The reduced effect of a single scattering with a low-mass particle via a point interaction

0.25 MB·English

by  Jeremy Clark

The reduced effect of a single scattering with a low-mass
particle via a point interaction
Jeremy Clark
[email protected]
9
0 Katholieke Universiteit Leuven, Instituut voor Theoretische Fysica
0
2 Celestijnenlaan 200D, 3001 Heverlee, Belgium

n
a January 15, 2009
J

5
1
Abstract
]
h
p In this article, we study a second-order expansion for the effect induced on a large
- quantum particle which undergoes a single scattering with a low-mass particle via a re-
h
t pulsive point interaction. We give an approximation with third-order error in λ to the
a
m map G → Tr2[(I ⊗ρ)Sλ∗(G⊗I)Sλ], where G ∈ B(L2(Rn)) is a heavy-particle observable,
ρ ∈ B (Rn) is the density matrix corresponding to the state of the light particle, λ = m
[ 1 M
is the mass ratio of the light particle to the heavy particle, Sλ ∈ B(L2(Rn)⊗ L2(Rn))
2
is the scattering matrix between the two particles due to a repulsive point interaction,
v
6 and the trace is over the light-particle Hilbert space. The third-order error is bounded in
1 operator norm for dimensions one and three using a weighted operator norm on G.
1
5
.
7 1 Introduction
0
8
0
In theoretical physics, many derivations of decoherence models begin with an analysis of the
:
v effect on a test particle of a scattering with a single particle from a background gas [9, 6, 8].
i
X A regime that the theorists have studied and which has generated interest in experimental
r physics [7] is when the test particle is much more massive than a single particle from the
a
gas. Mathematical progress towards justifying the scattering assumption made in the physical
literature in the regime where a test particle interacts with particles of comparatively low mass
can be found in [1, 3, 5]. In this article, we study a scattering map expressing the effect induced
on a test particle of mass M by an interaction with a particle of mass m = λM, λ ≪ 1. The
force interaction between the test particle and the gas particle is taken as a repulsive point
potential.
We work towards bounding the error ǫ(G,λ) in operator norm for G ∈ B(L2(Rn)), n = 1,3
of a second order approximation:
Tr [(I ⊗ρ)S∗(G⊗I)S ] = G+λM (G)+λ2M (G)+ǫ(G,λ), (1.1)
2 λ λ 1 2
where ρ ∈ B (L2(Rn)) is a density matrix (i.e. ρ ≥ 0 and Tr[ρ] = 1), G ∈ B(L2(Rn)),
1
S ∈ B(L2(Rn) ⊗ L2(Rn)) is the unitary scattering operator for a point interaction, and the
λ
partial trace is over the second component of the Hilbert space L2(Rn)⊗L2(Rn). M and M
1 2
1
are linear maps acting on a dense subspace of B(L2(Rn)) (M is unbounded). Our main result
2
is that there exists a c > 0 such that for all ρ, G, and 0 ≤ λ
kǫ(G,λ)k ≤ cλ3kρk kGk ,
wtn wn
where k·k is a weighted operator norm of the form
wn
~ ~
kGk = kGk+k|X|Gk+kG|X|k
wn
+ (kX P Gk+kGP X k)+ k|P~|e1G|P~|e2k,
i j j i
X X
0≤i,j≤d e1+e2≤3
and k·k is a weighted trace norm which will depend on the dimension. In the above, X~ and
wtn
~
P are the vector of position and momentum operators respectively: (X f)(x) = x f(x) and
j j
(P f)(x) = i( ∂ f)(x). Expressions of the type A∗GB for unbounded operators A and B are
j ∂xj
identified with the kernel of the densely defined quadric form F(ψ ;ψ ) = hAψ |GBψ i in the
1 2 1 2
case that F is bounded.
The scattering operator is defined as S = (Ω+)∗Ω−, where
λ
Ω± = s-limeitHtote−itHkin (1.2)
t→±∞
are the M¨oller wave operators, and H is the kinetic Hamiltonian and is the standard self-
kin
adjoint extension of the sum of the Laplacians − 1 ∆ − 1 ∆ , while the total Hamilto-
2M heavy 2m light
nianH includes anadditionalrepulsivepointinteractionbetween theparticles. Thedefinition
tot
of H is a little tricky for n > 1 since, in analogy to the Hamiltonian for a particle in a point
tot
potential [2], it can not be defined as a perturbation of H even in the sense of a quadratic
kin
form. Rather, it is defined as a self-adjoint extension of − 1 ∆ − 1 ∆ with a special
2M heavy 2m light
boundary condition. Going to center of mass coordinates, we can write
1 1 1 M +m
∆ + ∆ = ∆ + ∆ ,
heavy light cm dis
2M 2m 2(m+M) 2mM
so that the special boundary condition will be placed on the displacement coordinate corre-
sponding to ∆ and follows in analogy with that a single particle in a point potential as
dis
discussed in [2]. This also allows us to write down expressions for S . Non-trivial point po-
λ
tentials in dimensions > 3 do not exist and the main result of our analysis is restricted to
dimensions one and three.
