The Pennsylvania State University The Graduate School SEMICLASSICAL ANALYSIS OF QUANTUM SYSTEMS WITH CONSTRAINTS

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1 The Pennsylvania State University The Graduate School SEMICLASSICAL ANALYSIS OF QUANTUM SYSTEMS WITH CONSTRAINTS A Dissertation in Physics by Artur Tsobanjan c 2011 Artur Tsobanjan Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2011

2 The dissertation of Artur Tsobanjan was reviewed and approved by the following: Martin Bojowald Professor of Physics Dissertation Advisor, Chair of Committee Abhay Ashtekar Eberly Professor of Physics John Collins Professor of Physics Ping Xu Professor of Mathematics Richard Robinett Professor of Physics Associate Head, Director of Graduate Studies of the Department of Physics Signatures are on file in the Graduate School.

3 Abstract This dissertation addresses the problem of constructing true degrees of freedom in quantum mechanical systems with constraints. The method developed relies on assuming the behavior of the system of interest is nearly classical, or semiclassical, in a precise quantitative sense. The approximation is formulated as a perturbative hierarchy of the expectation values of quantum observables and their non-linear combinations, such as spreads and correlations. We formulate the constraint conditions and additional quantum gauge transformations that arise directly on the aforementioned quantities. We specialize this framework to several situations of particular interest. The first situation considered is the case of a constraint that commutes with all quantum observables, which is appropriate for quantization of a Lie algebra with a single Casimir polynomial. Through explicit order-by order counting argue that the true degrees of freedom are captured correctly at each order of the approximation. The rest of the models considered are motivated by the canonical approach to quantizing general relativity. The homogeneous sector of general relativity splits into classes of models, according to topology, which are described by a finite number of degrees of freedom and can therefore be quantized as quantum mechanical systems, making our construction directly applicable. In the Hamiltonian formulation the dynamics of these systems is governed by the Hamiltonian constraint, which, in the quantum theory, gives rise to several conceptual and technical issues collectively known as the Problem of Time. One of the aspects of this problem that does not possess a general solution is the dynamical interpretation of the theory. We use our construction, truncated at the leading order in quantum corrections, together with the intuition gained from dealing with constrained systems in classical mechanics, to define local notion of dynamics relative to a chosen configuration clock variable. We consider a class of models, where the chosen clock in not globally valid, eventually leading to singular dynamics. Within these models iii

4 we construct an explicit transformation between dynamical evolution relative to two distinct clocks, thus providing a consistent local dynamical interpretation of the true degrees of freedom of the quantum theory. iv

5 Table of Contents List of Figures List of Tables viii xi Chapter 1 Introduction Degrees of Freedom in Quantum Theory Approximate Treatment of Quantum Redundancy Hamiltonian Constraint and the Problem of Time Chapter 2 Effective Equations for Quantum Systems Classical Mechanics and Quantization Quantum Dynamics of Expectation Values Semiclassical Approximation Discussion Chapter 3 Effective Solution to Quantum Constraints Introduction Constraints in Classical and Quantum Mechanics Dirac Reduction on the Expectation Values Semiclassical truncation Counting Truncated Constraint Conditions Notation and preliminaries Constraints and truncation Counting degrees of freedom v

6 3.5.4 Independence of truncated constraints Details of the argument Auxiliary results Chapter 4 Deparameterizable Systems with a Hamiltonian Constraint Introduction Class of Models and Deparametrization Leading Order Quantum Corrections A Lesson from Classical Mechanics A Lesson from Quantum Mechanics Choice of a Clock and Partial Gauge-Fixing Consistent Physical Interpretation Example: Free Relativistic Particle Constraints at second order Gauge freedom Comparison with the Klein-Gordon solution Free massless particle Example: an Exotic Cosmological Model Classical behavior Quantum representation Effective constraints Comparison Long-term behavior of the state Short-term evolution with varying initial conditions Concluding Remarks Chapter 5 Effective Approach to the Problem of Time Introduction General Features of the Effective Treatment Constraints at Order and the Clock-Gauge Schrödinger regime for relativistic systems Complex time in deparameterizable systems What time is it? Failure of the Clock-Gauge and Transformation to a Different Clock A model of a bad internal clock Classical discussion Dirac quantization vi

7 5.3.3 Effective treatment Evolution in t and breakdown of the corresponding gauge Evolution through the extremal point of R[t] in a new gauge Switching gauges A timeless model: the 2D isotropic harmonic oscillator with fixed total energy Classical discussion Local relational evolution generated by physical Hamiltonians The quantum theory A local internal time Schrödinger regime Effective procedure Local evolution and comparison with the Schrödinger regime Gauge Transformations and Construction of Complete Orbits Concluding Remarks Appendix A Poisson Structure of Second Order Moments 142 Appendix B Positivity of truncated states 144 B.1 Discussion of positivity B.1.1 Algebraic positivity B.1.2 Positivity in the model of Section B Dynamics in the t-gauge B Dynamics in the q-gauge B Gauge transformation B.1.3 Positivity in the timeless model of Sec Appendix C Details of the Schrödinger Regime 153 C.1 Explicit moments for Sec Bibliography 155 vii

