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On coordinate independent state space of Matrix Theory. Yoji Michishita (Kagoshima Univ.) Based on JHEP09(2010)075 ( arXiv: 1008.2580[ hep-th ]) arXiv:1009.3256[math-ph]. Introduction. Matrix Theory (Banks- Fischler - Shenker -Susskind ‘96) SU(N) Quantum mechanics ← 10D SYM - PowerPoint PPT Presentation
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On coordinate independent state space of Matrix Theory
Yoji Michishita(Kagoshima Univ)
Based on JHEP09(2010)075 (arXiv10082580[hep-th])arXiv10093256[math-ph]
Introduction
Matrix Theory (Banks-Fischler-Shenker-Susskind lsquo96)
bull SU(N) Quantum mechanics larr 10D SYM bosonic coordinate matrices( SO(9) index) fermionic partners
componentsbull Nrarrinfin 11D M-theory
N finite DLCQ M-theory
bull describes N D0-branes = N KK modes of 1 unit of KK momentum
Multiparticle states rarr continuous spectrum(de Wit-Luumlscher-Nicolai lsquo89)
KK modes of 2 or more unitsrealized as bound states
rarr discrete spectrum
ConjectureSU(N) Matrix Theory has a unique normalizable zero
energy bound state
bull Calculation of Witten index seems consisitent(Yi lsquo97 Sethi-Stern rsquo97 etc)
bull Some information asymptotic form symmetry
(Hoppe et al etc)
bull No explicit expression is known(for zero energy bound state and any other gauge
invariant wavefunctions)
bull Why is it so difficultbasis of gauge invariant wavefunctions
rarr creation operators rarr states(16777216 states even for SU(2))
bosonic variables rarr
Schroumldinger eq rarr equations with variables
Enormous number of states and variablesrarr Even numerical calculation is difficult
systematic classification of these statesby representation of SU(N)timesSO(9)
Plan
Introduction2 Explicit construction of some coordinate
independent states in SU(2) case3 Number counting of representations in SU(2)
case4 SU(N) case5 Summary
Explicit construction of some coord indep states in SU(2) case
Wavefunction of zero energy bound stateGauge invariantSO(9) invariant (Hasler-Hoppe lsquo02)
Asymptotic form (SU(2))
Taylor expansion around the origin
coordinate independent
statesZero energy rarr
(supercharge )
rarr
Coord indep states for fixed rarr 256 states symmetric traceless (44) antisymmetric (84) vector-spinor (128)
Action of on these states
rarrfull states
etc
bull How do these states transform under gauge transformation
Not immediately clear
It is read off by acting generators of the gauge group
SU(2) case
bull The 1st term (SU(2)timesSO(9) singlet)of the expansion has been constructed and it is unique (up to rescaling) (Hoppe-Lundholm-Trzetrzelewski rsquo08 Hynek-Trzetrzelewski lsquo10)
bull 2nd term zero energy rarr
Let us construct coord Indep (adjoint)times(vector) representation of SU(2)timesSO(9) and see if it satisfies
First let us see the procedure of the construction of representations in a simpler case ie singlet case
bull Decomposition of SO(9) singlets into SU(2) representations
1 Enumerate SO(9) singlets 14 states
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
Introduction
Matrix Theory (Banks-Fischler-Shenker-Susskind lsquo96)
bull SU(N) Quantum mechanics larr 10D SYM bosonic coordinate matrices( SO(9) index) fermionic partners
componentsbull Nrarrinfin 11D M-theory
N finite DLCQ M-theory
bull describes N D0-branes = N KK modes of 1 unit of KK momentum
Multiparticle states rarr continuous spectrum(de Wit-Luumlscher-Nicolai lsquo89)
KK modes of 2 or more unitsrealized as bound states
rarr discrete spectrum
ConjectureSU(N) Matrix Theory has a unique normalizable zero
energy bound state
bull Calculation of Witten index seems consisitent(Yi lsquo97 Sethi-Stern rsquo97 etc)
bull Some information asymptotic form symmetry
(Hoppe et al etc)
bull No explicit expression is known(for zero energy bound state and any other gauge
invariant wavefunctions)
bull Why is it so difficultbasis of gauge invariant wavefunctions
rarr creation operators rarr states(16777216 states even for SU(2))
bosonic variables rarr
Schroumldinger eq rarr equations with variables
Enormous number of states and variablesrarr Even numerical calculation is difficult
systematic classification of these statesby representation of SU(N)timesSO(9)
Plan
Introduction2 Explicit construction of some coordinate
independent states in SU(2) case3 Number counting of representations in SU(2)
case4 SU(N) case5 Summary
Explicit construction of some coord indep states in SU(2) case
