th-matrix2014/talks/hudson_r.pdf · introduction. this talk is about the double product integral...

On a causal double product integral related to randommatrices.

Robin HudsonLoughborough University, UK

July 4, 2014, Krakow.

Robin Hudson Loughborough University, UK () July 4, 2014, Krakow. 1 / 28

Introduction.

This talk is about the double product integral

∏a<s<t<b

�1+

iλ2(dP (s) dQ (t)� dQ (s) dP (t))

�

Here P and Q are the "momentum" and "position" Brownianmotions of quantum stochastic calculus. Each of these on its own is astandard Brownian motion. But the two Brownian motions don�tcommute with each other; instead they satisfy the commutationrelation

[P (s) ,Q (t)] = �2is ^ t.I�ll have to tell you about:

(double) (quantum) (stochastic) product integralstheir properties; a condition for unitaritywhy I think this one may be related to random matrices.


Introduction.


∏a<s<t<b

�1+


�Here P and Q are the "momentum" and "position" Brownianmotions of quantum stochastic calculus. Each of these on its own is astandard Brownian motion. But the two Brownian motions don�tcommute with each other; instead they satisfy the commutationrelation

[P (s) ,Q (t)] = �2is ^ t.

I�ll have to tell you about:



Introduction.


∏a<s<t<b

�1+






Introduction.


∏a<s<t<b

�1+




(double) (quantum) (stochastic) product integrals

their properties; a condition for unitaritywhy I think this one may be related to random matrices.


Introduction.


∏a<s<t<b

�1+




(double) (quantum) (stochastic) product integralstheir properties; a condition for unitarity

why I think this one may be related to random matrices.


Introduction.


∏a<s<t<b

�1+






Stochastic product integrals.

Product integrals are "just a notation for solutions of di¤erentialequations".

So stochastic product integrals are "just a notation" for solutions ofstochastic di¤erential equations (sde�s).

So, for example, if B is Brownian motion,t

∏0(1+ dB) is the solution

X (t) at t of the sde dX = XdB, X (0) = 1which, following Itô calculus, is NOT eB (t) but eB (t)�

12 t .

A nice feature of the notation is thatt

∏0(1+ dL1)

t

∏0(1+ dL2) =

t

∏0(1+ dL1 + dL2 + dL1dL2)

for arbitrary linear combinations dL1, dL2 2 C hdB, dT i ofdi¤erentials of B and of time T , where dL1dL2 is worked out from the

Itô product tabledB dT

dB dT 0dT 0 0

.



Product integrals are "just a notation for solutions of di¤erentialequations".So stochastic product integrals are "just a notation" for solutions ofstochastic di¤erential equations (sde�s).




12 t .


∏0(1+ dL1)

t

∏0(1+ dL2) =

t

∏0(1+ dL1 + dL2 + dL1dL2)



dB dT 0dT 0 0

.






X (t) at t of the sde dX = XdB, X (0) = 1

which, following Itô calculus, is NOT eB (t) but eB (t)�12 t .


∏0(1+ dL1)

t

∏0(1+ dL2) =

t

∏0(1+ dL1 + dL2 + dL1dL2)



dB dT 0dT 0 0

.







12 t .


∏0(1+ dL1)

t

∏0(1+ dL2) =

t

∏0(1+ dL1 + dL2 + dL1dL2)



dB dT 0dT 0 0

.


Quantum stochastic calculus.

Quantum stochastic calculus is a theory of operator-valued stochasticintegrals like

bZa

(E (s) dP (s) + F (s) dQ (s) + G (s) dT (s))

.

P and Q are operator-valued Brownian motion processes, each one ofwhich can be simultaneously diagonalized to become multiplication bya classical Brownian motion B.

D�1P P (s)DP = mult B (s), D�1Q Q (s)DQ = mult B (s).

But because they don�t commute,

[P (s) ,Q (t)] = �2is ^ t.

they can�t be simultaneously diagonalized; DP 6= DQ .




bZa


.P and Q are operator-valued Brownian motion processes, each one ofwhich can be simultaneously diagonalized to become multiplication bya classical Brownian motion B.



[P (s) ,Q (t)] = �2is ^ t.

they can�t be simultaneously diagonalized; DP 6= DQ .




bZa


.P and Q are operator-valued Brownian motion processes, each one ofwhich can be simultaneously diagonalized to become multiplication bya classical Brownian motion B.



[P (s) ,Q (t)] = �2is ^ t.

they can�t be simultaneously diagonalized; DP 6= DQ .Robin Hudson Loughborough University, UK () July 4, 2014, Krakow. 4 / 28

Quantum stochastic product integrals.

