Metal-Insulator Transitions I.pdf

Metal-Insulator Transitions(I):
Basics for Non-relativistic Field
Theory of Many-Particle Systems
Detian Yang
Negele, J. W., and H. Orland, 1988, Quantum Many-Particle Systems (Addison Wesley, Redwood City, CA).

Quantum Mechanics: States, Dynamics and Measurements
(1) Physical states are described by vectors in Hilbert space.
Wavefunctions: functionals of quantum fields in 𝑥-representation
(2) The only observables are transition probability amplitudes Ψ Φ
(3) Physical processes are “artificially” divided into two “incompatible”
kinds (I) and (II)
(I) Unitary evolution processes that preserve information
(II) Non-unitary measurement processes that break time inversion
symmetry
𝑖𝜕! ⟩
|Ψ = (
𝐻 ⟩
|Ψ
Born’s Rule; Measurement Assumption;
Ψ Ψ , Ψ +
𝑂 Ψ , Ψ +
𝑂 Φ , 𝑃𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟 − 𝑣𝑎𝑙𝑢𝑒𝑑 𝑚𝑒𝑎𝑠𝑢𝑟𝑒(𝑃𝑂𝑉𝑀)
𝑖𝜕!𝜌 = [ (
𝐻, 𝜌]
⟩
|𝜑" = +
𝑃" ⟩
|Ψ
⟩
|Ψ
⟩
|Ψ(𝑡) = (
𝑈(𝑡, 𝑡#) ⟩
|Ψ(𝑡#)
ℏ = 𝟏
𝜌 𝑡 = (
𝑈(𝑡, 𝑡#)𝜌 𝑡#
(
𝑈†
(𝑡, 𝑡#)
𝑃(𝑖) = Ψ '
𝑃! Ψ
𝜌 𝜌$
=
+
𝑃"𝜌 +
𝑃"
𝑡𝑟[𝜌 +
𝑃"]
𝑡𝑟[𝜌 '
𝑃!]
'
𝑃! = ⟩
|𝜑! ⟨𝜑!|
𝜌 𝜌1 =
%
𝑀2𝜌 %
𝑀2
†
𝑡𝑟[ %
𝑀2𝜌 %
𝑀2
†]
𝑡𝑟[𝜌 '
𝐹!]
4
𝑀!
4
𝑀!
† = '
𝐹!

N-particle Hilbert Space and States
Hilbert space for N-particle system: ℋ% = ℋ⨂ℋ⨂ ⋯ ℋ
|𝛼& … 𝛼%) ≡ ⟩
|𝛼& ⨂ ⟩
|𝛼' ⨂ … ⨂ ⟩
|𝛼%
Orthonormal basis
{ ⟩
|𝛼 }
Wave function 𝜓% ⃑
𝑟&, ⃑
𝑟', … , ⃑
𝑟% = (⃑
𝑟&, ⃑
𝑟', … , ⃑
𝑟%|Ψ%): probability amplitude for
finding particles in N positions ⃑
𝑟&, ⃑
𝑟', … , ⃑
𝑟%
( )
!
⟩
|𝛼 ⟨𝛼| = 1
Wave function for ⟩
|𝛼& … 𝛼% : 𝜓0!…0"
⃑
𝑟&, ⃑
𝑟', … , ⃑
𝑟% = (⃑
𝑟&, ⃑
𝑟', … , ⃑
𝑟%|𝛼&𝛼' … 𝛼%)
= (⟨⃑
𝑟&|⨂⟨⃑
𝑟'|⨂ … ⨂⟨⃑
𝑟%|)( ⟩
|𝛼& ⨂ ⟩
|𝛼' ⨂ … ⨂ ⟩
|𝛼% )
= 𝜓0!
⃑
𝑟& 𝜓0#
⃑
𝑟' … 𝜓0"
⃑
𝑟%
Orthonormality: 𝛼6, 𝛼7, … , 𝛼8 𝛼1
6𝛼1
7 … 𝛼1
8 = 𝛼6 𝛼1
6 𝛼7 𝛼1
7 … 𝛼8 𝛼1
8
= 𝛿(𝛼6 − 𝛼1
6)𝛿(𝛼7 − 𝛼1
7) … 𝛿(𝛼8 − 𝛼1
8)
𝛼 𝛼" = 𝛿(𝛼 − 𝛼")
Completeness/Closure:
( )
!!,!",…,!#
|𝛼$, 𝛼%, … , 𝛼&)(𝛼$, 𝛼%, … , 𝛼&| = 1

Spin-Statistics Theorem
Identical half-integral spin particles(Fermions) satisfy Fermi–Dirac statistics
which permit no more than one particle per quantum state; identical integral
spin particles(Bosons) satisfy Bose–Einstein statistics which permits any
number of particles in each quantum state
Ian Duck and E. C. G. Sudarshan. American Journal of Physics 66, 284 (1998); doi: 10.1119/1.18860
𝜓 ⃑
𝑟!", ⃑
𝑟!#, … , ⃑
𝑟!$ = 𝜓 ⃑
𝑟", ⃑
𝑟#, … , ⃑
𝑟$
The wave function of N Bosons is totally symmetric relative to any permutation P
The wave function of N Fermions is totally antisymmetric relative to any permutation P
𝜓 ⃑
𝑟!", ⃑
𝑟!#, … , ⃑
𝑟!$ = (−1)!
𝜓 ⃑
𝑟", ⃑
𝑟#, … , ⃑
𝑟$
(−1)#
is the parity of permutation P: the number of switching two elements which bring (1,2, . . 𝑁) to (𝑃1, 𝑃2, . . 𝑃𝑁)
𝜓 ⃑
𝑟!", ⃑
𝑟!#, … , ⃑
𝑟!$ = 𝜉!
𝜓 ⃑
𝑟", ⃑
𝑟#, … , ⃑
𝑟$ 𝜉 = B
+1, 𝐵𝑜𝑠𝑜𝑛𝑠
−1. 𝐹𝑒𝑟𝑚𝑖𝑛𝑜𝑛𝑠

Define symmetrization operator 𝒫J and antisymmetrization operator 𝒫K
𝒫!/#𝜓 ⃑
𝑟", ⃑
𝑟#, … , ⃑
𝑟$ =
1
𝑁!
.
!
𝜉!
𝜓 ⃑
𝑟!", ⃑
𝑟!#, … , ⃑
𝑟!$
𝒫J𝜓 ⃑
𝑟6, ⃑
𝑟7 =
1
2
(𝜓 ⃑
𝑟6, ⃑
𝑟7 + 𝜓 ⃑
𝑟7, ⃑
𝑟6 ) 𝒫K𝜓 ⃑
𝑟6, ⃑
𝑟7 =
1
2
(𝜓 ⃑
𝑟6, ⃑
𝑟7 − 𝜓 ⃑
𝑟7, ⃑
𝑟6 )
𝜉 = B
1 𝐻𝑒𝑟𝑚𝑖𝑡𝑖𝑎𝑛
' 𝑑 ⃑
𝑟!𝑑 ⃑
𝑟" … 𝑑 ⃑
𝑟# 𝜙∗
⃑
𝑟!, ⃑
𝑟", … , ⃑
𝑟# 𝒫%/'𝜓 ⃑
𝑟!, ⃑
𝑟", … , ⃑
𝑟# = ' 𝑑 ⃑
𝑟!𝑑 ⃑
𝑟" … 𝑑 ⃑
𝑟# 𝜙∗
⃑
𝑟!, ⃑
𝑟", … , ⃑
𝑟#
1
𝑁!
2
(
𝜉(
𝜓 ⃑
𝑟(!, ⃑
𝑟(", … , ⃑
𝑟(#
=
!
#!
∑( 𝜉(
∫ 𝑑 ⃑
𝑟!𝑑 ⃑
𝑟" … 𝑑 ⃑
𝑟# 𝜙∗
⃑
𝑟!, ⃑
𝑟", … , ⃑
𝑟# 𝜓 , ⃑
𝑟(! ⃑
𝑟(", … , ⃑
𝑟(# =
!
#!
∑(! 𝜉(!
∫ 𝑑 ⃑
𝑟!!𝑑 ⃑
𝑟"! … 𝑑 ⃑
𝑟#! 𝜙∗
⃑
𝑟(!!!, ⃑
𝑟(!"!, … , ⃑
𝑟(!#! 𝜓 ⃑
𝑟!!, ⃑
𝑟"!, … , ⃑
𝑟#!
= ∫ 𝑑 ⃑
𝑟!!𝑑 ⃑
𝑟"! … 𝑑 ⃑
𝑟#!
!
#!
∑(! 𝜉(!
𝜙∗
⃑
𝑟(!!!, ⃑
𝑟(!"!, … , ⃑
𝑟(!#! 𝜓 ⃑
𝑟!!, ⃑
𝑟"!, … , ⃑
𝑟#! = ∫ 𝑑 ⃑
𝑟!!𝑑 ⃑
𝑟"! … 𝑑 ⃑
𝑟#! (𝒫%/'𝜙)∗
⃑
𝑟(!!!, ⃑
𝑟(!"!, … , ⃑
𝑟(!#! 𝜓 ⃑
𝑟!!, ⃑
𝑟"!, … , ⃑
𝑟#!
𝑃1 = 1!
, 𝑃2 = 2!
, … 𝑃𝑁 = 𝑁!
; 𝑃!
= 𝑃"#
; 𝜉$
= 𝜉$!
𝒫$/& = [𝒫$/&]†
2 𝑃𝑟𝑜𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑂𝑝𝑒𝑟𝑎𝑡𝑜𝑟 𝒫2
$/& = 𝒫$/&
𝒫2
%/'𝜓 ⃑
𝑟!, ⃑
𝑟", … , ⃑
𝑟# =
!
#!
∑( 𝜉(𝒫%/'𝜓 ⃑
𝑟(!, ⃑
𝑟(", … , ⃑
𝑟(# =
!
#!
∑( 𝜉( !
#!
∑(! 𝜉(!
𝜓 ⃑
𝑟(!(!, ⃑
𝑟(!(", … , ⃑
𝑟(!(#
=
!
#!
∑(
!
#!
∑(! 𝜉(*(!
𝜓 ⃑
𝑟(!(!, ⃑
𝑟(!(", … , ⃑
𝑟(!(# =
!
#!
∑(
!
#!
∑+ 𝜉+
𝜓 ⃑
𝑟+!, ⃑
𝑟+", … , ⃑
𝑟+# =
!
#!
∑( 𝒫%/'𝜓 ⃑
𝑟(!, ⃑
𝑟(", … , ⃑
𝑟(# = 𝒫%/'𝜓 ⃑
𝑟(!, ⃑
𝑟(", … , ⃑
𝑟(#
𝑄 = 𝑃𝑃!
, 𝜉$&$!
= 𝜉$$!
𝒫!ℋ" = ℬ" 𝒫#ℋ" = ℱ"
Hilbert Space of N-Boson & N-Fermion States

