∈L ,CWP =W ◦ Q × P . ↓ ( ,( , ),(Q,P)) L H C We denote by the space of all projective families of characteristic functions. C P H η 2 η η Proposition 2.3 We consider∗ the∗ same objects as in prop. 2.1. Letη∈ . On =L ( , �µ ) H C η η η η we define, for any (�, �)∈ × , a unitary operator T (�, �) on by: � −1 ∗,−1 η η η � � η �� � η � ∀ψ ∈ , ∀� ∈ ,[T (�, �)ψ](�) = exp � �(�)+ 2 � Ξ (�) ψ �+Ξ (�) , 4 D C C C η η as∞ well as a densely defined, essentially self-adjoint operator X (�, �) with dense domain := � η D C η C ( , ) (the space of smooth, co mpactly supported, complex-valued functions on ) by: ∗,−1 η η η � � � � η � ∀ψ ∈ , ∀� ∈ ,[X (�, �)ψ](�)=�(�)ψ(�)−� T ψ Ξ (�) ∗,−1 ∗ −1 −1 ∗ � � L C P η � η� � η � (whereT ψ denotes the differential ofψ at� and Ξ = Ξ = Ξ ). � ∗ ∗ η η For anyη η ∈ and any (H�, �)∈ × , we have: � � � −�1 � � η�→η � & η η →η η η →η T (� , � ) = Φ ◦ id H ⊗T (�, �) ◦Φ ,

� � � −�1 � � η�→η � η η →η η η →η X (� , � ) = Φ ◦ id ⊗X (�, �) ◦Φ , (2.3.1) � � ∗� ∗� L � η →η η →ηC � P where (� , � ) := Q (�),P (�) . ∗ ∗ η η η η Proof Letη∈ . For any (�, �)∈ × , it has been proven in prop. A.5, that T (�, �), resp. X (�, �), � L C P H is well-defined and is unitary, resp. essentially self-adjoint. �H C ∗ ∗ � � ∗� ∗� η η � η →η η →η � η η Letη� η ∈ � and (�, �)∈� ×� . Let (� , � ) := Q (�),P (�) . For� anyψ ∈ and η →η η →η 2 η →η η →η η →η anyψ ∈ C =LC ( , �µ ), we have, using the expression for Φ from [15, prop. 3.3]: � � � � � � −�1 � � η →η η η →η η η →η η →η η ∀(�, �)∈ × , Φ T (� , � )Φ (ψ ⊗ ψ ) (�, �)= � � � � −�1 � � −� 1 η η →η η →η η η →η = T (� , � )Φ (ψ ⊗ ψ ) ◦ φ (�, �) � � −� 1 � � −�1 � � −�1 � � −� 1 ∗,−� 1 � � η →η � η � η →η η →η η η →η η = exp � � ◦φ (�, �)+ 2 � Ξ (� ) Φ (ψ ⊗ψ ) φ (�, �)+Ξ (� ) � � −1 � � ∗,−1 � η � η →η η � η � = exp � �(�)+ 2 � Ξ (�) ψ (�)ψ �+Ξ (�) � η →η � η η� =ψ (�) T (�, �)ψ (�), C C

where we have used:� η →η η ∀(�, �)∈ × , � −� 1 & η →η η →η Q ◦ φ (�, �)=�, � � −� 1 ∗,−� 1 ∗� � � � ∗,−� 1 ∗� � η →η η →η η η →η η →η η η →η φ φ (�, �)+Ξ ◦ P (�) = �, �+Q ◦Ξ ◦ P (�) � η →η (from the definition ofφ in prop. 2.1) as well as: � ∗,−� 1 ∗� ∗,−1 η →η η η →η η Q ◦Ξ ◦ P = Ξ , −1 � � � � η � η →η η →η� η η as follows from the compatibility ofL ◦ Q × P ◦ L with the symplectic structures Ω η H H and Ω (see [14, def. 2.1] and [15, eq. (3.2.1)] ). Thus, we get: � � � � � � −�1 � � η η η →η η →η η →η η η →η η →η η η →η � η η� ∀ψ ∈ , ∀ψ ∈ ,Φ T (� , � )Φ (ψ ⊗ ψ ) =ψ ⊗ T (�, �)ψ , and therefore: H � � � −�1 � � η�→η � η η →η η η →η T (� , � ) = Φ ◦ id ⊗TD(�, �) ◦Φ . D C C

� � ∞ � −�1 � η η η →η η →η � η →η η →η η →η Similarly, we have, for anyψ ∈ and anyψ ∈ :=C ( , ), Φ (ψ ⊗ 5 D

� η η ψ )∈ and: � � � � −�1 � � η →η η η →η η →η η η →η � η η� Φ X (� , � )Φ (ψ ⊗ ψ ) =ψ ⊗ X (�, �)ψ , C C using: � η →η η ∀(�, �)∈ × , � −1 −�1 � � � � � � D φη�→ηD(�,�) η →η � η →η η� η � � η →η� η →η η →η � η η →η T Φ ψ ⊗ ψ =ψ (�) T ψ ◦ R +ψ (�)[T ψ ]◦Q . � η →η η D H Moreover, ⊗ (understood as a tensor product of vectorD spaces,D ie. without Dany completion) � � � � � � −�1 � η →η η η →η η � η →η � η � η →η� � η�→η η � η�→η is a dense subspace of Φ � � and Φ X (� , � )Φ � ⊗ = id � ⊗X (�, �) is H essentially self-adjoint (as a tensor product of essentially self-adjoint operators, see [23, theorem � � � � −�1 � η →η η η →η η →η η � VIII.33] ), hence the unique self-adjoint extensions of Φ X (� , � )Φ and id ⊗X (�, �) L coincide. S S H Proposition 2.4 We consider the same objects as in def. 2.2 and prop. 2.3. For anyη∈ and any η η η C η ρη ρ ∈ (withC Pthe space of non-negative, traceclass operators on ), we defineW by:

η ∗ ∗ H ρ η η W : × → η η η . S W S (�, �) L�→ Tr ρ TL(�, �) L H L C P L H ⊗ ⊗ η ↓ � η� � ρ � ( , ,Φ) ( ,( , ),(Q,P) ) ( , ,Φ) The mapW: ρ η∈ �→ W η∈ is a bijection → (where has been L W defined in [15, def. 2.2] ). C P η S W Proof For anyη∈ , we denote by the space of all continuous functions of positive type on ∗ ∗ η η η η η ρ η η × (prop. A.6). From props. A.7 and A.10, the mapW :ρ �→ W is a bijection → . S S & Moreover,L H we have: L � H ⊗ � � � −�1 � � η�→η � ( , ,Φ) � η�η∈ � η η η � η →η η η →η� = ρ � ∀η, ρ ∈ ∀η η , ρ = Tr Φ ρ Φ , W W & and: L C P L � ↓ � � � � ∗� ∗� � ( ,( , ),(Q,P) ) � η�η∈ � η η η η � η →η η →η� = W � ∀η, W� ∈ ∀η η S,W =W ◦ Q × P . � � � η η Now, from eq.H (2.3.1), we have, for anyη η and anyρ ∈ : � � −�1 � � ∗� ∗� � η�→η � η � η →η η η →η� η η � η →η η →η� W Tr Φ ρ Φ S=LWH (ρ )W◦ LQC P × P . � ⊗ ↓ ( , ,Φ) ( ,( , ),(Q,P) ) Thus,W is well-defined as a ma p → and is bijective. Asking for a state to have finite variances in all position and configuration variables is expressed at the level of its Wigner characteristic function by asking the latter to be twice differentiable at , and the corresponding covariance matrix can be obtained from the Hessian of the character- istic function (prop. A.8). In particular, the positivity requirement for the characteristic function (eq.0 (A.6.1) ) gives rise to inequalities that have to be satisfied by the covariance matrix (def. 2.6.2): these are nothing but the Heisenberg uncertainty relations (as is manifest from the rewriting in prop. 2.7.2 or 2.7.3, since and ). These inequalities will play a 2 2 η η V (�, �) = ΔXη(�, 0) U (�, �)=ΔXη(0, �) 6 crucial role for the result of the next subsection.

Moreover, the projective structure binding the partial characteristic functions of a projective state together goes down to its covariance matrices on the various labels , which arrange naturally into their own projective structure. η

L S L H L ⊗ � η� L ( , ,Φ) � η� C P Definition 2.5 Letρ= ρ η∈ ∈ and let W η∈ :=W(ρ). We say thatρ is a narrow (1) (2) ∗ ∗ η η η η state if there exist, for anyη∈ , a linear formW and a symmetric bilinear formW on × such that: C P 2 ∗ ∗ (1) (2) 2 η η η η τ η ∀(�, �)∈ × ,W (τ�, τ�)=1+�τW (�,L �)− W (�, �; �, �)+�(τ ) . (2.5.1) 2 H H From props. A.5 and A.8, we then have, for anyη∈ : η η C P η η 1. Tr ρ = 1 (so thatρH is a density matrix on ); ∗ ∗ η (1) 2. η η η η η ∀(�, �)∈ × C , Tr P ρ X H(�, �)=W (�, �); � � � � η η η η η η � � ∗ ∗ η (2) � � 3. η η X (�, �)ρ X (� , � ) + X (� , � )ρ X (�, �) η ∀(�, �),(� , � )∈ × , Tr S L H =W (�, �;� , � ). 2 ⊗ ( , ,Φ) We denote the space of all narrow states by ˆ . L Definition 2.6 We consider the same objects as in prop. 2.1. A projective family of variances is a � η ηL� C P family V ,U η∈ such that: ∗ ∗ L η η C P η η 1. for anyη∈ ,V , resp.U , is a strictly positive symmetric bilinear form�on , resp. ; ∗ ∗ −1 L η η η & η � η � 2. for anyη∈� and any (�, �)∈ × ,V (�, �) +U (�, �)−� Ξ (�) 0; � � ∗� ∗� � ∗� ∗� V η η η →η η →η η η η →η η →η 3. for anyη η ∈L C,VP =V ◦(Q × Q ) U =U ◦(P × P ). ↓ ( ,( , )L,(Q,P) ) We denote by the space of all projective families of variances.

C P η η C P Proposition∗ ∗ 2.7 Let−η∈1 and− let1 V , resp.U , be a strictly positive symmetric bilinear form on η η η η η η , resp. . LetV , resp.U , be the strictly positive symmetric bilinear form on , resp. , C characterized by: � ∗ −1 � � P η η � η η � η ∀�, � ∈ ,V V (�, ·),V (� ,·) =V (�, � ), � ∗ −1 � � η η � η C Cη � P η P resp. ∀�, � ∈ ,U U (�, ·),U (� ,·) =U (�, � ), ∗∗ ∗∗ η η η η with the canonical identification ≈ , resp. ≈ . C P � Then, the three following conditions are equivalent: ∗ ∗ −1 1. η η η η � η � ∀(�, �)C∈ × ,V (�, �) +U (�, �)−� Ξ �(�) 0; ∗ −1 −1 −1 η η 1 η � η η � 2. ∀� ∈ ,2V (�, �)− U Ξ (�),Ξ (�) 0; P 2 � ∗ −1 ∗,−1 ∗,−1 η η 1 η � η η � 3. ∀� ∈ ,2U (�, �)− V Ξ (�),Ξ (�) 0. 2 7 C C

∗ η η η Proof LetCV� be the linear mapC → C defined by: ∗ ∗∗ η η η η η ∀� ∈ , V� (�) =V (�, ·)∈ ≈ .

� η � η C C η η For any��= 0, we have� V� (�) =V (�, �)> 0 (forV is strictly positive), hence V� (�)�= 0. Thus, ∗ �η η η V is injective, and− therefore1 bijective, since and are finite-dimensional vector spaces of the η same dimension.C V is then defined by: � −1 � −1 −1 � η η η� η η � ∀�, � ∈ ,V (�, � ) :=V V� (�), V� (� ) , C P P P P R ∗ η �η η η and is− therefore1 a strictly positive symmetric bilinear form on . We have similarly U : → η η η andU : × &→ . C � 1 −1 1 2 1 3 1 2 �η 2.7.−1 ⇒ 2.7. 2.7.∗ ⇒ 2.7. . We assume that 2.7. holds. Applying with (�, �)= �, U ◦ η � η Ξ (�) for some�∈ yields: � −1 −1 −1 −1 −1 −1 η 1 η � η η � 1 η � η η � V (�, �) + U Ξ (�),Ξ (�) − U Ξ (�),Ξ (�) 0, 4 2 P where we have used: � −1 � −1 −1 � −1 � η � η � η� η η � η ∀�, � ∈ , U� (�) (� ) =U U� (�), U� (� ) =U (�, � ). & Thus, we obtain the condition 2.7.2 . Similarly, we can prove that 2.7.1 implies 2.7.3 . C 2 1 3 1 2 η 2.7.∗ ⇒ 2.7. 2.7. ⇒ 2.7. . We assumeC that−1 2.7.−1holds.V−1 provides a scalar product on η η � η η � , so the strictly positive symmetric bilinear formU Ξ (·),Ξ (·) can be diagonalized in a η � �∈{1,���,�} & η V -orthonormal basis (� ) (with� := dim ), ie. we have: −1 −1 −1 η � � �� η � η � η � � (�) �� ∀�, � ∈ {1,���,�}, V (� , � ) =δ U Ξ (� ),Ξ (� ) =λ δ , (�) � with ∀� ∈ {1, � � � , �} , λ > 0. Next, the condition 2.7.2 can be rewritten: (�) C ∀� ∈ {1, � � � , �} , λ 4. ∗ � η � � �∈{1,���,�} η Moreover, defining, for any�∈{1, � � � , �},� = Ξ (� ) with (� ) the basis in dual to � �∈{1,���,�} (� ) , we have: −1 −1 −1 (�) �� η � η � η � � η� (�) � (�) � � ∀�, � ∈ {1,���,�}, λ δ =U Ξ (� ),Ξ (� ) =U λ � , λ � −1 −1 η η � (�) � where we have used that, for any�∈{1, � � � , �}, U� ◦Ξ (� ) =λ � . Hence, we get: �� η� � � � δ ∀�, � ∈ {1,���,�},C U � , � P= (�) . λ � ∗ � ∗ � η � η Let�=� � ∈ and�=� � ∈ (with implicit summation). We have: � � −1 � � � � η η � η � � � V (�, �) +U (�,R �)−� Ξ (�) =� � + (�) − � � . λ Now, for anyσ,τ∈ , and anyλ∈ ]0, 4], we have:

8 � 2 2 τ σ + − στ 0, � λ 1 S 3 1 L so 2.7. is fulfilled. Similarly, weL can proveL H that 2.7. impliesL 2.7. . ⊗ � η� ( , ,Φ) � η� Proposition 2.8 LetC ρ= ρ η∈ ∈ ˆ and let W η∈ :=W(ρ). For anyη∈ , we define: � ∗ � (2) � (1) (1) � & P η η η η η ∀(�, � )∈ ,V (�, � ) :=W (�, 0;� , 0)−W (�, 0)W (� , 0) , � ∗ � (2) � (1) (1) � η η η η η ∀(�, � )∈ ,U (�, � ) :=W (0, �;0, � )−W (0, �)W (0, � ), (1) (2) η η V withW andW Las in def.L C 2.5.P ↓ � η η�η∈ ( ,( , ),(Q,P) ) Then, VS L,UH V∈ L C P . Accordingly, we define the mapV as: ⊗ ↓ ( , ,Φ) ( ,( , ),(Q,P)L) V: ˆ → � η η�η∈ . L ρ �→ CV ,U P R

∗ ∗ η η 1 Proof Letη∈ ,(�, �)∈ × , andτ∈ (1). Applying eq.(1) (2.2. ) for the points (0, 0),(τ�, 0), (0,τ�) −�τ Wη (�,0) −�τ Wη (0,�)

with respective coefficients−(1) (1 +�), � , � � (1) yields: −�τ Wη (�,0) −�τ Wη (0,�) � η η 4 + 2 Re (� − 1)� W (τ�, 0)− (1 +�)� � W (0, τ�)+ (1) τ2 −1 −�τ Wη (−�,�) � 2 �◦Ξη (�) η � +�� � W (−τ�, τ�) 0, C P � � ∗ ∗ � � � � η η η η where we have used thatW (0, 0) = 1 and that, for any (� , � )∈ × ,W (−� , −� ) = W(� , � ) C P (1) (2) η η (prop.� � A.6;∗ note∗ that this in particular accounts for the reality ofW andW ). Now, for any η η (� , � )∈ × , we have the expansion: (1) � � 2 2 −�τ Wη (� ,� ) � � (2) � � � � (1) � � 2 2 η τ η τ � η � � W (τ� , τ� ) = 1− W (� , � ;� , � ) + W (� , � ) +�(τ ). 2 2 Inserting in the previous inequality, we get: � 2 2 2 � −1 2 � τ η τ η τ η 4 + 2 −2 + V (�, �) + C U (�,P �)− � ◦Ξ (�) +�(τ ) 0. 2 2 2 ∗ ∗ η η Hence, we have, for any (�, �)∈ �× : −1 η η η V (�,C �) +U (�, �)−�◦Ξ (�) 0. P (2.8.1) ∗ −1 ∗ −1 η η � � η � η Let�∈ with��= 0. We have Ξ (�)�= 0, so there exists� ∈ such that� 1 ◦Ξ (�) = 1. � η � � � Applying eq. (2.8.1) with (0, � ) yields� :=U (� , � ) 0, allowing us to define� := 1+� � . Applying again eq. (2.8.1), now with� (�, �), we get:

η 1 V (�, �)− 2 0, P (1 +�) ∗ η η η soV (�, �)> 0. Similarly, we have, fo r any�∈ ,��= 0⇒U (�, �)> 0. Thus, def. 2.6. 1 and 2.6.2 � L are fulfilled (actually, we have shown that 2.6.1 is implied by 2.6.2). � � ∗� ∗� η η η →η η →η Letη η ∈ . FromW =W ◦(Q × P ) together with eq. (2.5.1) implies, for any 9 C P

∗ ∗ η η & (�, �)∈ × : (1) (1)� � � (2) (2)� � � � � η η η η W (�, �)=W (� , � ) W (�, �; �, �)=W (� , � ;� , � ), � � � ∗� ∗� � η →η η →η � with (� , � ) := Q (�),P (�) . Hence, def. 2.6.3 holds. The Heisenberg uncertainty relations that a projective family of covariance matrices need to satisfy if there exists a narrow quantum state with these covariances turn out to also be a sufficient condition for the existence of such a state. Indeed, we can construct a Gaussian operator combining the covariance matrix for the configuration variables with the one for the momentum variables, and the uncertainty relations are precisely what is required for this operator to be a positive semi- definite operator of unit trace, ie. a density matrix. Moreover, the projective conditions between the covariance matrices ensure that these Gaussian operators assemble into a projective quantum state. Although this is remarkably easy to check at the level of the characteristic functions (where the projections are simply restrictions so that the compatibility of the Gaussian states on various labels can be directly read out from the expression of their characteristic functions), it is instructive to understand how taking the partial trace of Gaussian states works, in particular how the form of factorization C C C conspires with the expression for the Gaussian states so that the correct projections� get� respectively applied on the covariance matrices for the positions and η η →η η momenta (in accordan≈ce with× the projection M M ).

