seminar May 5, 2005 Physics Institute, Prague 1
Non-relativistic and
relativisticSUSY constructions in QM
seminar May 5, 2005 Physics Institute, Prague 2
Collaborators:
•B. Bagchi and A. Banerjee (India)
•H. Bíla (Czechia)
•E. Caliceti and F. Cannata (Italy)
•H. B. Geyer (South Africa)
•V. Jakubský (Czechia)
•G. Lévai (Hungary)
•S. Mallik (India)
•C. Quesne (Belgium)
•R. Roychoudhury (India)
•A. Ventura (Italy)
•M. Znojil (Czechia)
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TABLE OF CONTENTS
I. ZOO OF SYMMETRIES
II. SUSY PLUS PT-SYMMETRY
III. NON-RELATIVISTIC MODELS
IV. RELATIVISTIC WITTEN
V. CONCLUSIONS
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idea (remember Darwin): from symmetries
towards supersymmetries towards PT symmetries
ie, for introduction: listie, for introduction: list
parallels between the concepts of symmetry, supersymmetry and (up to now less common) PT symmetry
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The concept of symmetries
(A) ORIGINS: LIE ALGEBRAS:
bound – states:
subspace with a symmetry, P |(z)> = z |
(z)>
+ Schroedigner equation, H |(z)> = E(z) |
(z)> = vanishing commutator: H P = P H
for illustration: parity P
admissible z = +1 or -1
doubly-indexed spectrum: n = 0, 1, ...
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The concept of symmetries
(A) ORIGINS: LIE ALGEBRAS:
bound – states:
subspace with a symmetry, P |(z)> = z |
(z)>
+ Schroedigner equation, H |(z)> = E(z) |
(z)> = vanishing commutator: H P = P H
for illustration: parity P
admissible z = +1 or -1
doubly-indexed spectrum: n = 0, 1, ...
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Symmetries
APPLICATIONS ctd:
beyond parity:
crystallographic studies and the like,
H = atomic, nuclear, ... physics
z = angular momenta etc
= field theory, particle physics
etc: symmetries = Lie groups
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Symmetries
(B) AN ALTERNATIVE IDEA: SUPERSYMMETRY:
cf. the graded Lie algebra sl(1|1):
three ‘graded’ generators: H plus Q and P
‘fermionic’ PP=QQ=0, plus a new compatibility:
H P - P H = H Q - Q H = 0
while P Q + Q P = H
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Symmetries SUPERSYMMETRY and sl(1|1) ctd.
specific representation of operators:
SUSY QM:
H = direct sum of the ‘left’ and ‘right’ H(L,R)
Q (lower, a) and P (upper, c) are nilpotent, D=2
a = upper (annih.), c = lower (creation), D=oo
related representation of |(z)>:
two components in |(z)>: z=(f,b), f = 0 or 1
conventional HO: f=1 if |b> = down, f=0 if up
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Symmetries SUPERSYMMETRY and sl(1|1) ctd.
specific representation of operators:
SUSY QM:
H = direct sum of the ‘left’ and ‘right’ H(L,R)
Q (lower, a) and P (upper, c) are nilpotent, D=2
a = upper (annih.), c = lower (creation), D=oo
related representation of |(z)>:
two components in |(z)>: z=(f,b), f = 0 or 1
conventional HO: f=1 if |b> = down, f=0 if up
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Symmetries
SUPERSYMMETRY, sl(1|1),
2 x 2 operators and 2-D |(f,b)> ctd.
• solvable HO, field theory (Fock space):
•non-vanishing Q |(0,b)> = z |(1,b-1)>
non-vanishing P |(1,b)> = z’ |
(0,b+1)>
• applications beyond HO:
two Schroedigner equations and a partnership E(L,R)
partners H -> shape invariance, exact solvability
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Symmetries
SUPERSYMMETRY, sl(1|1),
2 x 2 operators and 2-D |(f,b)> ctd.