https://math.mit.edu/events/stanley70/Site/Slides/Early.pdf
N-particle States as Irreducible Basis Vectors of Symmetric Group 𝑺𝑵
(1)(2)(3)(4) (12)(3)(4) (12)(34) (123)(4) (1234)
http://www.hep.caltech.edu/~fcp/math/groupTheory/young.pdf
D
𝑃MM
N+
=
𝑙O
𝑁P
G
Q
8,
𝐷MM
N+∗
𝑔 𝑃
Q (𝑚 = 1,2, . . 𝑙O)
𝑒M
N+
= D
𝑃MM
N+
𝛹 (𝑚 = 1,2, . . 𝑙O)
Any finite group 𝐺 = {𝑔} of order 𝑁Pwhose 𝑙O −dimension irreducible representations
𝛤
O(𝜈 = 1,2, . . ) are matrix group {𝐷MS
N+
}, act as operators group 𝑃P = {𝑃
Q} in a representation
space 𝑉, then the projection operators in 𝑉 corresponding to irreducible representation
𝛤
O(𝜈 = 1,2, . . ) are given by
And given any vector 𝛹 in 𝑉, the irreducible
basis vectors of 𝛤
O(𝜈 = 1,2, . . ) are
𝒫J/K𝜓 ⃑
𝑟6, ⃑
𝑟7, … , ⃑
𝑟8 =
1
𝑁!
G
U
𝜉U𝜓 ⃑
𝑟U6, ⃑
𝑟U7, … , ⃑
𝑟U8
𝜉 = B

𝛼6𝛼7 … 𝛼8 ≡ 𝑁! 𝒫J∕K 𝛼6𝛼7 … 𝛼8 =
1
𝑁!
G
U
𝜉U ⟩
|𝛼U6 ⨂ ⟩
|𝛼U7 ⨂ … ⨂ ⟩
|𝛼U8
A basis in ℬ8 & ℱ8 labeled by a general basis { ⟩
|𝛼 }in ℋ
𝛼$𝛼$ … 𝛼& = 𝑁! 𝒫8∕: 𝛼$𝛼$ … 𝛼& = − 𝑁! 𝒫8∕: 𝛼$𝛼$ … 𝛼&
𝜉 = -
For a N-fermion state, if 𝛼6 = 𝛼7
𝛼'𝛼' … 𝛼( = 0
Completeness/Closure: ( )
!!,!",…,!#
|𝛼$, 𝛼%, … , 𝛼&)(𝛼$, 𝛼%, … , 𝛼&| = 1
In ℋ%
( )
!!,!",…,!#
𝒫8∕:|𝛼$, 𝛼%, … , 𝛼&)(𝛼$, 𝛼%, … , 𝛼&|𝒫 ⁄
8 : = 𝒫2
⁄
8 : = 𝒫8∕:
( )
!!,!",…,!#
𝒫8∕:|𝛼$, 𝛼%, … , 𝛼&)(𝛼$, 𝛼%, … , 𝛼&|𝒫 ⁄
8 : =
1
𝑁!
( )
!!,!",…,!#
𝛼$, 𝛼%, … , 𝛼& {𝛼$, 𝛼%, … , 𝛼&| = 1
In ℬ% & ℱ%
ℋ(
ℬ( ℱ(
{ 𝛼'𝛼' … 𝛼( }
{𝒫, 𝛼#𝛼# … 𝛼- } {𝒫. 𝛼#𝛼# … 𝛼- }
𝒫$∕&
𝒫$∕& 𝛼'𝛼+ … 𝛼( = 𝛼'𝛼+ … 𝛼(

Orthogonality: {𝛼!, 𝛼", … , 𝛼#|𝛼$
!𝛼$
" … 𝛼$
#} = ,
%
𝜉%
𝛿(𝛼! − 𝛼$
%!)𝛿(𝛼" − 𝛼$
%") … 𝛿(𝛼# − 𝛼$
%#)
𝑁! (𝛼%, 𝛼&, … , 𝛼' 𝒫2
(∕* 𝛼+
%𝛼+
& … 𝛼+
') = 𝑁! (𝛼%, 𝛼&, … , 𝛼' 𝒫(∕* 𝛼+
%𝛼+
& … 𝛼+
')
= ∑, 𝜉, (𝛼% − 𝛼+
,%)𝛿(𝛼& − 𝛼+
,&) … 𝛿(𝛼' − 𝛼+
,')
(I) For fermions, one particle per state ⟩
|𝛼
𝛼!, 𝛼", … , 𝛼# 𝛼$
!𝛼$
" … 𝛼$
# = ∑% −1 % 𝛿 𝛼! − 𝛼$
%! 𝛿 𝛼" − 𝛼$
%" … 𝛿 𝛼# − 𝛼$
%#
= −1 %,, where 𝑃3 𝛼$
!𝛼$
" … 𝛼$
# = (𝛼!, 𝛼", … , 𝛼#)
(II) For bosons, 𝑛2 particles per state ⟩
|𝛼2 , 𝑖 = 1,2 … 𝑝 ≤ 𝑁
𝛼!, 𝛼", … , 𝛼# 𝛼$
!𝛼$
" … 𝛼$
# = ∑% −1 %
𝛿 𝛼! − 𝛼$
%! 𝛿 𝛼" − 𝛼$
%" … 𝛿 𝛼# − 𝛼$
%#
= 𝑛6! 𝑛7! … 𝑛^!, where ∑45!
6
𝑛< = 𝑁
𝛼&, 𝛼', … , 𝛼% 𝛼$
&𝛼$
' … 𝛼$
% = 𝜉;
T
0
𝑛0!
0! = 1; 𝜉 = B
a
-
𝑛- = 𝑁
𝑃 𝛼"
'𝛼"
+ … 𝛼"
( = (𝛼', 𝛼+, … , 𝛼()

Orthonormal basis in 𝓑𝑵 & 𝓕𝑵
⟩
|𝛼6𝛼7 … 𝛼8 ≡
1
∏c 𝑛c!
𝛼6𝛼7 … 𝛼8 =
1
𝑁! ∏c 𝑛c!
G
U
𝜉U ⟩
|𝛼U6 ⨂ ⟩
|𝛼U7 ⨂ … ⨂ ⟩
|𝛼U8
Completeness/Closure in 𝓑𝑵 & 𝓕𝑵 :
W X
𝜶𝟏,𝜶𝟐,…,𝜶𝑵
∏0 𝑛0!
𝑁!
⟩
|𝜶𝟏𝜶𝟐 … 𝜶𝑵 ⟨𝜶𝟏𝜶𝟐 … 𝜶𝑵| = 𝟏
(𝛽#, 𝛽/, … , 𝛽- ⟩
|𝛼#𝛼/ … 𝛼- =
1
𝑁! ∏0 𝑛0!
D
$
𝜉$
𝛽# 𝛼$# 𝛽/ 𝛼$/ … 𝛽- 𝛼$- =
1
𝑁! ∏0 𝑛0!
𝑆( 𝛽1 𝛼2 )
𝑆 𝛽1 𝛼2 ≡ D
$
𝜉$
𝛽# 𝛼# 𝛽/ 𝛼/ … 𝛽- 𝛼- = H
𝑃𝑒𝑟 𝛽1 𝛼2 , 𝐵𝑜𝑠𝑜𝑛𝑠
Det 𝛽1 𝛼2 , 𝐹𝑒𝑟𝑚𝑖𝑜𝑛𝑠
Permanent 𝑃𝑒𝑟 𝑀12 = ∑$ 𝑀#,$#𝑀/,$/ … 𝑀-,$-
Determinant 𝐷𝑒𝑡 𝑀12 = ∑$ −1 $
𝑀#,$#𝑀/,$/ … 𝑀-,$-
𝜷𝟏, 𝜷𝟐, … , 𝜷𝑵 𝜶𝟏𝜶𝟐 … 𝜶𝑵 =
#
-! ∏" 9"! ∏# 9#!
∑$,$! 𝜉$&$!
𝛽$!# 𝛼$# 𝛽$!/ 𝛼$/ … 𝛽$!- 𝛼$-
=
#
-! ∏" 9"! ∏# 9#!
∑$,: 𝜉:
𝛽#! 𝛼:#! 𝛽/! 𝛼:/! … 𝛽;! 𝛼:-! =
𝟏
∏𝜶 𝒏𝜶! ∏𝜷 𝒏𝜷!
𝑺 𝜷𝒊 𝜶𝒋
𝜉#.#-
= 𝜉##-
= 𝜉#(#-)./
; 𝑄 = 𝑃(𝑃"
)1'
𝑃′1 = 1!
, 𝑃′2 = 2!
, … 𝑃′𝑁 = 𝑁!
;
G
c
𝑛c = 𝑁

