Effective Casimir Conditions and Group Coherent States Martin Bojowald1∗ and Artur Tsobanjan1,2† 4 1 Institute for Gravitation and the Cosmos, The Pennsylvania State University, 1 104 Davey Lab, University Park, PA 16802, USA 0 2 2 American University, 4400 Massachusetts Ave NW, Washington, DC 20016, USA n a J 1 Abstract 2 Properties of group coherent states can be derived “effectively” without knowing ] h full wave functions. The procedureis detailed in this article as an example of general p methods for effective constraints. The role of constraints in the present context is - h played by a Casimir condition that puts states within an irreducible representation t of aLie group(or, equivalently, on aquantization of aco-adjoint orbitof thedualLie a m algebra). SimplificationsimpliedbyaCasimircondition,comparedwithgeneralfirst- [ class constraints, allows one to show that the correct number of degrees of freedom is obtained after imposing the condition. When combined with conditions to saturate 1 v uncertainty relations, moments of group coherent states can be derived. A detailed 2 example in quantum cosmology (cosmic forgetfulness) illustrates the usefulness of 5 the methods. 3 5 . 1 1 Introduction 0 4 1 There are several different definitions of coherent states based on group-theoretical proper- : v ties, whichallplayimportantrolesindiverse areasoftheoretical physics. Explicit examples i X of wave functions of coherent states can show aspects of quantum evolution and semiclassi- r a cal phenomena in a clean way, but it is not always possible to find such explicit realizations in general-enough terms. However, physical properties can still be extracted if one works with moments of a state instead of wave functions, a description which is the basis also of canonical effective theory [1, 2]. Dynamical equations for the moments follow from gener- alizations of Ehrenfest’s equations, while coherence is implemented by the condition that moments saturate uncertainty relations. The main link to group coherent states, studied in this article, is the imposition of an additional Casimir condition that restricts moments to states that belong to an irreducible representation of a Lie group describing the quantum system (or the quantization of a co-adjoint orbit in the dual Lie algebra of the group). Imposing the Casimir to be constant can be interpreted as a constraint on the origi- nal (non-symplectic) phase space with a Poisson structure given by the dual Lie algebra, ∗e-mail address: [email protected] †e-mail address: [email protected] 1 suggesting that methods for effective constraints [3, 4] can be used. Because a Casimir operator Cˆ commutes with all other operators, additional simplifications compared with ˆ ˆ general first-class constraints arise: (i) expectation values OC , which featureprominently in effective constraints, equal the symmetric version 1 OˆhCˆ +iCˆOˆ and are guaranteed to 2h i be real for self-adjoint Oˆ and Cˆ, and (ii) no gauge flow δ Oˆ /δǫ = [Oˆ,Cˆ] /i~ generated h i h i by a Casimir constraint need be considered. (On a non-symplectic phase space, first-class constraints do not necessarily generate gauge flows.) These simplifications allow us to confirm, to all orders in a semiclassical or moment expansion, that the correct number of quantum degrees of freedom is left after imposing the Casimir condition. In a second step, we then combine the Casimir condition with the requirement that uncertainty relations be saturated, restricting moments to those of a group coherent state. Higher orders of moments are more difficult to manage at this level, but we will be able to demonstrate several interesting relationships between the different conditions imposed. Quantum cosmology presents an example in which standard group coherent states are notalwaysavailableingeneralterms, thatiswithfullsqueezing, whiletheeffectivemethods elaborated here do apply. We end our article with a detailed discussion, clarifying the contentious issue of cosmic forgetfulness [5] which posits that certain pre-big bang models in loop quantum cosmology [6] suffer from a severe lack of control on the pre-big bang state. This issue has lost its urgency with the recent discovery that the same models, when embedded consistently in a setting with perturbative inhomogeneity, lead to signature change at high density [7, 8], so that no state can be evolved deterministically through the big bang. Nevertheless, the issue of cosmic forgetfulness may still be of interest from a mathematical perpective. 