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SINGULAR DYNAMICS IN QUANTUM MECHANICS AND QUANTUM FIELD THEORY By CHENGJUN ZHU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1994 Dedicated to My Parents and Friends ACKNOWLEDGMENTS I wish to thank my supervisor, Prof. J. R. Klauder, for his efforts, patience and encouragement throughout the time I have been at the University of Florida. He introduced me to most of the topics involved in this dissertation and always was ready for a discussion if needed. Over the years, his enthusiasm and encouragement have been constant sources of inspiration to push the project forward. I am grateful to the other members in my supervisory committee, Prof. Pierre Ramond for his valuable comments and suggestions, Prof. Charles Hooper, Prof. Robert Buchler, Prof. David Tanner and Prof. Christopher Stark, for patiently sitting through my oral exam and thesis defense. Special thanks should go to Prof. Buchler for his kindness, concern and heuristic suggestions when I was very depressed in jobhunting. Also I want to express my deep thanks to my fellow graduate students and colleagues, for their help and encouragement throughout the years. TABLE OF CONTENTS pages ACKN OW LEDGM ENTS ......................................................................................... iii A BSTR A CT ................................................................................................................ vi PART I CLASSICAL SYMPTOMS OF QUANTUM ILLNESSES CHAPTERS 1 INTRO DUCTION .................................................................................. 2 1.1 Basic Concepts of Operator Theory ...................................... 3 1.2 Symmetric Operators and Extensions ................................ 4 1.3 The Construction of SelfAdjoint Extensions .................... 8 2 CLASSICAL SYMPTOMS OF QUANTUM ILLNESSES ............. 10 2.1 Principal Assertions ............................................................. 10 2.2 Classical Symptoms of Quantum Illnesses .......................... 12 2.3 More General Examples ...................................................... 16 2.3.1 Examples of OneDimensional Hamiltonians ........ 16 2.3.2 Examples of ThreeDimensional Hamiltonians ....... 20 3 SELFADJOINTNESS OF HERMITIAN HAMILTONIANS .......... 22 3.1 Three Different Cases of Hermitian Hamiltonians.............. 22 3.2 The Application of Boundary Conditions and SelfAdjoint Extensions at Regular Points ............................................ 32 3.3 The Application of Boundary Conditions and SelfAdjoint Extensions at Singular Points ............................................ 34 3.4 Conclusion .............................................................................. 38 PART II OPERATOR ANALYSIS AND FUNCTIONAL INTEGRAL REPRESENTATION OF NONRENORMALIZABLE MULTI COMPONENT ULTRALOCAL MODELS 4 INTRO DUCTION .............................................................................. 44 5 CLASSICAL ULTRALOCAL MODEL AND THE STANDARD LATTICE APPROACH ....................................................................... 48 6 OPERATOR ANALYSIS OF MULTICOMPONENT ULTRALOCAL M O D ELS .............................................................................................. 53 6.1 Operator Analysis of SingleComponent Ultralocal Models ......................................................................................... 54 6.2 Operator Analysis of FiniteComponent Ultralocal Fields ......................................................................................... 60 6.3 Operator Analysis of Infinitecomponent Ultralocal Fields ......................................................................................... 65 7 THE CLASSICAL LIMIT OF ULTRALOCAL MODELS ................. 73 7.1 Ultralocal Fields and Its Associated Coherent States ......... 74 7.2 Selection of the Coherent States for Ultralocal Fields and the Classical Limit .......................................................... 77 8 PATHINTEGRAL FORMULATION OF ULTRALOCAL MODELS ........................................................................... 86 8.1 Operator Solutions of Ultralocal Models ........................ 87 8.2 EuclideanSpace PathIntegral Formulation of SingleComponent Ultralocal Fields .................................. 92 8.3 EuclideanSpace PathIntegral Formulation of MultiComponent Ultralocal Fields ................................... 96 REFERENCES ......................................................................................... 101 BIOGRAPHICAL SKETCH ...................................................................... 103 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SINGULAR DYNAMICS IN QUANTUM MECHANICS AND QUANTUM FIELD THEORY By Chengjun Zhu April, 1994 Chairman: Dr. John R. Klauder Major Department: Physics The classical viewpoint that the pathologies of quantum Hamiltonians are reflected in their classical counterparts and in the solution of the associated classical dynamics is illustrated through the properties of classical equations of motion. Rigorous mathematical proofs are provided by calculating the deficiency indices, which further help to analyze the possibility of extending the Hermitian Hamiltonian to a selfadjoint one. A few examples of how to add boundary conditions on wave functions in order to have selfadjoint extensions, as well as to cure the illness of certain Hamiltonians, are discussed. The nontrivial (nonGaussian), nonperturbative quantum field theory of O(N)invariant multicomponent, nonrenormalizable ultralocal models is presented. An indefinite nonclassical and singular potential has replaced the nonvanishing, positivedefinite nonclassical and singular potential appeared in the singlecomponent case. The operator theory of multicomponent ultralocal fields remains noncanonical even with the disappearance of the singular potential. The validity of this nontrivial solution is supported by the fact that the nontrivial quantum solution reduces to the correct classical theory in a suitable limit as h  0. The nontrivial (nonGaussian) path integral formulation constructed by the nonperturbative operator solution, involving the nonclassical and singular potential, replaces the standard lattice approach, which invariably leads to a Gaussian theory regardless of any nonlinear interactions. The appearance of the nonclasssical, singular potential suggests that we can not always place the classical Lagrangian or classical Hamiltonian directly into the pathintegral formulation, or in other words, a straightforward canonical quantization of fields with infinite degrees of freedom does not always apply. These differences suggest plausible modifications in the latticespace formulation of relativistic nonrenormalizable models that may lead to nontriviality for N 4 component models such as (%, n > 5 (and possibly n = 4), for N >! 1. PART I CLASSICAL SYMPTOMS OF QUANTUM ILLNESSES CHAPTER 1 INTRODUCTION The rules of quantization, as laid down in 1926 by Schrodinger, have stood the test of time and have provided the basis for applying (and teaching) nonrelativistic quantum mechanics1. Because of the overwhelming success of the theory, it is not surprising that physicists have little use for several mathematical niceties that go by the largely unfamiliar code words of "selfadjoint extension," "deficiency indices," operator domains," etc. (which we will briefly define in subsequent sections). Z3 These concepts typically become important when the quantization prescriptions are ambiguous or otherwise incomplete all by themselves4. Our goal is to show that such quantum technicalities are actually reflected in the classical theory by corresponding difficulties that are easy to see and understand. It will be clear that such an evident classical difficulty must lead to some kind of quantum difficulty, and thus the naturalness of the corresponding quantum technicality will become apparent. We may go so far as to assert that there are sufficient classical symptoms of any quantum illness that a complete diagnosis is possible already at the classical level (Chapter 2). Of course, resolving the problems and effecting a full cure, when one exists, can only take place at the quantum level (Chapter 3). This chapter contains basically a summary of the mathematical concepts and theorems prepared for later chapters. In Chapter 2, we will discuss the properties of classical solutions of given examples, whose "symptoms" can be used to diagnose any potential quantum "illnesses". In Chapter 3, we will provide the mathematical proof for all the assertions made in Chapter 2 and will also give a few examples of how to add boundary conditions on the wave functions in order to extend the Hamiltonians to be selfadjoint and so to cure the illness of certain Hamiltonians. 1.1 Basic Concepts of Operator Theory In this section we follow Ref. 3 closely. The operators we deal with in quantum mechanics are frequently unbounded operators which need to be defined on a domain D(A) to ensure that Aye r9 for V e D(A), where His Hilbert space. Generally, if A and B are two unbounded linear operators, D (aA + PB) = D(A) n D(B), a, 3 e C, (aA + PB)y = a (A y) + f(B y), Vye D (aA + +PB). For the product put D (AB) ={ V: ye D(B), By V D(A)), (AB)y= A (By), iye D (AB). In general D(AB) D(BA), and hence AB BA, and even A A'1 A'A, because their domains may not coincide. So extra care is needed in treating these operators. We recall that A is densely defined in Hif (and only if) His the closure of D(A), i. e., for y e H, there is a sequence of elements of D(A) converting to ig, [on V, n E D(A)]. For a densely defined operator A for H9 into ,4 there exists a unique adjoint operator At which satisfies (A W, 0) = ( Vp, A t), where Vy e D(A) and # e D(A t). If an adjoint operator At = A on D(A), and D(A) = D(At), then A is called a selfadjoint operator. 1.2 Symmetric Operators and Extensions If At = A on D (A), then D(A) a D (At) always holds (like Hermitian Hamiltonian operators). Now, (a) A is called a selfadjoint operator when D(A) = D(At), and one writes A = At. (b) A is called a symmetric operator when D(A) c D(At), and one writes A c At. Further, if there exists a symmetric extension As of A, such that A c As = Ast c At, then we say that A has the selfadjoint extension As (by enlarging D(A) to D(As) and contracting D(At) to D(Ast) = D(As)). Let At 0= i 4, with solutions 0 e N which are closed subspaces of 94 known as the deficiency spaces of A; and their dimensions, n respectively, are called the deficiency indices. We shall also say for brevity that A has deficiency indices (n+, n). Lemma 1. If A is symmetric, then D(At) = D(]A )N+@N., A denotes the closure of A. 5 Lemma 2 A symmetric operator with finite deficiency indices has a self adjoint extension if and only if its deficiency indices are equal. With the help of the two above