Quantization of the radiation-damped harmonic oscillator

ANNALS

OF PHYSICS

129,

Quantization

1-21

(1980)

of the Radiation-Damped BERTHOLD-GEORG Institut fiir Theoretische Physik Auf der Morgensteile 14, D-7400

Harmonic

Oscillator

ENGLERT der Universitiit Tiibingen Tiibingen, West German?

Received November 30, 1979; revised March 6, 1980

The Abraham-Lorentz equation for the harmonic oscillator is reconsidered. Quantization of the system is performed by applying Ostrogradsky’s formalism of generalized momenta to a Lagrangian which contains higher than first derivatives in time. The final six-dimensional phase-space does not allow the interpretation as phase-space of a threeboson-system but only as one of one boson, its antiboson and a “ghost”.

While studying classical electrodynamics we all learned that an accelerated charged particle must lose energy by emitting electro-magnetic waves (unless the acceleration is constant in magnitude and direction for all times which, of course, is a rather unphysical situation). In classical electrodynamics this phenomenon is usually described by introducing a self-force in the equation of motion of the charged particle. We then obtain the so-called Abraham-Lorentz equation [1] mi: = Fext +

m;i

,

(1)

where T, the characteristic time, depends on the particle’s mass following way: 2 e2 7-3x.

m

and charge e in the (2)

Within the framework of the Abraham-Lorentz model the equation of motion (I) can be derived under very simple and natural assumption. In particular, it turns out that the particle’s mass m is given just by its electro-magnetic self-energy and therefore must be renormalized if we have a point-particle. This last feature reminds us of what we do in QED. Again, when calculating radiation processes we use self-interaction terms and renormalized masses (and charges) quite in the same way as we do in the Abraham-Lorentz model. This is the main reason why the totally classical equation (1) describes some phenomena quite well. From this point of view we are justified in taking a further step and quantizing Eq. (1) in order to sum up all the radiation effects of QED in some way. A similar 1 OOO3-4916/8O/llOOOl-21$05.00/O All

Copyright 0 1980 by Academic Press, Inc. rights of reproduction in any form reserved.

2

BERTHOLD-GEORG

ENGLERT

attempt was made by Feshbach and Tikochinsky [2] to describe deep inelastic scattering by a dissipative system. Another aspect is the following: As we have to quantize an equation of motion that involves higher derivatives in time, we can hope to learn something about the problem of quantization of field theories with higher-order derivatives. Such field theories have been recently discussed in different contexts, such as confinement [3], general relativity [4] and even QED [5].

CLASSICAL

DESCRIPTION

Throughout this paper we shall confine ourselves to a one-dimensional oscillator. In this case (1) reads: mf = -kx

harmonic

+ m&f.

(3)

The general solution of the equation of motion is given by x(t)

where r, (-y

= qlePt cos(wt) + q2eMYtsin(d) + q,e+rt,

(4)

& iw) are the three solutions Q,,,,, of the characteristic equation mLP = -k

+ mr.Q3.

Using Vieta’s rules we easily find the subsequent relations: I-7 > 1,

r=&%-

1) >o,

w = & ((I% - 1)(3F7 + 1))1/2 > 0. In (4) the qi are arbitrary real constants depending only on the initial conditions. To avoid unphysical runaway solutions it is usually required that q3 vanish. This means that actually R + 29 + (1/Z+ w”) x = 0

(6)

would be the equation of motion. An equation of this type has been quantized by Feshbach and Tikochinsky [2]. But our aim is the quantization of (3) and therefore we do not generally make the assumption q3 = 0. For quantization we need pairs of canonical conjugate variables fulfilling a Jacobi algebra, and therefore first of all we need a Lagrangian which yields the equation of motion (3). As (3) contains the third derivative in time, there is no chance of finding a

RADIATION-DAMPED

HARMONIC

OSCILLATOR

3

“normal” Lagrangian, i.e., one that depends only upon t, X, k,..., (d/dt)Nx, because such a Lagrangian will always lead to an equation of motion where the highest derivative in time is even. An elegant way out is to consider both (3) and its time reverse. This trick was used in [2] to quantize (6) although it seems possible to use a Lagrangian of the form L = (F

3iz - $-

(w” + p) x) $?vt

for such a dissipative system [6]. However, in recent papers it was argued that such a Lagrangian does not describe a dissipative system at all [7]. Therefore we shall use a Lagrangian of the Feshbach-Tikochinsky type. It should be mentioned that their Lagrangian was already discussed, in the context of classical mechanics, by Bateman [8]. From this Lagrangian we obtain a Hamiltonian which is nothing more than the generator of the evolution in time. We cannot expect this Hamiltonian to teach us anything about the energy of the system. A Lagrangian yielding (3) and its time reverse mjj = -ky

- mrj;

(7)

is given by L = -$rnr(i+y

Obviously, appears:

- jijj) + m$$ - kxy.

we may also use an alternative L’ = -rnr%y

Lagrangian

+ rn@ - kxy.