8 List of Figures 4.1 Phase space trajectories, evolved for 0 β + 5α 0 : classical (dotted), coherent state (solid) and effective (dashed) Coherent state (solid) and effective (dashed) evolution of second order moments α = ( α) 2, p α = ( p α ) 2 and (αp α ) Square magnitude of the coherent state Newton-Wigner wave function evolved for 0 < β + < 15α Coherent state evolution of second order moments α = ( α) 2, p α = ( p α ) 2 and (αp α ) as well as the third order moment (ˆα ˆα ) Short-term (left) and long-term (right) coherent state evolution of the quantum uncertainty defined as ( α) 2 ( p α ) 2 ( (αp α )) 2 in units of Effective evolution of semiclassical states with different initial values of moments for 0 < β + < 3.5α 0. The initial values of moments are as follows. Dashed line the original coherent state: ( α) 2 = ( p α ) 2 =.005α0, 2 (αp α ) = 0. Dotted line: ( α) 2 =.025α0, 2 ( p α ) 2 =.001α0, 2 (αp α ) = 0. Thin solid line: ( α) 2 =.001α0, 2 ( p α ) 2 =.025α0, 2 (αp α ) = 0. Thick solid line:( α) 2 =.008α0, 2 ( p α ) 2 =.008α0, 2 (αp α ) =.005α Effective evolution of 2nd order moments ( α) 2 (dashed), (αp α ) (dotted) and ( p α ) 2 (solid), with a variety of initial values: upper left the original coherent state ( α) 2 = ( p α ) 2 =.005α0, 2 (αp α ) = 0; upper right ( α) 2 =.025α0, 2 ( p α ) 2 =.001α0, 2 (αp α ) = 0; lower left ( α) 2 =.001α0, 2 ( p α ) 2 =.025α0, 2 (αp α ) = 0; lower right ( α) 2 =.008α0, 2 ( p α ) 2 =.008α0, 2 (αp α ) =.005α viii

9 5.1 A typical classical configuration space trajectory is a parabola with the peak value of t dependent on p t 0 and the separation of branches dependent on p 0. The orientation of evolution, indicated by the arrows, is consistent with p 0 < 0 and p t 0 > 0. We refer to the left branch (solid) as incoming or evolving forward in t, the right branch (dashed) as outgoing or evolving backward in t Schematic plots of the real part of t (left) and the imaginary part of t (right) against the flow parameter s Left: evolution of moments ( q) 2 (solid) and (qp) (dashed) in t-gauge (( p) 2 = const). Somewhere after s = 2.3 the spread q := ( q) 2 becomes comparable to the expectation values, as q/η >.1, and the semiclassical approximation breaks down in t-gauge. Right: corresponding effective trajectory (solid) and the related classical trajectory (dashed); the effective trajectory quickly diverges after s = Plot of the semiclassical trajectory evolved past the extremal point in t-gauge (solid part of the trajectory), by temporarily switching to the q-gauge (dashed part of the trajectory). Dotted vertical lines indicate the points where gauges were switched Pictorial comparison of the classical relational Dirac observable q 2 (q 1 ) (full ellipse, blue curve) with the quantities q 2 (R[q 1 ]) calculated in the effective theory using the q 1 -gauge (violet dashed curve) and ˆq 2 (q 1 ) in the Schrödinger regime (yellow solid curve). Where valid, the three curves agree perfectly. The Schrödinger regime breaks down earlier than the q 1 -gauge of the effective framework. The initial data match in all three cases: we chose q 20 = 0.7 and p 20 = 0.7 for the Schrödinger regime, which via Eq. (C.1) yields ( q 2 ) 2 (q 1 = 0) = ( p 2 ) 2 (q 1 = 0) = 2 and (q 2 p 2 )(q 1 = 0) = 0. We have set M = 10 and, to amplify effects, = We take these values as initial data for the effective formalism as well, and, using Eq. (5.104), we determine the initial value for p 10 = (the minus sign is necessary here, since in Eq. (5.86) we quantized C + which evolves backwards in q 1 ). In the effective picture, due to the imaginary contribution to q 1 in the q 1 -gauge, we have set the initial value of the clock to q 1 = i 2p 10, but employ R[q 1 ] as relational clock (see also FIG. 5.7). The initial data for the classical curve has been chosen accordingly. As regards the axis labels: for the effective framework both q 1 and q 2 refer to the expectation values of the corresponding operators (for q 1 the real part), while for the internal time Schrödinger regime q 2 refers to the expectation value from Eq. (5.94) and q 1 is the real evolution parameter ix