Lemmas, the following theorem regarding deficiency indices is straightforward: THEOREM 1: Suppose the Hermitian operator A has deficiency indices (n+, n), then if (i) n, = n = 0, A is selfadjoint when D(A) is closed and the closure of A is self adjoint when D(A) is not closed. (ii) n. = n 0, A has selfadjoint extensions. (iii) n, n, A is not selfadjoint and has no selfadjoint extension. This theorem is used to classify Hermitian operators, and we see that not all Hermitian operators will generate fulltime quantum mechanical solutions. Example: A = i d/dx on L2(Q) A t= id/dx, At = fi idj <= +i k, 04(x) ~ exp(T x). (a) Q = ( oo, + ): Neither 0+ nor 0 is in L2( oo, + oo), so n+ = n = 0 and A is selfadjoint on its natural domain. This fact is consistent with its classical behavior (see Fig. 1(a)), where there is a fulltime evolution. (b) Q = [ 0, + *o): 0. = exp(x) is not in L2([ 0, + o)), so (n+, n) = (1, 0), A is not selfadjoint and possesses no selfadjoint extension. This fact is consistent with its classical behavior (see Fig. 1(b)), where there is no fulltime evolution. (c) 0 = [ 0, 1]: Both & are in 2([ 0, 1 ]), (n+, n) = (1, 1), A has selfadjoint extensions by adding the boundary condition Vy(1) = exp(i0) y(0), where 0 et. The spectrum of A = 0 + 2n ir, n = 0, 1, 2, .... For different 0, we get a different selfadjoint extension. The case of 0 = 0 is the usual case used in quantum mechanics and is called the periodic boundary condition. See Fig. 1(c) for its consistent classical behavior. The deficiency indices are clearly of central importance in the classification of Hermitian operators and in the construction of selfadjoint extensions. These may be found easily if explicit solutions of A tg = i y are known, but unless the coefficients of A are simple, the calculations will involve difficult special functions. In order to avoid those tedious, sometimes even impossible calculations, we will use a modified theorem of deficiency indices, especially when both end points of Q are singular. First, let us look at an example with quantum Hamiltonian H, where H y= (p2 + x3)y= (. + x3)V dx2 acting on functions defined on an interval S2 = ( oo, + oo). Both end points x = are singular. A direct calculation of the deficiency indices seems incredibly tedious since the behavior of a solution of Hy/ = i V at both end points must be known. To simplify the difficulties, we separate 2 = (a, b), where a and b are both singular, to two intervals with i2i = (a, c ] and Q2 = [ c, b), where c is a regular point. Here comes the second theorem regarding deficiency indices: THEOREM 2 Let Aa and Ab be symmetric operators associated with A on 121 and 2 respectively. If n, na, nb indicate the deficiency indices of A, Aa and Ab respectively, then n= na + nib 2, respectively (see Chapter 2 of Ref. 3). To apply this result to the above example, we have HtV d 2 + x3) =i dx2 Since H is a real operator, i. e., H = H, we have /. = y/+. So n+= n. and we only need to consider V+ and n+. Let i.+ = V, (d 2 +X3)W=iyl. dx2 The asymptotic behaviors at x + is given by f' x3y,= 0 lx ~ K/i5(2x5/2) ~ x 3/4exp( x5/2) ) 5 5 (. Ih/5() f 23/4cos ( 2/2. 2r) S(2 52) X /2 3/ si( 2), /2. ) "5 5 20 where Jv, Nv and Kv are the Bessel functions of the first and second kinds, respectively. Obviously, the integration l4(x) 2dx converges at x = + oo. Thus, n+, +.= 1. In the same way, (1, 2) 3/4 Wx ___ / so that both solutions of (1, satisfy JyO(x)2dx converges at x = oo. Therefore, n+, .= 2, and n+=n+, + + n+, 2 =1 + 22= 1 (n+, n.) = (1, 1). One boundary condition* is needed to extend H to be selfadjoint. The existence of bound states v. demonstrates that the energy spectrum is discrete. This result is exactly what is claimed in Chapter 2. Later in Chapter 3, we will see examples for three different cases of the deficiency indices: (i) n+= n. = 0, (ii) n+= n. 0, and (iii) n+& n.. 1.3 The Construction of SelfAdjoint Extensions Before we construct the selfadjoint extensions, we introduce some notation (see Chapter 10 of Ref. 3): (i) (, = (A t+, V) (b, At), where Ve D(A ); (ii) fh, f2 .. fn e D(At)= D(A) N+N. where fi = hi + i, i= 1,2,..., hie D(A), i eN+N.. If n+= n the number of boundary conditions = n+; see page 260 of Ref 3. We call fl, f2 *** fn linearly independent relative to D(A) if <, ..., are linearly independent. With the help of the preceding definitions, we introduce the following theorem: THEOREM 3 Let A be a symmetric operator with finite nonzero deficiency indices n+ = n. = n. Suppose that fl, f f2, e D(At) are linearly independent relative to D(A) and satisfy fj) = 0 (i, j = 1, 2,...,n). Then the subspace Mfof Hconsisting of all ye D(A ) such that (v, fi) = 0 ( i = 1, 2, ...., n) is the domain of a selfadjoint extension M of A, given by M iy = Atp for (V, fi) = 0, i = 1, 2, ...., n, are called the number of n boundary conditions on the wave function p. In chapter 3, we will give a few examples of how to use theorem 3 to obtain some of these selfadjoint extensions and thereby cure the illness of certain Hamiltonians. CHAPTER 2 CLASSICAL SYMPTOMS OF QUANTUM ILLNESSES Without using the precise definitions and techniques of operator theory, presented in chapter 1, we can simply show that the pathologies of quantum Hamiltonians are reflected in their classical counterparts and in the solutions of the associated classical equations of motion. This property permits one, on the one hand, to appreciate the role of certain technical requirements acceptable quantum Hamiltonian must satisfy, and, on the other hand, enables one to recognize potentially troublesome quantum systems merely by examining the classical systems. In 2.1, we will make the principal assertions that three categories of classical solutions correspond to three categories of the selfadjointness of quantum Hamiltonian operators. Examples are given in 2.2 and 2.3, while the mathematical proof will be presented in chapter 3. 