• solvable HO, field theory (Fock space):
•non-vanishing Q |(0,b)> = z |(1,b-1)>
non-vanishing P |(1,b)> = z’ |
(0,b+1)>
• applications beyond HO:
two Schroedigner equations and a partnership E(L,R)
partners H -> shape invariance, exact solvability
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Symmetries (C) Introduction of the third class of
antilinear,
so called PT - SYMMETRIES
• a prelude: a return to time-reversal T-symmetry
review QM
note that T-operation coincides with h.c.-operation
• pre-history:
BW (‘68): natural complexification of perturbations
CGM (‘80): real energies for imaginary cubic AHO
B (‘92): numerical experiments with V(x) = i x^3
BG (‘93): real spectra at quartic repulsion V=-x^4
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Symmetries (C) Introduction of the third class of
antilinear,
so called PT - SYMMETRIES
• a prelude: a return to time-reversal T-symmetry
review QM
note that T-operation coincides with h.c.-operation
• pre-history:
BW (‘68): natural complexification of perturbations
CGM (‘80): real energies for imaginary cubic AHO
B (‘92): numerical experiments with V(x) = i x^3
BG (‘93): real spectra at quartic repulsion V=-x^4
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Symmetries,use of PT symmetry:
NON-HERMITIAN QUANTUM THEORIES:
• idea: Hermiticity violated in pert.theory,
in phenomenology (condensed matter, biology) etc.
• the first implementations of PT symmetry:
BM (‘97): delta-expansions (in field theory)
BB (‘98): WKB (in quantum mechanics)
CJT (‘98): exact (Bessel functions)
FGRZ (‘98): perturbations (of imaginary cubic)
BB (‘98): QES quartic
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Symmetries
A RETURN TO CLASSICS:
• representations: Wigner in 1960’s (+rediscoveries)
• fights for an acceptance in physics:
indefinite metric in 1940’s (Dirac etc)
Gupta and Bleuler and their elimination trick
relativistic QM and 2 x 2 KG equation: FV (‘58)
field-models of Lie, Wick and Nishijima
MHD: U. G., cosmology: A. M.
exceptional points (BW singularities, measured !)
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Symmetries PT - SYMMETRY’s SUMMARY:
• an agreement on its consistency in physics:
PQ (quasi-parity): Z (‘99), BQZ (‘01)
CPT symmetry: BBJ (‘02)
metric eta plus: M (‘02)
a steady development of mathematics:
proofs [DDT (‘02), S (‘03)]
quasi-Hermiticity [SGH (‘92) + re-discoveries]
systematics and classifications: ES, QES ...
manybody, multidimensional (Calogero, Henon-Heiles)
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Symmetries
PT - SYMMETRY’s SUMMARY:
•
a steady development of mathematics:
proofs [DDT (‘02), S (‘03)]
quasi-Hermiticity [SGH (‘92) + re-discoveries]
systematics and classifications: ES, QES ...
manybody, multidimensional (Calogero, Henon-Heiles)
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An interplay of symmetries(appendix)
- 1 –
(A-B interface)
LIE vs. SUPER - SYMMETRY
•key motivation in field theory:
• multiplets, standard model
• absence of SUSY partners in experiments
• methodical laboratory: Witten’s SUSY QM
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An interplay of symmetries(a systematic review)
- 2 -
A-C interface between
LIE and PT - SYMMETRY
typical for solvable models: sl(2,2) [BQ (‘92)]:
• new models, new Hermitian limits [ZT (‘91)]
• methodical innovations: new boundary conditions
• for SUSY: regularization of singularities
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An interplay of symmetries(a systematic review)
- 3 -
B-C interface between
SUSY and PT - SYMMETRY
a core of our message - key points:
• new classes of representations
• efficient regularization recipes
• explicit constructions
• new models
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II.
Impact of PT on SUSY
Harmonic oscillators are,
after PT symmetrization,
numbered by (non-integer) parameter or =
= 1/2 in Hermitian case
sign-parameter q (= +1 or -1) added
q = -1 (fixed) in Hermitian case
•
energies E = 4n+2-2 q
Laguerre-polynomial wavefunctions
exceptional points = unavoided level crossings
•
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PT and SUSY representations:
SUSY HO example:
• initial superpotential W()
• partners:
H(L)= H()-2 -2, =
H(R)= H()-2 , =
• hierarchy E (+,) < E (+,) < E (-,) < E (-,),
ground E(L)=min(0,-4 ) (right: deg. and neg.)