Wavefunction of zero energy bound stateGauge invariantSO(9) invariant (Hasler-Hoppe lsquo02)
Asymptotic form (SU(2))
Taylor expansion around the origin
coordinate independent
statesZero energy rarr
(supercharge )
rarr
Coord indep states for fixed rarr 256 states symmetric traceless (44) antisymmetric (84) vector-spinor (128)
Action of on these states
rarrfull states
etc
bull How do these states transform under gauge transformation
Not immediately clear
It is read off by acting generators of the gauge group
SU(2) case
bull The 1st term (SU(2)timesSO(9) singlet)of the expansion has been constructed and it is unique (up to rescaling) (Hoppe-Lundholm-Trzetrzelewski rsquo08 Hynek-Trzetrzelewski lsquo10)
bull 2nd term zero energy rarr
Let us construct coord Indep (adjoint)times(vector) representation of SU(2)timesSO(9) and see if it satisfies
First let us see the procedure of the construction of representations in a simpler case ie singlet case
bull Decomposition of SO(9) singlets into SU(2) representations
1 Enumerate SO(9) singlets 14 states
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
bull describes N D0-branes = N KK modes of 1 unit of KK momentum
Multiparticle states rarr continuous spectrum(de Wit-Luumlscher-Nicolai lsquo89)
KK modes of 2 or more unitsrealized as bound states
rarr discrete spectrum
ConjectureSU(N) Matrix Theory has a unique normalizable zero
energy bound state
bull Calculation of Witten index seems consisitent(Yi lsquo97 Sethi-Stern rsquo97 etc)
bull Some information asymptotic form symmetry
(Hoppe et al etc)
bull No explicit expression is known(for zero energy bound state and any other gauge
invariant wavefunctions)
bull Why is it so difficultbasis of gauge invariant wavefunctions
rarr creation operators rarr states(16777216 states even for SU(2))
bosonic variables rarr
Schroumldinger eq rarr equations with variables
Enormous number of states and variablesrarr Even numerical calculation is difficult
systematic classification of these statesby representation of SU(N)timesSO(9)
Plan
Introduction2 Explicit construction of some coordinate
independent states in SU(2) case3 Number counting of representations in SU(2)
case4 SU(N) case5 Summary
Explicit construction of some coord indep states in SU(2) case
Wavefunction of zero energy bound stateGauge invariantSO(9) invariant (Hasler-Hoppe lsquo02)
Asymptotic form (SU(2))
Taylor expansion around the origin
coordinate independent
statesZero energy rarr
(supercharge )
rarr
Coord indep states for fixed rarr 256 states symmetric traceless (44) antisymmetric (84) vector-spinor (128)
Action of on these states
rarrfull states
etc
bull How do these states transform under gauge transformation
Not immediately clear
It is read off by acting generators of the gauge group
SU(2) case
bull The 1st term (SU(2)timesSO(9) singlet)of the expansion has been constructed and it is unique (up to rescaling) (Hoppe-Lundholm-Trzetrzelewski rsquo08 Hynek-Trzetrzelewski lsquo10)
bull 2nd term zero energy rarr
Let us construct coord Indep (adjoint)times(vector) representation of SU(2)timesSO(9) and see if it satisfies
First let us see the procedure of the construction of representations in a simpler case ie singlet case
bull Decomposition of SO(9) singlets into SU(2) representations
1 Enumerate SO(9) singlets 14 states
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
bull Calculation of Witten index seems consisitent(Yi lsquo97 Sethi-Stern rsquo97 etc)
bull Some information asymptotic form symmetry
(Hoppe et al etc)
bull No explicit expression is known(for zero energy bound state and any other gauge
invariant wavefunctions)
bull Why is it so difficultbasis of gauge invariant wavefunctions
rarr creation operators rarr states(16777216 states even for SU(2))
bosonic variables rarr
Schroumldinger eq rarr equations with variables
Enormous number of states and variablesrarr Even numerical calculation is difficult
systematic classification of these statesby representation of SU(N)timesSO(9)
Plan
Introduction2 Explicit construction of some coordinate
independent states in SU(2) case3 Number counting of representations in SU(2)
case4 SU(N) case5 Summary
Explicit construction of some coord indep states in SU(2) case
Wavefunction of zero energy bound stateGauge invariantSO(9) invariant (Hasler-Hoppe lsquo02)
Asymptotic form (SU(2))
Taylor expansion around the origin
coordinate independent
statesZero energy rarr
(supercharge )
rarr
Coord indep states for fixed rarr 256 states symmetric traceless (44) antisymmetric (84) vector-spinor (128)