(8)

in which only 2 but not p (9)

Both L and L’ possess advantages and disadvantages. L is symmetric in x and y, but involves an eight-dimensional phase-space, whereas L’ is not symmetric, but involves an only six-dimensional phase-space. As the general solutions of (3) and (7) contain six arbitrary constants, one might conclude that L’ is more advantageous for quantization. On the other hand the symmetry of L is quite attractive and promises easier calculations at the cost of too large a Hilbert space. The decision-which Lagrangian is the better one-cannot be made on this level, and therefore we shall deal with both Lagrangians. The fact that we have Lagrangians which give an equation of motion and its time reverse reminds us of the Lagrangian for the Dirac equation. Indeed both L and L’ are equivalent to d3 d2 y mrdt3 - mdt2 - k x 1

(

which has the structure of the Dirac-Lagrangian

4

BERTHOLD-GEORG

ENGLERT

In regard to this analogy it is obvious to interpret (7) as the equation of motion of the antiparticle when (3) describes the particle. However, a strong argument is that 7 does not change its sign if the particle’s charge does. So the antiparticle should obey the same Abraham-Lorentz equation as the particle. Nevertheless, after quantization we are led directly to a boson-antiboson interpretation.. It was Ostrogradsky who long, long ago (1850) showed what to do with a Lagrangian that depends on higher derivatives in time [9]. The equations of motions are obtained from the condition that the functional derivatives in the dynamical variables vanish. In our case

yields (7) and likewise 0 = &C/&y yields (3). The conjugate momenta for x and & are 6L PZ=8f=z

l!?L

6~

aL

pj:=J-g=ajE,

d aL ---9dt aa

and likewise for y and j. In particular we find p* = mj + rnTjj,

ps = -&mrj,

p9 = mk - mri?,

p+ = + imr.9,

(10)

when the Lagrangian (8) is considered, and

ph = mR - mrf,

when 15’ from (9) is used. The Hamiltonians

are given by

H = c Sp, - L = H(s, p,), 8

(11)

where s = x, 2, y, j or s = x, 3, y, respectively. As variables of the Hamiltonian the generalized coordinates x, &,... are to be understood as a priori totally independent. From now on we shall therefore have to distinguish between, e.g., 9 and (d/dt) x. Naturally 2 = (d/dt) x must eventually be fulfilled in some way. But in general there will be additional degrees of freedom that are obviously connected with the fact that the Lagrangian (8) involves a phase-space that is too large. To throw light on the situation we now concentrate on (8) and (10). Later on we shall see that there are great differences between the calculations using (8) or (9).

RADIATION-DAMPED

HARMONIC

OSCILLATOR

Equations (1 l), (10) and (8) give us the Hamiltonian

+ &rn~(iy -

in the raw form

@) - rn*j~ + kxy.

W)

In the first four terms we already expressed the fact that S in (11) means the time derivative of a generalized variable. These time derivatives have to be eliminated by

d

p=GPC

2

(14)

In (13) and above all in the last three terms of (12) several ji: and j appear which possess two different meanings: 2 = (d/dt) x and 3i as generalized variable. It is not known a priori which 3i has which meaning. Therefore we substitute all 3i and j by expressions of the type

with arbitrary real c. Substitutions at different places allow us to use three different c (not six, since substitutions for & and j can be made symmetrically in virtue of the symmetry of L). Two of the three c’s can be inserted uniquely, requiring only that the Hamiltonian equations of motion

lead to (10) after setting (d/dt) x = ff, etc. However, the third c is really arbitrary. Each choice yields a Hamiltonian that produces the correct equations of motion for x and y. In particular, we find that in the fifth term of (12) we must set

whereas in the sixth term

are the correct substitutions. (13) is to be modified in the following way

2(1 - c) mi-

Ps >

6

BERTHOLD-GEORG

ENGLERT

where c is arbitrary. The obtained Hamiltonian

is

(POP+ - PIP,) + Q$

H = & - v

(p&

Ps P+ + ; (PO3 + P,P)

- pj$) - mc@ + kxy.

