The Composite Particle Representation Theory

(CPRT)

Cheng-Li Wu2013 April

A microscopic Theory for Cluster Models

The CPRT provides a quantum representation that allows clusters in a many-body system to be described as elementary particles (bosons or fermions) with the correction due to internal motions being taken into account exactly, and the interactions between clusters could be calculated from the more fundamentalinteractions among their constituents.

Cheng-Li Wu2013 April

Outline

I. History

II. The Composite Particle Representations (CPR)

III. The Solutions of the CPR Schrödinger Equation

IV. The Test of the CPR & the Multi-Step CPR

V. The CPR for Few-body Problems

VI. Summary

I. History

1. The Nuclear Field Theory (NFT)The CPRT was proposed in 1982

That was initiated by the study of the Nuclear Field Theory (NFT)

The NFT is actually the predecessor of the CPRT

2. The Validity of the NFTThe NFT is an empirical theory. It is necessary to check

its validity. After the comparison with some solvable models, it was surprised that

The NFT Seems to be an Exact Theory!

3. The Boson –Fermion Hybrid Representation (BFHR)

Motivated by understanding why the NFT is exact, This work first time derived the NFT from first principles of QM 4. The propose of the Composite Particle

Representation Theory (CPRT)

CPRTNFT BFHR

1. The Nuclear Field Theory (NFT)

FB BFF V VVV 1

1

2

21 2( ) kk k

kFBV Z k k A B

†

k1

KZ

k2

k1

KZk2

1

1

2

21 2( ) kk k

kBFV Z k ABk

†

k1

K KV

k2

k’1 k’2

' '' 1 2 1 2K K FV k k V k k

1( )

1 22( )K Fk kZ V kZ k

0eff p pH V E

p eff pDV

10 ˆD E H q

1 2

1 21 2( ) kFB BFk k

kV V Z Ak k B

†† ( )1 21 2( ) FZ k V k kk

effV V VDV VDVDV VDVDVDV VDVDVDVDV

’’

+

’

’

+' ' effB B V B B

’

’

0

B. R. Mottelson, J. Phys. SOC. Jpn. Suppl. 24, 87 (1968)

The NFT is in fact the first one in nuclear physics that treats correlated fermion pairs as bosons, earlier than the IBM but in a different manner:

0 BH B B †0 0

NFT BFH VHH 0 ( ) ( )( )F FH V

Three Empirical Rules in the NFT

Problems: To include both boson and fermion states, the

basis must be over-complete, since the bosons are made of fermions.

To treat fermion pairs as bosons there must be violations of Pauli Principle since fermionpairs don’t satisfy the boson statistic.

How to solve the problems

??

1. Allow only boson states in the P space.

2. Discard all bubble Feynman Diagrams

3. Only states that are normalizable are physical.

All unnormalizable states are spurious

p eff pDV 21 0p eff eff pV D V

Bubble diagrams

,

2 1p eff eff pV D V

Because of Rule 2 it is possible that

1

It was hoped that the three empirical rules can correct the over-completeness & theviolation of the Pauli principle in the NFT.

2. The Validity of the NFT C. L. Wu & D. H . Feng, Phys. Rev. C21, 727 (1981).

Practically only the lowest order diagram in Veff was taken into account in the NFT applications (LNFT). The LNFT had been successful in the description of collective motions in heavier nuclear regions but fail in lighter nuclear regions. A summing method was proposed in 1980.

C. L. Wu & D. H . Feng, Phys. Rev. C21, 727 (1981).

It allows one to sum up the NFT Veff up to infinite orders for a 2-boson (4-fermion) system. Using this method, an exact NFT calculation of 4 nucleons moving in a single-j shell was first time being carried out, and by the comparison to the shell model results, one was able to test the validity ofthe NFT. It turns out that

The NFT is an Exact Theory!

3. The Boson-Fermion Hybrid Representation Formulation (BFHR)

C. L. Wu, Ann. 135, 7166 (1981). How can the NFT be an

exact theory ? This work demonstrated that it is possible to construct a special representation (BFHR) for quantum mechanics that allows treating fermion pairs as bosons, and yet remaining theoriginal fermion degrees of freedom unchanged.

The violation of the Pauli principle and the overcomplete-ness due to the extra boson degrees of freedom are wellcorrected in the BFHR.

By using the BFHR, the NFT was derived from the first principles with the 3 empirical rules emerged naturally.

The BFHR further demonstrates that the fermion states could also put in P space, as lone as they do not have overlap with any boson structure wavefunctions, thus extends the NFTapplication to odd fermion systems.

Can the NFT be derived from

the first principles

?