The quantum Itô product rule is

dP dQ dTdP dT �idT 0dQ idT dT 0dT 0 0 0

.

Notice that this contains the classical Itô product table twice over,once for (dP, dT ) and once for (dQ, dT ) . But it is noncommutative;dPdQ 6= dQdP.

Quantum stochastic product inegrals can be de�ned as before; for

each dL 2 C hdP, dQ, dT i ,b

∏a(1+ dL) is X (b) where X solves the

qsde dX = XdL, X (a) = I .Then once againb

∏a(1+ dL1)

b

∏a(1+ dL2) =

b

∏a(1+ dL1 + dL2 + dL1dL2) where

dL1dL2 is worked out using the quantum Itô product table above.





.

Notice that this contains the classical Itô product table twice over,once for (dP, dT ) and once for (dQ, dT ) . But it is noncommutative;dPdQ 6= dQdP.Quantum stochastic product inegrals can be de�ned as before; for



qsde dX = XdL, X (a) = I .

Then once againb

∏a(1+ dL1)

b

∏a(1+ dL2) =

b


dL1dL2 is worked out using the quantum Itô product table above.





.

Notice that this contains the classical Itô product table twice over,once for (dP, dT ) and once for (dQ, dT ) . But it is noncommutative;dPdQ 6= dQdP.Quantum stochastic product inegrals can be de�ned as before; for



qsde dX = XdL, X (a) = I .Then once againb

∏a(1+ dL1)

b

∏a(1+ dL2) =

b


dL1dL2 is worked out using the quantum Itô product table above.Robin Hudson Loughborough University, UK () July 4, 2014, Krakow. 5 / 28

When do discrete double products make sense?

We can agree that, in a noncommutative context,m

∏j=1xj means

x1x2...xm , not xmxm�1...x1 or any other permutation, but how can wede�ne double products such as ∏

(j ,k )2Nm�Nn

xj ,k or ∏1�j<k�N

yj ,k when

the rectangle Nm �Nn is not naturally well-ordered? The followingsimple theorem provides a partial answer.

Theorem: Let xj ,k , j = 1, 2, ...,m, k = 1, 2, ..., n andyj ,k , 1 � j < k � N be elements of an associative algebra with theproperty that whenever both j 6= j 0 and k 6= k 0 (but not necessarilywhen j = j 0 and k 6= k 0 or vice versa) each xj ,k commutes with eachxj 0,k 0 , and similarly each yj ,k commutes with each yj 0,k 0 . Then

m

∏j=1

(n

∏k=1

xj ,k

)=

n

∏k=1

(m

∏j=1xj ,k

)and

N�1∏j=1

(N

∏k=j+1

yj ,k

)=

N

∏k=2

(k�1∏j=1

yj ,k

).

eg when N = 4,fy1,2y1,3y1,4g fy2,3y2,4g y3,4 = y1,2 fy1,3y2,3g fy1,4y2,4y3,4g .


When do discrete double products make sense?

We can agree that, in a noncommutative context,m

∏j=1xj means

x1x2...xm , not xmxm�1...x1 or any other permutation, but how can wede�ne double products such as ∏

(j ,k )2Nm�Nn

xj ,k or ∏1�j<k�N

yj ,k when

the rectangle Nm �Nn is not naturally well-ordered? The followingsimple theorem provides a partial answer.Theorem: Let xj ,k , j = 1, 2, ...,m, k = 1, 2, ..., n andyj ,k , 1 � j < k � N be elements of an associative algebra with theproperty that whenever both j 6= j 0 and k 6= k 0 (but not necessarilywhen j = j 0 and k 6= k 0 or vice versa) each xj ,k commutes with eachxj 0,k 0 , and similarly each yj ,k commutes with each yj 0,k 0 . Then

m

∏j=1

(n

∏k=1

xj ,k

)=

n

∏k=1

(m

∏j=1xj ,k

)and

N�1∏j=1

(N

∏k=j+1

yj ,k

)=

N

∏k=2

(k�1∏j=1

yj ,k

).

eg when N = 4,fy1,2y1,3y1,4g fy2,3y2,4g y3,4 = y1,2 fy1,3y2,3g fy1,4y2,4y3,4g .