The first and second order expressions M (G) and M (G) respectively have the form
1 2
1 1 1
M (G) = i[V ,G] and M (G) = i[V + {A~,P~},G]+ϕ(G)− ϕ(I)G− Gϕ(I), (1.3)
1 1 2 2
2 2 2
where V , V , and (A~) for j = 1,...,n are bounded real-valued functions of the operator X~,
1 2 j
and ϕ is a completely positive map admitting a Kraus decomposition:
ϕ(G) = d~km∗ Gm , (1.4)
Z j,~k j,~k
X R3
j
with the m ’s being bounded multiplication operators in the X~-basis. Notice that terms
j,~k
in (1.3) are reminiscent of the form of a Lindblad generator [10]. In [4] the results of this
2
article are applied to the convergence of a quantum dynamical semigroup to a limiting form
with generator including the terms (1.3).
The explicit forms for V , V , A~, and ϕ are:
1 2
V = c s−1 dk|k|−1 d~v d~v ρ(~v ,~v )eiX~(~v1−~v2), (1.5)
1 n n Z Z 1 2 1 2
R+ |~v1|=|~v2|=k
V = c s−1 dk|k|−1 d~v d~v (~v +~v )∇ ρ(~v ,~v )eiX~(~v1−~v2), (1.6)
2 n n Z Z 1 2 1 2 T 1 2
R+ |~v1|=|~v2|=k
A~ = c s−1 dk|k|−1 d~v d~v ∇ ρ(~v ,~v )eiX~(~v1−~v2), (1.7)
n n Z Z 1 2 T 1 2
R+ |~v1|=|~v2|=k
ϕ(G) = c2s−2 d~k|~k|−2 d~v d~v ρ(~v ,~v )eiX~(−~v1+~k)Ge−iX~(−~v2+~k), (1.8)
n n Z Z 1 2 1 2
Rn |~v1|=|~v2|=|~k|
where s is surface area of a unit ballin Rn, c is a constant arising formthe scattering operator
n n
~ ~
S ,ρ(k ,k )istheintegralkernel ofρ, and∇ isthegradientofweakderivatives inthediagonal
λ 1 2 T
direction which is formally (∇ ρ(~k ,~k )) = lim h−1 ρ(~k + he ,~k + he ) − ρ(~k ,~k ) .
T 1 2 j h→0 1 j 2 j 1 2
The integral kernel ρ(~k ,~k ) is well defined since ρ is tra(cid:0)ceclass and hence Hilbert-Schmid(cid:1)t.
1 2
In dimension one, the integrals are replaced by discrete sums. In dimension two,
|v1|=|v2|=k
V has an additional term due Rto the logarithm in (4.4) which we did not write down in
2
the expression for V above. The multiplication operators m are defined as m (X~) =
2 j,~k j,~k
c s−1 β d~vf (~v)e−iX~(−~v+~k), where ρ = β |f ihf | is the diagonalized form of ρ. V ,
n n j |~v|=|~k| j j j j j 1
V , A~p, andRϕ are bounded under certain normPrestrictions on ρ, since, for example, kV k ≤
2 1
c k|P~|n−2ρk and kϕk = c k|P~|n−2ρ|P~|n−2k .
