Download Normal Orger & Graphs

Normal Order & Graphs
Combinatorial graffiti
Institute of Nuclear Physics, Polish Academy of Sciences
Collaboration:
Karol A. Penson, Université Paris VI
Allan I. Solomon, Open University (UK)
Gérard Duchamp, Université de Paris-Nord
Theme:
• Introduction
• Operators in Quantum Mechanics
• Occupation number representation
• Graphs
• Definitions and Examples
• Algebra of graph composition
• Operators vs Graphs
• Equivalence and calculus
• Number operator

N n
n n
• Basis in Fock space
0 , 1 ,
n+1 , ……
, …… ,
2
n1
annihilation
,
n
,
creation
• Operators
a
a
+
n
n
n+1
n
n+
[ a +, N ]   a
1 annihilation operator
[a,N] a
n  1creation operator
[ a , a +]  1
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+
ladder
structure
Heisenberg
algebra
3
• Creation and annihilation operators do not commute
a a +  a +a
[ a , a +] 1
• Order is important
aa+
1
normal form
aa+
a +a
a +a + 1
normal ordering
• Normal order
a a +a a a +a
unique
[ a , a +]
a +2a 4 + 4 a + a 3 + 2 a 2
1
a + - to the left
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a - to the right
4
A - bounded operator
• Matrix representation
A   Am ,n m n
Am,n  m A n
m,n
• Coherent state representation (Bargmann – Segal repr.)
A   1  d 2 z  ( z ) z z
• Normally ordered form
 ( z)  z A z
A    k ,l a † k a l
k ,l
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A - algebra of operators
G
- algebra of graphs
1:1
a+ k a l - basis in
A    k ,l a a
†k
A
1:1
l
k ,l
What about the product in
?
- basis in G
k
l
G    k ,l
k
l
k ,l
A
Can be transfered on graphs and is consistent
with the natural graph composition !!!
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G – graph :
vertices & lines + some rules
Basic building blocks:
output
• The vertex has two kinds of lines
• All lines are distinguishable
between each other
l ingoing
lines
k outgoing
lines
input
Picture of a process
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Composition rules:
CORRECT
• Vertices may be connected by joining
outgoing and ingoing lines,
i.e. lines keep their direction
&
1
WRONG
&
&,
4
2
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2
8
Labeling:
• Vertices are numbered with consecutive integers 1,2,3, …
and follow direction of the lines
3
2
1
CORRECT
5
2
1
8
3
3 2
1
WRONG
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7
4
1
2
6
3
WRONG
9
Labeling:
• Vertices are numbered with consecutive integers 1,2,3, …
and follow direction of the lines
5
2
1
8
3
7
4
6
Processes grow step by step
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Weights:

• Vertices may be attached weights
making up the overall weight of the graph
,
,



2





&
*
2

5
*

 2
3
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G1

G2
 No. of
G2 in are equal, resp.
,
,
4 lines
and
, …
in
G1
/

and
, …

5 lines
3 lines
2 lines
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1
2
1
2
No connection
(disconnected)
2
x
1
2
Twox connections
2
1
2
2
2
2
2
,
1
1
2
2
2
One connection
1
,
4
2
x
x
1
1
All two vertex graphs made of :
x
1
&
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All two vertex graphs made of :
&
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

G     i  i :  i  K ,  i  graph 
 i

Graph algebra:
Addition:
Multiplication:

 






  i i    i i    ( i  i )  i
i
 i
  i



 
  i  i      j  j  
 i
  j

Quotient graph algebra:
G  G

i
 i j
i, j

    k ,l
~
 k ,l
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j
k
l



15
1

,
2

1
2

2
1
2
2
*
=
2
1
+
4
+
2
1
1
2
*
=
2
1
+
2
1
Noncommutative !!
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Graph
Operator
a+
k outputs
a+
a+
a+ k
al
a+
1:1
a
a
a
l inputs
Basic quantum process and its picture.
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G - algebra of
graphs
A - algebra of operators
a+ k
al
- basis in
A    k ,l a † k a l
A
1:1
k
l
- basis in
G
G    k ,l
1:1
k ,l
k
l
k ,l
G
A
1:1
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G
18
[a , a ] 
†k
s
Commutator:
min{k , s}

i 0
a a a a 
†r
s
†k
min{k , s}

l
i 0
s !k !
a † k i a s i
i !( s  i )!(k  i )!
s !k !
a † r  k i a s  l i
i !( s  i )!(k  i )!
: rki
min{k , s}
r
s
*
k
l
=

2
i connections
i 0
1
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: sli
19
B  a †2 a 2
A  a †2 a
&
Product in A :
B A  a†2 a 2 a†2 a  a†4 a3  4 a†3a 2  2 a†2 a
Product in G
:
2
2
*
=
2
1
+
4
+
1
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2
1
20
a +2a 2 + a +2 a
= 2 a +4a 4 + 2 a +4a 3 + a +4a 2 + 4 a +3a 3 +
+ 6 a + 3 a 2 + 2 a + 3 a + 2 a + 2 a 2 + 2 a + 2a
+
2
=
1
2
+
22
1
+
44
2
6
1
1
2
+
1
+
2
2
4
1
+
+
1
2
2
22
1
2
+
2 2
1
2
2
+
+
2
2
1
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G
Summary:
a+ k
A
al
1:1
Gl
k
Conclusion:
Outlook:
• See operators as processes
• Quantum Mechanics in graph representation
• Alternative Language & Interpretation
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a +2a 2 + a +2 a
= 2 a +4a 4 + 2 a +4a 3 + a +4a 2 + 4 a +3a 3 +
+ 6 a + 3 a 2 + 2 a + 3 a + 2 a + 2 a 2 + 2 a + 2a
+
2
=
1
2
2
+
+
+
1
2
2
2
1
1
+
2
2
1
+
2
+
+
1
2
2
4
1
+
+
1
2
2
2
1
2
1
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k
l
• Composition rules:
Vertex & Lines
• Vertices may be connected by joining
lines keeping their direction
CORRECT
&
WRONG
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• Composition rules:
CORRECT
• Vertices may be connected by joining
outgoing and ingoing lines,
i.e. lines keep their direction
WRONG
&
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Labeling:
Vertices are numbered with consecutive
integers 1,2,3, … in accordance with the
direction of lines
3
2
1
CORRECT
3
1
2
WRONG
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1
2
3
WRONG
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&
• Composition rules:
• Vertices may be connected by joining
outgoing and ingoing lines,
i.e. lines keep their direction
CORRECT
1
4
WRONG
2
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2
27
G1

G2
 No. of
and
in
G1
G2
and
in are equal, resp.
,
5
2
1
/

,
8
3
7
4
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
10 lines
7 lines
6
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Labeling:
Vertices are numbered with consecutive
integers 1,2,3, … in accordance with the
direction of lines
5
2
1
8
3
7
4
6
Processes grow step by step
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B  a †2 a 2
A  a †2 a
&
Product in A :
B A  a†2 a 2 a†2 a  a†4 a3  4 a†3a 2  2 a†2 a
Product in G
:
*
=
+
4
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+
2
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B  a †2 a 2
A  a †2 a
&
Product in A :
AB A  a†2 a a†2 a 2 a†2 a  a†6 a 4  7 a†5a3  14 a†4 a 2  4 a†3a
Product in G
:
2
2
*
=
2
1
+
4
+
1
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2
1
31