Effective Constraints for Relativistic Quantum Systems
aa r X i v : . [ m a t h - ph ] D ec IGC–09/6–2
Effective Constraints for Relativistic Quantum Systems
Martin Bojowald ∗ and Artur Tsobanjan † Institute for Gravitation and the Cosmos, The Pennsylvania State University,104 Davey Lab, University Park, PA 16802, USA
Abstract
Determining the physical Hilbert space is often considered the most difficult butcrucial part of completing the quantization of a constrained system. In such a situ-ation it can be more economical to use effective constraint methods, which are ex-tended here to relativistic systems as they arise for instance in quantum cosmology.By side-stepping explicit constructions of states, such tools allow one to arrive muchmore feasibly at results for physical observables at least in semiclassical regimes. Sev-eral questions discussed recently regarding effective equations and state properties inquantum cosmology, including the spreading of states and quantum back-reaction,are addressed by the examples studied here.
One of the key issues in quantizations of fundamental theories, which due to their covarianceproperties are systems with gauge freedom generated by constraints, is the determinationof physical observables. They must satisfy the constraint equations and be invariant undergauge transformations. For canonical quantum theories, solving constraints is traditionallydone at the state level: one constructs a physical Hilbert space of states annihilated by theconstraint operator(s) and equipped with an invariant inner product. Explicit constructionscan be done in some special cases by different methods.Since explicit derivations are possible only in specific cases, it is not always clear whetherthe results are generic or mere artefacts of the simple models used. It is therefore impor-tant to have approximate methods for a wider range of cases, or at least to be able toperturb around known solvable ones while still ensuring that the constraints are solvedand the observables are gauge invariant. It turns out that such perturbation schemes aremost feasible if one deals with the observables directly, such as expectation values, side-stepping the computation and physical normalization of states. This procedure gives riseto canonical effective equations and constraints [1, 2].A procedure for effective constraints has been formulated in [2] and applied to param-eterized non-relativistic systems with a constraint p t + H = 0 where p t is the momentum ∗ e-mail address: [email protected] † e-mail address: [email protected]
1f time and H the Hamiltonian. (The concepts and results are reviewed briefly below.) Itwas shown that the physical observables in suitable regimes, including semiclassical ones,can be determined without making use of a physical inner product but instead throughimplementing reality conditions for quantum variables such as fluctuations, correlationsand higher moments. For applications of these methods to quantum gravity and cos-mology one has to extend them to relativistic systems, offering one additional subtlety:constraints would now be of the form p t − H = 0, requiring one to take a square rootand to make sign choices. Mathematically, for instance, the question is how to preciselydefine √ H = | H | at the operator level. This may not be obvious if the operator H has acomplicated spectrum or is not positive definite. Physically, one must decide how to treatand separate positive and negative frequency solutions corresponding to the two solutions p t = ±| H | . (See e.g. [3, 4] for discussions of relativistic systems.)For a time-independent Hamiltonian, it turns out that one can, at least for semiclassicalpurposes, simply use p t = ±h H i as the effective Hamiltonian [5, 6] without absolute values,even if H is not positive definite. One only has to ensure that the initial values used inthe effective equations of motion correspond to an initial state supported on a part of thespectrum of H with a definite sign. On such a state and with a self-adjoint H , h| H |i = h H i if the state is supported only on the positive part of the spectrum of H , and h| H |i = −h H i if the state is supported only on the negative part. Since the Hamiltonian is preserved,these statements will hold true throughout the whole evolution and there is no need foran absolute value in the effective Hamiltonian. This fact has been made use of in severalrecent derivations of effective equations in quantum cosmology, where the relevant versionsof H are not positive definite [7, 8].In those models, deparameterization was performed using t = φ as an internal timefrom a free, massless scalar φ . The same types of models also allow the construction of aphysical Hilbert space at the state level [9, 10], with the results in agreement with thoseobtained from effective equations. Most interesting from the cosmological perspective is,however, a system where the scalar φ has a non-trivial potential or at least a mass term.This has two immediate implications: in general, one can no longer deparameterize globallysince the solutions for the scalar would not be monotonic in the time coordinates, and theHamiltonian would no longer be (internal) time independent. Positivity can no longer beensured just by an initial condition, and using p φ = ±h H ( φ ) i as an effective Hamiltonianwithout an absolute value may then seem questionable. Explicitly using an absolute value,on the other hand, would make a derivation of the effective equations more complicated.At this stage, a direct treatment of effective constraints for relativistic systems withoutdeparameterization becomes relevant. This is what we present in the current paper.We will consider in detail models of relativistic particles and properties of observablesin their quantum theories. In the massive case, for instance, we are dealing with thequantization and implementation of a constraint C = p t − p − m . For physical states,the expectation value h C i = p t + (∆ p t ) − p − (∆ p ) − m must vanish and thus represents a quantum constraint. (We will notationally identify2lassical degrees of freedom with the expectation values to simplify the notation and toshow the relation between classical terms and quantum corrections. Thus, E = h E i and(∆ E ) = h ( E −h E i ) i = h E i− E .) As we will see in more detail below, there are additionalindependent constraints since expressions such as h qC i must also vanish for physical states.Allowing all possible factors to the left of C , this presents a constrained system of infinitelymany constraints for infinitely many quantum variables given by the moments of a state.The combined system of all constraints must be solved to find observable results, which isfeasible in semiclassical regimes where only a finite set of moments suffices to characterize astate approximately. The same kind of approximation also allows one to include potentialswithin the constraint, which may be explicitly time-dependent. We will exploit this tojustify the procedures used in quantum cosmology for deparameterized systems with time-dependent potentials, as developed in [11, 12, 13].Another question of interest is that of the spreading of states and quantum back-reaction of fluctuations and higher moments on the expectation values. If we compare theeffective constraint h C i written above, which contains only the second order moments inaddition to the expectation values, with the effective Hamiltonian of the correspondingdeparameterized system, h H i = h p p + m i = h p ( p + ( p − p )) + m = p p + m + ∞ X n =2 n ! ∂ n p p + m ∂p n h ( p − p ) n i = p p + m + m (∆ p ) ( p + m ) / − m m − p ( p + m ) / h ( p − p ) i + · · · (1)with a whole formal Taylor series that includes higher moments, different coupling