n 1 n 1
With the center of mass coordinate at the origin, the scattering operator S (neglecting the
index λ until it is explained) acts identically on the center-of-mass component of L2(R2n) =
L2(Rn)⊗L2(Rn) as
S = I +I ⊗(S(k)⊗|φihφ|), (1.9)
cm
where the right copy of L2(Rn) corresponds to the displacement variable and is decomposed in
the momentum basis into a radial and an angular component as L2(R+,rn−1dr)⊗L2(∂B (0)),
1
S(k) acts as a multiplication operator on the L2(R+) component, and φ = (sn)−211∂B1(0), is
the normalized indicator function over the whole surface ∂B (0). We call S(k) the scattering
1
coefficient, and it has the form
Dim−1 : Dim−2 : Dim−3 :
−iα −iπ −2ik
Sα(k) = k +i1α Sl(k) = l−1 +γ +ln(k)+iπ Sl(k) = l−1 +ik, (1.10)
2 2 2
where α is a resonance parameter defined for the one-dimensional case, l is the scattering length
in the two- and three-dimensional cases and γ ∼ .57721 is the Euler-Mascheroni constant. In
the one-dimensional case a scattering length l is sometimes defined as the negative inverse of
the resonance parameter α = µc, where c is the coupling constant of the interaction and µ is
~2
the relative mass mM = M λ . However, this contrasts with the two- and three-dimensional
m+M 1+λ
cases where the scattering length is proportional to the strength of the interaction. In the
context of this article, where the point interaction is between a light and an heavy particle,
3
we parameterize the resonance parameter as α = λ α in the one-dimensional case and the
1+λ 0
scattering length as l = λ l in the two- and three-dimensional cases for some fixed α and
1+λ 0 0
l. This corresponds to holding the strength of the interaction fixed. Thus S and S will be
λ λ
indexed by λ for the remainder of the article.
There are two main obstacles in attempting to find a bound for the error ǫ(G,λ) from (1.1).
The first obstacle is to find helpful expressions to facilitate making a Taylor expansion in λ of
Tr [(I ⊗ρ)S∗(G⊗I)S ]. Writing A = S −I, then
2 λ λ λ λ
Tr [(I⊗ρ)S∗(G⊗I)S ] = G+Tr [(I⊗ρ)A∗]G+GTr [(I⊗ρ)A ]+Tr [(I⊗ρ)A∗(G⊗I)A ],
2 λ λ 2 λ 2 λ 2 λ λ
and it turns out to be natural at all points of the analysis to approach the terms on the right
individually. Propositions 2.2 and 2.3 are directed towards finding expressions for
Tr [(I ⊗ρ)A∗] and Tr [(I ⊗ρ)A∗(G⊗I)A ] (1.11)
2 λ 2 λ λ
respectively (since Tr [(I ⊗ρ)A ] is merely the adjoint of Tr [(I ⊗ρ)A∗]). The expressions we
2 λ 2 λ
find in Propositions 2.2 and 2.3 are of the form
Tr [(I ⊗ρ)A∗]G = d~k dσU∗ f∗ G, and (1.12)
2 λ Z Z ~k,σ ~k,σ
Rn SOn
Tr [(I ⊗ρ)A∗(G⊗I)A ] = d~k dσ dσ U∗ h∗ Gh U , (1.13)
2 λ λ ZRn ZSOn×SOn 1 2 ~k,σ1,λ ~k,σ1,λ ~k,σ2,λ ~k,σ2,λ
for some unitaries U∗ , U and some bounded operators g , h∗ which are functions
~k,σ ~k,σ2,λ ~k,σ2,λ ~k,σ1,λ
~
of the vector of momentum operators P. In general, we will have the problem that
~ ~
dk dσkg k = ∞, and dk dσ dσ kh kkh k = ∞,
Z Z ~k,σ,λ Z Z 1 2 ~k,σ1,λ ~k,σ2,λ
Rn SOn Rn SOn×SOn
so the integrals of operators only have strong convergence. Propositions 3.1 and 3.3 make sense
of the integrals of operators such as (1.12) and (1.13) that arise and give operator norm bounds
for the limits. The basic pattern in the proof of Propositions 3.1 and 3.3 is an application of
the simple inequalities in Propositions A.2 and A.3 in addition to intertwining relations that
we have between the multiplication operators and the unitaries appearing in (1.12) and (1.13).
Bounding the third order error of the expansions in λ of the strongly convergent inte-
grals (1.12) and (1.13) brings up the second major obstacle. We will need to bound certain
strongly convergent integrals for all λ in a neighborhood of zero. For small λ there will be
unbounded expressions arising from the scattering coefficient S (k) that will have contrasting
λ
properties between the one- and three-dimensional cases. For example, In the limit λ → 0,
1S (k) becomes increasingly peaked in absolute value at k ∼ 0 in the one-dimensional case.
λ λ
For the three-dimensional case, 1S (k) becomes increasingly peaked at k = ∞. A difficulty
λ λ
with the two-dimensional case is the presence of the natural logarithm in the expression for
S (k) and the fact that 1S (k) is not peaked at a fixed point as λ varies. The peak point
λ λ 1+λλl0
does tend towards k ∼ 0 as λ → 0, but it is unknown how to attain the necessary inequalities
in this case.