10 5.6 Comparison of the effective (black dotted curves) and internal time Schrödinger regime results (blue dashed curves) for the observables in q 1 -time associated to moments: a) ( q 2 ) 2 (q 1 ), b) ( p 2 ) 2 (q 1 ) and c) (q 2 p 2 )(q 1 ). The curves agree perfectly to order. As explained in the main text, the Schrödinger regime breaks down earlier than the q 1 - gauge of the effective framework. The breakdown of the latter is clearly demonstrated by the divergence of the effective moments near q 1 = 3. The initial data is identical to the one for FIG Behavior of a) the real and b) the imaginary part of the local clock q 1 with respect to the gauge parameter s of C H for the effective configuration with initial data as given in the caption of FIG Clearly, while R[q 1 ] is monotonic along the flow of C H (as long as the q 1 -gauge is valid) and, therefore, constitutes a useful local clock, I[q 1 ] does not provide a suitable clock here. The divergence of both near s = 0.79 signifies the breakdown of the q 1 -gauge a) Reconstruction of a semiclassical physical state via gauge switching in the effective framework. The jumps between the q 1 -gauge (black dotted and dashed curves) and the q 2 -gauge (blue solid curves) are a consequence of the o( ) jumps in the gauge transformations (5.106). The final evolution in q 1 -Zeitgeist after the fourth clock change is given by the fat black dashed curve and coincides to o( ) with the initial evolution in q 1 -gauge prior to the first clock change. For convenience we have labeled the axes by q 1 and q 2. It should be noted that for the curves in q i -gauge, q i actually refers to R[q i ]. b) Comparison of (q 2 p 2 )(R[q 1 ]) in q 1 -gauge before (dashed curve) and after (dotted curve) the complete revolution around the ellipse. The difference between the two curves is clearly of o( 2 ) or smaller. Initial data for both a) and b): q 10 = i 2, p 10 = q 20 = p 20 = 1, ( q 2 ) 2 0 = ( p 2) 2 0 = 2. Furthermore, M = 3 and, to enhance effects, we have set = The initial value for (q 2 p 2 ) follows from Eq. (5.104) x

11 List of Tables 4.1 Poisson algebra of constraints for a free particle. First terms in the bracket are labeled by rows, second terms are labeled by columns Poisson algebra of gauge conditions (5.52) with the constraints (5.51). First terms in the bracket are labeled by rows, second terms are labeled by columns. Note that these results only hold on the gauge surface defined in (5.52) A.1 Poisson algebra of second order moments. First terms in the bracket are labeled by rows, second terms are labeled by columns xi

12 Chapter 1 Introduction The last hundred years or so have seen an incredible transformation in the methods employed within theoretical physics. Yet, for all the beauty and sophistication of the abstract mathematical constructions it presently employs, at the end of the day, it is still required to produce a set of real (or, probably more accurately, rational) numbers that need to match experimental results. Thus, for the purpose of computability, a theorist is at some level forced to describe the degrees of freedom of a system he studies through structures which behave more-or-less like multiple copies of the real line, which in the rest of this introductory chapter will be loosely termed linear (although affine may be a better term). Many examples of interesting systems are not naturally linear in the above sense; their properties are typically computed following one of the two broad methods of linearization The local method. Its main idea is to cut up degrees of freedom into pieces that individually can be described by linear spaces. The embedding method. Here one defines degrees of freedom as embedded in a higher-dimensional linear space. Consequently the linear degrees of freedom are over-complete. An illustration of these methods is given by the treatment of gauge theories: on the one hand they can be embedded in the space that allows all configurations of the gauge fields, keeping track of redundancy by representing the action of the

13 2 gauge group; on the other hand, physical quantities, such as scattering amplitudes are most easily computed if the gauge freedom is fixed, which may not be possible globally, thus leading to a local description. We would like to emphasize, however, that the situation occurs much more generally. In fact, both of the above approaches have a natural realization in the category of manifolds the quintessential weapon in the arsenal of a modern-day theorist. Local coordinate charts describe pieces of the manifold as pieces of a linear space, while embeddings of a manifold in higher dimensional linear spaces provide a global over-complete description. 1 Both methods have obvious advantages and disadvantages: the local method better captures true degrees of freedom, while the embedding method is typically better at handling global properties such as symmetries and invariants. Due to their complementary strengths they are most successful when employed in parallel. 1.1 Degrees of Freedom in Quantum Theory In quantum theory the issue of describing the true degrees of freedom of a system becomes even more acute. This is partially due to the fact that, somewhat contrary to intuition, quantum theories are constructed from their classical counterparts via the process of quantization, which is neither precisely defined nor completely general. The various quantization algorithms typically require a detailed knowledge of the structure of the degrees of freedom, so that quantizing true degrees of freedom of a theory may not be a viable option. Instead, one has a choice between quantizations of the local descriptions of the theory or of its over-complete embedded form. A major issue with the former approach arises because it is, in general, extremely difficult to relate the quantizations of different local formulations of a theory. Such a relation is, however, essential as e.g. the quantum superposition principle requires one to give a precise meaning to a superposition of states corresponding to different local quantum descriptions. Thus, in many cases, one opts for the latter approach, which has its own issues as one has to formulate and be able to (at least locally) solve the quantum version of the redundancy conditions. 1 These approaches are, of course, particular incarnations of the ideas of describing something new as built up of smaller known pieces or as being part of a larger object with known properties.