Action of on these states
rarrfull states
etc
bull How do these states transform under gauge transformation
Not immediately clear
It is read off by acting generators of the gauge group
SU(2) case
bull The 1st term (SU(2)timesSO(9) singlet)of the expansion has been constructed and it is unique (up to rescaling) (Hoppe-Lundholm-Trzetrzelewski rsquo08 Hynek-Trzetrzelewski lsquo10)
bull 2nd term zero energy rarr
Let us construct coord Indep (adjoint)times(vector) representation of SU(2)timesSO(9) and see if it satisfies
First let us see the procedure of the construction of representations in a simpler case ie singlet case
bull Decomposition of SO(9) singlets into SU(2) representations
1 Enumerate SO(9) singlets 14 states
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
bull Why is it so difficultbasis of gauge invariant wavefunctions
rarr creation operators rarr states(16777216 states even for SU(2))
bosonic variables rarr
Schroumldinger eq rarr equations with variables
Enormous number of states and variablesrarr Even numerical calculation is difficult
systematic classification of these statesby representation of SU(N)timesSO(9)
Plan
Introduction2 Explicit construction of some coordinate
independent states in SU(2) case3 Number counting of representations in SU(2)
case4 SU(N) case5 Summary
Explicit construction of some coord indep states in SU(2) case
Wavefunction of zero energy bound stateGauge invariantSO(9) invariant (Hasler-Hoppe lsquo02)
Asymptotic form (SU(2))
Taylor expansion around the origin
coordinate independent
statesZero energy rarr
(supercharge )
rarr
Coord indep states for fixed rarr 256 states symmetric traceless (44) antisymmetric (84) vector-spinor (128)
Action of on these states
rarrfull states
etc
bull How do these states transform under gauge transformation
Not immediately clear
It is read off by acting generators of the gauge group
SU(2) case
bull The 1st term (SU(2)timesSO(9) singlet)of the expansion has been constructed and it is unique (up to rescaling) (Hoppe-Lundholm-Trzetrzelewski rsquo08 Hynek-Trzetrzelewski lsquo10)
bull 2nd term zero energy rarr
Let us construct coord Indep (adjoint)times(vector) representation of SU(2)timesSO(9) and see if it satisfies
First let us see the procedure of the construction of representations in a simpler case ie singlet case
bull Decomposition of SO(9) singlets into SU(2) representations
1 Enumerate SO(9) singlets 14 states
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
Enormous number of states and variablesrarr Even numerical calculation is difficult
systematic classification of these statesby representation of SU(N)timesSO(9)
Plan
Introduction2 Explicit construction of some coordinate
independent states in SU(2) case3 Number counting of representations in SU(2)
case4 SU(N) case5 Summary
Explicit construction of some coord indep states in SU(2) case
Wavefunction of zero energy bound stateGauge invariantSO(9) invariant (Hasler-Hoppe lsquo02)
Asymptotic form (SU(2))
Taylor expansion around the origin
coordinate independent
statesZero energy rarr
(supercharge )
rarr
Coord indep states for fixed rarr 256 states symmetric traceless (44) antisymmetric (84) vector-spinor (128)
Action of on these states
rarrfull states
etc
bull How do these states transform under gauge transformation
Not immediately clear
It is read off by acting generators of the gauge group
SU(2) case
bull The 1st term (SU(2)timesSO(9) singlet)of the expansion has been constructed and it is unique (up to rescaling) (Hoppe-Lundholm-Trzetrzelewski rsquo08 Hynek-Trzetrzelewski lsquo10)
bull 2nd term zero energy rarr
Let us construct coord Indep (adjoint)times(vector) representation of SU(2)timesSO(9) and see if it satisfies
First let us see the procedure of the construction of representations in a simpler case ie singlet case
bull Decomposition of SO(9) singlets into SU(2) representations
1 Enumerate SO(9) singlets 14 states
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
Plan
Introduction2 Explicit construction of some coordinate
independent states in SU(2) case3 Number counting of representations in SU(2)
case4 SU(N) case5 Summary
Explicit construction of some coord indep states in SU(2) case
Wavefunction of zero energy bound stateGauge invariantSO(9) invariant (Hasler-Hoppe lsquo02)
Asymptotic form (SU(2))
Taylor expansion around the origin
coordinate independent
statesZero energy rarr
(supercharge )
rarr
Coord indep states for fixed rarr 256 states symmetric traceless (44) antisymmetric (84) vector-spinor (128)