Many-Body Operators
For an arbitrary operator +
𝑂 in ℬ% & ℱ% and any permutation 𝑃
(𝛼&, 𝛼', … , 𝛼%| +
𝑂|𝛼$
&𝛼$
' … 𝛼$
%) = (𝛼;&, 𝛼;', … , 𝛼;%| +
𝑂|𝛼$
;&𝛼$
;' … 𝛼$
;%)
(1) One-body Operator:
+
𝑂 𝛼&𝛼' … 𝛼% = X
"@&
%
+
𝑂" 𝛼&𝛼' … 𝛼%
Where D
𝑂2 only acts on particle 𝑖
D
𝑇 ⃑
𝑝6 ⃑
𝑝7 … ⃑
𝑝8 = G
2e6
8
`
⃑
𝑝2
7
2𝑚
⃑
𝑝6 ⃑
𝑝7 … ⃑
𝑝8
(𝛼!, 𝛼", … , 𝛼#| 6
𝑂|𝛽!, 𝛽", … , 𝛽#) = ,
45!
#
(𝛼!, 𝛼", … , 𝛼#| 6
𝑂4|𝛽!, 𝛽", … , 𝛽#) = ,
45!
#
9
784
𝛼7 𝛽7 𝛼4
6
𝑂4 𝛽4
(𝛼6, 𝛼7, … , 𝛼8| D
𝑂|𝛽6, 𝛽7, … , 𝛽8)
(𝛼6, 𝛼7, … , 𝛼8|𝛽6, 𝛽7, … , 𝛽8)
= G
2e6
8
𝛼2
D
𝑂2 𝛽2
𝛼2 𝛽2
For two non-orthogonal states

(2) Two-body Operator %
𝑉 :
Many-Body Operators
+
𝑉 𝛼&𝛼' … 𝛼% = X
&A"BCA%
%
+
𝑉"C 𝛼&𝛼' … 𝛼% =
1
2
X
&A"DCA%
%
+
𝑉"C 𝛼&𝛼' … 𝛼%
Where D
𝑉2f only acts on particles 𝑖 and 𝑗 D
𝑉2f = D
𝑉
f2
𝛼$, 𝛼%, … , 𝛼&
E
𝑉 𝛽$, 𝛽%, … , 𝛽& =
1
2
)
$=<>?=&
&
𝛼$, 𝛼%, … , 𝛼&
E
𝑉<? 𝛽$, 𝛽%, … , 𝛽& =
1
2
)
<>?
I
@><
@>?
𝛼@ 𝛽@ 𝛼<𝛼?
E
𝑉<? 𝛽<𝛽?
(𝛼6, 𝛼7, … , 𝛼8| D
𝑉|𝛽6, 𝛽7, … , 𝛽8)
(𝛼6, 𝛼7, … , 𝛼8|𝛽6, 𝛽7, … , 𝛽8)
=
1
2
G
2gf
𝛼2𝛼f
D
𝑉2f 𝛽2𝛽f
𝛼2 𝛽2 𝛼f 𝛽f
A local two-body operator satisfies ⃑
𝑟& ⃑
𝑟'
+
𝑉 ⃑
𝑟J ⃑
𝑟K = 𝛿 ⃑
𝑟& − ⃑
𝑟J 𝛿 ⃑
𝑟' − ⃑
𝑟K 𝑣(⃑
𝑟& − ⃑
𝑟')
+
𝑉 ⃑
𝑟&, ⃑
𝑟', … , ⃑
𝑟% =
1
2
X
&A"DCA%
%
𝑣(⃑
𝑟& − ⃑
𝑟') ⃑
𝑟&, ⃑
𝑟', … , ⃑
𝑟%

Many-Body Operators
(3) n-body Operator '
𝑹 :
+
𝑅 𝛼&𝛼' … 𝛼% =
1
𝑛!
X
&A"!D"#D⋯D"'A%
%
+
𝑅"!"#…"'
𝛼&𝛼' … 𝛼%
D
𝑅2T2U…2V
= D
𝑅U2T,U2U,…,U2V
(𝛼6, 𝛼7, … , 𝛼8| D
𝑅|𝛽6, 𝛽7, … , 𝛽8)
(𝛼6, 𝛼7, … , 𝛼8|𝛽6, 𝛽7, … , 𝛽8)
=
1
𝑛!
G
2Tg2Ug⋯g2V
𝛼2T
𝛼2U
… 𝛼2V
D
𝑅 𝛽2T
𝛽2U
… 𝛽2V
𝛼2T
𝛽2T
𝛼2U
𝛽2U
… 𝛼2V
𝛽2V
%
𝑅 on an N-particle state is the sum of the action of %
𝑅 on all distinct
subsets of n-particles
An n-body operator is entirely determined by its matrix elements
𝛼%!
𝛼%"
… 𝛼%#
%
𝑅 𝛽%!
𝛽%"
… 𝛽%#
in the Hilbert space ℋ& of n-particle
space.

Fock Space and !
𝒏-representation
Fock space is the direct sum of all 𝑛-particle Boson ℬ% = 𝒫M"
ℋ%(𝑁 =
0,1,2 … ) or Fermion spaces ℱ$ = 𝒫#$
ℋ'(𝑁 = 0,1,2 … ):
ℬ = ℬ(⨁ℬ)⨁ℬ*⨁ … = ⨁'+(
,
ℬ' ℱ = ℱ(⨁ℱ)⨁ℱ*⨁ … = ⨁'+(
,
ℱ'
ℬ: = ℱ: = ⟩
|0 ℬ" = ℱ" = ℋ
G
𝑵e𝟏
j
1
𝑁!
c G
d
c
𝑛c! ⟩
|𝜶𝟏𝜶𝟐 … 𝜶𝑵 ⟨𝜶𝟏𝜶𝟐 … 𝜶𝑵| + ⟩
|𝟎 ⟨𝟎| = 𝟏
ℬ8 or ℱ8 c G
∏c 𝑛c!
𝑁!
⟩
|𝜶𝟏𝜶𝟐 … 𝜶𝑵 ⟨𝜶𝟏𝜶𝟐 … 𝜶𝑵| = 𝟏
ℬ or ℱ
,
𝑵5𝟏
;
1
𝑁!
< ,
9
=
𝑛=! ⟩
|𝜶𝟏𝜶𝟐 … 𝜶𝑵 ⟨𝜶𝟏𝜶𝟐 … 𝜶𝑵| + ⟩
|𝟎 ⟨𝟎| = ,
𝑵5𝟏
;
𝒫?5/A5
+ ⟩
|𝟎 ⟨𝟎|
)
!
𝑛! = 𝑁
⨁#53
;
ℋ#

Fock Space and !
𝒏-representation
𝜶𝛼&𝛼' … 𝛼% = 𝑎0
† 𝛼&𝛼' … 𝛼%
For a general basis { ⟩
|𝛼 }in ℋ, define a Boson or Fermion creation operator
𝑎0
† on any symmetrized or antisymmetrized state 𝛼&𝛼' … 𝛼% of ℬ% &
ℱ%(𝑁 = 0,1,2, … )
𝑛𝜶 + 1 ⟩
|𝜶𝜶𝟏𝜶𝟐 … 𝜶𝑵 = 𝑎0
† ⟩
|𝜶𝟏𝜶𝟐 … 𝜶𝑵
Or
𝑎=
† ⟩
|𝜶𝟏𝜶𝟐 … 𝜶𝑵 = 𝑎=
† !
∏6 C6!
𝛼!𝛼" … 𝛼# =
!
∏6 C6!
𝜶𝛼!𝛼" … 𝛼# = 𝑛𝜶 + 1 ⟩
|𝜶𝜶𝟏𝜶𝟐 … 𝜶𝑵
⟩
|𝜶𝜶𝟏𝜶𝟐 … 𝜶𝑵 =
1
𝑛𝜶 + 1 ∏8 𝑛8!
𝜶𝛼'𝛼+ … 𝛼(
𝑎c
† ⟩
|𝛼6𝛼7 … 𝛼8 = 𝑛c + 1 ⟩
|𝛼𝛼6𝛼7 … 𝛼8 = 0, if 𝛼 ∈ {𝛼6, 𝛼7, … , 𝛼8}
(1) If 𝑁 = 0, 𝑎0
†
⟩
|𝟎 = ⟩
|𝛼 ;
(2) For Fermions
⟩
|𝟎 ≠ 𝟎
ℬ(.'/ℱ(.'
ℬ(/ℱ(
𝑎c
†
⟩
|𝜶𝟏𝜶𝟐 … 𝜶𝑵 ⟩
|𝜶𝜶𝟏𝜶𝟐 … 𝜶𝑵
𝑎c