14 3 Canonical approach to quantization uses the Hamiltonian form of the classical theory as the starting point. In this formulation, the over completeness of description is typically expressed by the so-called constraint functions. They simultaneously provide restrictions on degrees of freedom, like an embedding, and generate equivalence relations between configurations. For such systems, the quantum version of redundancy can be implemented by following Dirac s constraint quantization [1]. 2 Its local solution à la local gauge-fixing is generally not well-defined, which is, in part, due to the difficulty in appropriately defining what local means in quantum theory. In the present thesis, we utilize a precisely formulated semiclassical approximation, which provides us with a notion of locality, to construct a local analogue of the Dirac s quantum redundancy condition for systems with a finite number of classical degrees of freedom. 1.2 Approximate Treatment of Quantum Redundancy In addition to the previously mentioned linearization any theoretical framework needs a class of approximation schemes in order to computationally handle general situations of interest. Effective equations for quantum mechanical systems introduced in [28] and reviewed in Chapter 2, are based on assuming the system is in a semiclassical state, i.e. that it exhibits near-classical behavior. The semiclassicality condition is more naturally expressed using average values of physical observables and their moments quantities related to statistical spreads and correlations than using wavefunctions. Under this assumption, the quantum equations of motion can be expanded in the powers of Planck s constant. Truncating at a finite order of the expansion one obtains a reduced system of equations that is easier to solve numerically than the Schrödinger equation for the quantum wavefunction. In Chapter 3 we use the semiclassical expansion to tackle the problem of solving Dirac s quantum constraint conditions in quantum mechanics. We develop a 2 There are alternative approaches to implementing gauge symmetries within the quantum theory, most notably through the use of a path integral with an explicitly gauge-invariant measure. Dirac s method, however is prevalent in canonical treatments of quantum gravity and quantum cosmology, where defining invariant measure has proven rather difficult.

15 4 method for implementing the analogue of quantum redundancy within the semiclassical approximation. As in the case of the quantum equations of motion, the system can be truncated at a finite order making it more easily accessible for direct computations. 1.3 Hamiltonian Constraint and the Problem of Time A major motivation for the present work is the canonical approach to formulating a quantum theory of general relativity. The classical theory, formulated in terms of a metric field or an equivalent set of variables, has a large degree of redundancy, namely the diffeomorphisms of the space-time manifold. The Hamiltonian formulation of the general theory of relativity can be achieved by viewing it as a theory of Riemannian geometry on a three-dimensional space evolving in time. In this framework, diffeomorphisms take the form of the deformations of the three-dimensional space embedded in spacetime. Crucially, the Hamiltonian of the theory generates a particular class of such deformations and is itself a combination of constraint functions the Hamiltonian constraint (in addition it may contain boundary terms). In this situation, time evolution itself appears to follow along a redundant degree of freedom and the dynamical interpretation of the quantum version of the theory is difficult to define. This general situation leads to the various aspects of the problem of time in quantum gravity, some of which persist in symmetry-reduced homogeneous cosmological models of general relativity. These systems have a finite number of classical degrees of freedom and their quantum versions can be directly treated by the effective method we develop. They afford us an opportunity to study in isolation some of the issues associated specifically with this type of redundancy in dynamics. In Chapters 4 and 5 we perform effective analysis for a class of cosmology-inspired toy models, incorporating leading order quantum contributions. Our analysis agrees with the dynamical interpretations of the quantum constraint where they are available and extends to more general models where we are able to define local semiclassical dynamics and are able to relate different local dynamical frameworks at

16 5 the quantum level, giving those models a consistent local quantum dynamical description.

17 Chapter 2 Effective Equations for Quantum Systems Few systems of physical interest admit exact analytic solutions, instead, most realistic situations are described by a combination of perturbation around an analytic solution and numerical methods. In this chapter we describe a particular method for obtaining effective solutions to the equations of motion of ordinary quantum mechanics. This method has been introduced and developed in [28] and [29] the present chapter essentially serves as an overview of this technique, presented in a way that naturally sets up the scene for the subsequent chapters where the technique is extended to quantum systems with constraints. 2.1 Classical Mechanics and Quantization Canonical quantization takes the Hamiltonian formulation of a classical mechanical system as the starting point for building the quantum theory. In this setup, the state of a mechanical system is a point (or, more generally, a probability distribution) on a phase-space Γ, which is equivalent to specifying the system s configuration coordinates and their conjugate momenta. The phase-space has the structure of a symplectic manifold it possesses a two-form ω Ω 2 (Γ), that is non-degenerate at every point and closed, that is (i) p Γ, i X ω p = 0 implies X p = 0,

18 7 (ii) dω = 0. (2.1) Here and in the sequel i X denotes the natural contraction operator with the vector field X applied to a differential form, p designates that the expression is evaluated at a point p. Observable quantities correspond to real-valued functions on Γ and the symplectic structure encodes a relation between any pair of (differentiable) observables f and g their Poisson bracket that is most directly expressible using the inverse of ω: {f, g} := ω 1 (df, dg). (2.2) Non-degeneracy and closure of ω have exact analogues in terms of the Poisson bracket (i) p Γ, {f,.} p = 0 implies df p = 0, (ii) {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0. (2.3) This bracket naturally associates a vector field to each differentiable function X f (g) := {g, f}. (2.4) Condition (ii) above, also known as the Jacobi identity, ensures that this association is a Lie-algebra homomorphism between smooth functions and smooth vector-fields on Γ, where the functions are composed via the Poisson bracket and composition between vector fields is their usual Lie bracket. Dynamics of the mechanical system is given by the integral curves of the vector-field generated by the Hamiltonian function H a distinguished observable usually representing the energy of the system. Equations of motion written in a local basis of observables {x i } i=1,2...,n on Γ have the form d dt x i = {x i, H}, i = 1, 2..., N. (2.5) From the point of view of observables, the formulation of mechanics through the use of the Poisson bracket is more natural than using the symplectic form and this, indeed, is the point of view typically taken in canonical quantization. In