4. The propose of the Composite Particle Representation Theory (CPRT)

NFTAn Empirical

Theory

BFHRDerived from QM

CPRTDerived from QM

C. L. Wu & D. H. Feng, Commun. In Theor. Phys. 1, 705 (1982); 2, 811 (1983)

f f b + f

B B , f*

2f 2f

System

P space

Clusters

, , b* , f*FB

nb+2mf

nb+f+2mf

,

The b* and f* must not have any overlap with any clusters.

n & m could be any integers

The CPRT is the generalization of the BFHR

References

“The Composite Particle Representation Theory” C. L. Wu & D. H. Feng, Commun. In Theor. Phys. 1,

705 (1982); 2, 811 (1983) “The Boson-Fermion Hybrid Representation Formulation”

C. L. Wu, Ann. 135, 7166 (1981). “The Composite Particle Representation Method for Few –

Body Systems”, C. L. Wu (Unpublished).

Basic Theory & Techniques of the CPRT

Test of the CPRT

“The Composite Particle Representation Approach to Boson mapping” C. L. Wu, J. Mod. Phys. E, 2 83(1993).

“The Composite Particle Representation Calculations for odd Fermion Systems” A. L. Wang, K. X. Wang, R. F. Hui, C. X. Wu, C. L. Wu, Commun. In Theor. Phys. 5, 31 (1986).

“Test of the Composite Particle Representation Theory”

K. X. Wang, G. Z. Liu, C. L. Wu, Phys. Rev. 43, 2268 (1991).

“The Composite Particle Representation Method for Baryon Spectrum” Y. Y. Zhu at al., Commun. In Theor. Phys. 7, 149 (1987).

“Application of the Composite Particle Representation: Spin-polarized atomic hydrogen as a bosonic System” Y. Y. Zhu at al., Commun. In Theor. Phys. 7, 149 (1987).

Nucleon Systems

Qark Systems

Atomic System

s

II. The Composite Particle Representations (CPR)

1. Introduction 2. The Generalized Representation Transformation 3. The Composite Particle Transformation (CPR) 4. The Operators in the CPR 5. The Wave Functions in the CPR

1. Introduction

nucleon is a fermion

At nuclear level At quark level

Anucleon is a 3-quark cluster

nparticle is a boson

At Atomic level At nuclear level

Anparticle is a 2n-2p cluster

Elementary particles are not elementary!They are elementary only if the internal motions can be ignored.

There is a necessity to construct a theory which can describe a cluster like an elementary particle as a boson or a fermion, and yet can take into account the corrections due to the internal particle’s motion. The CPR theory is such a theory.

1. UNDERSTANDING

The Motivation of the CPR

To quantitatively understand under what conditions a cluster can behave like an elementary particle.

2. APPLICATIONS

Examples: an a-particle to be a boson; a nucleon to be a fermion; An atomic system to have Bose-Einstein condensation; …

Greatly simplifying many-body problems by treating clusters as elementary particles Examples: boson Model; -particle model; …. etc.

with the corrections due to the internal motions being taken into account

Calculating interactions between composite particles from more fundamental interactions among their constituents. Examples: nuclear force, interaction, …. etc.

2. The Generalized Representation Transformation

(GRT)The Usual unitary Transformation (URT)

' ' ' '[ ] 0 , , 0, , ,0k k k k k k k k kk

Alg † † † †

SA: Representation-A

Basis:

States:

Operators:

, ; 0k k k k † †

1 2 1 2

( ) ( ) 0n nA k k k k k k

k

C † † †

( )AL

SB: Representation-B

: k ku

( ) ( )( ) ( )B A u

( ) ( )B AL L u

( ) ( ) ( ) ( ) ( ) ( ) (α) (β) (α) (β)αβ A A A B B BL = Ψ L Ψ = Ψ L ΨEquivalence:

Key:Alg[]= Alg[u]

(1)

(2)

(3)

0 0 0

, ( ')[ ']

, ( ')

, ( ')

k k

i j ij

i j ij

i j ij

g g

g g f GG

g g f G

g g f G

Alg

†

†

† †

The Generalization to GRT

(1)

(2)

(3)

0 0 0

, ( )[ ]

, ( )

, ( )

k k

i j ij

i j ij

i j ij

g g

g g f GG

g g f G

g g f G

Alg

†

†

† †

The Basic Operator Set:

( ) ( )

( ), ( )

( ) ( )

( ) ( )

i j

A A

A A

G g g

G

L L G

†SA:

A Physical Space

SB :