� η η Note that this is the point where it is critical to→ be working on linear configuration spaces. While the Wigner transform machinery could be adapted, for example, to the case of a compact group [1] , the nice projection property of Gaussian distributions is what makes the construction of projective coherent states in the linear case much easier than in the case: even in the Geasiest – U – case, the equivalent of Gaussian states, namely Hall statesN [8] , do not have 2 � � such a nice behavior under partial trace. L G G= (1)

L V L C P S L H ↓ ⊗ � η η� ( ,( , ),(Q,P) ) ( , ,Φ) Proposition 2.9 LetL V ,U η∈ ∈ . Then, there existsρ∈ ˆ such that � η η� V(ρ) = V ,U η∈ , withV the map introduced in prop. 2.8. In other words, the mapV is L surjective. N � �∈{1,���,�} � �∈{1,���,�} � (�)� Proof Letη∈ . Let (� ) , (� ) and λ �∈{1,���,�} be as in the proof of prop. 2.7. For 1 � k1,���,k�

anyk , � � � , k ∈ C, we defineψ as: � 2 � (√� ) 2 λ � � − (�) � � 1 � � � � η k ,���,k 1 1/81 � k �1/4 η 1/4 k ∀�=� � ∈ N , ψ (�) := √ �=1 (�) � � � H (�) , α π λ k ! 2 λ k where forR anyk∈ ,H is thek-th Hermite polynomial: 2 k 2 k � −� k � ∀� ∈ ,H (�) := (−1) � k � , N H �� C � 1 � η η H � η 1 � k1,���,k� η andα is such that �µ (� � ) =α �� � � � �� . Then, for anyk , � � � , k ∈ ,ψ ∈ = 2 η η k1,���,k� η L ( , �µ ) with �ψ � = 1. Next, we define:

10 � ∞  � k  � (�) � � (�) � η � � 2 λ 2− λ k1,���,k� k1,���,k� ρ := 1 �  � (�) � (�)  |ψ � � ψ |. k ,���,k =0 �=1 2 + λ 2 + λ H (�) η Using that ∀� ∈ {1, � � � , �} , λ ∈ ]0, 4] (from the proof of prop. 2.7),ρ is a non-negative operator η on and we have: � � H ∞ � � k � � � (�) � � (�) � � 2 � �� � 1 � η η 1 � 2 λ 2− λ � k ,���,k �ρ � k1,���,k�=0 � �=1 � (�) � (�) � �ψ � = 1 , � 2 + λ 2 + λ � H 1 where�·� denotes the trace norm [23, theorem VI.20]. Hence,ρ is a (self-adjoint), positi ve semi- C η definite, traceclass operator on . Using Mehler formula [19], its kernel is given by: � � � � η ∀�=� � , �=� � ∈ , � ∞  � k  � (�) � � (�) � η � � 2 λ 2− λ k1,���,k� k1,���,k� ρ (�;�) := 1 �  � (�) � (�)  ψ (�) ψ (�) k ,���,k =0 �=1 2 + λ 2 + λ � � � � 2 � � � � � � 2 1 �/2 1 � +� 1 = η exp − + (�) (� − � ) α (2π) �=1 2 2 2λ � −1 � � ∗ ∗ � 1 �/2 1 η �+� �+� 1 η � η η � = ηC P exp − V , − U Ξ (� − �),Ξ (� − �) . α (2π) 2 2 2 2 ∗ ∗ H η η η Thus, for any (�, �)∈ × , we get, using the expression for T (�, �) from prop. 2.3: ρη η η η W (�, �) := Tr ρ T (�, �)

� � −1 1 �/2 η � � η � = η �µ (�) exp � �(�)+ � Ξ (�) + α (2π) 2 ∗,−1 ∗,−1 −1 � η η � � 1 η Ξ (�) Ξ (�) 1 η − V �+ , �+ − U (�, �) 2 2 2 2 � � 1 η 1 η = exp − V (�, �)− U (�,L �) . 2 � 2 � ∗� ∗� η η� S 3 H ρ ρ � η →ηL η →ηL H� From 2.6. , we have, for anyη η ∈ ,W =W ◦ Q × P ⊗ , hence, from the � � � −�1 η η →η � η →η Lη η →η� � η� ( , ,Φ) proof ofL prop. 2.4,ρ = Tr Φ ρ Φ . Soρ := ρ η∈ ∈ , and we have η � ρ �η∈ W =WC (ρ).P Moreover, for anyη∈ , we get the expansion: 2 2 ∗ ∗ η 2 η η ρ τ η τ η ∀(�, �)∈ S×L H ,W (τ�, τ�)=1− V (�, �)− U (�, �)+�(τ ). 2 2 ⊗ L ( , ,Φ) & Therefore,ρ∈ ˆ with: (1) (2) η η η η ∀η ∈ ,W (�, �)=0 W (�, �; �,L �)=V (�, �) +U (�, �). � η η� In particular, this impliesV(ρ)= V ,U η∈ .

11 � L H

� � � � −�1 H η η →η � η →η η η →η� Note. One can� also check directly that, for anyη η ∈ ,ρ = Tr Φ ρ Φ . Indeed, (η ) � � � −�1 η η →η � η →η η η →η� denoting byρC the integral kernel of Tr Φ ρ Φ , we have: � (η ) � � � −� 1 −� 1 η η η →η η � η →η η →η � ∀�, � ∈ , ρ C(�;�) = �µ C (z)ρ φ (z, �);φ (z, �) .

� � � ∗,−� 1 η η →η η →η η →η η Now, for any�,�∈ and anyz∈ , the definition ofφ together withQ ◦Ξ ◦ ∗� ∗,−1 η →η η P = Ξ (from the proof of prop. 2.3) yields: −� 1 � � η →η η →η η →η C C φ (z, �)=K (z)+J (�), � ∗,−� 1 ∗� ∗ � � � � η →η η η →η η η →η η →η η →η η withJ := Ξ ◦ P ◦Ξ andK the canonical injection of = KerQ in . So we get: � ∗� −� 1 −� 1 ∗� −� 1 −� 1 η � η � η →η η →η � η � η →η η →η �� U Ξ φ (z, �)−φ (z, �) ,Ξ φ (z, �)−φ (z, �) = � ∗� ∗ ∗� ∗ η � η →η η η →η η � =U P ◦Ξ (� − �),P ◦Ξ (� − �) ∗ ∗ η � η η � =U Ξ (� − �),Ξ (� − �) , as well as: � −� 1 −� 1 −� 1 −� 1 � −� 1 η →η η →η η →η η →η η φ (z, �)+φ (z, �) φ (z, �)+φ (z, �) V , = 2 2 � � � � (η ) � � (η ) � �� � (η ) � �� � � η η �+� η �+� �+� = Z� (z) (z) + 2 X� (z) + Y� , 2 2 2 & with: � � � (η ) ∗� −� 1 � (η ) ∗� −� 1 � (η ) ∗� −� 1 � η η →η η η →η η η →η η η →η η η →η η η →η X� :=K ◦ V� ◦ J , Y� :=J ◦ V� ◦ J C Z� :=K ◦ V� ◦ K , � � �η η →η 1 where V � is defined as in the proof of prop. 2.7. For anyz∈ \{0} , def. 2.6. implies: � (η ) � � � −� 1 � −� 1 � � η η η η →η η η →η Z� (z)C (z)=V C V� ◦ K (z), V� ◦ K (z) > 0, � (η ) � ∗� η η →η η →η hence Z� : → is invertible, and the above equation becomes: � −� 1 −� 1 −� 1 −� 1 � −� 1 η →η η →η η →η η →η η φ (z, �)+φ (z, �) φ (z, �)+φ (z, �) V , = 2 2 � � � � � � (η ) � (η ) −1 (η ) � ��� � (η ) −1 (η ) � �� η η η �+� η η �+� = Z� z+( Z� ) ◦ X� z+( Z� ) ◦ X� + � � � 2 � 2 �� (η ) (η ) ∗ (η ) −1 (η )� � �� � � η η η η �+� �+� + Y� −( X� ) ◦( Z� ) ◦ X� . 2 2 C & C On the other hand, using: � � � � � � � � � η →η η →η η →η η →η η η →η η →η η →η η →η η K ◦ R +J ◦ Q = id ,Q ◦ K = 0 Q ◦ J = id � −� 1 � � ∗� η →η η →η η η →η η η →η (as follows from the expressions forφ andφ ), together with V� =Q ◦ V� ◦ Q , we can check that:

12 C � � � � � (η ) (η ) ∗ (η ) −1 (η )� η η η η η η V� Y� −( X� ) ◦( Z� ) ◦ X� = id (2.9.1)

� � � ∗� ∗� η � η →η η →η� η � η →η η →η� (noting that V� is the lower right block of R ,Q ◦ V� ◦ R ,Q , which is the inverse ∗� ∗� −� 1 � � � η →η η →η� η � η →η η →η� of K ,J ◦ V� ◦ K ,J , eq. (2.9.1) can be visualized by performing a blockwise matrix inversion of the latter). Putting everything together and carrying out the Gaussian integration overz, we get:C � (η ) � −1 � � ∗ ∗ � η η 1 η �+� �+� 1 η � η η � ∀�, � ∈ , ρ (�;�)∝ exp − V , − U Ξ (� − �),Ξ (� − �) , H 2 2 2 2 � � � −�1 η →η � η →η η η →η� η so Tr H ΦH ρ Φ ∝ ρ , the proportionalityH factoH r being determined to be 1 from: � � � η � −�1 � η η →η � η →η η η →η� η η η � Tr Tr Φ ρ Φ = Tr ρ = 1 = Tr ρ .

R

2.2 A no-go result in theG= case To prove the advertised no-go result in the case of the holonomy-flux algebra with R (using the projective system set up in [13] ), it will be enough to concentrate on a certain (uncountable) subset of observables out of this algebra. So, we will choose an edge, that we will identifyG= with the line segment , and a continuous stack of surfaces intersecting this edge, one for each point in (see fig. 2.1). We will keep, for each surface, only its face looking at . Moreover, for all holonomies[0, include 1] d in the selected edge to have finite variance, it would be sufficient that all holonomies] 0, 1] ending at would have finite variance, since the holonomy between0 and can be expressed as a composition of the one between and and the one between and (hence� would have finite variance if 0those two had). � � � � 0 � 0 Thus, we will attach, to each point in , the holonomy starting at (and ending at ), as well as the flux that acts at the onset of this holonomy. It is manifest from the symplectic structure in prop. 2.10 that these pairs of variables� attac] 0,hed 1] to the various points in � are not independent0 canonically conjugate pairs: instead, the flux at some point acts on all holonomies that start at or above . As announced in section 1, these uncountably many non-zero] 0com, 1]mutators will play a decisive role in the proof. � � Now, suppose that it would be possible to construct a quantum state in which all those holonomies and fluxes would have finite variances. Then, the points in could be organized into countably many (overlapping) classes: a point would belong to the class indexed by some N if the variance of both the holonomy and flux attached to this point would] 0 be, 1] less than . The assumption of all those variances being finite would ensure that each point belongs at least toA one ∈ of those classes, so, as there are uncountably many points in , there should exist someA whose class K would be uncountable, hence infinite. ] 0, 1] A Finally given points in this infinite set K, we will, in the proof of lemma 2.12, combine the variances of the holonomies and fluxes attached to these points into a quadratic expression, that should, if all� these variances were bounded by a constant , be bounded independently of . However, we will show that, by virtue of the Heisenberg uncertainty relations, this quadratic A � 13 expression can be bounded below by a diverging expression of .

�

� In the present subsection, Σ will denote a finite-dimensional, analytic manifold (without boundary) L [17] and� its dimension (� 2). L (aux) Proposition 2.10 We define the label set as the set of all finite subsets of points in ] 0,1]: (aux)L L := {κ ⊂]0,1]|#κ < ∞}. (aux) (aux) 1 � We equip with the partial order⊂ (set inclusion). For anyκ= (� , � � � , � ) ∈ , with 1 C � R &P R � < � � � < � , we define: (aux) (aux) κ Pκ C := {�:κ→ } :={P:κ→ }, (aux) (aux) (aux),∗ κ κ κ as well asP the linear maCp Ξ : → given by: � �−1 (aux) (aux) (aux) κ κ κ � k k � k+1 k ∀P ∈ , ∀� ∈ ,Ξ (P)(�) := k=1 P(� )�(� )− k=1 P(� )�(� ).

(aux) (aux),−1 κ κ Ξ is invertible,C with Ξ such that: � � (aux),∗ (aux),−1 � � � � κ � κ � 1 if� � ∀� ∈M , ∀�C ∈ κ, ΞP (�) (�) =� � �→ . 0 else (aux) (aux) (aux) (aux) (aux) κ κ κ κ κ We equip :=L × with the symplectic structure Ω defined from Ξ as in prop. 2.1. � (aux) C C P P For anyκ⊂κ ∈ , we define:& (aux)� (aux)� (aux) (aux)� (aux)� (aux) κ →κ κ κ κ →κ κ κ Q : → P : → κ κ , � �→ �| L M P �→ P|

(aux)� (aux)� (aux)� (aux) (aux) (aux) ↓ κ →κ κ →κ κ →κ � L H� andπ =Q × P . Then, , , π is a projective⊗ system of phase spaces � (aux) (aux) (aux)� and fulfills prop. 2.1.1 and 2.1.2. We denote by , ,Φ the corresponding projective L system of quantum state spaces. (aux) L

Proof is a directed set, since any two finite(aux) subsets of ] 0, 1 ] are included(aux) in their union and 1 � 1 � κ this union is finite. Letκ= (� , � � � , � ) (aux)∈ ,−1 with� < � � � < � .Ξ(aux) being invertible can be κ κ checked from the expressL ion given for Ξ and thisM ensuresM that Ω is indeed a symplectic form (aka. an anti-symmetric, non-degenerate form). � (aux) (aux)� � κ →κ κ κ For anyκ⊂κ ∈ ,π is a surjective map → and is compatible with the symplectic structures, for we have: (aux),−1 (aux)� (aux)� ,−1 (aux)� ,∗ κ κ →κ κ Lκ →κ L M Ξ =P ◦Ξ ◦ Q . � �� (aux) (aux)�� (aux)� (aux)�� � (aux) (aux) (aux) κ →κ κ →κ κ →κ � � Moreover, for anyκ⊂κ ⊂ κ ∈ ,π =π ◦ π . Therefore, , , π is a projective system of phase spaces. Prop. 2.1.1 and 2.1.2 are fulfilled by construction with

14 L M �

(aux) κ κ V V ∀κ ∈ ,L = id . L C P ∅

↓ ↓ (aux) (aux) (aux) (aux) (aux) S S (aux) L H ∅ : ,( , ),(Q ,P ) Theorem 2.11 With the notations of def. 2.6 and prop. 2.10, = ( ) = . ⊗ ⊗ (aux) (aux) (aux) (aux) Hence, prop. 2.8K implies that ˆ := ˆ( , ,Φ ) = . V V K & LemmaK 2.12 Let be a countably infinite subset of ] 0, 1 ] and let�A> 0. We define: � ↓ � ↓ � [�] [�] [�] [�] � ( ,A) κ κ κ (aux) � & κ κ κ κ κ κ L:= (V ,U ) ∈ � ∀κ ⊂ , ∀� ∈ κ, V (h ,h ) A U (PV K,P )∅A , (aux) [�] [�] ↓ κ � � κ � � ( ,A) N with ∀κ ∈ , ∀� ∈ κ,h := � �→ �(�) P := P �→ P(�) . Then,V K = . K K ↓ κ κ κ ( ,A) Proof Proceeding by contradiction, we suppose that there exists (V ,U ) ∈ . Let�∈ . 1 � 1 � Since is infinite, there existκ= (� , � � � , � ) ⊂ with� < � � � < � . Then, we have: � � � [�k ] [�k ]� � [�k ] [�k ]� 1 � κ κ κ κ κ κ V h ,h +U P ,P 2A. � k=1 (�) � We define the� by� matrixM by: (�) [�k ] � (aux),−1 [��] � � k� κ κ � κ � 1 ifk � ∀k, � ∈ {1,���,�}, M := P Ξ h = . 0 else (�) (�)