19 8 quantum mechanics the space of states of the system of interest is an appropriately constructed Hilbert space H, while observables are represented by self-adjoint linear operators on H. The idea of quantization is to provide a map from real-valued functions on Γ to self-adjoint operators on H, henceforth ˆf will denote the operator assigned to the classical function f. The guiding principle for constructing such a map is the Dirac s quantization condition [ ˆf, ĝ] = i {f, g}, (2.6) where [.,.] denotes the commutator between linear operators. In practice, this condition is difficult to implement for all functions, moreover, the classical limit of the theory is insensitive to deviations from this prescription of the second order in the quantization scale. However, a large class of mechanical systems possesses a closed Poisson subalgebra of functions that resolve the points of Γ (i.e. form an (over-)complete global set of coordinates on Γ), for which it is possible to implement the above condition exactly. 1 Energy of the system is represented by a distinguished linear operator on H the Hamiltonian, which we will denote by Ĥ. Dynamical evolution of the system, in an analogy with the classical mechanics, is a differentiable one-parameter family of vectors in H, whose dependence on the time parameter is determined by the Schrödinger equation: d dt ψ t = Ĥψ t. (2.7) For the rest of the chapter and for much of the rest of this work we will focus on mechanical systems with Γ = R 2D, the most common example of such a system being a point particle moving in D-dimensions. The classical space of configurations for the particle is the D-dimensional Euclidean space R D, with coordinates q 1, q 2..., q D. In order to describe this system completely, in addition to the coordinates one needs to specify the values of their conjugate momenta p 1, p 2,..., p D for a point particle this is equivalent to specifying its velocity com- 1 Any mechanical system whose phase space is a cotangent bundle over some configuration space possesses such a closed Poisson subalgebra. This subalgebra consists of all functions of configuration variables alone as well as of all linear functions of momenta i.e. the functions whose Poisson bracket with any function of configuration variables only, yields another function that only depends on the configuration variables.

20 9 ponents. The phase space is thus a symplectic 2D-dimensional manifold R 2D, with the symplectic structure having the simple form: D ω = dq i dp i (2.8) i=1 and the corresponding Poisson bracket between a pair of differentiable functions on the phase-space is given by {f(q, p), g(q, p)} = D i=1 ( f g f ) g q i p i p i q i (2.9) whence the non-trivial brackets between the position and momentum coordinates are {q i, p j } = δ ij, where δ ij is the Kronecker delta. For the system at hand, the most commonly used Hilbert space of quantum states consists of the complex-valued functions on R D that are square-integrable with respect to the Euclidean measure dx 1 dx 2... dx D. Position operators act multiplicatively, momentum operators act by differentiation ˆq i ψ(q 1,..., q D ) = q i ψ(q 1,..., q D ), (2.10) ˆp i ψ(q 1,..., q D ) = ψ(q 1,..., q D ). (2.11) i q i Condition (2.6) then holds exactly on the subalgebra formed of positions and momenta. This quantum setup is quite a bit more general than one might first think. A theorem, due to Darboux, states that the symplectic structure on any 2Ddimensional symplectic manifold locally takes the form of (2.8) in some coordinate basis. So, in a precise way, this setup can approximate any finite dimensional mechanical system. In addition, at this stage this setting is only kinematical we have not yet chosen the Hamiltonian function, which provides the dynamics and thus the most interesting relations between observables from the Physical standpoint. We note in passing, that the distinction between kinematical and dynamical setting becomes even more important for theories with constraints.

21 2.2 Quantum Dynamics of Expectation Values Given a classical Hamiltonian, constructing its quantum counterpart is a nontrivial process. Here we will not concern ourselves with this step and assume that a suitable quantum Hamiltonian is given, and that it is polynomial in the position and momentum variables. We will write Ĥ = H(ˆq 1,..., ˆq D ; ˆp 1,..., ˆp D ), such that, as the abuse of notation suggests, H(q 1,..., q D ; p 1,..., p D ) is the classical Hamiltonian, with the additional reservation that the function H keeps track of the ordering of the products of the variables chosen during quantization. The quantum Hamiltonian is then a partial differential operator on the space of smooth square integrable wavefunctions on R D and the Schrödinger equation (2.7) becomes a partial differential equation (PDE). The total time derivative on the left-hand side may be substituted by a partial derivatiove in t if we view the 1-parameter family of wavefunctions in R D as a smooth function on R D+1. This equation can be cumbersome to solve numerically and is formulated in terms of the wavefunction, which does not directly yield information about observable quantities. The average value, or the expectation value, of an observable ˆf given a state ψ H is given by ψ, ˆfψ, where.,. denotes the usual L 2 inner product in H. Using the Schrödinger equation we obtain the evolution equation satisfied by this expectation value d dt ˆf = 1 [ ˆf, Ĥ] + ˆf i t 10 (2.12) where we have dropped the explicit reference to a particular state and allowed explicit functions of t in the definition of the observable ˆf. This is an ordinary differential equation (ODE), however, it cannot in general be solved on its own as its right-hand side will in general depend on expectation values of operators other than ˆf, which will require their own equations of motion; these, in turn, may depend on the expectation values of operators not already included. Appending the list of equations step-by-step in this fashion, one will in general end up with an infinite system of coupled ODEs of the form (2.12) in the place of the Schrödinger PDE. This replacement is not helpful, unless the system of ODEs can be made finite. One way this can happen is if the step-by-step procedure does stop after finite number of steps, i.e. there is some finite commutator-subalgebra

