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Letters m Muthematwal Physics 21' 237 244, 1991. I, 1991 Kluwer Academic Publishers. Prblted m the Netherlands. 237 A Golden-Thompson Inequality in Supersymmetric Quantum Mechanics* SLAWOMIR KLIMEK and ANDRZEJ LESNIEWSKI Departnlent o/ Ph)'mc,~, Hart ard Unirermty, Cambridge, MA 02138, U S A. (Recewed: 8 August 1990) Abstract. We establish a supersymmemc version of the Golden Thompson inequality AMS subject classification (1980). 81C05 I. I n t r o d u c t i o n The celebrated G o l d e n - T h o m p s o n inequality [8, 3, 9] states that the quantum partition function of a system does not exceed its classical partition function. A generalization of this inequality to systems in external magnetic fields was established in [1]. Apart from its usefulness in studying the classical limit of a quantum system, the Golden Thompson bound provides a useful criterion for the discreteness of energy levels of the system. In particular, it allows us to prove, in certain cases, that the Hamiltonian H = - h A + V ( x ) has a compact resolvent, even though the potential V ( x ) does not increase as Ixl ~ ov along certain directions. In this Letter, we study a supersymmetric version of this inequality. The Hamiltonian in supersymmetric quantum mechanics has the form /4 = - ~va ;+ ~89~' ~ v(x)[b*, bk] + ~lV'V(x)I-" where V is the superpotential. The main result of this Letter is the following inequality tr(e ,n) ~< (2xt) " 2 f d e t ( I + e ,v2,1~) e 1~:2~,[tw~)12 a~ ~x~]dx (see Section III for precise formulations). This inequality shows that H may have a compact resolvent even when V has flat directions. This improves on the result of Section IV of [4]. Our p r o o f of the inequality follows essentially [8] and [7], with some technical twists due to the presence of fermions. The Letter is organized as follows. Section 2 contains certain introductory results of technical character. In Section 3, we state and prove the inequality. * Supported m part by the Department of Energy under grant DE-FG02-88ER25065. 238 SLAWOMIR K L I M E K A N D A N D R Z E J LESNIEWSKI 2. Technical Preliminaries In this section, we prove two results (Proposition 2.2 and Proposition 2.3) of a technical character which will be used in Section 3 to prove our main result. The first of the two propositions establishes the convexity of a certain matrix function. The second proposition provides a criterion o f trace classness o f the heat kernel o f a positive operator on a Hilbert space. In the lemma below, Lfh(I/) denotes the space of Hermitian operators on a finite-dimensional complex vector space V. L E M M A 2.1. The function f: 5ah(C M) --+ ~ defined by f ( A ) .'= log det(I + e'~), (2.1) is convex. ProoJ~ We have the well known expansion det(I + e A) = I + ~ ~.v trA'"(cm m eA m~>l = I + ~] 1 ~.T tr/v "~c~'~(e*m')' (2.2) where Am(V) is the ruth exterior power of V, and where A~m~==A ^ I ^ . . . ^ I + I ^ A Using the inequality tr(e~S+~, (see [51, p. AI^...AI+...+IA...^I^A. (2.3) 27) ~w) ~< {tr(eS)}~{tr(er)}] - L (2.4) valid for S, T e ~ h ( V ) , and 0 ~< c~ ~< 1, we obtain from (2.2) det(I + e ~A+ I] - ~)8) ~< 1 + m~>l ~ m.V trA "~cM'(eA~m') . ~rA'mM)te ~J~ Applying HSlder's inequality with p = I/~, q = 1 / ( 1 - c~) (0 < ~ < 1) to the righthand side of (2.5) yields det(I + e ~a + (l - =)s) ~< 1 + ~ . trAm~e~(eA~"' ) 1 m.V trA mmM)(e~'~) = {det(l + eA) }~{det(I + eB)} ~- ~, i.e., f (~A + ( 1 - ~B) <~~tf(A) + ( 1 - ~)f (B), as claimed. D GOLDEN-THOMPSON INEQUALITY P R O P O S I T I O N 2.2. The function g: 5r 239 x R ~ ~ defined by g(A, x).'