Performing�(�) a singular value decomposition(�) ofM there exist two� by� orthogonal matricesO andO , and a diagonal matrix Δ with non-negative real entries, such that: (�) �(�)� (�) (�) C M =O Δ O , P � (aux),∗ k κ where (aux)( · ),∗ denotes the transpose matrix. Defining, for anyk∈{1, � � � , �},� ∈ and k κ � ∈ by: & � � (�) [��] �(�) [��] k � k� κ k � k� κ � := �=1 O h � := �=1 O P ,

we get: � � � � [��] [��]� � [��] [��]� 1 � κ k k κ k k 1 � κ κ κ κ κ κ V (� , � ) +U (� , � ) = V h ,h +U P ,P 2A. � k=1 � �=1

On the other hand, from def. 2.6.2, we have: � � � (aux),−1 1 � κ k k κ k k 1 � k � κ k � V (� , � ) +U (� , � ) � Ξ (� ) � k=1 � k=1 �(�) (�) (�)� (�) 1 � � 1 1 √ (�)� (�) = Tr O M O = Tr Δ = Tr M M . � � � (�) (�)� (�) 1 Now,N := 2 M M is given by: � � min(k,�) (�) (�) (�) � � k� 1 � �k �� 1 � 1 k � ∀k, � ∈ {1,���,�}, N = 2 M M = 2 1 = min , , � �=1 � �=1 � � � 15 � J √ (�) and the previous inequalities requiresN Tr N 2A. 2 � [0,1]� [0,1] On the complex Hilbert space :=L [0, 1] , �µ (with[�] �µ the usual, normalized measure

on [0, 1] ),J we define, for any�∈ , a bounded operatorN � by: � � [�] � � � � � � � � �−1 ∀ζ ∈ , ∀� ∈ [0, 1], N ζ (�) := min [�] , � �� ζ(�) �=1 � 1 � � � �� = 0 �� min [�] ,[�] ζ(�), J

[�] � 1 � with [�] := �� �� and�·� the ceiling function. For any�∈{1, � � � , �}, we define� ∈ by: � [�] (�) � √ � ���� ∀� ∈ [0, 1],� (�) := � O . J (�) 2 � [�] � [��] � � (�) (�)� � � (�) � � � �� Δ We have � � � =δ and, usingN =O O : � � (�) 2 [�] � �� � [�] [�] � Δ � � � � � � N = �� � � . �=1 � J [∞] Now, weJ define a bounded operatorN on by: 1 [∞] � � � ∀ζ ∈ , ∀� ∈ [0, 1], N ζ (�)= 0 �� min (�, �) ζ(�) � 1 � � N = 0 ��J � ζ(�) +� � �� ζ(�), [∞] � and, for any�∈ \{0}, we define� ∈ by: [∞] �� � � � √ 1 ∀� ∈ [0, 1],� (�) := 2 sin � − π � . J 2 � [∞] � [∞� ]� � � � � �� We have � � � =δ and:

[∞] [∞] [∞] � 4 � N � = 2 2 � . π (2� − 1) N ∞ � 2 Since =∞, there existsM∈ \{0} such that: �=1 π (2� − 1) M � 2 �=1N >2A+1. π (2� − 1) N 2 3 [�] [�] J � � � � � � Let�∈ such that�>8π M and complete � �∈{1,���,�} into an orthonormal basis � �∈ \{0} of . For any�∈{1, � � � , M}� , we have: � [∞] [∞] [�] [∞] [∞] [�] [∞] � � � � � � � � � 1 �N � − N � � �N − N � �� � . � 16 But we also have: ∞ 2 2 J 2 2 � 2 (�) � � [�] � [∞] [∞] [�] [∞] � (∞) � � � � � � � [∞]� � � � � 14 �� �� � � � � � � � � � �N � − N � � = � � Δ − Δ � � � � � � � � =1 � � � �

(∞) � (��) � �� 2� � � where Δ := and, for any� > �,Δ := 0. Thus, we get: π (2� − 1) N ∞ � 2 2 J 2 � � (∞) 2 (��) � �� � [��] [∞] � � � � � � � � 14 �� �� � � � � � � � � � 12 � ∈inf\{0} � Δ − Δ � � � � � � � , � � � � =1 � � �

and therefore:N � 2 2 � (∞) � (��) � � � � �� �� � � � � � ∈ \{0} � � inf � Δ − Δ � �. N

Hence, for any�∈{1, � � � , M} , there exists �� ∈ \{0}, such that: � � (�) 2� � � � � � �� � � 2 4 2 Δ � 2 � − � . �π (2�� − 1) � � � (�) 2 3 �� Using Δ 0 and�>8π M , this implies: � (�) � (�) 2� (�) −1 � � � � � � � � � �� � � �� � � �� � � 2 Δ � � 2 4 2 Δ � � 2 Δ � 1 2 � − � = � − � � + � . �π (2� − 1) � � �π (2� − 1) � � �π (2� − 1) � � 4πM � On the other hand, for any�� > ��, we have: � (��) � � � � � � � 2 Δ � 1 1 2 � − � , ��π (2� − 1) � � π M π M � so �� �. Next, for any��=� �∈ {1, � � � , M}, the inequality: � � � 2 2 � 1 � − � � 2 �π (2� − 1) π (2� − 1)� π M � holds, so �� �= �� . Therefore, we obtain: � (�) � M (�) � M �� �� � Δ � Δ � 2 1 �=1 �=1 �=1 − >2A, � � � π (2� − 1) 4πM (�) V 1 κ κ κ inK contradiction with Tr Δ 2A , whence the initial assu mption on the existence of (V ,U ) �∈ ↓ � ( ,A) is proven wrong. V Proof of theorem 2.11 Again, we proceed by contradiction and we suppose that there exists ↓ [�] � [�] [�]� [�] � [�] [�]� κ κ κ L (aux) L κ κ κ κ κ L κ (V ,U ) ∈ . For any�∈]0, 1 ] , we defineV :=V h ,h andU :=U P ,P (aux) � (aux) � � � (aux) for someκ∈ such that�∈κ. Ifκ ∈ is such that�∈κ , thenκ,κ ⊂ κ ∪ κ ∈ and:

17 � � [��] [��]� � � (aux)�,∗ � [��] (aux)�,∗ � [��] � � � [�] � [�] � � κ κ κ κ∪κ κ∪κ →κ � κ � κ∪κ →κ � κ � κ∪κ κ∪κ κ∪κ V h ,h =V Q h ,Q h =V h ,h � [�] [�]� [�] κ κ κ =V h ,h =V .

� � [��] [��]� [�] κ κ κ Similarly,U P ,P =U . Now,& we have: ∞ � � � � � [�] [�] � � ] 0,1]= A=1 � ∈]0,1] V A U A ,N & and since ] 0,1]is uncountably� infinite,� there existsA∈ \{0} such that: [�] [�] � � � K V � ∈]0,1] � V A U A K ↓ V K ∅ κ κ κ ( ,A) � is infinite, hence contains ↓some countably infinite subset . Then, we have (V ,U ) ∈ , but ( ,A) this is impossible sinceL = from the previous lemma . L CR R P R F R F M R �� �� PropositionCR 2.13 Let be theL directedR labelM setR definedM R in [13, def. 2.14]. For anyη∈ , let η � η η := {�:γ(η)→ }, :={P: (η)→ } (withγ(η) and (η) as in [13, def. 2.14] ), := ∗ � R � R � η R ��L M η →η η η T ( ) and, for anyη η ∈ , letπ : → be defined as in [13, prop. 3.8] in the � ↓ � �� η →η� special caseG= ( ,+ ). Then, , , π is a projective system of phase spaces fulfilling L P R R the hypotheses of prop. 2.1, with: �� η η � η η 1. ∀η ∈ , ∀P ∈ ,Ξ (P): � �→ P ◦ χ (�)�(�) (withχ as in [13, def. 2.14] ); L M R R �∈γ(η) F

−1 � η � �� η η � χ (F) � 2. ∀η ∈ , ∀(�, �)∈ ,L (�, �) := �, F �→ � δ (with, for anyF∈ (η) and any −1 χη (F) η � ∈ γ(η),δ (�) = 1 ifχR (�)R =F, and 0 otherwise); � � L C �η →η,� � � � � � � � � � 3. �� η η η →η η η →η,� η η →η,� ∀η η ∈ , ∀� ∈ ,Q (� ): � �→ k=1 ε (k)� ◦ � (k) (with the notations L P R R of [13,� prop. 3.5]); � � � � � � � � (1� ,3) 4. �� η η η →η η η η →η,F ∀η η ∈ , ∀P ∈ ,P (P ): F �→ � (1,3) P (F ) (withH also from [13, F ∈Hη�→η,F L H R R prop. 3.5]). R ⊗ � �� � TheL projective system ofCquantumR P stateR spaces , ,Φ provided by [13, prop. 3.10] (in the

special caseG= ( ,+ ) ) can,∗ be identified,∗ with the one provided by prop. 2.1. Moreover, for any R�� η η η ∈ and any (�, �)∈ �×R , � (�,id ) F (F,1) η � � η � F η X (�, �)= �∈γ(η) �(δ ) h − F∈ (η) �(δ ) P ,

using the densely defined, essentially self-adjoint operators introduced in [13, eqs. (3.11.1) and L R P C (3.11.2)] (the minus sign is the result of conflicting conventions in [15, prop. A.14] and prop. A.5). ∗ �� η η η Proof Assertions 2.1.1 and 2.1.2. Letη∈ .Ξ is an invertible linear map → with: 18 C R

R ∗ ,−1 −1 R R R η η � χη M(F) � C P ∀� ∈ ,Ξ (�): F �→ � δ .

η R R R η η η Next,L is anC invertibleP linear map → × , with: ,−1 � � η η η � � η � R R R ∀(�, P)∈ × ,L (�, P)=C �,P � �→ �∈γ(η) P ◦ χ (�)� (�) .

R R R η C P η η Let Ω be the symplectic structure on × defined by: � � � � � � η η η η η ∀�, � ∈ , ∀P, P ∈ ,Ω (�, P;� ,P ) := Ξ (P )(�)−Ξ (P)(� ). M R R R We have: R � � ,∗ � � � � R η � η η � M ∀(�, �),(� , � )∈ , L Ω (�, �;� , � ) =� (�)−�(� ). CR η η Hence,∗ L Ris a symplectomorphism (with respect to the canonical symplectic structure on = η 1 T ( ), see [14, eq. (2.16. )] ). Moreover, it coincides with the map definedR inR [13, prop. 3.8] in the � L C P caseG=( , +). R R� R R R � � � � �� η η η η Letη η ∈ . Using [13, prop. 3.8] we have, for any� ,P ∈ × : � � ,−1 � � � � � � η η � η η � η η →η η η →η η →η R R L ◦ π ◦ L (� ,P ) = Q (� ),P� (P L) . C C

R � � � � �� η →η η →η η →η Projective� of quantum state spaces. For anyη η ∈ , let M andφ , resp. and η →η R φ , be defined as in [13, prop. 3.6] (withG=( , +) ), resp. as in prop.� 2.1 . It follows from the η uniqueness part of [14,R prop. 2.10] and from the connectednessR of that the tangential lifts of � C� C η →η η →η the mapsφ andφ must coincide modulo a suitableR symplectomorphic identification of the ∗ � ∗ � C � C � η →η η →η η →η η →η cotangent bundlesT ( ) andT ( ). Hence, the mapsφ andφ themselves coincide � � η →η η →η moduloH aR diffeomorphicCR identificationH of theC manifolds and . Therefore, the associated

projective systems� of qua� ntum state� spaces coinci� de modulo a unitary identification of the Hilbert η →η 2 η →η R ηR→η 2 η →ηR R C R spaces :=L ( C) and P :=L ( D ). C C R ,∗ ,∗ R ∞ η η η � η η Observables. Let (�, �)∈ × andψ∈ :=C ( , ) . For any�∈ , we have: ,∗,−1 � η � � � η � X (�, �)ψ (�) :=�(�)ψ(�)−�[T ψ] Ξ (�) F −1 � � � F � χη (F) R = �∈γ(η) �(δ )�(�)ψ(�F)− F∈ (η) � �(δ )[T ψ](δ ),

R ,∗,−1 −1 η η F where we have used that, for anyF∈ (η), Ξ (�)(χ (F)) =�(δ ) . Now, specializing the R R � definitions from [13, prop. 3.11] to the� caseG=( , +), this can be rewritten as: (�,id ) F (F,1) � η � � � � η � � F � η � X (�, �)ψ (�) = �∈γ(η) �(δ ) h ψ (�)− F∈ (η) �(δ ) P ψ (�). �

S L HR R ∅ ⊗ �� Proposition 2.14 With the notationsL L of def. 2.5 and prop. 2.13, ˆ( , ,Φ ) = . (aux) �� Proof Directed pre-order on � . Let Ψ :U→V be an analytical coordi nate patch on Σ, 19 (�) � ε � Ψ B

1 2 3 τ τ τ 3/2 5/2 0 τ τ 1 2

Figure 2.1 – Constructing

κ η

R � � (�) ε withU an open neighborhood o f 0 in (recall that� := dim(Σ) 1). Letε such thatB ⊂ U. � R R R For anyτ�=τ ∈ [0, 2] we define: � � � �−1 τ,τ τ,τ �˘ :U ⊂ ≈ × → V � ε � � � ε � , R R(�, �R) �→Ψ 2 τ+�(τ − τ) , 2 �

� � � �−1 � � � τ,τ � � � R � 2 whereU = (�, �)∈ ≈ × � τ+�(τ − τ), � ∈ U is an open neighborhood of �−1 � � ε R τ,τ [0, 1]×{0} in . We denote by� be the corresponding edge. Next,(�− for1) anyτ∈� [0, 1] we τ τ,2 τ,2 define S˘ ≡ �˘ and we note thatU is an open neighborhood of{0} × B in . We denote τ L � byS the corresponding surface. (aux) 1 � 1 � L Letκ=(τ , � � � , τ )∈ , with 0<τ < � � � < τ 1 . We define (fig. 2.1): κ � τ� , τ�−1/2 � � � τ� , τ�+1/2 � � graphs γ := � � � ∈ {1, � � � , �} ∪ � � � ∈ {1, � � � , �} ∈ , 1 1/2 �+1/2 �+1/2 � �+1 whereτ := 0,τ := 2 and,L for any�∈{1, � � � , � −1},τ := 2 (τ +τ ) . We also define:

κ � τ � profls λ := {S | τ ∈ κ} ∼ ∈ , ∅ � � τ τ where∼ denotes the equivalence relation from [13, def. 2.12]. Since, for anyτ�=τ ,�(S )∩�(S ) = F , we have: τ κ � � � �� � � ↑ (λ ) = F (κ) � τ ∈ κ, � ∈ , ↓ , �& � � L ↑ where, for anyτ∈κ and any�∈ , ↓ : τ � � � edgesL τ τ F (κ) := {� ∈ | � � S ∀τ ∈ κ \ {τ} , � S }. κ κ κ �� & Thus,η := (γ , λ )∈ with: � � κ � �−1/2 τ κ � �+1/2 τ η τ , τ ↓ η τ , τ ↑ ∀� ∈ {1, � � � , �} , χ (� ) =F (κ) χ (� ) =F (κ).

Moreover, for any�∈{1, � � � , �} , we have: 20 � 1 1/2 −1 � �−1/2 τ ,0 τ , τ τ1 , τ3/2 τ , τ � =� ◦ � L ◦ � � � ◦ � , L τ� � � τ τ � � � � ↓ L faces edges � τ and F (κ)= F ∈ , withF := � ∈ � � ↓ S , ⊥ ⊥ edges where,� forF⊂ ,F and F :=F ◦ F are defined as in [13, prop. 3.3], and the equality τ τ� ↓ L L L L L F (κ)= F can be proved using prop. [13, 2.11.4]. (aux) (aux) L �� L �� Now, let := � and extend the pre-orders on and with: (aux) & � �� L L ∀η=(γ, λ)∈ , ∀κ ∈ , τ,0 τ L � L Lgraphs/� Lprofls/F � κ η⇔ ∀τ ∈ κ, γ ∈ λ∈ , L graphs/� edges profls/F faces where for�∈ , resp. for F ∈ , has been defined in [13, prop. 3.2], L � L L L resp. in [13, prop. 3.3]. To check that this defines a pre-order , we note that: � � L edges � L graphs L graphs/� L graphs/� ∀� ∈ , ∀γ γ ∈ , γ ∈ ⇒ γ ∈ , � � faces L profls profls/F profls/F L L ∀F ∈ , ∀λ λ ∈� , λ ∈ ⇒ λ ∈ . L (aux) κ κ �� Since, for anyκ∈ ,κ η withη the label constructed above, is cofinal in and, in L L L particular, is directed. � (aux) C R C �� Projective system on . For anyκ∈ and anyη=(γ, λ)∈ such thatκ η, we define: (aux) η→κ η κ Q : → �γ→�τ,0 � τ,0 τ,0 � �k=1 γ→� γ→� , � �→ τ �→ ε (k)�◦� (k) P R P as well as: (aux) η→κ η κ P : → � � � � �F ∈Hλ→Fτ , P �→ τ �→ P(F ) R τ,0 τ,0 τ,0 τ γ→� γ→� L γ→� L M M λ→F where� ,ε and� have been defined in [13, prop. 3.2], andH in [13, prop. 3.3]. (aux) R (aux) �� η→κ η κ Defining, for anyκ∈ and anyη∈ ,π : → as:

η→κ � η→κ η→κ� η R π = Q × P ◦ L , M R R and specializing the definitions from [13, prop. 3.9] to the caseG=(τ , +), we have: � (�τ,0 , id ) (F ,1) � η η→κ η η ∀(�,� �)∈ L, π (�, �) = τL �→h (�,� �), τ �→P (�, �) .

