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On a Quantum System with Memory.

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AnnaIen der Physik. 7. Folge, Band 46, Xeft 1, 1989, S. [email protected]
VEB J. 8. Barth, Leipzig
O n a Quantum System with Memory
Karl Marx University, Leipzig, GDR
Dedicated to Professor Dr. G. Vojta on the Occasion of his 60th Birthdtcy
A b s t r a c t . We consider the integro-differential equation for the classiciil trajectory of itn oscillator coupled t o another one. On the quantum level the elimination of the coordinate A of the "unvi9,
,u)for the associated imaginary time stochastic
sible" oscillator leads t o an effective path integral
process t (-co, 0 0 ) --f z ( t ) . We prove reflection positivit,y of the measure d p z B . d t , where d t
governes the free oscillator z and F is the counterpart of Feynman's influence functional. Finally,
realizing the Hamiltonian semigroup exp(- t H ) , t 2 0, in the physical Hilbert s p a c e 2 = L 2 ( X ,r,p ) ,
2+,we try t o understand what is memory.
aber ein Quantensystem mit Gedachtnis
I n h i t l t s i i b e r s i c h t . Wir untersuclien die Integro-Different,iaIgleichung fur die klassische Trajrktorie eines Oszillators, welcher an einen zweiten gekoppelt ist. Was passiert in der Quantenmechanik, menn man die Koordinate des ,,unsichtbazren" Oszillntors eliminiert ? In imilginiirer Zeit
erhiilten wir ein effektives Funktionelintegral ( X , S, p ) fiir den assoziierten stochastischen Prozel3
t E (-m,
0 0 ) -+ z(t). Formal gilt d p % E' . d:. Hierbei beschreibt das Man d t die Dynztmik des freien
Oszillittors ,,x" und 8' entspricht dem Feynmanschen EinfluRfunktionaI. Wir zeigen, daI3 dji reflexionspositiv ist und realisieren die Halbgruppe exp(-tH), t 2 0, in.& = L2((s,Z+,i t ) . Diibei versuchen
wir zu verstehen, wie in der Quantentheorie Gedaclitnis entsteht.
1. Introduction
Excited by the beauty of the P(p),-theory in terms of a Markov process [ 11 mathematicians attacked more realistic models for fundamental interact,ions. The concept
of imaginary time path integral for gauge and E'ermi fields requires a linking with differential geometry and non-commutative probability. Actually, the main effort is to
quantize gravity wit'hin a string theory.
A powerfull tool to solve those models is the method of effective action. As example
let u s look a t t'he classical equations of niotion
for a system with two degrees of freetloni. Following ideas of Feynniaii aiicl Wheeler
[2] one niay try to eliminate the variable A . It is possible to describe the above caricature
of QED in one space-time tlimension purely in ternis of a stochastic process
t E (-00,cu) -+ x ( t ) with memory. Unfortunately [3], the Haniiltonian does not admit
Ann. Physik Leipzig 46 (1989)1
a normalized ground state 9 (nith a bare particle potential the propagator is unknown)
and hence the model does not fit the Osteritalder-Schrader like axioms of Klein [4].
To learn something belon we analize to identical oscillators “z” a i d “A” with harmonic
instead of “Beak” coupling as i n ( I ) . Let coo denote their eigenfrequency. Our textbook
Lagrangean reads
- A ) 2 , 22 0 ,
11 here 8, corresponds to tlie chase 1 = 0. Of course, in normal coordinates the dynamics
factorizes. This on the quantuni level allows us to \\rite clown immediately the spectrum
of the Harniltonian H
9 = 9o- -((.
+ n6: m, n E {O} U N ) ,
-nhere a = (oo, b = + l o $ + 21 and N stands for the natural numbers. Let $2 denote
spec(Hi)= (mu
the grouiid state. N’e claim that the system IS completely determined by the amplitudes.
(Q, !PI%),
ul,,= d t l ) pl(t2) .‘ . Y(t,L)9, n E N ,
and p l ( t ) = e i t l f z e - I r H . For tl, t2, . . ., t, in an arbitrarily
> 0 the M ave functions Y/,
small time interval of length
carry “enough” information about the unvisible oscillator A
T h e o r e m 1.
Let M , be the Abelian algebra on 2 = L2(R2,dx d A ) of bounded continuous functions
f ( x ) , and let P , = exp(-tH), t 2 0. Assume 2 > 0. Then the system defined by the
Haniiltonian I1 satisf tes the following conditions :
+ H ) - l / l r ( l + H)-’/”l[ <
52 is cyclic for M , V {P,, t 2 O),
For f,(x) 2 0, j = 1, 2, . .., n, holds
(9,f,P,f,I’, . .. I’J,,Q) 2 0 .