(15)

With K = (4c - 1)/T

we can rewrite H and find H = ;

(P~P$ -P&P,)

+ &P,P,

+ ; (pa!* + PY.9)

+;(p+p9+$q+kxy

(15’)

This shows that the arbitrariness in H is closely connected with the fact that 2 # (a/&)x. For, if we set 2 = (&It) x and use the Hamiltonian equations of motion, then the term proportional to the arbitrary K vanishes in (15’). Obviously, reconstructing the Hamiltonian using equations of motion is not permissible. Let us now have a look at the equations of motion. From d i3H p=ap,>

d aH p=ap,,,

g=-E aY ’

we derive, after some simple manipulations, d pv=m~x-mm7~x, p+=mr

(

d2

d ;f7X--k”, (

d2 d3 mdt2-mrz+k

(16)

1 1

x=0

as well as

(&+K)(&-fx)

=o.

(17)

Equations (16) repeat what we know from (3) and (lo), whereas we learn from (17) that k(t) = J$ (t) + e-Kt (f(0)

- $

(0)) .

RADIATION-DAMPED

HARMONIC

OSCILLATOR

Hence, the initial condition 3(O) = g

(O),

or with (16)

4 k(0) - ;p$(o) = 0, suffices to guarantee that for arbitrary

K we

(18)

have for all t

Le.(t) = g

(t).

It turns out that it does not matter what we choose K to be equal to. The initial condition (18) must be made for every choice. Summing up what we have learned so far, we can say that the Hamiltonian (I 5) describes the same physics that are described by the Lagrangian (8) if we only remember to solve the equations of motion with respect to the initial condition (18) and likewise

; p*(O) + y-$(o) = 0.

(18’)

Within the framework of classical mechanics there are no problems left connected with the Lagrangian (8). But can we repeat the above procedure when dealing with the Lagrangian (9) ? L’ from (9) gives, together with the momenta from (lo’), the raw Hamiltonian (1% Trying to eliminate

the time derivatives, we immediately

notice that we can express

(d/dt) y and (d/dt) % by

whereas (d/dt) x camrot be written as a function of the momenta at all. Therefore we can only try to set (d/dt) x = pi and hope that the obtained Hamiltonian is meaningful. We get (22)

8

BERTHOLD-GEORG

The Hamiltonian

ENGLERT

equations of motions are

d 1 z n=---p;+;*,

1

(23) &p;

&=-&P;.

= -kx.

The set (23) indeed delivers (3) and (7), and above equations of motion. So H’ describes the system well. sical. No special conditions, like (18) above, must be dimensional as it should be. But we paid for all these canonical step from (19) to (22).

QUANTIZATION:

Quantization

APPROPRIATE

all 3i = (d/dt) x is one of the All solutions of (23) are phymade. The phase-space is sixadvantages with the not quite

COORDINATES

is performed by [s, s’] = 0,

b, P/l = if%,, 7

[P, 9Pdl = 0,

(24)

where s, s’ = x, $, y, j for H and s, s’ = x, 3i, y for H’. In (24) we do not follow Hayes [lo] who argued that one should replace fi by h/2 if the Lagrangian containing 8. But then Heisenberg’s equation of motion

$A =;[A,

HI

(25)

would not be formally identical with Hamiltonian’s equations, i.e., the correspondence principle would be violated. For this reason we quantize as in (24). For the subsequent considerations it is easy to use appropriate coordinates qj and pj . The qi are defined by (4) and in the case of H additionally by 1 dx i(t) = -$ (2) + e- h; q4 .

(26)

For H we find

where

(27)

+1

W=

0 SW7 ;;: +WT -1 \ -1 2yT 2yT +&,T +&wT i +1 +WT 0

+1 +rT -2yT

+*I-, +$rT -2yr +1

0 +1 0

-40 1 -4

RADIATION-DAMPED

HARMONIC

9

OSCILLATOR

and for H’

The conjugate momenta to the qj are given by

(27')

and

respectively. Obviously, with the help of (24) we have canonical commutators for the new variables:

[Pj ?PSI-= 0

(24')

(.j, k -= I, 2, 3, 4 resp.j, k = 1, 2, 3).