the first excited R-state: E=min(4,-4 )
the 2nd excited: max(0,-4 ) (right: nedeg. 0)
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PT and SUSY representations:
SUSY HO example:
• initial superpotential W()
• partners:
H(L)= H()-2 -2, =
H(R)= H()-2 , =
• hierarchy E (+,) < E (+,) < E (-,) < E (-,),
ground E(L)=min(0,-4 ) (right: deg. and neg.)
the first excited R-state: E=min(4,-4 )
the 2nd excited: max(0,-4 ) (right: nedeg. 0)
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PT and SUSY - representations
SUSY action:
A|,n+1)> = c |+1,n)>
A|,n)> = d |-1,n)>
C|+1,n)> = c |,n+1)>
C|-1,n)> = d |,n)>
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PT and SUSYregularizations:
• SUSY failure
Jevicki-Rodrigues =1/2 paradox:
E(L)=(-2,0,2,4,…)
E(R)=(4,8,12,…)
• Das-Pernice resolution:
delta-function in the origin and shift +2
• PT analytic resolution:
E(R)=(-2,2,4,6,…)
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PT and SUSYregularizations:
• SUSY failure
Jevicki-Rodrigues =1/2 paradox:
E(L)=(-2,0,2,4,…)
E(R)=(4,8,12,…)
• Das-Pernice resolution:
delta-function in the origin and shift +2
• PT analytic resolution:
E(R)=(-2,2,4,6,…)
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PT and SUSYregularizations:
• SUSY failure
Jevicki-Rodrigues =1/2 paradox:
E(L)=(-2,0,2,4,…)
E(R)=(4,8,12,…)
• Das-Pernice resolution:
delta-function in the origin and shift +2
• PT analytic resolution:
E(R)=(-2,2,4,6,…)
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PT and SUSYinterpretations
• singular HO in a new picture:
1. define the second-order operators:
A-1xA= A
CxC-1= C
2. derive q-dependent creation/annihilation:
A|,n+1)> = f |,n)>
C|,n)> = f |,n+1)>
seminar May 5, 2005 Physics Institute, Prague 30
PT and SUSYinterpretations
• singular HO in a new picture:
1. define the second-order operators:
A-1xA= A
CxC-1= C
2. derive q-dependent creation/annihilation:
A|,n+1)> = f |,n)>
C|,n)> = f |,n+1)>
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PT and SUSY SSUSY representation:
4. close the Lie algebra sl(2,R):
8 H = ACCA
4 A = AHHA
4 C = HCCH
5. get the new H(LH(L) ) = [H]^2 - 4 ^2
and the new H(RH(R) ) = [H]^2 - 4 ^2
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PT and SUSY SSUSY representation:
4. close the Lie algebra sl(2,R):
8 H = ACCA
4 A = AHHA
4 C = HCCH
5. get the new H(LH(L) ) = [H]^2 - 4 ^2
and the new H(RH(R) ) = [H]^2 - 4 ^2
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PT and SUSY
plans for future:
weaken PT symmetry to CPT symmetry
P -> P(gen) = C P, (PT broken: KM ‘91 etc)
C -> any differential operator (A ‘92, KS ‘04)
re-install SUSY (CCZV ‘04)
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III. Non-relativistic application:
A. Witten’s model
HAMILTONIANS? COUPLED CHANNELS!
(two-by-two matrices)
Example? Harmonic oscillator!
(no singularity)
(see standard reviews – skipped here)
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Non-relativistic application
HAMILTONIANS? PT COUPLED CHANNELS!
(complexified two-by-two matrices)
Example? Harmonic oscillator!
(with centrifugal singularity)
B. Beyond Witten
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IV. Relativistic application:
Klein-Gordon field
HAMILTONIANS? FESHBACH - VILLARS!
(two-by-two matrices)
Alternatively? Peano-Baker!