� (aux) � (aux)� �� κ →κ For anyκ κ ∈ and anyη∈ withκ η, we thus have, using the definition ofπ from prop. 2.10: (aux)� � η→κ κ →κ Lη→κ L π =π ◦ π . � � (aux) � �� Moreover, for anyκ∈ and anyηR η ∈ , withκ η, we have shown in [13, prop. 3.9] R R & R τ τ that: τ,0 τ,0 (� , id ) � (�� , id ) (F ,1) � (F� ,1) η η →η η η η →η η ∀τ ∈ κ,h ◦ π = h P ◦ π = P ,

therefore, we also have: 21 R

� � η →κ η→κ η →η π =π ◦ π L. � (aux) 1 � 1 � κ κ κ Letκ=(τ , � � � , τ )∈ with 0<τ < � � � < τ 1 and letη = (γ , λ ) be constructed as above.C WeR have: � �−1

κ � �−1/2 � �+1/2 ηκ � � η →κ � � � τ ,τ � τ ,τ ∀� ∈ , ∀τ ∈ κ, Q (�) (τ ) = �=1 �(� )− �=1 �(� ), P R as well as: � Rκ τ ηκ � � η →κ � � � ↓ � C ∀P ∈ , ∀τ ∈ κ, P (P) (τ ) =P F (κ) . ,−1 (aux),∗ ηκ κ Inserting the expressionR for Ξ from the proof of prop. 2.13, we get, for any�∈ : � ,−1 ∗ � � � � κ κ κ κ −1 τ� η →κ η η →κ � η →κ ηκ ↓ � � �k R χ �F (κ)� P ◦Ξ ◦ Q (�):τ �→ � ◦ Q δ =� τ �→ k=1 δ .

κ ,−1 ∗ (aux),−1 L κ η →κ ηκ ηκ →κ κ �η →κ � R R Thus,P ◦Ξ ◦ Q = Ξ , in other wordsπ is a projection compatibl� e with the �� κ symplectic structures.� Then, for anyη∈ such thatη κ, there existsη η,η such that � κ � η→κ η →η η →κ η →κ η →ηκ η→κ π ◦ π =π =π L ◦M π , soπ is aL projectionM R R compatible with the symplectic structures. (aux) (aux) (aux) ↓ ↓ L � � � �� �L H Thus, we can combine , , π and , , π into a⊗ projective system of phase spaces on , fulfilling prop. 2.1. 1 and 2.1.2. We denote by ( , ,Φ ) the corresponding L L projective system of quantum state spaces. (aux) L L L �� L L L S L H R R Mapping narrow states on to narrow states on . (aux)Applying [15, prop. 2.6] twice,⊗ to go from S ��L H �� A L H A L H R R ( ��, ,Φ ) to (using that is cofinal in ) and from to , there exist a mapσ: → ⊗ (aux) (aux) (aux) ⊗ (aux) (aux) (aux) ⊗ ( , ,Φ ) ( , ,Φ ) ( ��, ,Φ ) S L HandR R a mapα: A L H → such that: ⊗ ⊗ (aux) (aux) (aux) ( ��, ,Φ )L ( , ,Φ C) � P � � � ∀ρ ∈ , ∀A ∈ , Tr ρ α(A) = Tr σ(ρ)A . (aux) (aux),∗ (aux),∗ κ κ H 1 Moreover, for anyκ∈L and any (�, �)∈ × , we haveL from eq. (2.3. ): (aux) −1 (aux) � (aux) � � κ ηκ →κ κ � � κ � Rη →κ � κ � η →κ α T (�, �) ∼, = Φ ◦ id ⊗T (�, �) ◦Φ ∼, �� L � κ κ κ � η � η →κ η →κ� ∼, �� S L H R =R T � ◦ Q L , � ◦ P . ⊗ (aux) C ( ��P, ,Φ ) Thus, for anyρ∈ and anyκ∈ , we get: (aux),∗ (aux),∗ κ η κ κ κ κ σ(ρ) ρ κ � η →κ η →κ� ∀(�, �)S∈L HR R × S L ,WH (�, �)=∅ W � ◦ Q , � ◦ P , �

⊗ ⊗ (aux) (aux) (aux) � �� � henceσ ˆ( , ,Φ ) ⊂ ˆ( , ,Φ ) = . If we cannot have Gaussian states, the next best thing would be to have some quantum analogue of the classical probability distributions whose characteristic function takes the form:

, α/2 � −��(X) � � � � � � X = exp ��(�)− �(�) 22 with , a linear form and a (symmetric) non-negative bilinear form on the dual space. These distributions have the nice property that their marginal probabilities (the classical equivalent of theα partial ∈]0, 2] trace� of a quantum state)� are again of the same form, with the same (this is manifest by recalling that the characteristic function for a marginal probability is the restriction of the full characteristic function to the corresponding vector subspace of the dual space, inα complete analogy to def. 2.2.3). For the distribution is Gaussian, while for it is heavy-tailed (aka. has infinite variance), and these distributions generalize the Gaussian distribution in the sense that they appear as attractoα=2rs in heavy-tailed generalizations of the centα

However, as prop. 2.15 below shows, the previous arguments excludes in the same stroke a large class of states.

N S L H Proposition⊗ 2.15 We consider the same objects as in prop. 2.4 and we suppose that there exists a ( , L,Φ) 1 �+� stateρ∈ , integers �, � ∈ , and realsε , � � � , ε > 0 such that: C P (1) (�) 1. η η for anyη∈(1) there(�) exists�∗ linear∗ forms� , � � � , � and� (symmetric) non-negativ e bilinear η Lη η η forms� �, � � � , � on × ; � 2. for anyη η ∈ : (k) (k�) ∗� ∗� & η η η →η η →η ∀k ∈ {1, � � � , �} , � =� ◦(Q × P ) (�) (��) ∗� ∗� ∗� ∗� η η � η →η η →η η →η η →η � ∀�L ∈ {1, � � � , �} , � C=� ◦P (Q × P )×(Q × P ) ; ∗ ∗ η η 3. for anyη∈ and any (�, �)∈ × : & � (k) 2 (�) � � η � k η �+� ∀k ∈ {1, � � � , �} , �� (�, �)� < ε ∀� ∈ {1, � � � , �} , � (�, �; �, �)<ε

� ρη � 1 S L H ⇒∅�W (�, �)−1 � < . 2 ⊗ ( , ,Φ) L Then, ˆ �= . C P Proof For anyη∈ we define: � � ∗ ∗ η η ∀(�, �),(� , � )∈ × , � � � � � (k) (k) � � (�) � � � η 1 � η η � η B (�, �;� , � ) := 1 �+� � (�, �)� (� , � ) + � (�, �;� , � ) , 4π min(ε , � � � , ε ) k=1 �=1 C & P

as well as:� ∗ � � � ∗ � � η η η η C η P η L ∀�, � ∈ ,V (�, � ) :=B (�, 0;� , 0) ∀�, � ∈ ,U (�, � ) :=B (0, �;0, � ). � ∗ ∗ � η η η η V , resp.U , is a (symmetric) non-ne& gative bilinear form on , resp. , and for anyη η ∈ , we have: � ∗� ∗� � ∗� ∗� η η η →η η →η η η η →η η →η V =LV ◦(Q × Q ) U =U ◦(P × P ). C P � � ∗ ∗ η η Letη∈ . Reasoning by contradic tion, we suppose that there exists (� , � )∈ × such that: � � � � � −1 � η η � η � V (� , � ) +U (� , � )<� Ξ (� ) . 23 � −1 � � 2π � � 2π � & � η � ε ε In particular, this impliesε :=� Ξ (� ) > 0 . We define� := � and� := � , so that: −1 η η � η � V (�, �) +U (�, �)<2π � Ξ (�) = 2π. & Then, we have: η η η η B (�, 0; �, 0) =V (�, �)<4π B (0, �;0, �)=U (�, �)<4π, � as well as: η η η � η η � B (−�, �; −�, �) B (−�, �; −�, �)+B (�, �; �, �)=2 V (�, �) +U (�, �) <4π. Thus, we get: &

� ρη � 1 � ρη � 1 � ρη � 1 �W (�, 0)−1 � < , �W (0, �)−1 � < �W (−�, �)−1 � < . (2.15.1) 2 2 2 Now, the positivityC condition eq. (2.2.1) applied to the points (0, 0), (�, 0) and (0, �) can be rewritten as: � 1 2 � ∀z , z , z ∈ , 2 2 2 � 1 2 � � 1 ρη � 2 ρη S 1 2 ρη � |z | + |z | + |z | + 2 Re z z W (�, 0) + z z W (0, �)− z Lz HW (−�, �) H 0 (2.15.2) −1 η ⊗ η � η � ρ ( , ,Φ) η where we have used� Ξ (�) = 2π andW (0, 0) = 1 (forρ∈ , hence Tr ρ = 1). � � 1 2 Applying eq. (2.15.2) withz = 1 andz =z =−1 yields: � � ρη ρη ρη � 0 3− Re 2W (�, 0)−2W (0, �)−2W (−�, �) � ρη � � ρη � � ρη � 2 �1−W (�, 0)� + 2 �1−W (0, �)� −1− 2 Re W (−�, �) � ρη � � ρη � < Re 1−W (−�, �) − Re W (−�, �) . � ρη � � ρη � � ρη � Thus,−Re 1−W (−�, �) < Re W (−�, �) < Re 1−W (−�, �) , and since we1 also have � ρη � � ρη � � ρη � � ρη � Im W (−�, �) =−Im 1−W (−�, �) , this implies �W (−�, �)� < �1−W (−�, �)� < 2 < 1. Next, we apply eq. (2.15.2) with: 2 � � ρη � & z = 1− �W (−�, �)� , 1 ρη ρη ρη 2 ρη ρη ρη z =− W (�, 0)− W (0, �)W (−�, �) z =− W (0, �)− W (�, 0) W (−�, �) 1 2 � (these� values arise by optimizing onz , z , keepingz fixed). After simplifications, we get: 2 2 2 � η η � η η η � � � ρ � � ρ � � � ρ ρ ρ � 0 z − z �W (�, 0)� + �W (0, �)� −2z Re W (�, 0) W (0, �)W (−�, �) .

� ρη � � Since �W (−�, �)� < 1,z > 0, so this leads to: � 2 2 2 � ρη � � ρη � � ρη � � ρη ρη ρη � �W (−�, �)� + �W (�, 0)� + �W (0, �)� + 2 Re W (�, 0) W (0, �)W (−�, �) 1. � ρη 1 ρη Finally, we rewrite this inequality in terms of� :=W (−�, �)− 1,� :=W (�, 0)− 1 and 2 ρη � :=W� (0, �)−1: 2 � � 1 2 � 1 2 � 1 2 0 4 + 4Re (� +� + � ) + |� +� + � | + 2 Re (� � � ) 2 � 1 2 � 1 2 � 1 2 4−4 |� +� + � |+ |� +� + � | −2 |� � � |. � 1 2 � 1 2 However, this contradicts eq. (2.15.1) which requires |� +� + � | <3/2 and |� � � | <1/8, 24 2 � � since (� �→ � −4�) [0,3/2[ = ]−15/4, 0] . C P Thus, we have proven that: � ∗ ∗ −1 η η η C η � η � C ∀(�, �)∈ × ,V (�, �) +U (�, �)−� Ξ (�) 0. ∗ η η In particular,P for any��= 0∈ , there exists�∈ such that�(�)�= 0, so defining� := ∗ � � ∗ η � η Ξ R∈ , we have: �(�) � 2 η η ∀λ ∈ , λ V (�, �) +U (�, �)−λ 0, L V L C P η S L H ∅ η � hence the symmetric bilinear↓ formV is strictly positive.⊗ Similarly,U is also strictly positive. � η η� ( ,( , ),(Q,P)) ( , ,Φ) Therefore, V ,U η∈ ∈ , so by prop. 2.9, ˆ �= . The previous result can also be seen as excluding a different notion of state confinement, focus- ing on quantiles rather than moments (in other words, working with the cumulative distribution function rather than with the characteristic or moment-generating one).

C P

∗ ∗ C cyl cyl P Proposition 2.16 We consider the same objects a in prop. 2.4. LetL , resp.� , be the inductiveL � ∗ ∗� � � � ∗ L � η�η∈ � η →η�η η C � η�Cη∈ limit (without� any completion) of the inductive system , Q , resp. , ∗� � � ∗ ∗ ∗ ∗ � η →η�P P cyl→η cyl→η η cyl P η η . For anyη∈ , letQ , resp.P , be the canonical injection of in , ∗ ∗ η cyl S L H resp. of in . C P ⊗ L C P ( , ,Φ) We suppose that there∗ exists∗ a stateρ∈ , a real�>3/4 and a non-negative∗ ∗ (sy mmetric) cyl cyl cyl η η bilinearH formB on � × such that, for anyη∈ and any (�, �)∈ × : η � η η � Tr ρ Π (�, �) �, η η � � where Π (�, �), is the spectral projector of X (�, �) on − A, A with: � ∗ ∗ ∗ ∗ cyl� cyl→η cyl→η cyl→η cyl→η � A=S L H B ∅Q (�),P (�);Q (�),P (�) . ⊗ ( , ,Φ) L Then, ˆ �= . L Proof For anyη∈ , we define: � � (1) � � ∗ ∗ ∗ � ∗ � η cyl� cyl→η cyl→η cyl→η cyl→η � ∀(�, �),(� , � )∈ , � L (�, �;� , � ) :=B Q (�),P (�);Q (� ),P (� ) . (1) � η � By construction, � η∈ fulfills prop. 2.15.L 1 and 2.15.2. C P 3 2 ∗ ∗ (1) 1 � � η η η 1 Letε := 2�− 2 > 0. For anyη∈ and any (�, �)∈ × such that� (�, �; �, �)<ε , H � we have: � � � η � η η �� � � � (1) (1) � �Π (�, �) T (�, �)− id � η sup η �� −1 � �∈ −√� (�,�; �,�), √� (�,�; �,�)

25 � (1)  η  � (�, �; �, �) 3 = 2 sin   <2�− 2 , 2 H � H as well as: � η η � η � η � �T (�, �)− id � �T (�, �)�+ �id � = 2 . H H Thus, we get: � ρη � � η η � η η �� �W (�, �)−1 � = �T � ρ T (�, �)−H id � H

3 η η 2 η � η � 1 <2S�−L H +∅ 2T� ρ id −Π (�, �) < . � 2 ⊗ ( , ,Φ) Hence, prop. 2.15 implies ˆ �= .

3The Outlook intention to obtain better semi-classical states for Loop Quantum Gravity was an important motivation for the state space extension considered in [13]. However, as just shown, it turns out that going beyond the Ashtekar-Lewandowski sector is only one part of the solution, and that further obstructions to the existence of such states on the holonomy-flux algebra need to be addressed as well.

Note that, while the argument cannot be easily generalized beyond the R case (because the systematic study of narrow states in subsection 2.1 makes heavy use of the linearity, viz. the discussion preceding prop. 2.9), we can nevertheless take it as a hint on whatG= will be required to make the construction of semi-classical states easier – or even possible – in the general case. On the one hand, the negative result of subsection 2.2 casts serious doubts on the possibility to design admissible semi-classical states on the full holonomy-flux algebra, even in the compact group case: although diverging variances are of course excluded in this case (at least for the configuration variables), we expect that, if states very peaked around the group identity could be designed, the group structure should not matter much to them (heuristically, the portion of the configuration space seen by such states would be nearly linear). On the other hand, the approach we will develop in [16] to go around the obstruction should simplify the construction of projective states to the point that keeping arbitrary would become effortless.

The exclusion of large classes of states on the real-valued holonomy-flux algebra, discussed at G the end of subsection 2.2, also raises an additional question, namely whether the projective state space could possibly be empty in this case. While projective systems built on countable label sets always yield non-empty quantum state spaces (because projective states can then be constructed recursively, see [16, subsection 2.1]), the situation is less clear on uncountable labels sets (see [28] for an instructive example of a seemingly healthy, yet empty projective limit). In the case of the holonomy-flux algebra on a compact group , we have proved in [13, theorem 3.20] that all states on the corresponding AL Hilbert space belong to the projective state space, which is thus guaranteed to be non-empty, but we do not have suchGa proof if the group is non-compact, because the AL Hilbert space does not exist there.