Proof: Condition I ) is the (:limrri-Jaffe bound [j].To see ii) we only must consider
tlie action of P,, 2 0, on coherent states Y = {exp i x x } 9 E X o .Finally, iii) follows
from the positivity Y,(xA, T’A’)2 0 of the Mehler kernel. End of proof. We remark
that the vectors Y = f(z)Q, f E M,, span a proper subspace X 0 in the total physical
. one
Hilbert space 2‘.In canonical systems the vncnuni 52 is cyclic just for N o [GI, 1.e.
can take E = 0.
1. Effective Action
On the classical level the elniiiriat ion of the variable A leads t o an integro-differential
equation for the trajectory t + x ( l ) . Let us f i s sonie time interval [O, a] and a Green’s
function D(s, t) for the operator K = co2
here 2w2 = (c2
the trajectory t --+ A ( t ) satisfies the boundary conditions
&(t) =
z’ D(s, t ) A(s) = 0
+ b2. We assixme that
Quantum System with Memory
( t ) = A2
J ds D(s, t ) z(s),
t E [0, a],
for initial data x(0)and k ( 0 )has a unique solution. Together with a(t)E 0 it is equivalent
to the coupled Euler-Lagrange equations resulting from ( 2 ) . To convince ourselves that
this is true we apply the different,ial operator K on both sides, use co4 - I? = u2 . b2
and the identity
+ j- ds D(s, t ) KA(s),
A(t)= a(t)
t € [O, a].
Unfortunately, there is no universal action Seffgoverning the classical trajectory
t E [0, a ] -+ x(t) (independent of the boundary conditions for “ A ” ) . Because of this
obstacle we must go beyond the conventional schema of quantization. At real time and
in units where Ti = 1Feynman’s ansatz for the functional integral reads {exp iseff)
It is defined by the causal Green’s function which produces complex boundary conditions A m exp(&itw), as t -+ -&a. Roughly speaking, “A” prefers to live in its own
ground state y . Let U(s,t ) , t 2 s, denote the evolution operator for the time dependent
Hamiltonian J ( t ) = -(A2+ co2A2)- h ( t ) . A of the quantum oscillator “A”, driven
we find a solution
by the external force Ax(t).Applying the principle of least action to Seff
+ w 2 4(t)
A(t, [.(.)I)
y , 4,[%(.)I) U(0,a) y ) ,
t E [O, 01,
= U(0,a) AU(0, t)*. The above serniclassical equation makes the
physical meaning of Seffevident. We may check it passing to imaginary tinie. The existence of an associated stochastic process t (-00,oo) 3 x(t) on some probability space
( X , E, p ) follows from the properties of Y,= q(tl)q(t,) . . . q(tn)9 shown in theorem 1.
We briefly recall Klein’s construction. Let f i E Mo, t, 5; t, 5 ... 5 t, and t,+, - t, =
e 2 0. By the Riesz-Markov representation theorem me can rewrite (5) as an integral
Sccorcling to Kolmogorov’s theorem these measures admit a joint extension dp to
the smallest a-algebra B containing all 5 t l t 2 . .. t,. We observe that (11) holds not only
for f i E M, but just for polynomials. Indeed, the technical assumption guaranties finite
moments which via analytic continuation give the amplitudes (9,Y,J.One may argue
that dp M F . d l , where d t denotes the measure governing the free oscillator “x” and
F is the counterpart of Feynman’s influence functional. Since d t is ergotlic this cannot
be true, except for a finite time interval. Let
Ann. Physik Leipzig 46 (1989) 1
given 0 _< tl 5 t, 2 . . . 5 t, < 0, which is the Feynman-Kac formula. We remark
that Q, = y By is the ground state for H,, 1.e. when 1 = 0. Moreover, the semigroup
P,, t 2 0, is positivity preserving in the usual sense. So we can choose Q ( x , A ) 2 0,
antl (5) holds for f 7 = f i ( q A ) 2 0. The bounded continuous functions of both variables
2, A generate a maximal Abelian algebra M 2 M , V Pt, t 2 0} on Z . As well Iino\lIl
M w pt,t 2 O} = g(2-r).
Hence there is a Markov process t~ (-00,co) -+ ( ~ ( t A
) ,( t ) ) on (%(M),E X%,Y) such
that d v / E = dp. With other words “integrating out” the dummy variables A ( t ) ,
t E (-co,co),we recover precisely the measure d p of our model.