(3) Any basis vector could be generated by 𝑎𝜶𝒊
† (𝑖 = 1,2, … 𝑁) from ⟩
|𝟎
Fock Space and !
𝒏-representation
𝜶𝟏𝜶𝟐 … 𝜶𝑵 = 𝒂𝜶𝟏
† 𝒂𝜶𝟐
† … 𝒂𝜶𝑵
† ⟩
|𝟎
⟩
|𝜶𝟏𝜶𝟐 … 𝜶𝑵 =
1
∏𝜶 𝑛𝜶!
𝑎𝜶𝟏
† 𝑎𝜶𝟐
† … 𝑎𝜶𝑵
† ⟩
|𝟎
(4) Annihilation operator 𝑎0 = [𝑎0
†]†
𝑎c ⟩
|𝟎 = 𝟎
∀𝛼, 𝛼 𝑎c 0 = ( 0 𝑎c
† 𝛼 )∗= 0
⟨0|𝑎c
† = 0
𝑎0 𝛽#𝛽/ … 𝛽9 = ∑-?@
A #
-!
∫ ∑0&,0',…,0(
𝛼#𝛼/ … 𝛼- {𝛼#𝛼/ … 𝛼-| 𝑎0 𝛽#𝛽/ … 𝛽9 = ∑-?@
A #
-!
∫ ∑0&,0',…,0(
𝛼#𝛼/ … 𝛼- {𝛼𝛼#𝛼/ … 𝛼-| 𝛽#𝛽/ … 𝛽9}
=
#
(9"#)!
∫ ∑0&,0',…,0)*&
𝛼#𝛼/ … 𝛼9"# {𝛼𝛼#𝛼/ … 𝛼9"#| 𝛽#𝛽/ … 𝛽9} =
#
(9"#)!
∫ ∑0&,0',…,0)*&
𝛼#𝛼/ … 𝛼9"# ∑$ 𝜉$
𝛿 𝛼 − 𝛽$# 𝛿 𝛼# − 𝛽$/ … 𝛿(𝛼9"# − 𝛽$9)
=
#
(9"#)!
∫ ∑0&,0',…,0)*&
𝛼#𝛼/ … 𝛼9"# ∑1?#
9
𝜉 1"#
𝛿 𝛼 − 𝛽1 ) ∑$ 𝜉$
𝛿 𝛼# − 𝛽$# … 𝛿 𝛼1"# − 𝛽$(1"#) 𝛿 𝛼1 − 𝛽$(1&#) … 𝛿(𝛼9"# − 𝛽$9)
=
#
(9"#)!
∑1?#
9
𝜉 1"#
𝛿 𝛼 − 𝛽1 ) ∑$ 𝜉$
𝛽$# … 𝛽$(1"#)𝛽$(1&#) … 𝛽$9 =
#
(9"#)!
∑1?#
9
𝜉 1"#
𝛿 𝛼 − 𝛽1 ) ∑$ 𝜉/$
𝛽# … 𝛽(1"#)𝛽(1&#) … 𝛽9
=
#
(9"#)!
∑1?#
9
𝜉 1"#
𝛿 𝛼 − 𝛽1 )(𝑛 − 1)! 𝛽# … 𝛽(1"#)𝛽(1&#) … 𝛽9 = ∑1?#
9
𝜉 1"#
𝛿 𝛼 − 𝛽1 𝛽# … 𝛽(1"#)𝛽(1&#) … 𝛽9 = ∑1?#
9
𝜉 1"#
𝛽#𝛽/ … ]
𝛽1 … 𝛽9 𝛿 𝛼 − 𝛽1
D
𝑵?𝟏
A
1
𝑁!
^ D
𝜶𝟏𝜶𝟐 … 𝜶𝑵 {𝜶𝟏𝜶𝟐 … 𝜶𝑵| + ⟩
|𝟎 ⟨𝟎| = 𝟏
𝑎! 𝛽$𝛽% … 𝛽A = )
<B$
A
𝜉 <C$ 𝛽$𝛽% … P
𝛽< … 𝛽A 𝛿 𝛼 − 𝛽<
𝑎! ⟩
|𝛽$𝛽% … 𝛽A =
1
𝑛𝜶
)
<B$
A
𝜉 <C$ R
|𝛽$𝛽% … P
𝛽< … 𝛽A 𝛿 𝛼 − 𝛽<

Fock Space and !
𝒏-representation
(5) Commutation & Anticommutation Relation
𝑎0
†𝑎S
† 𝜶𝟏𝜶𝟐 … 𝜶𝑵 = 𝛼𝛽𝜶𝟏𝜶𝟐 … 𝜶𝑵 = 𝜉 𝛽𝛼𝜶𝟏𝜶𝟐 … 𝜶𝑵 = 𝜉𝑎S
†𝑎0
† 𝜶𝟏𝜶𝟐 … 𝜶𝑵
𝑎=, 𝑎D
†
EF
= 𝑎=𝑎D
† − 𝜉𝑎D
†𝑎= = 𝛿=D 𝑎=
†, 𝑎D
†
EF
= 𝑎=, 𝑎D EF
= 0
𝑎=𝑎D
† 𝜶𝟏𝜶𝟐 … 𝜶𝑵 = 𝑎= 𝛽𝜶𝟏𝜶𝟐 … 𝜶𝑵 = 𝛿=D 𝜶𝟏𝜶𝟐 … 𝜶𝑵 + ,
45!
#
𝜉4
𝛽𝜶!𝜶" … H
𝜶4 … 𝜶# 𝛿 𝛼 − 𝜶4
= 𝛿=D 𝜶𝟏𝜶𝟐 … 𝜶𝑵 + 𝜉 ∑45!
#
𝜉 4E! 𝑎D
† 𝜶!𝜶" … H
𝜶4 … 𝜶# 𝛿 𝛼 − 𝜶4
= 𝛿=D 𝜶𝟏𝜶𝟐 … 𝜶𝑵 + 𝜉𝑎D
†𝑎= 𝜶!𝜶" … 𝜶#
𝜉 = B
(6) Basis Transformation ( )
!
⟩
|𝛼 ⟨𝛼| = 1 ( )
E
!
⟩
| T
𝛼 ⟨ T
𝛼| = 1
⟩
| k
𝛼 = c G
c
⟩
|𝛼 𝛼 k
𝛼
𝑎G
=
† 𝛼!𝛼" … 𝛼# = I
𝛼𝛼!𝛼" … 𝛼# = < ,
=
𝛼 I
𝛼 𝜶𝛼!𝛼" … 𝛼# = < ,
=
𝛼 I
𝛼 𝑎=
† 𝛼!𝛼" … 𝛼#
𝑎n
c
† = c G
c
𝛼 k
𝛼 𝑎c
† 𝑎n
c = c G
c
k
𝛼 𝛼 𝑎c 𝑎E
!, 𝑎o
p
†
EF
= ,
=D
̃
𝛼 𝛼 𝛽 V
𝛽 𝑎=, 𝑎D
†
EF
= k
𝛼 l
𝛽

Fock Space and !
𝒏-representation
(7) h
𝒙-representation ⟩
|𝒙 = ⟩
|𝒓𝝈𝝉 𝒓: Space Coordinate; 𝝈: Spin; 𝝉: Internal degrees of freedom
Field Operators:
6
𝜓† 𝑥 = 𝑎H
† = < ,
=
𝛼 𝑥 𝑎=
† = < ,
=
𝜙=
∗
(𝑥) 𝑎=
† 6
𝜓(𝑥) = 𝑎H = < ,
=
𝑥 𝛼 𝑎= = < ,
=
𝜙=(𝑥) 𝑎=
6
𝜓(𝑥), 6
𝜓† 𝑦
EF
= 𝛿(𝑥 − 𝑦) 6
𝜓† 𝑥 , 6
𝜓† 𝑦
EF
= 6
𝜓(𝑥), 6
𝜓(𝑦) EF
= 0
(8) h
𝒏-representation q
𝒏𝜶 = 𝒂𝜶
†𝒂𝜶
t
𝑛c 𝛼6𝛼7 … 𝛼8 = 𝑎c
†𝑎c 𝛼6𝛼7 … 𝛼8 = 𝑎c
† ∑2e6
8
𝜉 2q6 𝛿 𝛼 − 𝛼2 𝛼6𝛼7 … t
𝛼2 … 𝛼8
= ∑2e6
8
𝛿 𝛼 − 𝛼2 𝛼6𝛼7 … 𝛼2 … 𝛼8
N
𝑵 = ,
𝜶
H
𝒏𝜶 = ,
𝜶
𝒂𝜶
†𝒂𝜶
q
𝒏𝜶 ⟩
|𝒏𝜶 = 𝒏𝜶 ⟩
|𝒏𝜶
(8.1) Hermitian operator
(8.2) Eigenstates and Eigenvalues
𝒏𝜶 = 𝒏𝜶 q
𝒏𝜶 𝒏𝜶 = 𝒏𝜶 𝒂𝜶
†𝒂𝜶 𝒏𝜶 = 𝒂𝜶 ⟩
|𝒏𝜶
𝟐 ≥ 𝟎 𝒏𝜶 = 𝟎, iff 𝒂𝜶 ⟩
|𝟎 = 𝟎
(8.2.1) Bosons
𝑎=, 𝑎=
†
E
= 1 𝒂𝜶
†𝒂𝜶𝒂𝜶
† = 𝒂𝜶
†(𝟏 + 𝒂𝜶
†𝒂𝜶)