Action of on these states
rarrfull states
etc
bull How do these states transform under gauge transformation
Not immediately clear
It is read off by acting generators of the gauge group
SU(2) case
bull The 1st term (SU(2)timesSO(9) singlet)of the expansion has been constructed and it is unique (up to rescaling) (Hoppe-Lundholm-Trzetrzelewski rsquo08 Hynek-Trzetrzelewski lsquo10)
bull 2nd term zero energy rarr
Let us construct coord Indep (adjoint)times(vector) representation of SU(2)timesSO(9) and see if it satisfies
First let us see the procedure of the construction of representations in a simpler case ie singlet case
bull Decomposition of SO(9) singlets into SU(2) representations
1 Enumerate SO(9) singlets 14 states
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
Explicit construction of some coord indep states in SU(2) case
Wavefunction of zero energy bound stateGauge invariantSO(9) invariant (Hasler-Hoppe lsquo02)
Asymptotic form (SU(2))
Taylor expansion around the origin
coordinate independent
statesZero energy rarr
(supercharge )
rarr
Coord indep states for fixed rarr 256 states symmetric traceless (44) antisymmetric (84) vector-spinor (128)
Action of on these states
rarrfull states
etc
bull How do these states transform under gauge transformation
Not immediately clear
It is read off by acting generators of the gauge group
SU(2) case
bull The 1st term (SU(2)timesSO(9) singlet)of the expansion has been constructed and it is unique (up to rescaling) (Hoppe-Lundholm-Trzetrzelewski rsquo08 Hynek-Trzetrzelewski lsquo10)
bull 2nd term zero energy rarr
Let us construct coord Indep (adjoint)times(vector) representation of SU(2)timesSO(9) and see if it satisfies
First let us see the procedure of the construction of representations in a simpler case ie singlet case
bull Decomposition of SO(9) singlets into SU(2) representations
1 Enumerate SO(9) singlets 14 states
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
coordinate independent
statesZero energy rarr
(supercharge )
rarr
Coord indep states for fixed rarr 256 states symmetric traceless (44) antisymmetric (84) vector-spinor (128)
Action of on these states
rarrfull states
etc
bull How do these states transform under gauge transformation
Not immediately clear
It is read off by acting generators of the gauge group
SU(2) case
bull The 1st term (SU(2)timesSO(9) singlet)of the expansion has been constructed and it is unique (up to rescaling) (Hoppe-Lundholm-Trzetrzelewski rsquo08 Hynek-Trzetrzelewski lsquo10)
bull 2nd term zero energy rarr
Let us construct coord Indep (adjoint)times(vector) representation of SU(2)timesSO(9) and see if it satisfies
First let us see the procedure of the construction of representations in a simpler case ie singlet case
bull Decomposition of SO(9) singlets into SU(2) representations
1 Enumerate SO(9) singlets 14 states
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
Coord indep states for fixed rarr 256 states symmetric traceless (44) antisymmetric (84) vector-spinor (128)
Action of on these states
rarrfull states
etc
bull How do these states transform under gauge transformation
Not immediately clear
It is read off by acting generators of the gauge group
SU(2) case
bull The 1st term (SU(2)timesSO(9) singlet)of the expansion has been constructed and it is unique (up to rescaling) (Hoppe-Lundholm-Trzetrzelewski rsquo08 Hynek-Trzetrzelewski lsquo10)
bull 2nd term zero energy rarr
Let us construct coord Indep (adjoint)times(vector) representation of SU(2)timesSO(9) and see if it satisfies
First let us see the procedure of the construction of representations in a simpler case ie singlet case
bull Decomposition of SO(9) singlets into SU(2) representations
1 Enumerate SO(9) singlets 14 states
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
rarrfull states
etc
bull How do these states transform under gauge transformation
Not immediately clear
It is read off by acting generators of the gauge group
SU(2) case
bull The 1st term (SU(2)timesSO(9) singlet)of the expansion has been constructed and it is unique (up to rescaling) (Hoppe-Lundholm-Trzetrzelewski rsquo08 Hynek-Trzetrzelewski lsquo10)
bull 2nd term zero energy rarr
Let us construct coord Indep (adjoint)times(vector) representation of SU(2)timesSO(9) and see if it satisfies
First let us see the procedure of the construction of representations in a simpler case ie singlet case
bull Decomposition of SO(9) singlets into SU(2) representations
