aa r X i v : . [ m a t h - ph ] O c t Discrete Linear Canonical Evolution
Jakub K´aninsk´y Charles University, Faculty of Mathematics and Physics, Institute of Theoretical Physics.E-mail address: [email protected]
October 7, 2020
This work builds on an existing model of discrete canonical evolution andapplies it to the general case of a linear dynamical system, i.e., a finite-dimensional system with configuration space isomorphic to R q and linearequations of motion. The system is assumed to evolve in discrete time steps.The most distinctive feature of the model is that the equations of motioncan be irregular. After an analysis of the arising constraints and the sym-plectic form, we introduce adjusted coordinates on the phase space whichuncover its internal structure and result in a trivial form of the Hamiltonianevolution map. For illustration, the formalism is applied to the example ofmassless scalar field on a two-dimensional spacetime lattice. Contents Introduction
Discrete dynamical systems have wide use in many areas of science including physics,engineering, biology, demography, finance and economics [1, 2, 3]. Sometimes this isbecause the problem at hand is naturally described in the discrete setting, sometimesrather because such formulation is far easier and better suited for an automated com-putation. Over the past decades, the latter reason have given rise to a vast body ofmathematics and computer science concerned with discretization [4]. In the most gen-eral regard, we can define a discrete dynamical system to be any system whose evolutionhappens in a series of discrete time steps. This is often realized by iterations of a singlefixed function on the phase space; where the function may be linear or non-linear inthe phase-space coordinates and the phase space may have one or more dimensions [5].However, one may consider a slightly richer setting, in which one allows the evolutionto be governed by a set of parameters which change in time.In this work we study discrete dynamical systems of arbitrary finite dimension withlinear evolution mappings and time-varying parameters. In doing so, we adopt an ap-proach which is profoundly physical. The evolution of our system will be defined by anaction functional, which gives rise to equations of motion in the canonical, Hamiltonianlanguage, and a phase space endowed with canonical coordinates as well as the sym-plectic form. The discrete linear canonical evolution advertised by the title is meant inexactly this context: as a classical discrete Hamiltonian evolution of a linear dynamicalsystem well known from physics. Our interest in this particular setting is motivated bydiscrete models of spacetime which represent an important complement to the standardformulation of general relativity based on continuous differential manifolds. These canbe used to study the behavior of classical and quantum fields on simplicial manifolds[6, 7, 8, 9, 10, 11]. Discrete spacetime models have also proven powerful for studyingcertain aspects of gravitation and are central to a few well-established approaches toquantum gravity [6, 12, 13, 14, 15, 16].Our starting point is the model of discrete canonical evolution introduced byB. Dittrich and P. A. H¨ohn in a series of articles [16, 17, 18] motivated by an ap-plication to simplicial gravity. Among other things, the authors provide two versions ofthe discrete canonical evolution, a global and a local one, and a key theorem about theconservation of the symplectic form. Notably, the formalism allows for irregular evolu-tion which need not always provide one-to-one correspondence between initial and finalstates. Thanks to that, it extends to systems with degenerate action or time-varyingnumber of degrees of freedom. The irregularity induces constraints as well as non-uniqueness of the classical evolution: a feature which is otherwise generally not verycommon in physics. Consequently, conservation of the symplectic form is only limited.The authors give an analysis of the arising constraints [18] which follows the path ofthe original Dirac’s classification [19].The problematics of discrete linear evolution was subsequently addressed in thearticle [20] where the author performed a detailed classification of constraints for thisspecial case. Therein, both constraints and degrees of freedom were classified into eighttypes according to their dynamical behavior, and a new basis on the phase space was de-fined which separates first class and second class constraints. A notion of reduced phasespace was introduced, followed by an analysis and classification of so-called effectiveactions . All these concepts are very useful in helping us understand the Hamiltonianevolution of discrete linear systems. The present work has the same aim, it only takes2 different path in pursuing it. Instead of constraints, we put the symplectic structureinto the foreground. We also limit our analysis to a single time-step, which allows fora much simpler viewpoint. There are number of places where the parallel between ourwork and [20] becomes significant, these will be pointed out in the text. Eventually, letus mention that the article [20] treats in its last section also the quantum case whichis beyond the scope of the present work.Within the paper, we only consider the global version of the discrete canonicalevolution. Its brief review can be found in Sec. 2. It will be applied strictly to the spe-cial case of linear dynamical system : a system of finite dimension, vector configurationspace and linear equations of motion. On one hand, the assumption of linearity is veryrestrictive; on the other hand, it allows us to analyze the evolution efficiently by meansof standard linear-algebraic tools. We recall the definition of linear dynamical systemalong with a couple of important details in Sec. 3. The only thing we alter about thisclassical definition is that we exchange the (implicit assumption of) continuous time fordiscrete time steps. The next Sec. 4 summarizes some essentials from linear algebra.We will take advantage of these throughout the work.Sec. 5 contains the main body of the work. As in [20], we use the assumptionof linearity to further develop the formalism given in the original papers [17, 18] andexplore the impact of irregularity on the dynamics of the system. However, unlikein [20], we employ singular value decomposition to describe the Hamiltonian evolutionmap and all the constraints explicitly. We provide an elementary rewriting of the linearcanonical evolution in terms of matrices, followed by an analysis of constraints withrespect to the symplectic structure. We further take the opportunity to build two spe-cial coordinate frames on the phase space which are fundamentally different from theframe described in [20]. These are adapted to the constraint surfaces of the time-step inquestion and result in a trivial evolution prescription. Both the previously existing andnewly introduced notions are given an explicit matrix form, and together constitute aneffective framework well suited for an immediate implementation. In the final part ofthe section, we discuss global solutions.Eventually, in Sec. 6 we provide a simple yet physically sound illustration of thetheoretical concepts introduced before. In order to do that, we consider a particularinstance of a discrete linear system: massless scalar field on a fixed two-dimensionalspacetime lattice. Similar models [6, 7, 11] have been studied since the pioneeringwork by Regge [21]. In our classical case, the formalism of discrete linear evolutionapplies straightforwardly. After a brief general description of the system, we offer a fewmini-examples of time-slice lattices and work out the corresponding evolution moves indetail. It is shown how the dynamics of the field is shaped by the geometry and causalstructure of the underlying spacetime, here described solemnly by its lattice represen-tation.
For a treatment of the canonical evolution of a discrete system we refer to the for-malism originally built to describe the evolution of simplicial gravity [16, 17, 18]. Itsfunctioning is briefly reviewed in this section. For more details, we suggest the readerconsult the original articles. 3et Q n be the configuration space of a discrete system at time-slice n with coordi-nates x nA (we will occasionally omit the index A ). Note that the configuration spacesat different time-slices need not be of the same dimension dim( Q n ) ≡ q n . The dynamicsof the system shall be described by the action S = t − X n =0 S n +1 ( x n , x n +1 ) (2.1)where the sum ranges over the individual time-slices n . The action contribution S n +1 governs the discrete time evolution move during the time-step between n and n + 1.We assume that the action is additive, so that the sum in (2.1) makes sense. (Thisis a restrictive condition since one could in principle consider non-additive actions; forinstance those that include interactions between more than two consecutive time-slices.These are ruled out by our additivity requirement.)Let us treat global time evolution moves, i.e., such moves that each of the variablesat a given time-slice is involved in the move and only occurs at this one time-slice, sothat neighboring time-slices do not overlap. For example, evolution moves in simplicialgravity which evolve between disjoint spacial hypersurfaces are global.Consider three consecutive time-slices n − , n, n + 1 and the boundary value prob-lem defined by the data at times n − n + 1. That is, we are given the boundarydata x n − and x n +1 and ought to extremize S = S n ( x n − , x n ) + S n +1 ( x n , x n +1 ) withrespect to x n . This yields the equations of motion0 = ∂S n ∂x n + ∂S n +1 ∂x n (2.2)which may or may not be uniquely solvable for x n as a function of x n − , x n +1 , depend-ing on whether the system under consideration is regular or irregular. An initial valueproblem can be treated in analogy by computing x n +1 from x n − , x n ; then the equa-tions of motion provide the Lagrangian time evolution L n : Q n − × Q n → Q n × Q n +1 .It may not, however, be defined on all of Q n − × Q n , nor map to all of Q n × Q n +1 , norbe unique in the presence of constraints.In order to describe the dynamics in canonical language, one may introduce discreteLegendre transformations . For an arbitrary time-slice n , we have S n : Q n − × Q n → R where Q n − × Q n is a fibre bundle. Pick a point q n − in Q n − . We denote the fibreover q n − by F n ( q n − ). Notice that F n ∼ = Q n . Choose a point f n in F n and a curve γ ( ε ) in F n such that γ (0) = f n . This allows us to define the post-Legendre transform F + S n : Q n − × Q n → T ∗ Q n by F + S n ( f n ) · dγ ( ε ) dε (cid:12)(cid:12)(cid:12)(cid:12) ε =0 = dS n ( γ ( ε )) dε (cid:12)(cid:12)(cid:12)(cid:12) ε =0 (2.3)Here it should be understood that dγ ( ε ) dε (cid:12)(cid:12)(cid:12) ε =0 is a vector in the tangent space T f n Q n ,and is being contracted with the covector F + S n ( f n ) belonging to the cotangent space T ∗ f n Q n . The point q n − is implicit in the equation, entering both the f n and γ ( ε ).Now exchange the roles of Q n − and Q n and choose f n − in F n − ( q n ). Let η ( ε ) be acurve in F n − such that η (0) = f n − . In analogy, we define the pre-Legendre transform − S n : Q n − × Q n → T ∗ Q n − by F − S n ( f n ) · dη ( ε ) dε (cid:12)(cid:12)(cid:12)(cid:12) ε =0 = − dS n ( η ( ε )) dε (cid:12)(cid:12)(cid:12)(cid:12) ε =0 (2.4)The cotangent bundles P n − = T ∗ Q n − and P n = T ∗ Q n are the phase spaces uponwhich we base the canonical formalism. The transforms are depicted in the followingdiagram. Q n P n = T ∗ Q n Q n − × Q n Q n − P n − = T ∗ Q n − F + S n F − S n The coordinate versions of the post- and pre-Legendre transformations are F + S n ( x n − , x n ) = ( x n , + p n ) where + p n = ∂S n ∂x n F − S n ( x n − , x n ) = ( x n − , − p n − ) where − p n − = − ∂S n ∂x n − (2.5)Note that in context of discrete evolution, the time-step action contribution S n +1 hasthe role of Lagrangian. Then, its variation enables one to define so-called Lagrangetwo-forms . They are given asΩ n +1 ( x n , x n +1 ) = − ∂ S n +1 ∂x nA ∂x n +1 B dx nA ∧ dx n +1 B (2.6)We remark that this form is