This article is organized as follows. Section 2 is concerned with proving Propositions 2.2
and 2.3 which give expressions for Tr[(I ⊗ ρ)A∗] and Tr[(I ⊗ ρ)A∗(G ⊗ I)A ]. In Section 3
λ λ λ
4
we prove Propositions 3.1 and 3.3 which give the primary tools for bounding the integrals of
operators which will arise in bounding the error term ǫ(G,λ) of our expansion (1.1). Section 4
contains the proof of Theorem 4.2 which is the main result of the article. This involves expand-
ing theexpressions in Propositions 2.2 and2.3 that we foundinSection 2 inλ and bounding the
error. The difficult parts of the proof are characterized by using the Propositions 3.1 and 3.3 to
translate unbounded expressions arising from the expansion of the scattering coefficient S into
λ
conditions on G and ρ through the weighted norms kGk and kρk being finite. Sections 2
wn wtn
and 3 apply to dimensions one through three (all dimensions where non-trivial point potentials
exist), while Section 4 does not treat dimension two.
2 Finding useful expressions for a single scattering
In this section, we will find expressions for Tr [ρA∗] and Tr [ρA∗GA ]. For notational conve-
2 λ 2 λ λ
nience, we will begin identifying I⊗ρ with ρ and G⊗I with G. Finding formulas for Tr [ρA∗],
2 λ
Tr [ρA∗GA ] begins with writing A = S −I in a convenient way. Let f,g ∈ L2(Rn ×Rn),
2 λ λ λ λ
where the first and second component of Rn×Rn correspond to the displacement and the center
of mass coordinate, then
∞ S (k)
hg|A fi = dK~ dk λ dkˆ g¯(kˆ ,K~ ) dkˆ f(kˆ ,K~ ) . (2.1)
λ Z cmZ s kn−1 Z 1 1 cm Z 2 2 cm
Rn 0 n (cid:0) ∂Bk(0) (cid:1)(cid:0) ∂Bk(0) (cid:1)
The above formula gives a quadratic form representation of A that involves integrating over a
λ
surface of 3n−1 degrees of freedom rather than 4n, since it acts identically over the center-of-
mass component of the Hilbert space and conserves energy forthe complementary displacement
coordinate. TheintegralkernelforA incenter-of-massmomentumcoordinatescanbeformally
λ
expressed as
S (|k |)
A (k ,K ;k ,K ) = λ dis,1 δ(|k |−|k |)δ(K −K )
λ dis,1 cm,1 dis,2 cm,2 s |k |n−1 dis,1 dis,2 cm,1 cm,2
n dis,1
However, for instance, this does not work directly towards finding even a formal expression for
(Tr [ρA∗]G)(K~ ,K~ ) = d~k d~k dK~ ρ(k ,k )A∗(~k ,K~ ;~k ,K~)G(K~,K~ ), (2.2)
2 λ 1 2 Z 1 2 1 2 λ 2 1 1 2
where we have written down a formal equation between integral kernel entries Tr [ρA∗G], G,
2 λ
andA∗ using momentum coordinates corresponding tothe heavy particle andthe light particle.
λ
In finding an expression for (2.2), it would be natural to have K~ as a parameterizing variable
2
since the expression above is just multiplication of G from the left by Tr [ρA∗].
2 λ
For λ = m, the center of mass coordinates are X~ = λ ~x + 1 X~ and x = ~x − X~,
M cm 1+λ 1+λ d
where ~x and X~ are the position vectors of the particle with mass m and M. The corresponding
momentum coordinates are ~k = 1 ~k − λ K~ and K~ = ~k + K~. The proposition below
d 1+λ 1+λ cm
gives two quadratic form representations of A using different parameterisations of the inte-
λ
gration in (2.1). (2.3) is directed towards finding an expression for Tr [ρA∗] and (2.4) is for
2 λ
Tr [ρA∗GA ]. The proof of the following proposition requires changes of integration.