Written in the new variables the Hamiltonians are

and

((A, B} : AB + BA). These Hamiltonians differ only in the additional last term in (29). This term is proportional to the arbitrary parameter K just like the last term in (15’). Indeed, these terms are the same in virtue of

Here we learn how to formulate the initial conditions (18) and (18’) in quantum mechanics. We must fulfill q4

= 0,

P4

=

0

(30)

10

BERTHOLD-GEORG

ENGLERT

which are, of course, weak identities. Fulfilling (30) means reducing the eight-dimensional phase-space to a six-dimesional one which contains only physical states. The facts that both Hand H’ describe the same physics and are connected by

H=H’+h,q,: suggest formulating (30) by simply throwing away the last term in (29) and totally forgetting about it afterwards. That in effect is what we would have had to do if we were working with Dirac’s calculus of constraint Hamiltonians (see, e.g., Ill]). However, let us discuss three possibilities of formulating (30). First, one could impose a Gupta-Bleuler condition on the states, i.e., one demands for all physical states 1 $) that they fulfill 94

I $J) = 0,

P4 I $>

=

0.

But as the commutator of q4 and p4 is i& it turns out that only 1 $) = 0 can satisfy these conditions. Hence, this first possibility is not sensible. Second, weaker conditions than the preceeding ones are

for all physical states 1 #). However, the states obtained in this way cannot be used as physical states too, because the set is not linear. If we have two such physical states 1 #> and 1 v) then an arbitrary linear combination 01I #) + p j y) need not be in general another physical state. Only in the case of

and

will this linear combination be a physical state. So, in virtue of the lack of linearity of the obtained set, this set cannot be used as a set of physical states. Third, since we learned from the above discussion that one should have

for any two physical states 1 $) and 1 v), we just require that. Obviously, it is impossible to decide with the help of (30a) alone whether a given state is a physical one or not. But there is a natural way of making the physical set X9 unique. Just require additionally that after evolving in time a physical state I 4) is still physical, i.e., for all t

RADIATION-DAMPED

HARMONIC

OSCILLATOR

11

and require that a certain class of states is physical. For physical reasons this class is to be taken as the q4 , p4-vacuum, i.e., if (PI - inwqdl 4) = 0, (3W I #)E=%D where w4 is a kind of internal frequency of the q4 ,p4-mode. So far we do not know how large w4 is, since the anticommutator {pI , q4) that occurs in H does not tell us anything about it. We shall be able to fix w4 in the next section when considering time reversal. Indeed, Eqs. (30) determine uniquely the linear subspace X9 of physical states. Hence, we have successfully eliminated the arbitrariness of the free choice of K in H. One might think that this arbitrariness implies a certain “gauge invariance,” K being the “gauge-fixing parameter”. However, it seems to us that the truth is much simpler: K is a Lagrangian multiplier for the constraints (18) and (18’) or (30). It may be misleading that we did not manage to fix the value of K, as it is usually possible for Lagrangian multipliers, however, this is insignificant for our reasoning. To affirm this interpretation of K let us mention that K shows up in H only but not, e.g., in the matrices of (27) to (28’). TIME

REVERSAL

In the second section we introduced y as the time reverse of x and obtained a Lagrangian which has in principle the structure of the Dirac-Lagrangian with x playing the role of $I and y that of $. Therefore the basic relations of time reversal (indicated by the tilde -) are f = y,

2 = -j,

2 = ji, etc.,

and, of course, r” = x,

j=-$

9 = 2, etc.

From (10) and (10’) we get the time reverse of the canonical momenta and a;: = -P:, 7

Due to these relations the Hamiltonians reversal, i.e., they are invariant. Furthermore Eqs. (27)-(28’) imply

$4 = nzr*‘.

H and H’ are obviously even under time

(33)

12

BERTHOLD-GEORG

ENGLERT

where D2 = (UT)” + (I% + r~)2. To diagonalize

(34)

(33) we use new coordinates Qj , Pi defined by (;I

=s(g:,

93 = Q3 3

P3

=

p3

’

t;:,

=S (2,

9

q4 = Q4 3

’ (35) P4

=

p4

>

where S=-

(D - CJ.IT)~/~ ( -(D + OJT)~/~

(&

(D + c.cT)~/~ (D - ,~)l/~ ) *

This is just a rotation in the l-2 plane and such rotations are generated by (plq2 p2ql), which is the first term in the Hamiltonians (29) and (29’) and commutes with H and H’. Therefore the Hamiltonians are left formally invariant by (35), i.e., H’ = w(P,Q2

-

P2QJ

y(QJ’,

-

+

P,Q,)

+

f

V’,

,

Qd

and

(36) H=

H’+{P,,Q,}.

Time reversal now means 81 =&PI, 0

e2=-&P2.