Fock Space and !
𝒏-representation
(8.2.1) Bosons 𝒂𝜶
𝒌 ⟩
|𝒏𝜶 = 𝑨(𝒏; 𝒌) ⟩
|𝒏𝜶 − 𝒌 (𝒂𝜶
†)𝒌 ⟩
|𝒏𝜶 = 𝑩(𝒏; 𝒌) ⟩
|𝒏𝜶 + 𝒌
Since 𝑛0 ≥ 0, 𝑛0 has to be an integer, or p
𝑛0 would have negative eigenvalues !
and the minimum eigenvalue is 0 with 𝒂𝜶 ⟩
|𝟎 = 𝟎.
𝑘 = 0,1,2, …
9
𝒏𝜶 ⟩
|𝒏𝜶 , 𝒏𝜶 = 𝟎, 𝟏, 𝟐, 𝟑, …
(𝒂𝜶
†)𝒌 ⟩
|𝟎 = 𝑩(𝒏; 𝒌) ⟩
|𝒌 𝟎 𝒂𝜶
𝒌
(𝒂𝜶
†)𝒌 ⟩
|𝟎 = 𝑩(𝒏; 𝒌) 𝟐
𝒌 𝒌 = 𝑩(𝒏; 𝒌) 𝟐
= 𝒌!
𝒂𝜶
𝒌
(𝒂𝜶
†)𝒌
= 𝒂𝜶
𝒌"𝟏
𝟏 + e
𝒏𝜶 𝒂𝜶
†
𝒌"𝟏
= 𝒂𝜶
𝒌"𝟏
(𝒂𝜶
†)𝒌"𝟏
+𝒂𝜶
𝒌"𝟏
e
𝒏𝜶(𝒂𝜶
†)𝒌"𝟏
= 𝒂𝜶
𝒌"𝟏
(𝒂𝜶
†)𝒌"𝟏
+𝒂𝜶
𝒌"𝟏
𝒂𝜶
†
𝒌"𝟏
e
𝒏𝜶 + 𝒌 − 𝟏 = 𝒌𝒂𝜶
𝒌"𝟏
(𝒂𝜶
†)𝒌"𝟏
+𝒂𝜶
𝒌"𝟏
𝒂𝜶
†
𝒌"𝟏
e
𝒏𝜶
𝒂𝜶
𝒌
(𝒂𝜶
†)𝒌
= 𝒌 𝒌 − 𝟏 𝒂𝜶
𝒌"𝟐
𝒂𝜶
†
𝒌"𝟐
+ 𝒂𝜶
𝒌"𝟐
𝒂𝜶
†
𝒌"𝟐
e
𝒏𝜶 + 𝒂𝜶
𝒌"𝟏
𝒂𝜶
†
𝒌"𝟏
e
𝒏𝜶 = 𝒌 𝒌 − 𝟏 𝒂𝜶
𝒌"𝟐
𝒂𝜶
†
𝒌"𝟐
+ 𝒌𝒂𝜶
𝒌"𝟐
𝒂𝜶
†
𝒌"𝟐
e
𝒏𝜶 + 𝒂𝜶
𝒌"𝟏
𝒂𝜶
†
𝒌"𝟏
e
𝒏𝜶
= ⋯ … =
𝒌!
𝟎!
+ D
𝒎?𝟎
𝒌"𝟏
𝒌!
(𝒎 + 𝟏)!
𝒂𝜶
𝒎
𝒂𝜶
†
𝒎
e
𝒏𝜶
⟩
|𝒏𝜶 =
𝟏
𝒏𝜶!
(𝒂𝜶
†)𝒏𝜶 ⟩
|𝟎
𝑩 𝒏; 𝒌 = 𝒌!
𝒂𝜶 ⟩
|𝒏𝜶 =
𝒏𝜶
(𝒏𝜶 − 𝟏)!
𝒂𝜶𝒊
†
𝒏𝜶𝒊
E𝟏
⟩
|𝟎 = 𝒏𝜶 ⟩
|𝒏𝜶 − 𝟏

Fock Space and !
𝒏-representation
(8.2.2) Fermions 𝒂𝜶, 𝒂𝜶
†
K
= 𝒂𝜶𝒂𝜶
† + 𝒂𝜶
†𝒂𝜶 = 𝟏
q
𝒏𝜶 𝒂𝜶 ⟩
†𝒂𝜶𝒂𝜶 ⟩
|𝒏𝜶 = 𝟎
q
𝒏𝜶 𝒂𝜶
† ⟩
†(𝟏 − 𝒂𝜶
†𝒂𝜶) ⟩
† ⟩
|𝒏𝜶
𝒂𝜶
𝟐 = (𝒂𝜶
†)𝟐= 𝟎
(i) 𝒂𝜶 ⟩
|𝒏𝜶 = 𝟎 q
𝒏𝜶 ⟩
|𝒏𝜶 = 𝟎 𝒏𝜶 = 𝟎 𝒂𝜶 ⟩
|𝟎 = 𝟎
𝒂𝜶
† ⟩
|𝟎 = 𝑪 ⟩
|𝟏
𝟎 𝒂𝜶𝒂𝜶
† ⟩
|𝟎 = 𝑪 𝟐
𝟏 𝟏 = 𝑪 𝟐
= 𝟏 𝑪 = 𝟏
𝒂𝜶
† ⟩
|𝟎 = ⟩
|𝟏 (𝒂𝜶
†)𝟐 ⟩
|𝟎 = 𝒂𝜶
† ⟩
|𝟏 = 𝟎
(ii) 𝒂𝜶 ⟩
|𝒏𝜶 = 𝑭 ⟩
|𝟎
𝟎 𝒂𝜶𝒂𝜶
† ⟩
|𝟎 = 𝑫 𝟐
𝟏 𝟏 = 𝑫 𝟐
= 𝟏 𝑫 = 𝟏
𝒂𝜶 ⟩
|𝟏 = 𝒂𝜶𝒂𝜶
† ⟩
|𝟎 = 𝟏 − q
𝒏𝜶 ⟩
|𝟎 = ⟩
|𝟎
𝒂𝜶 ⟩
|𝒏𝜶 = 𝟎 or 𝒂𝜶 ⟩
|𝒏𝜶 ∝ ⟩
|𝟎
𝒂𝜶
† ⟩
|𝒏𝜶 = 𝟎 or 𝒂𝜶
† ⟩
|𝒏𝜶 ∝ ⟩
|𝟏
𝒂𝜶
𝟐 ⟩
|𝒏𝜶 = 𝑫𝒂𝜶 ⟩
|𝟎 = 𝟎 𝒂𝜶 ⟩
|𝟎 = 𝟎
𝒂𝜶
† ⟩
|𝟎 = 𝑫 ⟩
|𝟏 𝒂𝜶
† ⟩
|𝟎 = ⟩
|𝟏 𝒂𝜶
† ⟩
|𝟏 = 𝟎 𝒂𝜶 ⟩
|𝟏 = ⟩
|𝟎 𝒏𝜶 = 𝟏, 𝑭 = 𝟏
9
𝒏𝜶 ⟩
|𝒏𝜶 , 𝒏𝜶 = 𝟎, 𝟏 ⟩
|𝒏𝜶 = (𝒂𝜶
†)𝒏𝜶 ⟩
|𝟎

