Time, classical and quantum
Paolo Aniello, Florio Maria Ciaglia, Fabio Di Cosmo, Giuseppe Marmo, Juan Manuel Pérez-Pardo
aa r X i v : . [ qu a n t - ph ] M a y Time, classical and quantum
P. Aniello , , F.M. Ciaglia , , F. Di Cosmo , , G. Marmo , , J.M. Perez-Pardo Dipartimento di Fisica “Ettore Pancini”, Universit`a di Napoli “Federico II”Complesso Universitario di Monte S. Angelo, via Cintia, I-80126 Napoli, Italy INFN - Sezione di NapoliComplesso Universitario di Monte S. Angelo, via Cintia, I-80126 Napoli, Italy
Abstract
We propose a new point of view regarding the problem of time in quantum mechanics,based on the idea of replacing the usual time operator T with a suitable real-valued function T on the space of physical states. The proper characterization of the function T relieson a particular relation with the dynamical evolution of the system rather than with theinfinitesimal generator of the dynamics (Hamiltonian). We first consider the case of classicalHamiltonian mechanics, where observables are functions on phase space and the tools ofdifferential geometry can be applied. The idea is then extended to the case of the unitaryevolution of pure states of finite-level quantum systems by means of the geometric formulationof quantum mechanics. It is found that T is a function on the space of pure states which isnot associated to any self-adjoint operator. The link between T and the dynamical evolutionis interpreted as defining a simultaneity relation for the states of the system with respectto the dynamical evolution itself. It turns out that different dynamical evolutions lead todifferent notions of simultaneity, i.e., the notion of simultaneity is a dynamical notion. The problem of time in quantum mechanics is a beautiful and subtle one. We can formalizeit with a simple question, namely, is there a self-adjoint operator we can associate to time inquantum mechanics? Or, even better, is time a quantum observable?There are many experimental instances in which this question makes sense because timeseems to acquire an observable character. For example, we can think of the time of arrival ofa particle in a detector, the time of occurence of a specific event, or the tunneling time of aparticle under the influence of a potential barrier.In standard quantum mechanics, observable quantities are described by means of self-adjointlinear operators on the Hilbert space of the system. In this setting, a time observable T wouldbe characterized as a self-adjoint operator T which is canonically conjugated to the Hamiltonianoperator H of the system: [ H , T ] = − ı ~ I . (1)In contrast with CCRs relating position and momentum, the commutation relation between T and H is plagued by severe technical difficulties. In [1], Pauli realized that a self-adjoint operator T canonically conjugated to the Hamiltonian operator H does not exist whenever the spectrum1f H is bounded from below. Pauli’s proof was not rigorous, and, to be fair, he never claimed itto be so. However, it took some time for the rigorous mathematical formulation of the problemto be settled (see [2], and [3]), and, in the meantime, different strategies to cope with the problemhave been proposed. For instance, attention has been given to the possibility of relaxing theself-adjointness condition for the time observable T . In this direction, of particular interest isthe construction of a maximally symmetric time operator T which is canonically conjugated tothe Hamiltonian operator H of the 1-dimensional free particle given by Ahronov and Bohm in[4]. This operator is a sort of canonical quantization of the classical passage time of Newtonianmechanics, and thus, its physical interpretation is related to the experimental concepts of passagetime, and of time of flight. Another change of perspective occured, and efforts were, and aremade to construct a positive operator-valued measure (POVM) having a particular covarianceproperty with respect to the dynamics, and that can be reasonably interpreted as a time POVM([5], [6], [7], and [8]). In this setting, the physical interpretation of the time POVM constructedin [9] is related to the experimental concept of time of occurrence. Finally, some interestingcounterexamples to Pauli’s theorem have been given. Among the most interesting ones is thecase of a phase operator constructed by Galindo [10] and Garrison and Wong [11], which is abounded, self-adjoint operator canonically conjugated to the number operator, and thus withthe Hamiltonian operator, of the 1-dimensional quantum harmonic oscillator. The physicalinterpretation of this operator is in some sense related to the quantum-mechanical formulationof the action-angle variables exposed by Dirac in [12].From this brief discussion we can extract two important facts. First of all, time in quantummechanics is a dynamical quantity which is intimately connected with the specific