2.1 Principal Assertions Specifically, for the classical system with a real Hamiltonian function H = 2m + V(q), the classical equation of motion 2m mq'(t) = V'[q(t)] admits two kinds of solution: (i) q(t) is global (which is defined through  < t < +0) and unique, e. g., H=p2+q4 ; (ii) q(t) is locally unique, but globally possibly nonexistent (escapes to infinity in finite time). This case may be also divided into two subcases: extendible (q(t) can be extended to a global solution) or nonextendible. For example, H = p2. q4, H = p2 q3 are extendible, but H = pq3 belongs to nonextendible situation. The corresponding quantum mechanical problem i ax, t) = HV(x, t) = [ d 2 + V(x) ] W(x, t) at 2m dx2 has similar cases. To conserve probability, the time evolution of the wave function must be effected by a unitary transformation. A unitary transformation gives a prescription for wave function evolution for all times t, < t < , or as we shall sometimes also say, for full time. The generator of such a unitary transformation times VT, which we identify with the Hamiltonian operator (up to constants), must satisfy one fundamental property, namely, that of being "self adjoint." Being selfadjoint is stronger, i.e., more restrictive, than being Hermitian, which is the generally accepted sufficient criterion. According to Theorem 1 on page 5, typically, there are three qualitatively different outcomes that may arise (i) no additional input is needed to make the Hamiltonian self adjoint; (ii) some additional input (i.e., the boundary conditions) is required to make the Hamiltonian selfadjoint, and qualitatively different Hamiltonians may thereby emerge depending on what choice is make for the needed input; and (iii) no amount of additional input can ever make the Hermitian Hamiltonian into a selfadjoint operator. Remember it is only selfadjoint Hamiltonians that have fully consistent dynamical solutions for all time. Thus in case (iii) no acceptable quantum dynamical solution exists in any conventional sense. Therefore we make the general assertions that there are three categories of classical solutions, which correspond to three categories of the selfadjointness of quantum Hamiltonian operators: q(t) Hamiltonian operator (i) global & unique selfadjoint & unique (ii) nonglobal but extendible selfadjoint extensions exist (iii) nonglobal & nonextendible no selfadjoint H exists 2.2 Classical Symptoms of Quantum Illnesses To be sure, the majority of classical systems one normally encountersas represented by their classical Hamiltonians, their Hamiltonian equations of motion, and the solution to these equations of motionare trouble free. But there are exceptions, classical systems which exhibit one or another kind of singularity in their solutions, and it is in these problematic examples that we will find the anomalous behavior we seek to illustrate. Let us consider three model classical Hamiltonians that will exhibit the three kinds of behavior alluded to above. Example a: H=lp2+iq4 The equations of motion lead to 4=2 2E 44 4 where E __ 0 is the conserved energy. Nontrivial motion requires E > 0 and E 2 q4/4 in which case qdq) + dq = t + c leads to a welldefined, unique solution valid for all time. This solution exhibits a periodic behavior with an energy dependent period T(E). See example a in Table I. Example b: H = 2 1 3. The equations of motion lead to 42 = 2E .q 3 and E can assume any real value. Since q3 < 3E, we find that + dq t+c. 2E 7 12, 3 This solution has the property that q diverges, q + for finite times, for any nonzero E; see the solid line of example b in the table. The indicator of this behavior is the observation that the integral converges as the upper limit goes to  oo leading to a finite value for the right side, namely, a finite value for the time. Actually, this same trajectory diverges at an earlier time as well. As b illustrates, the particle comes in from q = oo at some time, say t = to, and returns to q = 0 at a later time, say T + to, where .3E)1/3 dq T=2 j .q 2 v3 For any nonzero E, T < o0; in particular, by a change of variables, we learn that T= 1/2 1/3f dx 11/6 sL X3' where S(E) = E/ I E I, the sign of the energy. As I E I + 0 the particle spends more and more time near q = 0, until, at E = 0, it takes an infinite amount of time to reach (or leave) the origin. Example b in the table illustrates the generic situation for E 0 and finite T. How could one possibly expect the quantum theory to persist for all time when the generic classical solution diverges at finite times? The only possible way for this example to have a genuine quantum mechanics is for the particle to enjoy a fulltime classical solution. And to achieve thatand this is the important pointwhenever the particle reaches q = oo, we must launch the particle back toward the origin with the same energy, once again follow the trajectory inward, and then outward, until the particle again reaches q = oo, when we must again launch the particle back toward the origin with the same energy, ..., and so on, both forward and backward in time ad infinitum. In brief, to get fulltime classical solutions (as needed to parallel full time quantum solutions) we must recycle the same divergent trajectory over and over again in an endless periodic fashion (with period T(E) ); see b in the table with the solid and the dashed lines. Example c: H=pq3. The equations of motion, in this case, lead to 4 =q3, with the solution q(t) 1=1/2t. The solution is valid for t < c/2, diverges at t = c/2, and becomes imaginary for t > c/2. See example c in the table, where we chose the constant c = 0. In no way can an imaginary solution be acceptable as a classical trajectory. We cannot relaunch the particle once it has reached q = o at t = c/2 following the kind of solution we have found, the only solution there is, as we were able to do in case b. In short there is no possibility to have a fulltime classical solution in the present example. If there is no fulltime classical solution, then there should be no fulltime quantum solution, and that is exactly what happens. The corresponding quantum Hamiltonian may be chosen as Hermitian, but there are no technical tricks that can ever make it selfadjoint. Once again, the signal of this quantum behavior can be seen in the classical theory, i.e., a divergence of the classical solution followed by a change of that classical solution from real to imaginary. With such a classical symptom, it is no wonder that there is an incurable quantum illness. 2.3 More General Examples In what follows we discuss a number of hypothetical classical Hamiltonians, examine qualitatively the nature of their classical solutions, and address, based on the thesis illustrated in Sec. 2.1, the problem of making an Hermitian Hamiltonian into a selfadjoint one, if that is indeed possible. Let us dispense with the "healthy" cases at the outset. Whenever the classical equations of motion admit global solution for arbitrary initial conditions, then the quantization procedure is unambiguous (apart from classically unresolvable factorordering ambiguities). Observe that the existence of such global solutions is an intrinsic property of a Hamiltonian independent of any particular set of canonical coordinates. 2.3.1. Examples of OneDimensional Hamiltonians Example d: H=2 2q4 2 qf 2E = t + c. f V2/ +q4 E can be any real value. For E > 0, we notice that f dq / f2 E+q4 converges to, say T (E), which means when the particle travels from to + , the time interval is T (E). For nonzero E, T is finite; see the solid line of example d in the table. In order to get a fulltime classical behavior, we must send the particle back when it gets q = + ; see the two distinct dashed lines of d in the table. In quantum theory, this situation corresponds to adding two boundary conditions (at q = c), which are required to extend H to be selfadjoint. For E < 0, the paths do not cross the origin, but there is no essential difference in the quantum behavior from the case E > 0. That means we still need to add two boundary conditions to extend H to be selfadjoint. Example e: H = 2 q2 2 J d t + c, (1/f2)ln q+9T+ E= t +c. When the particle travels from to + o, the time spent by the particle spans the whole taxis (see example e in the table), so there is a global solution for an arbitrary initial condition. Thus the quantum Hamiltonian is unique and self adjoint. Since there is no periodic behavior, the spectrum of the quantum Hamiltonian is continuous. Example ft H=2pq2, 4 = 2q2, 2q = 1/( t + c). Global solutions exist (see example f in the table) and the quantum Hamiltonian is selfadjoint. Since there is no periodic behavior the spectrum is continuous. Example g H =2 p (1/q), consider 0 = ( o, 0) or (0, + o), 4 = 2(1/q), q2=4t+c. similar to Example c, q becomes imaginary for t < c /4, and there is no possibility of having a fulltime classical solution, just as in the case of example c. Therefore there is no way to extend the Hamiltonian to be selfadjoint. Example h: H=p2+2pq3, 4 =2 (p +q 3) =2 + E. Its classical solution is similar to example d, and is identical to H = p2 q6, to which it is canonically equivalent. Example i: H=p2+2p/q q 0, so Q = ( c, 0) or (0, + ), = 2(p + 1/q) = 2 E + 1/q2, 1 + Eq 2 = 2 E( t +c). The solution to the motion reads (2E)2(t + c)2 E q2 = 1. See example i in the table, where we chose E > 0. Since q = 0 is a singular point, we divide 9S into Q = ( , 0) and (0, + cc). For example, when the particle reaches q > 0+, we have to send it back or cross over q = 0 to get a fulltime classical behavior. By choosing c, the fulltime classical picture may be like the solid and dashed lines of i in the table. In quantum theory, one boundary condition is needed to extend H to be selfadjoint. Two additional examples (I and m ) appear in the table without discussion in the text. 2.3.2. Examples of ThreeDimensional Hamiltonians To simplify the problem, we consider here only central potentials V(r). In suitable units, the quantum Hamiltonian becomes ^= 2 H= V + V(r), V _i r2 UL r2dr dr r2 Set u(r) =) Ylm, then Hr =. + + V(r) for u(r). dr2 r2 The condition that I y 12 d3r converges now becomes the condition that folu(r) dr converges. For V(r) = A /rn, n = 1,2,3,..., A >0, Hr = pr2 (A /rn) + (1 ( +1)/r 2) and the classical equation of motion is t = 2 VE + (A /rn) (1 (I + 1)/r2). Under both cases of E < 0 and E > 0, we see the difference between n = 1 and n > 2. For