Stochastic processes associated with quantum systems

PHYSICS REPORTS (Review Section of Physics Letters) 77, No. 3(1981) 329—337. North-Holland Publishing Company

STOCHASTIC PROCESSES ASSOCIATED WITH QUANTUM SYSTEMS Abel KLEIN* Department of Mathematics, University of California, Irvine, California 92717, U.S.A.

Contents: 1. Quantum systems 2. Stochastic processes 3. The connection between quantum systems and stochastic processes 4. Perturbations of stochastically positive quantum systems

329 331

5. Positivity preserving semigroups Appendix: Symmetric local semigroups References

335 336 336

332 335

Abstract: Stochastic processes have been useful in constructing and studying states in Quantum Field Theory (e.g., the Erice Lectures [3], Simon [20], Glimm and Jaffe 16]) and in Quantum Statistical Mechanics (e.g., Ginibre [5],Høegh-Krohn [7], Fröhlich [4],Driessler, Landau and Perez [2]). By analytically continuing into imaginary time, we may in certain cases replace the non-commutative algebra of observables of the quantum system by a commutative algebra consisting of functions of a stochastic process. In this article we are going to discuss an appropriate mathematical framework for this connection between quantum systems and stochastic processes.

In 1 and 2 we introduce the appropriate mathematical structures. 3 is the heart of the article. In 4 we show how the associated stochastic process can be used to construct perturbations of the quantum system. In 5 we describe a class of quantum systems with associated stochastic processes. The appendix contains a brief account of symmetric local semigroups, which are needed for the reconstruction of quantum systems from stochastic processes in the non-zero temperature case.

1. Quantum systems In a mathematical model of a quantum system, the observables of the physical system are identified with the self-adjoint elements of a C*~algebra The dynamics is described by the time evolution at, a one-parameter group of automorphisms of The states of the physical system are identified with the states (i.e., normalized positive linear functionals) on Given that the system is in the state w, the basic objects are the correlation functions ~.

~.

~.

TBj

B,,(tl,

. . . ,

t~)= w(a

11(B1)

. .

.

whereB5,...,B~E~, t1,...,t~.ER. Stochastic processes have been associated with equilibrium states. A statek wis on ~ is said to be an 1, where Boltzmann’s constant equilibrium state at inverse temperature /3, 0< /3 (/3 (kT) Partially supported by the N.S.F. under grant MCS-7801433.

330

New stochastic methods in physics

and T is the temperature, 0 T < cc), relative to the time evolution a,, if for each A, B E ~ there exists a function FAB(z), analytic in the open strip {z E C; 0 < Tm z
. . . ,


.

. . . ,

. . .

. .

~‘,

. ,

~,

~,

~,

~,

F8

B~(tl,.

. . ,

t~)= ((1, ir(Bi) exp{i(t2

—

t1)H} ir(B2)

. .

-

exp{i(t~ t~_1)H}ir(B~)12). —

If to is a ground state (i.e., /3 = cc) then H 0, and the multiple-time analyticity is straightforward. If /3
OImZk÷1~~

~ImZ~ f3/2,

then F8 =

BJZ1,

. . . ,

Z~)

(exp{—iZkH} IT(Bk)* exp{i(Zk

—

exp{iZk+1H} IT(Bk±1)exp{i(Zk+2

Zk~l)H}Ir(Bk_l)*

. .

.

zk÷l)H} Ir(Bk±2). .

exp{i(z~ zi)H} Ir(Bi)*12, —

.

exp{i(z.

—

Z~_i)H} ir(B~)i’1).

2(R, dx), example, us consider Quantum Harmonic Oscillator mass m. So ~ = L andAsletanthe positionletoperator q bethe represented by multiplication by with x. Let .~1be the let C*~algebraof all bounded continuous functions of the position operator, so .sti = C(R), a commutative C*~algebra.Let H be the positive self-adjoint operator which is the closure of ~(— d2Idx2 + m m). Now let ~ be the C*~algebraof operators on ~Cgenerated by {ehtH A e~~’; A E t E R}, and let us define the time evolution a, on ~ by a,(B) = e~’B e~”. If we take ho = (m/IT)”4 exp(—mx2/2), then h2~is a unit vector in X such that Hf1 0 = 0, and (l~is the only such vector up to multiplication by scalars. If we let wo(B) = (Q~,B120), then to0 is a ground state for (~, a,). If IT is the identity representation of ~ (i.e., ir(B) = B), then (~, ir, h2~,H) is the GNS representation of (~, a,, too). —

~‘,

A. Klein, Stochastic processes associated with quantum systems

331

For /3
[Tr(e~’)]~

Tr(e~’~’B).

Here we use the fact that ~ is trace-class for 0
TB

for

t1,

. ..