2 λ λ
Proposition 2.1 (Quadratic form representations of A ). Let f,g ∈ L2(Rn ×Rn), then
λ
5
1. First Quadratic Form Representation
hg|A fi = d~kdK~ dσS (|~k|)
λ Z Z λ
Rn SOn
g¯(~k +λ(K~ +σ~k),K~ +(σ −I)~k)f(σ~k+λ(K~ +σ~k),K~), (2.3)
2. Second Quadratic Form Representation
I
hg|Al′fi = Z dK~2d~k1Z dσ det(I +λσ)−1Sλ(|I +λσ(~k1 −λK~2)|)
SOn
(σ −I) σ −I
g¯(~k ,K~ + (~k −λK~ ))f(~k + (~k −λK~ ),K~ ), (2.4)
1 2 1 2 1 1 2 2
1+λσ 1+λσ
where the total Haar measure on SO is normalized to be 1 (and for dimension one, the integral
n
over SO is replaced by a sum over {+,−}).
n
Theproofsof Propositions2.2and2.3work byusing thespectraldecomposition ofρ, special
cases of G, etc. so that the quadratic form representations (2.3) and (2.4) of A can be applied.
λ
Defining τ = ei~k·X~, recall that τ acts in the momentum basis as a shift: (τ f)(p~) = f(p~−~k).
~k ~k ~k
Proposition 2.2. Let ρ have continuous integral operator elements in momentum representa-
tion. Tr [ρA∗] has the integral form
2 λ
B˜∗ = d~k dστ τ∗ p S¯ (|~k|), (2.5)
λ Z Z ~k σ~k ~k,σ,λ λ
Rn SOn
where τ is a translation by ~a in the momentum P~ basis and p is a multiplication operator:
~a ~k,σ,λ
~ ~ ~ ~
p = ρ((1+λ)k +λP,(σ+λ)k +λP).
~k,σ,λ
Proof. The following equality holds:
Tr [ρA∗] = Tr [ β |f ihf |A∗] = β (id⊗hf |)A∗(id⊗|f i),
2 λ 2 j j j λ j j λ j
X X
j j
where the infinite sum on the right converges absolutely in the operator norm. If we take a
partial sum ρ = m β |f ihf |, then using (2.3),
m j=1 j j j
P
m m
hw|(id⊗hf |)A∗(id⊗|f i)vi = dK~ dK~ d~kS¯ (|~k|) dσ
j λ j Z 1 2Z λ Z
Xj=1 Xj=1 Rn×Rn
f¯(σ~k +λ(K~ +σ~k))w¯(K~ )f (~k +λ(K~ +σ~k))v(K~ +(σ −I)~k).
j 1 1 j 1 2
This has the form hw|[·]vi, where [·] is given by
d~kS¯ (~k) dστ∗ τ ρ ((1+λ)~k +λK~,(σ +λ)~k +λK~).
Z λ Z σ~k ~k m
Rn SOn
This converges in operator norm to the expression given by (2.5), since ρ → ρ in the trace
m
norm and by the bound given in Corollary 3.2.
6
Tr [ρA ] has a similar integral representation by taking the adjoint. Now we will delve
2 λ
into the form of Tr [ρA∗GA ]. In the following, the operator D acts on f ∈ L2(Rn) as
2 λ λ A
(DAf)(~k) = |det(A)|21f(A~k) for a element A ∈ GLn(R).
Proposition 2.3. Let β |f ihf | be the spectral decomposition of ρ. Tr [ρA∗GA ] can be
j j j j 2 λ λ
written in the form P
~ ~
k −λP
B˜ (G) = d~k dσ dσ U∗ m∗ S¯
λ Xj ZRn ZSOn×SOn 1 2 ~k,σ1,λ j,~k,σ1,λ λ(cid:0)(cid:12) 1+λ (cid:12)(cid:1)
(cid:12) (cid:12)
~k −λP~
GS m U , (2.6)
λ 1+λ j,~k,σ2,λ ~k,σ2,λ
(cid:0)(cid:12) (cid:12)(cid:1)
(cid:12) (cid:12)
where U = τ∗D τ ,. τ , τ , and D act on the momentum basis and m is a
~k,σ2,λ k 11++λλσ σ~k σk ~k 11++λλσ j,~k,σ,λ
~
function of the momentum operator P of the form
σ −I
βjdet(1+λσ)−21fj ~k + (~k −λP~) .