0

(37) g3

=

-L mm3

P,

)

Q4 =-‘P4. n-4

Here ~~,~.4are the internal frequencies of the different modes, in particular: w. = wD,

w3 = D/r,

w4 = l/T.

(38)

In (38) we find w4, which we needed in (30~). Time reversal was also considered by Feshbach and Tikochinsky [2], but their basic relations mean that all coordinates are even under time reversal and all canonical momenta are odd, i.e., P =

x,

j

=

Y,

10 = -Pa!,

5, = --par , etc.

RADIATION-DAMPED

HARMONIC

OSCILLATOR

13

These authors then found that their Hamiltonian consists of two parts, one even under time reversal, the other odd. A non-unitary transformation to another Hilbert space with a non-positive-definite metric, involving the time reverse of “bras” instead of “kets,” thus made the Hamiltonian non-Hermitian in this new Hilbert space. All this was done to obtain eigenstates of the Hamiltonian which do not exist in the proper Hilbert space as we shall see in the next section. These eigenstates belong to complex eigenvalues whereby the imaginary part can be interpreted to describe damping. However, since these eigenstates cannot be transformed back into the Hilbert space, which one had in the beginning, there is no one-to-one-identification possible. Hence, it seems to us that such a non-unitary transformation is not really useful and should not be performed. Further, it should be remarked that the assumption of just even coordinates and odd momenta is not even self-consistent, since it implies the relation

in [2] showing that f = (d/dt) x is not odd as the time derivative of an even quantity should be. Similar self-consistency arguments forbid even coordinates and odd momenta in the present paper. But there are good reasons that x and y should interchange under time reversal, as shown in (31).

TIME

EVOLUTION

For the calculation of the evolution in time of an arbitrary state it is fitting to introduce creation and annihilation operators. Following the method of Feshbach and Tikochinsky [2] we define a,=--

1

2(&?WJ,)1/2 ((Pl - imw,,QJ + i(P, - imq-,QJ),

~2 - 2~fimio,~12((PI

-

i/~woQJ

-

V,

-

hw,Qz)),

(39) ~3 = ~2hm)031,2V’3

-

imw3Q3>,

where w 0,3,4are the internal frequencies of the different modes from (38). In the sequel the wj in (39) are quite arbitrary as they never appear but if we want the aj to have simple relations under time reversal, we must take wj from (38). Obviously we have

[Uj, a,J = [Ujf, &+I = 0,

[Uj) qc+]= 6j, .

(40)

14

BERTHOLD-GEORG

The Hamiltonians

ENGLERT

(29) and (29’) now are

H’ = hw(u,+a,

- uz+uz) + hyi(u,+u,+

- u1u2) + $

(a;” - us”).

(41) H = H’ - $

(a:” - uJ2).

H’ acts on the Fock space to al, a,, us, whereas H acts on the Fock space to a, , a, , u3, u4 . In the second case the physical subspace .%?!, defined in Eqs. (30), is found to be built up by the states k, I, m, n = 0, 1, 2,...,

where I 0) is the Fock space vacuum. Let us now consider the following operators x, = ; (q+a2+ + up,),

x; = ; (a;” + u,2),

x4 = ; (uf + u42),

y, = ; (Ul’U2’

Y, = f (a;” - u32),

y4 = f (a:” - Use), (42)

- U1U2)’

z, = ; (Ul’Ul +

with commutation

a2a2+),

23

= ;

(%+a3

+

z4

u,u,+>,

= ;

(u4+u4

+ u4u4+>

relations

K , Y7cl= h2G

3

[Yj , Z,] = -i&,X,,

[Zj , X,] = -i&

Y,

(43)

(j, k = 0, 3, 4). Hence, we have a O(2, 1) @ O(2, 1) @ O(2, l)-algebra. The Casimir operators are c, = Zk2 - x,2 - Y,2; accordingly c, = (+&zl’Ul - u2+u2))2 - $ = ho2 - 4 (44) and c, = c, = -& = ($)” - $. So we can rewrite the Hamiltonians

as

H’ = 2Awh, + 2fzyY,, - 2WY3 ,

H=

H’+2hKY4.