G
𝒂𝜶
†|𝒏𝜶𝟏
𝒏𝜶𝟐
… 𝒏𝜶𝒌
… = ∑𝒊 𝜹𝜶𝜶𝒊
𝟏
𝒏𝜶𝟏
!𝒏𝜶𝟐
!…𝒏𝜶𝒊
!…
(𝒂𝜶𝟏
† )𝒏𝜶𝟏 (𝒂𝜶𝟐
† )𝒏𝜶𝟐 … 𝒂𝜶𝒊
†
𝒏𝜶𝒊
*𝟏
… ⟩
|𝟎
= ∑𝒊 𝜹𝜶𝜶𝒊
𝒏𝜶𝒊
*𝟏
𝒏𝜶𝟏
!𝒏𝜶𝟐
!…(𝒏𝜶𝒊
*𝟏)!…
(𝒂𝜶𝟏
† )𝒏𝜶𝟏 (𝒂𝜶𝟐
† )𝒏𝜶𝟐 … (𝒂𝜶𝒊
† )𝒏𝜶𝒊
*𝟏
… ⟩
|𝟎 = ∑𝒊 𝜹𝜶𝜶𝒊
𝒏𝜶𝒊
+ 𝟏 P
|𝒏𝜶𝟏
𝒏𝜶𝟐
… (𝒏𝜶𝒊
+ 𝟏) …
Fock Space and !
𝒏-representation
(8.3) Orthonormal Basis in t
𝑛-representation
𝒏𝜶𝒊
(𝒊 = 𝟏, 𝟐, 𝟑, … ) are the particle numbers in state 𝜶𝒊; 𝒏𝜶𝒊
= 𝟎, 𝟏 for Fermions
•
|𝒏𝜶𝟏
𝒏𝜶𝟐
… 𝒏𝜶𝒌
… =
𝟏
𝒏𝜶𝟏
! 𝒏𝜶𝟐
! … 𝒏𝜶𝒌
! …
(𝒂𝜶𝟏
† )𝒏𝜶𝟏 (𝒂𝜶𝟐
† )𝒏𝜶𝟐 … (𝒂𝜶𝒌
† )𝒏𝜶𝒌 … ⟩
|𝟎
P
𝒂𝜶|𝒏𝜶𝟏
𝒏𝜶𝟐
… 𝒏𝜶𝒌
… = ∑𝒊 𝜹𝜶𝜶𝒊
𝒂𝜶𝒊
𝒏𝜶𝟏
!𝒏𝜶𝟐
!…𝒏𝜶𝒊
!…
(𝒂𝜶𝟏
† )𝒏𝜶𝟏 (𝒂𝜶𝟐
†
𝒏𝜶𝒊
… ⟩
|𝟎
= ∑𝒊 𝜹𝜶𝜶𝒊
𝒏𝜶𝒊
𝒏𝜶𝒊
𝒏𝜶𝟏
!𝒏𝜶𝟐
!…(𝒏𝜶𝒊
8𝟏)!…
(𝒂𝜶𝟏
† )𝒏𝜶𝟏 (𝒂𝜶𝟐
†
𝒏𝜶𝒊
8𝟏
… ⟩
|𝟎 𝒐𝒓 ∑𝒊 𝜹𝜶𝜶𝒊
𝜹𝟏𝒏𝜶𝒊
𝟏
𝒏𝜶𝟏
!𝒏𝜶𝟐
!…(𝒏𝜶𝒊
8𝟏)!…
(𝒂𝜶𝟏
† )𝒏𝜶𝟏 (𝒂𝜶𝟐
†
𝒏𝜶𝒊
8𝟏
… ⟩
|𝟎
= S
∑𝒊 𝜹𝜶𝜶𝒊
𝒏𝜶𝒊
P
|𝒏𝜶𝟏
𝒏𝜶𝟐
… (𝒏𝜶𝒊
− 𝟏) … , 𝑩𝒐𝒔𝒐𝒏
∑𝒊 𝜹𝜶𝜶𝒊
𝜹𝟏𝒏𝜶𝒊
P
|𝒏𝜶𝟏
𝒏𝜶𝟐
… (𝒏𝜶𝒊
− 𝟏) … , 𝑭𝒆𝒓𝒎𝒊𝒐𝒏 𝒂𝜶𝒊
𝒂𝜶𝒊
†
𝒏𝜶𝒊
= 𝒏𝜶𝒊
𝒂𝜶𝒊
†
𝒏𝜶𝒊
8𝟏
+ 𝒂𝜶𝒊
†
𝒏𝜶𝒊
𝒂𝜶𝒊
For 𝑩𝒐𝒔𝒐𝒏
∑𝑵:𝟎
< ∑𝒏𝜶𝟏
.𝒏𝜶𝟐
.⋯.𝒏𝜶𝒌
.⋯:𝑵 u
|𝒏𝜶𝟏
𝒏𝜶𝟐
… 𝒏𝜶𝒌
… v𝒏𝜶𝟏
𝒏𝜶𝟐
… 𝒏𝜶𝒌
… | = ∑𝒏𝜶𝟏
𝒏𝜶𝟐
…𝒏𝜶𝒌
…:𝟎
<
u
|𝒏𝜶𝟏
𝒏𝜶𝟐
… 𝒏𝜶𝒌
… v𝒏𝜶𝟏
𝒏𝜶𝟐
… 𝒏𝜶𝒌
… | = 𝟏
Completeness in Fock Space:
⟩
|𝜶 ≡ { ⟩
|𝜶𝟏 , ⟩
|𝜶𝟐 , … , ⟩
|𝜶𝒌 , … }

Fock Space and !
𝒏-representation
(9) Creation and Annihilation Operators as a Basis for Fock Space Operators
(9.1) One-body Operator
t
𝑛c 𝛼6𝛼7 … 𝛼8 = 𝑎c
†𝑎c 𝛼6𝛼7 … 𝛼8 = 𝑎c
† ∑2e6
8
𝜉 2q6 𝛿 𝛼 − 𝛼2 𝛼6𝛼7 … t
𝛼2 … 𝛼8
= ∑2e6
8
𝛿 𝛼 − 𝛼2 𝛼6𝛼7 … 𝛼2 … 𝛼8
Assume one-body operator %
𝑈 is diagonal in the orthonormal basis { ⟩
|𝛼 }
%
𝑈 ⟩
|𝛼 = 𝑈c ⟩
|𝛼 = 𝛼 %
𝑈 𝛼 ⟩
|𝛼
𝛼', 𝛼+, … , 𝛼(
4
𝑈 𝛽', 𝛽+, … , 𝛽( = a
#
𝜉#
a
!:'
(
y
@A!
𝛼#@ 𝛽@ 𝛼#!
'
𝑂! 𝛽! = a
!:'
(
𝑈-;
𝛼', 𝛼+, … , 𝛼(
4
𝑈 𝛽', 𝛽+, … , 𝛽(
= ∑- ∑!:'
(
𝑈- 𝛿 𝛼 − 𝛼! 𝛼', 𝛼+, … , 𝛼(|𝛽', 𝛽+, … , 𝛽( = 𝛼', 𝛼+, … , 𝛼( ∑- ∑!:'
(
𝑈- 𝛿 𝛼 − 𝛼! 𝛽', 𝛽+, … , 𝛽(
= 𝛼', 𝛼+, … , 𝛼( ∑- 𝑈- {
𝑛- 𝛽', 𝛽+, … , 𝛽(
%
𝑈 = G
c
𝑈c t
𝑛c = G
c
𝛼 %
𝑈 𝛼 𝑎c
†𝑎c 𝑎E
!
† = ( )
!
𝛼 T
𝛼 𝑎!
† 𝑎]
E
!
† = ( )
!
𝛼 T̀
𝛼 𝑎!
†
Diagonal Basis
General Basis %
𝑼 = G
𝜶
𝑼𝜶q
𝒏𝜶 = G
n
𝜶
G
o
n
𝜶
‚
𝜶 𝑼 ƒ
‚
𝜶 𝒂n
𝜶
†𝒂o
n
𝜶
n
𝜶 𝑼 o
n
𝜶 = a
-
}
𝛼 𝛼 𝑼𝜶 𝛼 ~
}
𝛼
Kinetic Energy Operator
'
𝑇 = −
ℏ+
2𝑚
• 𝑑3𝑥 '
𝜓† 𝑥 ∇+ '
𝜓 𝑥 = • 𝑑3𝑝 '
𝜓† ⃑
𝑝
ˆ
⃑
𝑝+
2𝑚
'
𝜓 ⃑
𝑝

Fock Space and !
𝒏-representation
(9) Creation and Annihilation Operators as a Basis for Fock Space Operators
(9.2) Two-body Operator
Assume two-body operator D
𝑉 is diagonal in the orthonormal basis { ⟩
|𝛼 }
D
𝑉 ⟩
|𝛼𝛽 = 𝑉cp|𝛼𝛽) = (𝛼𝛽| D
𝑉|𝛼𝛽) |𝛼𝛽)
𝛼', 𝛼+, … , 𝛼(
'
𝑉 𝛽', 𝛽+, … , 𝛽( = a
#
𝜉#
1
2
a
!AB
y
@A!
@AB
𝛼#@ 𝛽@ 𝛼#!𝛼#B
'
𝑉 𝛽!𝛽B =
1
2
a
'C!ABC(
(
'
𝑉
-;-<
{𝛼', 𝛼+, … , 𝛼(|𝛽', 𝛽+, … , 𝛽(}
$
%
∑$=<>?=&
& E
𝑉
!.!/
is the sum over all distinct pairs of particles in the state |𝛼$, 𝛼%, … , 𝛼&}
D
𝑃cp = t
𝑛c t
𝑛p − 𝛿cp t
𝑛c count pairs of particles
D
𝑃cp = 𝑎c
†𝑎c𝑎p
†𝑎p − 𝛿cp𝑎c
†𝑎c = 𝑎c
†(𝛿cp + 𝜉𝑎p
†𝑎c)𝑎p − 𝛿cp𝑎c
†𝑎c = 𝑎c
†𝜉𝑎p
†𝑎c𝑎p
= 𝑎c
†𝑎p
†𝑎p𝑎c
𝑎-, 𝑎8
† = 𝑎-𝑎8
† − 𝜉𝑎8
†𝑎- = 𝛿-8
𝑎-
†, 𝑎8
†
1D
= 𝑎-, 𝑎8 1D
= 0
𝜉 = B