I +λ
p (cid:0) (cid:1)
Proof. Equation (2.4) tells us how A acts as a quadratic form. In order to use (2.4), we will
λ
lookathv|Tr [ρA∗GA ]wiin thespecial case where G = G⊗I = |yihy|⊗I is a one-dimensional
2 λ λ
projection tensored with the identity over the light-particle Hilbert space. Formally, this allows
us to write
hv|Tr [ρA∗GA ]wi = β hv⊗f |A∗|y ⊗φ ihy ⊗φ |A |w⊗f i,
2 λ λ j j λ l l λ j
XX
j l
where (φ ) is some orthonormal basis over the light-particle Hilbert space allowing a repre-
m
sentation of the identity operator as a sum of one-dimensional projections , and the spectral
decomposition of ρ has been used. Once (2.4) has been applied, we build up to an expres-
sion (2.6), taking care with respect to the limits involved. By Corollary 3.4, the expression (2.6)
defines a bounded completely positive map (c.p.m.). Since Tr [ρA∗GA ] defines a c.p.m. and
2 λ λ
agrees with (2.6) for one-dimensional orthogonal projections, it follows that the two expressions
are equal on B(L2(Rd)). This follows because c.p.m.’s are strongly continuous and the span of
one-dimensional orthogonal projections is strongly dense.
The following holds, where the right-hand side converges in the operator norm:
Tr [ρA∗GA ] = β (id⊗hf |)A∗GA (id⊗|f i).
2 λ λ j j λ λ j
X
j
For G = |yihy|, (id ⊗hf |)A∗GA (id ⊗ |f i) = ϕ (I), where ϕ is the completely positive
j λ λ j y,j y,j
map such that for H ∈ B(H)
ϕ (H) = (id⊗hf |)A∗(|yihy|⊗H)A (id⊗|f i).
y,j j λ λ j
Since ϕ is completely positive, ϕ ( m |φ ihφ |) converges strongly to ϕ (I). ϕ (I) is
y,j y,j l=1 l l y,j y,j
determined by its expectations hv|ϕ (I)Pvi, and moreover
y
N
hv|ϕ (I)vi = lim hv|ϕ ( |φ ihφ |)vi
y,j y,j m m
N→∞ X
m=1
N
= lim hφ |υ ihυ |φ i = kυ k2 = d~kυ¯ (~k)υ (~k), (2.7)
N→∞X m v,j,y v,j,y m v,j,y ZRn v,j,y v,j,y
m=1
7
where υ is defined as the vector υ = (hy|⊗id)A (|vi⊗|f i). Using (2.4), hφ |υ i can
v,j,y v,j,y λ j m v,j,y
be expressed as
I
hφ |υ i = dK~ dσS (~k −λK~) det(I +λσ)−1
m v,j,y Z Z λ I +λσ
SOn (cid:0)(cid:12) (cid:12)(cid:1)
(cid:12) σ −I (cid:12) σ −I
φ¯ (~k)y¯ K~ + (~k −λK~) f ~k + (~k −λK~ v(K~). (2.8)
m j
I +λσ I +λσ
(cid:0) (cid:1) (cid:0) (cid:1)
By (2.7), we can evaluate Tr [(|f ihf |)A∗(|yihy| ⊗ I)A ] = hv|ϕ (I)vi through expression
2 j j λ λ y
~ ~ ~ ~
dkυ¯ (k)υ (k). Through (2.8) we have an a.e. defined expression for the values υ (k).
Rn v,j,y v,j,y v,j,y
NR ow, writing down d~kυ¯ (~k)υ (~k) using the expression for υ (~k), the result can be
Rn v,j,y v,j,y v,j,y
viewed as an integraRl of operators acting from the left and the right on |yihy|, followed by an
evaluation hv|(·)vi. Using the intertwining relation:
σ −I
m(P~)τ∗D τ = τ∗D τ m(P~ − (~k −λP~)),
~k 11++λλσ σ~k ~k 11++λλσ σ~k 1+λσ
for a function m(P~) of the momentum operators P~ and the fact that σ+λ = σI+λσ−1 is an
I+λσ I+λσ
isometry for 0 ≤ λ < 1, the expression can be written:
~k −λP~
hv|ϕ (I)vi = hv| d~k dσ dσ [U∗ m∗ S¯
y,j ZRn ZSOn×SOn 1 2 ~k,σ1,λ j,~k,σ1,λ λ(cid:0)(cid:12) 1+λ (cid:12)(cid:1)
(cid:12) (cid:12)
~k −λP~
(|yihy|)S m U ]|vi.
λ 1+λ j,~k,σ2,λ ~k,σ2,λ
(cid:0)(cid:12) (cid:12)(cid:1)
(cid:12) (cid:12)
So ϕ (I) = Tr [(|f ihf |)A∗(|vihv|)A ] agrees with the expression (2.6) for a fixed j and for
y,j 2 j j λ λ
G = |vihv|forallv,andhencebyourobservationatthebeginningoftheproof,Tr [(|f ihf |)A∗GA ]
2 j j λ λ
is equal to the expression (2.6) for a single fixed j and all G ∈ B(L2(Rn)). However, if we take
the limit m → ∞ for ρ = m β |f ihf |, then the expression (2.6) converges in the operator
m j=1 j j j
norm and Tr [ρ A∗GA ] coPnverges to Tr [ρA∗GA ]. Hence we have equality for all trace class
2 m λ λ 2 λ λ
ρ.