(45)

Before proceeding let us have a look at the time reverse of uj , Xj , etc. Using (37) we easily find iTI = -ial+,

4 = -ia,+,

and

ri; = -ia,+ (46)

ii4 = +iua+

RADIATION-DAMPED

HARMONIC

15

OSCILLATOR

as well as xj = -xj,

q=

+yj,

zj = +zj

and

(47) I-& = +A,

affirming that the Hamiltonians in (45) are even under time reversal. Since (46) differs from the relations that are usually obtained for creation and annihilation operators, i.e., a” = --a’, we expect that difficulties will occur when interpreting the aj . This will become clear in the next section. In the 0(2, l)-algebras there exist common eigenstates to Z and C. No eigenstates exist to X and Y. Therefore the Hamiltonians, which consist of 2 resp. 3 parts acting on separate orthogonal subspaces, do not possess eigenstates either. This can be expected when one remembers that the classical problem has no periodic solutions. Moreover, the Hamiltonians are not bound, due to the part involving h, . So we have a rather unusual situation. This is probably connected with the fact that our Hamiltonians are just the generators of the time evolution but not the energy operators. The common eigenstates to C and Z in an arbitrary 0(2, I)-algebra are labeled byj and m in the sense of

Z I.Am>= (m+ $)Ij, m>,

(48)

C 1j, m) = (jz - *)I j, m).

In virtue of the relations [Z, x f iyl = qx

* iY)

(491

we see that for a certain phase-choice (X - iY)lj,

m) = ((m + 1)2 -.j2)1/2 lj, in + I),

(X + iY)lj,

m) = (m2 -,j2)1/2 / j, m - 1).

This shows that there is either a minimal value M or a maximal obeys M2 =j2 or (N + 1)2 =j2. Further, for all k = I, 2, 3 ,... the relations

lM+kl>lMl

or

must be valid, from which we learn M>-&>N. 595/129/I-z

IN+1--kl>lN+1I

(501

value N for m which

16

BERTHOLD-GEORG

ENGLERT

Therefore we cannot have (unitary) representations with both a minimal and a maximal value for m. Here it is obvious that 0(2, 1) is a noncompact group. Looking at the Z, in (42) we see that in our case we have only representations with a minimal value for m. In particular, for the common eigenstates for Z,,,,,, and C,,,,,, , for which we use the notation

Ijo,

*

mo9J3,m3,J4,

*

~4>~lljo,~o)Olj~,~,>OIj4,m4~,

the subsequent eigenvalues show up jo = 0, zk;, ztl, It%...;

Mo=ljo/;

j, = M3 = *t);

j, = Al4 = *i;

(51)

always mk = hfk, &fk + 1, ikfk $ 2,... (k = 0, 3, 4). The choice of phase which was made in (50) leads to the following relation between the 0(2, 1) eigenstates and the familiar Fock space base: (4 9 n2 9 123 3 n41j0,mo,j,,m,,j4,m,) --s n,.m,+j, s?+rn,4~ s n,.2m,+1/2

(521

Sn,.zm,+lle

.

Hence, we are still using the old Fock space base, but the states are labeled in a much more fitting way. In this notation the physical subspace .x& is characterized simply by j, = M4 = -a. Then, due to (52), only states with even n4 contribute. After these preparations we can now ask how an arbitrary state propagates in time. The main quantities are the time-dependent matrix elements

Go,ml,,A ,mi,.i4, m~lexp(-~~~)ljo,mo,j3,m3,i4,m,~ =e -2i50wt(j, , mh 1exp(--2iyfY,)I x
j, , m,)

(53)

j, , m3>.

We see that we have achieved our goal when we know in an arbitrary representation of the 0(2, I)-algebra the transition amplitudes %k*,m(u) = (j, m’ 1exp(--2iuY)(j,

m).

(54)

The calculation can be performed using methods known from the analogue for O(3) [12]. In the phase-choice (50) and for representations with m bound from below we obtain the result + m’ + 1) 4&&J) = (- Iy+ ((m - M,)! .F(m + Mi F(m + l)(m’ - MJ! f(m’ + Mi + 1))112 x (sinh o)~+~‘-‘~J (cash CT)~+~‘+~

F(Mj - m, Mi - m’; -m

- m’; coth2 0).

(55)

RADIATION-DAMPED

With the formal polynomials

identity

HARMONIC

between the Hypergeometric

Ppyx) = (” -; q(q)”

F(--n,

17

OSCILLATOR

functions and the Jacobi

--n - ,r; j3 + 1; *)

)

it is (m - Mj)! Qm’ + Mj + 1) lP ) (sinh u)~‘-*~ (cash cr)m+n’+l

’

phnd+&.

-(m+m’i1)

(cash 2~).