2 Effective Casimir constraints For simplicity, we consider the case of a single Casimir condition C, imposed on a non- symplectic phase space so that the submanifold C = const is symplectic. The phase-space function C can be seen as one coordinate of a Poisson manifold, such that C itself is a Casimir function in the Poisson sense, and submanifolds C = const are the symplectic leaves. Since C, by definition, has a vanishing Poisson bracket with any other function on the Poisson manifold, it does not generate a Hamiltonian gauge flow when viewed as a constraint. Nevertheless, it can be identified as a first-class constraint owing to the non- symplectic nature of the phase space. (Notions equivalent for symplectic geometry and usually associated with first-class constraints, such as the properties of non-trivial gauge flows and symplectic properties of constraint sets, may no longer be equivalent for Poisson manifolds. The generalization of standard definitions therefore requires some care [9].) As an example, we may look at a 3-dimensional manifold equipped with coordinates (or basic functions) V,J ,J and Poisson brackets V,J = J , V,J = J and + − + − − + { } { } − J ,J = V, for instance interpreted as the (dual) Lie algebra of sl(2,R). The Casimir + − { } − function reads C = J2 + J2 V2. Another way to interpret the same system is to use + − − partially complex variables V with J = J +iJ and J∗ = J iJ . One can then realize + − + − − 2 the Poisson brackets V,J = iJ , V,J∗ = iJ∗ , J,J∗ = 2iV (1) { } − { } { } by functions V and J := V exp( iP) of canonical variables V and P with V,P = 1. − { } SettingtheCasimir functiontozero, C = JJ∗ V2 = 0, thenamountstoarealitycondition − for P. In this form, the Casimir condition plays a role in some cosmological models [10, 11]. Upon passing on to the quantum treatment of the system, the role of C as a first-class constraint (though one on a non-symplectic phase space) allows us to employ the effective- constraintmethodsdevelopedin[3,4,12]. Tothisend, weviewthecorresponding quantum systemalgebraically, basedonthecommutatorsofbasicoperatorsquantizing(1). Thestate space of the algebra can be formulated geometrically by making use of the expectation- value functional , applied to all polynomials in basic operators. Expectation values of h·i products of operators are identified as moments ∆(VaJbJc) := (Vˆ Vˆ )a(Jˆ Jˆ )b(Jˆ Jˆ )c (2) + − h −h i + −h +i − −h −i iWeyl−ordered of the basic operators in a state considered. Any product that is not Weyl-ordered can be rearranged as a sum of Weyl-ordered terms, some of which with explicit ~-factors. The set (2) therefore prescribes the expectation values for all polynomials in the basic variables. A Poisson geometry of the quantum phase space, given by expectation values and moments of basic operators, is provided by the commutator, defining a Poisson bracket on expectation-value functionals by [Aˆ,Bˆ] Aˆ , Bˆ = h i , (3) {h i h i} i~ extended to all functions by the Leibniz rule. One element of this algebra is the Casimir or constraint operator Cˆ to be imposed as a constraint, Cˆ ψ = 0. (For a non-zero Casimir, we can simply redefine the constraint | i operatorasCˆ const1ˆ without changing commutators.) Evaluatedinanystateannihilated − by Cˆ, the expectation values polCˆ must all vanish for arbitrary polynomials pol in the h i basic operators. For one quantum constraint operator we obtain an infinite number of constraint functions on the qucantum phase space. For multiple ones, if thecquantum