Thus under the assumption that [xj ,k , xj 0,k 0 ] = 0 whenever j 6= j 0 andk 6= k 0 we may de�ne the discrete rectangular double product

∏(j ,k )2Nm�Nn

xj ,k , and the discrete triangular double product

∏1�j<k�N

xj ,k by

∏(j ,k )2Nm�Nn

xj ,k =m

∏j=1

(n

∏k=1

xj ,k

)=

n

∏k=1

(m

∏j=1xj ,k

), (1)

∏1�j<k�N

xj ,k =N�1∏j=1

(N

∏k=j+1

xj ,k

)=

N

∏k=2

(k�1∏j=1

xj ,k

). (2)

This assumption holds if xj ,k = 1+ iλ (δjP δkQ � δjQ δkP)where the δjP and δkQ are increments over disjoint intervals forj 6= k. Increments over disjoint subintervals commute.For example if [s, t[\[u, v [= ∅ so that either t � u or v � s,[P (t)� P (s) ,Q (v)�Q (u)] = �2i (t ^ v � t ^ u � s ^ v + s ^ u)= either �2i(t � t � s + s) = 0, or �2i (v � u � v + u) = 0.



∏(j ,k )2Nm�Nn


∏1�j<k�N

xj ,k by

∏(j ,k )2Nm�Nn

xj ,k =m

∏j=1

(n

∏k=1

xj ,k

)=

n

∏k=1

(m

∏j=1xj ,k

), (1)

∏1�j<k�N

xj ,k =N�1∏j=1

(N

∏k=j+1

xj ,k

)=

N

∏k=2

(k�1∏j=1

xj ,k

). (2)

This assumption holds if xj ,k = 1+ iλ (δjP δkQ � δjQ δkP)where the δjP and δkQ are increments over disjoint intervals forj 6= k. Increments over disjoint subintervals commute.

For example if [s, t[\[u, v [= ∅ so that either t � u or v � s,[P (t)� P (s) ,Q (v)�Q (u)] = �2i (t ^ v � t ^ u � s ^ v + s ^ u)= either �2i(t � t � s + s) = 0, or �2i (v � u � v + u) = 0.



∏(j ,k )2Nm�Nn


∏1�j<k�N

xj ,k by

∏(j ,k )2Nm�Nn

xj ,k =m

∏j=1

(n

∏k=1

xj ,k

)=

n

∏k=1

(m

∏j=1xj ,k

), (1)

∏1�j<k�N

xj ,k =N�1∏j=1

(N

∏k=j+1

xj ,k

)=

N

∏k=2

(k�1∏j=1

xj ,k

). (2)

This assumption holds if xj ,k = 1+ iλ (δjP δkQ � δjQ δkP)where the δjP and δkQ are increments over disjoint intervals forj 6= k. Increments over disjoint subintervals commute.For example if [s, t[\[u, v [= ∅ so that either t � u or v � s,[P (t)� P (s) ,Q (v)�Q (u)] = �2i (t ^ v � t ^ u � s ^ v + s ^ u)= either �2i(t � t � s + s) = 0, or �2i (v � u � v + u) = 0.


More generally, given arbitrary dr =3

∑u,v=1

αu,vdLu dLv 2 I I ,

where (dL1, dL2, dL3) = (dP, dQ, dT ) , we can construct the discrete

product ∏(j ,k )2Nm�Nn

1+

3

∑u,v=1

αu,v δjLu δkLv

!where, having

�xed two time intervals [a, b[ and [c , d [ δjLu is the increment of Luover the jth subinterval of the partition of [a, b[ into m subintervals ofequal length and similarly δkLv is the increment of Lv over the kthsubinterval of the partition of [c , d [ into n subintervals of equal

length. Similarly we can form ∏1�j<k�N

1+

3

∑u,v=1

αu,v δjLuδkLv

!where now, having �xed a single time interval [a, b[, δjLu and δkLuare the increments of Lu over the partition of [a, b[ into Nsubintervals of equal length.

Thus we can construct discrete approximations to double productintegrals. Now to de�ne them!


More generally, given arbitrary dr =3

∑u,v=1

αu,vdLu dLv 2 I I ,

where (dL1, dL2, dL3) = (dP, dQ, dT ) , we can construct the discrete

product ∏(j ,k )2Nm�Nn

1+

3

∑u,v=1

αu,v δjLu δkLv

!where, having

�xed two time intervals [a, b[ and [c , d [ δjLu is the increment of Luover the jth subinterval of the partition of [a, b[ into m subintervals ofequal length and similarly δkLv is the increment of Lv over the kthsubinterval of the partition of [c , d [ into n subintervals of equal

length. Similarly we can form ∏1�j<k�N

1+

3

∑u,v=1

αu,v δjLuδkLv

!where now, having �xed a single time interval [a, b[, δjLu and δkLuare the increments of Lu over the partition of [a, b[ into Nsubintervals of equal length.Thus we can construct discrete approximations to double productintegrals. Now to de�ne them!


De�nition of double product integrals.