3. Imaginary Time Axioms
To control the Peynman-Kac formula (13) in the limit (T -+ +co me will use the famous
reflection positivity condition of Osterwalder and Schratler.
ex:(t)e* = +t), t E (-c*3, q,
tlenot,e time reflection. Clearly, 8, = 1. 8 lifts to a unitary operator on L 2 (X ,3,p).
By construction B is generat’ed by E(, t E (-00,oo).Moreover, the time translat.ion
group {Tt,t E (-00, m)}acts ergotlically, i.e. the only invariant sets in Gare @, antl X .
Let E+ denote the @-subalgebrain G generated by Et, t 2 0, and E+ = E ( . 1 E+)contlitional expectation. Then t,he crucial property in question reads E+BE+2 0, on L2(x,E, p ) .
Since the measure dp is Gaussian everything is hidden in the covariance C(s, t ) . It
defines the inverse of a iion-local posit’ive operat’or on L2(IR,d t ) called Euclidean action.
Indeed, for smoothed linear rancloni functions 1 = :c(h), h E S(R+),using
( 16)
we get
Above 01 = h ( i u ) antl p = h(ib) is the Fourier transform. To understand why dp is
reflection positive we may refer to dv. But we forget about the oscillator “A”. The idea
is t o introduce a functional F(-a, u) for the symmetric interval [0,@]V O(0, CT] and to
choose appropriate boundary conditions [S].
Theorem 2.
Let G(s, t ) be the Dirichlet Green’s function for the elliptic differential operator
(w2 -
$) 2 0 on the imaginary time interval
function f holds
a]. Then for any E(0,@)-measurable
Quantum System with Memory
may rewrite - ( x , Gx)=
2 . 0 2 , with functions Y and 2 measurable with respect to E(0,a) 2 Z+.
Now we apply a standard trick [Y]: Expanding the exponent and using 1, 2 0 we get
for the integral in question a sum of positive terms. End of proof. We remark that the
functional P(-a, a) given by the nieinory is n o t multiplicative and hence breakes the
Markov property. Nelson’s construction does not work. Due to [lo]we redefine the physical Hilbert space as the subspace
Proof: Decomposing the square [0, a]x [0, a]
L ~ X , lu)
in L 2 ( X ,E+, p), where W = (E+,OE+)l’ais strictly positive. Let c o= E+ A E- which
is contained in I’.Precisely for f E L2(x,Eo,p) holds Wf = f . R E So,
so that we may
iclentjfyHo with L”(X, Zo,p). However, in general the mapping W which is a selfadjoint
contraction on P ( X , 9,p), acts nontrivially. It intertwines imaginary with “physical”
time evolution according to the formula
W flx(t7)
. . . , tn) E %,
where t, 2 t, 5 . . . 5 t,. As remarked the existence of those vectors is ensured only by
additional assumptions on the regularity of the trajectories t E (-00,co) -+ x(t), 1.e.
dp. W commutes with second quantization and hence is completely determined on linear
functions 1 = x ( h ) , hE S(R+). T,et k ( t ) = 7 - x ( t ) exp(--tco), s h e r e 7 = - anti
denotes the Heaviside step function. Using (16) one can show that
Y = Wx(h)
NN (ba
+ w . ( p - a )2 9 ,
where x = x(O), z = x ( k ) and 0 5 w(A)2 1, span the two-particle subspace 9 in &‘.
From the relation dp = IQ(x, z ) ,1 dx dz we are able to read off the ground state
wave function R in the new variables 2, z. Similarly as on the classical level the coordinate
A of the univisible oscillator disappeared. This seems to be in conflict with a remark of
Feyninan concerning a primitive model of &ED as described by equation (1): “No wave
function Y(t)can be defined to give the amplitude that the particle is a t the place
x a t a particular time t. Such an amplitude would be insufficient for continuing calculations into the future, since a t any time one must also know what the oscillator “A”
is doing” [ 1I].
4. Hamiltonian Semigroup
To convince ourselves that there is no contradiction we analyze the semigroup
P, = exp(-tH), t 2 0, for the process t E (-co,co) -+ z ( t )on ( X , 9,p ) in more detail.
By (3) the matrix pt = exp(--tU)/B, t 2 0, diagonalizes in the natural basis
Ann. Physik Leipzig 46 (1989) 1
We observe that W does not. Let en denote the projection operator in 9t onto the subspace spanned by Y = x Sz. We claim that enplen,t
0, do not build a semigroup.