termsbetween the expectation values and the moments seem to be implied. Then, back-reactionmight seem different in these two treatments, apparently making them incompatible. Byour specific constructions in this paper we will reconcile these apparent disagreements,and provide an illustration by numerical solutions in a specific example (App. B). Thisis also important for quantum cosmology, where quantum back-reaction is crucial to theunderstanding of how a quantum state evolves toward and possibly through the big bangand how much of the pre-big bang state can be reconstructed [14, 15]. In a canonical effective description, the dynamics of a quantum system with n degrees offreedom is formulated in terms of the expectation values, i.e. the evaluations of a statefunctional in the elements of an algebra A generated by 2 n basic operators q i and p i , i = 1 , . . . , n . This whole set of infinitely many variables can be conveniently split into the2 n expectation values of the basic operators, such as q i = h q i i and p i = h p i i , i = 1 , . . . , n ,together with the infinitely many moments of the form G { a j } , { b j } = * n Y i =1 ( q i − h q i i ) a i ( p i − h p i i ) b i + Weyl {h A i , h B i} = h [ A , B ] i i ~ (2)extended using linearity and the Leibniz rule.At second order, P ni =1 ( a i + b i ) = 2, this set of moments includes all fluctuations andcovariances. In semiclassical regimes, moments fall off at least as ~ P ni =1 ( a i + b i ) such thatonly low orders need to be considered for the first approximation to quantum effects.Below we will employ the notation (∆ a ) = h ( a − a ) i for fluctuations and ∆( ab ) = h ( a − a )( b − b ) i Weyl = h ( a − a )( b − b ) + ( b − b )( a − a ) i for covariances, which may makesecond order equations easier to interpret.Any operator C = C ( q i , p j ) gives rise to a function on the space of states, which canbe expressed in terms of the moments by Taylor expanding h C ( q i , p j ) i = h C ( q i + ( q i − q i ) , p j + ( p j − p j )) i (3)in q i − q i and p j − p j as in (1). If C ( q i , p j ) is a constraint operator, (3) must vanish onphysical states and thus is a constraint on the quantum phase space.A single constraint on the phase space removes one pair of variables, but not the wholetower of moments associated with it in the quantum phase space. For a complete reductionone has to make use of additional constraints, provided by the set of phase space functions h f ( q i , p i ) C ( q j , p k ) i which must also vanish on physical states. These functions are ingeneral independent from the quantum constraint h C ( q i , p j ) i as functions of expectationvalues and moments. As shown in [2], this set of constraints remains first class. (Note thatwe do not order the products of the operators in the constraints symmetrically to ensurethat the constraint operator acts directly on the state. As a result, in some cases one has todeal with complex-valued constraints requiring reality conditions for physical observables.This has been discussed in [2] and will also be seen in more detail in the examples below.)Given such a system of constraints on the quantum phase space, one can proceed inthe classical way and find the reduced quantum phase space of observables or solve theconstraints and fix the gauge. At this stage, it is convenient (but not required) to decidewhich internal time variables ( t, p t ) among the ( q i , p j ) should be used. Since a quantizationof the corresponding deparameterized system, if it exists, would not give rise to any momentinvolving an operator of t or p t , solving the constraints must remove all moments includingat least one factor of t or p t from the original quantum phase space. That this indeedhappens was verified to second order of the parameterized non-relativistic particle in [2].In that case, there was a single linear term p t in the constraint, such that all p t -moments(∆ p t ) , ∆( p t t ), ∆( p t q ) and ∆( p t p ) can be removed by solving the system of constraints, to4econd order: h ( q − q ) C i = ∆( qp t ) + pM i ~ pM ∆( qp ) = 0 h ( t − t ) C i = pM ∆( pt ) + ∆( tp t ) + i ~ h ( p t − p t ) C i = (∆ p t ) + pM ∆( pp t ) = 0 h ( p − p ) C i = ∆( pp t ) + pM (∆ p ) = 0 . This leaves the moments (∆ t ) , ∆( tq ) and (∆ tp ), which are removed by factoring outthe gauge flow, or simply by setting them to zero as a well-defined set of gauge-fixingconditions. (Note that a smaller number of gauge fixing conditions than constraints isrequired because the second order moments satisfy a Poisson algebra which is degeneratefrom the symplectic point of view; see [16] for some notions of constrained systems in thenon-symplectic case. Additionally, setting the fluctuation (∆ t ) to zero is consistent withthe generalized uncertainty relation(∆ t ) (∆ p t ) − ∆( tp t ) ≥ ~ t − p t -correlations ∆( tp t ) = − i ~ / h ( t − t ) C i = 0 with the gauge fixing condition ∆( pt ) = 0, to be imaginary and of just theright size to saturate the uncertainty relation.) After solving the constraints and fixingthe gauge, observable moments are recovered on which physical reality conditions can beimposed. Classically a free relativistic particle is described by a single constraint C = p t − p − m on the phase space coordinatized by two canonical pairs t, p t and q, p . For the quantumversion we consider the unital associative algebra generated by four basic elements t , p t , q , p subject to the canonical commutator relations. That is, A consists of (countable) sums ofpolynomials of the form t i p jt q k p l ; terms with a different ordering may be expressed usingthe commutation relations [ t , p t ] = i ~ , [ q , p ] = i ~ . There is no product-ordering ambiguity in the case of the above constraint and it is nat-urally identified with an element C = p t − p − m of A . As the equivalent of Dirac’s We assume the units have been chosen so that both length and momentum have the units of the squareroot of action (e.g. geometrized units). C ψ = 0 we demand that the constraint has a vanishing right action on thestates, which in our case are complex linear functions α : A → C , this implies that α ( aC ) = 0 , ∀ a ∈ A ; henceforth we drop explicit reference to α and write this conditionas h aC i = 0 for the expectation value in a physical state α . In order to impose all theseconditions systematically we take the previously mentioned basis in A and impose theconstraint via an infinite (but countable) set of conditions h t i p jt q k p l C i = 0 . (4)We reduce the above infinite system of equations using the same method that waspreviously employed for a Newtonian particle [2]—a semiclassical expansion based on thehierarchy (cid:10) ( t − h t i ) i ( p t − h p t i ) j ( q − h q i ) k ( p − h p i ) l (cid:11) Weyl ∝ ~ ( i + j + k + l ) of moments. In what follows, we assume a semiclassical state and drop the terms of order ~ and above,keeping the terms of order ~ and below. This will suffice to demonstrate the feasibilityof our methods for relativistic systems. To this order our system is described by fourteenindependent functions: four expectation values of the form a = h a i ; four spreads of theform (∆ a ) = h ( a − a ) i and six covariances of the form ∆( ab ) = h ( a − a )( b − b ) i Weyl .