Download A Golden-Thompson inequality in supersymmetric quantum

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.
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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
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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.
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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
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New York, 1975.
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