dynamicalevolution of the system and with specific experimental questions. Second, it seems that self-adjoint operators are simply not enough to handle the problem of time in quantum mechanics,and different mathematical objects may be appropriate to treat different aspects of time. Inthis article, we focus on the simultaneity aspect of time in quantum mechanics, and propose todescribe it by means of a real-valued function T on the space of physical states, which we calla time function, satisfying a particular equivariance condition with respect to the dynamicalevolution of the system.In accordance with Einstein’s theory of special relativity, we recognize two different butrelated aspects of our common perception of time in physical phenomena. On the one hand,time appears as an evolution parameter, a sort of ordering label by means of which we formalizethe perception of the causal aspect of “before” and “after”. Following an heuristic argument, themathematical object that captures this aspect of time in a spacetime framework is a vector field,say ∂∂t . Given an integral curve γ m ( τ ) of ∂∂t starting at m = γ m (0), the parameter τ “measures”causality in the sense that m = γ m ( τ ) casually precedes m = γ m ( τ ) if and only if τ < τ , andthus, all the events lying on γ m ( τ ) are interpreted as causally connected through the spacetimeevolution determined by ∂∂t . It is clear that this causal aspect of time is meaningful only inrelation to events lying on the same integral curve γ m ( τ ) of ∂∂t .On the other hand, time is naturally associated to the concept of simultaneity, which is aparticular relation between different events that need not to lie on the same integral curve of ∂∂t .The purpose of simultaneity is to provide a way to compare the evolution through ∂∂t of differentinitial events so that a relational notion of “before” and “after” is meaningful. Consequently, thesimultaneity aspect of time can not be described by means of the vector field ∂∂t , and the correctmathematical object is an integrable differential one-form, say dt . The integrability condition2mplies that dt defines a codimension-one foliation F of the spacetime M . Of course, dt cannot be completely arbitrary since simultaneity must take into account the specific spacetimeevolution described by ∂∂t . Specifically, dt must satisfy: dt (cid:18) ∂∂t (cid:19) = 1 . (2)This condition implies that the leaves of the foliation F induced by dt are transversal to theintegral curves of ∂∂t , and it is precisely this transversality condition that motivates the inter-pretation of the events on the same leaf as simultaneous events. For an extensive and rigoroustreatment of this notion of simultaneity in a spacetime framework we refer to [13] and [14].The causality and simultaneity aspects of time encoded in the couple (cid:0) ∂∂t , dt (cid:1) are purelykinematical, as they are defined with respect to a fixed spacetime background. This meansthat we have to lift these considerations to a dynamical setting in order to make contact withquantum mechanics, where time seems to acquire a purely dynamical flavour, and spacetimedoes not enter directly in the formulation of the theory.In quantum mechanics, as well as in other dynamical theories, we have two objects that wecan use to set up a dynamical framework for simultaneity. There is the space of states P of thesystem and its dynamical evolution { φ τ } . The space of states P plays a role analogous to thatof spacetime M in our previous discussion, while the trajectories of the dynamical evolution { φ τ } represent the causality aspect of time in analogy with the integral curves of the vectorfield ∂∂t . What is missing is a mathematical object describing simultaneity with respect to { φ τ } ,and we propose to identify it with a function T defined on a suitable subset P ∗ of the space ofstates P . In analogy with the one-form dt , the time function T satisfies a suitable equivariancecondition with respect to { φ τ } which resembles the transversality condition between dt and ∂∂t .Specifically, T is such that its level sets are mapped into each other by { φ τ } , and thus, all thestates in a level set of T are interpreted as simultaneous states.Following this line of thought, we will analyze the case of unitary evolution of the purestates of a finite-level quantum system by means of the so-called geometric formulation of quan-tum mechanics, according to which, the tools of differential geometry characteristic of classicalmechanics can be used in the quantum setting. Consequently, in Section 2 the idea of a timefunction T is presented in the classical setting. A precise definition for the equivariance conditionof T with respect to the dynamical evolution is given, and different examples of time functionfor well-known physical systems are presented.In Section 3 we briefly recall the main points of the geometric formulation of