26 To analyze this question more closely in the R case, one can observe that any state on the algebra A L H (aka. positive linear form of norm , see [7, part III, def. 2.2.8] and [15, prop. 2.4]) G= yields a C⊗-valued function on the inductive limit C P , which satisfies suitable positivity ( , ,Φ) 1 requirements (in accordance with def. 2.2.2). Since the∗ space∗ of states on a -algebra is never cyl cyl empty [25, theorem I.9.18] , we are assured that there do× exist such functions of∗ positive type on

C P . However, a function of positive type comes from a projective stateC in S L H if and only∗ if it∗ is continuous with respect to the inductive limit topology on C P ⊗(this reflects cyl × cyl ( , ,Φ) the characterization of normal states over a -algebra, see [25, corollary III.3.11]).∗ ∗ What we have cyl cyl established above is that, in the case of the projective∗ structure constructed as in× [13, subsection 3.1] for the holonomy-flux algebra with R : W there do not exist any function of positive type twice-differentiable at with respect to the G= • inductive limit topology; given some scalar product on C P , there do not exist any function0 of positive type

• continuous at with respect to the∗ corresponding∗ (strong) topology. cyl cyl This still leaves a fairly large window for× projective quantum states to be found, however, it would be reassuring to have0 an actual proof of their existence, and, even better, constructive techniques to obtain them.

On the other hand, the first of these two points demonstrates that the failure to find admissible semi-classical states would not be solved by looking for an even larger state space: if there would exist such states on the algebra they would be twice-differentiable, hence continuous, at , and they would belong to the projective state space (thanks to the positivity condition, continuity at zero implies continuity everywhere). In other words, a state on the algebra that would have0 finite variances would automatically be regular enough to be representable as a projective family of density matrices (aka. projective quantum state). As announced in section 1, the problem we uncovered here has its roots in the holonomy-flux algebra itself. Therefore, the resolution we will propose in [16] is meant to act directly on the structure of this algebra. At the same time, it has the potential to bypass the concerns just raised about the projective state space being possibly empty, since it should provide us with a systematic procedure to construct arbitrary projective quantum states, no matter whether the gauge group is compact or not [16, subsection 2.1].

Acknowledgements

This work has been financially supported by the Université François Rabelais, Tours, France (via a 3-years doctoral stipend from the French Ministry of Education – Contrat Doctoral Normalien), and by the Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany (via the Bavarian Equal Opportunities Sponsorship – Förderung von Frauen in Forschung und Lehre (FFL) – Promoting Equal Opportunities for Women in Research and Teaching).

This research project has also been supported by funds to Emerging Field Project “Quantum Geometry” from the FAU Erlangen-Nürnberg within its Emerging Fields Initiative.

27 AIn Appendix: this appendix, weWignergive a brief characteristic overview of the Wigner-Weyl functionstransform [29] , that allows to represent quantum density matrices as functions on the phase space, mimicking the probability distributions of classical statistical physics. The presentation below follows very closely [11] , merely adapting of the notations and definitions to the ones we will be using in section 2.

Note that, in view of applying this tool to projective systems of quantum state spaces (subsec- tion 2.1), we will be doing only ‘half’ of the Wigner transform: going from the density matrix to the Wigner characteristic function. The second half would be to perform a Fourier transform yielding the Wigner quasi-probability distribution, in analogy to the reconstruction of a classical probability distribution from its characteristic function. However, Wigner characteristic functions are more convenient when working with partial traces of the underlying density matrices, exactly like classical characteristic functions are convenient for computing marginal probabilities.

A.1 Weakly continuous representations of the Weyl algebra We begin by recalling basic facts regarding the Weyl algebra and weakly continuous represen- tations thereof. These representation-theoretic considerations will come into play to prove that the Wigner transform is surjective onto a suitably defined space of functions.

C P P C

C C C P ∗

Definition A.1 Let , be two∗ finite-dimension al real vector spaces and let Ξ : → be an C P invertible linear map (with the dual of ). We equip × with the symplectic form Ω given by: � � � � � � C C P C C ∀�, � ∈ , ∀�, � ∈ ,Ω(�, �;� , � ) := Ξ(� )(�)−Ξ(�)(� ). ∗ ∗ The mapM := id × Ξ is then a symplectomorphism × →T ( ) (T ( ) being equipped with I C P C its canonical symplectic structure, see eg. [14, eq. (2.16.1)] ). ∗ ∗ PropositionI A.2 LetC be theP complex vector space of functions × → with support on finitely many points, ie: ∗ C ∗ P I ∀ι ∈ ,#{(�, �)∈ × | ι(�, �)�= 0} < ∞. ∗ ∗ (�,�) In particular, forC anyP (�, �)∈ × , we defineδ ∈ by: � � � � ∗ ∗ � � � (�,�) 1 if (� , � ) = (�, �) ∀(� , � )∈ I × , δ (� , � ) := I . 0 else 1 2 C P 1 2 Forι , ι ∈ , we define their productι � ι ∈ by: ∗ ∗ C P �ξ(�1,�1;�2,�2) 1 2 � 1 1 1 2 2 2 ∗ ∗ ∀(�, �)∈ × , (ι � ι ) (�, �) :=(�1,�1),(�2,�2)∈ × � ι (� , � )ι (� , � ), C P �=�1+�2, �=�1+�2

∗ ∗ 1 1 2 2 where, for any (� , � ),(� , � )∈ × : 28 −1 −1 1� 2 � 2� 1 � 1 1 2 2 � Ξ (� ) − � Ξ (� ) ξ(� , � ;� , � ) := I . 2 C P ∗ We also define, for anyι∈ , its conjugateι by: ∗ ∗ ∗ ∀(�, �)∈ × , ι (�, �) := ι(−�, −�), I

where ∗· stands∗ for complex conjugation. I (0,0) , �, is a -algebra [7, section III.2.2] , with unitδ . I 1 2 1 2 1 2 Proof Letι , ι ∈ . Sinceι andIι are supported on finitely many points, the sum definingι � ι 1 2 is finite andι � ι is again supported on finitely many points, ie. it is in . The operation� is C P 1 2 3 bilinear, and, for anyι , ι , ι ∈ , we have: ∗ ∗ � 1 2 3 � ∀(�, �)∈ × , ι ��(ι� � ι ) (�, �)= � ξ(�1,�1;�1,�1) � ξ(�2,�2;�3,�3) � 1 1 1 � 2 2 2 3 3 3 � � = �=�1+�1 � ι (� , � ) �1=�2+�3 � ι (� , � )ι (� , � ) � � �=�1+�1 �1=�2+�3

� ξ(�1,�1;�2,�2)+� ξ(�1,�1;�3,�3)+� ξ(�2,�2;�3,�3) � 1 1 1 2 2 2 3 3 3 = �=�1+�2+�3 � ι (� , � )ι (� , � )ι (� , � ) �=�1+�2+�3

� 1 2 3� I = (ι � ι )�ι (�, �), I ∗ (0,0) so it is associative as well. We can check thatδ is a unit for ,�. Finally, the operation ( · ) is C P 1 2 an antilinear involution by construction, and, for anyι , ι ∈ , we have: ∗ ∗ ∗ −� ξ(�1,�1;�2,�2) ∗ ∗ ∗ ∗ 1 2 � 1 1 1 2 2 2 2 1 ∀(�, �)∈ × ,(ι � ι ) (�, �)= �=�1+�2 � ι (� , � )ι (� , � ) = (ι � ι )(�, �). �=�1+�2 �

Note that the elementary observables in a representation are labeled by elements of the dual space C P of the phase space M C P , because they are actually labeled by the linear κ observables of∗ which∗ they are the quantizatiX (�,on �) (in the spirit of [15,κ def. A.3], up to a difference in sign convention).× Therefore, the unitary≈ operators× , which are obtained by exponentiating these observables, are labeled by elements of the dual as well. This observation will be important κ for the development of subsection 2.1. T (�, �)

H C P H C P A A κ Definition∗ H ∗ A.3 Let be a complex∗ ∗ Hilbert space. A weakly continuous, unitary representation of κ κ κ κ × on is a map T : × → (where is the algebra of bounded linear operators κ C P H over ), satisfying∗ ∗ the following properties: C κ P κ 1. ∀(�, �)∈ × ,T (�, �) is a unitary operator on ; ∗ ∗ �ξ(�1,�1;�2,�2) 1 1 2 H2 κ 1 1 κ 2 2 κ 1 2 1 2 2. ∀(� , � ),(� , � )∈ × ,T (� , � )T (� , � ) =� T (� +� , � +� ); κ H κ C P C 3. T (0, 0) = id ; � � ∗ ∗ κ κ 4. ∀φ, φ ∈ , the map (�, �) �→ �φ |T (�, �)φ� is a continuous function × → . 29 I H

∗ I κ This provides a representation of the -algebra on : C P κ � κ : ∗ ∗ ∀ι ∈ ,T {ι} = (�,�)∈ × ι(�, �)T (�, �) C P

∗ ∗ −1 1 3 κ κ (using† in particular that points A.3. to A.3. imply∀(�, �)∈ × ,T (−�, −�)=T (�, �) = κ H C P T (�, �) ). H ∗ ∗ κ κ Proposition A.4 Let ,T be as in def. A.3. Then, there exists, for any (�, �)∈ × a densely R κ κ defined (possibly unbounded), self-adjoint operator X (�, �) on such that: κ κ ∀τ ∈ ,T (τ�, τ�) = exp(� τX (�, �)) D H (the exponential being defined via spectral resolution [23, theorem VIII.5] ). C P D κ κ Moreover, there exists a dense vector subspace ⊂ such that: ∗ ∗ D κ � κ � � κ � 1. for each (�, �)∈ × , ⊂ Dom X (�, �) (where Dom X (�, �) denotes the dense domain κ C P κ κ D D of X (�, �)) and X (�, �)| is essentially self-adjoint; ∗ ∗ C κP κ R κ 2. ∀(�, �)∈ × ,X (�, �)� � ⊂ ; D D D � � ∗ ∗ � � � � � � � � Rκ κ κ κ κ κ 3. ∀(�, �),(� , � )∈C ×P , ∀τ, τ ∈ ,X (τ�+τ � , τ�+τ � )| =τX (�, �)| +τ X (� , � )| H . ∗ ∗ R κ κ κ Proof Let (�, �)∈ × . For anyτ∈ ,T (τ�, τ�) is unitary (def. A.3.1), and T (0, 0) = id 1 2 (def. A.3.3). Moreover, using def. A.3.2, we have, for anyτ , τ ∈ : κ 1 1 κ 2 2 κ� 1 2 1 2 � T (τ �, τ �)T (τ �, τ �)=T (τ +τ )�,(τ +τ )� , (A.4.1) 1H 1 2 2 C 1 P2 sinceξ(τ �, τ �;τ ��,� τ �)=∗τ τ ∗ξC(�, �; �,P �) = 0. � � � κ κ 4 Letφ∈ ,(� , � )∈ � ×� and∗ ε>∗ 0. Letφ := T (� , � )φ. From def. A.3. , there exists an open neighborhoodU of (� , � ) in × , such that: �� �� � � �� �� � 2� � κ � ε ∀(� , � )∈U, ��φ |T (� , � ) φ� − �φ � � < . C P �� �� 2 �� �� ∗ ∗ κ Hence, thanks to the unitarity of T (� , � ) for any� (� , � )∈ × , we get: �� �� �� �� � � 2 � � �� �� � 2� κ κ � κ � ∀(� ,H � )∈U,�TR (� , � )φ−T (� , � )φ� 2 ��φ |T (� , � ) φ� − �φ � � < ε . (A.4.2) � � κ � � � � Letφ∈ ,τ ∈ andε> 0. Applying the previous point with (� , � ) = (τ �, τ �), there exists ε > 0 such that: � � 2 �−ε �+ε κ κ � � ∀τ ∈]τ , τ [,�T (τ�, τ�)φ−T (τ �, τ �)φ� < ε. H κ Thus, τ �→TR (τ�, τ�) is a strongly continuous one-parameter unitary group, so, by Stone’s theorem κ κ [23, theorem VIII.8], there exists a densely defined, self-adjoint operator X (�, �) on such that, κ � κ � C P for anyτ∈ ,T (τ�, τ�) = exp � τX (�, �) . C P C C P C ∗ ∗ Let ∞�µ be∗ a Lebesgue∗ C measurP∞ e∗ on the∗ finite-dimensioH nal real vectorH space × . For any � � � ∈ C ( × , ) (where∗ C ∗( × , ) denotes the space of sm ooth, complex valued, compactly κ κ supported functions on × ), and anyφ∈ , we define φ {�} ∈ as: C P � � � � � � � ∗ ∗ κ φ {�} := × ��µ(� , � )�(� , � )T (� , � )φ. 30 C P H

� � � � C P H ∗ ∗ � � C � �P � � κ κ κ As shown above (� , � ) �→T (� , � )φ is a continuous∗ ∗ map × H→ , so (� , � ) �→ �(�∗ , � )T∗ (� , � )φ κ is a continuous, compactly supported map × → , hence it is integrable on × , �µ [10, D H κ section III.1]. Therefore, φ {�} is well-defined as an element of . D C κ P Cκ H We define the vector subspace ⊂ as: ∞ ∗ ∗ κH � κ C:= VectP {φ {�} | � ∈ C ( × , ),φ∈ }. κ 3 2 Letφ∈ ∗ and∗ ε> 0. From def. A.3. and eq. (A.4. ), there exists an open neighborhoodU of (0, 0) in × such that: � � � � 2 C P κC ∀(� , � )∈U,�T (� , � )φ−φ� < ε. ∞ ∗ ∗ � Let�∈C� ( × , &) be a bump function wit h: �C P ∗ ∗ � 0, supp(�)⊂U × ��µ �=1,

where supp(�) denotes� the support of�. Then, we have: C P � � � � � � � ∗ ∗ κ �φD − φ {�}� × ��µ(� , �H)�(� , � ) �φ −T (� , � )φ�<ε. C R κ H C P κ C P Thus, is a dense subspace∞ ∗ of∗ . ∗ ∗ κ � Letφ∈ ,�∈C ( × , ) and (�, �)∈ × . We have, for anyτ∈ : C P � � � � � � � κ ∗ ∗ κ κ T (τ�, τ�) φ {�}= × ��µ(� , � )�(� , � )T (τ�, τ�)T (� , � )φ C P �� �� � �� �� �� �� �τ ξ(�,�;� ,� ) �� �� ∗ ∗ κ = × ��µ(� , � )�(� − τ�, � − τ�)� T (� , � )φ, C P C P C �� ��

where∗ we∗ have used�� the translation�� invariance of�� the�� Lebesgue measure(τ) ∞ and∗the fact∗ that∀(� , � )∈ C P (�,�) � × , ξ(τ�, τ�;� − τ�, � − τ�)=τξ(�, �;�� � , � ). Defining� ∈ C ( × , ) by: � � ∗ ∗ (τ) � � �τ ξ(�,�;� ,� ) � � (�,�) ∀(� , � )∈ × , � (� , � ) :=� �(� − τ�, � − τ�), � (τ) � κ R (�,�) C we thus get T (τ�, τ�) φ {�}=φ � . C P 1,(τ) 2,(τ) ∞ ∗ ∗ (�,�) (�,�) � We now define,C forP anyτ∈ ,� , � ∈ C ( × &, ) by: � � � � ∗ ∗ 1,(τ) � � (τ ) � � � 2,(τ) � � 1,(τ ) � � � (�,�) � � (�,�) � � (�,�) � � (�,�) � � ∀(� , � )∈ × , � (� , � ) := � (� , � )�τ =τ � (� , � ) := � (� , � )�τ =τ . �τ � �τ � R C P Then, we have: � � ∗ ∗

∀τ ∈ , ∀(� , � )∈ × , � (τ) � � � � τ τ �� (τ) � � (�,�) 1,(0) � � � � � �� 2,(τ ) � � (�,�) � (� , � )−�(� , � ) (�,�) 1 (�,�) � (� , � ) := − � (� , � ) = 0 �τ 0 �τ � (� , � ). τ τ � � � � LetK :={(� + τ�, � +τ�)|(� , � )∈ supp(�), τ ∈[−1, 1]}.K is compact as the contin uous image of the compact supp(�)×[−1, 1] and we have: � � (τ) � � (�,�) ∀τ ∈[−1,1],∀(� , � ) /∈K, � (� , � ) = 0 .