constraints are first class, the system of effective constraints is first-class in the phase- space sense [3]. In the special case of a Casimir constraint, all effective constraints are Casimir functions on the Poisson manifold defined by (3): it is straightforward to see that a constraint of the form polCˆ = 0 weakly Poisson commutes with all quantum ˆ h i phase-space functions if C commutes with all operators. For first-class constraints onscymplectic phase spaces, the viability of these methods has already been demonstrated, addressing also the problem of time [13, 14, 15]. In general, the ordering of effective constraints in the specific form C := polCˆ (4) pol h i c 3 is important for the system of constraints in order to remain first class and to vanish in physical states. We may assume symmetric polynomials without loss of generality, because re-ordering terms would just contribute quantities proportional to lower-order constraints. However, even if the basic operators and Cˆ are assumed to be self-adjoint with respect to some -relationon the basic algebra, which we will do in what follows, the ordering in (4) is ∗ in general not symmetric, leading to the possibility of complex-valued effective constraints. For Casimir conditions, on the other hand, we have [pol,Cˆ] = 0 by definition, so that we can substitute the symmetric ordering 1 polCˆ +Cˆpol without changing the expectation 2h i value. We are therefore dealing with real-valued effectcive constraints for all polynomial or moment orders. Moreover, Casimir conditcions are ecasier to implement because no gauge flows need be considered: we have a vanishing [Oˆ,pol]Cˆ Oˆ , polCˆ = h i 0 {h i h i} i~ ≈ c on the solution space of the effectivecconstraints, again using the commutation property of a Casimir. This simplification is the main reason that allows us to directly test effective methods at higher orders in moments. 2.1 Removing degrees of freedom In classical systems, Casimir constraints remove phase-space degrees of freedom so that, in the absence of a gauge flow, the constraint surface is symplectic. The dimension of a symplectic manifold is restricted to be even, and its quantization corresponds to a fixed pattern of expectation values and moments: for every canonical pair (q,p) there are two independent expectation values and, starting with n = 2 reaching ad infinitum, a tower of n+1 moments for every integer n, defined as in (2) for a single canonical pair of basic operators: ∆(qapb) := (qˆ qˆ )a(pˆ pˆ )b . (5) Weyl−ordered h −h i −h i i Semiclassically, a moment of order n behaves like O(~n/2), giving rise to a general semi- classical expansion with a finite number of degrees of freedom per canonical pair at any fixed order n. If this pattern is violated, one cannot interpret the degrees of freedom in the usual way of quantum mechanics, indicating either extra constraints if therearenot enough free moments or spurious degrees of freedom if too many variables remain unrestricted. In particular, for a single Casimir constraint we must, by imposing C = 0, eliminate suffi- pol ciently many variables to leave a certain number of canonical pairs with their characteristic moments. For the example of one quadratic Casimir constraint on a 3-dimensional Poisson man- ifold such as the one of (V,J ,J ), one can heuristically see that the reduction is correct, + − working locally and assuming that all effective constraints are independent. Taking an expectation value Cˆ = 0 restricts the basic expectation values to two independent ones h i ˆˆ (for fixed second-order moments). The next order of constraints is obtained for AC = 0 h i 4 with Aˆ one of the three basic operators (or a linear combination of them). Fixing third- order moments, we obtain three constraints for the six second-order moments of three independent basic operators, just enough to restrict the second-order moments to three independent ones: two fluctuations and one correlation. To see the correct reduction in numbers for all orders, we note that one