Throughthe formula Tr [ρS∗GS ] = G+B˜∗G+GB˜+B˜(G), it is clear thatB˜∗+B˜ = −B˜(I)
2 λ λ
by plugging in G = I. However, it is not at all obvious that this equality takes place through
the expressions (2.5) and (2.6) for B˜∗ and B˜(I), respectively, since the operators U appear
~k,σ,λ
only in form for B˜(I).
It is convenient to notice the intertwining relation h(~k−λP~)U = U h( 1+λ ~k−λP~)).
~k,σ,λ ~k,σ,λ I+λσ
Let g ∈ L2(Rn), then gˆ = B˜(I)g can be written:
gˆ(p~) = d~k dσ dσ U∗ m∗ U (p~)
Xj ZRn ZSOn×SOn 1 2(cid:0) ~k,σ1,λ j,~k,σ1,λ ~k,σ1,λ(cid:1)
I
|S |2 (~k −λp~) U∗ m U (p~)(U∗ U g)(p~), (2.9)
λ I +λσ ~k,σ1,λ j,~k,σ2,λ ~k,σ1,λ ~k,σ1,λ ~k,σ1,λ
(cid:0)(cid:12) 1 (cid:12)(cid:1)(cid:0) (cid:1)
(cid:12) (cid:12)
where we have intertwined U∗ from the left to the right, and
~k,σ1,λ
σ −I
U~k∗,σ1,λm∗j,~k,σ1,λU~k,σ1,λ (p~) = βj det(I +λσ1)−21 f¯j ~k + I +1 λσ (~k −λp~) ,
(cid:0) (cid:1) p (cid:0) 1 (cid:1)
8
U~k∗,σ1,λmj,~k,σ2,λU~k,σ1,λ (p~) = βj det(I +λσ2)−21 fj ~k +(σ2 −I)(I +λσ1)−1(~k −λp~) ,
(cid:0) (cid:1) p (cid:0) (cid:1)
U∗ U g (p~) = det 1+λ 12 det I +λσ2 21 g(p~+(σ −σ )(I +λσ )−1(~k −λp~)).
~k,σ1,λ ~k,σ1,λ I +λσ 1+λ 1 2 1
(cid:0) (cid:1) (cid:0) 1(cid:1) (cid:0) (cid:1)
Making the change of variables σ1 (~k −λp~) → ~k, the resulting expression has only angular
I+λσ1
dependance of σ σ−1 = σ, and integrating out the other angular degrees of freedom yields B˜.
2 1
3 Bounding integrals of non-commuting operators
Now we move on to proving Propositions 3.1 and 3.3 below which are proved in much greater
generality thanneeded forthis section, but they will serve asthe principle toolsin Section 4. To
state these propositions we will need to generalize the concept of a multiplication operator. Let
H , H be Hilbert spaces. Given a bounded function M : Rn → B(H ,H ) we can construct an
1 2 1 2
element M ∈ B(L2(Rn)⊗H ,L2(Rn)⊗H ) using the equivalence L2(Rn)⊗H ∼= L2(Rn,H ),
1 2 1 1
where for f ∈ L2(Rn)⊗H
1
M(f)(~x) = M(~x)f(~x).
We will call these multiplication operators.
Proposition 3.1. Define B : L2(Rn)⊗H → L2(Rn)⊗H , s.t.
1 2
B = d~k dστ∗τ q , (3.1)
ZRn ZSOn ~k aσ~k ~k,σ
where q is a multiplication operator in the P~ basis of the form:
~k,σ
q = n (P~)η(x ~k +y P~,x ~k +y P~),
~k,σ ~k,σ 1,σ σ 2,σ σ
where η(~k ,~k ) is continuous and defines a trace class integral operator on L2(Rn),
1 2
a ,x ,x ,y ∈ M (R), and n ∈ B(L2(Rn)⊗H ,L2(Rn)⊗H ) is a multiplication operator.
