In this paper we present a theory of reduction of quantum systems in the
presence of symmetries and constraints. The language used is that of
Lie--Jordan Banach algebras, which are discussed in some detail together with
spectrum properties and the space of states. The reduced Lie--Jordan Banach
algebra is characterized together with the Dirac states on the physical algebra
of observables

Falceto, F.

Ferro, L.

Ibort, A.

Marmo, G.

English

arXiv

In this paper we present a theory of reduction of quantum systems in the presence of symmetries and constraints. The language used is that of Lie-Jordan Banach algebras, which are discussed in some detail together with spectrum properties and the space of states. The reduced Lie-Jordan Banach algebra is characterized together with the Dirac states on the physical algebra of observables

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DOI 10.1393/ncc/i2013-11527-1Colloquia: MSQS 2012IL NUOVO CIMENTO Vol. 36 C, N. 3 Maggio-Giugno 2013Reduction of Lie-Jordan algebras: QuantumF. Falceto(1), L. Ferro(2)(3), A. Ibort(2) and G. Marmo(2)(3)(1) Departamento de F´ısica Teo´rica, Universidad de ZaragozaPlaza San Francisco s/n, 50009 Zaragoza, Spain(2) Departamento de Matema´ticas, Universidad Carlos III de MadridAvda. de la Universidad 30, 28911 Legane´s, Madrid, Spain(3) Dipartimento di Scienze Fisiche, INFN, Sezione di NapoliUniversita` di Napoli “Federico II”, Via Cintia Edificio 6, I–80126 Napoli, Italyricevuto l’11 Gennaio 2013Summary. — In this paper we present a theory of reduction of quantum systems inthe presence of symmetries and constraints. The language used is that of Lie-JordanBanach algebras, which are discussed in some detail together with spectrum proper-ties and the space of states. The reduced Lie-Jordan Banach algebra is characterizedtogether with the Dirac states on the physical algebra of observables.PACS 03.65.Fd – Algebraic methods.PACS 03.65.Ca – Formalism.PACS 03.65.Ta – Foundations of quantum mechanics; measurement theory.PACS 02.10.Hh – Rings and algebras.1. – IntroductionThis paper is the second part of two that jointly present a theory of reduction ofLie-Jordan algebras that can be used as an alternative to deal with symmetries and localconstraints in quantum physics and quantum field theories.The algebraic approach to quantum systems [1] has had a profound influence in boththe foundations and applications of quantum physics. The background for that approachis to consider a quantum system as described by a C∗-algebra A whose real part are theobservables of the system [2,3], and its quantum states are normalized positive complexfunctionals on it.A geometrical approach to Quantum Mechanics [4-7] has also been developed in thelast twenty years and has provided useful insight into such properties of quantum systemslike integrability [8], the intrinsic nature of different measures of entanglement [9,10], etc.Moreover a geometrical description of dynamical systems provides a natural settingto describe symmetries and/or constraints [11]. For instance, if the system carries asymplectic or Poisson structure several procedures were introduced along the years, likec© Societa` Italiana di Fisica 117118 F. FALCETO, L. FERRO, A. IBORT and G. MARMOMarsden-Weinstein reduction, symplectic reduction, Poisson reduction [12], reductionof contact structures, etc. However, it was soon realized that the algebraic approach toreduction provided a convenient setting to deal with reduction of classical systems [13,14].In the standard approach to quantum mechanics, constraints are imposed on thesystem by selecting subspaces determined by the quantum operators corresponding tothe constraints of the theory, Dirac states, and equivalence of quantum states were dealtwith by using the representation theory of the corresponding group of symmetries. Thispaper is devoted to present a fully algebraic approach to the reduction of quantumsystems.While the first part of this contribution is focused mainly on classical mechanics, thissecond part is devoted to the study of the quantum case. Lie-Jordan algebras are thecommon algebraic language in both parts.In the first two sections we introduce Lie-Jordan-Banach algebras and discuss someuseful spectral properties and the Cauchy-Schwarz inequalities. Then we will addressthe problem of the reduction of quantum systems as the reduction of Lie-Jordan-Banachalgebras. We will first consider the reduction in the presence of symmetries of the systemand then in the presence of constraints. A precise characterization of the reduced algebraand an explicit description of the reduced states will be obtained. In the quantum case,the reduction procedure is more subtle than the classical one, due to the requirementimposed on the algebra to be non (Jordan) associative.2. – Lie-Jordan-Banach algebrasWe recall here the definition of Lie-Jordan algebra already given in the first part ofthis contribution and then supplement the definition with a Banach structure. A Lie-Jordan algebra is a real vector space L equipped with the symmetric Jordan product ◦and the antisymmetric Lie bracket [·, ·] satisfying the Jacobi identity[[a, b], c] + [[c, a], b] + [[b, c], a] = 0,(1)Leibniz’s identity[a, b ◦ c] = [a, b] ◦ c + b ◦ [a, c],(2)and the associator identity(a ◦ b) ◦ c− a ◦ (b ◦ c) = 2 [b, [c, a] ] ,(3)for some  ∈ R.Combining (2) and (3) we obtain the weak associative property of Jordan algebras(a2 ◦ b) ◦ a = a2 ◦ (b ◦ a).(4)Definition 1 (Lie-Jordan-Banach algebra). A Lie-Jordan algebra L, complete with re-spect to a norm ‖ · ‖ that satisfies1) ‖a ◦ b‖ ≤ ‖a‖ ‖b‖,2) ‖[a, b]‖ ≤ ||−1‖a‖ ‖b‖,REDUCTION OF LIE-JORDAN ALGEBRAS: QUANTUM 1193) ‖a2‖ = ‖a‖2,4) ‖a2‖ ≤ ‖a2 + b2‖,sfor all, a, b ∈ L is called a Lie-Jordan-Banach algebra, LJB-algebra for short.Notice that if we are given a LJB-algebra L, by taking combinations of the twoproducts we can define an associative product on the complexification LC = L ⊕ iL.Specifically, we definexy = x ◦ y − i[x, y].(5)Such associative algebra equipped with the involution x∗ = (a + ib)∗ = a − ib and thenorm ‖x‖ = ‖x∗x‖1/2 is the unique C∗-algebra whose real part is precisely L [15].3. – Spectrum and states of Lie-Jordan-Banach algebrasDefinition 2. Let L be a LJB-algebra. The spectrum σ(a) of a ∈ L is defined as the setof those z ∈ R for which a− z1 has no inverse in L.Note that a LJB-algebra L is a complete order unit space with respect to the positiveconeL+ = {a2 | a ∈ L},(6)or equivalently an element is positive if its spectrum is positive.We shall in this section prove some useful properties of the spectrum and then theCauchy-Schwarz–like inequalities.Lemma 1.