= det(I + e A) e ~, (2.6) is convex. Proof. Since R ~ ) , ~ e '' is a monotonically increasing convex function, and the composition of a convex function with a monotonically increasing convex function is convex, it suffices to show that log g(A, x) = log det(I + e 4) + x, is convex. But this is a consequence of Lemma 2.1. (2.7) [] In the proposition below, ~'~ denotes a complex Hilbert space. If H is a symmetric operator, then D(H) denotes its domain and Q(H) denotes the domain of the corresponding bilinear form. If H is positive and self-adjoint, then Q(H) is closed in the topology defined by the norm Ux U,,= Ux ]] + (x, HX) ''2 P R O P O S I T I O N 2.3. Let H and U be two positive self-adjoint operators on ~f, and let H,, e > O, be the form sum H,..=H +eU. (2.8) Assume that (i) Q(H)c~ Q(U) is dense in Q(H) in the topology of Q(H), (ii) for some t > 0, tr(e ,m) ~< C, (2.9) where C is a constant independent of e. Then the heat kernel of H is trace class and lim tr(e m,) = tr(e ,n). (2.10) \0 Proof. Let {#,,(H~)},;~= i be the sequence of numbers given by the minimax principle ([6], Chapter XIII.1). Since by (ii) H~ has a compact resolvent,/~,(H~:) are the eigenvalues of H,. Furthermore, as a consequence of (i) and Theorem 10.2 in [2], we have #,(H~.) ".~It,(H), (2.11) as ~ ~ 0. Therefore, by (2.9), e ,~,~i~, C ~> lim t r ( e - " , ) = lim ~ '~0 ~,0 ./> I = ~ e n~ tltn(H) I Consequently, #,,(H) ~ o~, as n --* ~ , and so by Theorem XIII.64 of [6], H has a 240 SLAWOMIR KLIMEK AND ANDRZEJ LESNIEWSKI compact resolvent, and its spectrum consists of the numbers #,,(H), n >/1. But then (2.10) follows from (2.12). [] 3. The Inequality In this section we formulate and prove our supersymmetric Golden-Thompson inequality. We begin with a brief review of supersymmetric quantum mechanics. The Hilbert space of supersymmetric quantum mechanics is d f , = L 2 ( ~ n ) | A(C"), where A(C") denotes the Grassmann algebra over C ". We choose an orthonormal basis for C ~, and introduce the fermionic creation and annihilation operators b*, bj, i ~<j ~<n, in the usual fashion. Let V: ~"--* ~ be a polynomial of algebraic degree N >/2; V is called a superpotential (we will be concerned with polynomial superpotentials only). The supercharge Q is a symmetric operator defined on C~ (~") | A(C") by i i Q ' = ~ 2 ~ (b'V, + b , V , ) + 7 ~ (b*V,V(x)-b,V,V(x)). (3.1) Its square is the Hamiltonian, H , = Q 2 = - 89 + 89~" V~kV(x)[b*, bk] + 89 ]Vj V(x)] 2. I.k (3.2) I Clearly, H >~0 on C(~(E") | The Friedrichs extension of H defines a positive self-adjoint operator which we will also denote by H. The main result of this paper is the following theorem. T H E O R E M 3.1. Assume that f det(I + exp{ -tV~-V(x)}) exp{ - ~_t([VV(x)[2 - AV)} d"x < oc, Jbr some t > O. Then, exp{ - tH} is trace class, and tr(e '") ~< (2~t) ,,.2 ~det(I + e ,v:)(,)) e-l.2,(Ivv(,)l~" A) ) d"x. (3.3) Proof. F o r e > 0, we set H~ = H + eU, where U.'=89 2N. This defines an essentially self-adjoint operator on D(H) c~D(U). Indeed, C0~(~") | A(C") c D(H) c~ D(U) and the operator Tr :z i l - ~_A + 89 V(x)l 2 - AV) + _~elx[ 2N, (3.4) is essentially self-adjoint of Co~(~") | A(C"). Since s ,= y. v~k Vlx)b* bk I,k (3.5) GOLDEN 241 THOMPSON INEQUALITY is an infinitesimal perturbation (in the sense of Kato) of T,:, it follows that H~ is essentially self-adjoint on C~ ( R " ) | A(C"). We let H, denote its closure. A simple perturbation theory argument shows also that e x p { - tH, } is trace class for t > 0. We now claim that the following Duhamel expansion converges in the first Schatten norm I1" I1,: e tH, : e t(T, + S) = E (--t)m;~ m >1 0 e 'I)T' ' " e 'lLSe-('2 (''- '-, ~T~Se (' ,.,)r, d ' s . i}r (3.6) where , o,,,==Is~ ~,,: O<<.s~. "'<~s,,<<.tl. Indeed, using H61der's inequality on first Schatten class, f~ e ~ ' r ' S . . . e (',,, s,,, ,)r~Se (l--,,,,)r, d ' s I <" F a~ where so = 0, 1-[ s,,, t]e , z,,, ~, ~,r,S e ,/m,+, ~,,r, I,'(~; + I st ) dms, + ~ = 1. But since +S<~C(T~ _+_/)l 2 (3.7) it follows that for 0 < a, T < l, lie ~ , s e ~r, II -< c( T) ,,4 (3.8) and, consequently, rI ]]e-1'2(,, ,, ,,r, S e ,'2%+, s, ,r; ]l,.