Fock Space and !
𝒏-representation E
𝑃!^ = [
𝑛! [
𝑛^ − 𝛿!^ [
𝑛! = 𝑎!
†𝑎^
†𝑎^𝑎!
E
𝑃!^ 𝛼$𝛼% … 𝛼& = ([
𝑛! [
𝑛^ − 𝛿!^ [
𝑛!) 𝛼$𝛼% … 𝛼& = [∑<,?B$
&
𝛿 𝛼 − 𝛼< 𝛿 𝛽 − 𝛼< − ∑@B$
&
𝛿!^𝛿(
)
𝛼 −
𝛼@ ] 𝛼$𝛼% … 𝛼&
1
2
a
'C!ABC(
(
'
𝑉
-;-<
=
1
2
[a
-
a
8
𝑉-8 a
!,B:'
(
𝛿 𝛼 − 𝛼! 𝛿 𝛽 − 𝛼! − a
-
a
8
𝑉-8 a
@:'
(
𝛿-8𝛿 𝛼 − 𝛼@ ]
𝛼', 𝛼+, … , 𝛼(
'
𝑉 𝛽', 𝛽+, … , 𝛽( =
'
+
∑'C!ABC(
( '
𝑉
-;-<
{𝛼', 𝛼+, … , 𝛼(|𝛽', 𝛽+, … , 𝛽(}
=
'
+
∑-8 𝑉-8 [∑!,B:'
(
𝛿 𝛼 − 𝛼! 𝛿 𝛽 − 𝛼! − ∑@:'
(
𝛿-8𝛿 𝛼 − 𝛼@ ] {𝛼', 𝛼+, … , 𝛼(|𝛽', 𝛽+, … , 𝛽(}
= 𝛼', 𝛼+, … , 𝛼(
'
+
∑-8 𝑉-8
'
𝑃-8 𝛽', 𝛽+, … , 𝛽(
N
𝑽 =
𝟏
𝟐
,
𝜶𝜷
𝑽𝜶𝜷
N
𝑷𝜶𝜷 =
𝟏
𝟐
,
𝜶𝜷
(𝜶𝜷|N
𝑽|𝜶𝜷)𝒂𝜶
†𝒂𝜷
†𝒂𝜷𝒂𝜶
𝑎E
!
† = ( )
!
𝛼 T
𝛼 𝑎!
† 𝑎]
E
!
† = ( )
!
𝛼 T̀
𝛼 𝑎!
†
Diagonal Basis
General Basis N
𝑽 =
𝟏
𝟐
,
E
!]
𝜷]
]
𝜷]
E
!
( k
𝛼ƒ
𝜷|N
𝑽| l
k
𝛼ƒ
ƒ
𝜷)𝒂E
!
†
𝒂]
𝜷
†
𝒂o
o
𝜷
𝒂o
n
c
=
𝟏
𝟒
,
E
!]
𝜷]
E
!]
]
𝜷
{ k
𝛼ƒ
𝜷 N
𝑽 l
k
𝛼ƒ
ƒ
𝜷}𝒂E
!
†
𝒂]
𝜷
†
𝒂o
o
𝜷
𝒂o
n
c
𝑎-, 𝑎8 1D
= 0
N
𝑽 =
𝟏
𝟐
c 𝑑3𝑥𝑑3𝑦𝑣(𝑥 − 𝑦) %
𝜓† 𝑥 %
𝜓† 𝑦 %
𝜓 𝑦 %
𝜓 𝑥
(9.3) n-body Operator
N
𝑹 =
𝟏
𝒏!
,
𝝀𝟏𝝀𝟐…𝝀𝒏;𝝁𝟏𝝁𝟐…𝝁𝒏
𝝀𝟏𝝀𝟐 … 𝝀𝒏
N
𝑽 𝝁𝟏𝝁𝟐 … 𝝁𝒏 𝒂𝝀𝟏
†
… 𝒂𝝀𝒏
†
𝒂𝝁𝒏
… 𝒂𝝁𝟏

Coherent States: Eigenstates of 𝑎! 𝑎-, 𝑎8
†
1D
= 𝑎-𝑎8
† − 𝜉𝑎8
†𝑎- = 𝛿-8
Assume 𝑎c
†
⟩
|𝜑 = 𝜑 ⟩
|𝜑 , normalized ⟩
|𝜑 ≠ 0
𝜑 𝑎c𝑎c
†
𝜑 = |𝜑|7 = 𝜑 1 + 𝜉𝑎c
†
𝑎c 𝜑 = 1 + 𝜉 𝜑 𝑎c
†
𝑎c 𝜑 = 1 + 𝜉[ 𝜑 7 + || ⟩
|𝜑 Œ||7]
𝜑 𝑎c
†
𝜑 = 𝜑 𝜑 𝑎c 𝜑 = 𝜑∗
𝑎c ⟩
|𝜑 = 𝜑∗ ⟩
|𝜑 + ⟩
|𝜑 Œ
For Fermions, 𝑎c
†
𝑎c
†
⟩
|𝜑 = 𝜑7 ⟩
|𝜑 = 0 𝜑 = 0 𝑎c
†
⟩
|𝜑 = 0
⟨𝜑 ⟩
|𝜑 Œ = 0
(1) Fermions,
|𝜑|7 = 1 − [ 𝜑 7 + || ⟩
|𝜑 Œ||7] | ⟩
|𝜑 Œ||7 = 1
(2) Bosons,
|𝜑|7 = 1 + [ 𝜑 7 + || ⟩
|𝜑 Œ||7] 0 = 1 + || ⟩
|𝜑 Œ||7 Impossible !
⟩
|𝜙 = G
Se•
j
G
cT,cU,…,cV
𝜙cT,cU,…,ck
⟩
|𝛼6𝛼7 … 𝛼S
e
𝑎!
†
|𝜙 ∝ ⟩
|𝜙
⟩
𝑎!|𝜙 ∝ ⟩
|𝜙

⟩
|𝝓 = )
A0!A0"…A0.
…B`
𝝓𝜶𝟏
𝒏𝜶𝟏
𝒏𝜶𝟏
!
𝝓𝜶𝟐
𝒏𝜶𝟐
𝒏𝜶𝟐
!
…
𝝓𝜶𝒊
𝒏𝜶𝒊
𝒏𝜶𝒊
!
R
|𝑛!!
𝑛!"
… 𝑛!.
…
l
|𝒏𝜶𝟏
𝒏𝜶𝟐
… 𝒏𝜶𝒌
… =
𝟏
𝒏𝜶𝟏
! 𝒏𝜶𝟐
! … 𝒏𝜶𝒌
! …
(𝒂𝜶𝟏
† )𝒏𝜶𝟏 (𝒂𝜶𝟐
† )𝒏𝜶𝟐 … (𝒂𝜶𝒌
† )𝒏𝜶𝒌 … ⟩
|𝟎
= )
A0!A0"…A0.
…B`
𝝓𝜶𝟏
𝒏𝜶𝟏
𝒏𝜶𝟏
!
𝝓𝜶𝟐
𝒏𝜶𝟐
𝒏𝜶𝟐
!
…
𝝓𝜶𝒊
𝒏𝜶𝒊
𝒏𝜶𝒊
!
𝟏
𝒏𝜶𝟏
! 𝒏𝜶𝟐
! … 𝒏𝜶𝒊
! …
(𝒂𝜶𝟏
† )𝒏𝜶𝟏 (𝒂𝜶𝟐
† )𝒏𝜶𝟐 … (𝒂𝜶𝒊
† )𝒏𝜶𝒊 … ⟩
|𝟎
= )
A0!A0"…A0.
…B`
𝟏
𝒏𝜶𝟏
! 𝒏𝜶𝟐
! … 𝒏𝜶𝒊
! …
(𝜙!!
𝒂𝜶𝟏
† )𝒏𝜶𝟏 (𝜙!"
𝒂𝜶𝟐
† )𝒏𝜶𝟐 … (𝜙!.
𝒂𝜶𝒊
† )𝒏𝜶𝒊 … ⟩
|𝟎
= 𝑒a0!𝒂𝜶𝟏
†
𝑒a0"𝒂𝜶𝟐
†
… 𝑒a0.
𝒂𝜶𝒊
†
… ⟩
|𝟎
= 𝒆∑𝜶 𝝓𝜶𝒂𝜶
†
⟩
|𝟎
𝜙0, 𝜙G "
= 0
⟩
|𝜶 ≡ { ⟩
|𝜶𝟏 , ⟩
|𝜶𝟐 , … , ⟩
|𝜶𝒌 , … }
⟨𝝓| = ⟨0|𝒆∑𝜶 𝝓𝜶
∗ c0
𝑎- ⟩
|𝜙 = 𝜙- ⟩
|𝜙
⟨𝝓|𝒂𝜶
† = ⟨𝝓| 𝝓𝜶
∗
𝒂𝜶
† ⟩
|𝝓 = 𝒂𝜶
†𝒆
∑𝜶- 𝝓𝜶-𝒂𝜶-
†
⟩
|𝟎 =
𝝏
𝝏𝝓𝜶
⟩
|𝝓
⟨𝝓|𝑎- = ⟨0|𝒆
∑𝜶- 𝝓𝜶-
∗
L𝜶-
𝑎- =
𝝏
𝝏𝝓𝜶
∗ ⟨𝝓|
𝝓 𝝓" = 𝒆∑𝜶 𝝓𝜶
∗ 𝝓𝜶
-
𝑛-/
𝑛->
… 𝑛-;
… 𝑛′-/
𝑛′->
… 𝑛′-;
… = 𝛿F=/F"=/
𝛿F=>F"=>
… 𝛿F=;
F"=;