n > 2, near the origin, i.e., r ~ 0, i= 2 VE + A /rn, so r could go as near as r ~ 0. See example k in the table. In quantum theory, one condition is required to extend the Hamiltonians to be selfadjoint. It corresponds to one parameter (e.g., see parameter B in Ref. 4), which is needed to specify quantum solutions. But for n=1, i = 2 E + A/r (/ + 1)/r2; with E < 0 and I > 0, the motion oscillates between the two roots of the expression under the radical, namely, + A 2 1(1+1) r+ + 2(+1) 21EI 4E2 2 E 2E 4E2 IE So it avoids the singular point r = 0, and therefore it has a fulltime classical evolution (see example j in the table), thus its quantum Hamiltonian is a self adjoint operator. That is why the hydrogen atom does not have any problems. CHAPTER 3 SELFADJOINTNESS OF HERMITIAN HAMILTONIANS Having recalled that simple and physically natural classical "symptoms" are available to diagnose any potential quantum "illness," we now discuss some of the standard techniques in Chapter 1 used to analyze Hamiltonian operators and confirm that the classical viewpoint presented in Chapter 2 is fully in agreement with conventional analyses. In particular, we analyze in this chapter the selfadjointness of several Hamiltonians which appeared in Chapter 2 by calculating their deficiency indices directly. Therefore, we can verify the consistency of solutions of classical and quantum Hamiltonians. According to Theorem 1 on page 5, we have three different cases of the deficiency indices: (i) n+= n = 0, (ii) n,= n. 0, and (iii) nl+ n.. Cases (i) and (iii) are simple, where either a densely defined operator is or is not selfadjoint, but case (ii) is much more complicated. Discussions such as how to add boundary conditions on the wave functions in order to extend the Hamiltonians to be selfadjoint, and what kind of physical interpretation those extensions imply, etc., will be presented in 3.2 and 3.3. 3.1 Three Different Cases of Hermitian Hamiltonians In this section we will apply Theorem sl & 2 to four general types of Hamiltonian, just as we did in 1.2 for H = p 2 + X3. (1) H = p2 bxm, (2) H = p xm + xm p, (3) H = p2+pxm +xmp, (4) H=p2+xpm +pmnx where m is a positive integer and b is a real constant. Those four types of H cover all the Hamiltonians discussed in chapter 2. 3.1.1 H=p2bxm Obviously, H is Hermitian. Similarly to what we did for H = p2 + x3 in 1.2, let with p = id /dx, j+ bx m)=Tiv. As both b and m are real, V. = Thus, n+ = n. = n. So consider Vy+ only; we write The asymptotic behavior of this equation as x is given by The asymptotic behavior of this equation as x > is given by since we require v (oo) = 0. The solutions are given by Bessel functions: (X) ~ Z1/(m + 2)( 2 f m+2) m+2 where Zv ~ Jv, Nv, Iv, or Kv, which are Bessel functions of the first and second kinds, respectively. To examine whether or not J I Vy(x) I 2dx converges at x oo, we need to use the following asymptotic forms of the Bessel functions: lyl 4+oo, v(y) ~ cos [y (v +1] Nv(y) 2sin [y (v+ Iv(y) 1 exp(y) Kv(y) exp(y) If J I yix) 12dx converges, then V(x) e D(Ht) c A9. Examples: (a) H=p2 +x4 : near x =ithus, near I xl ~ + 0; thus, V x4y= O. Then b = 1 and m = 4. We should choose Z ~ I, K (with y = 1/3 1x 13), the Bessel functions of the second kind. For x ~ o, p(x) ~ K1/6(a), 3 so n+,L.0 = 1 n+ =n+,+. + n+, 2 =1 +12=0= n., i. e., (n+, n.) = (0, 0). Thus, H is selfadjoint in L2(( oo, + oo)), which makes possible the global solution of the Schrodinger evolution equation (see Sections 9.6 and 9.7 of Ref. 3); the existence of the bound states Vp and Vy. implies a discrete spectrum of energy. (b) H=p2 x4 : y" +x4, = iW Here b = +1 and m = 4; thus 2)(X) ~ xJ1/6 (3) and AN1 6(1x3) ~ (1/x)cos or sin (1x3 r). 3 3 3 3 It is easy to check that both of the solutions are in L2, so n+, + = 2 = n+, ., as is symmetric on x. n+= 2 + 22 = 2 = n., i.e., (n+, n.) = (2, 2). Two boundary conditions are required to extend H to be selfadjoint, and a discrete energy spectrum is expected. Indeed, even a WKB analysis leads to a discrete spectrum. Iv will make the integral diverge, and therefore is not allowed. (c) H = p2 x2 (p2 + x2 is similar to p2 + x4 ) v/ +x2 = i where b = 1 and m = 2. 2)X) ~ 1/4(2) and INi,4(ix2) ~ l12os or sin (4 ) IYo 4 4 4 8 but neither of them are in 2(0f); therefore, (n+, n.) = (0, 0) and H is selfadjoint, but its spectrum is continuous because no bound states exist. Again, this conclusion follows from WKB arguments as well. (d) H = p2 + x (p2 x is basically the same ) V+ xV+ =i+ v+ (x i))v+ = 0. For x > 0, b = 1 and m = 1, IY+(x) ~ iTK1/3(2x i)3/2), which is in L2. But for x <0, b = + 1, and m = 1, V1," 2(x) 1~ [1TTi/3(34W +i)32)3, Ixr1 +Ni/3(2jWx +i)3/2) 1/W1/4 where we have ignored the unimportant phase term. Both of the solutions are not in L2 ( the square integral of i.+ diverges at x ~ o). That means there is no solution for x < 0, and no solution could be connected to iV+, x >0 either. Therefore, we have (i) (n, n.) = (0,0) for = ( 0), (ii) (n+, n) = (0, 0) for = (oo, +oo), (iii) (n+, n.) = (1, 1) for 12 = [ 0, + ). The spectrum is expected to be continuous in (i) and (ii), but discrete in (iii). This is an interesting system. In section 3.2, we apply it to a particle in a gravitational field and a uniform electrical field. 3.1.2 H=pxrm +x mp By substituting p ~ i d/dx and using [ p, xm ] = imx m 1, we get (p xm + x mp) /p = (2p xm imx m1 )I/= V 2xm V = (T1 mx m 1) (i) m = 1, 2x Vf = ( T1 1)I+, x .