can, at least locally, view the Casimir function as a coordinate on the Poisson manifold transversal to symplectic leaves. Instead of using basic phase-space functions such as V, J and J , we can locally transform to Casimir– + − Darboux coordinates of a canonical pair (q,p) on the symplectic leaf and C transversally. This decomposition is relevant also for the counting of degrees of freedom of the corre- sponding quantum system. As for constraints, in addition to the expectation value of Cˆ, ˆ we must remove all moments of C with itself (such as the fluctuation ∆C) as well as all cross moments with qˆand pˆ. These cross moments are nothing but the effective constraints polCˆ = 0, with pol a polynomial of degree n to restrict C-moments of order n+1. Since h i there are as many cross-moments as constraints of this form, we obtain the correct number ofcdegrees of freedocm provided the constraints are all independent (and the local argument remains valid in quantum theory). In most cases, and always if the coordinate change to Casimir–Darboux coordinates is not global, the transformation from (V,J ,J ) to (q,p,C) is non-linear and relationships + − between (V,J ,J )-moments and (q,p,C) are non-trivial. For instance, a moment of low + − order in one system may involve moments of all orders in the other. For a global statement about parameter counting, more-detailed considerations must be performed. 2.2 Counting truncated constraint conditions For more generality, we work with a general finite-dimensional algebra of generators xˆ , i = i 1,2,...M, denoting their expectation values (or, occasionally, the corresponding classical values) by x , i = 1,2,...M. To define the moments, we introduce i ∆x := xˆ x 1ˆ i i i − and their Weyl-ordered products c i1 i2 iM eˆ = eˆ := ∆x ∆x ...∆x , ~i (i1,i2,...iM) 1 2 M Weyl−ordered (cid:16) (cid:17) which form a linear basis for the (extendced)calgebra.cWe use a compact notation in which ~i is an M-tuple of non-negative integers. Expectation values ∆(~x~i) := ∆(xi1 xiM) = eˆ . (6) 1 ··· M h ~ii of the basis elements are the moments. For later use we define the degree ~i := M i | | n=1 n and a partial ordering~i ~j if i j for all n, so that~i >~j if~i ~j and~i =~j. We will use ~i! to denote (i !i !...i ≥!) and,nas≥alrneady defined in (6), ~x~i = xi≥1 xiM. B6 y constPruction, 1 2 M 1 ··· M ∆(~x~i) = 0 for all~i with ~i = 1. | | 5 Interpreting the xˆ as basic operators of a quantum system, we assume that their i commutator algebra is linear (and follows from a direct quantization of Poisson brackets of the corresponding classical functions): [xˆ ,xˆ ] = i~ǫ kxˆ (7) i j ij k where ǫ k are structure constants, identical to the structure constants of the classical ij Poisson algebra. For a semisimple Lie algebra, which we will assume in an example later in thisarticle, ǫ withthethirdindexpulleddownbycontractionwiththeKillingmetric, are ijk totally antisymmetric. The case of the Weyl algebra [qˆ,pˆ] = i~1ˆ for a single canonical pair, as another important example, does not seem to fall within the current setting. However, we could treat qˆ, pˆand 1ˆ as three ‘generators’ or, alternatively, start from the Heisenberg algebra [qˆ,pˆ] = i~zˆ and enforce a constraint Cˆ = zˆ 1ˆ in the way described below. − It follows that ∆x ,∆x = i~ ǫ kxˆ = i~ ǫ k ∆x +x 1ˆ (8) i j ij k ij k k h i Xk Xk (cid:16) (cid:17) d d d with the two right-hand-side terms respectively of polynomial degree 1 and 0 in the oper- ators ∆x . (This relation is not formulated for linear operators, owing to the presence of i expectation values that depend on a state. Nevertheless, the usual rules can be applied if one tredats x , during a calculation, as some real number and identifies it with the expec- k tation value of xˆ only in the final results.) In this form, the relation plays an important k role in considerations of orderings: it allows one to apply commutators to symmetrically order any product of ∆x -s by adding terms of lower polynomial degree and proportional i to powers of ~, c