For dr 2 I I , corresponding to the two de�nitions of the discrete"rectangular" double product as a product of row or column products

∏(j ,k )2Nm�Nn

xj ,k =m

∏j=1

(n

∏k=1

xj ,k

)(1),

" " " "" " " "

=n

∏k=1

(m

∏j=1xj ,k

)(2),

! ! ! !! ! ! !

we de�ne the rectangular product integral over [a, b[�[c , d [ byba ∏ d

c (1+ dr) = ba ∏

n1+ c∏ d

c (1+ dr)o(1)

= ∏ dc

n1+ba

c∏ (1+ dr)o(2).

Theorem: These two de�nitions agree.Notation: ba ∏

�∏ d

c

�denotes a product integral a with a system

algebra on the right (left), and c∏ denotes a decapitated productintegral where the system algebra I has no identity.




∏(j ,k )2Nm�Nn

xj ,k =m

∏j=1

(n

∏k=1

xj ,k

)(1),

" " " "" " " "

=n

∏k=1

(m

∏j=1xj ,k

)(2),

! ! ! !! ! ! !


c (1+ dr) = ba ∏

n1+ c∏ d

c (1+ dr)o(1)

= ∏ dc

n1+ba

c∏ (1+ dr)o(2).

Theorem: These two de�nitions agree.

Notation: ba ∏�∏ d

c






∏(j ,k )2Nm�Nn

xj ,k =m

∏j=1

(n

∏k=1

xj ,k

)(1),

" " " "" " " "

=n

∏k=1

(m

∏j=1xj ,k

)(2),

! ! ! !! ! ! !


c (1+ dr) = ba ∏

n1+ c∏ d

c (1+ dr)o(1)

= ∏ dc

n1+ba

c∏ (1+ dr)o(2).

Theorem: These two de�nitions agree.Notation: ba ∏

�∏ d

c




Similarly, corresponding to the two de�nitions of the discrete"triangular" double product

∏1�j<k�N

xj ,k =N�1∏j=1

(N

∏k=j+1

xj ,k

)(1)#

=N

∏k=2

(k�1∏j=1

xj ,k

)(2)! "" "

" " "we de�ne the triangular product integral over [a, b[ by

∏a<s<t<b

(1+ dr (s, t)) =n1+ (1+ dr (s, t))bs

c∏o b

∏a(1)

=b

∏a

n1+ c∏ t

a (1+ dr (s, t))o(2)

Theorem: These two de�nitions agree.Notation: In (1) the symbols ∏ and c∏ are written to the right ofthe integrand because we use backward adapted stochastic calculus.



∏1�j<k�N

xj ,k =N�1∏j=1

(N

∏k=j+1

xj ,k

)(1)#

=N

∏k=2

(k�1∏j=1

xj ,k

)(2)! "" "


∏a<s<t<b

(1+ dr (s, t)) =n1+ (1+ dr (s, t))bs

c∏o b

∏a(1)

=b

∏a

n1+ c∏ t

a (1+ dr (s, t))o(2)

Theorem: These two de�nitions agree.

Notation: In (1) the symbols ∏ and c∏ are written to the right ofthe integrand because we use backward adapted stochastic calculus.



∏1�j<k�N

xj ,k =N�1∏j=1

(N

∏k=j+1

xj ,k

)(1)#

=N

∏k=2

(k�1∏j=1

xj ,k

)(2)! "" "


∏a<s<t<b

(1+ dr (s, t)) =n1+ (1+ dr (s, t))bs

c∏o b

∏a(1)

=b

∏a

n1+ c∏ t

a (1+ dr (s, t))o(2)

Theorem: These two de�nitions agree.Notation: In (1) the symbols ∏ and c∏ are written to the right ofthe integrand because we use backward adapted stochastic calculus.


Properties of double products.

Theorem: For a < b < c and s < t�ba ∏ t

s (1+ dr)� �c

b ∏ ts (1+ dr)

�=ca ∏ t

s (1+ dr) .

Proof. Use ba ∏ ts (1+ dr) =

ba ∏

n1+ c∏ t

s (1+ dr)o.�

Theorem: For a < b and s < t < u�ba ∏ t

s (1+ dr)� �

ba ∏ u

t (1+ dr)�=ba ∏ u

s (1+ dr)

Proof. Use ba ∏ ts (1+ dr) = ∏ t

s

n1+ba

c∏ (1+ dr)o.�

It can be shown that these bimultiplicativity properties, whencombined with adaptedness and covariance under time shifts,characterize rectangular double products.

But�ba ∏ t

s (1+ dr1)� �

ba ∏ t

s (1+ dr2)�6=�

ba ∏ t

s (1+ dr1 + dr2 + dr1dr2)�.


There is a relation between triangular and rectangular double products:Theorem: For a < b < c

∏a<s<t<c

(1+ dr (s, t)) =

∏

a<s<t<b(1+ dr (s, t)) I[b,c [

!�ba ∏ c

b (1+ dr)�

I[a,b[ ∏

b<s<t<c(1+ dr (s, t))

!.