=0 with Vi 1/x., which is not in 2 since it diverges too rapidly at x ~ 0, and Y = const., which is not in L2 either. Therefore, (n., n.) = (0, 0) Thus, H = p x + x p is selfadjoint and its spectrum is continuous. (ii) m > 1, 2// v/ = (T1 mx m )/xm ~t m/2exp ( 1 xl m) As x ~ o, ~ 1/x" m/2, which are square integrable, so o are regular points. But now x = 0 is a singular point. Checking V near x = 0, we find m = odd, n. = 1, n+ = 0, which has no selfadjoint extension. H is not self adjoint and has no complete evolution, just as was the case for H = p x3 + x3 p in Chapter 2. m = even, n. = n+ = 0, H is selfadjoint with continuous spectrum. See the example of H = p x2 + x2 p in the previous chapter. It is interesting to point out that for 0 = ( , 0), (n+, n.) = (0, 1) if m = odd and (1, 0) if m = even; for 1 = ( 0, + c), (n+, n.) = (0, 1) if m = odd and (0, 1) if m = even. Therefore, H is not selfadjoint in both 2 = ( o, 0) and 1 = ( 0, + ), no matter whether m is even or odd. To explain that by the classical picture (see example c in the table), we say that neither of those intervals could contain a complete temporal evolution. 3.1.3H=p2+ pxm +xmp In the above section, we found that p xm + xm p is not selfadjoint if m = odd. Here we will show that by adding p2 to it, the situation will be changed dramatically. First let us examine the classical solutions:* H=p2+pqm +q mp=p 2+2pqm 4=2p+2qm=2 VE+q2m (i) If m = 1, the solution is like that for H = p 2 q2, where H is selfadjoint. (ii) If m > 1, the solution is similar to that for H = p2 q4, where boundary conditions must be added to make H selfadjoint. Second, let us see that quantum features of H: H=p2+pxm +xmp=p 2+2pxmimx m1 (p2+ 2p x m imx m 1)V = i + 2i x m + imx m 1 / = T iy The asymptotic for near x ~ is V + 2i xmy/+ imx ml1 i = 0. * H =p2+2pqm,2p=2qm1+2 7E+q2m 2p+2pm=+ 2VE+q2m We introduce an equation with the solution given by the Bessel functions, i.e., 0 +[(1 2a)/x T 2ifyxry1]0 + [(a2 vy2)/x2. ij( 0 = xaexp (ipx 7) Zv(Px x). Note that V just fits this equation by choosing y=m+1, =l1/7, v = 1/2(m + 1), ~ ~VWexp(i1xm +1) Zv( 1 ( xm+ 1)~ 1/ m +1 m +1 which is square integrable when m > 1*. So n+, = n., ,. = 2, and (n+, n) = (2, 2), when m > 1. So by adding p2, H always has selfadjoint extensions no matter whether m is even or odd. This is exactly consistent with what we have just discussed from the classical point of view. As for the case m = 1, i+ ~ 1/if, which is not square integrable, so (n+, n.) = (0, 0), which again agrees with what follows from the classical point of view. 3.1.4H=p2+ xpm +p mx Classical method: * Here Zv Jn, Nv. a = 1/2, p = aH/q = 2 p m which is similar to 4 = 2 q m based on H = 2 p qm; therefore, when m = even, H is selfadjoint; when m = odd, H is not selfadjoint. Quantum method: x 4 i d/dp, [x, pm] = imp m1 (p2 + x p m + p mx )(p) = i /(p) 2ip my, = (i p2 imp m 1)y+ ~ (1/V ) plm)xexp (m 2(m 3) p 3m (3.1) For m > 1: (n+, n.) = (0, 0), if m = even, (n+, n.) = (1, 0), if m = odd. For m = 1, (n+, n.) = (0, 0) which has already been discussed in Section 3.1.3. We see again that the classical picture is consistent with the operator analysis. It is heuristic to observe that the last factor in Eq. (3.1) comes from the part p2 in H, and it only induces a phase in the solution, therefore it would not change the convergence or divergence of the square integral, i. e., the selfadjointness of p 2 + x pm + pmx is exactly the same as x pm + pmx, and the latter one turns out to be the same as p xm + xmp. In other words, unlike the p2 in p2+ p xm +xmp, the p2 in p2 + x p m + p mx will not change the properties of selfadjointness. 3.2 The Application of Boundary Conditions and SelfAdjoint Extensions at Regular Points Examples: (a) A = p = i d/dx, with 2 = [b, a] p+= iV V = exp (T x). Thus, n. = n = 1, apply theorem 3 to this case, we have fl= C1i+ + C2V, C1, C2 C C 0 = (f, f = ( dfi l/dx, fl) (f, i dfl/dx) = i [ fifl* ]I = i [ f (b) f(b) f (a) fi(a)]. Set fi(b)/fi(a) = z; we have z z* = 1, so z = exp(i0), 0 R. For selfadjoint D(M) = [Vx): ( V, f) = 0 ],i.e., Wp(b) fi(b) W(a) fi(a) = 0 (b) / (a) = exp (iO) (3.2) let us look at the eigenfunctions of p:  ~ exp (iAx) Vi;(b)/ V(a) = exp (iA( b a)) Vf, has to satisfy (3.2) to be in D(M); so we get A(ba)=0+22rn n = 0, 1, 2, ... ,A=(0+2rnn )/(ba). The case of 0 = 0 is what is usually considered in quantum mechanics, and is called the periodic boundary condition. (b) H = p2 + x, with S2 = [0, + o) As we already discussed in (d) of Section 3.1.1, n+ = n = 1 for 2 = [0, + c). Therefore* 0 = (fl, f) = ( p2f +xf, fl) (fl, p 2fl +xf) = (p2f, fl1) (f, p2f1) = (ft f fI f (0)f(0) fl(0)fl (0). fi(0)/f(0) = [h(0)//f(0)] = tan 0 0 E 9t, D(M) = [*K(x): ( V, f)= 0 ], so the boundary condition at x = 0 is V(0)cosO V'(0)sin0 = 0 limx [fif/ flfl* = 0, since when x + , v/+ = f K1 /3(2(x )3/2), 3 h= c pV+ + c2Vf (3.3) In fact, at any regular point, the form is the same. Consider a particle with 1/2 unit mass moving vertically in the earth's gravitational field and set x = 0 at the surface of the earth, D = [0, + ). Then its Hamiltonian is given by H = p2 + mgx =p2 +x, by setting g = 2. It is interesting to note that this system has a natural selfadjoint extension set by y(0) = 0. That is the case of q = 0 in the above boundary condition (3.3). Additionally consider a physical system of an electron in a uniform electric field E applied in the + x direction. The form of the Hamiltonian is then H = p2 + x with S2 = ( o, + oo). For such a system, no special boundary condition is needed since n+ = n. = 0. 