(55’)

(55’) is a more formal result as P:@(x) is well defined only for CX,/3 ;- - 1 and j x / _ I, and therefore one has to be careful. With the help of (55) we can now describe the evolution in time for every given arbitrary state. Tn particular, the behavior for very large t is of great interest. Let us have a look at the time dependence of a base state Ij,, , m, ,,j3 , m3 , j4 , ma>. We have

exp(- ~Hf)~.i”.1)1,.i2,m2,.i,,1)14)

For large (J we have from (55’): l)r(m’+A4+

(- l)“--M’ ( r(m+Mj+

“!!;,,&J) ““I,

I)

lie )

(m - M,)! (m’ - Mj)!

z2Mg+l 4?Mj+l)o

x q2A4j + 1) e Making

.

use of .<.j. ,l,’

1 +20X

/ j,

m)

=

(j,

m’

1 e-i(n/2)Ze-i20Ye+i(n!z)z

j,

,,,‘

we get, in particular, i

j.

,)1

i

e.

(n

2)

y

I,j,

M,)

--- (-

l)m-A'j

(j,

r(m

x

(

Mj

i e -(n'2)X,

j,

,)l:

+ Mj + 1)

(t?? - Mj) ! r(2

Mj

~.

2”jtl/?

1/Z

+ 1). )

and can write /I/i,,,,,,(g)

-+

USI

e- (2nrjt*)u

(1:

m’

/ &(n’2)x

) .j, Mj)(‘;

Mj

/ e-m(n ‘2)X / ,j, nl\.

18

BERTHOLD-GEORG

ENGLERT

Likewise we obtain q?&&J)

2-l

e@fj+l)o (j, iyz’ 1e-2)x

1j, &&)(j,

Mi j ef(n’2)x j j, m).

Here and below expressions like (j, m’ ( e(lr/zlX I j, m> must always be understood to be purely formal. They are not scalar products in the Hilbert space, but only the numbers that one obtains when one regards @i,,J u ) as an analytic function of cr and takes the value at u = i(z-/4). Indeed, CZ*(~/~)X1j, m) is not a vector in the Hilbert space as its norm turned out to be infinite: 11e*(nlzJx I j, m)l12 = (j, m 1 einx 1j, m) = UA,,

(

fi

$

1

= GO.

However, these pseudo-vectors are useful abbreviations. They allow us to put down the time evolution at large t in a very simple way. Namely, we obtain for large t (the upper/lower sign refers to K > O/K < 0)

I+= ----t

+(a’2)(xo-X3*x4) I.& , I.h I ,A ,.L , h , .jA

(56)

where R = (2 I h I + 1) y + G$, + 1) r + Pj, + 111K I > 0 and a& = (j, Mj / e+cnj2)x j j, m). We see that in a given representation, i.e., for fixed j, , for large t the system will always be described by the same pseudo-state and the same time dependence, independent of the initial state. Only the amplitudes a2m. depend on the initial state, i.e., on the mk , The pseudo-states e*(n/2)x I j, m) may be interpreted as pseudo-eigenstates to Y, as the relations Ye*(nlzJx j j, m) = +i(m

+ 4) e*(n/2)X I j, m)

can be easily proved [2]. 3ut formal manipulations based, for example, on the formal identity e+(mP)X 1j, m)(j, m j e-(n/2)X Q=C j.m

are at least doubtful because both these pseudo-states have infinite norm and the pseudo-eigenvalues of the Hermitian Y are imaginary. However, by such manipulations (56) is obtained in a straightforward way.

RADIATION-DAMPED

HARMONIC OSCILLATOR

NUMBER

19

OPERATORS

So far we have not touched on the problem of interpretation of the creation and annihilation operators defined in (39). We know that the Fock space built up by these operators is meaningful and useful for the description of the time evolution of an arbitrary state. Let us now consider number operators to count the “particles” created by the aj from (39). We denote them by Nj = aj+aj and notice the following relations: Nl N, N, N4

= = = =

h, + 2, - 4, 4, + 2, - 4, 22, - &, 22, - 4.