22 11 that includes the observables of interest and the Hamiltonian. This may also hold only on some subspace of states that is preserved by the dynamics, e.g. dynamical coherent states, in which case one may need to impose some algebraic conditions on the expectation values in order to decouple the infinite system of equations. This suggests another way one may reduce the system of equations to a finite size, namely, by finding a class of states for which the equations decouple approximately as perturbative series in some small parameter. One such well-motivated and efficient perturbation method is based on the semiclassical approximation and will be introduced in the following section. Before introducing the approximation, let us more closely examine the properties of the expectation values of observables for the D-component canonical quantum system introduced earlier. The quantum operators that correspond to the position and momentum coordinates generate an associative algebra generating new operators through operator addition and multiplication. If we allow multiplication by complex numbers, the algebra generated will consist of all ordered complex polynomials in {ˆq 1,..., ˆq D ; ˆp 1,..., ˆp D }. Not all of these correspond to independent linear operators as antisymmetric products between the generators are subject to the canonical commutation relations (CCRs) [ˆq i, ˆp j ] = i δ ijˆ1 (2.13) Thus, we can write a countable basis for the algebra by selecting an ordering e.g. any element can be written as a complex linear combination of the standardordered terms ˆq n 1 1,..., ˆq n D D ; ˆp m , ˆp m D D and the identity, products can be reordered iteratively by the CCRs and the addition of lower order products. We will refer to this unital, associative, complex algebra of finite-order polynomials in {ˆq 1,..., ˆq D ; ˆp 1,..., ˆp D } as the algebra of quantum observables and denote it A, even though this is clearly an abuse of the term observable as A includes polynomials that would not correspond to self-adjoint operators on H. The inner product on H provides a notion of convergence on A as a subalgebra of linear operators on H, and one could complete A by including convergent power series as well as more sophisticated functions on the spectra of ˆq i and ˆp i. Here we will, for the most part, restrict our attention to polynomials avoiding subtle ana-

23 12 lytic issues a large class of interesting observables can be at least approximated by A. We also note that the observable algebra possesses a star-structure that is represented by passing to the adjoint operators on H and classically corresponds to taking complex conjugates. It is completely determined on A by requiring ˆq i = ˆq i, ˆp i = ˆp i and (câˆb) = cˆb â. States in H can be viewed as linear assignments of complex numbers to elements of A, via taking the expectation values and this is indeed a more general notion of a state, that includes e.g. density states evaluated via a trace. For vectors in H as well as for the so-called density matrices, this assignment of values to A is not completely arbitrary, but satisfies the positivity condition ââ 0 (2.14) and by the Gel fand-naimark-segal construction, specifying a positive linear state on an associative star-algebra defines a Hilbert space representation of the algebra, where star-invariant elements correspond to self-adjoint operators. In this sense, linear functionals on the algebra carry equivalent information about the representation as the Hilbert space vectors. This is the point of view that we take from now on: polynomial algebra generated by the set of basic observables will play the central role instead of the wavefunctions. The term states will refer to linear functionals on A, which are elements of its vector-space dual denoted A, while the elements of H will be referred to as vectors or wavefunctions. By the duality, elements of A can also be viewed as linear functionals on A : â will denote the function induced by â on A by the natural pairing â (α) := α(â) (2.15) We will refer to this function as the expectation value of â which is consistent with our earlier use of the term if we restrict to states constructed through the use of a Hilbert space inner product. Since a linear state is completely determined by the values it assigns to a basis, the functions induced by a linear basis on A (e.g. the standard ordered polynomials) induce a complete set of coordinates on A. These functions naturally possess an antisymmetric bracket that follows from the

24 13 commutator on A { â, ˆb } := 1 [ â, i ˆb ] (2.16) Since it is defined on a complete set of coordinates, the bracket can be extended to all differentiable functions on A by demanding linearity and the Leibnitz rule in each argument. It is simple to verify that the bracket satisfies the Jacobi identity (ii) of (2.3), by first verifying it for the expectation values and then showing that it is preserved by the proposed extension. However, in general, this bracket is degenerate, i.e. it does not generally satisfy condition (i) of (2.3). We will refer to it as the quantum Poisson bracket or simply Poisson bracket when there is no possibility of confusion. With this definition, the dynamical equations for expectation values can be recast in a distinctly classical form d dt f = {f, Ĥ } + t f (2.17) where f is any function on A, that may also explicitly depend on t. This view of quantum dynamics is related, but in a sense dual to the geometrical formulation of quantum mechanics of [12, 13, 14], which focuses on the symplectic geometry of the projective space of H induced by the inner product, rather than the above Poisson geometry induced by the algebra commutators. 2.3 Semiclassical Approximation Intuitively, a semiclassical state yields probability distributions that are sharply peaked about their average values. In practice, this notion clearly depends on the choice of observables to be sharply defined by the semiclassical states. A (pure) classical state corresponds to a point on the phase-space and we would like to say that a state that is semiclassical is, in some sense, close to some classical state, we therefore want our semiclassical states to be simultaneously sharp on a set of observables that classically correspond to a (over-) complete set of coordinates. For D-dimensional particle, we demand that the set of linear observables {ˆq 1,..., ˆq D ; ˆp 1,..., ˆp D } be sharply defined in such a state. In particular, all the deviations from the average values ( ˆq i 2 ˆq i 2 ) and ( ˆp 2 i ˆp i 2 ) should be small. Of course the product of deviations of non-commuting variables is bounded from