(,, + ' ',' 1 ~/~<m ~<C~tr(e 1:4r,) 1-[ I ~l~m (s,+i-s,) ,2 This implies that t'" I m >I o e 'Jr, S " ' e ("-. . . . . ')r, S e ~' ..... )r, dins da,~, l ~<tr(e-r':4) >J ~ E ~ m H (SI+I--S,)I/2dmS<OO' nit [ <~l<~,qt i.e. the series on the right-hand side of (3.6) converges in H" II,. The identity (3.6) is a standard consequence of this convergence. 242 SLAWOMIR KLIMEK AND ANDRZEJ LESNIEWSKI We can now write tr(e "%)= ~ tr(b*~bk,---b*b~ m) • (-1) m ~ m/> 0 xf [j} {k} tr(e -~lr~ Vflk, ,d,'m X e-" V(x) ' ' ' e -r -'m I) T, Vjmk m V(x) x ~m)r,) dms. (3.9) By standard arguments (see e.g., [7], Chapter III), ~r"~e ~, T, ,j,k,V2 V ( x ) 9 "" e r - ,,,, 2 , )r, VSml,," V(x) e (' s m )T~) = f d"x f I~ V';d,, ~ V(e~(s,)) exp {I--~ for []VV(o(s))[ 2 - A V ( o ( s ) ) I~t~m + + elo(s)l 2N] ds} d#t~.~(o)), where d~,y(O)) denotes the conditional Wiener measure. This and the elementary identity \ t,j,k valid for A e 5eh(C'), yield tr(e '"O=fd"xfdet(I+exp{-fo'V"V(~o(s))ds})x 1 ' x exp{-5f ~ [[VV(~176 9 (3.11) Interchanging the sum and the integral is legitimate, since for arbitrary C > O, ; ; {';o' [Ivv(~(s))l-~+~l~(s)l =~] &x exp - 5 + C2 ds + IV,2kg(~o(s)) I ds d~2,x(~) < oo, l,k and Lebesque's dominated convergence theorem can be applied. Observe now that ~" EIW(~,(s, 12- Av(~,(s, + ~l~,(s~l~'l ds t expt- ~1 Jo ) exp{- t, 243 GOLDEN-THOMPSON INEQUALITY as e ~ 0. Therefore, using Lebesque's monotone convergence theorem, we obtain the following identity: lim tr(e-'H~) = e'~,0 f;( d"x {; det I + exp - V2V(w(s)) ds }) x • e x p { - ~ 1 f0' [IVV(co(s))lz--AV(co(s))] ds} dp'~.x(co). (3.12) We now use Proposition 2.2 and Jensen's inequality to bound the right-hand side of (3.12) by 'fo'ff - ds d"x det(I+exp{-tV2V(co(s))})• t • exp{ - 89 IVV(co(s)) [2 - A V(co(s))] } d/~,x (co). Writing co(s)= x + &(s), where ~3(s) is the Wiener process with c~(0)= c3(t)= 0, and then substituting x + ~(s) -~ x, we rewrite the above integral as fdet(Z + exp{-tV2V(x)})exp{- 89 2 - AV(x)]} d"x fd,~),o(~b) = (2rot)-./2 f det(I + e x p { - tV 2 V(x) }) exp{- 89 2- A V(x)]} dnx. As a consequence, lim tr(e-m0 ~<(21rt) -,/2 f d e t ( I + e.\0 3 e - t v z v ( x ) ) e -O/2)t[Nv(x)12 - AV(x)]d"x. Since C~ ( ~ " ) | c Q ( H ) n Q(U) and is dense in Q(H), we can use Proposition 2.3 to conclude the proof. [] COROLLARY 3.2. Assume that fI } exp - ~ [IVV(x)I2 - tr(IV2V(x)])] dmx < oo, (3.13) where IAI.'= (A'A)~/2denotes the absolute value o f the matrix A. Then, e x p { - t H } is trace class, and ;{' tr(e - m ) ~<(2/~t) "/2 exp - ~ [IVV(x)I 2 P r o o f Let 2~. . . . . tr(N~g(x)l)l ) d"x. (3.14) 2,, be the eigenvalues of A 9 ~(,ah(V). Then d e t ( I + e - A ) e'/2'~(~)--- 1--[ ( e(m)a~ + e (,/2)a~)< 2 . exp{89 (3.15) 1 ~J~n The claim is a consequence of this inequality and Theorem 3.1. [] 244 SLAWOMIR KLIMEK AND ANDRZEJ LESNIEWSKI References 1. Combes, J. M., Schrader, R., and Seiler, R., Ann. Phys. 111, 1 18 (1978) 2 Farts, W. G., Self-adjoint operators, Lecture Notes in Mathematics 433, Springer, Berlin, Heidelberg, New York, 1975. 3. Golden, S., Phys. Rev. B137, 1127 1128 (1965). 4. Klimek, S. and Lesniewskt, A., Comm. Math. Phys,, to appear. 5. Ruelle, D., Statistical Mechamcs, Benjamin, New York, 1969. 6. Reed, M. and Simon, B., Methods of Modern Mathematwal Phystcs, IV, Academic Press, New York, 1978, 7. Simon, B., Functional Integration and Quantum Physics, Academic Press, New York, 1979, 8. Symanzik, K., J. Math. Phys. 6, 1155-1156 (1965). 9. Thompson, C. T., J, Math. Phys. 6, 1812-1813 (1965).