Their equations of motion are $Nl

=&N2

=2yX

$ N3 = --4rX,, ; N4 = +41cX., , with the solutions N,(t) = h,, - 4 + Z, cosh(2yt) + X0 sinh(2yt), N,(t) = --h, - & + Z, cosh(2yt) -C X,, sinh(2yt) as well as N3(t) = -4 + 22, cosh(2rt) - 2X, sinh(2I’t), N4(t) = -4 + 22, cosh(2rct) + 2X, sinh(2Kt). If we initially had exactly IZ~ ... n4 particles, I nl , n2 , n3, n,>, then the expectation values

i.e., if the initial

<# I Nj(t)l #> = n&l would be for large t: ndt> = n2(t) w $(n, -t n2 + 1) ezvt, n,(t) w *(n3 + +) e2rt, n,(t) w +(nl + 3) e21Klt.

state / #> was

20

BERTHOLD-GEORG

ENGLERT

So, even if there were no particles at all at the beginning, we would always have very many particles if we only waited long enough. This shows that such a straightforward interpretation is not reasonable. But at least for NI and N2 we can rescue the interpretation as N,(t) - N,(t) = NI - N, = const. The new interpretation is meant to understand the particles created by uz+ as the antiparticles to those created by a li. Then counting antiparticles by negative integers, the total number of particles is constant and an arbitrary number of particle-antiparticle pairs is created in time. However, for the objects created by u3imand a4+ such a reinterpretation is impossible. They are a kind of “ghost” particle which should not be observable. A theory describing only the physical boson-antiboson system is obtained when Eq. (6) is quantized instead of Eq. (3) [2]. Hence, in the end the physics reduce again to Eq. (3). Nevertheless, the “ghosts” are there and their influence on the time evolution shows up in the amplitudes a+, 3 and aj,,mI and in the contributions (2j, + 1) I’ + (2j, + 1)1K 1 to R in (56). As it is possible to do all the calculations starting with the Lagrangian L' and dealing from the beginning only with a six-dimensional phase-space, we can say that the a,-ghosts are artifacts of the formalism, whereas the as-ghosts are physical and cannot be avoided. Moreover, we can eliminate all artifacts stemming from the a,ghosts by the choice of K = 0 or equivalently by the use of the above mentioned formalism of “constraint Hamiltonians.” Hence, the quantum mechanical phase-space of a radiation-damped harmonic oscillator and its time reverse turns out to be the Fock space of one boson, its antiboson and a non-observable bosonic “ghost.” Of course, it is mathematically identical with the phase-space of the three-dimensional harmonic oscillator, but the interpretation is totally different. ACKNOWLEDGMENTS The author acknowledges the hospitality of the Institute of Physics, Jagellonian University, Cracow where the main part of the calculations was done. Discussions with Prof. Jerzy Rayski and the members of his staff were very helpful. This work was supported in part by the Studienstiftung des deutschen Volkes. REEERBNCE~ 1. J. D. JACKSON, “Classical Electrodynamics,” 2nd ed., Wiley, New York, 1975. 2. H. FESHBACH AND Y. TIKOCHINSKY, Trans. N. Y. Acad. Sci. 38 (1977), 44. 3. H. NARNHOFER AND W. THIRRING, Phys. Lett. B 76 (1978), 428; B.-G. ENGLERT, AND J. M. RAYSKI, JR., Phys. Lett. BS3 (1979), 399. 4. S. FERRARA AND B. ZUMINO, Nucl. Phys. B 134 (1978), 301. 5. D. ZWANZIGER, Phys. Rev. D 17 (1978), 457; 19 (1979), 3614. 6. For example, V. V. DODONOV AND V. I. MAN’KO, Nuovo Cimento B 44 (1978), several methods is given in R. W. HASW, .I. Math. Phys. 16 (1975), 2005.

J. KARKOWSKI

265; a review of

RADIATION-DAMPED

I-IARHONIC OSCILLATOR

21

J. Math. Phys. 20 (1979), 762; J. R. RAY, Amer. J. Pkys. 47 (1979), 626. Rev. 38 (1931), 815. Mem. Acad. St. Petersburg 6 (1850), 385; E. T. WHITTAKER, “A Treatise Dynamics of Particles and Rigid Bodies,” Cambridge Univ. Press, London/

7. D. M. GREENBERGER, 8. H. BATEMAN, Pkys. 9. M. OSTROGRADSKY,

on the Analytical New York, 1964. 10. C. C. HAYES, J. Math. Pkys. 10 (1969), 1555.

T. REGGE, AND C. TEITELBOOM, “Constraint Hamiltonian dei Lincei, Rome, 1976. 12. A. R. EDMONDS, “Angular Momentum in Quantum Mechanics,” Princeton 1975; J. SCHWINGER, On angular momentum, in “Quantum Theory of (L. C. Biederharn and H. Van Dam, Eds.), Academic Press, New York,

11. A.

HANSON,

Systems,” Academia

Nazionale

Univ. Press, Princeton Angular Momentum” 1965.