(3) Completeness/Closure Relation
c d
𝜶
𝒅𝝓𝜶
∗ 𝒅𝝓𝜶
𝟐𝒊𝝅
𝒆q ∑𝜶 𝝓𝜶
∗ 𝝓𝜶 ⟩
|𝝓 ⟨𝝓| = 𝟏
𝒅𝝓𝜶
∗ 𝒅𝝓𝜶
𝟐𝒊𝝅
=
𝒅(𝑹𝒆𝝓𝜶)𝒅(𝑰𝒎𝝓𝜶)
𝝅
(3.1) Directly Integrate the l.h.s
(3.2) Schur’s lemma: if an operator commutes with all 𝑎c and 𝑎c
†
, then it is proportional to
the unit operator in Fock Space
𝒂𝜶, ⟩
|𝝓 ⟨𝝓| = 𝑎- ⟩
|𝜙 ⟨𝜙| − ⟩
|𝜙 ⟨𝜙|𝑎- = (𝝓𝜶 −
𝝏
𝝏𝝓𝜶
∗ ) ⟩
|𝝓 ⟨𝝓|
𝑎- ⟩
|𝜙 = 𝜙- ⟩
|𝜙
⟨𝝓|𝑎- =
𝝏
𝝏𝝓𝜶
∗ ⟨𝝓|
𝒂𝜷, • y
𝜶
𝒅𝝓𝜶
∗ 𝒅𝝓𝜶
𝟐𝒊𝝅
𝒆1 ∑𝜶 𝝓𝜶
∗ 𝝓𝜶 ⟩
|𝝓 ⟨𝝓| = • y
𝜶
𝒅𝝓𝜶
∗ 𝒅𝝓𝜶
𝟐𝒊𝝅
∗ 𝝓𝜶 (𝝓𝜷 −
𝝏
𝝏𝝓𝜷
∗ ) ⟩
|𝝓 ⟨𝝓|
= • y
𝜶
𝒅𝝓𝜶
∗
𝒅𝝓𝜶
𝟐𝒊𝝅
∗ 𝝓𝜶 𝝓𝜷 ⟩
|𝝓 ⟨𝝓| − • y
𝜶
𝒅𝝓𝜶
∗
𝒅𝝓𝜶
𝟐𝒊𝝅
𝝏
𝝏𝝓𝜷
∗ 𝒆1 ∑𝜶 𝝓𝜶
∗ 𝝓𝜶 ⟩
|𝝓 ⟨𝝓| + 𝒆1 ∑𝜶 𝝓𝜶
∗ 𝝓𝜶𝝓𝜷 ⟩
|𝝓 ⟨𝝓| = 𝟎
𝑎8
†
, • y
𝜶
𝒅𝝓𝜶
∗ 𝒅𝝓𝜶
𝟐𝒊𝝅
∗ 𝝓𝜶 ⟩
|𝝓 ⟨𝝓| = 𝟎 • y
𝜶
𝒅𝝓𝜶
∗ 𝒅𝝓𝜶
𝟐𝒊𝝅
∗ 𝝓𝜶 𝟎 𝝓 𝝓 𝟎 = • y
𝜶
𝒅𝝓𝜶
∗ 𝒅𝝓𝜶
𝟐𝒊𝝅
∗ 𝝓𝜶 = 𝟏
Set 𝝓𝟎𝟎…𝟎… = 𝟎 𝝓 = 𝟏
• 𝒅𝒙𝒆1𝒙𝟐
= 𝝅

(4) 𝝓 –representation/Holomorphic representation
Trace of operators: 𝑻𝒓4
𝑨 = • a
𝝀
𝝀 4
𝑨 𝝀 = • y
𝜶
𝒅𝝓𝜶
∗
𝒅𝝓𝜶
𝟐𝒊𝝅
∗ 𝝓𝜶 • a
𝝀
𝝀 𝝓 𝝓 4
𝑨 𝝀
( )
𝝀
⟩
|𝝀 ⟨𝝀| = 1
= • y
𝜶
𝒅𝝓𝜶
∗ 𝒅𝝓𝜶
𝟐𝒊𝝅
∗ 𝝓𝜶 • a
𝝀
𝝓 4
𝑨 𝝀 𝝀 𝝓 = • y
𝜶
𝒅𝝓𝜶
∗ 𝒅𝝓𝜶
𝟐𝒊𝝅
∗ 𝝓𝜶 𝝓 4
𝑨 𝝓
States: ⟩
|𝝍 = • y
𝜶
𝒅𝝓𝜶
∗ 𝒅𝝓𝜶
𝟐𝒊𝝅
∗ 𝝓𝜶 ⟩
|𝝓 𝝓 𝝍
Holomorphic function 𝝍(𝝓∗
) = 𝝓 𝝍
𝝍 𝝓∗ = 𝝓 𝝍 = • y
𝜶
𝒅𝝓′𝜶
∗ 𝒅𝝓′𝜶
𝟐𝒊𝝅
𝒆1 ∑𝜶 𝝓"𝜶
∗ 𝝓"𝜶 𝝓 𝝓′ 𝝓′ 𝝍
= • y
𝜶
𝒅𝝓′𝜶
∗
𝒅𝝓′𝜶
𝟐𝒊𝝅
𝒆1 ∑𝜶 𝝓"𝜶
∗ 𝝓"𝜶 𝒆∑𝜶 𝝓𝜶
∗ 𝝓𝜶
-
𝝍 𝝓"∗
𝝓 𝝓" = 𝒆∑𝜶 𝝓𝜶
∗ 𝝓𝜶
-
= • y
𝜶
𝒅𝝓′𝜶
∗ 𝒅𝝓′𝜶
𝟐𝒊𝝅
𝒆1 ∑𝜶(𝝓-
𝜶
∗
1𝝓𝜶
∗ )𝝓"𝜶 𝝍 𝝓"∗
Particle number in a coherent state
| 𝑛cT
𝑛cU
… 𝑛cp
… 𝜙 |7 = d
c
|𝝓𝜶|𝟐
𝒏𝜶! Poisson distribution with |𝝓𝜶|𝟐 as mean value

Coherent States: Bosons ⟨𝝓|𝑎- =
𝝏
𝝏𝝓𝜶
∗ ⟨𝝓|
⟨𝝓|𝒂𝜶
† = ⟨𝝓| 𝝓𝜶
∗
Operators: 𝝓 𝑎- 𝝍 =
𝝏
𝝏𝝓𝜶
∗ 𝝓 𝝍 =
𝝏
𝝏𝝓𝜶
∗ 𝝍(𝝓∗)
𝝓 𝒂𝜶
† 𝝍 = 𝝓𝜶
∗
𝝓 𝝍 = 𝝓𝜶
∗
𝝍(𝝓∗
)
𝒂𝜶 =
𝝏
𝝏𝝓𝜶
∗
𝒂𝜶
† = 𝝓𝜶
∗
𝑎-, 𝑎8
†
1
= 𝑎-𝑎8
† − 𝑎8
†𝑎- = 𝛿-8
𝑎-
†, 𝑎8
†
1
= 𝑎-, 𝑎8 1
= 0
𝝓𝜶
∗
, 𝝓𝜷
∗
1
=
𝝏
𝝏𝝓𝜶
∗ ,
𝝏
𝝏𝝓𝜷
∗
1
= 𝟎
𝝏
𝝏𝝓𝜶
∗ , 𝝓𝜷
∗
1
= 𝜹𝜶𝜷
4
𝐻 ⟩
|𝝍 = 𝑯 𝒂𝜶
†, 𝑎- ⟩
|𝝍 = 𝑬 ⟩
|𝝍
Schr ̈
odinger Equation 𝑯 𝝓𝜶
∗
,
𝝏
𝝏𝝓𝜶
∗ 𝝍 𝝓∗
= 𝑬𝝍(𝝓∗
)
a
𝜶,𝜷
𝑻𝜶𝜷𝝓𝜶
∗
𝝏
𝝏𝝓𝜶
∗ +
𝟏
𝟐
a
𝜶,𝜷,𝜸,𝜹
(𝜶𝜷|𝒗|𝜸𝜹)𝝓𝜶
∗ 𝝓𝜷
∗ 𝝏
𝝏𝝓𝜹
∗
𝝏
𝝏𝝓𝜸
∗ 𝝍 𝝓∗ = 𝑬𝝍(𝝓∗)
4
𝐻 = '
𝑇 + '
𝑉
Matrix Elements of Normal-ordered Operators
𝝓 𝐴(𝒂𝜶
†, 𝑎-) 𝝓" = 𝐴(𝝓𝜶
∗ , 𝝓′𝜶)𝒆∑𝜶 𝝓𝜶
∗ 𝝓𝜶
-
𝝓 𝝓"
= 𝒆∑𝜶 𝝓𝜶
∗ 𝝓𝜶
-
Two-body Operators
𝝓 r
𝑽 𝝓f =
𝟏
𝟐
∑ghij(𝝀𝝁|r
𝑽|𝜈𝜌) 𝝓 𝒂𝝀
†
𝒂𝝁
†
𝒂j𝒂i 𝝓f =
𝟏
𝟐
∑ghij(𝝀𝝁|r
𝑽|𝜈𝜌)𝝓𝝀
∗
𝝓𝝁
∗ 𝝓j
f 𝝓i
f 𝒆∑𝜶 𝝓𝜶
∗ 𝝓𝜶
6
The distribution of particle numbers
Q
𝑁 =
𝝓 S
𝑁 𝝓
𝝓 𝝓
=
∑𝜶 𝝓 𝒂𝜶
†𝒂𝜶 𝝓
𝝓 𝝓
= U
𝜶
𝝓𝜶
∗ 𝝓𝜶
4
𝑵 = a
𝜶
©
𝒏𝜶 = a
𝜶
𝒂𝜶
†𝒂𝜶
Variance 𝜎& =
𝝓 S
𝑁𝟐 𝝓
𝝓 𝝓
− Q
𝑁& = ∑𝜶 𝝓𝜶
∗ 𝝓𝜶 = Q
𝑁 lim
S
(→<
𝜎
®
𝑁
= lim
S
(→<
1
®
𝑁
= 0

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