31 C P R �� � � �� 2,(τ ) � � ∗ ∗ (�,�) Moreover, the map� , � ;τ �→ � (� , � ) is continuous on × × , hence bounded onK×[−1, 1], C P so there existsM> 0 such that: � � � ∗ ∗ � (τ) � � � � � � (�,�) � K ∀τ ∈[−1,1],∀(� , � )∈ × , �� (� , � )� M1 (� , � ),

K where1 denotes the indicator function ofK. So, the dominated conv ergence theorem yields: C P � � � � (τ) � � � � � ∗ ∗ � (�,�) κ � τ→lim0 × ��µ(� , � ) �� (� , � )T (� , � )φ � = 0 ,

and therefore: � κ � �T (τ�, τ�) φ {�} − φ {�} (�,�) � τ→lim0 � − φ {� }� = 0 , � τ � C P where: � � ∗ ∗ � � 1,(0) � � � � � � � � (�,�) (�,�) (� ,� ) ∀(� , � )∈ × , � (� , � ) :=� (� , � ) =�ξ(�, �;� , � )�(� , � )−[T �](�, �) � � � � (� ,� ) (withT � the differential of� at (� , � ) ). κ & By definition of X (�, �), this implies [23, theorem VIII.7]: D � κ � κ C (�,�)P C C P C φ {�} ∈ Dom X (�, �) X (�, �) φ {�}= −� φ {� }. D D ∞ ∗ ∗ ∞ ∗ ∗ κ � κ � D (�,�) � � Hence, ⊂ DomD X (�, �) , and, since� ∈ C ( × , ) for any�∈C ( × , ), κ κ κ κ κ X (�, �)� � ⊂� . Next, X (�, �)| is symmetric (as the restriction of a self-adjoint operator), and κ we have, forφD ∈ : � � � κ κ κ T (τ�, τ�)φ − φ X (�, �)| φ = −� τ→lim0 , τ so: D † �� κ κ � � φ ∈ Ker X (�,D �)| ± � ⇔ � � � κ � κ �φ |T (τ�, τ�)φ � − �φ | φ � ⇔ ∀φ ∈D , −� τ→lim0 ∓ � �φ | φ �=0 τ � � � � � κ � κ � ⇔ ∀φ ∈ , �φR |T (τ�, τ�)φ � τ=0 = ∓ �φ | φ � D �τ � � � � � � � � Dκ D � � κ � κ ⇔ ∀φ ∈ , ∀τ ∈ , �φ |T (τ �, τ �)φ ��τ =τ = ∓ �φ |T (τ�, τ�)φ � �τ � D R κ κ κ 1 (using T (τ�,� τ�)� � ⊂ and eq. (A.4. )) � ∓τ � D κ κ ⇔ ∀φ� ∈ , ∀τ ∈ � , �φ |T (τ�, τ�)φ �=� �φ | φ � κ D R ⇔ ∀φ ∈ � , �φ | φ �=0 � κ κ κ (since, for eachφ ∈ , τ �→ �φ |T (τ�, τ�)φ � is bounded, T (τ�, τ�) being unitary for any D D H τ ∈ ). D D � † � ⊥ � κ κ � κ κ κ κ κ Thus, Ker X (�, �)| ± � = ={0} (for is dense in ), and therefore X (�, �)| is C P C C P R

essentially self-adjoint [2∞3, theorem∗ ∗ VIII.3]. � � ∗ ∗ � � Finally, for any�∈C ( × , ), (�, �),(� , � )∈ × andτ,τ ∈ , we have: 32 � � � � � � � (τ�+τ � ,τ�+τ � ) (�,�) (� ,� ) � � =τ� +τ � , which proves point A.4.3. Of course, the usual Schrödinger representation is such a weakly continuous representation of the Weyl algebra, and, in fact, the Stone-von Neumann theorem [27] , which we will recall below (prop. A.10, following the proof given in [11, theorem 15a] ), tells us that it is the only one (more precisely, it is the only irreducible one: an arbitrary weakly continuous representation thus decomposes as a direct sum of independent copies of the Schrödinger representation).

H C C C P � 2 Proposition A.5 Let :=L ( ,∗ �µ) with∗ µ a Lebesgue measure on the finite-dimensional real H C � vector space . For any (�, �)∈ × , we define� −1 a unitary operator T (�, �) by: � �(�)+ 2 � Ξ (�) ∗,−1 C � P � � � ∀φ ∈ , ∀� ∈ ,[T (�, �)φ](�) :=� ( ) φ �+Ξ (�) , ∗ ∗ ∗ ∗,−1C P∗ −1 � −1� ( ) where Ξ : → is the dual of the map Ξ and Ξ =∗ Ξ ∗ = Ξ . � D C D T is a weakly continuous, unitary representation of × . H C P � � Let be the space of smooth, compactly supported, complex-valued functions on . is a D ∗ ∗ � � dense vector subspace in , and, for any (�, �)∈ × , the linear operator X (�, �) defined on � by: D C � � ∗,−1 � � � � � ∀φ ∈ , ∀� ∈ , X (�, �)φ (�) :=�(�)φ(�)−�[T φ] Ξ (�) , C P D � & is essentially self-adjoint (withT φ the differential ofφ at�). Moreover,D we have: ∗ ∗ C � P � � � � � � ∀(�, �)∈ × , ⊂ Dom X (�, �) X (�, �)=X (�, �)| , ∗ ∗ H C � P where, for any (�, �)∈ × ,X (�, �) is the self-adjoint generator introduced in prop. A.4. ∗ ∗ � C � Proof Letφ∈ and (�, �)∈� −1 × . Letφ be given by: 2 � � �(�)+ � Ξ (�) � ∗,−1 � ∀� ∈ , φ (�) :=� ( ) φ �+Ξ (�) . Using the translation invariance of the Lebesgue measure, we have: C C � � 2 � � � 2 2 � � �� �µ(�) |φ (�)| = �µ(� ) �φ � � = �φ� H, H

� H � � C P hence T (�, �) is well-defined as a linear operator → and is∗ isometric.∗ H C � � 1 1 2 2 Moreover, we have T (0, 0) = id and, for any (� , � ),(� , � )∈ × : � ∀φ ∈ , ∀� ∈ , � −1 � �1(�)+ 2 �1 Ξ (�1) ∗,−1 � 1 1 � 2 2 ( ) � 2 2 � 1 � [T (� , � )T (� , � )φ](�)=� � [T−(1� , � )φ] �−+Ξ1 (−�1 ) �(� 1+�2)(�)+ 2 (�1+�2) Ξ (�1+�2) −�2 Ξ (�1) +�1 Ξ (�2) ∗,−1 � 1 2 � =� [ ( ) ( ) ( )] φ �+Ξ (� +� ) � ξ(�1,�1;� 2 ,�2) � 1 2 1 2 C P =� [T (� +� , � +� )φ](�), ∗,−1 −1 H ∗ ∗ 2 � 1 � 1 � 2 � where we have used� Ξ (� ) =� Ξ (� ) . In particular, we thus have, for any (�, �)∈ × , � � � � � � T (�, �)T (−�, −�)=T (−�, −�)T (�, �) = id , so T (�, �) is invertible, and, being isometric, it is 33 H

D C � P a unitary operator on . ∗ ∗ � Letφ∈ ,(�, �)∈ × andε> 0. In particular,φ is bounded and has com pact support, so: �C M := �µ(�) |φ(�)| < ∞. C

1 Moreover,φ is also absolutely contin uous, so there exists an open neighborhoodU of 0 in such C that: � � 1 ε ∀� ∈ , ∀� ∈ U , |φ(�+� )−φ(�)| < . P 4M+1 ∗,−1 P ∗ The map∗ Ξ being linear on theC finite-dimensional∗ vector space , it is continuous, therefore 2 1 C C P U := Ξ �U � is an open neighborhoo∗ ∗ d of 0 in . C C P Next, the mapz: × × → , defined by: � � � � � −1 � � � ∗ C∗ � � −�ξ(�,�;� ,� )+� � (�)+ 2 � Ξ (� ) C ∀� ∈ , ∀(� , � )∈ × , z(�;� , � ) :=� ( ) , C P � is continuous, so,(�) for any�∈ ∗, there∗ exists an open n eighborhoodV of� in and an open 3 neighborhoodU of (0, 0) in × such that: � � (�) � � � 3 ε ∀� ∈ V , ∀(� , � )∈U , |z(�;� , � )−1|< 2 . N C 4 �φ� + 1 1 � Since the support supp(φ) ofφ is compact, there exist(��) �∈ and� , � � � , � ∈ such that C P � � � ��=1 � 3 ��=1 3 3 supp(∗ φ)∗⊂ V . Thus, definingU := U ,U is an open neighborhood of (0, 0) in × , and: � � � � 3 ε2 ∀� ∈ supp(Cφ), ∀(� , � )∈U , |z(�;� , � )−1| < . C P 4 �φ� + 1 ∗ ∗ ∗ � � 4 2 3 4 Then,U := ( × U )∩U is an open neighborhood of (0, 0) in × , and, for any (� , � )∈U : � � 2 � � �T (�+� , �+� )φ−T (�, �)φ� � � 2 � � −�ξ(�,�;� ,� ) � � � � � = 2 �φ� − 2 Re φ � � T (� , � )φ� � � � −�ξ(�,�;� ,� ) � � � � � � (where we� have� used T (�+� , �+� ) =� T (�, �)T (� , � ) and the fact that both T (�, �) �� and T (� , � ) are unitary)� � �� � −�ξ(�,�;� ,� ) � � � 2� � � � � 2�� φ � � T (� , � )φ − �φ� � C � � � ∗,−1 � � � � � � � 2� �µ(�) |φ(�)| z(�;� , � )φ �+Ξ (� ) − φ(�) C � ∗,−1 � � 2 � � � � � � 2 C�µ(�) |φ(�)| �φ �+Ξ (� ) − φ(�)�+2 supp(φ) �µ(�) |φ(�)| |z(�;� , � )−1| � � (for ∀� ∈ , |z(�;� , � )| = 1) D C P H < ε. ∗ ∗ � � H � H Hence, for anyφ∈ , the map (�, �) �→T (�, �)φ is continuous × → . � D � � Now, the smooth, compactly supported functions are dense in , so for any φ,φ ∈ , and � anyε> 0, there exist φ� ∈ such that: 34 ε �φ − φ�� < C� P . C P 3 �φ �+1 ∗ ∗ ∗ ∗ 5 Next, for any (�, �)∈ � , there exists an open neighborhoodU of (�, �) in � such that: �� �� �� �� 5 � � ε ∀� , � ∈ U , �T (� , � ) φ� −T (�, �) φ�� < � . 3 �φ �+1 �� �� 5 � Thus, for any� , � ∈ U , � �� �� � � �� �� � � � � � � |�φ |T (� , � ) φ� − �φ |T (�, �) φ�| �φ � 2 �φ − φ�� +�T (� , � ) φ� −T (�, �) φ�� < ε �� �� � D � C P R D (since T (�, �) and T (� , � ) are both unitary), which proves def. A.3.4. ∗ ∗ (τ) C � � (�,�) � Letφ∈ and (�, �)∈ ×� 2 . For−1 anyτ∈ , we have [T (τ�, τ�)φ]=φ ∈ , where: (τ) � τ �(�)+ 2 τ � Ξ (�) ∗,−1 (�,�) � � ∀� ∈ , φ (�) :=� D ( ) φ �+τΞ (�) ,

� (�,�) � and X (�,C �)φ= −� φ ∈ , where: (τ) � (�,�) � (�,�) � ∀� ∈ , φ (�) := φ (�H)�τ=0 . D �τ �

� � � � In particular, X (�, �)φ belongs to , thus X (�, �) is well-defined as a linear operator on . Moreover, like in the proof of prop. A.4, the dominated convergence theorem yields: (τ) 2 C � � � � (�,�) � �φ (�)−φ(�) (�,�) � τ→lim0 �µ(�) � − φ (�)� = 0 , � τ �

and therefore: � � � �T (τ�, τ�)φ−φ � � τ→lim0 � − �X (�, �)φ � = 0 . � τ � & Hence, we have [23, theorem VIII.7]: D � � � φ ∈ Dom (X (�, �)) X (�, �)φ=X (�,D �)φ, � � � � � � so ⊂ Dom (X (�, �)) and X (�, �)=X (�, �)| . D Finally, using: � � � R T (τ�, τ�)φ−φ ∀φ ∈ ,X (�, �)φ= −� limτ→0 D D , D H τ � � � � � � � together with ∀τ ∈ ,T (τ�, τ�) ⊂ and the density of in , we get, via the same � argument as in the proof of prop. A.4, that X is essentially self-adjoint.

A.2 The Wigner-Weyl transform We now come to the Wigner transform itself. The Wigner characteristic function associated to a quantum state maps to a point the expectation value of in this state. In particular, it is defined as a function on the dual space C P , since, as underlined above, the operators κ (�, �) ∗ ∗ T (�, �) 35× are labeled by elements of the dual. Note that if we would complete the Wigner transform, by Fourier-transforming this characteristic function, we would pass to the dual again and obtain a κ Tquasi-probability(�, �) defined on the phase space, in correct analogy with the probability distributions considered in classical statistical physics.

Insisting on a strict distinction between M and its dual, may seems an unnecessary care in the case of a linear space, since we could simply choose some identification between the two. However, it allows us to keep track of the nature of the objects we are considering, and in particular of the natural direction of their transformations under morphisms. Probability measures are naturally push-forwarded along a projection, while functions on M should be pull-backed, but functions on the dual, being pull-backed by the dual map, effectively flow in the same direction as probabil- ity measures (viz. def. 2.2): thus, defining characteristic functions as a functions on the dual is consistent with the fact that they encode the same information as probability measures.

We begin by characterizing the image of the (first half of) the Wigner transform. The positivity condition eq. (A.6.1) reflects the requirement for density matrices to be non-negative operators and it manifests the truly quantum nature of the Wigner quasi-probability: a similar equation for a clas- sical characteristic function, reflecting the positivity of its associated probability distribution, would not have the twisting phase . In other words, eq. (A.6.1) captures the non-commutation � � of the position and momentum�ξ(�,� variables;� ,� ) (viz. prop. A.2), and it will play a central role in the derivation of the negative result� in subsection 2.2. Finally, the continuity of the Wigner charac- teristic function expresses the fact that it comes from a density matrix on a weakly continuous representation (this is closely related to the characterization of normal states on a -algebra, see [25, corollary III.3.11]). ∗ W

C P C I

∗ ∗ � Proposition A.6 A continuous function� � W: × → is said of positive type i f, for anyι∈ : C P � � � �ξ(�,�;� ,� ) � � � � ∗ ∗ 1 (�,�),(� ,� )∈ × ι(� , � )ι(�, �)� W(� − � , � − � ) 0. (A.6. )

C P & In particular, this implies: � ∗ ∗ W ∀(�, �)∈ × ,W(−�, −�)= W(�, �) |W(�, �)| W (0, 0) . C P C I We denote by the space of all continuo us functions of positive type. ∗ ∗ (0,0) (�,�) Proof Let (�, �)∈ × and letλ∈ . Applying� eq. (A.6. 1) toδ +λδ ∈ yields: � 2� 1 + |λ| W (0, 0) +λW(�, �)+ λ W(−�, −�) 0, C where we have usedξ(0, 0; 0, 0) =ξ(�, �;0, 0) =ξ(0, 0; �, �)=ξ(�, �; �, �) = 0. In particular,W (0, 0)C

is real positive (as follows from settingλ =2 0). Thus,λW(�, �)+ λ W(−�, −�) is real for anyλ∈ , � � � � so W(�, �) =W(−�, −�). Now, 1 + |λ| W (0, 0) + 2 Re [λ W(�, �)] is positive for anyλ∈ , H hence |W(�, �)| W (0, 0). H C P C κ κ Proposition A.7 Let ,T be as in def. A.3∗ and∗ letρ be a traceclass, (self-a djoint) positive C P κ ρ semi-definite operator on . The mapH W : × → , defined by: ∗ ∗ ρ κ � κ � ∀(�, �)∈ × ,W (�, �) := Tr ρT (�, �) , 36 C P is a continuous function of positive type. ∗ ∗ N κ κ Proof ForH any (�, �)∈ × ,T (�, �) is a bounded operator, henceρT (�, �) is traceclass, therefore ρ � � W is well-defined. From the spectral theorem, there existN∈ ∪ {∞}, an orthonormal family k k N κ & k k N (φ ) in and a family of non-negativeH reals (� ) such that: � � � k k k � k κ ρ= � |φ � � φ | � = Tr ρ. kCN P k N N � ∗ ∗ Let (�, �)∈ × andε> 0. LetK∈ withK N such that: � � k ε � < . K

Letι∈ . We have: � � C P C P � � �ξ(�,�;� ,� ) � � ∗ �� �� �� �� � ρ � ρ � � ∗ ∗ �� �� ∗ ∗ (�,�),(� ,� )∈ × ι(� , � )ι(�, �)� W (� − � , � − � ) = (� ,�H )∈ × [ι � ι](� , � )W (� , � ) ∗ I κ κ � C P � =Tr [ρT {ι � ι}] κ (where T {ι } has been defined∗ for∗ ι ∈ in def. A.3, as a finite linear combination of un itary κ operators belonging to T � × �) � � � � † � � k k � κ κ k = � φ T {ι} T {ι} φ 0. k N � �

ρ Therefore,W fulfills eq. (A.6.1). Like its classical analogue, the Wigner characteristic function is related to the moment-generating function (precisely, the latter, when it exists, is the restriction to the complex axis of the analytical expansion of the former). In particular, a quantum state exhibits finite variances for the elementary

37 observables if its characteristic function is twice-differentiable at , and the covariance matrix can then be recovered from the corresponding Hessian. κ X (�, �) 0

H

κ κ ρ C P Proposition A.8(1) Let ,T , ρ andW be as in prop.(2) A.7.∗ We∗ moreover assume that there exist a ρ ρ linear formWC andP a symmetric bilinear formW on × such that: 2 ∗ ∗ (1) (2) 2 ρ ρ ρ τ ρ ∀(�, �)∈ × ,W (τ�,C τ�)=PW (0, 0) +�τW (�, �)− W (�, �; �, �)+�(τ ) . (A.8.1) 2 � � ∗ ∗ κ � κ � � � � � Then,κ forκ any (�,κ �),(� ,κ � )∈ × , the operatorsρX (�, �) (densely defined on Dom X (�, �) ) X D(�,�)ρX (H� ,� )+X (� ,� )ρX (�,�) H and 2 (which can be defined at least as a sesquilinear form on the dense κ κ & κ subset H ⊂ ) admit a traceclass exteH nsion on , with: � � � � κ κ κ κ κ (1) κ (2) � � κ ρ X (�, �)ρX (� , � ) + X (� , � )ρX (�, �) ρ Tr ρX (�, �)=W (�, �) Tr =W (�, �;� , � ). C P 2 ∗ ∗ H Proof Let (�, �)∈ × . For anyτ�= 0, we define: κ † (τ) κ κ κ 2 id −T (τ�, τ�)−T (τ�, τ�) Y (�, �) := 2 . τ H H (τ) κ κ Since T (τ�, τ�) is a unitary operator, Y (�, �) is a bounded,† (self-adjoint) positive semi-definite κ κ κ κ κ operatorH on . Using T (0, 0) = id and T (τ�, τ�) = T (−τ�, −τ�), we get: ρ ρ ρ κ (τ) κ 2W (0, 0)−W (τ�, τ�)−W (−τ�, −τ�) Tr ρY (�, �)= 2 . τ Hence, eq.H (A.8.1) implies: κ (τ) (2) κ ρ τ→lim0 Tr ρY (�, �)=W (�, �; �, �).