25 14 below by the well-known Heisenberg s uncertainty principle (which incidentally can be derived starting from (2.14)) ( ˆq 2 i ˆq i 2 )( ˆp 2 i ˆp i 2 ) 2 4 (2.18) Thus for a state with minimal uncertainty, sharply peaked in both ˆq i and ˆp i, we have deviations in each of order. 2 This assumption only sets bounds on combinations of expectation values of e.g. ˆq i and ˆq i 2, but not ˆq i 3 and so on, which could still differ significantly from ˆq i 3 and so forth. We therefore need a systematic set of conditions involving all expectation values expressing their sharpness. These conditions are formulated most naturally not on the set of expectation values of a basis of A but rather on the set of moments of ˆq i and ˆp i ; these moments are similar to the Hamburger momenta of probability distributions, except they are taken over a collection of variables that do not in general commute and therefore ordering is important. We define the moments, or quantum variables as they are termed in [28], and their semiclassical properties as follows (ˆq 1 ˆq 1 ) n 1... (ˆq D ˆq D ) n D (ˆp 1 ˆp 1 ) m 1... (ˆp D ˆp D ) n D Weyl i (n i+m i )/2 (2.19) where the subscript Weyl indicates that the product inside is ordered completely symmetrically ( Weyl-ordered ), e.g. (abc) Weyl = 1 (abc + bca + cab + cba + bac + acb). (2.20) 6 We refer to the quantity on the left-hand side of (2.19) as a moment of order i (n i + m i ) it follows that the order of moments is larger than or equal to two. Clearly the deviations ( ˆq 2 i ˆq i 2 ) = (ˆq i ˆq i ) 2 and ( ˆp 2 i ˆp i 2 ) = (ˆp i ˆp i ) 2 are themselves second order moments. The entire set of moments together with the expectation values of the generators form an alternative complete set of coordinate functions on A, which is not hard to see as they are polynomial functions of the expectation values of symmetrically ordered products of the generators of A: a moment of order N can be expressed in terms of expectation values of products of 2 We are implicitly assuming that the units of q i and p i both have dimension of the square-root of action, which can be accomplished by a canonical transformation.

26 15 the generators of total order N or less and vice versa. Let us examine the expression for an expectation value in terms of moments in a little more detail. Any polynomial ˆf A can be written using the symmetric basis ˆf = n 1,...,m 1... α n1,...,m 1... (ˆq n ˆq n D D ˆp m ˆp n D D ) Weyl =: f sym (ˆq n 1 1,..., ˆq n D D ; ˆp m 1 1,..., ˆp n D D (2.21) where the map f sym is defined so that it keeps track of the ordering. Note: if ˆf is obtained by taking a classical polynomial function f as the basis and prescribing some ordering, then f sym (q 1,..., q D ; p 1,..., p D ) = f(q 1,..., q D ; p 1,..., p D ) + o( ). By writing ˆq i = (q i + (ˆq i q i )) we deduce ˆf = f sym (q 1,..., q D ; p 1,..., p D ) (2.22) + where n 1,...,m n 1!... n D!m 1!... m D! (n i +m i ) f sym (q 1,..., q D ; p 1,..., p D ) q n q n D D p m p m D D ê (n1,...,m 1...) ê (n1,...,m D ) := ((ˆq 1 ˆq 1 ) n 1... (ˆq D ˆq D ) n D (ˆp 1 ˆp 1 ) m 1... (ˆp D ˆp D ) n D ) Weyl (2.23) It is not difficult to verify (2.22) in the case where ˆf consists of a single totally symmetric product, the rest immediately follows by linearity. And since ê (n1,...,m 1...) are precisely the moments of (2.19), the expansion of ˆf follows. The above hierarchy is realized in the archetypal semiclassical state the Gaussian wavefunction: for D = 1 the wavefunction ψ(q) = ke ip 0 q e (q q 0 )2 2, with expectation values taken via the usual L 2 (R, dq) inner product, is peaked about ˆq = q 0 and ˆp = p 0. 3 The moments in this function can be computed explicitly [41] at 3 Here the units have been adjusted so that both q and p have the units of the square-root of action and the spreads in each variable are identical. This can be accomplished by a rescaling canonical transformation, given a small length or momentum scale.