3.3 The Application of Boundary Conditions and SelfAdjoint Extensions at Singular Points In this section, we will look at the effect of an attractive central potential. We discussed in the classical point of view that there is no trouble for systems with nonrelativistic Coulomb potentials, since they have a full time evolution. Here we will see that this holds because they are selfadjoint. We will also find selfadjoint extensions for the superattractive potentials [V(r) = A/r n, n > 2]. After separating the angular part from the radial part, we have Hr = d 2/dr2 +1 (I + 1)/r 2 + V(r) which acts on u(r) with the requirement that Jo u(r)r dr must converge. (i) V(r) = A/r, A > 0 Htr = iu , becomes, at small r, S+ /r (I + 1)/r2) =0 u ~ Z (2+ 1)(2 r) ~ (21f) (2+ 1). Thus, with I > 0, nt0o = 1. At large r, u iu = 0 and its is not hard to find that n+,+o= 1. Therefore, (n+, n.) = (0, 0) for H with a Coulomb potential. (ii) V(r) = A/r2 u + ( +(i+A/r2 1 (1 + 1)/r2)u = 0. At small r, the eigenfunction UE and u+ have the same form: u +(i+/r (1)/r2) =0 u+ + ( i + I /r I (I + 1)/r2)u+ = 0 SA l(l+1) u + r2 u= 0 72 u, uErP, p=1/2 1/4(A(l +1)) For an swave (I = 0) with A > 1/4 or A 1(I + 1) > 1/4, we have u, UE r 1/2sin ('ln r ), r1/2cos(A'ln r ) where A' = [A (I + 1) 1/4]1/2. Then n, 0.= 2. With n,. = 1, we get (n+, n) = (1, 1). Now let us construct selfadjoint extensions: at small r, fh = cl rl/2cos(A'ln r ) + c2r 1/2sin(A.'n r). Using ( ,yr = limr 0 o [ ''* V'], and noting that ( ) is a bilinear form on D(Ht) xD(Ht), and(g,, h )= (h,g ), we have (rl/2sin('ln r), r 1/2sin(A'ln r) y = 0 (r1/2cos(A'ln r ), r 1/2cos(A'ln r) )b = 0 (r 1/2os(A'ln r), r 1/2sin(A'ln r) y = A' Thus, 0 = (fl, f )= (c2 ClC2)', C1C2= real. Consequently c2/c1 is also real. Set c2/cl = tan 0, where 0 e 9t; then fi ~ rl/2cos(;'ln r 0) The eigenfunctions have the same form at small r, i. e., UE = CErl/2os(9'ln r 0) in order to satisfy (uE, fl )= 0. In Ref. 4, the author found the dependence of eigenenergy on the choice of 0. Because of 0, we have a oneparameter family of selfadjoint extensions. Further, D(M) = [ u(r): (u, fl) = 0 ], and therefore the boundary condition at r = 0 is given by limr o u'(r)r 1/2cos('ln r 0) u(r) (r/2cos(ln r 0)) =0 L dr J (iii) V(r) =A/r", n > 2, uE +(E +A/rn (1 + 1)/r2)UE = 0. at small r, UE +(a/rn)UE = 0 uE~ #Z.1/(n+1 i (2)/2 Using the Bessel asymptotic forms, we get UE rn/4sin(yr (n 2)/2) rn/4 cos(yr(n2)/2) where y= 2 /(n 2). u have the same form, so nt 0o. = 2, (n+, n.) = (1, 1). The boundary condition is similar to (ii) fi = ci rn/4 sin(yr (n 2)/2 ) + c2rn/4 cos(yr (n 2)/2) using Thus, ( rn/4 sin(yr("n2)/2 ), rn/4 cos(yr ("2)/2). = A1/2 (f, f, )=0 gives C2/C1= tan 8, f ~ rn/4 sin(yr (n 2)/2 + 0) uE = CE rn/4 sin(yr(n2)/2 + 0) D(M)=[u(r): (u, f)= 0 ], gives the requirement on u(r) at r + 0 : limr +o u'(r)rn/4 sin(yr(n"2)/2 + 0)) u(r) dr n/4 sin(yr(n2)/2 + 0))] = 0 L dr J 3.4. Conclusion By calculating the deficiency indices for several representative examples of Hermitian Hamiltonians, we have analyzed the possibilities in each case of extending the operator to a selfadjoint one. In so doing we have confirmed the 6e91 connection proposed in the previous chapter that relates possible extensions to associated characteristics in the solutions of the classical equations of motion. The purpose of this chapter has been twofold: on the one hand, by demonstrating that anomalies in the classical solutions to a dynamical system are reflected in anomalies of the quantum Hamiltonian, certain technical issues ( such as deficiency indices, etc. ) are brought into the realm of everyday experience. On the other hand, the intimate connection between classical and quantum properties should enable one to examine a given system at a classical level in order to assess what problems, if any, are expected to arise at the quantum level. 40 Table I, Summary of classical and quantum highlights associated with several model problems, Each model has a label, a classical li miltonian, typical solution trajectories, nature of those solutions, character of quantum Ilamlltonlan. spectral properties, and. In some cases. related examples as well, 1 he solid curve in the figure portion represents a typical trajectory or part of a trajectory In the case of a periodic orbit The dotted curve denotes an alternative typical trajectory, and the dashed curve denotes a periodic extension of the basic orbit. In d. the figure illustrates two possibly distinct periodic extensions. h is omitted because it is a related example of d. Example Classical Qualitative Graph Nature of Selfadjoint Spectral Related Examples Label Hamiltonian t(q) of equations Classical Quantum Properties m = 1,2,3,... H of motion solutions Hamiltonian t a P' + 4 global unique discrete p2 + 9 "* t ) one parameter I6 p' .  periodic family of discrete p9 qs s. 1 solution (one p' 92'"+ ) boundary condition) t c 2pq' = partially nonexistent none pq" ,m > 1 ,m = odd complex t two parameter family of d p'  periodic solution (two discrete q'" m>2 ' boundary p2 + 2pq," m > 2 conditions) pe 2 globall unique continuous p9 + 2pq ___^__ Clatonian Hamiltonian 21,/q"' ' = 0,I4 p' + 2p/q'" I x .,ip 1r thrnole 1 nhie I (Co'nutied } Table I. (Continued.)  IA ^1 Fig. 1. Classical trajectories for the simple Hamiltonian H=p in three separate coordinate domains: (a) a (b) O 