quantummechanics, and then pass to analyze finite-dimensional systems providing the construction ofa family of time functions for all the unitary evolutions generated by a Hamiltonian operator H with at least two different eigenvalues. This is interesting because the construction of atime operator T canonically conjugated to a Hamiltonian operator H for a finite-level quantumsystem is forbidden because of dimensional reasons. We will find that the time function T is notthe expectation value function of some linear operator on the Hilbert space H of the system.Hence, T is not associated to an observable in the canonical sense, i.e., a self-adjoint linearoperator on H , rather, it is more like other functions on the space of states, such as Entropyor Purity. We want to stress that the time functions for the unitary evolutions of finite-levelquantum systems are not the quantization of some time function defined for a classical system.Indeed, the geometrical definition of time function we give in Section 2 is general enough to3ncompass all those physical system described using the tools of differential geometry. In the description of a classical system S , the space of states is a finite-dimensional differentialmanifold P , while the algebra of observables O of S is realized as an algebra of functions on P . Essentially, we will take O to be the algebra C ∞ ( P ) of real-valued, smooth functions on P endowed with the pointwise product. The expectation value of an observable f ∈ O on the state p ∈ P is just the evaluation f ( p ) of the function f on the state p . The dynamical evolution ofthe system is described by the one-parameter group { φ τ } τ ∈ R generated by a complete vectorfield Γ referred to as the dynamical vector field, and the image φ τ ( p ) ⊂ P of p ∈ P throughthe dynamical evolution is the dynamical trajectory of the initial state p . In the Hamiltonianformulation P is endowed with a Poisson tensor Λ, and the dynamical vector field Γ is theHamiltonian vector field associated to a Hamiltonian function H :Γ = Λ(d H ) . (3)We can divide dynamical evolutions into three different classes according to the nature of theirdynamical trajectories. First of all there are periodic dynamical evolutions, for which all thedynamical trajectories are periodic. Then, there are non-periodic dynamical evolutions, forwhich none of the dynamical trajectories are periodic, and finally, there are mixed dynamicalevolutions, for which some dynamical trajectories are periodic and some are not.The time function we want to describe explicitely depends on the dynamical evolution of thesystem, and, in general, different dynamics will lead to different time functions. We will nowintroduce two different types of time function and associated simultaneity relations, one whichis well-suited for non-periodic dynamical evolutions, and one which is well-suited for periodicdynamical evolutions (periodic time function). In both cases, we need to introduce a reducedspace of states P ∗ , i.e., an open dense subset of P which is invariant with respect to the dynamicalevolution and which will be the domain of definition of the time function T . The fact that P ∗ is in general different from the whole space of states P is related to the existence of fixed pointsfor the dynamical evolution { φ τ } in consideration, i.e., states that are completely unaffected by { φ τ } . As we will see in Section 3, the case of mixed dynamical evolution can be handled usinga periodic time function.In the case of a non-periodic dynamical evolution { φ τ } , we define a function T : P ∗ → R tobe a time function if the following conditions are satisfied: T ( p ) = T ( φ τ ( p )) ∀ τ = 0 , (4) T ( p ) = T ( p ) = ⇒ T ( φ τ ( p )) = T ( φ τ ( p )) . (5)The function T naturally induces an equivalence relation ∼ T on P ∗ given by: p ∼ p iff T ( p ) = T ( p ) . (6)An equivalence class of ∼ T is denoted as F t , with t ∈ R . As a set, F t is given by: F t := { p ∈ P : T ( p ) = t } . (7)4quation 5 implies that the dynamical evolution of the system is perfectly transversal to theequivalence relation ∼ T , that is, it “moves” the states in the equivalence class F t into an equiv-alence class F t ′ which is different from the initial one because of equation 4. Accordingly, ∼ T isinterpreted as a simultaneity relation relative to the dynamical evolution { φ τ } , and the statesin F t are interpreted as the simultaneous states defined by ∼ T .Note that a similar approach appears in [15], where a classical dynamical time is defined asa function T on the phase-space of the system such that { T , H } = L Γ T = 1, where { , } denotesthe Poisson brackets, H is the Hamiltonian function of the system, and Γ the dynamical vectorfield associated to H . Furthermore, the idea of dynamical time, both in classical and quantumtheory, is analyzed in [16].For the sake of simplicity, here we will only give the explicit form for the time functionsof some simple systems without entering into a discussion of the explicit construction of suchfunctions. However, the detailed construction we will give in Section 3, which is based on theaction-angle variables formulation of the dynamics of the system, can easily be adapted to theexamples presented here.Let us consider a point particle in a constant force field. The space of states of the systemis P = T ∗ R ∼ = R with global Cartesian coordinates ( q , p ), and the dynamical evolution { φ τ } is generated by the complete vector field:Γ = X j =1 (cid:18) p j m ∂∂q j + F j ∂∂p j (cid:19) . (8)It is clear that { φ τ } has no fixed points since Γ has no zeros. The expressions of the dynamicaltrajectories of the system in the coordinates system ( q , p ) read: φ τ ( q j , p j ) = (cid:18) F j m τ + p j m τ + q j , F j τ + p j (cid:19) with j = 1 , , . (9)Let T : P → R be the function: T ( q , p ) = F · p F . (10)An explicit calculation shows that: T ( φ τ ( q , p )) = τ + F · p F , (11)which means that T is a time function for the dynamical evolution considered.Now, let us consider the free point particle. The space of states of the system is P = T ∗ R ∼ = R with global Cartesian coordinates ( q , p ), and the dynamical evolution is generated by thecomplete vector field: Γ = X j =1 p j ∂∂q j . (12)The explicit form of the dynamical trajectories of the system in the coordinates system ( q , p )reads: φ τ ( q j , p j ) = ( p j τ + q j , p j ) with j = 1 , . (13)5n this case, the dynamical evolution presents fixed points since Γ has zeros. Specifically, everystate ( q , ) is a fixed point of { φ τ } .Consequently, we have to define the reduced space of states P ∗ as the space of states P without the fixed points: P ∗ := { ( q , p ) ∈ P : ( q , p ) = ( q , ) } . (14)In this case, a possible time function for the system is given by the time of arrival of Newtonianmechanics: T ( q , p ) = p · q p = ⇒ T ( φ τ ( q , p )) = τ + p · q p . (15)Note that this result is in accordance with [15].In the case of a periodic dynamical evolution { φ τ } , we need to choose a different target spacefor the function T in order to handle the periodic trajectories of the system. Specifically, wechose the target space to be the one-dimensional torus T , and define a function T : P ∗ → T tobe a periodic time function T with period τ T if the following conditions are satisfied: T ( φ τ ( p )) = T ( φ τ ( p )) iff τ = τ + kτ T , k ∈ Z , (16) T ( p ) = T ( p ) = ⇒ T ( φ τ ( p )) = T ( φ τ ( p )) . (17)Because of the global non-trivial topology of the torus T , the simultaneity relation ∼ T associatedto T becomes periodic, and the closed trajectories of the system can be handled accordingly.Furthermore, the periodicity of T needs not to be the periodicity of { φ τ } , and thus we canmanage systems admitting periodic trajectories with different periods using the same periodictime function.The paradigmatic system for which a periodic time function is needed is the one-dimensionalharmonic oscillator on P = T ∗ R ∼ = R . In Cartesian coordinates ( q , p ) the dynamical evolutionis generated by the complete vector field Γ:Γ = pm ∂∂q − mν q ∂∂p . (18)The dynamical trajectories of the system are: φ τ ( q , p ) = (cid:16) q cos( ντ ) − pmν sin( ντ ) , − mνq sin( ντ ) − p cos( ντ ) (cid:17) , (19)and the only fixed point of the dynamical evolution is the origin (0 ,
0) itself.We define the reduced space of states as P ∗ := P − { (0 , } , and note that there is a naturaldiffeomorphism: Ψ : P ∗ −→ T × R + . (20)Using a local coordinates system ( ϑ , H ) on T × R + , the local expression of Ψ reads:( q , p ) Ψ( q , p ) = (cid:18) ϑ = arctan (cid:18) νm qp (cid:19) , H = p m + mν q (cid:19) . (21)6 direct calculation shows that the local expression of the dynamical vector field e Γ = Ψ ∗ Γ on T × R + with respect to ( ϑ , H ) is: e Γ = ν ∂∂ϑ , (22)and thus the local expression of the dynamical trajectories is:Ψ ( φ τ ( q , p )) = ( ντ + ϑ , H ) . (23)It is then clear that the function T : P ∗ → T defined as: T := pr T ◦ Ψ , (24)is a periodic simultaneity function for { φ τ } with period τ T = ν π . Note that T is a submersion.The simultaneous states associated to ∼ T can be described by means of a particular one-formΘ on P . To define Θ, denote with θ the differential one-form on the torus T which is dual tothe globally defined vector field on T generating the action of the torus on itself. The one-form θ is a closed but not exact one-form. Then, define Θ as the pullback of θ by means of T , i.e.:Θ := T ∗ θ = 1 H ( pdq − qdp ) . (25)This is a closed but not exact one-form on P ∗ , hence, it gives rise to a foliation F of P ∗ , andthe leaves of this foliation are precisely the simultaneous states defined by T . In this case, theleaves are just the radial lines in P ∗ approaching the origin (0 ,
0) without ever reaching it.
We will now extend the ideas exposed in the previous section to the finite-dimensional quantumcase. Our approach is based on the so-called geometric formulation of quantum mechanics([17], [18], [19], [20]), according to which the mathematical methods of differential geometrycharacteristic of classical mechanics, can be used in the quantum context.The space of pure states of a finite-level quantum system with Hilbert space
H ∼ = C n is thecomplex projective space P ( H ) associated to H . An element p ψ ∈ P ( H ) is an equivalence classof non-null vectors in H with respect to the equivalence relation: | ψ i ∼ | ϕ i iff | ψ i = α | ϕ i , α ∈ C . (26)The set P ( H ) is endowed with the quotient topology, and we denote by π the continuousprojection from H onto P ( H ).From a geometrical point of view, P ( H ) is a real, 2( n − P ( H ) is encoded in three geometrical objects, namely, a symplecticstructure ω , a Riemannian metric g , and a complex structure J satisfying the compatibilitycondition: g ( X , Y ) = ω ( J ( X ) , Y ) ∀ X, Y ∈ X ( P ( H )) . (27)In this framework, to every self-adjoint operator A ∈ B ( H ) there is associated a real-valuedfunction e A : P ( H ) → R by means of: 7 A ( p ψ ) := h ψ | A | ψ ih ψ | ψ i . (28)Accordingly, e A is nothing but the expectation value of A on the state p ψ . In this way, ob-servables of quantum mechanics are represented by means of expectation value functions on thecomplex projective space. The symplectic structure ω and the Riemannian metric g allow forthe definition of two vector fields naturally associated to e A , namely: X A := i d e A Λ Y A := i d e A G , (29)where Λ = ω − and G = g − . Furthermore, the tensor Λ allows to define of a Poisson bracket { , } on the algebra of smooth functions. On the expectation value functions the bracket reads: { e A , e B } = Λ (d e A , d e B ) = e ı [ A , B ] , (30)where [ , ] denotes the commutator of linear operators.The action | ψ i 7→ U | ψ i of the unitary group U ( H ) ∼ = U ( n ) on H induces the action p ψ f p ψ = p U ψ of U ( H ) on P ( H ), and it turns out that the fundamental vector fields of this action areprecisely the Hamiltonian vector fields X A associated to each e A , with A ∈ B ( H ) and such that U = exp( − ı A ). Accordingly, the dynamical evolution generated by the self-adjoint Hamiltonianoperator H is written in geometrical language as the one-parameter group of diffeomorphisms { φ τ } of P ( H ) generated by this dynamical vector field:Γ ≡ X H := i d e H Λ . (31)From the physical point of view, this dynamical evolution describes a closed quantum system.In general, the unitary evolution can be periodic or mixed, and we will show that every suchdynamical evolution admits a family of periodic simultaneity functions. Each of which is givenby a submersion T : P ∗ → T , where P ∗ is an open submanifold of P = P ( H ) and is invariantwith respect to the dynamical evolution. This is intimately connected with the fact that thedynamical system is integrable in the sense of Liouville-Arnol’d, i.e., it always admits ( n − P ∗ of P which is invariant for the dynamical evolution, and for which aformulation in terms of action-angle variables is possible. The family of periodic simultaneityfunctions is related to the submersions arising from the projection onto one of the ( n − H denote the Hamiltonian operator of the system. Suppose its spectrum σ ( H ) contains atleast two different eigenvalues, and construct an orthonormal basis {| j i} j =1 ,...,n on H consistingof eigenvectors of H . Write H = P j ν j E j , with ν j the eigenvalues of H , and E j the projectoronto the subspace in H spanned by the j -th eigenvector. The operator E j is self-adjoint for all j , and [ E j , E k ] = for all j and k . Consequently, the functions e E j ≡ e j are constants of themotion in involution, i.e., { e j , H } = 0 for all j , and { e j , e k } = 0 for all j and k . Moreover, thevector fields X j := X E j are pairwise commuting, and the dynamical vector field Γ reads:Γ = X j ν j X j . (32)8he vector fields X j are complete and, since their flows are periodic, each of them separatelydefines an action of a torus T on P . However, they are not all independent because P j E j = I ,which means that P j X j = 0, and the same holds true for the functions e j , that is, P j e j = 1.Nevertheless, we can always find ( n −
1) independent vector fields and functions among them.Let us choose the first ( n −
1) vector fields and the first ( n −
1) functions, that is, set: X n = − n − X j =1 X j e n = 1 − n − X j =1 e j , (33)and write the dynamical vector field as:Γ = n − X j =1 ( ν j − ν n ) X j . (34)Using the ( n −
1) constants of the motion we will now show that there is an open submanifold P ∗ ⊂ P for which a global action-angle variables formulation of the problem is possible. At thispurpose, note that the critical points of e j , that is, the points p ψ ∈ P for which d e j ( p ψ ) = 0,are just the zeros of the vector field X j , and these are all those states p ψ for which h j | ψ i = 0or h k | ψ i = 0 for all k = j , where | j i and | k i denote, respectively, the j -th and k -th normalizedeigenvector of H . From this it follows that the set: P ∗ n := { p ψ ∈ P : d e ( p ψ ) ∧ d e ( p ψ ) ∧ · · · ∧ d e n − ( p ψ ) = 0 } (35)consists of all those states p ψ such that h j | ψ i 6 = 0 for all j , that is, the vector ψ of which p ψ is the associated ray, must have non-zero components with respect to every normalizedeigenvector of H . Had we started with a different choice of ( n −
1) functions and vector fields,say, X k = − P j = k X j and e k = 1 − P j = k e j , the set P ∗ k would have been the same as P ∗ n ,therefore, the explicit choice of ( n −
1) independent functions and vector fields is irrelevant, andwe will simply write: P ∗ := { p ψ ∈ P : h j | ψ i 6 = 0 ∀ j = 1 , ...n } . (36)The set P ∗ is an open subset of P , and thus an open submanifold of P on which there are( n −
1) linearly independent constants of the motion in involution. Let F : P → R n − be givenby p ψ ( e ( p ψ ) , · · · , e n − ( p ψ )). Every a = F ( p ψ ) with p ψ ∈ P ∗ is a regular value, hence, F − ( a ) is a closed submanifold of P .Since F − ( a ) is a closed subset of P , and P is a compact manifold, we have that F − ( a ) isa compact submanifold of P . Then, according to the Liouville-Arnold’s theorem, we have thediffeomorphism: Ψ : P ∗ −→ [ α T n − α ! × I n − , (37)with I = (0 , α is an index labelling the connected components of P ∗ . It is easy tosee that P ∗ is connected, and thus α = 1. Indeed, writing the vector | ψ i of which p ψ is theassociated pure state as 9 ψ i = n X j =1 r j e ıϑ j | j i , (38)the condition p ψ ∈ P ∗ implies r j = 0 for all j . From this, it follows that the set of vectors | ψ i such that their associated ray p ψ is in P ∗ is connected. Then, since the projection π from H to P ( H ) = P is continuous, we conclude that P ∗ is connected.The explicit form for the diffeomorphism Ψ is:Ψ( p ψ ) = (cid:18) e ı ( ϑ − ϑ n ) , · · · , e ı ( ϑ n − − ϑ n ) , ( r ) P nk =1 ( r k ) , · · · , ( r n − ) P nk =1 ( r k ) (cid:19) . (39)The ( n −
1) vector fields X j are tangent to T n − and are precisely the fundamental vector fieldsof the ( n −
1) tori composing T n − . The dynamical vector field is the linear combination:Γ = n − X j =1 ( ν j − ν n ) X j (40)of the canonical vector fields X j with constant coefficients ν j . Consequently, Γ is tangent to T n − , and the dynamical evolution on P ∗ is the result of ( n −
1) uncoupled uniform motions oneach of the tori. If ν j = ν n , the projection pr T j onto the j -th torus provides us with a periodictime function T j : P ∗ → T given by: T j := pr T j ◦ Ψ , T j ◦ φ τ ( p ψ ) = e ı (( ϑ j − ϑ n )+( ν j − ν n ) τ ) . (41)Had we started with a different choice of the ( n −
1) vector fields X j and functions f j , wewould have got another diffeomorphism:Φ : P ∗ −→ T n − × R n − (42)and another family of periodic simultaneity functions: e T j := pr T j ◦ Φ . (43)The relation between T j and e T j can easily be understood. To see this, let us define the inter-twining diffeomorphisms: I ΨΦ : P ∗ −→ P ∗ , I ΨΦ := Φ − ◦ Ψ , (44) I ΦΨ : P ∗ −→ P ∗ , I ΦΨ := Ψ − ◦ Φ . (45)Clearly, I − = I ΦΨ and I − = I ΨΦ . Consequently: T j = pr T j ◦ Ψ = pr T j ◦ Ψ ◦ I ΦΨ ◦ I ΨΦ = pr T j ◦ Φ ◦ I ΨΦ = e T j ◦ I ΨΦ = I ∗ ΨΦ ( e T j ) . (46)Let us now illustrate the above construction in the case of a 2-level quantum system, that is,in the case of the Qubit, where H ∼ = C and the complex projective space is the 2-dimensionalsphere, that is, P ( H ) ∼ = S .Let H be the Hamiltonian operator of the system, ν and ν its eigenvalues, and | i , | i itsnormalized eigenvectors. It is clear that the only meaningful dynamical situation corresponds to10he case in which H has a non-degenerate spectrum, otherwise, H is proportional to the identityoperator, its associated dynamical vector field Γ on P ( H ) is the null vector field, and there isno dynamical evolution at all. Consequently, we assume ν = ν .The pure states p | i and p | i corresponding to the normalized eigenvectors of H are antipodalpoints on the sphere P ( H ) ∼ = S , and they are the only fixed points of the dynamical evolution ofthe system. The dynamical trajectories are circles on the sphere with center on the axis passingthrough p | i and p | i . In this case, the reduced space of states P ∗ is the space of states withoutthe fixed points, hence, it has the topology of a cylinder.If we choose to consider the constant of the motion e E ≡ e associated to E = | ih | , wecan write e = 1 − e and: Γ = ( ν − ν ) X , (47)with X the Hamiltonian vector field associated to e . Then, there is the isomorphism Ψ : P ∗ → T × I : Ψ( p ψ ) = (cid:18) e ı ( ϑ − ϑ ) , ( r ) ( r ) + ( r ) (cid:19) , (48)where | ψ i = r e ıϑ | i + r e ıϑ | i , and r , r = 0. The periodic time function associated to Ψreads: T ◦ φ τ ( p ψ ) = e ı (( ϑ − ϑ )+( ν − ν ) τ ) . (49)On the other hand, if we choose the constant of the motion e E ≡ e , we can writeΓ = ( ν − ν ) X , (50)and we obtain the isomorphism Φ : P ∗ → T × I :Φ( p ψ ) = (cid:18) e ı ( ϑ − ϑ ) , ( r ) ( r ) + ( r ) (cid:19) , (51)In this case, the periodic time function associated to Φ reads: e T ◦ φ τ ( p ψ ) = e ı (( ϑ − ϑ )+( ν − ν ) τ ) . (52)In both cases, the sets of simultaneous states are just the meridians on the 2-dimensional sphere. In this contribution, a different approach toward the problem of time in quantum mechanicsis proposed. Motivated by spacetime considerations, we have investigated the possibility ofdefining a notion of simultaneity in the dynamical context of quantum mechanics. The mainidea is to describe the simultaneity aspect of time for a physical system subject to a dynamicalevolution { φ τ } by specifying all those states that can be interpreted as simultaneous states withrespect to the dynamics. The essential ingredient in this description is a function T defined onthe space of physical states of the system with values in the real numbers R or in the circlegroup T , which is equivariant with respect to the dynamical evolution { φ τ } of the system itself.11he sets of simultaneous states are then defined as the level sets of T . Accordingly, the notionof simultaneity encoded in T is of dynamical nature.The time function T is introduced in the classical setting, where the space of states P ofthe system is a finite-dimensional differential manifold, and the dynamical evolution { φ τ } isthe one-parameter group of diffeomorphisms of P generated by a complete vector field Γ. Twodefinitions for a time function T are given, one which is well-suited for dynamical evolutionspresenting no periodic orbits, and one which allows to handle periodic orbits. Some simpleexamples are then briefly illustrated.By means of the so-called geometric formulation of quantum mechanics ([17], [18], [19], [20]),the unitary evolutions of pure states of a finite-level quantum system are analyzed. It is proventhat every finite-level quantum system subject to a unitary evolution allows for the definition of afamily of periodic time functions. These functions are intimately connected with the geometricalstructure of the system, specifically, to the action-angle variables formulation of the dynamics.We want to stress two facts concerning these time functions. First of all, they are not thequantization of some classical time function. The geometrical definition of the time functiongiven in section 2 applies to classical systems as well as to quantum systems. Then, they arenot the expectation value functions of some operator on the Hilbert space of the system, hence,they are not observables in the canonical sense. In this regard, the time function introducedhere is more similar to the various notions of Entropy, or Purity, or to the various measures ofentanglement.It is worth noting that the canonical commutation relation clearly forbids the existence ofa time operator T for finite-level quantum sytems, while the time function T introduced herealways exists for such systems. This makes T particularly relevant in the context of quantuminformation theory, where finite-level quantum systems are extensively used. For instance, aninteresting perspective would be to analyze the geometrical properties of the sets of simultaneousstates in the case of composite systems, in order to understand if the time function of one ofthe component systems can be used as a sort of internal clock for monitoring the dynamicalevolution of the others in analogy with [21]. We will consider this situation in the future.Using the results in [22] and [23], the geometrical definition of the time function T couldbe extended to the case in which the Hilbert space of the system is infinite-dimensional. Thissubject is under current investigation.Finally, the definition of T makes no reference to the infinitesimal generator of the dynamicalevolution, hence, a generalization to the case of dissipative dynamics is possible. Funding: G. M. and J.M. P.P. would like to acknowledge the partial support by the SpanishMINECO grant MTM2014-54692-P and QUITEMAD+, S2013/ICE-2801.