(Poisson brackets between all these variables are listed in App. A.) The infinite system ofconstraint functions is reduced to just five non-trivial conditions C = h C i = p t − p − m + (∆ p t ) − (∆ p ) = 0 C t = h ( t − h t i ) C i = 2 p t ∆( tp t ) + i ~ p t − p ∆( tp ) = 0 C p t = h ( p t − h p t i ) C i = 2 p t (∆ p t ) − p ∆( p t p ) = 0 C q = h ( q − h q i ) C i = 2 p t ∆( p t q ) − p ∆( qp ) − i ~ p = 0 C p = h ( p − h p i ) C i = 2 p t ∆( p t p ) − p (∆ p ) = 0 . (5)Note that the semiclassical hierarchy of variables is critical to the above reduction inthe number of constraint conditions. In particular, C = 0 implies that p t − p − m =(∆ p ) − (∆ p t ) , which in turn implies that the combination of the expectation values C Class := p t − p − m is of order ~ on the constraint surface. In other words, the classicalconstraint is satisfied to order ~ . The terms of the form (∆ a ) C Class and ∆( ab ) C Class arethen of order ~ and should be dropped in our present treatment. The complete infinitesystem of constraint functions is a closed Poisson algebra—or, in the language of classicalconstraint analysis, a first-class system [17]. In general, due to the nature of the abovetruncation one would expect the reduced system of constraints to remain closed only toorder ~ . In our case the Poisson algebra of the truncated set of constraint functions—displayed in Table 1—is exactly closed with respect to the bracket.6able 1: Poisson algebra of constraints for a free particle. First terms in the bracket arelabeled by rows, second terms are labeled by columns. C C t C pt C q C p C − C pt − C p C t C pt p t C pt − pC p pC t +2 p t C q p t C p C pt pC p − p t C pt pC pt C q pC p − pC t − p t C q − pC pt p t C pt − pC p C p − p t C p pC p − p t C pt To solve the constraint system we eliminate five variables using the five conditionsfrom (5). Specifically, we eliminate the five quantum variables associated with p t , havingin mind that t will be chosen as time in a deparameterized treatment. We start by notingthat p t C p t = 0 gives 0 = p t (∆ p t ) − pp t ∆( p t p ) . (6)However C p = 0 implies p t ∆( p t p ) = p (∆ p ) and substituted in (6) gives 0 = p t (∆ p t ) − p (∆ p ) . Finally, eliminating p t through C = 0 we obtain a quadratic equation in (∆ p t ) (cid:0) (∆ p t ) (cid:1) − (∆ p t ) ( p + m + (∆ p ) ) + p (∆ p ) = 0with two solutions(∆ p t ) = 12 (cid:16) p + m + (∆ p ) ± p ( p + m + (∆ p ) ) − p (∆ p ) (cid:17) . In order to see whether either solution is compatible with the hierarchy assumed by thesemiclassical approximation, we expand the solution to order ~ . One finds(∆ p t ) = 12 ( p + m ) p ) p + m ± s p ) (2 m − p + (∆ p ) )( p + m ) ! = 12 ( p + m ) (cid:18) ± p ) ( p + m ) ( p + m ± ( m − p )) + O (cid:0) (∆ p ) (cid:1)(cid:19) . Looking at the solution with the “+” sign we see the following leading order behavior(∆ p t ) = p + m + O ( ~ )which is inconsistent with the assumption that (∆ p t ) is of order ~ . The solution with the“ − ” sign leads to (∆ p t ) = p (∆ p ) p + m + O (cid:0) (∆ p ) (cid:1) ~ and therefore consistent with the semiclassical approximation. Sub-stituting the latter solution back into the constraint conditions (5) we obtain two sets ofsolutions p t = ± E ∆( tp t ) = ± pE ∆( tp ) − i ~ p t ) = p + m + (∆ p ) − E ∆( p t q ) = ± pE (cid:18) ∆( qp ) + i ~ (cid:19) ∆( p t p ) = ± pE (∆ p ) (7)where E = 1 √ (cid:16) p + m + (∆ p ) + p ( p + m + (∆ p ) ) − p (∆ p ) (cid:17) . By rearranging the terms in the above expression one can easily verify that E ≥ p p + m if one assumes a“momentum eigenstate”, that is if one sets the spread in momentum (∆ p ) = 0. The truncated system of constraints (5) is equivalent (when consistency with the semi-classical approximation is evoked) to the restriction to two disjoint surfaces, each onecorresponding to a choice of sign in (7). Each surface is described by an equivalent set of“linearized” constraints C ± = p t ± EC ± = (∆ p t ) − p − m − (∆ p ) + E C ± = ∆( p t p ) ± pE (∆ p ) C ± = ∆( p t q ) ± pE (cid:18) ∆( qp ) + i ~ (cid:19) C ± = ∆( tp t ) ± pE ∆( tp ) + i ~ . (8)The above constraint conditions can be expressed as sums of the conditions in (5) andtherefore form a first class system. Additionally, for the calculations to follow it is usefulto note that p t , p , (∆ p t ) , ∆( p t p ) and (∆ p ) are first class functions with respect to eitherset of constraints. This can ultimately be traced back to the fact that [ p t , C ] = 0 = [ p , C ].It follows that E , which is a function of p and (∆ p ) only, is also first class.On the constraint surfaces the “linearized” constraints can be used to eliminate thefive variables p t , (∆ p t ) , ∆( tp t ), ∆( p t p ), ∆( p t q ). At this stage there remain four degrees8f freedom associated with the algebra elements generated by t . These will be treatedas gauge parameters associated with the time evolution of the system. From this pointof view, one has a four-parameter space to choose from when it comes to the evolutionof the “physical variables” (i.e. those associated with the algebra generated by q and p alone). Viewing our system expanded to second order in quantum variables as a classicalconstrained system, the evolution on the “physical variables” — q , p , (∆ q ) , ∆( qp ), (∆ p ) — may be generated by any constraint function of the form C Ham = X i µ i C i ± (9)where the multipliers µ i are arbitrary functions of the physical variables. The presence ofseveral constraints, all associated with the classical Hamiltonian, means that a priori thereis no unique time parameter. Depending on the choice of gauge, any combination of t withmoments involving t can play the role of time.At this stage we would like to restrict the gauge freedom down to a single parameterand to interpret the first class flow in the direction of t as the dynamical evolution of thesystem. This may be accomplished by introducing three gauge choices φ = (∆ t ) − f ( q, p, (∆ q ) , ∆( qp ) , (∆ p ) ) = 0 φ = ∆( tq ) − f ( q, p, (∆ q ) , ∆( qp ) , (∆ p ) ) = 0 φ = ∆( tp ) − f ( q, p, (∆ q ) , ∆( qp ) , (∆ p ) ) = 0with functions f , f and f to be determined. We define C Ham as the first class function(9) that remains after the gauge conditions have been introduced. It must therefore be acombination of the original constraint functions as in equation (9) that in addition satisfies { C Ham , φ i } ≈ , i = 1 , , ≈ ’ denotes equality on the surface defined by imposing both the con-straints and the gauge conditions. A simple set of such conditions that was also used inRef. [2] to recover the deparameterized dynamics of a Newtonian particle is provided by φ = (∆ t ) = 0 φ = ∆( tq ) = 0 φ = ∆( tp ) = 0 . (10)(Again, (∆ t ) = 0 is consistent with the uncertainty relation since ∆( tp t ) = − i ~ / C ± .)Let Σ ± be the surfaces defined by simultaneously imposing the constraints { C i ± } andthe above gauge conditions. These are coordinatized by the physical variables— q , p , (∆ q ) ,∆( qp ), (∆ p ) —and the one remaining gauge degree of freedom— t . It is straightforward toverify that the variables (∆ t ) , ∆( tq ) and ∆( tp ) generate a Poisson ideal of the algebra ofphysical and gauge variables (i.e. the variables that do not involve p t ). That is, { φ i , X }
9s a sum of gauge conditions with some coefficients. It follows that on Σ ± the gauge-fixingconditions have a trivial Poisson algebra { φ i , φ j } ≈ C ± remains first class on Σ ± .Since E is a function of the “physical” variables only, { E, φ i } ≈ { φ i , C ± } = { φ i , p t ± E } ≈ . Furthermore, writing C ± = ∆( tp t ) ± φ p/E + const . one can quickly establish that C ± also remains first class but has a vanishing Poisson flow on Σ ± .The remaining set of surface-defining conditions composed of C ± , C ± , C ± and { φ i } i =1 , , is second class for all admissible values of the physical variables. This canbe seen by relabeling the conditions as χ = C ± , χ = C ± , χ = C ± , χ = φ , χ = φ , χ = φ and looking at the Poisson bracket matrix ∆ ij := { χ i , χ j } . On Σ ± the componentsof the matrix are ∆ ≈ i ~ ± pE ( i ~ + 2∆( qp )) ± pE (∆ p ) i ~ − ∆( qp ) − (∆ p ) − (∆ q ) − i ~ − ∆( qp ) − i ~ ∓ pE ( i ~ + 2∆( qp )) ∆( qp ) − i ~ (∆ q ) ∓ pE (∆ p ) (∆ p ) i ~ + ∆( qp ) 0 0 0 . Calculating the determinant one obtains the same result for both choices of the signdet[ ∆ ] ≈ − ~ (cid:20) ~