1 H � Letτ > 0 such that: κ (τ) (2) 1 κ ρ ∀τ ∈ ]0, τ [, Tr ρY (�, �) W (�, �; �, �)+1=: A.

κ κ Applying the spectral theorem to the self-adjoint operator X (�, �), we denote by�Π (�, �) its projection-valued measure, with: +∞ � κ κ X (�, �)= −∞ σ �Π (�, �)[σ].

κ κ Then, for anyτ�= 0, we have, using T (τ�, τ�) = exp (� τX (�, �)) : +∞ (τ) � κ 22 κ Y (�, �) := −∞ [1− cos(τσ)]�Π (�, �)[σ]. τ There existsε> 0 such that: 2 � ∀� ∈]−ε, ε[,1− cos(�)> , 4 hence, for anyτ�= 0, the operator:

38 ε/τ 2 (τ) � κ 22 (τσ) κ Y (�, �)− −ε/τ �Π (�, �)[σ], H � R τ 4 � κ � � is a bounded, (self-adjoin[B,Bt)] positive semi-definiteB operator on . Defining, for anyB B ∈ , κ � κ the spectral projectorH Π (�, �) := B �Π (�, �)[σ] , we� thus have:

κ [−B,B] 2 [−B,B] ε κ κ κ 2 ∀B > 1 , Tr ρΠ (�, �)X (�, �) Π (�,N �) 2A. (A.8.H ) τ � � k k N κ Now, from the spectral theorem, there existN∈ ∪ {∞}, an orthonormal family (φ ) in & k k N and a family of strictly positive reals (� ) H such that: � � � k k k � k κ ρ= � |φ � � φ | � = Tr ρ. k�N � k N

For eachk N and eachB 0, we have: � [−B,B] 2 [−B,B] [−B,B] 2 � k � κ κ �κ k � � κ κ k � φ � Π (�, �)X (�, �) Π (�, �)φ = �X (�, �)Π (�, �)φ � 0, 1 so, for anyB>ε/τ and anyk� N, [−B,B] 2 � κ κ k � 2A �X (�, �)Π (�, �)φ � k . � k � κ � Therefore,φ ∈ Dom X (�, �) . Moreover, we have [23, theorem VIII.7]: κ k k T (τ�, τ�)φ − φ κ k τ→lim0 =�X (�, �)φ , τ hence, we get: 2 (τ) � κ k k � 2 � k � κ k � �T (τ�, τ�)φ − φ � κ k τ→lim0 φ � Y (�, �)φ =τ→ lim0 � � =�X (�, �)φ � . � τ � Fatou’s lemma then yields:� � 2 � (τ) (2) � k κ k � k � k � κ k � ρ τ→0 � k N � �X (�, �)φ � lim inf k N � φ Y (�, �)φ =W (�, �; �, �).

Next, this provides the bound:� H � � (2) � k k κ k ρ κ k N � �φ � �X (�, �)φ � W (�, �; �, �) Tr ρ, � H

H κ k �k N k where we have used �φ � = 1 and the Cauchy-Schwarz inequality (with � = Tr ρ). κ κ Hence, we can define a bounded operator {ρX (�, �)} on by: � κ � k k κ k {ρX (�, �)} := k N � |φ � �X (�, �)φ |,

κ � κ � κ and this operator coincides withρX (�, �) on Dom X (�, �) , since X (�, �) is self-adjoint. In addition, � we have: � � �� � � �� � κ �1 k � k κ k �1 k k κ k {ρX (�, �)} k N � |φ � �X (�, �)φ | = k N � �φ � �X (�, �)φ � < ∞, 39 1 κ where�·� denotes the trace norm [23, theorem VI.20], so {ρX (�, �)} is traceclass. Its trace is given by:H � � κ κ � k κ k k � k k κ k Tr {ρX (�, �)}= k N � �X (�, �)φ | φ �= k N � �φ |X (�, �)φ �.

R � Now, from: � �� � ∀� ∈ , �� −1 � |�|, κ together with the spectral� decomposition of X (�, �), we get, for anyτ�= 0: � κ k k � �T (τ�, τ�)φ − φ � κ k � � �X (�, �)φ �. � τ � Thus, the dominated convergence theorem yields: � � � � κ k k � � k k κ k � k k � T (τ�, τ�)φ − φ � � �φ |X (�, �)φ �= � τ→lim0 φ � k N k N � τ � � � κ k k � � k k � T (τ�, τ�)φ − φ =τ→ lim0 � φ � , k N � τ

H which can be rewritten, using eq. (A.8.1), as: κ (1) κ H ρ H Tr {ρX (�, �)}=W (�, �). ρ κ κ κ (note thatW (0, 0) = Tr ρ for T (0, 0) = id ). Similarly, the inequality:� � 2 (2) � k κ k ρ k N � �X (�, �)φ � W (�, �; �, �)<∞, H

κ κ κ ensures that we can define a traceclass operator{X (�, �)ρX (�, �)} on by: � κ κ � k κ k κ k {X (�, �)ρX (�, �)} := k N � |X (�, �)φ � �X (�, �)φ |,

and we have: � �� �� � �� � � κ � κ κ κ κ ∀φ , φ ∈ Dom X (�, �) , �φ | {X (�, �)ρX (�, �)} φ �=�X (�, �)φ | ρX (�, �)φ �. Moreover, using again the dominated convergence theorem, with: 2 � (τ) � κ k k � 2 � k � κ k � �T (τ�, τ�)φ − φ � κ k ∀τ �= 0, φ � Y (�, �)φ = � � �X (�, �)φ � , � τ � we get:H H κ κ (τ) (2) κ κ κ ρ Tr {X (�, �)ρX (�, �)} =C τ→ lim0 TrP ρY (�, �)=W (�, �; �, �). D � � ∗ ∗ Finally, for any (�, �),(� , � )∈ × , we have, thanks to prop. A.4.3: � �� � � � � � �� � κ κ κ κ κ ∀φ, φ ∈ �� , �X (�, �)φ� | ρX� (� , � )φ �+��X (�� , � )φ� | ρX (�, �)φ �= κ κ = �φ |�� {X (�+� , �+� )ρX (��+� , �+�� � )} φ��+� � � � κ κ κ κ − �φ | {X (�, �)ρX (�, �)} φ � − �φ40 | {X (� , � )ρX (� , � )} φ �. � � � � � � � � κ κ κ κ κ κ {X (�+� ,�+� )ρX (�+� ,�+� )}−{X (�,�)ρX (�,�)}−{XD(� ,� )ρX (� ,� )} � � 2 � � Hence, Xκ (�,�)ρX κ (� ,� )+Xκ (� ,� )ρX κ (�,�) provides a traceclass extension of the κ H 2 sesquilinear form� � � � on � and� its� � trace is given by: κ κ κ κ κ κ κ {X (�+� ,�+� )ρX (�+� ,�+� )}−{X (�,�)ρX (�,�)}−{X (� ,� )ρX (� ,� )} (2) � � ρ � Tr 2 =W (�, �;� , � ).

As a first step toward recovering the density matrix from its Wigner characteristic function, we observe that the positivity condition eq. (A.6.1) is exactly what we need to turn a function on C P into a state (aka. a normalized positive linear functional, see [7, part III, def. 2.2.8] ) on the∗ Weyl∗ algebra. Then, from any state we can reconstruct a representation of the algebra, via the GNS× construction [5] , and the continuity of the characteristic function ensures that this representation will be weakly continuous.

W H C P H

Proposition A.9∗ Let∗ W∈ . Then, there exist a w eakly continuous, unitary representation WH W WC PW H ,T of × and a vectorζ ∈ such that: ∗ ∗ W C PW W W W W 1. = Vect{T (�, �)ζ |(�, �)∈ × } (ie.Hζ is a cyclic vector for ,T ); ∗ ∗ W W W W 2. ∀(�, �)∈ ×I ,W(�, �)= �ζ |T (�, �)ζ � .

1 2 Proof Letι , ι ∈ . We define: C P ∗ 1 2 � 1 2 W ∗ ∗ �ι | ι � := (�,�)∈ × (ι � ι )(�, �)W(�, �). I

∗ 1 2 This is well defined, for the sumI has only finitely many non-zero contributions (ι � ι ∈ ), 1 W and eq.C (A.6. )P ensures that�·|·� provides an Hermitian, positive∗ semi-definite∗ ∗ sesquilinear 1 2 2 1 form onN the∗ complex∗ I vector space (the hermiticityI comes fromN (ι � ι ) =ι � ι together with ∀(�, �)∈ × ,W(−�, −�)= W(�, �), as was proven in prop. A.6). WC W W Let := {υ ∈ | �υ | υ� = 0}. For anyι∈ and anyυ∈� , we have: � � I W N ∀λ ∈ , �ι | ι� + 2 Re λ �ι | υ� = �ι+λυ|ι+λυ� 0, 1 2 1 2 W therefore �ι | υ� = 0. Hence, for anyι , ι ∈ and anyυ , υ ∈ : 1 1 2 2 W 1 2 W I N H �ι +υ | ι +υ � = �ι | ι � , I N W W W so�·|·� induces an inner product on the quotient / . We denote by the completion of W C P I / with respect∗ to∗ the corresponding norm. 1 2 Let (�, �)∈ × . For anyι , ι ∈ , we have: (�,�) 1 (�,�) 2 1 2 W N N �δ � ι | δ � ι �W = �ι | ι � ∗ ∗ H∗ (�,�) 1 (�,�) 2 1 (−�,−�) (�,�) 2 1 2 (�,�) W W (for (δ � ι ) �(δ � ι ) =ι �(δ � δ )�ι =ι � ι ), so in particular [δ � ·]� � ⊂ , (�,�) C P W W andδ � · induces a unitary operator T (�, �) on . Then, def. A.3.2 and A.3.3 come from: ∗ ∗ �ξ(�1,�1;�2 ,�2) 1 1 2 2 (�1,�1) (�2,�2) I (�1+�2, �1+�2) ∀(� , � ),(� , � )∈ × , δ � δ =� δ , ∗ I (0,0) C P and from the fact thatδ is the unit∗ of∗ the -algebra . 1 2 Letι , ι ∈ . For any (�, �)∈ × , we have:41 �ξ(�2,�2;�1,�1)+�ξ(�,�;� 1+�2,�1+�2) 1 (�,�) 2 � 1 1 1 2 2 2 2 1 2 1 W C P �ι | δ � ι � = (�1,�1),(�2,�2�) ι (� , � )ι (� , � )W(�+� − � , �+� − � ). ∗ ∗ ∈ ×

W I N H 1 (�,�) 2 W Thus, the map (�, �) �→I N �ι | δ � ι � is aH finite linear combination of translationsH ofW , and those 1 2 W 1 W 2 W are continuousI N by definition of . So, for any ι�, ι� ∈ / , the map (�, �) �→ �ι� |T (�, �) ι�� W W 1 2 W is continuous. Now, / being dense in , there exist, for anyφ , φ ∈ and anyε> 0, 1 2 W & ι�, ι� ∈ / suchH that: H H H 1 1 W ε 2 2 W ε �φ − ι�� < C 2 P W √ �φ − ι�� < 1 W √ .C P 6 �φ � + 6ε 6 �φ � + 6ε ∗ ∗ ∗ ∗ Next, for any (�, �)∈ × , thereH exists an open neighborhoodH U of (�, �) in × such that: � � � � � 1 W 2 W 1 W 2 W � ε ∀(� , � )∈U, ��ι� |T (� , � ) ι�� − �ι� |T (�, �) ι�� � < . � � 3 Hence, for any (� , � )∈U, we have: � � 1 W 2 1 W 2 |�φ |T (� , � )φ � − �φ |T (�, �)φ �| < ε, � � I N C P W W 4 where we have used that T (�, �) and T (� , � ) are unitary. This proves def. A.3.I N. ∗ ∗ W (0,0) W I Finally, we defineζC to beP the equivalence class ofδ in / . For any (�, �)∈ × , (�,�) (0,0) (�,�) W W (�,�) W we haveδ � δ =δ ∗ , so∗ T (�, �)ζ is the equivalence class ofδ in / . But since I N (�,�) C P = Vect {δ |(�, �)∈ × }, we have: ∗ ∗ W W W / = Vect{T (�, �)ζ |(�, �)∈ × }, H which proves point A.9.1. Moreover, we also have: W W W W (0,0) (�,�) W � �ζ |T (�, �)ζ � = �δ | δ � =W(�, �), which proves point A.9.2. Finally, the invertibility of Wigner transform can be seen as a consequence of the Stone-von Neumann theorem [27] (which we implicitly prove below, following closely [11, theorem 15a] ). In- deed, the Schrödinger representation being the unique irreducible, weakly continuous representation of the Weyl algebra, the Hilbert space H constructed from above can be written as a direct sum of independent copies of H . Projecting the vector representing in H on each of W these copies, we obtain a collection of vectors in H , and weW can reconstruct a density matrix � W W on H as the statistical superposition of the correspondingζ pure states (thisW is consistent with the � general result that a state is pure if and only if its GNS representation is irreducible, see [7, part � III, theorem 2.2.17] ).

W H Proposition A.10 LetW∈ . Then, there exists a unique traceclass, (self-adjoint ) positive � ρC ρ C semi-definite operatorρ on such thatW=W (withW defined as in prop. A.7). C P C ∗

Lemma A.11 LetC( ·∗ | ·P) be a real inner product on .∗ Using the resulting ide ntification of with

andC identifyingP ∗ with∗ through the dual map Ξ , we are provided with a correspondingC realP

inner∗ product∗ on × , which we will denote by ( · , · | · , · ). Let �µ be the Lebesgue measu∗ re∗ on × normalized with respect to this Euclidean structure. We define a map Ψ on × 42 through: C P ∗ ∗ � � 1 (�, � | �, �) ∀(�, �)∈ × ,Ψ(�, �) := exp − , C P α 4 (�,� | �,�) � ∗ ∗ � � C P whereα := × ��µ(�, �) exp − 2 . ∗ ∗ LetW be a bounded,C continuoP us function on × such that: C P ∗ ∗ � � ξ(�,�;� 1,�1) � � 1 1 ∗ ∗ � � 1 ∀(� , � ),(� , � )∈ × , × ��µ(�, �)Ψ(�, �)� W(� + �, � +�) = 0 . (A.11. ) C Then,W≡ 0. C P � �∈{1,���,�} � �∈{1,���,�} Proof Let (∗� ) be an orthonormal basis of ∗, ( ·,· ) , let (� ) be the corresponding dual � �∈{1,���,�} basis in and let (� ) be the basis in given by: ∗ C P� � ∀� ∈ {1, � � � , �} , � := Ξ (� ). ∗ ∗ � � � � For any (�, �)∈ × , we will write�= : � � and�= : � � (with implicit summation). In particular, we thenC have: P & � � � � � � � � ∗ ∗ � � � � � � � � � � � � − � � � � ∀(�, �),(� , � )∈ × , ξ(�, �;� , � ) = �, � | � , � =� � +� � , 2 1 � 1 C � P �

as well as� �µ(�, �)= �� � � � �� �� � � � �� ∗. Therefore,∗ α= (2π) . � � Now, for anyβ> 0 and any (� , � )∈ × , we have: C P C P � � � �ξ(�,�;� 1,�1) � ∗ ∗ 1 1 ∗ ∗ � 1 1 � � � × ��µ(� , � ) × ��µ(�, �) �Ψ(β� , β� ) Ψ(�, �)� W (� + �, � +� ) � � 2� � � 2 ∞ �W � < ∞, β ∞ where �W � is the sup norm of the bounded functionW . Thus, Fubini’s theorem , together with eq. (A.11.1) yields: C P C P � � �ξ(�,�;� 1,�1) ∗ ∗ � � ∗ ∗ 1 1 1 1 0 = × ��µ(�, �)Ψ(�, �)W (� + �, � +� ) × ��µ(� , � )Ψ(β� , β� )� .

Performing the Gaussian integration, we get: � �C P � 2 � � 2 � ∗ ∗ γ −1 γ � � 0 = × ��µ(�, �) exp − (�, � | �, �) W (� + �, � +� ) , π 4 2 � 1 β whereγ := 1 + / .