1 Enumerate SO(9) singlets 14 states
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
SU(2) case
bull The 1st term (SU(2)timesSO(9) singlet)of the expansion has been constructed and it is unique (up to rescaling) (Hoppe-Lundholm-Trzetrzelewski rsquo08 Hynek-Trzetrzelewski lsquo10)
bull 2nd term zero energy rarr
Let us construct coord Indep (adjoint)times(vector) representation of SU(2)timesSO(9) and see if it satisfies
First let us see the procedure of the construction of representations in a simpler case ie singlet case
bull Decomposition of SO(9) singlets into SU(2) representations
1 Enumerate SO(9) singlets 14 states
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
bull 2nd term zero energy rarr
Let us construct coord Indep (adjoint)times(vector) representation of SU(2)timesSO(9) and see if it satisfies
First let us see the procedure of the construction of representations in a simpler case ie singlet case
bull Decomposition of SO(9) singlets into SU(2) representations
1 Enumerate SO(9) singlets 14 states
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
bull Decomposition of SO(9) singlets into SU(2) representations
1 Enumerate SO(9) singlets 14 states
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
2 Compute representation matrix of
3 Eigenvalue spectrum0 0 plusmn1 plusmn2 plusmn3 plusmn4 plusmn5 plusmn6rarr 1 singlet and 1``spin 6rdquo representation(SU(2) rarr )
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
Ladder operator Eigenvalue 6 rarr unique eigenvector
darr
darr
darr
rarr orthogonal rarr
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
bull Decomposition of SO(9) vectors into SU(2) representations1 Enumerate SO(9) vectors 36 states
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
2 Representation matrix of
3 Eigenvalue spectrum
rarr1 ``spin 1rdquo 1 ``spin 3rdquo 1 ``spin 5rdquo 1 ``spin 7rdquo
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
bull Spin 1 (adjoint) repr
This is the unique candidate for the 1st order termof the expansion of
Does this satisfy YES
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
Number counting of representations in SU(2) case
Explicit construction rarr too cumbersome
Number counting can be done more efficiently by using characters in group theory
Character for repr
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
bull Orthogonality relation
Consider the following quantity
and decompose it into SU(2)timesSO(9) characters
rarr repr multiplicity
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
bull Computation of the character
(Cartan subalgebra part)
States
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
uarrdecomposition into SU(2) characters
decomposition into SO(9) characters rarr orthogonality relations
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
bull result SO(9) representations are indicated by Dynkin labels (72 representations)
SU(2) representations are indicated by spins The unique SU(2)timesSO(9) singletOther states we have constructed
This means is automatically satisfied
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
As in SU(2) case cannot be separated into creation and annihilation operators without spoiling mainifest SU(N)timesSO(9) symmetry
Several different ways of respecting symmetries
SU(N) case
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
bull SU(3) case( =18446744073709551616 states)
Decomposing into SU(3) characters first
1454 singlets
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
Decomposing into SO(9) characters first
1454 singlets
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
bull Many SU(3)timesSO(9) singlets rarr does not mean that there are many bound states
(The power series may give nonnormalizable states or
and other equations may not have nontrivial solution)
bull SU(4) SU(5) SU(6) hellip
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
Summary
bull In SU(2) case we explicitly constructed some coord Indep states in lower repr of SU(2)timesSO(9) rarrgive lower terms in Taylor expansion of zero energy normalizable wavefunction
bull In SU(2) case we counted the multiplicity of representations of SU(2)timesSO(9) by computing the character
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
bull In SU(N) case we computed the character and saw some multiplicities rarrmany singlets
bull Exact expression of zero energy bound state or other states
bull Application to scattering or decay process
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