Here we use the continuous tensor product structure of the Fock space towrite F[a,c [ = F[a,b[ F[b,c [ so both sides are operators in F[a,c [.Proof. The statement is true when b = a or when b = c . We apply theoperator

�!d id+ id �d to the rhs tensor product, where

�!d is the

usual forward adapted stochastic di¤erential (so that�!d id does not see

the third term) and �d is the backward di¤erential. Using the Leibniz-Itô

formula twice, everything cancels. So the rhs does not depend on b. �


Unitarity

For dr 2 I I denote by dr its quasi-inverse, so that dr + dr + drdr = 0.In fact since the algebra I is nilpotent, dr = �dr + 1

2dr2. De�ne the

reversed rectangular double product ba

�∏ d

c (1+ dr) by reversing theorder of all simple products, so that stochastic di¤erential equations aredriven from the left, dX = dk X , rather than from the right, dX = Xdk.

Theorem ba

�∏ d

c (1+ dr) is a right inverse toba ∏ d

c (1+ dr) ;�ba ∏ d

c (1+ dr)��

ba

�∏ d

c (1+ dr)�= I .

Proof. Denote the two double products by X and X . Using De�nition (1)of the double product��!

d id�X = XY ,

��!d id

�X = Y X ,�

id�!d�Y =

�Y + I 2

�dr ,�

id�!d�Y = dr

�Y + I 2

�where Y = c∏ d

c (1+ dr) and Y =c �∏ d

c (1+ dr) .Robin Hudson Loughborough University, UK () July 4, 2014, Krakow. 13 / 28

By the Leibniz-Itô formula��!d id

� �XX

�= X

�Y + Y + Y Y

�X ,�

id�!d� �Y + Y + Y Y

�= 0.

But Y + Y + Y Y = 0 when d = c . So Y + Y + Y Y = 0 for all d .So��!d id

� �XX

�= 0. But XX = I when b = a. So XX = I for

all b.�

If we use De�nition (2) of the double product we get exactly the sameresult. We do NOT prove:

Theorem ba

�∏ d

c (1+ dr) is a left inverse toba ∏ d

c (1+ dr)�ba

�∏ d

c (1+ dr)��

ba ∏ d

c (1+ dr)�= I .

Proof. Use backward adapted quantum stochastic calcuus instead.Corollary: ba ∏ d

c (1+ dr) is unitary valued if and only if

dr + dr † + drdr † = 0

where † denotes the natural product involution in I I .



� �XX

�= X

�Y + Y + Y Y

�X ,�

id�!d� �Y + Y + Y Y

�= 0.


� �XX


all b.�If we use De�nition (2) of the double product we get exactly the sameresult. We do NOT prove:

Theorem ba

�∏ d


c (1+ dr)�ba

�∏ d

c (1+ dr)��

ba ∏ d

c (1+ dr)�= I .



dr + dr † + drdr † = 0




� �XX

�= X

�Y + Y + Y Y

�X ,�

id�!d� �Y + Y + Y Y

�= 0.


� �XX



Theorem ba

�∏ d


c (1+ dr)�ba

�∏ d

c (1+ dr)��

ba ∏ d

c (1+ dr)�= I .

Proof. Use backward adapted quantum stochastic calcuus instead.

Corollary: ba ∏ dc (1+ dr) is unitary valued if and only if

dr + dr † + drdr † = 0




� �XX

�= X

�Y + Y + Y Y

�X ,�

id�!d� �Y + Y + Y Y

�= 0.


� �XX



Theorem ba

�∏ d


c (1+ dr)�ba

�∏ d

c (1+ dr)��

ba ∏ d

c (1+ dr)�= I .



dr + dr † + drdr † = 0

where † denotes the natural product involution in I I .Robin Hudson Loughborough University, UK () July 4, 2014, Krakow. 14 / 28

The corresponding theorem for triangular product integrals is provedsimilarly. Again backward-adapted quantum stochastic calculus is required.

Theorem: ∏a<s<t<b

(1+ dr (s, t)) is unitary-valued if and only if

dr + dr † + drdr † = 0.

In particular ∏a<s<t<b

�1+ 1

2 iλ (dP (s) dQ (t)� dQ (s) dP (t))�is

unitary-valued for each real number λ. Indeed, fordr = 1

2 iλ (dP dQ � dQ dP) , evidently dr + dr † = 0, but alsofrom the Itô table


,

drdr † =14

λ2 (dP dQ � dQ dP) (dP dQ � dQ dP)

=14

λ2 (dT dT + dT dT )� 14

λ2 (dT dT + dT dT )= 0


How to construct the discrete approximations.

From now on we consider the unitary triangular double productintegral

R[a,b[ =: ∏a<s<t<b

�1+

12iλ (dP (s) dQ (t)� dQ (s) dP (t))

�.

For N = 1, 2, ... we consider the discrete approximations

R (N )[a,b[ =: ∏

1�j<k�N

�1+

12iλ (δjPδkQ � δjQδkP)

�where δjP and δjQ are the increments of P and Q over thesubinterval [a+ (j � 1) b�aN , a+ j b�aN [.

Thus R (N )[a,b[ = ∏

1�j<k�N

�1+ 1

2 iλb�aN (pjqk � qjpk )

�where

pj =

rNb� a

�P�a+ j

b� aN

�� P

�a+ (j � 1) b� a

N

��and qj is de�ned similarly with P replaced by Q.


These satisfy the standard canonical commutation relations

[pj , qk ] = �2iδj .k , [pj , pk ] = [qj , qk ] = 0so that we can interpret each pjqk � qjpk as an angular momentum

and hence as the in�nitesimal generator of a one-parameter group ofrotations.For example, if we realise (p1, q1) and (p2, q2) in L2

�R2�as

(pj f ) (x1, x2) = �p2i

∂

∂xjf (x1, x2) , (qj f ) (x1, x2) =

p2xj f (x1, x2) ,

then we can compute the in�nitesimal generator of the one parameter

group of rotations Rθf (x1, x2) = f�(x1, x2)

�cos θ � sin θsin θ cos θ

��by

the chain rule, as �i dd θRθf (x1, x2)��θ=0

= �i ((x1 sin θ + x2 cos θ) ∂1 + (�x1 cos θ + x2 sin θ) ∂2)

f�(x1, x2)


��θ=0

=

�12(p1q2 � q1p2) f

�(x1, x2) .



[pj , qk ] = �2iδj .k , [pj , pk ] = [qj , qk ] = 0so that we can interpret each pjqk � qjpk as an angular momentumand hence as the in�nitesimal generator of a one-parameter group ofrotations.

For example, if we realise (p1, q1) and (p2, q2) in L2�R2�as

(pj f ) (x1, x2) = �p2i

∂

∂xjf (x1, x2) , (qj f ) (x1, x2) =

p2xj f (x1, x2) ,




��by



f�(x1, x2)


��θ=0

=

�12(p1q2 � q1p2) f

�(x1, x2) .



[pj , qk ] = �2iδj .k , [pj , pk ] = [qj , qk ] = 0so that we can interpret each pjqk � qjpk as an angular momentumand hence as the in�nitesimal generator of a one-parameter group ofrotations.For example, if we realise (p1, q1) and (p2, q2) in L2

�R2�as

(pj f ) (x1, x2) = �p2i

∂

∂xjf (x1, x2) , (qj f ) (x1, x2) =

p2xj f (x1, x2) ,




��by



f�(x1, x2)


��θ=0

=

�12(p1q2 � q1p2) f

�(x1, x2) .


But P and Q and hence also the pj and qj�s are realised in the Fockspace F

�L2 (R)

�.

For each unitary operator W on L2 (R) there is a correspondingunitary operator Γ (W ) on F

�L2 (R)

�, called its second

quantization, which may be de�ned by its action on the exponentialvectors e (f ) , f 2 L2 (R) by

Γ (W ) e (f ) = e (Wf ) .

Alternatively, if F�L2 (R)

�is realised as the direct sum of n-particle

spaces

F�L2 (R)

�=

∞Mn=0

nsymL2 (R) ,

then Γ (W ) is the direct sum operator

Γ (W ) =∞Mn=0

nW .

Either way, one deduces that second quantization is multiplicative;

Γ (W1W2) = Γ (W1) Γ (W2)


We return to the discrete approximation to

R[a,b[ = ∏a<s<t<b

�1+


�' R (N )

[a,b[ = ∏1�j<k�N

�1+


�= ∏

1�j<k�N

�1+

12iλb� aN

(pjqk � qjpk )�

'N!∞

∏1�j<k�N

exp�iλ (b� a)2N

(pjqk � qjpk )�

We introduce the orthonormal family in L2 (R) of normalizedindicator functions

χj (x) =

( qNb�a if a+

(j�1)N (b� a) � x < a+ j

N (b� a)[0 otherwise

j = 1, 2, ...,N

together with their span KN and its orthocomplement K?N .


We return to the discrete approximation to

R[a,b[ = ∏a<s<t<b

�1+


�' R (N )

[a,b[ = ∏1�j<k�N

�1+


�= ∏

1�j<k�N

�1+

12iλb� aN

(pjqk � qjpk )�

'N!∞

∏1�j<k�N


(pjqk � qjpk )�

We introduce the orthonormal family in L2 (R) of normalizedindicator functions

χj (x) =

( qNb�a if a+

(j�1)N (b� a) � x < a+ j

N (b� a)[0 otherwise

j = 1, 2, ...,N

together with their span KN and its orthocomplement K?N .Robin Hudson Loughborough University, UK () July 4, 2014, Krakow. 19 / 28

The operator exp�iλ(b�a)2N (pjqk � qjpk )

�is the second quantization of

the unitary operator on L2 (R) which acts on KN through the action onthis basis of the rotation matrix

R (j ,k )N

=

26666666666666664

(j) (k)1 � � � 0 � � � 0 � � � 0.... . .

... � � � ... � � � ...

(j) 0 � � � cos�

λ(b�a)2N

�� sin

�λ(b�a)2N

�� 0

... � � � .... . .

... � � � ...

(k) 0 � � � sin�

λ(b�a)2N

�� cos

�λ(b�a)2N

�� 0

... � � � ... � � � .... . . � � �

0 � � � 0 � � � 0 � � � 1

37777777777777775and as the identity on K?N .


Thus

∏1�j<k�N


(pjqk � qjpk )�

= ∏1�j<k�N

Γ�R (j ,k )N � IK?N

�= Γ

∏

1�j<k�N

�R (j ,k )N � IK?N

�!

= Γ

∏

1�j<k�NR (j ,k )N

!� IK?N

!.

Thus we want to evaluate the product ∏1�j<k�N

R (j ,k )N and �nd

limN!∞

∏

1�j<k�NR (j ,k )N

!� IK?N .


We want to evaluate the limiting form of the double product of rotationmatrices

∏1�j<k�N

26666666666666664

(j) (k)1 � � � 0 � � � 0 � � � 0.... . .

... � � � ... � � � ...

(j) 0 � � � cos�

λ(b�a)2N

�� sin

�λ(b�a)2N

�� 0

... � � � .... . .

... � � � ...

(k) 0 � � � sin�

λ(b�a)2N

�� cos

�λ(b�a)2N

�� 0

... � � � ... � � � .... . . � � �

0 � � � 0 � � � 0 � � � 1

37777777777777775in the form I +K where K is an integral operator on L2 ([a, b[) . Theexplicit computation of the kernel of K will be the subject of Yuchen Pei�spresentation and my lecture in Bedlewo next week. It involves countinggeneralized Dyck paths using Catalan numbers and their derivatives, theso-called Catalan triangle.


One must then show, after all this approximation and handwaving,that I +K really is unitary, and that Γ (I +K ) really does satisfy theqsde de�ningR[a,b[ = ∏

a<s<t<b

�1+ 1

2 iλ (dP (s) dQ (t)� dQ (s) dP (t))�.

All of this is surprisingly di¢ cult. The di¢ culty is surprising becausethe programme has been completed without di¢ culties for thecorresponding rectangular double product integral; see R L Hudsonand P Jones, Explicit construction of a unitary double productintegral, pp 215-236, in Noncommutative harmonic analysis withapplications to probability III, eds M Bo·zeko, A Krystek and LWojakowski, Banach Center Publications 96 (2012).

Perhaps it is more di¢ cult becausec

∏a

r (1+ dr)

=

b

∏a

r (1+ dr) I!�

ba ∏ c

b� (1+ dr)

� I

c

∏b

r (1+ dr)

!.

Compare this with f (x + y) = f (x)ω (x , y) f (y) .


Connection with random matrices.

1+ 12 iλ (dP (s) dQ (t)� dQ (s) dP (t)) is a random in�nitesimal

rotation; maybe ∏a<s<t<b

�1+ 1


the limiting form of a large random rotation matrix with randomnessbuilt in via the noncommuting Brownian motions P,Q.

The Euler factorization expresses every 3� 3 rotation R essentiallyuniquely in the form R = R12 (φ)R13 (θ)R12 (ψ) .From it we deducethat R can also be expressed essentially uniquely in the form

R = R12 (θ12)R13 (θ13)R23 (θ23)

by applying the Euler decomposition to R = RR13�� 12π

�and writing

the third factor as R12 (ψ) = R13� 12π�R23 (ψ)R13

�� 12π

�.

More generally every N �N rotation (ie proper orthogonal) matrix Rcan be expressed essentially uniquely as

R = ∏1�j<k�N

R jk (θjk ) .

The product integral looks like a continuum limiting form of this.





�1+ 1


the limiting form of a large random rotation matrix with randomnessbuilt in via the noncommuting Brownian motions P,Q.The Euler factorization expresses every 3� 3 rotation R essentiallyuniquely in the form R = R12 (φ)R13 (θ)R12 (ψ) .From it we deducethat R can also be expressed essentially uniquely in the form

R = R12 (θ12)R13 (θ13)R23 (θ23)


�and writing


�� 12π

�.


R = ∏1�j<k�N

R jk (θjk ) .

The product integral looks like a continuum limiting form of this.





�1+ 1


the limiting form of a large random rotation matrix with randomnessbuilt in via the noncommuting Brownian motions P,Q.The Euler factorization expresses every 3� 3 rotation R essentiallyuniquely in the form R = R12 (φ)R13 (θ)R12 (ψ) .From it we deducethat R can also be expressed essentially uniquely in the form

R = R12 (θ12)R13 (θ13)R23 (θ23)


�and writing


�� 12π

�.


R = ∏1�j<k�N

R jk (θjk ) .

The product integral looks like a continuum limiting form of this.Robin Hudson Loughborough University, UK () July 4, 2014, Krakow. 24 / 28

A �rst problem with this idea is that

∏a<s<t<b

�1+


�= Γ (I +K )

is a second quantization, acting on symmetric tensor powers ofL2 (R) as tensor powers of I +K . The spectrum must becorrespondingly highly structured; it is hard to imagine anythingrelated to the semicircle distribution emerging.

This di¢ culty may perhaps be overcome by employing "�nitetemperature" quantum Brownian motion (Pσ,Qσ) where each of Pσ

and Qσ is a standard Brownian motion but the commutation relationbecomes

[Pσ (s) ,Qσ (t)] = �2is ^ t

σ2.

One would expect there to be a quantum invariance principle, statingthat for an arbitrary sequence of iid canonical pairs ((pn, qn))n2N ofmean zero and variance σ2 the discrete product

∏1�j< k�N

�1+ iλ

2N (pjqk � qjpk )�converges in distribution to this

limit. But this is not proved.



∏a<s<t<b

�1+


�= Γ (I +K )

is a second quantization, acting on symmetric tensor powers ofL2 (R) as tensor powers of I +K . The spectrum must becorrespondingly highly structured; it is hard to imagine anythingrelated to the semicircle distribution emerging.This di¢ culty may perhaps be overcome by employing "�nitetemperature" quantum Brownian motion (Pσ,Qσ) where each of Pσ


[Pσ (s) ,Qσ (t)] = �2is ^ t

σ2.


∏1�j< k�N

�1+ iλ


limit. But this is not proved.



∏a<s<t<b

�1+


�= Γ (I +K )

is a second quantization, acting on symmetric tensor powers ofL2 (R) as tensor powers of I +K . The spectrum must becorrespondingly highly structured; it is hard to imagine anythingrelated to the semicircle distribution emerging.This di¢ culty may perhaps be overcome by employing "�nitetemperature" quantum Brownian motion (Pσ,Qσ) where each of Pσ


[Pσ (s) ,Qσ (t)] = �2is ^ t

σ2.


∏1�j< k�N

�1+ iλ


limit. But this is not proved.Robin Hudson Loughborough University, UK () July 4, 2014, Krakow. 25 / 28

We may replace 12 iλ (dP dQ � dQ dP) by other generators dr

satisfying the unitarity condition dr + dr † + drdr † = 0, for example

dr = iλdP dQ + 12

λ2dT dT ,

dr = iλdQ dP + 12

λ2dT dT ,dr = iλ (dP dQ + dQ dP) + dT dT

These double products are no longer second quantizations. They may bedescribed as the unique covariant implementors of families of Bogolubovtransformations, that is invertible real-linear transformations whichpreserve the imaginary part of the inner product in L2 (R). The Bogolubovtransformations can be evaluated explicitly, at least in the rectangularcase. Additional terms of form dP dT or dT dQ can be included;then the Bogolubov transformations become inhomogeneous.


Some references.

[1] R L Hudson and P Jones, Explicit construction of a unitary doubleproduct integral, pp 215-236, in Noncommutative harmonic analysis withapplications to probability III, eds M Bo·zejko, A Krystek and LWojakowski, Banach Center Publications 96 (2012).[2] R L Hudson, Quantum Lévy area as a quantum martingale limit, pp169-188, in Quantum probability and Related Topics XXIX, eds L Accardiand F Fagnola, World Scienti�c (2013).[3] S Chen and R L Hudson, Some properties of quantum Lévy area inFock and non-Fock quantum stochastic calculus, Probability andMathematical Statistics 33, 425-434 (2013).


Thank you for your attention.


th-matrix2014/talks/hudson_r.pdf · introduction. this talk is about the double product integral...

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