not fully analogical to the classical Lagrange two-form,see e.g. Chap. 7 of [22], since it is built purely out of coordinates and does not containvelocities (at least not explicitly). This is convenient, since there is no canonical notionof velocity to start with. One can check that the Lagrange two-forms arise from pullingback the canonical two-forms ω n = q n X A =1 dx nA ∧ dp nA (2.7)with the Legendre transformation,Ω n = F + S ∗ n ω n , Ω n +1 = F − S ∗ n +1 ω n (2.8)In (2.7), p n stands either for p + n (in case we use F + S ∗ n ) or p − n (in case we use F − S ∗ n +1 ).Now that we have passed to the canonical language, we may go on and use it togive a formulation of the equations of motion. To do that, we only need the abovegiven equations − p n − = − ∂S n ∂x n − , + p n = ∂S n ∂x n (2.9)We shall refer to − p as pre-momenta and to + p as post-momenta . By virtue of theequations (2.9), we implicitly define the global one-step Hamiltonian evolution map H n : P n − → P n , acting as H n ( x n − , − p n − ) = ( x n , + p n ) (2.10)In practice, one can use the equation for pre-momenta in (2.9) to determine x n (ifpossible) and use this result in the post-momenta equation in (2.9) to determine + p n .5he definition of H n does not require equations of motion. However, the equations ofmotion are present in the canonical picture as the momentum-matching of pre- andpost-momenta, + p n = − p n (2.11)We can omit the superindices + and − , assuming that momentum-matching holds.In the latter, we will discuss the special case in which the dimensions of the config-uration spaces at consecutive steps are equal, dim( Q n ) = dim( Q n ) = q , but the systemis irregular, meaning that det ∂ S n +1 ∂x n ∂x n +1 = 0 (2.12)Then, an equal number s n of left and right null vectors L nI , R n +1 J , respectively, of theLagrange two-form arises in an open neighborhood in Q n × Q n +1 , L nI ∂ S n +1 ∂x nI ∂x n +1 J = 0 , ∂ S n +1 ∂x nI ∂x n +1 J R n +1 J = 0 (2.13)Above, I, J = 1 , ..., s n . This case is of interest because it is possible to treat systemswith temporally varying numbers of degrees of freedom in this irregular fashion.The definitions of pre- and post-Legendre transforms are directly applicable tosingular systems. The rank of both is 2 q − s n . The Legendre transformations thusfail to be onto, and their images form submanifolds of dimension 2 q − s n in the phasespaces P n , P n +1 . We call C − n = Im( F − S n +1 ) ⊂ P n the pre-constraint surface and C + n +1 = Im( F + S n +1 ) ⊂ P n +1 the post-constraint surface .We choose s n irreducible pre-constraints − C nL , L = 1 , ..., s n to describe C − n . Theyare automatically satisfied by the pre-momenta − p n (because these arise from the corre-sponding Legendre transform), but impose (through momentum-matching) non-trivialconditions on post-momenta + p n resulting from the previous evolution move between n − n . Analogically, the s n irreducible post-constraints + C Rn +1 , R = 1 , ..., s n are automatically satisfied by the post-momenta + p n +1 but will provide non-trivialconditions for the pre-momenta − p n +1 by momentum-matching. The pre- and post-constraint surfaces at a time-step n generally do not coincide, we thus have to restrictthe evolution to C n = C − n ∩ C + n at each time-slice. We remark that the constraintscan be subjected to analysis analogical to the standard Dirac’s classification [19]; theauthors provide such analysis in [18].For singular systems, the Hamiltonian evolution map can be only defined betweenthe corresponding constraint surfaces, that is, H n +1 : C − n → C + n +1 . On the other hand,as a consequence of the right null vectors in (2.13), we are also no longer able to solvethe pre-momentum equation in (2.9) uniquely for x n +1 . Therefore, we must specify s n additional a priori free parameters λ n +1 R to uniquely determine x n +1 ( x n , − p n , λ n +1 ).The post-momenta + p n +1 will in general depend on the parameters λ n +1 R . Likewise,on account of the left null vectors in (2.13), we can no longer uniquely express x n as afunction of x n +1 and + p n +1 . Then, if we want to evolve the system backwards, we mustspecify s n a posteriori free parameters µ nL in order to determine x n ( x n +1 , + p n +1 , µ n ).We have one a priori free λ nR per post-constraint and one a posteriori free µ nL perpre-constraint. The arbitrariness arising in the form of these free parameters may be,however, later reduced by subsequent constraints at later time-slices.6n consequence of restricting the Hamiltonian evolution to the constraint hyper-surfaces, it cannot be a symplectic map. Nonetheless, a slightly weaker assertion holds: Theorem 2.1.
Let ι − n : C − n → P n and ι + n +1 : C + n +1 → P n +1 be embeddings ofthe constraint surfaces into the corresponding phase spaces, and ( ι − n ) ∗ , ( ι + n +1 ) ∗ theassociated pullbacks. Then ( ι − n ) ∗ ω n = H ∗ n +1 ( ι + n +1 ) ∗ ω n +1 (2.14) Proof.
See the proof of Theorem 6.1 in [17].
Before giving a definition of the linear dynamical system, let us remind that for a gen-eral classical system of finite dimension, the configuration space Q is a q -dimensionalmanifold, and one may define the phase space P = T ∗ ( Q ) to be the cotangent bun-dle over Q which is a symplectic manifold of dimension 2 q . It is equipped with thesymplectic two-form ω = q X A =1 dx A ∧ dp A (3.1)where ( x , ..., x q ) are some local coordinates on Q and ( p , ..., p q ) are components of thecotangent vectors in the coordinate basis associated with ( x , ..., x q ). So far the generalcase. Now, borrowing from Chap. 2 of [23], we give the anticipated definition. Definition 3.1. A linear dynamical system satisfies the following two conditions:(1) Its configuration space Q has a natural vector space structure. Then we canchoose a basis in Q and use the basis components of vectors in Q to define linearcoordinates ( x , ..., x q ) globally on Q . These coordinates then give rise to globallywell defined linear canonical coordinates ( x , ..., x q , p , ..., p q ) on P .(2) The Hamiltonian H is a quadratic function on P so that the equations of motion˙ x A = ∂H∂p A , ˙ p A = − ∂H∂x A are linear in the linear canonical coordinates.For simplicity we shall denote the canonical (symplectic) basis of P as { e I } qI =1 and the linear canonical coordinates of a point y ∈ P by ( y , ..., y q , y q +1 , ..., y q ). Thenit is understood that y = y I e I with an implicit summation over I = 1 , ..., q .A key consequence of the vector space structure of P given by the property (1) isthat one may identify the tangent space T x P at any point x ∈ P with P itself. Underthis identification, the symplectic form ω becomes a bilinear function ω : P × P → R .Furthermore, since the components of the symplectic form are constant in the canonicalcoordinate basis, the corresponding bilinear map is independent of the choice of point x used to make the identification. We shall refer to ω : P × P → R as the symplecticstructure on P . For y, u ∈ P , it is given by ω ( y, u ) = y A u A + q − y A + q u A (3.2)with an implicit summation over A = 1 , ..., q ; and we will also take advantage of analternative form ω ( y, u ) = y T σu = y I σ IJ u J with summation over I, J = 1 , ..., q andthe matrix σ = (cid:18) − (cid:19) (3.3)7lthough the phase space P is generally a symplectic manifold, we see that in caseof a linear dynamical system, it may be viewed as a symplectic vector space ( P , ω ),i.e., a vector space on which is defined a non-degenerate, antisymmetric, bilinear map ω .Thanks to the property (2) of the linear dynamical system, the Hamiltonian func-tion H on P takes the form H ( t, y ) = y I K ( t ) IJ y J (3.4)(again, summing over I, J = 1 , ..., q ) where K ( t ) is a symmetric 2 q × q matrix. TheHamilton’s equations of motion then turn out as˙ y I = σ IJ ∂H∂y J = σ IJ K JK y K (3.5)now, let y ( t ), u ( t ) be two solutions of the equations of motion (3.5) and let s ( t ) = ω ( y ( t ) , u ( t )) = σ IJ y ( t ) I u ( t ) J (3.6)Then we have ˙ s = 0 (3.7)Thus, for a linear dynamical system, the symplectic product s of two solutions is con-served. (This result is really a consequence of a much more general fact for nonlinearsystems that dynamical evolution defines a canonical transformation on phase space.)Thus, the symplectic structure ω : P × P → R gives rise to a natural symplecticstructure ω on the vector space of solutions S to the equations of motion, since thesymplectic structure on S obtained by identifying P and S does not depend upon thechoice of initial time t under which the identification is made. Let us take a moment to remind a few basic tools from linear algebra which will provevery useful for our work. The following is an adaptation of Section 3 of [24]. We firstlook at the singular value decomposition and Moore-Penrose pseudoinverse, as definedin [25]. Then we shortly mention symplectic matrices.
Theorem 4.1.
Let A ∈ R m × n be an m × n matrix with m ≥ n . Then there existorthogonal matrices U ∈ R m × m and V ∈ R n × n and a matrix Σ = (cid:18) diag( σ , ..., σ n )0 (cid:19) ∈ R m × n with σ ≥ σ ≥ ... ≥ σ n ≥
0, such that A = U Σ V T (4.1)The numbers σ , ..., σ n are called singular values of A . If σ r > A has rank r .The singular value decomposition exists for any matrix, we use m ≥ n for sim-plicity. Note that the decomposition is not unique—only the matrix Σ is in generaluniquely determined by A . In the following, we will adopt the notation U = ( U U )8nd V = ( V V ) with U ∈ R m × r , U ∈ R m × m − r , V ∈ R n × r and V ∈ R n × n − r . Wefurther denote Σ r = diag( σ , ..., σ r ) ∈ R r × r . Then one can write A = (cid:0) U U (cid:1) (cid:18) Σ r
00 0 (cid:19) (cid:18) V T V T (cid:19) = U Σ r V T (4.2)This form will be called narrowed singular value decomposition . It is useful for definingthe pseudoinverse: Definition 4.1.
Let A ∈ R m × n be an m × n matrix and A = U Σ V T = U Σ r V T its(narrowed) singular value decomposition. Then the matrix A + = V Σ + U T = V Σ + r U T with Σ + = (cid:18) Σ + r
00 0 (cid:19) ∈ R n × m and Σ + r = diag( σ − , ..., σ − r ) ∈ R r × r is called the Moore-Penrose pseudoinverse of A .In the next we remind the fundamental spaces associated to the matrix A . Definition 4.2.
We define the following fundamental spaces:(1) R ( A ) = { y | ∃ x ∈ R n : y = Ax } ⊂ R m is the range or column space .(2) R ( A T ) = { z | ∃ y ∈ R n : z = A T y } ⊂ R n is the row space .(3) N ( A ) = { x | Ax = 0 } ⊂ R n is the null space .Note that since R ( A ) ⊥ = N ( A T ), it holds that R m = R ( A ) ⊕ N ( A T ). Ana-logically, one has R ( A T ) ⊥ = N ( A ), therefore R n = R ( A T ) ⊕ N ( A ). With the helpof the narrowed singular value decomposition A = U Σ r V T and the Moore-Penrosepseudoinverse, one can easily write down the projectors to these spaces: Theorem 4.2.
The projectors to the spaces of Definition 4.2 are given by P R ( A ) = AA + = U U T P R ( A T ) = A + A = V V T P N ( A T ) = − AA + = U U T P N ( A ) = − A + A = V V T (4.3)The above described tools can be readily used to write an explicit solution to ageneral linear set of equations. Consider the matrix problem Ax = b (4.4)with A ∈ R m × n a matrix, x ∈ R n and b ∈ R m . Obviously, the equation only has asolution for x if b ∈ R ( A ). This condition (or constraint ) can be equivalently stated as U U T b = 0 (4.5)where we assumed A = U Σ r V T and projected the right hand side of (4.4) onto N ( A T )with P N ( A T ) = U U T . If the constraint (4.5) holds, there is a family of solutions for x of the form x = A + b + V c (4.6)where c ∈ R s is an arbitrary vector of dimension s ≡ n − r . By the way, this is thesolution of the linear least squares problem Ax ≈ b which comes around by projectingthe right-hand side of (4.4) onto R ( A ) and thus solving the equation Ax = AA + b rather than (4.4). Considering the linear least squares problem is equivalent to simply9gnoring the constraint (4.5).In the rest, we shall very briefly recall the definition of a symplectic matrix andreview its elementary properties. For more information, we refer to [26]. Definition 4.3. A symplectic matrix W is a real 2 q × q matrix satisfying W T σW = σ (4.7)with σ = (cid:18) − (cid:19) (4.8) Theorem 4.3.
Let us denote W = (cid:18) E FG H (cid:19) (4.9)where
E, F, G, H are real q × q matrices. Then the following conditions are equivalent:1. The matrix W is symplectic.2. E T G , F T H are symmetric and E T H − G T F = EF T , GH T are symmetric and EH T − F G T = It follows from condition 2. that the inverse of a symplectic matrix W is W − = (cid:18) H T − F T − G T E T (cid:19) (4.10) We shall now apply the discrete canonical evolution formalism to the case of a lineardynamical system, using a matrix formulation of the problem. All the distinctive fea-tures of the (irregular) discrete evolution then become very clear, translated into theproperties of the corresponding matrices.We will consider the set { , ..., t } ∋ n of finite number of time-slices. We requirethat the maximum number q of degrees of freedom at any time-slice is finite, therefore,the system in question has a finite number N of degrees of freedom bounded by q ( t + 1).We define a solution to be a point in the total phase space P t = N tn =0 P n satisfyingthe evolution equations; in particular, it must satisfy all the constraints. The set ofsolutions shall be denoted by S t . To obtain the time-slice data of a solution, one mayuse the projection P n : P t → P n , given naturally as P n y = y n . Often the projectionwill be implicit. It may of course happen that the constraints are so severe that nosolution on the whole interval { , ..., t } exists; we are however more interested in thecase when the solutions are plentiful.Let us give a short remark on the action of the discussed system. In accordancewith (2.1), we are assuming that it is additive so that S ( x ) = P t − n =0 S n +1 ( x n , x n +1 )10here S n +1 ( x n , x n +1 ) is the action contribution corresponding to the time-step between n and n + 1. We know that the variation of S n +1 ( x n , x n +1 ) produces the equations ofmotion in the form of momentum-matching (2.11). However, if we do not limit ourselvesto solutions, we may as well consider any coordinate configuration and rewrite the actionin terms of coordinates, pre-momenta and post-momenta. The single time-step actioncontribution then amounts to (in matrix notation) S n +1 = (cid:0) x Tn +1 + p n +1 − x Tn − p n (cid:1) (5.1)The validity of this formula will become clear in a moment. In the terms used, theoverall action between time-slices 0 and t is S = (cid:0) x Tt + p t − x T − p (cid:1) + P t − n =1 x Tn ( + p n − − p n ) (5.2)For solutions y ∈ S t , we can rewrite the action as a function on the solution space S , S ( y ) = (cid:0) x Tt p t − x T p (cid:1) (5.3)which, upon plugging in for momenta, can be seen as a discrete analogue of the Hamil-ton’s principal function. Note that the Hamilton’s principal function is given attentionin Sections 7 and 9 of [20].Eventually let us define the symplectic structure ω n on P n × P n by the usualprescription ω n ( y n , z n ) = x n ( y ) T p n ( z ) − x n ( z ) T p n ( y ) (5.4)i.e., we confirm that ( x n , p n ) are canonical coordinates on P n . For a linear dynamical system, the equations (2.9) can be written in matrix form − p n = − ∂S n +1 ∂x n = L n x n + R n x n +1+ p n +1 = ∂S n +1 ∂x n +1 = ¯ L n +1 x n + ¯ R n +1 x n +1 (5.5)where − p n , + p n +1 , x n , x n +1 are coordinate vectors of dimension q and L n , R n , ¯ L n +1 , ¯ R n +1 are q × q matrices. If we consider a solution to the equations of motion, we may dropthe − and + indices of the momenta, enforcing momentum matching. We remark that(5.5) are identical to the equations (3.3) in [20] although we employ a slightly differentnotation for the matrices.In connection to (5.5), one should recall a well known fact that for linear systems,the second partial derivatives of action are symmetric; this is merely a consequenceof the Young’s (or Clairaut’s, or Schwarz’s) Theorem [27]. Taking this into account,one asserts that ∂S n +1 ∂x nA ∂x n +1 B = ∂S n +1 ∂x n +1 B ∂x nA . Computing ∂∂x nA ∂S n +1 ∂x n +1 B = ¯ L n +1 BA and ∂∂x n +1 B ∂S n +1 ∂x nA = R nAB , one finds ¯ L n +1 = − R Tn (5.6)11he similar assertions ∂S n +1 ∂x nA ∂x nB = ∂S n +1 ∂x nB ∂x nA , ∂S n +1 ∂x n +1 A ∂x n +1 B = ∂S n +1 ∂x n +1 B ∂x n +1 A imply L n = L Tn , ¯ R n +1 = ¯ R Tn +1 (5.7)i.e., the matrices L n and ¯ R n +1 are symmetric . This will make our later work substan-tially easier.One may deduce from (5.5) the general form of the action contribution S n +1 ( x n , x n +1 ), S n +1 = (cid:0) x Tn +1 ¯ L n +1 x n + x Tn +1 ¯ R n +1 x n +1 − x Tn L n x n − x Tn R n x n +1 (cid:1) == − (cid:0) x Tn L n x n + 2 x Tn R n x n +1 − x Tn +1 ¯ R n +1 x n +1 (cid:1) (5.8)where we have used the identity (5.6) to simplify the expression. Turning once againto (5.5), one can easily express this quantity in mixed variables as (5.1). Let us stressout that although quite convenient, the form (5.1) is not relevant form the perspectiveof Hamiltonian mechanics, since it is not given in terms of canonical coordinates. Notethat we have not provided any Hamiltonian function at all. Therefore, the best wecan do is to employ (5.8) which can be viewed as the Lagrangian and express theHamiltonian evolution map directly from it. This is what we are going to do in thenext paragraph. Meanwhile, let us point out that (5.8) has its equivalent in eq. (3.1)of [20]. The two expressions for action are related via L n ≡ − a n +1 , R n ≡ − c n +1 and¯ R n +1 ≡ b n .We will now give an explicit formulation of the (forward) Hamiltonian evolutionin terms of the matrices introduced in (5.5). We start by solving the pre-momentumequation R n x n +1 = p n − L n x n (5.9)The matrix R n is generally singular. Upon using the narrowed singular value decom-position, we find that the solution exists when P N ( R Tn ) ( p n − L n x n ) = 0 (5.10)with the projector to the null space of R Tn given by P N ( R Tn ) = U ( R n ) U ( R n ) T (5.11)Next, if the solution does exist, it is of the form x n +1 = R + n ( p n − L n x n ) + V ( R n ) λ n +1 (5.12)where R + n = V ( R n ) Σ r ( R n ) − U ( R n ) T (5.13)is the Moore-Penrose pseudoinverse of R n and λ n +1 is an arbitrary vector of dimension s n ≡ q − r n with r n ≡ rank( R n ). Note that the equation (5.10) is a matrix form of thepre-constraint, defining the linear pre-constraint subspace C − n ⊂ P n . We can rewriteit in a more compact form C n y n = 0 , C n = (cid:0) − P N ( R Tn ) L n P N ( R Tn ) (cid:1) (5.14)where y n = (cid:18) x n p n (cid:19) (5.15)12s a coordinate vector of dimension 2 q representing a point y n ∈ P n and the two-block matrix C n has dimensions q × q . Upon plugging the solution (5.12) into thepost-momenta equation in (5.5), one finds y n +1 = E n y n + F n +1 λ n +1 (5.16)with E n = (cid:18) − R + n L n R + n ¯ L n +1 − ¯ R n +1 R + n L n ¯ R n +1 R + n (cid:19) = (cid:18) − R + n L n R + n − R Tn − ¯ R n +1 R + n L n ¯ R n +1 R + n (cid:19) (5.17)and F n +1 = (cid:18) V ( R n )¯ R n +1 V ( R n ) (cid:19) (5.18)The matrix E n has dimensions 2 q × q and the matrix F n +1 has dimensions 2 q × s n . The coordinate vectors y n +1 are elements of the post-constraint surface C + n +1 ⊂ P n +1 . The matrix formulation has made explicit the dependence of the evolution ona free parameter λ n +1 ∈ R s n . We shall acknowledge this by denoting the Hamiltonianevolution map , originally introduced in (2.10), by H n +1 ( λ n +1 ) : C − n → C + n +1 . Viewed inthis way, the map is unique, but not onto , because the parameter λ n +1 picks a particularsubspace C +( λ n +1 ) n +1 ≡ H n +1 ( λ n +1 ) C − n ⊂ C + n +1 as the image. Thanks to the linearity ofthe evolution equations, the post-constraint surface C + n +1 can be viewed as an affinespace over C +(0) n +1 ≡ H n +1 (0) C − n with the point-set Λ n +1 ≡ { F n +1 λ n +1 | λ n +1 ∈ R s n } . Itis very much correct to treat it this way, distinguishing between vectors E n y n ∈ C +(0) n +1 and points F n +1 λ n +1 ∈ Λ n +1 . Note that both C +(0) n +1 and Λ n +1 are linear subspaces of P n +1 . However, one can say more: Observation 5.1.
The subspaces C +(0) n +1 and Λ n +1 have zero intersection. Proof.
It suffices to look at the coordinate parts of E n y n and F n +1 λ n +1 . These aregiven simply by the two terms in (5.12). Since R + n = V ( R n ) Σ r ( R n ) − U ( R n ) T andthe columns of V ( R n ) are by definition orthogonal to the columns of V ( R n ), it followsthat the two parts are orthogonal. The assertion is implied.As a consequence of the above observation, the post-constraint surface C + n +1 canbe viewed as a linear subspace C + n +1 = C +(0) n +1 ⊕ Λ n +1 of P n +1 , rather than an affinespace. This further simplifies our work.For completeness, we provide also the explicit form of the backward Hamiltonianevolution. It is fully analogical to the preceding case. To express the evolution fromtime-step n + 1 to time-step n , one has to solve the post-momentum equation in (5.5),which yields a constraint P N (¯ L Tn +1 ) (cid:0) p n +1 − ¯ R n +1 x n +1 (cid:1) = 0 (5.19)with the projector P N (¯ L Tn +1 ) = U ( ¯ L n +1 ) U ( ¯ L n +1 ) T (5.20)and a solution x n = ¯ L + n +1 (cid:0) p n +1 − ¯ R n +1 x n +1 (cid:1) + V ( ¯ L n +1 ) µ n (5.21)13ith an arbitrary vector µ n ∈ R ¯ s n where we denote ¯ s n = q − ¯ r n with ¯ r n = rank( ¯ L n +1 ).The constraint (5.19) can be rewritten compactly as¯ C n +1 y n +1 = 0 , ¯ C n +1 = (cid:16) − P N (¯ L Tn +1 ) ¯ R n +1 P N (¯ L Tn +1 ) (cid:17) (5.22)When it is satisfied, a solution to the backwards evolution exists and is given by y n = ¯ E n +1 y n +1 + ¯ F n µ n (5.23)with ¯ E n +1 = (cid:18) − ¯ L + n +1 ¯ R n +1 ¯ L + n +1 R n − L n ¯ L + n +1 ¯ R n +1 L n ¯ L + n +1 (cid:19) (5.24)and ¯ F n = (cid:18) V ( ¯ L n +1 ) L n V ( ¯ L n +1 ) (cid:19) (5.25) We devote this subsection to the study of constraint surfaces and the general propertiesof the Hamiltonian evolution map H n +1 ( λ n +1 ) with respect to the symplectic structures ω n and ω n +1 given by (5.4). We start with Theorem 2.1, which is nothing but a variationon the equation (3.7), saying that “symplectic product of solutions is conserved”. Inour context, Theorem 2.1 can be given in this exact simple wording. Assume that y n , z n ∈ C − n . From (5.5), one gets ω n ( y n , z n ) = x n ( y ) T L n x n ( z ) + x n ( y ) T R n x n +1 ( z ) − x n ( z ) T L n x n ( y ) − x n ( z ) T R n x n +1 ( y ) == x n ( y ) T R n x n +1 ( z ) − x n ( z ) T R n x n +1 ( y ) (5.26)where we used the symmetry of L n . Forward Hamiltonian evolution with initial condi-tions y n , z n then yields ω n +1 ( y n +1 , z n +1 ) = x n +1 ( y ) T ¯ L n +1 x n ( z ) + x n +1 ( y ) T ¯ R n +1 x n +1 ( z ) − x n +1 ( z ) T ¯ L n +1 x n ( y ) − x n +1 ( z ) T ¯ R n +1 x n +1 ( y ) == x n +1 ( y ) T ¯ L n +1 x n ( z ) − x n +1 ( z ) T ¯ L n +1 x n ( y ) == − x n ( z ) T R n x n +1 ( y ) + x n ( y ) T R n x n +1 ( z ) == ω n ( y n , z n ) (5.27)where we used symmetry of ¯ R n +1 , exploited the relation (5.6) and compared with(5.26). Similarly for backwards evolution: if y n , z n ∈ C + n , then ω n ( y n , z n ) = x n ( y ) T ¯ L n x n − ( z ) − x n ( z ) T ¯ L n x n − ( y ) (5.28)and we find ω n − ( y n − , z n − ) = x n − ( y ) T R n − x n ( z ) − x n − ( z ) T R n − x n ( y ) == − x n ( z ) T ¯ L n x n − ( y ) + x n ( y ) T ¯ L n x n − ( z ) == ω n ( y n , z n ) (5.29)14e have thus explicitly checked that the symplectic product of solutions is in-deed conserved, understanding that a solution y n of the equations of motion satisfiesmomentum-matching as well as all the constraints . We will later extend our discussionof solutions to the case of multiple time-steps. At this point we can move on to theanalysis of the constraint surfaces.It is essential to understand that while ω n is symplectic on P n , it is generally notsymplectic on C − n ⊂ P n because there it may be degenerate. In other words, the con-straint surface C − n is in general not a symplectic subspace of P n . This is unfortunatefor many applications which require to have a well defined evolution map between sym-plectic spaces, as is the case of canonical quantization. Luckily, we can easily reducethe constraint spaces by a standard procedure to make them symplectic and establishsuch an evolution map. We do this below, first formally and then explicitly.Let us describe the procedure in general terms for an arbitrary linear subspace C of the symplectic space ( P , ω ). We first define the corresponding null space N ω ( C ) as N ω ( C ) = { z ∈ C | ω ( z, u ) = 0 ∀ u ∈ C } (5.30)The reader may notice that N ω ( C ) = C ∩ C ω where C ω = { z ∈ P | ω ( z, u ) = 0 ∀ u ∈ C } is so-called skew-orthogonal set to C , see Sec. 1.2 in [26] for details. For us, N ω ( C )is simply the subspace of C which makes ω degenerate. If N ω ( C ) = { } , then C issymplectic and we are done. Otherwise we proceed with a second step, in which we getrid of the degeneracy: we take ˜ C = C / N ω ( C ). This is the space of equivalence classes[ y ] = { y + z | z ∈ N ω ( C ) } of equivalent y ∈ C . This means that if y, ˜ y ∈ C are suchthat ˜ y = y + z with z ∈ N ω ( C ), then [ y ] = [˜ y ]. One can also put it differently: Definition 5.1.
We say that y, ˜ y ∈ C are symplectically equivalent on C w.r.t. ω andwrite y C ∼ ˜ y , if ω ( y, u ) = ω (˜ y, u ) for all u ∈ C . Observation 5.2.
The space ˜ C is composed of equivalence classes [ y ] = { ˜ y ∈ C | y C ∼ ˜ y } of symplectically equivalent vectors in C . Proof.
A simple exercise.The point of this procedure is captured by the following observation.
Observation 5.3.
Let ω : ˜ C × ˜ C → R be defined by ω ([ y ] , [ u ]) = ω ( y, u ) where y ∈ [ y ], u ∈ [ u ] are arbitrarily chosen representatives of their classes. Then ( ˜ C , ω ) is symplectic. Proof.
First of all, note that according to Observation 5.2, the definition of ω : ˜ C × ˜ C → R is consistent. Next, by assigning [ y ] + c [ u ] = [ y + cu ] (for y, u ∈ C and c ∈ R ), welet ˜ C inherit the natural vector-space structure of C . Suppose that ω ([ y ] , [ z ]) = 0 forall [ y ] ∈ ˜ C . Then it holds ω ( y, z ) = 0 for all y ∈ C . In other words, ω ( y, z ) = ω ( y, y ∈ C , i.e., z C ∼ z ] = [0]. It follows that ω on ˜ C is non-degenerate. Sinceit is also bilinear and antisymmetric, it is symplectic.Because equivalence classes are not very practical for computations, we add anoptional third step, which is to consider a representative space ˙ C ⊂ C such that each y ∈ ˙ C corresponds to a class [ y ] ∈ ˜ C . In other words, ˙ C is a set of symplecticallyinequivalent vectors y ∈ C originating by picking one particular element from each class.We strongly prefer a choice of ˙ C which is a linear subspace of C so that the vector-spaceoperations on ˙ C align with those on C ; we shall therefore assume this. A possible way tofind such ˙ C is to pick a maximal linearly independent set of symplectically inequivalent15ectors in C and generating ˙ C as linear span of this set. Needless to say, ω naturallycarries over from C to ˙ C , making ( ˙ C , ω ) symplectic. One can make the followingobservation. Observation 5.4.
It holds C = ˙ C ⊕ N ω ( C ). Proof.
Any ˜ y ∈ C gives rise to a unique [˜ y ] ∈ ˜ C , which is in turn uniquely representedby y ∈ ˙ C . We denote ˜ y − y ≡ z , then by definition z ∈ N ω ( C ). It follows that thereis a unique decomposition ˜ y = y + z of the vector ˜ y ∈ C into y ∈ ˙ C and z ∈ N ω ( C ),hence the assertion.The above observation suggests a special choice of ˙ C which is uniquely fixed by aninner product on C . It is of course ˙ C = N ω ( C ) ⊥ . We use this choice below on someoccasions with the canonical inner product.We will now use the described procedure for the pre-constraint surface C − n , result-ing in the symplectic space ˙ C − n ⊂ P n . Once we have it defined, we would like to seehow ˙ C − n evolves to the next time-slice. In what comes next, we would like to offer a particular choice of the representativespaces ˙ C − n , ˙ C + n +1 and a construction of symplectic bases of P n , P n +1 that will beadapted to this choice. Before we start, let us give a short comment on how ourapproach differs from that of [20]. In Sec. 6 of that reference, a special choice of ba-sis on the phase space is introduced. As it will be in our case, the associated lineartransformation given by eq. (6.1) is canonical (so the new basis is symplectic). Its pur-pose is to separate the primary and secondary constraints. With a subsequent secondtransformation, it is then possible to trivialize the constraints, which opens the door toa classification of the propagating degrees of freedom constituted by observables andfree parameters. Further analysis describes how these propagate from time-slice n − n to time-slice n + 1.Our approach is quite different, and perhaps more lightweight. We focus on a singletime-step from n to n + 1 and find symplectic bases in both the involved phase spaceswhich disentangle the most important subspaces induced by the symplectic structure,as discussed in the previous section. In doing so, we also choose the representativespaces of symplectically equivalent time-slice data and build their symplectic bases.Incidentally (due to our rather natural choice), the corresponding symplectic (or tosay, canonical) transformation will turn out to trivialize the one-step Hamiltonian evo-lution map.First let us parametrize C − n . To do that, we need to solve the pre-constraint (5.14)or, equivalently, (5.10). The latter can be solved trivially by writing p n = L n x n + θ n (5.31)with an arbitrary θ n ∈ R ( R n ) in the column space of R n ; recall that R ( R n ) ⊥ N ( R Tn ).Then we have a general form of y n ∈ C − n , y n = (cid:18) x n L n x n + θ n (cid:19) (5.32)16nd we see that dim C − n = q + r n with r n = dim R ( R n ) = rank( R n ). Now we take amoment to parametrize N ω ( C − n ). Recall that z n ∈ N ω ( C − n ) must be in C − n and mustsatisfy ω n ( y n , z n ) = 0 for all y n ∈ C − n . These two conditions give us z n = (cid:18) µ n L Tn µ n (cid:19) (5.33)with µ n ∈ N ( R Tn ) such that ( L Tn − L n ) µ n ∈ R ( R n ). However, the second condition issatisfied automatically, thanks to the symmetry of L n . We then see that dim N ω ( C − n ) =dim N ( R Tn ) = s n with s n ≡ q − r n . Now we can already guess the possible structure of˙ C − n . Let us write the relation ˜ y n = y n + z n in the parametrized form, (cid:18) ˜ x n L n ˜ x n + ˜ θ n (cid:19) = (cid:18) x n L n x n + θ n (cid:19) + (cid:18) µ n L Tn µ n (cid:19) = (cid:18) x n + µ n L n ( x n + µ n ) + θ n (cid:19) (5.34)and we see that ˜ x n = x n + µ n and ˜ θ n = θ n . Therefore it is natural to pick a unique y n from each class [ y n ] solemnly by fixing x n = ̺ n ∈ R ( R n ) orthogonal to µ n . That is,we chose y n ∈ ˙ C − n to be of the form y n = (cid:18) ̺ n L n ̺ n + θ n (cid:19) (5.35)with ̺ n ∈ R ( R n ) and still θ n ∈ R ( R n ). Clearly, this is a linear subspace of C − n , as werequire. Also, dim ˙ C − n = 2 r n is indeed even, as expected from a symplectic space.Thanks to the very simple structure of ˜ C − n and subsequently of ˙ C − n , it is possibleto construct a natural symplectic basis for ˙ C − n . We shall denote it by { ˙ e nX } with X = 1 , ..., r n , q + 1 , ..., q + r n . Knowing that the basis vectors have the structure˙ e nX = (cid:18) ̺ nX L n ̺ nX + θ nX (cid:19) (5.36)(the indices do not represent components here, but give names to vectors) one expressesthe requirement of symplecticity as σ XY = ω n ( ˙ e nX , ˙ e nY ) = ̺ TnX θ nY − θ TnX ̺ nY (5.37)Now recall the singular value decomposition R n = U ( R n )Σ( R n ) V ( R n ) T . Theorthogonal matrices U ( R n ) , V ( R n ) are not unique; one can use any suitable choice ofthese two. In the following, we shall cease to write arguments of all the matrices arisingfrom the singular value decomposition of R n , i.e., we denote U ≡ U ( R n ), V ≡ V ( R n ),Σ ≡ Σ( R n ) and so on. We will occasionally use this condensed notation.We shall satisfy the equation (5.37) by the deliberate choice ̺ nE = 0, ̺ nE + q = − U ε nE , θ nE = − ̺ nE + q , θ nE + q = 0 for all E = 1 , ..., r n . Here we use the canonicalbasis { ε nE } r n E =1 in R r n to generate the set { U ε nE } r n E =1 which is an orthonormal basisof R ( R n ). Recall that U is a q × r n submatrix of U = (cid:0) U U (cid:1) . We end up with˙ e nE = (cid:18) U ε nE (cid:19) , ˙ e nE + q = − (cid:18) U ε nE L n U ε nE (cid:19) (5.38)Surely we could have chosen a different arrangement of the vectors, but this one isthe most suitable for our cause. In any way, we have a symplectic basis of ˙ C − n . Since˙ C − n ⊂ P n , we know by virtue of the “symplectic Gram–Schmidt theorem”, see Sec. 1.217f [26], that it can be extended to a full symplectic basis of P n . We shall do this byfixing ˙ e nM = (cid:18) U ι nM (cid:19) , ˙ e nM + q = − (cid:18) U ι nM L n U ι nM (cid:19) (5.39)where { ι nM } qM = r n +1 is the (oddly numbered) canonical basis of R s n and obviously M = r n + 1 , ..., q . It is clear that the vectors ˙ e nI are all linearly independent. Thuswe have fixed a basis { ˙ e nI } qI =1 of P n . The reader can easily check, using the analogueof (5.37), that it is indeed symplectic. We will verify this one step later in anotherway. Before that however, it is worthy noticing that { ˙ e nM + q } qM = r n +1 is a basis of N ω ( C − n ). This is a reminder of the fact that C − n = ˙ C − n ⊕ N ω ( C − n ). On the other hand, { ˙ e nM } qM = r n +1 is a basis of C −⊥ n (the orthogonal complement in P n is taken w.r.t. thecanonical inner product here). This is the space of vectors that do not satisfy thepre-constraint.We formalize the basis transformation as e nJ = ˙ e nI ˙ W nIJ (5.40)and read out from (5.38) and (5.39) the inverse matrix˙ W − n = (cid:18) − UU − L n U (cid:19) (5.41)Now the promised verification: because U T L n U is symmetric and U T U = , it followsfrom Theorem 4.3 that ˙ W − n is a symplectic matrix . Its inverse is also symplectic, andusing (4.10), we find that it is ˙ W n = (cid:18) − U T L n U T − U T (cid:19) (5.42)Thus we can take for granted that (5.40) is a symplectic transformation passing fromthe new symplectic basis to the canonical one.We will shortly continue our discourse by evolving the described subspaces of P n to P n +1 using the prescription (5.16). Before we do so, we want to remark thatbecause of the conservation of the symplectic structure, one automatically obtains H n +1 ( λ n +1 ) N ω ( C − n ) = N ω ( H n +1 ( λ n +1 ) C − n ) = N ω ( C +( λ n +1 ) n +1 ) and consequently also H n +1 ( λ n +1 ) ˜ C − n = ˜ C +( λ n +1 ) n +1 for arbitrary λ n +1 . It therefore seems natural to generateour representative space ˙ C + n +1 in the post-constraint surface by evolving the represen-tative space in the pre-constrain surface, i.e., fix ˙ C + n +1 = H n +1 ( λ n +1 ) ˙ C − n with some λ n +1 . This is indeed possible. The symplectic structure of ˙ C − n will not be touched by H n +1 ( λ n +1 ), and it is therefore guaranteed that ˙ C + n +1 will be also a symplectic spaceof the same dimension. For simplicity, we shall opt for ˙ C + n +1 = H n +1 (0) ˙ C − n .Now let us proceed with the calculation. We start with z n ∈ N ω ( C − n ) and recall(5.17) so that we can evolve it by H n +1 (0), only to find E n ˙ e nM + q = (cid:18) − R + n L n R + n − R Tn − ¯ R n +1 R + n L n ¯ R n +1 R + n (cid:19) (cid:18) − U ι nM − L n U ι nM (cid:19) = 0 (5.43)because R n = U Σ r V T and so R Tn U = V Σ r U T U , which is annihilated by U T U = 0.Now, (5.43) tells us that H n +1 (0) z n = 0. As a result, the image of N ω ( C − n ) under18he Hamiltonian evolution map H n +1 ( λ n +1 ) is a single point H n +1 ( λ n +1 ) N ω ( C − n ) = { F n +1 λ n +1 } of the affine space C + n +1 . In particular, we have H n +1 (0) N ω ( C − n ) = { } .From the conservation of the symplectic structure, it follows that H n +1 (0) N ω ( C − n ) = N ω ( H n +1 (0) C − n ) = N ω ( C +(0) n +1 ). Therefore, we get that N ω ( C +(0) n +1 ) = { } , i.e., the sub-space C +(0) n +1 of the post-constraint surface C + n +1 turns out to be symplectic.Next we would like to evolve the vectors in ˙ C − n . For this purpose we define a set { ¨ e n +1 X } , again with X = 1 , ..., r n , q + 1 , ..., q + r n , by¨ e n +1 X = H n +1 (0) ˙ e nX = E n ˙ e nX (5.44)This results in ¨ e n +1 E = (cid:18) R + n U ε nE ¯ R n +1 R + n U ε nE (cid:19) = (cid:18) V Σ − r ε nE ¯ R n +1 V Σ − r ε nE (cid:19) (5.45)¨ e n +1 E + q = (cid:18) R Tn U ε nE (cid:19) = (cid:18) V Σ r ε nE (cid:19) (5.46)with E = 1 , ..., r n . Here we again recalled R n = U Σ r V T and R + n = V Σ − r U T , whichgives R Tn U = V Σ r and R + n U = V Σ − r . One can see that the set { ¨ e n +1 X } is linearlyindependent. From the conservation of the symplectic structure σ XY = ω n ( ˙ e nX , ˙ e nY ) = ω n +1 (¨ e n +1 X , ¨ e n +1 Y ) (5.47)it subsequently follows that { ¨ e n +1 X } is a symplectic basis of H n +1 (0) ˙ C − n . Since thenull space N ω ( C − n ) was mapped to zero by H n +1 (0) and there is nothing else to map,we get H n +1 (0) ˙ C − n = H n +1 (0) C − n = C +(0) n +1 . At the beginning of this paragraph, wehave decided that ˙ C + n +1 = H n +1 (0) ˙ C − n . It follows ˙ C + n +1 = C +(0) n +1 . This makes perfectsense: C +(0) n +1 is a linear, symplectic subspace of C + n +1 such that C +(0) n +1 ⊕ Λ n +1 = C + n +1 and we will see in a moment that Λ n +1 = N ω ( C + n +1 ) so that the direct sum compliesto Observation 5.4. In summary, we have specified the two advertised representativespaces ˙ C − n , ˙ C + n +1 as well as the symplectic Hamiltonian map H n +1 (0) : ˙ C − n → ˙ C + n +1 .Our last task is to extend { ¨ e n +1 X } to a symplectic basis { ¨ e n +1 I } qI =1 of P n +1 , whichwe know is possible. The extension should span the point-space Λ n +1 as well as the restof P n +1 , but there is not much information on how it should look. Therefore we haveno other choice but to employ our creativity. Our strategy for finding the extension isthe following: we again assume the transformation e n +1 J = ¨ e n +1 I ¨ W n +1 IJ (5.48)and write down everything we know about the matrix ¨ W − n +1 into the block form¨ W − n +1 = (cid:18) V Σ − r A B ¯ R n +1 V Σ − r C V Σ r D (cid:19) (5.49)with sought-for q × s n matrices A, B, C, D . We require that ¨ W n +1 is symplectic, whichimplies a set of conditions on the unknown matrices through Theorem 4.3. One findsa class of solutions to these conditions parametrized as A = V bB = V aC = ¯ R n +1 V b − V dD = ¯ R n +1 V a − V c (5.50)19here a, b, c, d are s n × s n matrices such that w ≡ (cid:18) a bc d (cid:19) (5.51)is symplectic.Now, there are multiple choices of w which could be considered convenient, but wefound after some trial and error that our purpose is best served by w = (cid:18) − (cid:19) (5.52)which yields perhaps the most simple and natural form of the matrix (5.49). The restof the basis is then fixed to¨ e n +1 M = (cid:18) V ι nM ¯ R n +1 V ι nM (cid:19) , ¨ e n +1 M + q = (cid:18) V ι nM (cid:19) (5.53)which reminds of the structure encountered in (5.39). Now, recalling (5.18), one mayobserve that the set { ¨ e n +1 M } qM = r n +1 is in fact a basis of the point-space Λ n +1 , whoseelements are F n +1 λ n +1 = (cid:18) V ¯ R n +1 V (cid:19) λ n +1 = ¨ e n +1 M λ n +1 M − r n (5.54)At this point we can prove the preconceived relation N ω ( C + n +1 ) = Λ n +1 with ease, bynoticing that the general properties of symplectic bases imply ω n +1 (¨ e n +1 N , ¨ e n +1 M ) = 0and ω n +1 (¨ e n +1 N , ¨ e n +1 X ) = 0 for all N, M = r n +1 , ..., q and all X = 1 , ..., r n , q + 1 , ..., q + r n . These prove the point, since ¨ e n +1 X form the basis of C +(0) n +1 and ¨ e n +1 M form thebasis of Λ n +1 , as we know. The reader can check the above symplectic products ex-plicitly by plugging from (5.45), (5.46) and (5.53) and using the fact that V T V = 0.We can rewrite (5.49) with (5.50) and (5.52) as¨ W − n +1 = (cid:18) V Σ − r V R n +1 V Σ − r ¯ R n +1 V V Σ r V (cid:19) (5.55)where it must be still understood that V ≡ V ( R n ), V ≡ V ( R n ) and Σ r ≡ Σ r ( R n ).We repeat for clarity that the odd block columns of (5.55) have width r n and the evenblock columns have width s n . We can further simplify the form of (5.55) by introducingthe q × q matrix ¯Σ ≡ (cid:18) Σ r (cid:19) (5.56)With that, we can write ¨ W − n +1 = (cid:18) V ¯Σ − R n +1 V ¯Σ − V ¯Σ (cid:19) (5.57)The inverse is easily found to be¨ W n +1 = (cid:18) ¯Σ V T − ¯Σ − V T ¯ R n +1 ¯Σ − V T (cid:19) (5.58)Eventually, we have at our hand the whole new symplectic basis { ¨ e n +1 I } qI =1 of P n +1 as desired. 20et us summarize our findings and explain their significance for the coordinatedescription of the one-step evolution. In doing so, we will continue to view the post-constraint surface C + n +1 as a linear subspace of P n +1 . As we have made clear before,this is possible thanks to the fact that Λ n +1 has zero intersection with C +(0) n +1 . Weremind that we have fixed the representative space ˙ C − n to be the space spanned by { ˙ e nE , ˙ e nE + q } r n E =1 . On the other hand, the representative space on the post-constraintsurface was fixed as ˙ C + n +1 = C +(0) n +1 , which is spanned by { ¨ e n +1 E , ¨ e n +1 E + q } r n E =1 . Wealso know that N ω ( C + n +1 ) = Λ n +1 and that C +(0) n +1 ⊕ Λ n +1 = C + n +1 . As a linearsubspace of P n +1 , the post-constraint surface C + n +1 has basis { ¨ e n +1 E , ¨ e n +1 E + q } r n E =1 ∪{ ¨ e n +1 M } qM = r n +1 .We can now pass to the adapted coordinates and use them to describe vectors in P n and P n +1 . A vector u n ∈ P n can be written in coordinates w.r.t. the new basis { ˙ e nI } qI =1 as u n = ˙ u nI ˙ e nI . Similarly, a vector v n +1 ∈ P n +1 can be written in coordinatesw.r.t. the new basis { ¨ e n +1 I } qI =1 as v n +1 = ¨ v n +1 I ¨ e n +1 I . Thanks to our construction, itis now exceptionally easy to make judgments about their nature: Observation 5.5.
For u n ∈ P n , the following statements hold.(i) u n ∈ C − n if and only if ˙ u nM = 0 for all M = r n + 1 , ..., q (ii) u n ∈ N ω ( C − n ) if and only if ˙ u nE = ˙ u nE + q = ˙ u nM = 0 for all E = 1 , ..., r n and all M = r n + 1 , ..., q (i.e., only ˙ u nM + q can be nonzero)(iii) u n ∈ ˙ C − n if and only if ˙ u nM = ˙ u nM + q = 0 for all M = r n + 1 , ..., q Observation 5.6.
For v n +1 ∈ P n +1 , the following statements hold.(i) v n +1 ∈ C + n +1 if and only if ¨ v n +1 M = 0 for all M = r n + 1 , ..., q (ii) N ω ( C + n +1 ) = Λ n +1 , i.e., v n +1 ∈ N ω ( C + n +1 ) if and only if ¨ v n +1 E = ¨ v n +1 E + q =¨ v n +1 M + q = 0 for all E = 1 , ..., r n and all M = r n + 1 , ..., q (iii) v n +1 ∈ ˙ C + n +1 if and only if ¨ v n +1 M = ¨ v n +1 M + q = 0 for all M = r n + 1 , ..., q Proof.
Follows directly from the preceding discussion.As for the general form of the evolution map, we know that all vectors u n ∈ C − n can be evolved into u n +1 = H n +1 ( λ n +1 ) u n ∈ C + n +1 . In canonical coordinates, this isrepresented (in matrix form) as u n +1 = E n u n + F n +1 λ n +1 (5.59)On the other hand, upon using our adapted coordinates—recall the relations (5.44) and(5.54)—, this prescription simplifies substantially. In particular, the vector componentsassociated to the symplectic bases { ˙ e nX } and { ¨ e n +1 X } , X = 1 , ..., r n , q + 1 , ..., q + r n ,of ˙ C − n and ˙ C + n +1 , respectively, are conserved by the evolution,¨ u n +1 X = ˙ u nX (5.60)while the zero components ˙ u nM are updated with an arbitrary constant contributionfrom the point-set part F n +1 λ n +1 ∈ Λ n +1 ,¨ u n +1 M = λ n +1 M − r n (5.61)21nd the null-space components ˙ u nM + q are annihilated,¨ u n +1 M + q = ˙ u nM = 0 (5.62)Let us remark that we could have introduced a more logical transformation whichwould instead result in ¨ u n +1 M = ˙ u nM = 0 and ¨ u n +1 M + q = λ n +1 M − r n . However, thistransformation—try to use an identity matrix instead of (5.52) to see it come out—would have a less practical matrix (5.58), so we decided to proceed this way instead.Because of providing the Hamiltonian evolution with such a beautifully simple form,the adapted coordinates defined solemnly by the two symplectic matrices (5.42) and(5.58) can be very helpful not only in classifying vectors, but also in describing theevolution. With this we close our discussion of the constraint surfaces. In this section we briefly discuss solutions spanning over the whole considered timeinterval from n = 0 to n = t , i.e., elements of the solution space S t . Recall that a solution y ∈ S t is a point in P t which satisfies momentum-matching − p n = + p n aswell as all the constraints originating in the irregularity of the system. At every time-slice n we can identify two kinds of pre-constraints: there is the forward pre-constraint C n y n = 0 which has to be satisfied by y n so that the solution continues to time-slice n + 1, and there is the backwards pre-constraint ¯ C n y n = 0 which has to be satisfiedshould the solution continue to time-slice n − y n ∈ P n satisfy the pre-constraint, i.e., y n ∈ C − n ,there is no guarantee that the evolved configuration y n +1 will be in C − n +1 . We thereforedefine the constraint surfaces D n = { y n ∈ P n | ∃ solution y ∈ S t such that y n = P n y } (5.63)For a linear system like ours, one can check that D n are linear subspaces of P n . Wemay also consider the total constraint space D t = N tn =0 D n . By definition, each solu-tion is in D t but not all points in D t are solutions, i.e., S t ⊂ D t . We must keep inmind that because of the free parameters of the Hamiltonian evolution map, y ∈ S t isin general not uniquely defined by y n ∈ D n .The previously given argument for conservation of symplectic product can be ex-tended by induction to arbitrary combination of evolution steps. We can be thereforesure that if y, z are two solutions, then ω n ( y n , z n ) is independent of n . This motivatesa definition of product of solutions ω : S t × S t → R with ω ( y, z ) = ω n ( y n , z n ) for anarbitrary n ∈ { , ..., t } . This product is not generally symplectic.Having established ω on S t , we can treat S t in the same way we treated anarbitrary subspace C of a symplectic space ( P , ω ) in Sec. 5.2 and classify the solutionsby their product structure. Let us say that two solutions y, ˜ y ∈ S t are symplecticallyequivalent if ω ( y, z ) = ω (˜ y, z ) for all z ∈ S t , and write y ∼ ˜ y . Then we render theequivalence classes [ y ] of all symplectically equivalent solutions [ y ] = { ˜ y | ˜ y ∼ y } . Thespace of such equivalence classes shall be denoted by ˜ S t . There is a naturally inducedproduct ω : ˜ S t × ˜ S t → R , ω ([ y ] , [ z ]) = ω ( y, z ). This is worth the effort for the followingreason: 22 bservation 5.7. The space ( ˜ S t , ω ) is symplectic. Proof.
The proof is analogical to that of Observation 5.3.The construction of Sec. 5.2 can be straightforwardly applied to D n which is asubspace of the symplectic space P n . Thus we get the space ˜ D n of equivalence classes[ y n ] = { ˜ y n ∈ D n | ˜ y n ∼ y n } with y n ∼ ˜ y n defined by ω n ( y n , z n ) = ω n (˜ y n , z n ) for all z n ∈ D n . The space ˜ D n is equal to D n / N n with N n = { z n ∈ D n | z n ∼ } . We ofcourse set ω n : ˜ D n × ˜ D n → R to act as ω n ([ y n ] , [ u n ]) = ω n ( y n , u n ), resulting in thesymplectic vector space ( ˜ D n , ω n ).The relationship between ˜ D n and ˜ S t is particularly simple: Observation 5.8.
For every initial condition [ y n ] ∈ ˜ D n , n ∈ { , ..., t } exists a solution[ y ] ∈ ˜ S t such that [ P n y ] = [ y n ]. This solution is unique. Proof.
By definition of D n , there is a solution y ∈ S t for each y n ∈ D n such that P n y = y n . Next, ω ( y, u ) = ω n ( P n y, P n u ), therefore y ∼ u ⇔ P n y ∼ P n u . It follows that[ y n ] = [ P n y ]. Assume there are two solutions [ y ] , [ u ] ∈ ˜ S such that y n ∼ u n , then y ∼ u and [ y ] = [ u ].We conclude our discussion by: Observation 5.9.
The spaces ( ˜ D n , ω n ) for each n ∈ { , ..., t } and ( ˜ S t , ω ) are allmutually symplectomorphic. Proof.
The symplectomorphism of ( ˜ D n , ω n ) (for arbitrary n ) and ( ˜ S t , ω ) is given byObservation 5.8. Since the symplectomorphic relation is transitive, it follows that forany n, m ∈ { , ..., t } , ( ˜ D n , ω n ) is symplectomorphic to ( ˜ D m , ω m ).For practical purposes, we can go one more step and represent each class of sym-plectically equivalent solutions [ y ] ∈ ˜ S t by a single solution y ∈ [ y ]. The space ofthese representative solutions shall be denoted by ˙ S t . We require that ˙ S t is a linearsubspace of S t . Once it is chosen, it fixes uniquely the spaces ˙ D n = { P n y | y ∈ ˙ S t } ofthe corresponding initial data. We let ˙ S t , ˙ D n inherit the symplectic structures ω , ω n of ˜ S t , ˜ D n , respectively. Note that ( ˜ S t , ω ) and ( ˙ S t , ω ) are trivially symplectomorphic.Then ( ˙ S t , ω ), ( ˙ D n , ω n ) become symplectic spaces and all the tildes in the statement ofObservation 5.9 can be replaced by dots. In this section we look at a particular example of a discrete linear dynamical system anduse it to demonstrate the application of the above introduced formalism. We considermassless scalar field on a Regge triangulation corresponding to a flat spacetime regionwith 1 space and 1 time dimension. This will truly be a toy model, since we keep thebackground fixed and only care about the field’s dynamics. On the other hand, oneshould add that in two spacetime dimensions the Einstein tensor vanishes identically[28], and this behavior carries over from continuum to lattice [6], so it would suffice toconsider a conformally flat spacetime to get the full theory including gravitation. How-ever, this is not our objective now, as the present toy model will serve its illustrativepurpose well. 23ccording to our previous assumption, the triangulation shall be composed of afinite number of spacelike slices indexed by n ∈ { , ..., t } , such that every slice includesa finite number of vertices (at most q ) and every vertex is a member of exactly one slice.For simplicity, we shall consider triangulation with only two kinds of edges: spacelikeand timelike, and suppose that all edges of each of these families have identical geom-etry. Edges between vertices which belong to the same slice are spacelike , while edgesbetween vertices which inhabit neighboring slices are timelike . We do not allow for anyother kind of edge.Let us say more about the scalar field. One can describe it easily by associating afield value ϕ i ∈ R to every vertex i . The corresponding scalar field action can be founde.g. in Sec. 6.12 of [6]; in our case it will take the form S t = 12 X edges ij w ij ( ϕ i − ϕ j ) (6.1)Here the sum runs over all edges ij in the triangulation. We do not endow edges withany orientation and assume that ij is the same edge as ji , so we can equivalently write P edges ij ≡ P ij δ eij with δ eij = ( i and j are connected by an edge0 otherwise (6.2)We assume that edges only connect distinct vertices, i.e., δ eii = 0 for all i . We remarkthat the action (6.1) is similar to that used in Example 2.1. of [18], with the majordifference that here we consider Lorentzian lattice, and not Euclidean. This is thereason why we need to include a coefficient w ij providing weight to every edge. It isproportional to the dual edge volume (here, area) and inversely proportional to thesquared edge length: w ij = V ij l ij (6.3)By definition, w ij = w ji is symmetric.We shall assume for simplicity that all our triangles are identical. In result, therewill be only two kinds of triangles in our lattice: (2,1) type triangles, which have twovertices at the sooner time-slice and one vertex at the later time-slice, and (1,2) typetriangles whose configuration is the opposite. Note that all triangles have one spacelikeedge and two timelike edges, regardless of their type. The dual edge volume is alsothe same for both the types. It may be fixed as V ij = mA where A is a constantcontribution from one triangle (i.e., 1 / m is the number of triangleswhich contain the given edge. Our triangulation will be mostly built of interior edgeswhich belong to exactly two triangles; we therefore decide to divide the action by theoverall constant 2 A . Occasionally it will be useful to consider boundary edges whichbelong to only one triangle (typically edges on a boundary); for these we shall includea factor of 1 / w ij to have the numbers right.Because of the Lorentzian nature of our lattice, the squared edge lengths l ij mustbe taken in account too. In a triangulation of a flat spacetime, they are given simply by(squared) spacetime intervals between points which correspond to the two vertices ofthe edge in question. We shall fix them as follows. First we provide our flat spacetimeregion with an orthogonal frame consisting of a time coordinate t and a space coordinate24 . Then we draw a triangular lattice such that every triangle has one edge aligned withthe x direction and all triangles are equilateral with unit side in the Euclidean metricinduced by the frame, see Fig. 6.1. x t Figure 6.1: Triangulation of flat 2-dimensional spacetime.Spacelike edges are drawn in full line, timelike edges aredrawn in dashed line.It follows that all spacelike edges have squared length 1 and all timelike edges havesquared length − /