16 + (∆( qp )) + 2 (cid:18) (∆ p ) (∆ q ) − ~ (cid:19) (cid:0) (∆ p ) (∆ q ) − (∆( qp )) (cid:1)(cid:21) . The determinant is non-zero in the region where reality, positivity and uncertainty condi-tions are imposed on the state—that is, if one demands q, p, (∆ q ) , ∆( qp ) , (∆ p ) ∈ R (∆ p ) , (∆ q ) ≥ p ) (∆ q ) − (∆( qp )) ≥ ~ . With these conditions in place, the sum of the terms inside the square bracket in theexpression for the determinant is strictly positive, which means that the determinant itselfis strictly negative.There is one important check that one needs to perform. The introduction of φ i = 0, i = 1 , , ± a mixture of first and second class and one is required toadjust the Poisson structure of the functions parameterizing the surfaces through the useof the Dirac bracket. Before we can identify q , p , (∆ q ) , ∆( qp ), (∆ p ) as the expectationvalues and moments of a physical canonical pair of operators, we need to verify thattheir Dirac brackets on Σ ± are identical to the Poisson brackets one would obtain for the10uantum variables associated with a single canonical pair. The bracket may be computedas follows { f, g } Dirac := { f, g } − { f, χ i } (∆ − ) ij { χ j , g } . (11)Using the fact that ∆ ij (and hence also (∆ − ) ij ) is off-block-diagonal and that the physicalvariables have vanishing Poisson brackets with the gauge fixing conditions, one can easilyverify that the second term in the above definition vanishes for the brackets between q , p ,(∆ q ) , ∆( qp ), (∆ p ) (as well as t ), and therefore their Poisson structure is unchanged asrequired by our interpretation. These variables are the remaining physical quantities upto second order on the reduced quantum phase space.To summarize, we impose the gauge-fixing conditions φ i = 0, i = 1 , ,
3, interpret q , p , (∆ q ) , ∆( qp ), (∆ p ) as the physical expectation values and moments and as a resultdemand reality, positivity and quantum uncertainty. With all of these conditions takentogether, φ i = 0, i = 1 , , C ± (recall that C ± generates no flow on Σ ± ). This means that the time evolution is given by C Ham = µ C ± . Finally, we fix the remaining Lagrange multiplier µ by demanding t —the last remaininggauge variable—to be the time parameter. That is, we demand that { t, C Ham } ≈
1, whichleaves us with C Ham = C ± = p t ± E .
Taking the non-relativistic limit of E we recover the results for a deparameterized freeNewtonian particle. Specifically, if we formally take p /m to be of order δ , it follows thatin a semiclassical state (∆ p ) /m is of order ~ δ . We expand the expression for E to theleading order in δ : E = m √ (cid:18) p + (∆ p ) m (cid:19) s − p (∆ p ) ( p + m + (∆ p ) ) ! = m √ (cid:18) p + (∆ p ) m + O ( δ ) (cid:19) (cid:0) O ( ~ δ ) (cid:1) = m + p + (∆ p ) m + O ( δ ) . The standard positive frequency solutions to the Klein-Gordon equation (see for exam-ple [18]) form a Hilbert space of momentum-space wave-functions square integrable withrespect to the Lorentz-invariant measure: H = L (cid:18) R , d k ǫ k (cid:19) , where ǫ k = √ k + m . H as p = k and q = i ~ (cid:18) ∂∂k + ǫ k (cid:18) ∂∂k ǫ k (cid:19)(cid:19) . The time evolution is generated by the Hamiltonian H = ( p + m ) . One can evaluatethe evolution equations for the expectation values of the observables using Ehrenfest’stheorem dd t h O i = 1 i ~ h [ O , H ] i + ∂ h O i ∂t . In our formalism, the right-hand side is equivalent to the quantum Poisson bracket betweenthe expectation values, thusdd t h O i = {h O i , h H i} + ∂ h O i ∂t = n h O i , h (cid:0) p + m (cid:1) i o + ∂ h O i ∂t . We recall that our procedure at order ~ together with the gauge fixing conditions for thepositive frequency solutions resulted indd t h O i = {h O i , p t + E } = {h O i , E } + ∂ h O i ∂t . In order to see whether the methods agree, we only need to compare h ( p + m ) i and E to order ~ . To verify this explicitly we expand the operator in terms of its moments,assuming the expectation value to be taken in a semiclassical state: D ( p + m ) E = * ( p + m ) + p ( p + m ) ( p − p ) + m p + m ) ( p − p ) + + (higher moments)= p p + m (cid:18) m (∆ p ) p + m ) (cid:19) + O ( ~ ) . For comparison, we expand E in powers of (∆ p ) , which we assume to be of order ~ . E = 1 √ p p + m (cid:18) p ) p + m + 1 + ( m − p )(∆ p ) ( p + m ) + O (cid:0) (∆ p ) (cid:1)(cid:19) = p p + m s m (∆ p ) ( p + m ) + O ((∆ p ) )= p p + m (cid:18) m (∆ p ) p + m ) (cid:19) + O (cid:0) (∆ p ) (cid:1) . Thus, up to the terms of order ~ the two results agree.Unlike the exact Klein-Gordon solution, our approach avoids explicit reference to arepresentation. The action of the Lorentz group on our variables can be understood through12ts action on the algebra of observables. In particular, we assume that the pairs ( p t , p )and ( t , q ) transform as components of a contravariant and a covariant vector respectively.Looking at the truncated system of constraints (5) under a Lorentz transformation onefinds that C remains invariant, while the pairs ( C p t , C p ) and ( C t , C q ) themselves transformas components of a contravariant and a covariant vector respectively, so that the wholetruncated system of constraints is preserved. For a massless particle, the constraint operator takes the form C = p t − p . To second order in moments, the constraint functions produced remain as in equation (5),except for C , which now reads C = h C i = p t − p + (∆ p t ) − (∆ p ) = 0 . The disappearance of a constant term from C does not affect the Poisson algebra of theconstraints, so that the table of Section 3.1 still applies. The solution to the constraints,however takes on a simpler form: following the same steps as previously and eliminating(∆ p t ) in a way compatible with the semiclassical approximation we obtain(∆ p t ) = (∆ p ) . Together with C = 0 this implies p t = p . As we see, the classical constraint is satisfied exactly by the expectation values. We solvethis via p t = ±| p | . There are two related reasons for taking the absolute value of p in the above solution.Firstly, to emphasize the sign of the energy. Secondly, to match the limit as m is set tozero of the solution obtained in Section 3.1. The full solutions read p t = ±| p | ∆( tp t ) = ± p | p | ∆( tp ) − i ~ p t ) = (∆ p ) ∆( p t q ) = ± p | p | (cid:18) ∆( qp ) + i ~ (cid:19) ∆( p t p ) = ± p | p | (∆ p ) . (12)13he steps of Section 3.2 can be repeated exactly for the m = 0 case with | p | playing therole of E . With the gauges fixed in an identical way, this results in evolution on q , p ,(∆ q ) , ∆( qp ), (∆ p ) generated by the constraint C Ham = p t ± | p | . (13)The implications will be discussed further in the conclusions. In this section we consider the consequences of adding a potential term to the quantumconstraint. We consider a quadratic time-independent potential in Section 4.1 followedby a homogeneous time-dependent potential in Section 4.2. The systems considered inthis section have the same kinematical degrees of freedom as the free relativistic particle;however, the additional terms in the constraint element break Lorentz invariance. On theother hand, certain structural properties of the constraint element remain very similar tothe free particle case, which makes extension of the calculations performed in Section 3fairly simple. As we will see, the constraints are still straightforward to solve, but theirPoisson algebra is only approximately closed and in the case of the time-dependent po-tential, the gauge analysis requires more subtlety. These examples show that the effectiveprocedure used here is feasibly applicable to a wider range of models than the existingexplicit constructions of a physical inner product.