σ(a21 + a22 + λ[a1, a2]) ∪ {0} = σ(a21 + a22 − λ[a1, a2]) ∪ {0},∀ a1, a2 ∈ L and ∀λ ∈ R.Proof. For z 	= 0 the invertibility of a21 + a22 + λ[a1, a2] − z1 implies the invertibility ofa21 + a22 − λ[a1, a2]− z1. Namely, one computes that(a21 + a22 + λ[a1, a2]− z1)−1 = z−1{2a1 ◦ (b ◦ a1)− a21 ◦ b + 2a2 ◦ (b ◦ a2) +−a22 ◦ b + 2[a1, b ◦ a2] + 2a1 ◦ [b, a2]− 1}with b = {a21 + a22 + λ[a1, a2]− z1}−1.Lemma 2. If the spectrum σ(a21+a22+λ[a1, a2]) is negative, then a21+a22+λ[a1, a2] = 0∀ a1, a2 ∈ L and ∀λ ∈ R.Proof. Note that a21+a22−λ[a1, a2] = 2a21+2a22− (a21+a22+λ[a1, a2]) and then under theassumptions of the lemma σ(a21 + a22 − λ[a1, a2]) ⊂ R+. This implies, using the previouslemma, that σ(a21 + a22 + λ[a1, a2]) = {0} and therefore it is 0.120 F. FALCETO, L. FERRO, A. IBORT and G. MARMOTheorem 1.X = a21 + a22 − λ[a1, a2] ∈ L+,∀ a1, a2 ∈ L and ∀λ ∈ R.Proof. By applying the continuous functional calculus, it is well known that every X ∈ Lhas the decomposition X = X+ + X− [16, 17], where X+,X− ∈ L+ and X+ ◦ X− =[X+,X−] = 0. It follows that X3− = −(b21+b22−λ[b1, b2]) ≥ 0 with b1 = a1◦X−+λ[a2,X−]and b2 = λ[a1,X−] + a2 ◦ X−. But X3− = −2b21 − 2b22 + (b21 + b22 + λ[b1, b2]) which is anegative quantity and then in turn implies that X− = 0 and then X = X+ ≥ 0.The space of states S(L) of a Jordan-Banach algebra consists of all real normalizedpositive linear functionals on L, i.e.ρ : L → R,(7)such that ρ(1) = 1 and ρ(a2) ≥ 0, ∀ a ∈ L. The state space is convex and compact withrespect to the w∗–topology. We shall now prove the Lie-Jordan algebra version of theCauchy-Schwarz inequalities.Theorem 2. Let L be a LJB-algebra and ρ a state on L. Then if a, b ∈ L we haveρ(a ◦ b)2 ≤ ρ(a2)ρ(b2),(8)andρ([a, b])2 ≤ 12ρ(a2)ρ(b2).(9)Proof. Let λ ∈ R, then we have0 ≤ ρ((λa + b)2) = λ2ρ(a2) + 2λρ(a ◦ b) + ρ(b2).If ρ(a2) = 0 then ρ(a ◦ b) = 0 since λ is arbitrary. If ρ(a2) 	= 0, let λ = −ρ(a ◦ b)ρ(a2)−1,and the first proof is immediate.The second inequality is proved similarly by using the positivity of a2 + b2 + 2[a, b]as stated in Thm. 1.Example 1. The self-adjoint subalgebra Bsa of the algebra B(H) of bounded linear opera-tors on a Hilbert space H with the operator norm is a Lie-Jordan-Banach algebra and thestates are the positive linear functional on Bsa. Let ϕ be a continuous state with respectto the ultrastrong topology on B(H), then there is a positive linear trace class operatorρˆ ∈ Bsa such thatϕ(a) = Tr(ρˆa)(10)for all a ∈ Bsa.Conversely, if ρˆ is a positive trace class operator, then the functional a → Tr(ρˆa)defines an ultrastrongly continuous positive linear functional on Bsa.REDUCTION OF LIE-JORDAN ALGEBRAS: QUANTUM 1214. – Reduction by symmetriesWe can consider a symmetry associated to a Lie group G with Lie algebra g as a mapgˆ : G → Aut(L, ◦)(11)which assigns to each element g of the group, an automorphism U(g) of the Lie-Jordanalgebra algebra L. Let ξ ∈ g, then the infinitesimal generator of the symmetry is definedasξˆ(a) =ddsU(expsξ)(a)∣∣∣∣s=0(12)and is a Jordan derivation, in fact ∀ a, b ∈ Lξˆ(a ◦ b) = ξˆ(a) ◦ b + a ◦ ξˆ(b)(13)i.e. ξˆ ∈ Der(L, ◦). In addition ξˆ is a skew derivation [15,18] since it preserves the positivecone of the algebra ξˆ(L+) ⊂ L+ and hence, by a theorem on Jordan derivations [15],there exists J ∈ L such that ∀ a ∈ L,ξˆ(a) = [J, a].(14)It follows that ξˆ is also a Lie derivation.Then if D is a set of derivations describing the symmetries of the LJB-algebra, wecan introduce the subspaceFD = {a ∈ L | ξˆ(a) = 0 ∀ ξ ∈ D},(15)which is easily seen to be the Lie-Jordan subalgebra representing the symmetrical physicalalgebra of observables.The states on the reduced algebra are simply given by the restriction of the states ofthe full algebra on the subalgebra:S(FD) = {ρ|FD s.t. ρ ∈ S(L)}.