Indeed, in the basis {XQ, zSz> of W we find
and hence
for some convex function 0 2 z(t) 1. We conclude that, for A > 0, the information in
EoTtEn,t 2 0, is insufficient to describe the stochastic process t E (-00, co)-+ x(t)
governed by the measure dp. Above 01, /3 stand shortly for exp(-tu) respectively
exp(-ttb), t 2 0. A t real time we have the following result. The wave function Y(t)E W
with components u = (ZQ, Y ( t ) )and v = (zQ, Y(t))satisfies the equation
= - (6 - a ) 2 0. To check it we only must calculate the infinitesimal gene-
rator of p l , t 2 0. Via second quantization we obtain the Schroclinger equation for
arbitrary Y(t)E X . There is a quantum counterpart of (7).
Theorem 3.
Let D+(s,t) = i . x(t
denote Green's function for
- iZ) which vanishes for t 2 s. Then TL =
s) e-i(t-s)c, where c =
the first order differential operator L = (C
(xQ, Y(t))is governed by the integro-differentia1 equation
ds D+(s, t ) u ( s ) , t E (-w,co).
- i $ ) u(t)= E~ ' cw
Proof: Applying the operator on the left once more we find the correct solutions ~ ' ( t=
exp(-itu) antl esp(-itb). End of proof. Some years ago Krolikowski and Rzewuski
tleriretl a one-time equation for a similar problem [12]. We may compare the time evolution ot the quantum state Y ( t)= exp(-itH)x SZ f .% with the corresponding classical
beating solution. Let E = - ( b -- a ) antl 6 = - . Then the probability aniplitude
to find the system a t time t 2 0 again in the initial state Y(0)E W A X,, is given
The realization of the process t 6 (--00, 00) -+ z ( t ) on the probability space (%'(it!),
3x8, Y ) simplifies everything. Thei (lea to recover the Markov property is not new.
Karwowski [13] tried t o lift E+BE+ 2 0 to a projection operator in the larger Hilbert
space X = L2(%(N),
EX%, Y). We claini that the natural embedding
J :L 2 ( X ,Z, p )
Quantum System with Memory
is isometric and local. It intertwines 0 and Tt with the corresponding operators living
on Y. Applying Nelson’s reconstruction to the two-dimensional Markov process t E
(-00,oo) -+ ( ~ ( t A
) ,( t ) )we get a semigroup (Q, = z T t , t 2 O} in L2(%?(N(EX Y ) m
2 , where 7t denotes the projection operator. Let V = n J . We find that
L2(x, s“+, p ) -+ 8
has polar decomposition V = y W , where the unitary part y identifies the variables
q , r with the normal coordinates built u p from x and A . The consistency condition
reads y/L2(x,So,p ) = J. Moreover, for g E M,, and arbitrary f L2(x,E+, p) holds
W ( g . Ttf) = y*@g * T t J f )
= y*Jg * ~ *TtJf
=y .
y*QtyWf, t 2 0 .
6. Conclusions
Our aim was to understand the mechanism by which a quantum system acquires
memory. We learned this within the simple model of two coupled oscillators distinguishing “ L ” . But “A” remained “unvisible”. I n Thirring’s book we read: “Wohl kann man
messen, was man will, doch nicht alles auf einmal. Man miBt in Wirklichkeit doch nur
ein kleines Subsystem, so claB es von Interesse ist, das Gesamtsystem in ein zu beobachtencles (offenes) System und einen Rest, der als Warmebad fungiert, zu teilen .. . Nun
wirkt das System auf das Bad und dieses wieder auf das System zuriick. Diese Ruckwirkung beeinflu B t das System aber erst spiiter, so daB die momentane Zeitentwicklung
des Systems also von seiner ganzen Vorgeschichte abhangt” [ 141.
A t the end let us calculate the imaginary time amplitude (@, exp(-tH) U(O)),
t 2 0, where Y(0)= x 0 as above and @ = A D . Using the embedding we find immediately the expression
which means ferromagnetic coupling of ‘(x”and “A”. Consequently, the correlation of
xi = x(ti), j = 1, 2 , . . .,n increases with parameter 0 5 I < 1, too. For the cylinder
measures defined by (11)we can rewrite
We claim that the matrix G“, i, j = 1, 2, . . .,n giving long-range forces, satisfies reflection positivity which is the known condition for the existence of a transfer matrix
in classical spin systems.
A c k n o w l e d g e m e n t . I thank Prof. G. Vojta for bringing my attentlion to ref. [I51
and for some interesting discussions.
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Bei der Redaktion eingegangen am 13. Juni 1988.
Anschr. d. Verf.: Dr. J. LOFFELHOLZ
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