2, i.e., our distinction of the edges has the expected geometricalmeaning. Altogether, we put w ij = ij is an interior spacelike edge − ij is an interior timelike edge1 / ij is a boundary spacelike edge − ij is a boundary timelike edge (6.4)If there is no edge between the vertices i and j , the weight w ij is undefined.The last thing we need to decide before we go on to employ the discrete canonicalevolution is the topology of the lattice. Since we need each time-slice to have a limitednumber of vertices, we opt for the tube topology with time direction along the tube.More precisely, we will consider a lattice in which each time-slice is formed by a singleclosed loop of vertices connected by spacelike edges, and the individual neighboringtime-slices are connected by timelike edges so that the resulting lattice is triangular.Since we want to consider only a finite number of time-slices, we cut the tube so thatit starts with time-slice 0 and ends with time-slice t . Consequently, all the space-like edges at time-steps 0 and t will be boundary edges. Then we can say that (6.1) isthe action of the field corresponding to the region between 0 and t , as one would expect.Suppose we are given an instance of the above defined triangular lattice with thescalar field living on it. In the general case, the number of vertices in a time-slice will bevarying with time. As discussed in [17], one can use the formalism of discrete canonicalevolution to describe the field’s dynamics. Suppose that the largest time-slice has q vertices; then one simply provides all other time-slices with additional virtual verticesso that every time-slice has q vertices in the end. After that, the total number of ver-tices is N ≡ q ( t + 1). Virtual vertices are not connected by any edges, and thereforehave no influence on the geometry. If we include into our consideration the field valuesat virtual vertices, we can say that our field has q degrees of freedom at each time-slice.The obvious implication of this trick is that the field values at virtual vertices have nosignificance for the action (6.1). In consequence, the system at hand will be irregular.25ow we can straightforwardly apply the formalism of discrete canonical evolution.Since the action (6.1) is quadratic in field values, the system satisfies our additional as-sumption of linearity, so we can take advantage of our treatment given in the precedingsection. We rewrite S t = ϕ T Kϕ (6.5)where ϕ ∈ R N and K is a real, symmetric N × N matrix which we call the dynamicalmatrix . By comparison of (6.1) and (6.5), one finds that K ij = δ ij X k δ eik w ik − δ eij w ij (6.6)From this expression it is clear that all the row and column sums of K are zero. Let usalso point out that since virtual vertices are by definition not associated to any edges, K ij = 0 whenever i or j is a virtual vertex.Next we need to split up the action into individual time-step contributions. Weshall do that simply by splitting the lattice into t individual time-steps (between 0 and1, ..., between t − t ). The splitting of lattice induces a corresponding splittingof the matrix K . We let K ( n ) be the q × q submatrix of K corresponding to time-slice n and K ( n,n +1) the q × q submatrix of K with rows corresponding to variables attime-slice n and columns corresponding to variables at time-slice n + 1. Thanks to thesymmetry of K , the submatrix K ( n ) is symmetric and K T ( n,n +1) = K ( n +1 ,n ) . Moreover,our splitting of the lattice results in further decomposition K ( n ) = K − ( n ) + K +( n ) (for n = 1 , ..., t −
1) where K − ( n ) and K +( n ) describe the boundary time-slice n of the twoseparated time steps: one between n − n ( − ), other between n and n + 1 (+).These matrices are given by the same formula (6.6) (with i, j both belonging to n ) towhich one plugs the lattice of the appropriate individual time-step. Note that the onlyquantities which change in splitting the lattice are the dual volumes of spacelike edges.An interior spacelike edge ij has dual volume V ij = V − ij + V + ij whose one part V − ij lies in the time-slice between n − n and the other part V + ij lies in the time-slicebetween n and n + 1. The splitting of K ( n ) therefore corresponds to splitting of thesedual volumes according to the given geometry, so that one gets the correct time-stepaction contribution S n +1 . In our simplified setting, the splitting is done very easilyby turning all (originally interior) spacelike edges into boundary spacelike edges, i.e.,dividing their edge weight by the factor of two. See Fig. 6.2 for an illustration.Figure 6.2: Illustration of the splitting procedure. Spacelikeinterior edges (drawn in bold line) turn into spacelike bound-ary edges (drawn in normal line). Timelike edges (drawn indashed line) remain unchanged.The action contribution from the time-step between n and n + 1 takes the form S n +1 = (cid:16) x Tn K +( n ) x n + 2 x Tn K ( n,n +1) x n +1 + x Tn +1 K − ( n +1) x n +1 (cid:17) (6.7)26here x n ∈ R q is the q -tuple of field values ϕ i in vertices i (including the virtual ones)belonging to time-slice n . The matrix K ( n,n +1) describes the interaction along timelikeedges so it does not come with any additional factor. With this, one easily identifiesthe matrices in (5.5) as L n = K +( n ) R n = K ( n,n +1) ¯ L n +1 = − R Tn = − K ( n +1 ,n ) ¯ R n +1 = − K − ( n +1) (6.8)We see that (5.6) and (5.7) indeed hold. The reader can also easily check that theindividual contributions (6.7) give the action (6.5), S t = t − X n =0 S n +1 ( x n , x n +1 ) (6.9)as we desire. Let us remark that thanks to our assumption of a closed loop topology ofeach time-slice, there will be no timelike boundary, and all timelike edges will be foundin the interior. On the other hand, due to our splitting of the lattice into individualtime-steps, all spacelike edges will be on the boundary. Therefore (6.4) simplifies to w ij = ( − ij is an interior timelike edge1 / ij is a boundary spacelike edge (6.10) We can move on to discuss particular examples of lattice time-steps. We craft them sothat they are as simple as possible and at the same time make visible the full range ofthe model’s behavior. On the most basic level, there are three situations with differentimplications for the evolution. First, the vertices of the lattice are equally distributedamongst time-slices and well connected; then the system turns out regular. Second,the number of well connected vertices decreases in a time-step which results in a pre-constraint. Third, the number of well connected vertices increases in a time-step whichresults in a free parameter of the evolution. The first three examples are supposed toillustrate these cases. Last but not least, we should remark that the regularity of theevolution is not only dependent on the numbers of vertices at subsequent time-slices,but also on their connectivity. If the connectivity is poor, we intuitively feel that thesystem will be irregular, because the lattice will obstruct propagation of degrees offreedom. However, there are occasions on which our intuition can be misleading. Wedemonstrate this fact by one bonus example.
Example 6.1.
First we consider a time-step between time-slices 0 and 1 with exactlythree vertices at each time-slice. The lattice is depicted in Fig. 6.3.We have q = 3, t = 1 and N = 6. The dynamical matrix (6.6) is K = − − / − / − − / − − − / − / − − / − (6.11)27 2 34 5 6 n = 0 n = 1Figure 6.3: Diagram of the time-step lattice of Example 6.1.The fragments of edges on the right are meant to be con-nected to the fragments on the left, so that each time-slice isa closed loop.Because the matrix is symmetric, we only write the upper triangle. One can check thatthe row and column sums of K are indeed zero. It is easy to read out the matrices of(6.8). Since we have a single time-step, no splitting is needed. We get L = − , R = , ¯ R = − L (6.12)At this point we can easily express the canonical evolution between time-slices 0 and 1by plugging into (5.14)—(5.18). Since R is regular, there is no pre-constraint, and thepoint-space Λ has dimension zero. In other words, the present single-time-step systemis regular. For the evolution we get simply y = E y with E = 14 − − − − − − / / − − − / / − / − / − (6.13)This solves uniquely any initial-value problem. For instance, the canonical initial vec-tor y = (cid:0) (cid:1) T evolves into y = (cid:0) − / − / (cid:1) T . Thereader can check that the symplectic form is fully conserved.Eventually, let us switch to the adapted coordinates. First we perform the singularvalue decomposition R = U Σ V T with the result U = 1 √ √ − √ −√ √ √ , Σ = , V = 1 √ √ −√ − √ √ √ − (6.14)From (5.42) and (5.58) we have˙ W = (cid:18) − U T L U T − U T (cid:19) , ¨ W = (cid:18) ¯Σ V T − ¯Σ − V T ¯ R ¯Σ − V T (cid:19) (6.15)These give ˙ W = 12 √ √ √ √ √ √ √ − √ √ − √ √ −
10 5 5 − − √ − √ − √ √ − √ − − (6.16)28nd ¨ W = 14 √ √ √ √ − √ √ − − − √ − √ − √ √ √ √ √ − √ − √ √ −
10 5 − − (6.17)If we express the above vectors y and y in the adapted bases by transforming themwith the matrices (6.16) and (6.17), we get ˙ y = ˙ W y = √ (cid:0) √ − −√ (cid:1) T and ¨ y = ¨ W y = ˙ y , i.e., the adapted coordinates of the vector are conserved as ex-pected. Example 6.2.
Now let us consider a different triangular lattice with three vertices attime-slice 0 but only one vertex at time-slice 1, as illustrated by Fig. 6.4. We presumethat because of the loss of degrees of freedom in the time-step from 0 to 1, the systemwill be irregular and a non-trivial pre-constraint will arise.1 2 34 5 6 n = 0 n = 1Figure 6.4: Diagram of the time-step lattice of Example 6.2.It is made of three identical type 2-1 triangles. Vertices 4and 6 are virtual. Dashed edges are timelike.We have q = 3, t = 1 and N = 6 as before. We read out the dynamical matrix K = − − / − / − − / − − (6.18)and the matrices governing the evolution L = − , R = , ¯ R = (6.19)As we presumed, R is singular. We see that r = 1 and s = 2. According to (5.14),there is a pre-constraint C y = 0 which must be satisfied if we want to evolve y to thenext time-slice. To express the pre-constraint, we may take advantage of the singularvalue decomposition R = U Σ V T with U = 1 √ √ −√ − √ √ √ − , Σ = √ , V = (6.20)29e use P N ( R Tn ) = U U T where U = 1 √ −√ −
10 2 √ − , P N ( R Tn ) = − − − − (6.21)and compute C = − − − − − − − (6.22)The pre-constraint surface C − is identified as the null space of this matrix; it hasdimension four. This is because the total dimension of the phase space is six and thepre-constraint has dimension two. The latter corresponds to the number of virtualvertices. A state y can be evolved to time-slice 1 if and only if it belongs to C − . Theevolved state is never unique, since it is given by y = E y + F λ with an arbitrary λ ∈ R . A quick calculation reveals E = 16 , F = (6.23)One can observe that F λ adds an arbitrary contribution to the field values at virtualvertices 4 and 6. This makes perfect sense, because the virtual vertices have no phys-ical meaning, so it would be strange if their associated field values were in any waydetermined. On the other hand, the momenta of the virtual vertices are fixed to zero,with no contribution from F λ .Let us switch to the adapted coordinates. We can straightforwardly calculate thetransformation matrices given as in (6.15), obtaining˙ W = 1 √ √ √ √ √ √ √ −√ / √ / −√ √ − / − / − − −√ −√ −√ √ −√ − (6.24)and ¨ W = 12 √ √ √ − √
30 0 0 2 √ (6.25)If one needs the inversed versions of these matrices, which give explicitly the adaptedbases (in language of the canonical ones), one can easily take the inverse by using(4.10). Recalling Observations 5.5 and 5.6, we can classify state-space vectors on bothtime-slices based on their adapted coordinates. Thus we know that ˙ y belongs to thepre-constraint surface C − if and only if its second and third component are zero. The30rst and fourth component of ˙ y represent field values and momenta, respectively, ofvectors in the representative space ˙ C − . The fifth and sixth component of ˙ y parametrizethe null space N ω ( C − ), which is a subspace of the pre-constraint surface C − .For example, the vector y = (cid:0) − (cid:1) T clearly satisfies C y = 0,and thus belongs to the pre-constraint surface. Its adapted coordinates are ˙ y =˙ W y = √ (cid:0) √ − √ √ (cid:1) T . We see that this form confirms that y ∈ C − . Moreover, it tells us that the vector has nonzero intersection with thenull space N ω ( C − ), and therefore does not belong to the representative space ˙ C − .Nevertheless, we can easily evolve it to time-slice 1 by using (5.60)—(5.62) with theresult ¨ y = √ (cid:0) √ λ λ − √ (cid:1) T . Note that because of our choice ofthe adapted coordinates, the null space N ω ( C +1 ) is parametrized by the second andthird coordinate of ¨ y (which take the free parameters λ , λ ), while the complement P r C +1 of the post-constraint surface to the full phase space is parametrized bythe fifth and the sixth component. The vector y automatically belongs to the post-constraint surface C +1 , thus the two zeros at these positions. The representative-spacecomponents are conserved. One can easily check that ¨ y = ¨ W y with y computedfrom (6.23). Example 6.3.