,

. . . ,

[Tr(e~’1)]’ Tr(e_$H exp{i(t1 — t~)H}B1 exp{i(t2— t1)H} x B2~~exp{i(t~—

t~)

4, E R. Since H

F8,

B,(Z1,

. . . ,

~)

0, we have the analytic continuation =

[Tr(e”)]’ X

to the region Tm Z1

S

Tr(e~’exp{i(z1

—

Z~)H}BI exp{i(Z2— Z1)H}

B2.~ exp{i(Zn — Z~_1)H}B~) .

Tm z2

.

Im Z~ Im Z1

.

+

/3.

2. Stochastic processes A stochastic process with values in the locally compact Hausdorif space K is a family {X,},ER of random variables on a probability space (Q, £, ~u)with values in K. We will call K the value space of the process, and (0, £, ~s) the underlying probability space. We will denote by ( ) the expectation in the underlying probability space, i.e., (F) = fF d~sfor an integrable function F on (0, ia). Two stochastic processes {X1,},ER and {X2,},ER with the same value space K (but perhaps with different underlying probability spaces) are said to be equivalent if ,~,

(f1(X ~

. .

f, (X ~

=

(f1(X’2,1)

. . .

~

for all f’, f~E C(K), ti, tn E R. Let {X,},ER be a stochastic process with values in K. The processis stationary if the processes {X,},ER and are equivalent for all s E R; symmetric if the processes {X,},ER and {X_,},ER are equivalent; weakly stochastically continuous if f(X,) is continuous in measure, as a function of t E R, for all! E C(K). A stochastic process {X,},w~will be called periodic with period/3, 0 <8
. . . ,

X_,,) F(X,1,

.

. .

,

X,~)) 0

for all t1,. 4. E [0,/3/2], for any bounded measurable function F on K”. Osterwalder—Schrader (OS) positivity was introduced in the context of Euclidean Field Theory (Osterwalder and Schrader [18]).For periodic processes with period /3
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New stochastic methods in physics

3. The connection between quantum systems and stochastic processes Typically, stochastic processes have been associated with quantum systems (~,a,, to), in thermal equilibrium at inverse temperature /3, 0
.~‘

. ,

,~‘,

to purely imaginary times, and the functions TA A,,(1S1 isa), with s1 computed explicitly. (iii) a stochastic process {X,},ER with values in K is identified so that TA

A,(151

isa)

=

(A1(X~,).

.

.

S2 ~

~

Sn

s1 + /3, are

(*)

An(Xsj)

forallA1,...,A~Es21, s1~s2~.s~s1+/3. To illustrate this procedure, let us go back to the quantum harmonic oscillator. In this case = C(R). If we think of the quantum harmonic oscillator as a one space-time scalar Bose quantum field theory, the von Neumann algebra closure of is the “time-zero algebra”. The TA A,(1SI,. ,is4 can be computed explicitly and we find that if we take {X,},ER to be the Gaussian process with mean zero and covariance .~‘

. .

(X, X3) = (2m)’

if /3 =

e_I~~m

cc,

m+e~_~m) iff3
. . ,

. ,

.~“

~‘

A. Klein, Stochastic processes associated with quantum systems

333

These two requirements turn out to be also sufficient for the existence of an associated stochastic

process. Theorem (Klein [11; Theorem 2.4] for /3 = cc, Klein and Landau [17; Theorem 6.1] for /3
~,

TA1

A,,(1Si,

. .

.

,

isa) = (A1(X51)

. . .

An(Xs,,))

(*)

for all A1,. A~ E si s2~ 5,, Si + /3. Moreover, the stochastic process is periodic with period /3, stationary, symmetric, weakly stochastically continuous, and OS-positive. The process is unique up to equivalence. Conversely, let {X,},ER be a stochastic process with values in the locally compact Hausdorif space K, which is periodic with period /3, 0
,

(A 1(x~1). . for all A1,. phisms.

. . ,

~,

.

A,,(.~k’5,,))= F.~(A1)

E C(K) and

A,,

~(A,)(is1,

. . .

,

15n)

5,, S s~ +

S~~

/3. The quantum system is unique up to isomor-

We will now make some comments about the proof. Given the stochastically positive quantum system, the stochastic process is constructed from The underlying probability space is constructed as a “path space” where the paths take values in the spectrum of ,s~.The stochastic positivity condition is the crucial condition for the construction of a probability measure on the “path space”. The stationarity of the stochastic process follows from the invariance of the equilibrium state under the time evolution. The symmetry comes from the self-adjointness of the Hamiltonian H in the GNS representation. The periodicity of the process for /3
~‘,

(*),

2) = (A(X~)2) + (A(X ((A(X~) — A(X0)) 2)

2)

— 2(A(X 0) A(X~))

0)

—

(A(X

= 2{(A(X0)

0) A(X~))} = 2{FAA(0,

0)— TAA(0, i$)} = 0,

since FA.B(0, 0) = w(AB) and FAB(0, i$) w(BA) by the KMS condition. From the stationarity it follows that for each t E R, A(X,±~)= A(X,) a.e., for all A E The one property of the stochastic process that is specific to stochastic processes associated with quantum systems is OS-positivity. To see how it follows from (*) let .~‘.