σ 1,σ 2,σ σ n ~k,σ 1 2
Let
|det(x +y (a −I))|,|det(x +y (a −I))|,|det(x )|, and |det(x )|
1,σ σ σ 2,σ σ σ 1,σ 2,σ
be uniformly bounded from below by 1 for some c > 0. Finally, let the family of maps n (K~) ∈
c ~k,σ
B(H ,H ) satisfy the norm bound:
1 2
supkn k ≤ r.
~k,σ
~k,σ
Then B is well defined as a strong limit and is bounded in operator norm by
kBk ≤ crkηk .
1
Proof. We check the conditions for Proposition A.2 (applied for integrals rather than sums).
Due to the intertwining relations between the unitaries τ∗τ and the multiplication operators
~k aσ~k
q , we will then have a bound from above by an integral of multiplication operators. We must
~k,σ
show that 1(G +G ) is bounded, where
2 1 2
G = d~k dσ|τ∗τ q | and G = d~k dσ|q∗ τ∗ τ |.
1 Z Z ~k aσ~k ~k2,σ2 2 Z Z ~k1,σ1 aσ~k ~k
SOn SOn
9
The integrand of G is the multiplication operator
1
|τ∗τ q | = |n (P~)||η(x ~k +y P~,x ~k +y P~)|.
~k a~k ~k,σ ~k,σ 1,σ σ 2,σ σ
and the integrand of G is
2
|q∗ τ∗ τ | = τ∗τ∗ |n (P~)||η(x ~k +y P~,x ~k +y P~)|τ∗ τ
~k,σ aσ~k ~k ~k aσ~k ~k,σ 1,σ σ 2,σ σ aσ~k ~k
= |n (P~ +σ~k −~k)||η(x′ ~k +y P~,x′ ~k +y P~)| (3.2)
~k,σ 1,σ σ 2,σ σ
where x′ = x +y (a −I), and we have used that τ M(P~) = M(P~ −k)τ .
j,σ j,σ σ σ k k
However since the operators in the integrand of G are all multiplication operators in P~,
1
bounding a sum on them in the operator norm can becomputed as a supremum in the following
way:
kG k ≤ supk d~k dσ|n (P~)||η(x ~k +y P~,x ~k +y P~)|k
1 Z Z ~k,σ 1,σ σ 2,σ σ B(H1)
P~ SOn
≤ supkn (P~)k sup d~k dσ|η(x ~k +y P~,x ~k +y P~)| (3.3)
~k,σ B(H1) Z Z 1,σ σ 2,σ σ
(cid:0) P~ (cid:1) P~ (cid:0) SOn (cid:1)
A similar result holds for G . Now applying Lemma A.1 to (3.3) along with our conditions
2
on x , x , and n (P~) we get the bound kG k ≤ rckηk .
1,σ 2,σ ~k,σ 1 1
Corollary 3.2. The integral of operators (2.5) converges strongly to a bounded operator with
norm less than or equal to 1 kρk .
(1−λ)n 1
The bound in the above corollary in not sharp, since in Proposition (2.2) we show that
B˜ = Tr [ρA∗]. Thus kB˜k ≤ kρk kS −Ik ≤ 2kρk, since S is unitary.
2 λ 1 λ λ
Proof. We apply Proposition (3.1) with n (P~) = S (|~k|), η = ρ, a = σ, x = 1 + λ,
~k,σ λ σ 1,σ
x = I+σ, and y = λ. |n (P~)| ≤ 1, so we can take r = 1. All determinants involved are of
2,σ σ ~k,σ
operators of the form σ +λσ where σ ,σ ∈ SO , so these determinants have a lower bound
1 2 1 2 n
of (1−λ)n. Hence we can take c = (1−λ)−n.
Proposition 3.3. Let G ∈ B(H ⊗ L2(Rn),H ⊗ L2(Rn)), and ϕ : B(H ⊗ L2(Rn),H ⊗
l r l r
L2(Rn)) → B(H0 ⊗L2(Rn),H0 ⊗L2(Rn)) has the form
l r
ϕ(G) = d~k dσ dσ U∗ h∗ Gg U ,
Xj Z ZSOn×SOn 1 2 ~k,σ1 j,~k,σ1 j,~k,σ2 ~k,σ2
where U~k,σ acts on the L2(Rn) tensor as U~k,σ = τ~kDbσ τa∗σ~k, and hj,~k,σ and gj,~k,σ are multiplica-
~
tion operators in P of the form:
h = n(1) (P~)η(1)(x ~k +x P~), and g = n(2) (P~)η(2)(x ~k +x P~).
j,~k,σ j,~k,σ j 1,σ 2,σ j,~k,σ j,~k,σ j 1,σ 2,σ
10