eˆeˆ = eˆ +~ (1)β ~k eˆ +~2 (2)β ~k eˆ + ~i ~j (~i+~j) ~i,~j ~k ~i,~j ~k ··· ~kX<~i+~j ~kX<~i+~j |~k|≤|~i+~j|−1 |~k|≤|~i+~j|−2 where (n)β ~k are polynomials in the expectation values x . ~i,~j i 2.2.1 Constraints and truncation Following the general procedure of effective constraints, the Casimir condition of an oper- ator Cˆ is imposed by demanding f(xˆ ,xˆ ,...,xˆ )Cˆ = 0 for all polynomial functions f. 1 2 M h i These are infinitely many conditions for infinitely many moments, but working order by order in the moments (or in a semiclassical expansion) one can truncate these conditions systematically using the basis eˆ . We introduce { ~i} C := eˆ Cˆ = 0, ~i ZM ~i h ~i i ∀ ∈ + with the convention eˆ = 1ˆ, so that C := Cˆ . In order for the truncation to be consistent, 0 0 h i we must suitably combine the truncation of constraint functions according to the degree ~i with a truncation of variables that feature in the system. | | 6 Assume that we truncate at some order N 2 of the semiclassical expansion: we drop moments of degree greater than N, that is all≥∆(~x~i) with ~i > N. The truncation of the | | system of constraints is more subtle. All C are linear functions of the moments ∆(~x~j), ~i but they contain terms of three different types. It is useful to assign orders to these terms according to their type, as follows: A term of the form f(x ,x ...x )∆(~x~i), where f is a polynomial in the expectation 1 2 M • values not proportional to the classical constraint, is assigned semiclassical order equal to ~i . | | A term of the form C(x ,x ...x )∆(~x~i), where C is the classical polynomial expres- 1 2 M • sion for the constraint, requires a special treatment. As an exception to the previous point, it is assigned semiclassical order ~i + 2. This exception can be understood | | from the fact that C(x ,x ...x ), although it may not vanish exactly when quan- 1 2 M tum corrections are included in C = C +O(~), is of the order ~ when the quantum 0 constraint C = 0 is imposed. See also [3]. 0 Terms such as ~nf(x ,x ...x )∆(~x~i), which may arise upon reordering algebra el- 1 2 M • ements, are assigned semiclassical order equal to ~i +2n. | | All these orders are consistent with the correspondence of order n to terms O(~n/2) as it applies to the moments. To truncate, terms in any expression for a constraint function are dropped when they are of semiclassical order higher than N. 2.2.2 Counting degrees of freedom We begin by counting the degrees of freedom of an unconstrained system generated by polynomials in M basic variables (for a phase space of dimension M). We have M expec- tationvalues and, ateach semiclassical order N 2, thedegrees offreedom arerepresented by the independent functions ∆(~x~i) with ~i = N≥. | | The task of counting degrees of freedom is simplified by rephrasing the problem: The number of such variables is given by the number of M-tuples of non-negative integers with ~i = N, aquantitythatwewillcall (N). Eachsuch M-tupleisproducedbyconsidering M | | N a row of N +M 1 identical objects, marking M 1 of them to serve as partitions. The − − value of i is then the number of unmarked objects between partition (n 1) and partition n − n. (Partitions 0 and M are assumed to be at the ends.) With this rephrasing we directly obtain N +M 1 (N) = − . M N M 1 (cid:18) − (cid:19) Each constraint, in the absence of gauge flows, removes a single classical degree of freedom, and we expect the tower of effective constraint conditions to remove the corre- sponding moments. After the constraints are imposed, there should be as many degrees of freedom as for a system with M 1 generators, with (N) = N+M−2 free variables − NM−1 M−2 (cid:0) (cid:1) 7 at each semiclassical order. Using the identity a a−1 = a−1 , we conclude that the b − b b−1 required number of independent conditions at each order should be N+M−2 . (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) M−1 In order to show consistency of the effective procedure, we now proceed to counting the (cid:0) (cid:1) number of conditions independently. This question is more difficult to answer because con- straint conditions C generally mix terms of different orders. As a first step, we show that, ~i truncated at a given order, the system of constraints is finite. The number of constraints at each order is then the number of additional non-trivial constraint conditions that arise when we raise the truncation order by one. A general constraint function has the form 1 ∂|~j−~i|C C = ∆(~x~j)+~ (1)α ~j ∆(~x~j) ~i (~j ~i)!