F(X~,.

where c1,

. . , X~)

- - . ,

c,,

= ~

E C,

c1A11(Xj.

A~,1E ~

for i

=

1,.

. . ,

m;

j

=

1,

. . . ,

n,

and 0

s1

S2 ~

s,,

~ /3/2. Using the

New stochastic methods in physics

334

GNS representation, it follows from (*) that (A,,(X~5,). . . A~(.X~,) B1(X~,)

.

=

(exp{—s1H} A exp{—(s2 ~

exp{—siH} B1 exp{—(s2

—

—

. .

s1)H} A~

s~)H}B2

. .

.

.

.

exp{—(s,, — s,,1)H} A~11,

exp{—(s,, — s,,_1)H} B,,12).

Hence (F*(X_~,...,~ =

c1 exp{—s1H} AE.l

exp{—(s2

—

s1)H} A12~. exp{—(s,,

—

A1n1211

s,,~)H}

0.

Let us now sketch the construction of the quantum system from the stochastic process. The quantum system is constructed in the GNS representation. Let (0, ~z)be the underlying probability space, and let 1+ be the a--algebra generated by {X,; 0 t /3/2}, and2(Q, E± be~z)defined the conditional expectation with by respect to 1±.Let U(t) and R be the unitary operators on L U(t) F(X 5, Xx,) = F(X~+, X~,,.,) .~,

.~,

RF(XSI

X~,,)=F(X.

where F is a bounded measurable function on K”. {U(t), t E R} 2(Q, forms aia). strongly oneLet ~ continuous be the range of parameter unitary group. Now let V = E+RE± as an operator on L V. Then ~‘÷,

(V(F), V(G)) = (F*(RG)) gives a positive definite inner product on due to OS-positivity. Let ~ be the completion of ~o in this inner product. Notice that V is a contraction from L2(Q, £+, ~a) into with dense range. Let 12 = V(1). A representation y of C(K) on is obtained by y(A) V(F) = V(A(X F) for A E C(K), 2(Q, ,X+, ii). We let be the C*~closureof y(C(K)). To construct the0)Hamiltonian H weFE let L P(t) 71(F) = V(U(t) F). In the /3 = cc case we can define P(t) for all t 0 and prove that P(t) is a strongly continuous self-adjoint contraction semigroup, hence P(t) = e”~ for a unique seif-adjoint operator H 0. If /3
~,

~

~

.~±,

£~?,

~,

~‘,

,~,

~‘,

.~,

A. Klein, Stochastic processes associated with quantum systems

4.

335

Perturbations of stochastically positive quantum systems

Let (~, a,, co) be a stochastically positive quantum system at inverse temperature /3, 0
.~,

.~‘.

.~‘,

P~(t)V(F)=

V(exp{_J V(X~)ds}U(t)F).

Theorem (Klein and Landau [17; Theorem 16.4]). If either VEL2~(K,dv)and e~”EL1(K,dv)for some e, 5 > 0, or V E L2(K, di.’) and V 0, then the Feynman—Kac—Nelson formula gives a symmetric local semigroup with self-adjoint generator H~= (H + V). In particular H + V is essentially selfadjoint. The standard results of this kind require hypercontractivity (e.g., Klein and Landau [14]). By using symmetric local semigroups (see appendix) we can drop the requirement of hypercontractivity. In general H + V will be unbounded below.

5. Positivity preserving semigroups Let ~W= L2(X, dx), and let H 0 be a self-adjoint operator on ~ such that e_tH is positivity preserving, i.e., if ~ E ~ 0, then ~t1~ç~ 0 for all t 0. Let .c~’ be a C*.algebra such that = L~(X,dx) (e.g., = C 0(X) if X is locally compact). Define ~ as the C*~algebragenerated by {e~’A e”; A E .~i,I E R}, and let a,(B) = e~’B e”~’for BE Suppose H has a positve ground state 1l~,i.e., 12~is a unit vector in ~Wsuch that 11~ 0 and Hf)0 = 0. Then, if we let wo(B) = (hl~,B120), a,, to0) forms a stochastically positive quantum system at /3 = cc Moreover the associated stochastic process theif Markov property (e.g., Klein [11]). 81’ is trace-class for some 0<13
.~‘

~.