A relativistic particle in a quadratic potential is subject to the constraint C = p t − p − q − m . (A coupling constant in the potential could be absorbed by rescaling.) This gives rise tothe following set of constraint functions truncated at second order C = p t − p − q − m + (∆ p t ) − (∆ p ) − (∆ q ) = 0 C t = 2 p t ∆( tp t ) + i ~ p t − p ∆( tp ) − q ∆( tq ) = 0 C p t = 2 p t (∆ p t ) − p ∆( p t p ) − q ∆( p t q ) = 0 C q = 2 p t ∆( p t q ) − p ∆( qp ) − i ~ p − q (∆ q ) = 0 C p = 2 p t ∆( p t p ) − p (∆ p ) − q ∆( qp ) + i ~ q = 0 . (14)The above system of constraints is first class only to order ~ as can be seen from theirPoisson algebra in Table 2.The system of constraints may be solved following the same steps that have beenemployed to solve the constraints for the free particle. We use C , C q , and C p to eliminate14able 2: Poisson algebra of constraints for the particle in a quadratic potential. First termsin the bracket are labeled by rows, second terms are labeled by columns. C C t C pt C q C p C − C pt C p − C q p t C pt − pC p − qC q p t C q +2 pC t p t C p +2 qC t C t C pt p t p )∆( tq ) +4∆( tq ) ( ∆( qp )+ i ~ ) +4∆( tq )(∆ p ) − p t q )∆( tp ) +4(∆ q ) ∆( tp ) − tp ) ( ∆( qp ) − i ~ ) − p t C pt +2 pC p +2 qC q pC pt − qC pt C pt − p t p )∆( tq ) 0 − q ) ∆( p t p ) +4(∆ p ) ∆( p t q )+4∆( p t q )∆( tp ) − ∆( p t q ) ( ∆( qp ) − i ~ ) +4∆( p t p ) ( ∆( qp ) − i ~ ) − p t C q − pC t − pC pt p t C pt − pC p − qC q C q − C p − tq ) ( ∆( qp )+ i ~ ) +4(∆ q ) ∆( p t p ) 0 +4 “ (∆ q ) (∆ p ) − ~ ” − tp )(∆ q ) +4∆( p t q ) ( ∆( qp ) − i ~ ) − qp )) − p t C p +2 qC t qC pt pC p +4 qC q − p t C pt C p C q − tq )(∆ p ) − p ) ∆( p t q ) − “ (∆ q ) (∆ p ) − ~ ” tp ) ( ∆( qp ) − i ~ ) +4∆( p t p ) ( ∆( qp ) − i ~ ) − qp )) p t , ∆( p t q ) and ∆( p t p ) respectively. Substituted into C p t this yields a quadratic equationin (∆ p t ) :0 = (cid:0) (∆ p t ) (cid:1) − (cid:0) p + q + m + (∆ p ) + (∆ q ) (cid:1) (∆ p t ) + (cid:0) p (∆ p ) + 2 qp ∆( qp ) + q (∆ q ) (cid:1) . The solution compatible with the semiclassical approximation has the form p t = ± E ∆( tp t ) = ± E ( p ∆( tp ) + q ∆( tq )) − i ~ p t ) = p + q + m + (∆ p ) + (∆ q ) − E ∆( p t q ) = ± E (cid:18) p ∆( qp ) + i ~ p + q (∆ q ) (cid:19) ∆( p t p ) = ± E (cid:18) p (∆ p ) + q ∆( qp ) − i ~ q (cid:19) (15)where E = 1 √ " p + q + m + (∆ p ) + (∆ q ) (16)+ (cid:18) ( p + q + m + (∆ p ) + (∆ q ) ) − (cid:0) p (∆ p ) + 2 qp ∆( qp ) + q (∆ q ) (cid:1)(cid:19) . We note that as C = p t − E and p t are both exactly first class, E must also be first class,since it does not vanish on the constraint surface.15f one applies the semiclassical approximation once again to drop the terms of ordershigher than ~ , the constraint system may be treated as first class. The linearized versionsof the constraints take the form C ± = p t ± EC ± = (∆ p t ) − p − q − m − (∆ p ) − (∆ q ) + E C ± = ∆( p t p ) ± E (cid:18) p (∆ p ) + q ∆( qp ) − q i ~ (cid:19) C ± = ∆( p t q ) ± E (cid:18) p ∆( qp ) + p i ~ q (∆ q ) (cid:19) C ± = ∆( tp t ) ± E ( p ∆( tp ) + q ∆( tq )) + i ~ . (17)One can follow the process outlined in the Section 3.2 and impose the set of gauge-fixingconditions (10). Once again C ± is first-class on the gauge-fixed surfaces Σ ± and writing C ± = ∆( tp t ) ± pE φ ± qE φ it is not difficult to see that, once again, C ± remains first-classbut has a vanishing flow on Σ ± . We recall that the gauge conditions have a vanishing flowon the remaining free variables; therefore only the first term in the expressions for eachof the constraints C i ± above has a non-vanishing Poisson bracket with the conditions φ i .As a result, the Poisson bracket matrix ∆ remains as in Section 3.2 up to entries of order ~ . Imposing reality, positivity and quantum uncertainty and demanding { t, C Ham } ≈ C Ham = p t ± E .