27 16 all orders and have the form ((ˆq ˆq ) a (ˆp ˆp ) b) W eyl ψ = (a+b) a!b! 2 a+b ( a 2)!( 2)! b 2, for even a and b 0, otherwise (2.24) where the hierarchy of (2.19) is satisfied explicitly. The quantum Poisson bracket (2.16) extends to moments as they are functions on A and they evolve according to (2.17). The Poisson bracket between the moments and expectation values ˆq i or ˆp i vanish. For the moments generated by a single canonical pair the explicit formula for the bracket can be found in Section II.A of [41]. The situation, is combinatorially more involved for the D- component system, however brackets between low order moments are not difficult to compute directly. We define the notion of a total semiclassical order for functions that are products between different moments and powers of : Order (fg n ) := Order(f) + Order(g) + 2n. (2.25) The key feature, that we will shortly need for consistency, is that the semiclassical order of a bracket between two moments has the same order as the sum of the orders of the moments involved minus two (recall that the bracket between the moments and expectation values of the generators vanish). For the one-component system this can be seen directly from the formula in [41] and proven inductively in a more general situation. Since moments have an order of at least two, it follows that the functions of total order N and above for N 2 form a Poisson ideal i.e. for any function f of order N and any other function g of arbitrary semiclassical order Order({f, g}) 2. The prescription for truncating the potentially infinite set of ODEs of the form (2.12) is to first write them for and in terms of the moments and the expectation values of the generators, which can be accomplished using (2.17); this is followed by dropping all terms of total semiclassical order N or above. The above mentioned feature of the quantum Poisson bracket means that brackets between two functions can be taken before or after the above truncation giving the same

28 17 result. In addition to the consistency of the truncation, this leads to a great practical simplification, as brackets between truncated functions are typically much simpler to compute. Only a finite number of moments will remain below order N and, consequently, only a finite number of ODEs will be required to evolve the system. In particular, according to (2.17), the time derivative of a discarded moment is proportional to its bracket with the Hamiltonian expectation value, and, due to the above result, is of at least the same semiclassical order as the discarded moment in question; in this sense, the evolution of discarded moments is slow and therefore they will likely remain small for a certain portion of the evolution. 2.4 Discussion Advantages of the above technique for setting up effective evolution of a quantum mechanical system are its simplicity and flexibility. A disadvantage is the difficulty in controlling the growth of the moments beyond considered semiclassical order, which can at the moment only be inferred indirectly by observing whether the hierarchy (2.19) persists amongst the moments that have been retained or by explicitly computing the dynamics at higher orders. A limitation is set by the semiclassical regime: it may be possible to construct a similar perturbation based on a different approximation, however this does not appear straightforward. 4 Finally, a drawback that will be partially addressed in the next section, is the extension of this setup to systems that have a phase-space other than R 2D ; to obtain a set of global phase-space coordinates one then usually has to use an over-complete system of functions. If the observable algebra is generated by the quantized coordinates, one needs to deal with the redundancy associated with the over-completeness of the system. We would like emphasize that the effective method is not meant as a substitute for, but merely as a supplement to the ordinary methods of quantum mechanics. Specifically, it offers a prescription for recovering approximate semiclassical dynamics given a quantum system and its initial state that satisfy certain criteria. 4 In [28] and [29] the semiclassical expansion was combined with an adiabatic approximation which assigned a higher order to time-derivatives of moments, still, the semiclassical expansion appeared to be essential.

29 18 The phenomenon that is captured best by this approximate scheme is the quantum back-reaction the deviation of the trajectory traced by the expectation values of coordinates and momenta from the classical phase-space trajectory due to the distortion and spreading of the wavefunction (see Section 4.5 for an explicit example). On the other hand, the above expansion in moments is rather difficult to use when it comes to the study interference, as the magnitude of these phenomena is generically of the order of the spread of the wavefunction, which then should not be considered small. Another area that is physically important but difficult to study by any algebraic means is the spectra of operators, which depend sensitively on the precise details of the Cauchy-complete Hilbert space representation, including boundary conditions. In addition, there are wavefunctions of physical interest, which do not assign a convergent expectation value to every element of the polynomial algebra A (e.g. wavefunctions with a polynomial fall-off), which is necessary in order to replace the Scroödinger equation by the infinite system of ODEs. The feasibility of treating some or all of these subtleties through algebraic means merits a separate study, which, however, lies outside of the scope of the present work.

30 Chapter 3 Effective Solution to Quantum Constraints 3.1 Introduction Constraints can arise in classical Hamiltonian systems in a variety of ways and indicate redundancy in the degrees of freedom used to describe the system. Most common way of constructing a Hamiltonian theory is through the Legendere transformation of a Lagrangian description of a system. This also provides the most common way the constraints get introduced as, in some cases, the Legendre transformation is degenerate: not all phase-space states are possible, configurations and momenta are restricted by a number of conditions or constraints they must satisfy. On the surface satisfied by the constraints the flows they generate through the Poisson bracket identify equivalent configurations and the degrees of freedom need to be reduced further. This is the situation in the Hamiltonian gauge-field theories and general relativity. Here we only consider systems with finite number of classical degrees of freedom, with immediate applications provided by homogeneous cosmological models. The reduction of degrees of freedom can be postponed until after quantization. They are removed by analogy with symmetries by promoting the constraint conditions to operators and restricting to the states that are annihilated by their action this is known as the Dirac s constraint quantization. The full reduction