A Generalized Rep. space

' ( ), ( )i jG g g †

, i i i ig g g g † †

[ ] [ ']Alg G Alg G

Alg[]= Alg[u] Alg[G(] = Alg[G’(]

GRT: i ig g

( ) ( )( ') ( ')B AG G G

( ') ( ')B AL G L G G

G G’

( ) ( ) ( ) ( ') ( ') ( ')G G G G G G(α) (β) (α) (β)αβ A A A B B BL = Ψ L Ψ = Ψ L ΨEquivalence:

,B B †SB: The Boson

Space 1 2 1 2 1 2

1 2 1 2 1 2 1 2

1 2 1 2 1 2

' , , ,

,

k k k k k k

k k k k k k k k

k k k k k k t tt

G N A N A A

N C C B B A C B

A C B C C C B B B

†

†

† † † †

( ') ( ')

( ') ( ')

' { ', '}

G G G

G G G

G G N N A A

(β) (β)B A

B A

Ψ Ψ

L = L

' '

(1)' '

(2)' '

(3)' '

(4)' '

' '

(1' '

0 0 0 0

, , 0

, ( )

0 0 0 0

, , 0

lg[ ] , ( )

, ( )

,

,lg[ ']

( )

K K K

K K K K

K K KK

K K

K K K

K K K K

K K

KK

K K KK

K K K

KK

K

A N A

A A A A

A A f GA G

N A f G

N A f G

N

A N A

A A A A

A A fA

G

G

N f

†

†

†

†

†

†

†

†

†

)

(2)' '

(3)' '

(4)' '

( )

, ( )

, ( )

, )

(

K K KK

K K KK

K K KK

G

N A f G

N A f G

N N f G

†

[ ] '] [AAl lgG Gg The meaning of

SA: The Physical

Space

1 2 1 2 1 2

1 2 1 2

1 2 1 2 1 2 1 2

, , ,

,

,

k k k k k k

k k k k

k k k k k k k k

G N A N A A

N a a

A a a A a a

†

†

† † †

Alg[G] =

' '

(1)' '

(2)' '

(3)' '

(4)' '

0 0 0 0

, , 0

, ( )

, ( )

, ( )

, ( )

K K K

K K K K

K K KK

K K KK

K K KK

K K KK

A N A

A A A A

A A f G

N A f G

N A f G

N N f G

†

† †

†

†

K≡k1k2

SA: The Physical

Space

1 2 1 2 1 2

1 2 1 2

1 2 1 2 1 2 1 2

, , ,

,

,

k k k k k k

k k k k

k k k k k k k k

G N A N A A

N a a

A a a A a a

†

†

† † †

Alg[G] =

' '

(1)' '

(2)' '

(3)' '

(4)' '

0 0 0 0

, , 0

, ( )

, ( )

, ( )

, ( )

K K K

K K K K

K K KK

K K KK

K K KK

K K KK

A N A

A A A A

A A f G

N A f G

N A f G

N N f G

†

† †

†

†

K≡k1k2

1 2 1 2

1 2

k k k kk k

C a a † † † B

†

Fermion pairs Bosons

Boson Mapping An Example of the GRT The Dyson & Holstein-Primakoff Boson Expansions

[ ] '] [AAl lgG Gg GRT: G(a) G’

(B)

,B B †SB: The Boson

Space 1 2 1 2 1 2

1 2 1 2 1 2 1 2

1 2 1 2 1 2

' , , ,

,

k k k k k k

k k k k k k k k

k k k k k k t tt

G N A N A A

N C C B B A C B

A C B C C C B B B

†

†

† † † †

( ) ( ) ( )

( ') ( ') ( ')

G G G

G G G

(α) (β)αβ A A A

(α) (β)B B B

L = Ψ L Ψ

= Ψ L Ψ

and are related, by a unitary trans-formation, therefore, SA and SB are actually the same space just using different basis: SA = SB

and could be unrelated, thus SA and SB are

generally different spaces: SA SB

The URTThe GRT

The Usual Unitary

Representation

Transformation

What are the Generalization of the GRT?

In GRT, G{andG’ have one to one correspondence but are not necessary to be equal.

