The Composite Particle Representation Theory (CPRT) Cheng-Li Wu 2013 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 fundamental interactions among their constituents. Cheng-Li Wu 2013 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 NFT The 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 ) k1k1  k2k2 k1k1  k2k2 k1k1 k2k2 k’ 1 k’ 2  ’’ ’’  +  ’’  ’’ +  ’’  ’’ 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:

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 fermion pairs 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 Bubble diagrams, Because of Rule 2 it is possible that It was hoped that the three empirical rules can correct the over-completeness & the violation 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 V eff 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 C. L. Wu & D. H. Feng, Phys. Rev. C21, 727 (1981). It allows one to sum up the NFT V eff 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 of the 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 the original fermion degrees of freedom unchanged.  The violation of the Pauli principle and the overcomplete- ness due to the extra boson degrees of freedom are well corrected 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 NFT application to odd fermion systems. Can the NFT be derived from the first principles ?

4. The propose of the Composite Particle Representation Theory (CPRT) NFT An Empirical Theory BFHR Derived from QM CPRT Derived from QM C. L. Wu & D. H. Feng, Commun. In Theor. Phys. 1, 705 (1982); 2, 811 (1983) ffb + f BB, f* 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 Systems

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 levelAt quark level A  nucleon is a 3- quark cluster  n  particle is a boson At Atomic levelAt nuclear level An  particle 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) SA: Representation-A Basis: States: Operators: S B : Representation-B Equivalence: Key: Alg[  ]= Alg[u  ]

The Generalization to GRT The Basic Operator Set: S A : A Physical Space S B : A Generalized Rep. space Alg[  ]= Alg[u  ]Alg[G(  ] = Alg[G’(  ] GRT: G  G’ Equivalence:

S B : The Boson Space The meaning of S A : The Physical Space Alg[G] = K≡k1k2K≡k1k2 S A : The Physical Space Alg[G] = K≡k1k2K≡k1k2 Fermion pairs Bosons Boson Mapping An Example of the GRT The Dyson & Holstein-Primakoff Boson Expansions GRT: G(a) G’(B) S B : The Boson Space

 and  are related, by a unitary trans-formation, therefore, S A and S B are actually the same space just using different basis: S A = S B  and  could be unrelated, thus S A and S B are generally different spaces: S A  S B The URTThe GRT The Usual Unitary Representation Transformation What are the Generalization of the GRT? In GRT, G{  and  G’  have one to one correspondence but are not necessary to be equal. S B :G  A Boson space S A :  A fermion space G (  ) A subspace of S A Boson Mapping G’ (  ) A subspace of S B S B :  S A :G  Over completed basis S A :  The Physical space

Reason: since {  } already describe the whole physics in S A, and {C} are extra degrees of freedom, no subspace containing {C} in S B can be equivalent to S A. G’(  )=? No G’(  ) can be found to satisfy Alg[G]=Alg[G’] unless G’(  )= G(  )! G’ (  ) S B :  SA :GSA :G S A :  3. The CPR Transformation The CPR Alg [  ]: +/- = Fermi/Bose k  state index t  particle index Cluster Structures Alg[  ] ~ Alg[  ] Alg[G’(  )]= Alg[G(  )] Fine G’(  ) To Cancel the extra degrees of freedom It is necessary to introduce “Negative Composite particles”

Physical space C  O The Physical Space : { ,O} Definition : The Negative Composite Particles Properties : The Physical Space : { ,O} Even only one O will make O makes no contribution to observables. C  O Physical space

The CPR Basic Operator Set Replace ===== Check

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 {  eg: Three Steps in the CPR Transformations S CPR : SA:SA: The Physical Space Step 1. Determine the cluster structures Because we need to know for and by solving Step 2. Rewrite operators in terms of G : 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’ :

4. The Operators in the CPR NonUnchanged Type 1-body operators N-body operators Non Unchanged Non

The CPR Hamiltonian

5. The Wave Functions in the CPR A Problem: is not unique! q =1 q =2 q =3 An Example Of non- uniqueness The f q ’s are arbitrary 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 Unlike the URT, the can not be obtained by the CPRT from ; It must be determined by solving the CPR Schrödinger Equation: