Unitary causal quantum stochastic double
products as universal interactions I.
Robin L Hudson,
School of Mathematics,
University of Loughborough,
Leicestershire LE11 3TU,
Great Britain.
Yuchen Pei,
Mathematics Research Centre,
University of Warwick,
Coventry CV4 7AL,
Great Britain.
.
January 31, 2015
Abstract
After reviewing the theory of trianguar(causal) and rectangular quantum stochastic double product integrals, we consider examples when these
consist of unitary operators. We …nd an explicit form for all such rectangular product integrals which can be described as second quantizations.
Causal products are proposed as paradigm limits of large random matrices in which the randomness is explicitly quantum or noncommutative in
character.
PACS 02.50Fz, 42.50Lc
1
Introduction: from discrete to continuous double products.
Double products of triangular (or causal) and rectangular forms
Y
Y
xj;k ;
xj;k
1 j<k N
(1)
(j;k)2Nm Nn
are problematical when the elements xj;k do not commute with
Y each other because there is no preferred ordering for them. Thus, while
xj is naturally
1 j N
de…ned as x1 x2 :::xN ; respecting the natural ordering of the integers, one might
de…ne the triangular product in (1) either as
8
9
8
9
=
=
Y < Y
Y < Y
xj;k or as
xj;k :
(2)
:
;
:
;
1 j<N
j<k N
1<k N
1
1 j<k
Similarly one might take either
8
Y < Y
1 j m
:
1 k n
xj;k
9
=
;
or
8
Y < Y
1 k n
:
1 j m
xj;k
9
=
(3)
;
to de…ne the rectangular product in (1).
These ambiguities are resolved if the quantities xj;k have the property which
we call weak commutativity, that xj;k commutes with xj 0 ;k0 whenever both j 6= j 0
and k 6= k 0 (but not necessarily when j = j 0 but k 6= k 0 or vice versa). Under the assumption of weak commutativity the two alternatives (2) are equal
as are the two alternatives (3). More generally weak commutativity
implies
Y
xjr ;kr where
the equality of all triangular products of the form
1 r
1
2 N (N
1)
is any ordering of the 12 N (N 1)
(j1 ; k1 ) ; (j2 ; k2 ) ; :::; j 12 N (N 1) ; k 12 N (N 1)
indices f(1; 2) ; (1; 3) ; :::; (N 1; N )g with the property that (jr ; kr ) always precedes (js ; ks ) whenever both jr js Y
and kr ks : Similarly, given weak commutativity, all rectangular products
xjr ;kr have the same value for any or1 r mn
dering of the mn indices f(1; 1) ; :::; (m; 1) ; (1; 2) ; :::; (m; 2) ; :::; (1; n) ; :::; (m; n)g
having this same property.
An example where weak commutativity arises is when each xj;k is obtained
by embedding a 2 2 matrix, in the triangular case at the intersections of the
j-th and k-th rows and columns of an N N matrix , and in the rectangular
case in the intersections of the j-th and (m + k)-th rows and columns of an
(m + n) (m + n) matrix, and then completing by adding 1’s and 0’s in the
remaining diagonal and nondiagonal places respectively. A second example is
when, in the triangular case, each xj;k is the ampliation, to the N -fold tensor
product N H of a Hilbert space H with itself, of an operator on the 2-fold
tensor product H H embedded in the j-th and k-th copies of H within N H,
and similarly in the rectangular case if it is the ampliation to m+n H of such an
operator embedded in the j-th and (m + k)-th copies of H within m+n H =
( m H) ( n H) :
This paper concerns double product integrals, denoted
Y
Y
(1 + dr) ;
(1 + dr) ;
(4)
<[a;b[
[a;b[ [c;d[
which are continuous analogs of such discrete triangular and rectangular double
products de…ned over either an interval [a; b[ of the real line or the Cartesian
product [a; b[ [c; d[ of two such intervals. Here we denote by <[a;b[ the set
(x; y) 2 R2 : a x < y < b : In analogy and by extension of the second example above, they consist of operators in continuous tensor products of Hilbert
spaces such as Fock spaces. Each such product is characterize by a stochastic
di¤ erential generator dr: This is an element of I I where I = C hdP; dQ; dT i
2
is the complex vector space of linear combinations of the stochastic di¤erentials
of the components of a quantum planar Brownian motion (P; Q) and of the time
process T:
The quantum Itô product formula equips the space I; and hence also I I;
with an associative multiplication so that both become y-algebras under natural
involutions in which dP; dQ and dT are all self-adjoint.
Alternative notations which we will …nd useful are
Y
Y
(1 + dr) =
(1 + dr (x; y))
<[a;b[
Y
a x<y<b
(1 + dr)
Y
=
(1 + dr (x; y)) :
a x<b;c y <d
[a;b[ [c;d[
We also write for example when
dr = dP
Y
(1 + dr)
<[a;b[
Y
Y
=
dQ
dQ
dP;
(1 + dP (x) dQ (y)
(5)
dQ (x) dP (y))
a x<y<b
(1 + dr)
Y
=
(1 + dP (x)
dQ (y)
dQ (x)
dP (y)) :
a x<b;c y <d
[a;b[ [c;d[
The latter rectangular product integral lives in a double continuous tensor product, namely the tensor product of two Fock spaces.
In both cases the product integrals (4) can be approximated heuristically by
partitioning the underlying intervals and replacing each di¤erential in I contributing to dr by increments of the corresponding process over the subintervals
of the partition, so that
Y
Y
Y
Y
(1 + j;m+k r)
(1 + dr) '
(1 + j;k r) ;
(1 + dr) '
<[a;b[
1 j<k N
[a;b[ [c;d[
(j;k)2Nm Nn
where, for example, in the case (5), with the partition
[a; b[
=
N
[
[xj
1 ; xj [
with xj = a +
j=1
j;k r
=
j
(b
N
a) ;
(P (xj ) P (xj 1 )) (Q (xk ) Q (xk 1 ))
(Q (xj ) Q (xj 1 )) (P (xk ) P (xk 1 )) :
Because of the commutation relations
[P (s) ; Q (t)] =
2i min fs; tg I; [P (s) ; P (t)] = [Q (s) ; Q (t)] = 0
(6)
satis…ed by the processes P and Q; we have weak commutativity of the j;k r and
the approximating discrete double products are thus well de…ned in all cases.
3
In some cases this form of approximation can be used to explicitly evaluate
the corresponding continuous double product as a limit. Indeed, if one follows
the original philosophy of Volterra [12], that product integrals are direct multiplicative analogues of additive integrals such as those of Riemann, Lebesgue
and Itô [11], then our double products should properly be de…ned as such limits.
However there are formidable technical di¢ culties in making such a de…nition
general and rigorous in the triple generalization involved in quantum, stochastic,
double product integrals. So we follow an alternative de…nition.
We de…ne our double product integrals as solutions of quantum stochastic
di¤erential equations (qsde’s) which are themselves driven by solutions of qsde’s,
using quantum stochastic calculus . There are two de…nitions in the rectangular case, corresponding to the two equivalent forms (3) of the approximations,
namely
Y
(1 + dr)
=
[a;b[ [c;d[
=
b
a
Y
Y
d
c
1+
Y
cd
1 +ba
(1 + dr)
(7)
(1 + dr) :
(8)
c
Y
c
Here the single product integrals are all de…ned as solutions of quantum stoY
cd
chastic di¤erential equations (qsde’s). For example
c (1 + dr) is the value
X (d) at d of the solution of the qsde
dX = (X + 1) dr; X (c) = 0
in which the second copy of I in I I is operative in the sde, the …rst copy
being an initial system algebra in which the basis elements dP; dQ and dT are
multiplied according to the quantum Itô product rule. Note that the algebra I
is nonunital so that an initial condition X (c) = 1 is meaningless; the 1 in the
Y
cd
qsde is the identity operator in the Fock space.
c (1 + dr) is thus of form
X (d) = dP
Jcd ( ) + dQ
Jcd ( ) + dT
Jcd ( ) :
where the operators Jcd ( ) ; Jcd ( ) and Jcd ( ) are …nite sums of iterated quantum stochastic integrals over [c:d[ of at most second order because I 3 = 0: The
right hand side of (7) becomes
b
a
Y
1+
Y
cd
c
(1 + dr)
=ba
Y
1 + dP
Jcd ( ) + dQ
Jcd ( ) + dT
Jcd ( )
which is, by de…nition, the value Y (b) at b of the solution of the qsde
dY = Y dP
Jcd ( ) + dQ
Jcd ( ) + dT
Jcd ( ) ; Y (a) = I:
Here the system algebra consists of …nite sums of iterated stochastic integrals
over [c:d[ and is on the right, so that the solution Y consists of operators on
4
the tensor product of two Fock spaces. Existence and uniqueness follow using
results of Fagnola [2].
More details of the construction of both rectangular and triangular double
product integrals, in particular proof of the equivalence of the two forms (7) and
(8), can be found in [3]. It is also proved in [3] that a necessary and su¢ cient
condition for either a rectangular or triangular double product integral to be
unitary valued is that the di¤erential generator dr satisfy the condition
dr + dry + drdry = 0:
(9)
In the present paper we analyze some examples of generators dr satisfying the
unitarity condition (9) and show how the corresponding unitary double product
integrals can be constructed as second quantizations of unitary operators which
can be determined explicitly as limits of discrete products. Finally we discuss the
possibility that such product integrals provide alternative paradigms to models
of random evolutions in terms of large random matrices.
2
Analysis of the unitarity condition.
The components P and Q of quantum planar Brownian motion [1] (P; Q) are
individually classical one-dimensional Brownian motions, so that for example,
for each t > 0; P (t) is a normally distributed random variable of zero mean
and variance t: But P and Q do not commute with each other, instead they
satisfy the Heisenberg commutation relations (6) in the rigorous Weyl sense.
The pair (P; Q) is realised [1] as processes of self-adjoint operators in the Fock
space F =
L2 [0; 1[ over the Hilbert space L2 [0; 1[ (some de…nitions are
given in Section 4 below). Probabilities are determined in the usual quantum
probabilistic way [9][8] using the vacuum as state vector.
Corresponding to each natural direct sum decomposition such as
L2 [0; 1[= L2 [0; s1 [
N
M1
j=1
L2 [sj ; sj+1 t[ L2 [sN ; 1[;
the Fock space F decomposes into a tensor product
F = F s01
N
O1
j=1
Fssjj+1
Fs1
N
in such a way that each exponential vector is a product vector:
e (f ) = e (f0s1 )
e fss12
e fss23
e fssNN
1
1
e (fN
)
formed from the restrictions of the function f 2 L2 [0; 1[ to the intervals [0; s1 [;
[s1 ; s2 [; [s2 ; s3 [; :::; [sN 11 ; sN [; [sN ; 1[.
5
The quantum Itô product table for planar Brownian motion (P; Q) enables
quantum stochastic integrals [6] to be multiplied according to the Leibniz-Itô
formula for the stochastic di¤erential of their product:
d (M N ) = (dM ) N + M dN + dM dN
where the product dM dN is evaluated from the Itô multiplication rule for the
basic di¤erentials, namely
dP
dT
idT
0
dP
dQ
dT
dQ
idT
dT
0
dT
0
0
0
:
Here the time process T consists of multiples of the identity operator I, T (t) =
tI:
The Itô algebra I is nilpotent, I 3 = 0; and I 2 = CdT is one-dimensional.
It follows that all products in I I are scalar multiples dT dT; where the
scalars for products of the basis elements drj;k = dRj dRk ; j; k = 1; 2; 3
where (dR1 ; dR2 ; dR3 ) = (dP ; dQ ; dT ) ; are given by the table
dr11
1
i
i
1
dr11
dr12
dr21
dr22
dr12
i
1
1
i
dr21
i
1
1
i
dr22
1
i
i
1
(10)
for j; k 2 f1; 2g ; while all such products involving j = 3 vanish.
Let a general element of I I be expanded in the form
dr = i
3
X
j;k drj;k
j;k=1
for complex scalars
j;k :
Theorem 1 The unitarity condition (9) holds if and only if
(j; k) 6= (3; 3) ; and
i(
33
Proof. We have dry =
33 )
i
+(
3
X
2
22 )
11
u;v dru;v
+(
21
+
2
12 )
j;k
is real for all
= 0:
(11)
and so, using the table (10),
u;v=1
n
2
drdry =
j 11 j + j
+2i Im (( 12 +
+2 Re ( 12 21
2
12 j
2
2
+ j 21 j + j 22 j
21 ) ( 11
22 ))
)g
dT
dT
22 11
6
:
(12)
Thus, for the condition (9) to hold we require …rstly that uv
uv = 0 for all
(u; v) 6= (3; 3) ; so that these uv must all be real numbers. When this is the
case (12) becomes
drdry = (
2
22 )
11
+(
21
+
2
12 )
:
Thus for (9) to hold we must have (11). Conversely these conditions imply (9).
Thus the unitary generators without any time terms, that is, with
if either u or v = 3; so that in particular 33 = 0; are of form
where
3
dr
;
= i ( (dP
=
12
=
21
dQ
and
dQ
=
11
dP ) + (dP
=
dP + dQ
dQ))
uv
=0
(13)
are real parameters.
22
Approximation by discrete double products.
Y
We approximate the rectangular and triangular double product integrals
[a;b[ [c;d[
Y
(1 + dr ; ) and
(1 + dr ; ) where dr ; is the generator (13) as follows.
<[a;b[
We partition the intervals [a; b[ and [c; d[ into m and n equal sized subintervals,
[a; b[=
m
G
[xj
1 ; xj [; [c; d[=
j=1
where xj = a +
(13) as
i ( (dP
dQ
j
m
(b
dQ
n
G
[yk
1 ; yk [
(14)
k=1
a) ; yk = c +
dP ) + (dP
k
n
(d
c) : Writing the right hand side of
dP + dQ
dQ)) = i
2
X
uv dRu
dRv
0
k
!!
u;v=1
we then make the approximation
Y
(1 + dr
;
)
=
[a;b[ [c;d[
Y
1+i
uv dRu
u;v=1
[a;b[ [c;d[
'
2
X
Y
1+i
2
X
uv j
dRv
!
(Ru )
(Rv ) (15)
u;v=1
(j;k)2Nm Nn
where
j
(Ru ) = Ru (xj )
Ru (xj
0
1) ; k
7
(Rv ) = Rv (yk )
Rv (yk
1) :
Because of the commutation relation (6) each of the pairs (R1 (xj )
R2 (xj ) R2 (xj 1 )) satis…es the commutation relation
[R1 (xj )
R1 (xj
1 ) ; R2
(xj )
R2 (xj
1 )]
=
2i (xj
xj
1)
=
R1 (xj
2i
b
1) ;
a
m
and each of R1 (xj ) R1 (xj 1 ) and R2 (xj ) R2 (xj 1 ) commutes with each of
m
R1 (xk ) R1 (xk 1 ) and R2 (xk ) R2 (xk 1 ) : Hence the operators (pj ; qj )j=1
de…ned by
r
r
m
m
pj = i 1
(R1 (xj ) R1 (xj 1 )) ; qj = i 1
(R2 (xj ) R2 (xj 1 ))
b a
b a
satisfy the standard1 canonical commutation relations
[pj ; qk ] =
2i
jk ; [pj ; pk ]
= [qj ; qk ] = 0; j; k = 1; 2; :::; m:
m
Similarly, the operators (p0k ; qk0 )k=1 de…ned by
r
r
n
n
(R1 (yk ) R1 (yk 1 )) ; qk0 = i 1
(R2 (yk )
p0k = i 1
d c
d c
R2 (yk
1 )) :
satisfy the standard canonical commutation relations. The approximation (15) becomes
Y
(1 + dr ; )
[a;b[ [c;d[
'
where
4
Y
1+i
mn
pj
qk0
qj0
pk + (pj
p0k + qj
qk0 )(16)
(j;k)2Nm Nn
mn
=
q
(b a)(d c)
:
mn
A one-parameter unitary group in two degrees
of freedom.
We consider the selfadjoint operator
L( ; ) =
(pq 0
qp0 ) + (pp0 + qq 0 )
(17)
where (p; q) and (p0 q 0 ) are mutually commuting standard canonical pairs, together with the one-parameter unitary group eixL( ; ) x2R of which L ( ; ) is
the in…nitesimal generator. It is convenient to make use of two di¤erent unitarily equivalent standard representations of the quantum system of two degrees
of freedom described by (p; q) ; (p0 q 0 ) :
1 In
quantum probability it is most convenient to normalise Planck’s constant to the value
4 :
8
The Schrödinger representation is in the Hilbert space L2 R2 and is given
informally by
p
p=
2i
p
p
@
; q = 2s; p0 =
@s
2i
p
@
; q = 2t
@t
in terms of the components of the vector2 (s; t) in R2 : More rigourously the
0
corresponding one parameter unitary groups eixp x2R ; eixq x2R ; eixp
x2R
and eixq
0
x2R
are given by the actions
eixp f ((s; t) )
0
eixp f ((s; t) )
= f
= f
s+
p
; eixq f ((s; t) ) = ei
2x; t
p
s; t + 2x
0
p
; eixq f ((s; t) ) = ei
2xs
p
f ((s; t) )
2xt
f ((s; t) ) :
By contrast the Fock representation, which we will now describe, is in the
Fock space
C2 over the Hilbert space C2 : It is convenient to de…ne the Fock
space (H) over an arbitrary Hilbert space H abstractly as another Hilbert
space generated by a family (e (f ))f 2H of so-called exponential vectors satisfying
he (f ) :e (g)i = ehf;gi
for arbitrary f; g 2 H: Any two candidate Fock spaces are then unitarily equivalent under a unique unitary operator which exchanges the families of exponential vectors, and can thus be identi…ed. If R is a unitary operator on H then
there is a unique unitary operator (R) on (H) ; called the second quantization of R; such that for each f 2 H; (R) e (f ) = e (Rf ). If H = H1 H2 is
a direct sum we identify (H) with the tensor product (H1 )
(H2 ) with
e (f1 f2 ) = e (f ) e (f2 ) :
The family of Weyl operators (W (f ))f 2H consists of the unitary operators
for which, for arbitrary f; g 2 H;
W (f ) e (g) = e
hf;gi
2
1
2 kf k
e (f + g) :
They satisfy the Weyl relation
W (f ) W (g) = e
i Imhf;gi
W (f + g) :
For arbitrary unitary R on H and f 2 H we have
(R) W (f ) (R)
We realise (p; q) and (p0 ; q 0 ) in
eixp
0
eixp
2
1
= W (Rf )
(18)
C2 in terms of Weyl operators by de…ning
= W ((x; 0) ) ; eixq = W (( ix; 0) ) ;
0
= W ((0; x) ) ; eixq = W ((0; ix) ) :
denotes transposition
9
These satisfy the canonical commutation relations because, for example, the
Weyl relation implies that
eixp eiyq = e2ixy eiyq eixp
which is the rigorous form of the commutation relation
[p; q] =
2i:
To establish the unitary equivalence with the Schrödinger realisation we …rst
make the chain of identi…cations
C2 =
(C
C) =
(C)
(C) = l2
l2
where (C) is identi…ed with the standard Hilbert space l2 of square-summable
sequences, with
z2
z
e (z) = 1; p ; p ; ::: :
1!
2!
1
Next we map the space l2 onto L2 (R) by mapping the sequence (zn )n=0 2
1
X
1
l2 to
zn hn where (hn )n=0 is the orthonormal basis of Hermite functions
n=0
in L2 (R) : Finally we identify L2 (R) L2 (R) with L2 R2 by extending the
identi…cation (f g) ((s; t) ) = f (s) g (t) ; and thereby map
C2 = l 2 l 2
onto L2 (R) L2 (R) = L R2 : The Fock vacuum vector e (0C2 ) = e (0) e (0) is
mapped to h0 h0 which is the Gaussian function
G (s; t) =
1
e
1
2
(s2 +t2 ) :
Note that in the Schrödinger representation,
L ( ; ) (h0
h00 ) = 0
(19)
since
((pq 0
qp0 ) G) (s; t)
=
((pp0 + qq 0 ) G) (s; t)
=
2i
2
2
2
1
@
@
t s
e 2 (s +t ) = 0;
@s
@t
2
2
2
1
@
+ st e 2 (s +t ) = 0:
@s@t
Let us now realise eixL( ; ) in the Fock representation as the second quantization of a unitary operator given on C2 by left multiplication by a unitary
2 2 matrix. We introduce the notation
=
+ i = ei j j :
10
Theorem 2 In the Fock representation,
eixL(
; )
e i sin (2x jvj)
cos (2x jvj)
cos (2x jvj)
ei sin (2x jvj)
=
:
(20)
Proof. We use the notation
( ; )
Note …rst that
( ; )
(x)
consider
e i sin (2x jvj)
cos (2x jvj)
cos (2x jvj)
ei sin (2x jvj)
(x) =
:
is indeed a one-parameter unitary group. Now
x2R
1
( ; )
(x) eiyp
( ; )
(x) W ((y; 0) )
( ; )
(x)
1
=
( ; )
= W
( ; )
= W
cos (2x jvj) y; ei sin (2x jvj) y
(x)
(x) (y; 0)
0; ei sin (2x jvj) y
= W ((cos (2x jvj) y; 0) ) W
:
Since
0; ei sin (x jvj) y
W
= W ((0; cos sin (2x jvj) y + i sin sin (2x jvj) y) )
= ei cos
sin
sin(2xjvj)2 y 2
i cos
sin
sin(2xjvj)2 y 2 iy cos
= e
= ei sin(2xjvj)y(cos
W ((0; cos sin (2x jvj) y) ) W ((0; i sin sin (2x jvj) y) )
sin(2xjvj)p0
e
e
iy sin
sin(2xjvj)q 0
p0 sin q 0 )
we get
( ; )
Forming
(x) eiyp
d
i dy
y=0
1
( ; )
(x)
Forming
(x)
d
i dx
p
x=0
p0 sin q 0 ))
:
we deduce that
1
( ; )
= eiy(cos(2xjvj)p+sin(2xjvj)(cos
( ; )
(x)
= cos (2x jvj) p + sin (2x jvj) (cos p0
sin q 0 ) :
in turn we get
[iK; p] = 2 jvj (cos p0
sin q 0 ) = p0
q 0 = [iL ( ; ) ; p]
where K is the selfadjoint in…nitesimal generator of the one-parameter unitary
group
: Similarly, we …nd that
( ; ) (x)
x2R
[iK; q] = [iL ( ; ) ; q] ; [iK; p0 ] = [iL ( ; ) ; p0 ] ; [iK; q 0 ] = [iL ( ; ) ; q 0 ]
11
Since the Fock representation is irreducible we deduce that K di¤ers from
L ( ; ) by at most a multiple of the identity. But the second quantizations
e (0) to itself, hence
( ; ) (x) all map the vacuum vector e (0C2 ) = e (0)
K annihilates e (0) e (0) : Moreover L ( ; ) also annihilates e (0) e (0) ;
since by (19) its Schrödinger equivalent annihilates the equivalent vector in
the Schrödinger representation. It follows that K = L ( ; ) :
5
Realisation of rectangular double products as
second quantizations.
Since
mn
=
q
(b a)(d c)
;
mn
for large m and n we can make the approximation
1 + i mn L ( ; )
' exp (i mn L ( ; )) ;
for mutually commuting pairs (p; q) and (p0 ; q 0 ), to to obtain from (16) the
further approximation
Y
Y
(1 + dr ; ) '
exp (i mn ( (pj qk0 qj p0k ) + (pj p0k + qj qk0 ))) :
[a;b[ [c;d[
(j;k)2Nm Nn
(21)
We embed the Hilbert space Cm+n = Cm Cn into L2 ([a; b[) L2 ([c; d[) by
mapping each element "l of the canonical orthonormal basis of Cm+n to the
vector l ; where
8 q
m
<
0 if l = 1; 2; :::; m
b a [xl 1 ;xl [
q
=
:
l
n
: 0
d c [xl 1 ;xl [ if l = m + 1; m + 2; :::; m + n
We regard each complex m n matrix M = [Mj;k ]j;k2Nm N acting on Cm+n
by left multiplication as an operator on L2 ([a; b[) L2 ([s; t[) given by
m+n
X
Mj;k
j;k=1
12
j
h
kj
Then using (20), the approximation (21) becomes
Y
Y
(1 + dr ; ) '
[a;b[ [s;t[
(j;k)2Nm Nn
02
B6 1
B6 ..
B6 .
B6
B6 (j) 0
B6
B6 .
B6 ..
B6
B6 (m+k) 0
B6
B6 .
@4 ..
..
.
(j)
(m+k)
0
..
.
0
..
.
cos (2
..
.
ei sin (2
..
.
0
jvj)
mn
..
e i sin (2 mn jvj)
..
.
cos (2 mn jvj)
..
.
.
mn jvj)
0
..
.
0
31
0 7C
.. 7C
C
. 7
7C
7
0 7C
C
C
.. 7
7
. 7C
C
C
0 7
7C
7
.. 5C
. A
1
which is equal to the second quantization of the matrix operator
2
6 1
6 ..
6 .
6
6 (j) 0
Y
6
6 .
6 ..
(j;k)2Nm Nn 6
6 (k) 0
6
6 .
4 ..
..
.
(j)
(m+k)
0
..
.
0
..
.
cos (2
..
.
ei sin (2
..
.
0
jvj)
mn
..
e i sin (2 mn jvj)
..
.
cos (2 mn jvj)
..
.
.
mn jvj)
0
0
..
.
3
0 7
.. 7
. 7
7
0 7
7
.. 7
:
. 7
7
7
0 7
.. 7
. 5
1
We denote by this matrix operator by Wm;n : The limit
lim Wm;n may then
m;n!1
be found in two stages as follows. First consider Wm;1 and W1;n ; which are
respectively given by
2
6
6
6
6
6
m
Y6
6
6
6
j=1 6
6
6
6
4
(j)
1
0
..
.
(j)
0
..
.
0
0
1
..
.
0
0
0
..
.
0
(m+1)
0 0
..
.
(m+1)
0
0
..
.
0
0
..
.
0
0
..
.
0
0
..
.
0
..
.
0
1
..
.
0
0
..
.
0
..
.
0
13
0
0
..
.
1
0
0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
2
6
6
6
6
6
=6
6
6
6
4
and
2
m 3
0
0
..
.
0
..
.
0
0
0
0
6
6
6
6
n 6
Y
6
6
6
k=1 6
6
6
4
m 4
m 3
m 2
m 5
m 4
m 3
..
.
.
m 1
..
.
..
.
0
2
m 2
m 1
m
3
k+1
0
1
..
.
0
..
.
(k+1)
..
.
0
..
.
0
0
..
.
0
..
.
0
0
..
.
0
..
.
0
1
..
.
0
0
0
0
n
2
..
.
0
0
..
.
..
0
0
..
.
.
1
n 2
n 1
n 3
n 2
n 2
n 4
n 3
n 3
0
0
..
.
0
..
.
0
0
0
0
n 5
..
.
0
0
a)(d c)
mn
..
n 4
..
.
.
..
.
0
jvj and
=
=
(b
a) (d
mn
e
i
Using the limits
lim
m!1
m
7
7
7
7
7
7
7
7
7
7
7
5
n 1
q
= cos 2 (b
=
..
2
6
6
6
6
6
=6
6 ..
6 .
6
4
where
..
.
0
0
m 2
=
0
lim @1
m!1
= e
1
2
2
r
2(b a)(d c)jvj2 =n
;
c)
3
7
7
7
7
7
7:
7
7
7
5
(22)
3
7
7
7
7
7
7
7
7
7
5
sin 2
q
(23)
(b a)(d c)
mn
!2 1m
jvj A
the limit of the operator (22) is the matrix operator on L2 ([a; b[) C
2
E 3
(d c)=n
I + An
e i f[a;b[
D
5:
lim Wm;1 = 4
2
(d c)=n
m!1
ei g[a;b[
e 2(b a)(d c)j j =n
(d c)=n
(24)
D
(d c)=n
and g[a;b[
denote respectively the map C !L2 ([a; b[) ; z 7!
D
E
(d c)=n
and the covector L2 ([a; b[) ! C; h 7! g[a;b[
; h ; where for s 2
(d c)=n
Here f[a;b[
zf[a;b[
[a; b[
jvj :
(d c)=n
f[a;b[
E
(s) = e
2(d c)j j2 (s a)=n
(d c)=n
; g[a;b[
14
(s) = e
2(d c)j j2 (b s)=n
;
and An is the integral operator on L2 ([a; b[) whose kernel is
2
(s; t) 7!
2j j
<[a;b[ e
2j j2 (d c)(t s)=n
:
Similarly, since
lim
n!1
n
2(b a)(d c)jvj2 =m
=e
;
the limit of the operator (23) is the matrix operator on C L2 ([c; d[)
3
2
D
2
(b a)=m
e 2(b a)(d c)jvj =m
g[c;d[
5:
lim W1;n = 4 (b a)=m E
n!1
f[c;d[
I + Dm
(25)
The matrix product rules (22) and (23) continue to hold even if the scalars
; ; and are replaced by a scalar, a covector, a vector and an operator on
L2 (R) in the …rst case, and by an operator, a vector , a covector and a scalar
respectively in the second. It is thus possible to evaluate lim
lim Wm;n
m!1
and lim
n!1
n!1
lim Wm;n by substituting into (22) and (23) respectively the four
m!1
entries in the matrix (25) and in (24). Denoting by V[a;b[ the integral operator
on L2 ([a; b[)
Z b
V[a;b[ f (s) =
f (t) dt; s 2 [a; b[
s
and using the limits
m
lim (I + Dm )
m!1
=e
jvj2 (b a)V[c;d
jvj2 (d c)V[a;b[
n
; lim (I + An ) = e
n!1
one …nds after some computation that the two iterated limits are equal, and
that their common value is
I +A
B
C
I +D
lim Wm;n =
m;n!1
;
(26)
where A; B; C and D are respectively integral operators from L2 ([a; b[) to itself,
from L2 ([c; d[) to L2 ([a; b[) and vice versa, and from L2 ([c; d[) to itself, whose
kernels are given repectively by
Ker A (s; t)
=
<[a;b[ (s; t)
1 (t
X
N
Ker B (s; t)
=
[a;b[ [c;d[
jvj (d
Ker C (s; t)
=
[c;d[ [a;b[
Ker D (s; t)
=
<[c;d[ (s; t)
N =0
15
s) (t
c)
;
2
(N !)
2
jvj (c
N
s) (t
a)
;
2
(N !)
N =0
1 (t
X
;
N
2
jvj (b
N =0
1
X
c)
N ! (N + 1)!
N =0
1
X
N +1
2
s)
N
s)
2
jvj (b
N ! (N + 1)!
N +1
a)
:
See [4] for more details of essentially this argument in the case = 0 and [5] for
an alternative derivation where also it is shown rigorously that the limit (26)
is indeed a unitary operator and that these heuristics are rigorously justi…ed.
Namely,
I +A
B
C
I +D
=
Y
(1 + i (dP
dQ
dQ
dP ))
[a;b[ [c;d[
in so far as the relevant qsde’s are satis…ed.
An explicit form similar to (26) for the corresponding unitary operator
W<[a;b[ such that
Y
W<[a;b[ =
(1 + i ( (dP dQ dQ dP ) + (dP dP dP dP )))
<[a;b[
is more di¢ cult to obtain. However in the case = 0 it has been found provisionally [10]. At present it remains to be veri…ed that the operator W<[a;b[ is
rigourously unitary and that
W<[a;b[ satis…es the de…ning qsde.
6
Universality.
The unitary generators of form dr ; of form (13) can be characterized among
all such generators by their invariance under rotational automorphisms of the
form
(dP; dQ) 7! (cos dP sin dQ; sin dP + cos dQ)
of the Ito algebra I: The corresponding double products are likewise invariant
under gauge transformations
(P; Q) 7! (cos P
sin Q; sin P + cos Q)
of the underlying quantum planar Browian motion.
The generators dr ; are complemented by a second real two parameter
family
dr0 ; = i ( (dP
dQ + dQ
dP ) + (dP
dP
dQ
dQ)) +
;
dT
dT
of unitary generators which require the additional non-zero time term of form
dT to satisfy the unitarity condition (9). The corresponding dou; dT
ble products are no longer given by second quantizations. However they can
be characterized as unitary implementors of explicit Bogolubov transformations, that is, invertible real-linear transformations of the complex Hilbert space
L2 (R+ ) L2 (R+ ) which preserve the imaginary part of the inner product. See
[7] for the case when = 0:
It is expected that the full 4-dimensional manifold of unitary double products, with generators of form
d
; ;
0; 0
= dr
;
+ dr0 0 ;
16
0
+
; ;
0; 0
dT
dT;
admits a similar characterization as unitary implementors of explicit Bogolubov
transformations, but this has yet to be established in full generality.
Y
We propose that triangular (or causal) double products of form
1+d
<[a;b[
provide a randomized model for unitary time evolution of complex non-relativistic
quantum systems, and conjecture that these products o¤er paradigm universal
limits for large random unitary matrices.
References
[1] A M Cockcroft and R L Hudson, Quantum mechanical Wiener processes,
Journal of Multivariate Analysis 7, 107-124(1977).
[2] Fagnola F. On quantum stochastic di¤erential equations with unbounded
coe¢ cients, Probab. Th. and Rel. Fields 86 (1990) 501-516.
[3] R L Hudson, Forward and backward adapted quantum stochastic calculus
and double product integrals, Russian Journal of Mathematical Physics 21,
No3 (V P Belavkin Memorial Volume), 348-361 (2014).
[4] R L Hudson, A double dilation constructed from a double product of rotations, Markov Processes and Related Fields 13, (J T Lewis mememorial
volume) 169-190 (2007).
[5] R L Hudson and P Jones, Explicit construction of a unitary double product integral, pp 215-236, in Noncommutative harmonic analysis with applications to probability III, eds M Boz·eko, A Krystek and L Wojakowski,
Banach Center Publications 96 (2012).
[6] R L Hudson and K R Parthasarathy, Quantum Ito’s formula and stochastic
evolutions, Communications in Mathematical Physics 93, 301-323 (1984).
[7] P Jones, Unitary double product integrals as implementors of Bogolubov
transformations, Loughborough University PhD thesis (2014).
[8] P-A Meyer, Quantum probability for probabilists, Springer (1992).
[9] K R Parthasarathy, An introduction to quantum stochastic calculus,
Birkhäuser, Basel (1992).
[10] Y Pei, Warwick University PhD thesis, to be completed July 2015.
¼ Center for
[11] A Slavik, Product integration:its history and applications, NEC
Mathematical Modelling, Volume 1, History of Mathematics, Volume 29,
MATFYZPRESS, Praha (2007).
17
; ;
0; 0
[12] V Volterra, Sui fondamenti delle equazioni di¤erenziaqli lineari, Memorie
della Società Italiana della Scienze (3)VI (1887).
18