∂xj1−i1...∂xjM−iM C~i X~j≥~i − 1 M |~j|X≥|~i|−1 +~2 (2)α ~j ∆(~x~j)+...+~|~i| (|~i|)α ~j ∆(~x~j) (9) C~i C~i |~j|X≥|~i|−2 X~j Here(n)α ~j arecoefficients of semiclassical order zero, polynomial intheexpectation values C~i x . The first sum comes from the Weyl-symmetric part of the element eˆ Cˆ. Subsequent i ~i sums arise from its components that are antisymmetric in one, two and more adjacent pairs of moment-generating elements ∆x . Each antisymmetric pair can be reduced by i using the commutation relations, producing the powers of ~. For a Casimir constraint, Cˆ ~i is guaranteed to be real; therefore, therce must be an even number of commutators applied in each re-ordering step and we have only even powers of ~, or (n)α ~j = 0 for odd n. C~i The important feature of the above expansion is that the lowest semiclassical order terms are C∆(~x~i) and ∂C C¯ := ∆(~x(i1,...,ik−1,ik+1,ik+1,...,iM)). (10) ~i ∂x k k X The latter term has the lowest order, ~i + 1, keeping in mind that for the purposes of | | truncating the constraints, C is of order 2. After truncation at order N, constraints C = 0 ~i are satisfied identically for all ~i > N 1. This observation allows us again to rephrase | | − the counting problem: The number of non-trivial conditions up to order N is the same as the number of non-negative integer M-tuples of degree N 1 and less. As we go from − truncation at order N 1 to truncation at order N, this number changes by the number of − M-tuples of degree N 1, which is exactly the quantity (N 1) = N+M−2 required − NM − M−1 to be eliminated by the counting of degrees of freedom. (cid:0) (cid:1) Thus, provided the non-trivial constraint conditions remaining after truncation are functionally independent, they remove precisely one combinatorial degree of freedom in the quantum mechanical sense. In order to show independence, which is done in detail in [16], one considers the gradients d C¯ on the space of expectation values and { ∆ ~i}1≤|~i|≤N−1 moments. While the operator d takes the gradient with respect to the quantum moments ∆ only, thefunctional independence conditionsderived depend predominantly ontheclassical 8 form of the constraint function. For a semiclassical state, applying careful truncations as before, one can conclude: (i) The truncated constraints are independent as long as dC (the gradient taken with respect to the expectation values only) is not comparable to ~ or the moments in at least one coordinate direction, assuming that expectation values satisfy the classical constraint. This condition can be interpreted as the semiclassical analog of regularity of the constraints. (ii) For expectation values off the classical constraint surface, theconstraintfunctionsarefunctionallyindependent aslongas, forsomek,neither∂C/∂x k nor ∂N−2(C−1)/∂xN−2 are comparable to ~ or the moments. k The conditions may clearly be violated somewhere on the quantum phase space, but near the classical constraint surface terms such as 1/C and its derivatives diverge and ap- pearinthegradients. Thegradients, unlike theconstraints, canthusbeconsidered “large.” These conditions are sufficient, but not necessary, so that in some cases the constraints may be independent even if the conditions do not hold. To summarize, so long as the classical constraint is sufficiently regular, the truncated set of gradients d C¯ { ∆ ~i}1≤|~i|≤N−1 is linearly independent for expectation values x lying in some neighborhood of the clas- i sical constraint surface C = 0, leading to functional independence of the truncated set of constraint functions in that region. 