Directly expanding E in powers of (∆ q ) , ∆( qp ) and (∆ p ) we get the expression to order ~ E = p p + q + m " q + m )(∆ p ) − qp ∆( qp ) + ( p + m )(∆ q ) p + q + m ) + O (cid:0) (∆ p ) (cid:1) + O (cid:0) (∆ q ) (cid:1) + O (cid:0) (∆( qp )) (cid:1) . (18)The specific constraint considered in this section can be implemented quite completelyat the level of physical states, although specifics of the dynamics are more difficult toextract than with effective methods. Below we briefly describe the solution and compareit to our effective treatment. The algebra elements may be represented kinematically asdifferential operators on the space of square-integrable wave-functions in two variables x and x in the usual way t = x , p t = ~ i ∂∂x , q = x , p = ~ i ∂∂x . The constraint operator splits into a sum of commuting, and therefore simultaneouslydiagonalizable, components: 16 p t = − ~ ∂ ∂x has infinite-norm eigenstates of the form exp( ik ~ x ) φ ( x ), with eigenval-ues k . • p + q = − ~ ∂ ∂x + x = 2 H harm , where H harm is precisely the standard Hamiltonian ofthe harmonic oscillator on x (with mass and frequency set to unity). This operatorhas normalizable eigenstates of the form ψ ( x ) ϕ n ( x ), where ϕ n ( x ) is the usualnormalized n -th eigenstate of the harmonic oscillator, the corresponding eigenvaluesare 2( n + ) ~ . • Every wavefunction is an eigenstate of m with the eigenvalue m .The eigenfunctions of the constraint operator are Ψ k,n ( x , x ) = exp( ik ~ x ) ϕ n ( x ), thecorresponding eigenvalues are (cid:0) k − n + ) ~ − m (cid:1) . The space of solutions to the quan-tum constraint equation is therefore spanned by the wavefunctions Ψ k,n for which k = ± q n + ) ~ + m . These states are not normalizable with respect to the square integra-tion in both x and x , however they have unit norm when the integration is taken withrespect to x alone.As is usually done for such systems we decompose the solution space into two segmentsone belongs to the positive part, the other one to the negative part of the spectrum of p t (e.g. the separation of positive and negative frequencies of the solutions to the Klein-Gordon equation). The general element of the solution space is a linear combination ofeither positive or negative frequency null eigenfunctions of C , denoted by Ψ + and Ψ − respectively Ψ ± phys = ∞ X n =0 α n exp ∓ ix q n + ) ~ + m ~ ϕ n ( x ) . The separation into two components allows us to define a positive-definite physical innerproduct on each one of them individually h Ψ | Φ i phys := Z ∞−∞ Ψ( x , x )Φ( x , x )d x (19)where the bar denotes a complex-conjugate and both states Ψ, Φ belong to the samecomponent. On the space of solutions, the above inner product is independent of thevalue taken by x . Furthermore this inner product is consistent with interpreting t astime, since we can formally write h Ψ | t | Ψ i phys = x . (This equation requires some care inits interpretation since t is not a physical operator. But just viewing the integration onthe right hand side of (19) easily allows us to introduce an operator t by multiplication.The dependence on x of the result is then in agreement with the fact that t is not aphysical observable.) It is also consistent with the gauge choices of equation (10), which isstraightforward to verify using the fact that for any operator A , polynomial in t , p t , q , p h Ψ | tA | Ψ i phys = x h Ψ | A | Ψ i phys . − ~ i dd x Ψ ± = ± (cid:0) p + q + m (cid:1) Ψ ± . That is, an ordinary quantum mechanical system with time evolution in the variable x gen-erated by the self-adjoint, positive square-root Hamiltonian H = ( p + q + m ) , whichis defined through its action on the basis of eigenstates: H ϕ n ( x ) = q n + ) ~ + m ϕ n ( x ).To compare the physical states and the effective solutions, we proceed as we have donebefore, in Section 3.3. We expand the expectation value of the square-root hamiltonian ina semiclassical state h H i = *(cid:10) p + q + m (cid:11) + ( p + q + m ) − h p + q + m i h p + q + m i − (( p + q + m ) − h p + q + m i ) h p + q + m i + + (higher moments) . Proceeding with the above expansion and keeping only the terms up to order ~ one obtainsthe expression that is identical to the one for E in equation (18). Thus, to leading order inthe semiclassical regime, the effective solution to the constraint is consistent with physicalstate evolution, and the gauge choice of equation (10) is consistent with the physical innerproduct defined above together with the interpretation of h t i as measuring the physicaltime. For a direct comparison between fully quantum and effective time evolutions for aspecific semiclassical state of this system see App. B.If we replace ( p + q + m ) in the constraint by any positive operator f ( q , p ) analyticin q and p , physical states can in principle be found in a similar way. One could find thespectrum of f ( q , p ) and construct the solutions out of its simultaneous eigenfunctions with p t . Finding the spectrum of a given operator is in general a complicated task. Further, wewere helped in this example by the fact that the spectrum of ( p + q + m ) is discreteand the eigenfunctions are normalizable with respect to square integration over x alone.Defining the physical inner product is more difficult if parts of the spectrum of f ( q , p )are continuous. Finally, determining suitable coherent states for semiclassical purposes, asdone for this model in App. B, can be a difficult task. The leading order effective solution,on the other hand, can be obtained in much the same way as was done for the aboveexample, without explicit in-depth knowledge of the spectrum of q or the exact form of itseigenstates. As mentioned in the introduction, time-dependent potentials are of interest in quantumcosmology. Another example where time-dependent terms arise is a relativistic particlemoving on a non-static curved background space-time. In such cases, our methods can be18sed as well, but additional subtleties do arise. Adding a “potential” V ( t ) = λt to theclassical constraint gives the second order quantum constraints C = p t − p − m + (∆ p t ) − (∆ p ) + λt = 0 C t = 2 p t ∆( tp t ) + i ~ p t − p ∆( tp ) + λ (∆ t ) = 0 C p t = 2 p t (∆ p t ) − p ∆( p t p ) + λ ∆( p t t ) − iλ ~ = 0 C q = 2 p t ∆( p t q ) − p ∆( qp ) − i ~ p + λ ∆( qt ) = 0 C p = 2 p t ∆( p t p ) − p (∆ p ) + λ ∆( tp ) = 0 . (20)These constraints, once again, form a closed Poisson algebra only up to order ~ . Weproceed to solve the above set of polynomial equations explicitly— p t C p t = 0 implies p t (∆ p t ) − p t p ∆( p t p ) + 12 λp t ∆( tp t ) − i ~ λp t = 0 . Using pC p = 0 and λC t = 0 to eliminate p t p ∆( p t p ) and λp t ∆( tp t ) respectively we obtain p t (∆ p t ) − p (∆ p ) + λp ∆( tp ) − λ (∆ t ) − i ~ λp t = 0 . Finally, we eliminate (∆ p t ) using C = 0 to obtain a quartic equation in p t p t − (cid:0) p + m − λt + (∆ p ) (cid:1) p t + i ~ λp t + (cid:18) p (∆ p ) + 14 λ (∆ t ) − λp ∆( tp ) (cid:19) . The exact solutions to the above quartic equation are of course readily available, thoughthey involve long algebraic expressions and are not particularly illuminating. Furthermore,due to the linear term in p t in the equation, the gauge choices we have employed previouslytogether with the reality conditions would lead us to conclude that p t is complex. A moresubtle gauge analysis is required to solve the constraint without further approximations.For instance, to respect reality conditions one would use a gauge relating moments to theexpectation values, for example by making ∆( tp ) dependent on p t .However, assuming the potential changes very slowly allows one to move forward withthe standard gauge choice. Treating λ as a second small parameter in addition to ~ / ,such that λ ~ is of higher than second order and discarding the terms of order higher than ~ we are left with p t − (cid:0) p + m − λt + (∆ p ) (cid:1) p t + p (∆ p ) = 0 . This equation could also be obtained by directly dropping products of λ and second ordermoments or ~ in the expressions for the constraint functions (20) and solving them. It is aquadratic equation in p t , with solutions that are much easier to interpret. Compatibilitywith the semiclassical approximation once again selects for us a set of solutions that havea very similar form to those in (7), (12), and (15), with E = 1 √ " p + m − λt + (∆ p ) + (cid:0) ( p + m − λt + (∆ p ) ) − p (∆ p ) (cid:1) . C Ham = p t ± E , for a slowly varying potential in a semiclassical state. Thus, to thesemiclassical order considered, the system behaves as a non-relativistic quantum particlein one dimension subject to a time-dependent hamiltonian H = ( p + m − λt ) .In fact, a more general “slowly varying” potential may be treated to order ~ in ananalogous manner. We assume that the potential has the form V ( t ) = V (0) + λ ˜ V ( t ),where λ is small in the sense discussed earlier, and ˜ V ( t ) is a polynomial in t . This implies,in particular, that h λ ˜ V ( t ) i = λ ˜ V ( t ) + O ( ~ ). We absorb the constant part of the potentialinto the definition of m and the constraint functions to order ~ look exactly as they do fora free relativistic particle (5), except for C , which acquires an extra term C = p t − p − m + (∆ p t ) − (∆ p ) + λ ˜ V ( t ) = 0 . The resulting system of constraint functions may be solved, gauge-fixed and interpreteddirectly following the methods employed throughout Sections 3 and 4. This demonstratesthe flexibility of the constructions, confirming the methods of [11, 12, 13], where also slowlyvarying potentials were assumed. In contrast to this earlier work, the general methodspresented here are in principle applicable to arbitrary potentials, but the gauge fixingwould have to be considered in each case in detail. This provides access to questions aboutthe role of time when potentials forbid a global monotonic internal time t , resulting in anew perspective to be followed elsewhere. One of the main hurdles for quantum gravity is the physical Hilbert space issue. At leastfor semiclassical questions, technical and conceptual difficulties can be circumvented byusing expectation values and moments directly rather than states. Other advantages ofthis method are that the specification of semiclassical regimes is easier via moments (whilesemiclassical wave functions are often difficult to formulate, even simple-looking Gaussianones not always being semiclassical at all in some models of quantum cosmology as pointedout in [19]) and that density states are automatically included.We have extended the methods for effective constraints of [2] to relativistic systems,clarifying several physically relevant issues of effective equations: • Square root effective Hamiltonians, which so far were strictly justified only for time-independent potentials, are valid even in the time dependent case provided the po-tential varies slowly in time. No extra conditions on the dynamics are implied by thepositivity conditions. • Quantum back-reaction follows reliably from square-root Hamiltonians. Our exam-ples of relativistic systems have shown three different cases: – Massless particles do not contain moments in their reduced Hamiltonian (13)and thus are not subject to quantum back-reaction.20
Free massive particles do have quantum back-reaction from moments of all or-ders, which is initially unexpected since the quantum constraint has only a linearterm of (∆ p ) with no coupling to the expectation values. (Our formulas, donehere only to second order, do not show this explicitly.) – Particles subject to a q -dependent potential receive quantum back-reaction fromthe covariance of their wave function as seen in (16). This is also unexpectedsince the expectation value of the constraint does not contain mixed moments.As in the case of massive particles, the unexpected results are explained by thepresence of higher order constraints. • Higher order constraints, which are crucial for the effective procedure, also affect theamount of spreading of wave functions, or the time dependence of moments. For afree, massless particle wave packets do not spread, but they do in the other cases.Effective equations obtained in the way developed here reliably describe the physical be-havior of dynamical wave packets. With these considerations, effective methods as theyhave been used in quantum cosmology are established even in the case of φ -dependentpotentials, as studied specifically for instance in [13, 8]. A further application would beto reconsider the appearance of certain future singularities which have been shown notto be removed by the tree-level approximation (disregarding all moments) [20] but wherequantum back-reaction is expected to be strong.Especially in the presence of potentials, calculations performed here are much morefeasible than the methods involving constructions of physical Hilbert spaces followed bycomputations of the expectation values in explicit physical states. They can be expected tobe of far more general applicability, including full quantum gravity. For such an extension,several other issues remain open: describing field theories and handling situations of manyclassical degrees of freedom. But there is already a promise that effective techniques allowone to evade difficult obstacles from physical inner product issues which so far have impededprogress. Especially the semiclassical regime of canonical quantum gravity and potentiallyobservable effects can be brought under much higher control. Acknowledgements
We thank Don Marolf and Madhavan Varadarajan for discussions. This work was sup-ported in part by NSF grant PHY0748336.