(~,

co(B) = [Tr(e~]’

12,

Tr(e~”B)

for B E then to satisfies the KMS condition at inverse temperature /3, and (9k, a,, to) is a stochastically positive quantum system. Moreover, the associated stochastic process satisfies the two-sided Markov property for semicircies (Klein and Landau [17; Theorem 18.4]). In case e” is positivity improving, i.e., ~ 0, ~‘~0 implies e~’w>0 for all t>0, our result on perturbations of stochastically positive quantum systems can be used to prove the following result: ~,

~‘,

New stochastic methods in physics

336

Theorem (Klein and Landau [17; Theorem 19.1]). Let e~”be a positivity improving semigroup on L2(X, dx) such that e~H is trace-class. Let V be a self-adjoint operator on L2(X, dx) given by multiplication by a real measurable function. If either Tr(eti11 I T~/~2~)0, or Tr(e~”V2)
Appendix. Symmetric local semigroups A symmetric local semigroup (P(t), ~,, T) on the Hilbert space consists of: (i) for each I, 0 t~-T (T>0), a linear subset 9l~,of the Hilbert space ~ such that p2’, D ~. if ts, and ~ = UO
~‘.

~.

Theorem (Klein and Landau [16]). Let (P(t), ~ T) be a symmetric local semigroup on the Hilbert space Then there exists a unique self-adjoint operator H on ~ such that ~, C ~(e”) and P(t) is the restriction of e~’to ~, for all 0 ~ I T. Moreover, if ~‘.

U

~=

O
where 0< S

U P(s)~,, 0
T, then ~ C ~ and ~ is a core for H.

For a full account see Klein and Landau [16].

References [1] H. Araki, Relative Hamiltonian for faithful normal states of a von Neumann algebra, PubI. RIMS, Kyoto Univ. 9 (1973) 165—209. [2] W. Driessler, L. Landau and J. Perez, Estimates of critical lengths and critical temperatures for classical and quantum lattice systems, J. Stat. Phys. 20 (1979) 123—162. [3] Ence Lectures, Constructive Quantum Field Theory, eds. G. Velo and A. Wightman (Springer-Verlag, Berlin, 1973). [4] J. Frtihlich, The reconstruction of quantum fields from Euclidean Green’s functions at arbitrary temperatures, Helv. Phys. Acta 48 (1975) 355—363.

J. Ginibre, Some applications of functional integration in statistical mechanics, in: Statisticalmechanics and quantum field theory, eds. C. DeWitt and R. Stora (Gordon and Breach, New York, 1971) pp. 327—428. [61J. Glimm and A. Jaffe, Quantum Physics. A functional integral point of view (Springer-Verlag, Berlin, 1981). [7] R. Høegh-Krohn, Relativistic quantum statistical mechanics in two-dimensional space-time, Comm. Math. Phys. 38 (1974) 195—224. [5]

A. Klein, Stochastic processes associated with quantum systems

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[81N.M. Hugenholtz, States and

representations in statistical mechanics, in: Mathematics of Contemporary Physics, ed. R.F. Streater (Academic Press, London, 1972) pp. 145—182. [91A. Klein, A characterization of Osterwalder-Schrader path spaces by the associated semigroup, Bull. Amer. Math. Soc. 82 (1976) 762—764. [101A. Klein, When do Euclidean fields exists?, Lett. Math. Phys. 1(1976)131—133. [11] A. Klein, The semigroup characterization of Osterwalder—Schrader path spaces and the construction of Euclidean fields, J. Functional Analysis 27 (1978) 277—291.

[12] A. Klein, A generalization of Markov processes, Ann. Probability 6 (1978) 128—132. [13] A. Klein, Gaussian OS-positive processes, Z. Wahrscheinlichkeitstheorie 4~(1977) 115—124. [14] A. Klein and L. Landau, Singular perturbations of positivity preserving semigroups via path space techniques, J. Functional Analysis 20 (1975) 44—82. [15] A. Klein and L. Landau, Periodic Gaussian Osterwalder—Schrader positive processes and the two-sided Markov property on the circle, Pacific J. Math., to appear. [161A. Klein and L. Landau, Construction of a unique self-adjoint generator for a symmetric local semigroup, J. Functional Analysis, to appear. [17] A. Klein and L. Landau, Stochastic processes associated with KMS states, J. Functional Analysis, to appear. [18] K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions, I, Comm. Math. Phys. 31(1973)83-112; II, Comm. Math. Phys. 42 (1975) 281—305.

[19] D. Robinson, C*~algebrasand quantum statistical mechanics, in: C*~algebrasand their applications to statistical mechanics and quantum field theory, ed. D. Kastler (North-Holland, Amsterdam, 1976) pp. 235—252. [20] B. Simon, The P(4~Euclidean (quantum) field theory (Princeton University Press, Princeton, N.J., 1974).