References [1] W. Pauli.
General Principles of Quantum Mechanics . Springer-Verlag Berlin Heidelberg,1980.[2] E. A. Galapon. Pauli’s theorem and quantum canonical pairs.
Proc. R. Soc. Lond. A ,458:451–472, 2002. 123] M. D. Srinivas and R. Vijayalakshmi. The time of occurrence in quantum mechanics.
Pramana , 16(3):173–199, 1981.[4] Y. Aharonov and D. Bohm. Time in the quantum theory and the uncertainty relation fortime and energy.
Phys. Rev. , 122(5):1649–1658, 1961.[5] R. Brunetti, K. Fredenhagen, and M. Hoge. Time in quantum physics: From an externalparameter to an intrinsic observable.
Foundations of Physics , 40(9):1368–1378, 2009.[6] P. Busch, M. Grabowski, and Lahti P.J. Time observables in quantum theory.
PhysicsLetters A , 191(5-6):357–361, 1994.[7] G. Muga, R. Sala Mayato, and I. Egusquiza, editors.
Time in Quantum Mechanics , volume734 of
Lecture notes in Physics . Springer-Verlag Berlin Heidelberg, 2008.[8] G. Muga, R. Sala Mayato, and I. Egusquiza, editors.
Time in Quantum Mechanics - Vol.2 ,volume 789 of
Lecture notes in Physics . Springer-Verlag Berlin Heidelberg, 2009.[9] R. Brunetti and K. Fredenhagen. Time of occurrence observable in quantum mechanics.
Phys. Rev. A , 66(4):044101, 2002.[10] A. Galindo. Phase and Number.
Letters in Mathematical Physics , 8:495 – 500, 1984.[11] J. C. Garrison and J. Wong. Canonically Conjugate Pairs, Uncertainty Relations, andPhase Operators.
Journal of Mathematical Physics , 11(8):2242 – 2249, 1970.[12] P. A. M. Dirac. The elimination of the nodes in quantum mechanics.
Proceedings of the RoyalSociety of London A: Mathematical, Physical and Engineering Sciences , 111(757):281–305,1926.[13] R. de Ritis, G. Marmo, and B. Preziosi. A New Look at Relativity Transformations.
GeneralRelativity and Gravitation , 31(10):1501–1517, 1999.[14] G. Marmo and B. Preziosi. The Structure of space-time: Relativity groups.
Int.J.Geom.Meth.Mod.Phys. , 3:591–603, 2006.[15] G. Esposito, G. Marmo, and E.C.G. Sudarshan.
From Classical to Quantum Mechanics .Cambridge University Press, 2004.[16] G. Bhamathi and E.C.G. Sudarshan. Time as a dynamical variable.
Physics Letters A ,317:359–364, 2003.[17] A. Ashtekar and T. A. Schilling.
On Einstein’s Path: Essays in Honor of Engelbert Schuck-ing , chapter Geometrical Formulation of Quantum Mechanics, pages 23–65. Springer NewYork, NY, 1999.[18] E. Ercolessi, G. Marmo, and G. Morandi. From the equations of motion to the canonicalcommutation relations.
La rivista del Nuovo Cimento della Societ`a Italiana di Fisica , 33(8-9), 2010.[19] J. Grabowski, M. Kus, and G. Marmo. Geometry of quantum systems: density states andentanglement.
J.Phys. A: Math. Gen. , 38(47), 2005.1320] J. F. Carinena, A. Ibort, G. Marmo, and G. Morandi.
Geometry from dynamics, classicaland quantum . Springer Netherlands, 2015.[21] D. N. Page and W. K. Wootters. Evolution without evolution: Dynamics described bystationary observables.
Phys. Rev. D , 27(12):2885–2892, 1983.[22] R. Cirelli, P. Lanzavecchia, and A. Mania. Normal pure states of the von Neumann algebraof bounded operators as Kaehler manifold.
J. Phys. A: Math. Gen. , (16):3829–3835, 1983.[23] R. Cirelli, A. Mania, and L. Pizzocchero. Quantum mechanics as an infinite dimen-sional Hamiltonian system with uncertainty structure.