(16)Example 2. A simple example is the quantum reduction of a free particle in R3 tothe S2 sphere. Consider the formal Lie-Jordan algebra L generated by the operators{rˆ, pˆr, θˆ, pˆθ, φˆ, pˆφ} and the symmetries D = {[rˆ, ·], [pˆr, ·]}. Then the reduced Lie-Jordanalgebra according to the above prescription isFD = {a ∈ L | [rˆ, a], [pˆr, a] = 0},that is the algebra of observables of a particle on the sphere which is generated by{θˆ, pˆθ, φˆ, pˆφ}.122 F. FALCETO, L. FERRO, A. IBORT and G. MARMO5. – Reduction by constraintsA quantum system with constraints is a pair (L, C) where the field algebra (L, ◦, [·, ·])is a unital LJB-algebra containing the constraint set C [19, 20]. The constraints selectthe physical state space, also called Dirac statesSD = {ω ∈ S(L) | ω(c2) = 0, ∀ c ∈ C},(17)where S(L) is the state space of L. We define the vanishing subalgebra V asV = {a ∈ L | ω(a2) = 0, ∀ω ∈ SD}.(18)Proposition 1. V is a non-unital LJB-subalgebra.Proof. Let a, b ∈ V. From (3) it follows:(a ◦ b)2 = 2 [b, [a ◦ b, a]] + a ◦ (b ◦ (a ◦ b)).(19)If we introduce c = [a ◦ b, a] and d = b ◦ (a ◦ b), eq. (19) becomes(a ◦ b)2 = 2 [b, c] + (a ◦ d).(20)From the inequalities (8), (9) it is easy to show that if ω(a2) = 0 thenω(a ◦ b) = 0 = ω([a, b]), ∀ b ∈ L.(21)Then if we apply the state ω to the expression (20), from (21) it follows:ω((a ◦ b)2) = 2 ω([b, c]) + ω(a ◦ d) = 0.(22)By definition of V, this means that ∀ a, b ∈ V, a ◦ b ∈ V.By applying the state ω to the relation(a ◦ b)2 − 2 [a, b]2 = a ◦ (b ◦ (a ◦ b))− 2 a ◦ [b, [a, b]],(23)we obtain ω([a, b]2) = ω((a ◦ b)2) = 0, that is [a, b] ∈ V, ∀ a, b ∈ V. Hence V is aLie-Jordan subalgebra and by definition of state ω(1) = 1, hence 1 /∈ V.V also inherits the Banach structure since it is defined as the intersection of closedsubspaces.We can use the vanishing subalgebra to give an alternative useful description of theDirac states.Proposition 2. With the previous definitions we haveSD = {ω ∈ S(L) | ω(a) = 0, ∀ a ∈ V}.(24)REDUCTION OF LIE-JORDAN ALGEBRAS: QUANTUM 123Proof. As V is a subalgebra and it contains C it is clear that the right-hand side isincluded into SD.To see the other inclusion it is enough to consider that for any state ω(a)2 ≤ ω(a2),therefore any Dirac state should vanish on V.Define now the Lie normalizer asNV = {a ∈ L | [a,V] ⊂ V},(25)which corresponds roughly to Dirac’s concept of “first class variables” [21].Proposition 3. NV is a unital LJB-algebra and V is a Lie-Jordan ideal of NV .Proof. Let a, b ∈ NV and v ∈ V. Then by definition of normalizer it immediately follows:[[a, b], v] = [[a, v], b] + [[v, b], a] ∈ V.Let us now prove that ∀ v ∈ V, v ◦ a ∈ V, this is V is a Jordan ideal of NV :ω((v ◦ a)2) = 2 ω([a, [v ◦ a, v]]) + ω(v ◦ (a ◦ (a ◦ v))),which gives zero by repeated use of properties (8) and (9).Then it becomes easy to prove that NV is a Jordan subalgebra:[a ◦ b, v] = [a, v] ◦ b + a ◦ [b, v] ∈ V.Finally, since the Lie bracket is continuous with respect to the Banach structure, it alsofollows that NV inherits the Banach structure by completeness.In the spirit of Dirac, the physical algebra of observables in the presence of the con-straint set C is represented by the LJB-algebra NV which can be reduced by the closedLie-Jordan ideal V which induces a canonical Lie-Jordan algebra structure in the quo-tient:L˜ = NV/V.(26)We will denote in the following the elements of L˜ by a˜.The quotient Lie-Jordan algebra L˜ carries the quotient norm,‖a˜‖ = ‖[a]‖ = infb∈V‖a + b‖,(27)where a ∈ NV is an element of the equivalence class [a] of NV with respect to the ideal V.The quotient norm provides a LJB-algebra structure to L˜.Example 3. Consider again the quantum reduction of the free particle constrained witha fixed value of the angular momentum. The constraint operator is expressed by c =Lˆ2 − 2l(l + 1)1. The set Hl of Dirac states is given by the convex span of the states{|m, l〉〈m, l|, m = 0,±1, . . . ,±l} and the vanishing subalgebra V consists of the operatorsvanishing on Hl:V = {aˆ ∈ L | aˆ|Hl = 0}.124 F. FALCETO, L. FERRO, A. IBORT and G. MARMOIt follows that the Lie normalizer NV is given by the operators preserving Hl:NV = {aˆ ∈ L | aˆ(Hl) ⊂ Hl},and the reduced algebra L˜ = NV/V = {˜ˆa : Hl → Hl}, which gives rise to a reducednonlinear dynamics out of an unreduced linear one.5.1. The space of states of the reduced LJB-algebra . – The purpose of this last sectionis to discuss the structure of the space of states of the reduced LJB-algebra with respectto the space of states of the unreduced one.Let L be a LJB-algebra and V its vanishing subalgebra with respect to a constraintset C and NV the Lie normalizer of V. Then we will denote as before by L˜ the reducedLie-Jordan-Banach algebra NV/V and its elements by a˜.Let S˜ = S(L˜) be the state space of the reduced LJB-algebra L˜, i.e. ω˜ ∈ S˜ meansthat ω˜(a˜2) ≥ 0 ∀ a˜ ∈ L˜, and ω˜ is normalized. Notice that if L is unital, then 1 ∈ NV and1+ V is the unit element of L˜. We will denote it by 1˜.We have the following:Lemma 3. There is a one-to-one correspondence between normalized positive linearfunctionals on L˜ and normalized positive linear functionals on NV vanishing on V.Proof. Let ω′ : NV → R be positive. The positive cone on L˜ consists of elements of theform a˜2 = (a + V)2 = a2 + V, i.e.K+eL = {a2 + V | a ∈ NV} = K+NV + V.(28)Thus if ω′ is positive on NV , ω′(a2) ≥ 0, henceω′(a2 + V) = ω′(a2) + ω′(V),(29)and if ω′ vanishes on the closed ideal V, then ω′ induces a positive linear functional on L˜.Clearly ω′ is normalized then the induced functional is normalized too because 1˜ = 1+V.Conversely, if ω˜ : L˜ → R is positive and we defineω′(a) = ω˜(a + V),(30)then ω′ is well-defined, positive, normalized and ω′|V = 0.Notice also that given a positive linear functional on NV there exists an extension ofit to L which is positive too.Lemma 4. Given a closed Jordan subalgebra Z of a LJB-algebra L such that 1 ∈ Z andω′ is a normalized positive linear functional on Z, then there exists ω ∈ S(L) such thatω(a) = ω′(a), ∀ a ∈ Z.Proof. Since L is a JB-algebra, it is also a Banach space. Due to the Hahn-Banachextension theorem [17], there exists a continuous extension ω of ω′, i.e. ω(a) = ω′(a),∀ a ∈ Z, and moreover ‖ω‖ = ‖ω′‖.From the equality of norms and the fact that ω′ is positive we have ‖ω‖ = ω′(1), butω is an extension of ω′ then ‖ω‖ = ω(1), which implies that ω is a positive functionaland satisfies all the requirements stated in the lemma.REDUCTION OF LIE-JORDAN ALGEBRAS: QUANTUM 125We can now prove the following:Theorem 3. The set SD(NV) of Dirac states on L restricted to NV is in one-to-onecorrespondence with the space of states of the reduced LJB-algebra L˜.Proof. In Prop. 2 we characterised the Dirac states as those that vanish on V. Combiningthis result with that of Lemma 3 the proof follows.∗ ∗ ∗This work was partially supported by MEC grants FPA–2009-09638, MTM2010-21186-C02-02, QUITEMAD programme and DGA-E24/2. 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Reduction of Lie-Jordan algebras: Quantum

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Reduction of Lie--Jordan algebras: Quantum

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