Now we provide the third promised instance of a triangular one-steplattice, which is the time-reversed version of that from Example 6.2. Its depiction isgiven in Fig. 6.5. We again expect to find the system irregular, but since degreesof freedom are added, the irregularity should give rise to a nontrivial space of freeparameters. 1 2 34 5 6 n = 0 n = 1Figure 6.5: Diagram of the time-step lattice of Example 6.3.It is again made of three identical triangles, this time of type1-2. Vertices 1 and 3 are virtual.We have K = − − − / − / − − / − (6.26)and thus L = − , R = , ¯ R = 12 (6.27)31he computation is fully analogical to the preceding case. We find U = , Σ = √ , V = 1 √ √ −√ − √ √ √ − (6.28)which gives C = (6.29)There is a nontrivial pre-constraint, even though the number of physical degrees offreedom increases. The pre-constraint surface C − has dimension four—the same as inthe previous example. However, observe that the pre-constraint is only concerned withthe momenta at virtual vertices. It reflects the fact that the field at virtual vertices,however it looks, does not propagate to the future, therefore its momenta must be zero.The evolution of vectors y ∈ C − is described by E = 16 , F = 12 √ − √ −
20 42 √ − −√ −
10 2 √ − (6.30)As one expects, the evolution only takes into account the variables at vertex 2, thefield values and momenta at virtual vertices 1 and 3 are irrelevant. This time, the freeparameters λ , λ will influence all resulting field values and momenta.We can go on to the adapted bases. The transformation matrices are˙ W = − − − (6.31)and ¨ W = 16 √ √ √ √ − √ √ − − − √ − √ − √ √ √ √ √ − √ − √ √ − − − (6.32)The classification of vectors in adapted coordinates is the same as before. To try itout, take for example the initial vector y = (cid:0) (cid:1) T . It clearly satisfiesthe pre-constraint C y = 0. In the adapted coordinates, it looks like ˙ y = ˙ W y = (cid:0) − (cid:1) T . The zeros at positions two and three confirm that y ∈ C − .The zeros at positions five and six are the result of our arbitrary choice, and mean that y has zero intersection with the null space N ω ( C − ) and so y ∈ ˙ C − . Let us evolve thisvector to time-slice 1. According to our trivial evolution prescription, we write ¨ y = (cid:0) λ λ − (cid:1) T . As always, it holds ¨ y = ¨ W y . The last two components of32 y tell us that we are in the post-constraint surface C +1 . There are two free parametersentering the evolution just as in Example 6.2. However, unlike in Example 6.2, thepresent free parameters have physical significance, since they contribute to the fieldvalues and momenta at real vertices 4, 5 and 6 of the lattice. Example 6.4.
Eventually, let us consider a one-step lattice analogical to the lattice ofExample 6.1, but with only two vertices per time-slice. The diagram is given in Figure6.6. Because of the smaller number of vertices, the spacelike edges at both time-slicesare doubled (we keep two edges between the two vertices of each time-slice in order tosatisfy our assumption that each time-slice is a closed loop), and every vertex shares atleast one edge with every other. This makes the lattice slightly unusual; nevertheless,it still formally complies to our assumptions.1 23 4 n = 0 n = 1Figure 6.6: Diagram of the time-step lattice of Example 6.4.It is formed by four triangles (one doubled type 2-1 triangleand one doubled type 1-2 triangle).Let us work out the corresponding matrices. We can take t = 1, q = 2, and so N = 4. The dynamical matrix is K = − − − − − − (6.33)We have implemented the double edges simply by summing up the weights. It holds L = − (cid:18) (cid:19) , R = (cid:18) (cid:19) , ¯ R = (cid:18) (cid:19) (6.34)Now the catch is clear: the matrix R is not regular, instead r = 1 and s = 1. Thesingular value decomposition of R results in U = 1 √ (cid:18) −
11 1 (cid:19) , Σ = (cid:18) (cid:19) , V = 1 √ (cid:18) −
11 1 (cid:19) (6.35)(note that U = V , this is because R is symmetric) which gives C = 12 (cid:18) − − − − (cid:19) (6.36)The pre-constraint surface C − is not the whole P , it has dimension three. For vectors y in C − , the evolution is fixed by E = 18 , F = 1 √ − − (6.37)33ooking at F , one can see that the dimension of the null space N ω ( C +1 ) is one, i.e.,there is one free parameter λ of the evolution. The transformation matrices to theadapted bases are˙ W = 1 √ − − − − − − , ¨ W = 1 √ − − − − / / − − (6.38)Take for example the vector y = (cid:0) (cid:1) T , which satisfies the pre-constraint. Itsadapted coordinates are ˙ y = √ (cid:0) − (cid:1) T . The zero at position two signifiesthat we are on the pre-constraint surface. The other three components parametrizethe pre-constraint surface. In particular, the last component parametrizes N ω ( C − ).Evolving to time-slice 1, we write ¨ y = √ (cid:0) λ − (cid:1) T . All this is a standarduse of the formalism. It shows us that in spite of a constant number of degrees offreedom and high connectivity, the system we obtain is irregular. We interpret thisbehavior by saying that the lattice is overconnected . Eventually let us briefly comment on lattices with multiple time-steps. There is reallynothing new to these, since they are but individual time-step lattices stacked on topof each other, forming a system arbitrarily extended in time. The evolution of suchsystem is given simply as a series of the individual one-step evolution moves. Thingscan get more complicated if one asks questions about global properties of the system,e.g. when one wants to find initial data y which give rise to a global solution. In thatcase, one needs to trace back all the pre-constraints arising anywhere in the lattice.This is why we say that constraints propagate in time , both to future and past. Onthe other hand, given a vector y ∈ P n , one may evolve it by a series of local one-stepevolution moves and in this way find its later versions. The solution (if it exists) maybranch out with an increasing number of free parameters or tail off (meaning that itis restricted or even ceases to exist) due to pre-constraints. At all cases, we know wellthat the symplectic structure of solutions will be conserved in time.To illustrate some of the possible behavior of multistep systems, we offer two ex-amples. Both are built up from the time-step lattices of Examples 6.1—6.3 and extendover three time-steps. In other aspects, they are quite different. Example 6.5.
The first case is depicted in Fig. 6.7. It starts and ends with a singlevertex, but widens in between. Nevertheless, this widening has little effect on the prop-agating degrees of freedom, since the free parameters arising during time-step between0 and 1 will be eventually diminished by the pre-constraint at time-slice 2.
Example 6.6.
The second case of a multistep system is depicted in Fig. 6.8. Thistime the lattice begins and ends with three vertices which carry three degrees of free-dom. The evolution between time-slices 0 and 1 is regular, but at time-slice 2 thelattice narrows down to a single vertex, thus obstructing the propagation of degreesof freedom. In result, the number of propagating degrees of freedom is restricted to one.34 2 34 5 67 8 910 11 12 n = 0 n = 1 n = 2 n = 3Figure 6.7: Diagram of a lattice with a temporal widening.1 2 34 5 67 8 910 11 12 n = 0 n = 1 n = 2 n = 3Figure 6.8: Diagram of a lattice with a temporal narrowing. In this work, the existing formalism of discrete canonical evolution was revisited andapplied to the case of linear dynamical system, i.e., a system with vector configurationspace and quadratic action. Thanks to the very strong assumption of linearity, wecould rewrite the one-step evolution into a simple matrix form. The key object in thisformulation is the matrix R n describing interaction of variables between time-steps n and n + 1. One can easily obtain the explicit Hamiltonian evolution map, all it takesis a singular value decomposition of R n . For an irregular system, the evolution map isonly defined on a subset C − n of the phase space P n called the pre-constraint surfaceand is neither unique nor symplectic.In order to understand the symplectic structure of the model, we performed ananalysis of the constraint surfaces in relation to the symplectic form. Then we con-structed two special bases of the phase space P n at each discrete time n which explic-itly separate the constraint surfaces, the null spaces and the subspaces of propagatingdegrees of freedom. The corresponding transformations were given in terms of twosymplectic matrices (5.42) and (5.58). Thanks to this construction, we were able tointroduce a reduced evolution map H n +1 (0) : ˙ C − n → ˙ C + n +1 which is symplectic . More-over, it was shown that in the adapted coordinates given by the new bases, the generalone-step evolution map assumes a trivial form. We also gave some theoretical back-ground for considering global solutions.In comparison with a previously published article [20] on the topic, we made a num-ber of decisions that lead to a significantly different approach. We gave much moreattention to the symplectic structure since we consider it to be the most importantobject of the model. We limited our analysis of the Hamiltonian evolution to a single35ime-step, which resulted in a less complex and arguably more straightforward treat-ment. Unlike in [20], the construction is performed explicitly in terms of matricespresent in the action or their singular value decomposition. This allows for a smoothimplementation and an easy application of the formalism to any problem dealing witha linear discretely evolving system.In the last section we provided a fully worked-out example of discrete linear evo-lution of massless scalar field on a fixed two-dimensional spacetime lattice. Althougha toy model, it has a sound physical base, and one can easily think of generalizations.One can observe how the scalar field is shaped by the lattice, shaped by its geometryand causal structure. The example is closely related to the intended application of thepresent work, which is the description of matter or gauge fields on a fixed spacetimelattice in the manner similar to (quantum) field theory on curved spacetime. Withthis example we also demonstrated in simple fashion the most important features ofthe irregular linear evolution as well as its overall utility, and illustrated the previouslyintroduced formalism.The present analysis is supposed to serve one more purpose, namely to providethe necessary tools for a subsequent treatment of a quantum version of the consideredsystem. This is an interesting and relevant problem, addressed before in Sec. 10 of[20] and more generally in the preceding work [29, 30]. The pursuit craves for specialpreparation since the standard quantization procedure typically requires a one-to-onesymplectic evolution map on the phase space which can be used to induce the corre-sponding unitary evolution map on the Hilbert space describing the system. However,the case of discrete linear canonical evolution does not meet this requirement. Thereare of course ways of surpassing this problem, yet they are easier to follow with anappropriate set of tools and good understanding of the classical design. Within thispaper, we have spent effort to increase our understanding and prepare grounds for thefollowing work concerning the quantum analogue. It is currently under preparation andlikely to appear sometime in the near future. Acknowledgments
This work was supported by Charles University Grant Agency [Project No. 906419].
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