: ; :

( ) '( )

( ) ( ) '( )

( ) ( ) '( )

A B

B A

B A

S S

G G

G G

L L G G

:

G( ) , '( )

( ) ( )

( ) ( )B A

B A

Unitary u

G u

u

L L u

G( ) '( )

( )i i ij jj

G

u u

( ) '( )

( ) ' ( )i i

G G

g g

{ } { }

{ } { }, { } { '}

{ } { '}

Generally Dim Dim

Dim Dim G Dim Dim G

but Dim G Dim G

'

Dim Dim

Dim G Dim G

SB :GA Boson

space

SA: A fermion

space

G ()A subspace

of SA

Boson Mapping

G’ ()A subspace

of SB

SB:

SA :GOver completed b

asis

SA :The Physical

space

: ; :

( ) '( )

( ) ( ) '( )

( ) ( ) '( )

A B

B A

B A

S S

G G

G G

L L G G

:

G( ) , '( )

( ) ( )

( ) ( )B A

B A

Unitary u

G u

u

L L u

G( ) '( )

( )i i ij jj

G

u u

( ) '( )

( ) ' ( )i i

G G

g g

Reason: since {} already describe the whole physics in SA , and {C} are extra degrees of freedom, no subspace containing {C} in SB can be equivalent to SA.

G’()=?

No G’() can be found to satisfy

Alg[G]=Alg[G’]unless G’()= G()!

G’ ()SB

:

SA :G

SA :

3. The CPR Transformation

The CPR( ) † C †

Alg []:+/- = Fermi/Bosek state indext particle index

Cluster Structures( )* ( ' ')

' '

( )* ( )' '

K KK

K K KK

C C

C C

, , ,kt kt C C† †

Alg[] ~ Alg[]

( ) ( )( ') ( ')CPR AG G G

( ') ( ')CPR AL G L G G

'( ) { , , , , }tK K kt kt kkG A A N † †

Alg[G’()]= Alg[G()]

Fine G’()

To Cancel the extra degrees of freedom

It is necessary to introduce

“Negative Composite particles”

, C C†

, , , C C C C† †

' ' ' '

' ' ' ' ' '

, , , , 0

, , , 0, '

kt k t kk kt k t kt k t

kt k t kt k t kt k t t t

† † †

† † †

1 2 1 2

( ) ( )

1 1 1 1

1

( ) 0 ( ) 0

, , , , , ,

( ) , i i

A K KK

n n

nt

K K k t k k k t k ti

C A n

K k k k t t t

A n N

† †

† † †N

Physical spaceC

C

O C C

The Physical Space :

{,O}

Definition :

The Negative Composite Particles

( ) ( , ) ( ) ( , )

( ) ( , ) ( ) , )CPR B A

CPR B A

O O

L L O L O

L(

, ,,, , kt kt C C C C†† †

' ' ' '

' ' ' ' ' '

, , , , 0

, , , 0, '

C C C C C C

C C C C C C

† † †

† † †

Properties :

, [ , ] 0, lg[ , ] 0O O O A OCC †

The Physical Space :

{,O}( ) ( ) ( ) ( ) ( ) ( )CPR CPR CPR A A AL L

( , ) ( , ) '( , ) 0O L O O Even only one O will makeO makes no contribution to observables.

C

O

Physical spaceC

The CPR Basic Operator Set

( ) O † † C † †

C C † †† CC

† ††

( ) Correc iC t ons ††( ( )) Nothing †† ( ) ( ))( () O CC C † †† †† †††

( )( ) 0 0 ,A K K AK

C A † † , ,,, , kt kt C C C C†† †

''( ) { , , , , }tkt kt kk K KG N A A † †

'( ) { , , , , },tkt kt kk K KG N A A † †

( ) ( ) O † † † ) )( (K KK K

K K

C OAA C †† †

( )* ( )' 'K K KKC C

( )K K KA A C O

† † †

( )K K KA A C O

† † †

( ) ( )' ( ) ( )N n

, , N C C N C C † † ,N N n C C C C n

† †

( ) ( ) ,ktkt

N n

' ,kt ktkt kt

N N N N N n

( ) ( )kt kt kt

kt kt kt

N N C C C C N

† † ( ) ( )

kt kt ktN N C C C C N

† † Replace ' t

kt kkN Nby

( ) ( )' ' '

t t tkk kk kkN N C C C C N

† †

( ) ( )' ' '

t t tkk kk kkN N C C C C N

† †

lg[ '] lg[ '] lg[ 'lg[ , '] lg[ , ']lg[ ][ , ']lg '] A N A AA A N AA A N A AG lg[ ]A G

lg ,A N A

==

lgA N

=

lgA A

=

lg ,A N=

lg[ , ]A ACheck

We may convert only some ‘s with specific ’s into composite particles, then only the in these ‘s are of {A}; and could be just number operators as well, then All the other A’s and with k k’ remain to be of {

( )( )

'tkkN

,K KA A †

eg:{ }N

Three Steps in the CPR Transformations

, , , ,, kt kt CC CC † ††

( ) ( ) ' ' '

t t tkk kk kkN N C C C C N

† †

{ , , }C C SCPR:{ } { , }kt kt †SA:

The Physical Space '( ) { , , , , }t

kt kt kk K KG N A A † †'

'( ) { , , , , }kk

tkt kt K KG N A A † †

{ , , }N A { , ', '}N A

( ) ( )K K KA A C C C

† † † †

Step 1. Determine the cluster structures ( ) ( ) ( ) 0 0 A K K A

K

C A † †

Because we need to know for and ( )

KC KA † '

tkkN( ) ( ) ( )H

by solving

Step 2. Rewrite operators in terms of G :( ) ( , , ) , ( ) ( , , ), H H N A L L N A

N and A should be predefined, and in G={N,A} are those in the operators that cannot be written in the form of N and A.

Note:

Step 3. G G’ : ( ) , ', '( ) , ( )( ) , ', ' , CPR CPRH N A L N AH L

4. The Operators in the CPR

Non Unchanged

tkN

Typ

e1-

body

op

erat

ors

N-b

ody

op

erat

ors

CL

( )A

L

'LK

tkk

kt

N

'

( )'

'

LKK K K

KK

A A

†

( )L ( ) K K K

K

A A

†

L t tk k

kt

N

L ( )tk k k

kt

†

( )L ( )C C

† ( )

'L ( )K K KK

A A

†

( )L C C

† ( )L K

KK KA AC C

† †( )' '

'

LKK K KKK

A A

†

pCLpLpCPR p CCL L L L

( )L C C

† Lt

k ktkt

N

Non

Non

NonUnchanged

( )L ,AL ( ) (1)L K AL K

( )L 0 ,AL

Non

( ) { , , , , }kt kt K KG A A † † '( ) { , , , , }t

kt kt kk K KG N A A † †

( )L C C

†

' 'Ltkk kk

kt

N Non

0CPRΗ

The CPR Hamiltonian

( )L AL

' + LL t

k ktkk t

CPRL C NC

†

(( ) ) )' '

'

(L L LKKK

K KK K KKK

CPRL C C C A A A AC

† †† †

() )(V ,K V K

0 ( )' '

'

V kt kt KK K Kkt KK

H H V N A A

†

( ) (( ))V ,V

0 0 ( )+V C C CH H V H C C

†

CPRV( )

'V ' ,tKK Kt V K t

pCPR pCCH H H H

CPRH

0CPRH 0Hkt

( )V

( )V K

( )VKK

0H0H

VCPR

0CH 0

pH

0H

CV pVpCV

CPR C pCH H HV N

0 0 ( ) ) (( ) ( ), H V H N H

0 pCPR pCHH V V 0

0 =p kt ktkt

CH H H NN

5. The Wave Functions in the CPR

( )CPR ( ) (G G')(α)

AΨ η = ΨA Problem: is not unique!

1 2 3 1 2 3

1 2 3

( ) ( ) 0 , A k k k k k kk k k

C

† † † ( )' 'kk k kC † †

q =1 2 3 1k k kA † †2 3k kA†

2 3k kA†2 3k kC q =2

3 1 2k k kA † †

3 1k kA†3 1k kA†

3 1k kCq =31 2 3k k kA † †

1 2k kA†1 2k kA†

1 2k kC

{ } { } { }q q qk k k k k kA A C O

† † †

1 2 3

1 2 3

( ){ }( ) 0 ,

q qq k k k k k kk k k

G C A

† †

( ) ( )( ) ( ), 1A q q qq q

f G f ( ) ( ) ( ')CPR q qq

f G G

An ExampleOf non-

uniqueness

The fq’s arearbitrary

1 2 3

1 2 3

1 2 3

1 2 3

( ){ } { }

( ) ( ){ }

0

( ) 0 ( ) ( )

q q q

q q

CPR q k k k k k k k kq k k k

A q k k k k k k Aq k k k

f C A C O

f C C O O

† † †

† †

So, is undetermined, and is not unique, but they are all physically equivalent, since contributes nothing to the observables. But for describing a system as a composite particle system we need to know , since

( )O ( )CPR

( )O

( )O ( )( )( ) .C CO

1 2 3

1 2 3

( )}

({

) ( ) ( ), ( ) 0q qCPR q k k k k k k

q k k k

O O f C C O

† †

Unlike the URT, the can not be obtained by the CPRT from ; It must be determined by solving the CPR Schrödinger Equation:

( )CPR ( )

A

( ) ( ) ( ) ,CPR CPR CPRH E ( ) ( ) () ) ,( CPR AO ( ) ( )C C ( ) ( )Practically, we don't need to know , we can directly obtain by solving the CPR Schrodinger Eq.A C