A Second order Poisson algebra
The expectation values obey the classical Poisson algebra for two canonical pairs, wherethe non trivial brackets are { t, p t } = 1 and { q, p } = 1 . (∆ t ) ∆( tp t ) (∆ p t ) (∆ q ) ∆( qp ) (∆ p ) ∆( tq ) ∆( p t p ) ∆( tp ) ∆( p t q )(∆ t ) t ) tp t ) 0 0 0 0 2∆( tp ) 0 2∆( tq )∆( tp t ) − t ) p t ) − ∆( tq ) ∆( p t p ) − ∆( tp ) ∆( p t q )(∆ p t ) − tp t ) − p t ) − p t q ) 0 − p t p ) 0(∆ q ) q ) qp ) 0 2∆( p t q ) 2∆( tq ) 0∆( qp ) 0 0 0 − q ) p ) − ∆( tq ) ∆( p t p ) ∆( tp ) − ∆( p t q )(∆ p ) − qp ) − p ) − tp ) 0 0 − p t p )∆( tq ) 0 ∆( tq ) 2∆( p t q ) 0 ∆( tq ) 2∆( tp ) 0 ∆( tp t ) (∆ t ) (∆ q ) +∆( qp )∆( p t p ) − tp ) − ∆( p t p ) 0 − p t q ) − ∆( p t p ) 0 − ∆( tp t ) 0 − (∆ p ) − (∆ p t ) − ∆( qp )∆( tp ) 0 ∆( tp ) 2∆( p t p ) − tq ) − ∆( tp ) 0 − (∆ t ) (∆ p ) qp ) − ∆( tp t )∆( p t q ) − tq ) − ∆( p t q ) 0 0 ∆( p t q ) 2∆( p t p ) − (∆ q ) (∆ p t ) ∆( tp t ) 0 − ∆( qp ) The Poisson brackets between the expectation values and the moments are zero. In Ta-ble 3 we provide the Poisson brackets for the second order moments of quantum variablesassociated with two canonical pairs t , p t ; q , p . B Coherent state and effective evolution
Here we consider the constrained system of Section 4.1 in a specific semiclassical state andcompare the evolution of “classical” quantities given by the effective quantum theory toproperties of the state. This appendix is included to address the riliability of an effectivesolution to a constrained quantum system through comparison within a specific example.Recall the constraint: C = p t − p − q − m . Formally, a positive frequency solution of the constraint described in Section 4.1 reducesthe system to one canonical degree of freedom that evolves subject to the Hamiltonian H = ( p + q + m ) . We begin by providing the classical trajectory: a canonical pair q , p subject to the Hamiltonian function H = ( p + q + m ) evolves according to theequations of motion: dd t q = { q, H } = p (cid:0) p + q + m (cid:1) − = pH − dd t p = { p, H } = − q (cid:0) p + q + m (cid:1) − = qH − . H itself is a constant of motion we differentiate the first equation abovewith respect to time to obtain the second order equationd d t q = H − dd t p = − H − q . This equation has the general solution q ( t ) = A sin (cid:18) tH (cid:19) + B cos (cid:18) tH (cid:19) , where A and B are constants. It follows that p ( t ) = H dd t q = A cos (cid:18) tH (cid:19) − B sin (cid:18) tH (cid:19) and thus H = √ A + B + m . The classical phase-space trajectory is therefore a circle ofradius √ A + B traversed with the angular frequency ( A + B + m ) − .Effective equations of motion are set up in much the same way as their classical coun-terparts. Phase-space degrees of freedom are q , p , (∆ q ) , (∆ p ) , ∆( qp ). The time evolutionis generated by the function E of equation (18) through the quantum Poisson bracketdd t q = p p p + q + m + p (∆ q ) (2 q − p − m ) + q ∆( qp ) (4 p − q − m ) − p (∆ p ) ( q + m )2 ( p + q + m ) dd t p = − q p p + q + m + 3 q (∆ q ) ( p + m ) − p ∆( qp ) (4 q − p − m ) − q (∆ p ) (2 p − q − m )2 ( p + q + m ) dd t (∆ q ) = 2∆( qp ) ( q + m ) − q ) qp ( p + q + m ) , dd t (∆ p ) = 2(∆ p ) qp − qp ) ( q + m )( p + q + m ) dd t ∆( qp ) = (∆ p ) ( q + m ) − (∆ q ) ( p + m )( p + q + m ) . The first term in each of the first two equations is identical to the classical equations ofmotion, the extra terms constitute the leading order quantum corrections. The systemof equations is straightforward to evolve numerically for a sufficiently short time startingfrom a state that is initially semiclassical.The quantum evolution of positive frequency solutions is governed by the Schr¨odingerequation − ~ i dd t Ψ( x, t ) = (cid:0) p + q + m (cid:1) Ψ( x, t ) . The square root operator has the same eigenstates { ϕ n ( x ) } n =0 ,... ∞ ; as the one dimensionalquantum harmonic oscillator, with eigenvalues λ n = q n + ) ~ + m . A wavefunctioncan be evolved by decomposing it into the eigenstates of the Hamiltonian. If the state at23 = 0 is given by Ψ( x,
0) = P ∞ n =0 c n ϕ n ( x ), where c n are constant complex numbers, thenat any other time, the wavefunction isΨ( x, t ) = ∞ X n =0 c n exp (cid:18) − i λ n t ~ (cid:19) ϕ n ( x ) . (21)To compute the expectation values we write q = p ~ / a ∗ + ˆ a ), p = i p ~ / a ∗ − ˆ a ),where ˆ a ∗ and ˆ a are the usual creation and annihilation operators of the quantum harmonicoscillator. One finds h Ψ , q Ψ i ( t ) = ∞ X n =0 p ~ ( n + 1)Re (cid:20) ¯ c n c n +1 exp (cid:18) − it λ n +1 − λ n ~ (cid:19)(cid:21) h Ψ , p Ψ i ( t ) = ∞ X n =0 p ~ ( n + 1)Im (cid:20) ¯ c n c n +1 exp (cid:18) − it λ n +1 − λ n ~ (cid:19)(cid:21) In a similar manner one can obtain expressions for the moments of q and p .Figure 1: Classical (dotted), coherent state (solid) and effective (dashed) phase spacetrajectories, evolved for 0 ≤ t ≤ q .In order to complete the comparison we select a semiclassical state with a known de-composition into the eigenstates { ϕ n ( x ) } . A simple choice is to set the initial wavefunctionto be a coherent state of the harmonic oscillator c n = exp (cid:18) − | α | (cid:19) α n √ n ! , α ∈ C (22)which we can consider as a kinematical coherent state for our system. For the non-relativistic harmonic oscillator, as time goes on α changes, but the shape of the state24s preserved. Clearly, this is not the case for our relativistic evolution: Combining (22)with (21), we have an evolving state which, expanded in harmonic oscillator stationarystates, has coefficients c n e − iλ n t/ ~ = 1 √ n ! e −| α | / α n e − i √ n +1+ m / ~ t . Due to the square root in this relativistic model, these coefficients are not of the coherentstate form (22) unless t = 0. The physical states we obtain are not dynamical coherentstates; quantum back-reaction ensues which in the effective treatment is captured by thecoupling terms between moments and expectation values in (18).For a specific numerical example, we set α = q √ ~ so that at t = 0 the state is a Gaussianpeaked about q = q and p = 0, with zero covariance and minimal spread(∆ q ) = (∆ p ) = ~ , ∆( qp ) = 0 . We take these as initial values for the numerical evolution of the effective equations for thesystem. We assume the two physical scales to be separated by a single order of magnitudeby setting q √ ~ = 10. For simplicity we set m = 0. Depicted in FIG. 1 are the classical,coherent and effective quantum phase-space trajectories starting from the same initialstate. One can see that the effective equations describe the correct semiclassical trajectoryfor much of the evolution displayed. An internal measure of consistency is the size ofsecond order moments. From FIG. 2, we see that the semiclassical approximation clearlybreaks down after approximately t = 2 q , as (∆ p ) becomes too large. The same figuredemonstrates that up until that time the evolution of the moments themselves is very wellapproximated by the effective equations. The other two moments display similar behavior.Figure 2: Coherent state (solid) and effective (dashed) evolution of the second order mo-ment ∆ p = p (∆ p ) in units of q . 25 eferences [1] M. Bojowald and A. Skirzewski, Effective Equations of Motion for Quantum Systems, Rev. Math. Phys.
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