Letε>∞ 0. There exists�2 A> 0 such that: ∞ �2 � 2�−1 − 4 � 2�−1 − 4 ε A �� � � < ∞ 0 �� � � . 4 �W � + 1 W being continuous,C P there also exists� δ∈]0, A [ such that: ∗ ∗ 2 � � � � � ε ∀(�, �)∈ × (�, � | �, �) δ , |W(� + �, � +�)−W(� , � )| < . 2 43 δ Hence, applying the previous equality withβ= √ 2 2 , we have: 2 A − δ C P γ � − 4 (�,� | �,�) � � ∗ ∗ |W (� , � )| × ��µ(�, �)� = 2 C P γ �� − 4 (�,� | �,�) � � � � ∗ ∗ � � � � � = � × ��µ(�, �)� [W (� + �, � +� ) − W (� , � )]� γ2 γ2 � � � − 4 (�,� | �,�) � − 4 (�,� | �,�) ε 2 ∞ 2 < (�,� | �,�) δ ��µ(�, �)� + 2 �W � (�,� | �,�) δ ��µ(�, �)� . 2 Changing to spherical coordinates and dropping an overall constant, this can be rewritten as: ∞ �2 γδ �2 ∞ �2 � 2�−1 − 4 � 2�−1 − 4 � 2�−1 − 4 � � ε ∞ |W (� , � )| 0 �� � � < 0 �� � � + 2 �W � γδ �� � � 2 ∞ �2 � 2�−1 − 4 < ε 0 �� � � . C P ∗ ∗ � � � where we have used that γδ=A. Thus, we have, for an y (� , � )∈ × and anyε> 0, � � W H |W(� , � )| < ε, and thereforeW≡ 0. H W W Proof of prop. A.10 Existence. LetW∈ and let ,T be the weakly� continuous, unitary H W W representation introduce� d in prop. A.9, with cyclic vectorζ . For any φ,φ ∈ , the map W W (�, �) �→Ψ(�, �) �φ |T (�, �)φ� is continuous,H � hence meaH surable,H and we have: C P � � � � W W W ∗ ∗ � W � × ��µ(�, �) �Ψ(�, �) �φ |T (�, �)φ� � 2 H�φ � �φ� < ∞. H � W W W Hence, there exists aH bounded operator T {Ψ} on such that,H for any φ,φ ∈ : C P � � � W W W ∗ ∗ W �φ |T {Ψ} φ� := × ��µ(�, �)Ψ(�, �) �φ |T (C�, �)φ� P . ∗ ∗ W In particular,† T {Ψ} is symmetric (since, for any (�, �)∈ × , Ψ(�, �) = Ψ(�, �) = Ψ(−�, −�) W W C P H and T (�, �) = T (−�, −�∗)). Being∗ bounded, it� is therefore self-adjoint. 1 1 W Now, for any (� , � )∈ × Hand any φ,φ ∈ , we have: H � � � � � W W 1 1 � � W � � W � W 1 1 �� W φ � T {Ψ}T (� , � ) φ = φ � T {Ψ} T (� , � )φ H C P � � W ∗ ∗ W W 1 1 = × ��µ(�, �)Ψ(�, �) �φ |T (�, �)T (� , � )φ� H C P � �ξ(�,�;� 1,�1) � W ∗ ∗ W 1 1 = × ��µ(�, �)Ψ(�, �)� �φ |T (�+� , �+� )φ� H C P � �ξ(�,�;� 1,�1) � W ∗ ∗ 1 1 W = C× ��µ(�,P �)Ψ(� − � , � − � )� �φ |T (�, �)φ� , ∗ ∗ 2 1 1 1 1 1 1 where we† have used def. A.3. and∀† (�, �)∈ × , ξ(� − � , � − � ;� , � ) =ξ(�, �;� , � ). Since W W W 1 1 W 1 1 T {Ψ} = T {Ψ} and T (� , �H) = T (−� , −� ), we also have: H C P � � −�ξ(�,�;� 1,�1) � W � � � W 1 1 W � � W ∗ ∗ 1 1 W φ � T (� , � )T {Ψ} φ = × ��µ(�, �)Ψ(�−� , �−� )� �φ |T (�, �)φ� .

44 C P H

∗ ∗ � 1 1 W Applying these two formulas successively, we get, for any (� , � )∈ × and anyφ,φ ∈ : � � � � W W 1 1 W � � φ � T {Ψ}T (� , � )T {Ψ} φ = � � � W W 1 1 � W �� = φ � T {Ψ}T (� , � ) T {Ψ} φ H C P � �ξ(�,�;� 1,�1) � W ∗ ∗ 1 1 W W = × ��µ(�, �)Ψ(� − � , � − � )� �φ |T (�, �)T {Ψ} φ� H C P C P � � � � � � � � �ξ(�,�;� 1+� ,�1+� ) � � � W ∗ ∗ ∗ ∗ 1 1 W = × ��µ(�, �) × ��µ(� , � ) Ψ(�−� , �−� ) Ψ(� −�, � −�)� �φ |T (� , � )φ� .

Invoking Fubini’s theorem and performing the Gaussian integration, like in the proof of lemma A.11, we obtain: � � � � W W 1 1 W � � φ � T {Ψ}T (� , � )T {Ψ} φ = H C P � � � � 1 1 1 1 � � � � � � W ∗ ∗ (� , � | � , � ) W = × ��µ(� , � ) exp − Ψ(� , � ) �φ |T (� , � )φ� 4 H � 1 1 1 1 � � (� , � | � , � ) W W = exp − H �φ |T {Ψ} φ� . 4 � W Since this holdsC forP any φ,φ ∈ , we have: ∗ ∗ � � W W W (�, � | �, �) W 1 ∀(�, �)∈ × ,T {Ψ}T (�, �)T {Ψ} = exp − HT {Ψ} . (A.10. ) 4 W W In particular, applying with (�, �) = (0, 0), and using that T (0, 0) = id , this implies: W W W T {Ψ}T {Ψ}=T {Ψ}. W H T {Ψ} being idempotent and self-adjoint, it is an orthogonal projection. � � Next, weC defineψ ∈ by: � � � 1 (� | �) ∀� ∈ , ψ (�)= �/4 exp − H π 2 C � (choosing the Lebesgue measureµ, entering the definition of in prop. A.5, to be normalized with respect to the scalar product ( · | · ) on ; this normalization can be chosen without loss of generality since the HilbertC spacesP obtained using different normalizations ofµ are unitarily identified in a natural way through∗ a∗ corresponding renormalization of the wave-functions). We

can check that, for anyH (�, �)∈ × : � � � � � � (�, � | �, �) �ψ |T (�, �)ψ � = exp − . (A.10.2) 4 � (�)� W C P W Let ψ � be an orthonormal basis of the image of T {Ψ}. Then, for any �, � and any � � ∗ ∗ W (�, �),(� , � )∈ × , we have, using the hermiticity and idempotence of T {Ψ} together with W � the properties of the representationsH T and T : � � � (�) � (�)� W W � W W W T (� , � )ψ � T (�, �)ψ = H � (�) � � � (�)� W W � W W W W W = T {Ψ} ψ � T (−� , −� )T (�, �)T {Ψ} ψ 45 � � H �ξ(�,�;� ,� ) � (�) � � � (�)� W � W W W W W =� ψ � T {Ψ}T (� − �H, � − � )T {Ψ} ψ � � H �ξ(�,�;� ,� ) � � (�) (�) � � � � � � � W � W W W =� �ψ |T (� − � , � − � )ψ � ψ � T {Ψ} ψ 1 2 H (combining eqs. (A.10.� � ) and (A.10. )) �� � � � � � =δ �T (�H, � )ψ |T (�, �)ψ � . C P H (�) � (�) � ∗ ∗� (�) C P W W W � H W For any�, we define := Vect T (�, �)ψ � (�, �)∈ × . Each is thus stable ∗ ∗ H H (�) W W under T � × �. The previous computation shows(�) that(�) the are mutually orthogonal and W � that thereI exists, for each�, an isometric injectionI : → such that: (�) � (�)� W W � � ∀ι ∈ , I T {ι} ψ = T {ι} ψ . I & Then, using that: 1 2 W 1 W 2 W 1 2 � 1 � 2 � 1 2 ∀ι , ι ∈ ,T {ι }T {ι }=T {ι � ι }I T {ι }T {ι }=T {ι � ιC}, P H � (�) � � � (�) � ∗ ∗� (�) W W � W W � W together with the density of T {ι} ψ � ι ∈ = Vect T (�, �)ψ � (�, �)∈ × in , I we get: (�) (�) W � ∀ι ∈ , I T {ι}=T {ι} I .

Now, we define: H H ⊥ rest � (�)� W � W := � ,

H H H so that: rest (�) W W � W = ⊕ � H C P H (�) ∗ ∗ W W (as canrest be seen as a consequence of Riesz lemma). Since each is stable under T � × �, W so is . Next, we write:

rest � (�)� W W � W ζ =ζH + � ζ , H

rest rest (�) (�) W W W W withζ ∈ and ∀�, ζ ∈ . In particular, we have: 2 2 2 � rest� � � (�)� W � W � � W � �ζ � = ζ + � �ζ � , H (�) W W so there can be only(�) countably many(�) non-zeroζ (actually, the cyclicity ofζ together with the W W stability of each requires allζ to be non-zero, so this even implies that� takes value in a C P H countable set). Using prop. A.9.2, we have: ∗ ∗ W W W W ∀(�, �)∈ × ,W(�, �)= �ζ |T (�, �)ζ �

46 H H rest rest (�) � (�) � � W � W � � W � W � W W � W W =Hζ T (�, �)ζ + � ζ � T (�, �)ζ .

(�) (�) (�) (�) � W � Thus, definingζ :=I ζ ∈ for each�, the isometric and inter twining properties ofI C P yield: H H ∗ ∗ rest rest (�) (�) � � W � W � � � � � � W � W � � � ∀(�, �)∈ × ,W(�, �)= ζ T (�, �)ζ + � ζ T (�, �)ζ .

The bound: H � 2H (�) 2 (�) 2 � � � � � � � � � � � W � W W � ζ = � �ζ � �ζ � < ∞, H

� allows us to define a traceclass, (self-adjoint) positive semi-definite operatorρ on by: � � (�) � � (�)� � � � � ρ := � ζ ζ .

C P H Then, we get: ∗ ∗ rest rest C P � � ρ ∀(�, �)∈ × ,W(�, �)=W (�, �)+Tr [ρT (�, �)] =W H (�, �)+W (�, �), ∗ ∗ rest rest rest � W � W CW � PW H where, for any (�, �)∈ × ,W (�, �) := ζ � T (�, �)ζ . C ∗ ∗ rest W C P H W C P T being a weakly continuous, unitary representation of × on , the functionW ∗ ∗ rest 2 � � ∗ ∗ H � W � W H is continuous × → and it is bounded by �ζ � . Now, for any (� , � )∈ × , � � rest rest (�) (�) (�) W W W W W W T (� , � )ζ ∈ , so it is in particular orthogonal to eachψ , asψ ∈ . But since � (�)� W W ψ � is anC orthonormalP basis of the image of the orthogonal projection T {Ψ} , this implies: � � ∗ ∗ � � rest W W W ∀(� , � )∈ × ,T {Ψ}T (� , � )ζ = 0 , C P H and therefore: � � �� �� ∗ ∗ �� �� rest � � rest � W WH � W W W � W ∀(� , � ),(� , � )∈ × , T (� , � )ζ � T {Ψ}T (� , � )ζ = 0 . � �� C P W On the other hand, we have, for anyφ , φ ∈ : H � � �� �� ∗ ∗ �� �� �� � � � W W W W ∀(� , � ),(� , � )∈ × , �T (� , � )φ |T {Ψ}T (� , � )φ H� = C P � �� �� �� � � � W ∗ ∗ W W W = × ��µ(�, �)Ψ(�, �)�T (� , � )φ |T (�, �)T (� , � )φ � H C P �� �� �� �� � � � −�ξ(� ,� ;�,�) �ξ(�−� ,�−� ;� ,� ) �� � �� � �� � W ∗ ∗ W = × ��µ(�, �)Ψ(�, �)� � �φ |T (�+� − � , �+� − � )φ � H �� �� � � C P � �� � �� −�ξ(� ,� ;� ,� ) � �ξ(�,�;� +� ,� +� ) �� � �� � �� � W ∗ ∗ W =� × ��µ(�, �)Ψ(�, �)� �φ |T (�+� − � , �+� − � )φ � .

Thus, we have: C P C P ∗ ∗ � �ξ(�,�;� 1,�1) rest � � 1 1 ∗ ∗ � � ∀(� , � ),(� , � )∈ × , × ��µ(�, �)Ψ(�, �)� W (�+� , �+� ) = 0 , rest ρ so, from lemma A.11,W ≡ 0, henceW=W . 47 H H 1 2 � Uniqueness. HLetρ , ρ be two traceclass, (self-adjoint) positive semi-definite operators on such ρ1 ρ2 � � thatW =W . Like above, we define a bounded linear operator T {Ψ} on satisfying, for 1 2 � anyφ , φ ∈ : H H �C P � � 1 � 2 ∗ ∗ 1 � 2 �φ |T {Ψ} φ � := × ��µ(�, �)Ψ(�, �) �φ |T (�, �)φ � .

� Using the bases introduced in the proof of lemma A.11 together with the expression for T (�, �) from H prop. A.5, this can be rewritten as: 1 � 2 � �φ |T {Ψ} φ � = � 1 � 1 � � 1 � � � � 1 1 1 1 1 � 2� 1 �� = � �� � � � �� �� � � � �� �� � � � �� φ (� � )φ (� +� )� × (2π) � � � � � � � � � � �� � � +� � 1 � × exp − exp � � � + � � 4 2 � � � 1 � 1 � � � � � � 1 1 1 1 1 � 2� 1 �� � � = � �� � � � �� �� � � � �� φ (� � )φ (� +� )� exp − × (2π) � � 4 � 1 � � � � � �� � � 1 � × �� � � � �� exp − +�� � + � � 4 2 (using Fubini’s theorem as the integral is absolutely convergent) � � � 1 � 1 � � � � � � 1 � 1 1 1 1 � 2� 1 �� � � = √ �� � � � �� �� � � � �� φ (� � )φ (� +� )� exp − × π � � � � 4 � 1 1 � (2� +� ) (2� +� ) × exp − 4 � � � � � 1 � 1 � � � � 1 1 2 2 � 1 � 1 1 2 2 1 1 � 2 2 � � � +� � = √ �� � � � �� �� � � � �� φ (� � )φ (� � ) exp − π 2 � � � H H 2 1 (with the change of variables� :=� +� ) 1 � � � 2 � H = �φ | ψ � �ψ | φ � . 1 2 � � N � � Since this holds for anyφ , φ ∈ , we have T {Ψ}= |ψ � � ψ |. � (1)� 1 k HNow, from the spectral theorem, there existN ∈ ∪ {∞}, an orthonormal family φ k

∗ ∗ 1 1 2 H2 Thus, we get, for any (� , � ),(� , � )∈ × : � 1 1 � 1 � 2 2 � � �T (� , � )ψ | ρ T (� , � )ψ � = H � (1) � � (1)� � (1) � � k � 1 1 � � k k � � 2 2 � � = k

48 C P H (1) � −�ξ(�,�;� 1+�2,�1+�2)+�ξ(�1,�1;� 2,�2) (1) (1) � � � 1 1 � k ∗ ∗ k � � 2−� 2−� k � = k

(using Fubini’s theorem with the discrete measure on H{k < N}) C P 1 2 1 2 1 1 2 2 � −�ξ(�,�;� +� ,� +� )+�ξ(� ,� ;� ,� ) � 1 1 ∗ ∗ 1 � 2−� 2−� = × ��µ(�, �)Ψ(�, �)� Tr [ρ T (�+� , �+� )]

1 1 2 2 C P 1 2 1 2 �ξ(� ,� ;� ,� ) � −�ξ(�,�;� +� ,� +� ) 1 1 1 ∗ ∗ ρ 2−� 2−� =� × ��µ(�, �)Ψ(�, �)� W (�+� , �+� ).

H 2 ρ1 ρ2 As the same holds forρ ,W H=W implies:H � (0) � � H � 1 � C 2 P� ∀φ, φ ∈ , �φ | ρ φ� = �φ | ρ φ� , (0) ∗ ∗ � H� � C P where := Vect{T (�, �)ψ |(�, �)∈ × }. (0) ⊥ � � �� �� ∗ ∗ � � � Finally, letφ∈ H. Like above, we thenH have, for any (� , � ),(� , � )∈ × : �� �� � � � � � � � � 0 =�T (� , � )ψ | φ� �φ |T (� , � )ψ � H � � �� �� C P � �� � �� −�ξ(� ,� ;� ,� ) � −�ξ(�,�;� +� ,� +� ) � �� � �� � ∗ ∗ � =� × ��µ(�, �)Ψ(�, �)� �φ |T (�+� − � , �+� −H � )φ� ,

C P � � so applying lemma A.11 to the bounded,H continuous function (�, �) �→ �φ |T (�, �)φ� yields: ∗ ∗ � � H H ∀(�, �)∈ × , �φ |T (�, �)φ� =H 0, 2 (0) � � � � � and in particular �φ� = �φ |T (0, 0)φ� = 0, soφ = 0. Hence, = and, therefore, 1 2 ρ =ρ .

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