3 Uncertainty relations For moments of a group coherent state, in a quantization of a co-adjoint orbit on the dual of the Lie algebra corresponding to (7), we require that uncertainty relations be saturated. Together with the Casimir condition, several equations are then to be solved. We first derive uncertainty relations, beginning the usual procedure familiar from text- books on quantum mechanics. We pick a pair (xˆ ,xˆ ) of basic operators, assumed self- i j adjoint as noted before. Starting with a generic state ψ in a Hilbert-space representa- tion of the basic algebra, we introduce three new states v := ∆x ψ, w := ∆x ψ and xi i xj j u := w (v w / v 2)v . The uncertainty relation for our pair of operators then xi,xj xj− xi· xj || xi|| xi follows from d d v w 2 0 u 2 = w 2 | xi · xj| , (11) ≤ || xi,xj|| || xj|| − v 2 || xi|| with saturation if and only if u = 0. xi,xj Inserting our specific expressions for u and w in terms of ψ, we can express each xi xj term in the Schwarz inequality v 2 w 2 v w 2 in terms of moments. We easily || xi|| || xj|| ≥ | xi · xj| obtain v 2 = ∆(x2) and w 2 = ∆(x2) and, with a little more re-ordering work, || xi|| i || xj|| j ~2 v w 2 = (xˆ x )(xˆ x ) 2 = ∆(x x )+ 1i~ǫ kx 2 = ∆(x x )2 + (ǫ kx )2. | xi · xj| |h i − i j − j i| | i j 2 ij k| i j 4 ij k Uncertainty relations ~2 ∆(x2)∆(x2) ∆(x x )2 (ǫ kx )2 (12) i j − i j ≥ 4 ij k in standard form then follow. 9 3.1 Higher orders Higher-order moments are restricted by higher-order uncertainty relations. We can derive thembyusing non-linearpolynomialsinthe∆x todefinestatesv := pol ψ andw := i pol1 1 pol2 \ pol ψ and proceeding as before. Without loss of generality, we require pol to be Weyl- 2 1/2 ordered. Unlike (12), these higher-order reldations mix moments of diffedrent orders. (For rdecent work on higher-order uncertainty relations with canonical basic operators, see [17].) The first ones beyond (12), for instance, involve moments of second, third and fourth order, obtained when pol is linear and pol quadratic (or vice versa). Using the relations 1 2 in the appendix, we compute w 2 = ∆(x2x2) (13) || xjxk|| j k 1 3 ~2 ǫ lǫ m∆(x x ) ǫ lǫ m∆(x x ) ~2ǫ lǫ m∆(x x ) jk jl k m jk kl j m jk jk l m −6 − − 4 (cid:0) (cid:1) (no sum over repeated lower indices) and 2 1 v ,w 2 = ∆(x x x ) ~2 ǫ lǫ m +ǫ lǫ m x (14) |h xi xjxki| i j k − 12 ik lj ij lk m (cid:18) (cid:19) 1 (cid:0) (cid:1) + ~2 ǫ l∆(x x )+ǫ l∆(x x ) 2 . ij k l ik j l 4 (cid:0) (cid:1) If we only consider terms of lowest order (six), this third-order uncertainty relation becomes ∆(x2x2)∆(x2) ∆(x x x )2 0. (15) j k i − i j k ≥ Unless there are third-order correlations between the basic variables, the only implication is that fourth-order moments of the form ∆(x2x2) must be positive, which already follows j k from their definition. The next order of uncertainty relations is more interesting and bounds fourth-order moments by a positive number. Defining ~xˆ~i = xˆi1xˆi2...xˆiM , in general, the 1 2 M Weyl−ordered leading-order contribution to the uncertainty relation implied by pol = ~xˆ~i and pol = ~xˆ~j (cid:0) (cid:1) 1 2 is of the form ∆(~x2~i)∆(~x2~j) ∆(~x2~i+2~j) U d d (16) − ≥ where U follows from the squared imaginary part of ~xˆ~i~xˆ~j ∆(~x~i+~j). The leading order is h i− obtained if exactly one commutator is applied in the re-ordering required to bring ~xˆ~i~xˆ~j h i into ∆(~x~i+~j). It has two contributions, one of degree 2(~i + ~j +1)(after taking the square) | | | | from the ∆x -term in (8), and one of order 2(~i + ~j ) from the x -term. The latter is k k | | | | always of the same order as the leading contribution on the left-hand side of (16), except when it hadppens to vanish. It always vanishes for third-order uncertainty relations (15) because it contains only the vanishing ∆(~x~i) with ~i = 1. For higher orders, however, U | | in (16) is non-zero to leading order, so that the familiar form of uncertainty relations is obtained, with the right-hand side non-zero and proportional to ~2. 10