(stronger asymmetry)
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Klein Gordon
FESHBACH - VILLARS HAMILTONIANS
(two-by-two matrices)
POTENTIALS? SIMPLE MODELS m=m(x,E)
(exactly solvable examples)
interpretation: transitions to lower energies
(energy-dependence!)
meaning: non-Hermitian D admitted
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Klein Gordon FESHBACH - VILLARS
SCALAR POTENTIALS m=m(x,E)
equation: -y‘‘=Dy and abbreviations equation: -y‘‘=Dy and abbreviations iy‘=u, y=viy‘=u, y=v
re-written iu‘=Dv, iv‘=u, i.e., two-re-written iu‘=Dv, iv‘=u, i.e., two-dimensionaldimensional
i Y‘=H Yi Y‘=H Y
oror
| u‘ | | 0 D | | u | i | | = | | . | | | v‘ | | I 0 | | v |
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Klein Gordon FESHBACH - VILLARS
SCALAR POTENTIALS m=m(x,E)
SURPRIZE: PT-SYMMETRY! .
i.e., two-dimensional PT-symmetry rulei.e., two-dimensional PT-symmetry rule
where the ‘metric‘ is anti-Hermitian,where the ‘metric‘ is anti-Hermitian,
| 0 I | = | | | -I 0 |
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Klein Gordon FESHBACH - VILLARS
PT-SYMMETRY.
energy-representationenergy-representation
| u | | 0 D | | u | E | | = | | . | | | v | | I 0 | | v |
and solution (Dv=Eu, u = Ev, Du = E^2 and solution (Dv=Eu, u = Ev, Du = E^2 u):u):
| u | | E u | | E | | | = | | = | | . u | v | | u | | 1 |
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Klein Gordon FESHBACH - VILLARS
PT-SYMMETRY.
D-spectrum positive (Du = E^2 u),D-spectrum positive (Du = E^2 u),
H-spectrum in the positive/negative H-spectrum in the positive/negative pairs,pairs,
with remarkable wavefunctions,with remarkable wavefunctions,
| E u | | - E u | | | , | | | u | | u |
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Klein Gordon FESHBACH - VILLARS
NEW REQUIREMENT - SUPERSYMMETRY,
formed by L + R direct sum:formed by L + R direct sum:
| H(L) 0 | G = | | = P Q + Q P | 0 H(R) |
where
| 0 0 | | 0 C | Q = | | , P = | | | A 0 | | 0 0 |
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Klein Gordon FESHBACH - VILLARS
CONSEQUENCES OF PT-SUPERSYMMETRY.
submatricessubmatrices A and C are two-by- A and C are two-by-
two:two:
| 0 | | 0 | A = | | , C = | | | 0 | | 0 |
SUSY requires that SUSY requires that
D(L), D(L), D(R), D(R), and and
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Klein Gordon FESHBACH - VILLARS
PT-SUPERSYMMETRY.
the first two rules = non-relativistic the first two rules = non-relativistic SUSY:SUSY:
D(L), D(L), D(R),D(R),i.e., annihilation and creation, i.e., annihilation and creation,
respectively:respectively:
| 0 | | 0 | AC = | | x | | | 0 | | 0 |
the latter two conditions must be the latter two conditions must be weakened,weakened,
seminar May 5, 2005 Physics Institute, Prague 45
Klein Gordon FESHBACH - VILLARS
PT-SUPERSYMMETRY.
the first two rules = non-relativistic the first two rules = non-relativistic SUSY:SUSY:
D(L), D(L), D(R),D(R),i.e., annihilation and creation, i.e., annihilation and creation,
respectively:respectively:
| 0 | | 0 | AC = | | x | | | 0 | | 0 |
the latter two conditions must be the latter two conditions must be weakened,weakened,
seminar May 5, 2005 Physics Institute, Prague 46
Klein Gordon PT-SUPERSYMMETRY.
We may emphasize:We may emphasize:there are three levels of Hilbert there are three levels of Hilbert
spaces:spaces:level of action of level of action of D(L) and D(L) and D(R),D(R),plus their 2-component direct sum:plus their 2-component direct sum:level of action of level of action of H(L) and H(L) and H(R),H(R),plus their 4-component direct sum:plus their 4-component direct sum:SUSY level of the action of SUSY level of the action of GG
on all of them we have kets |uon all of them we have kets |u i.e., i.e., right eigenstates u = |u(1)> of right eigenstates u = |u(1)> of D,D,
FV 2-component right eigenstates of HFV 2-component right eigenstates of H SUSY 4-component right eigenstates of GSUSY 4-component right eigenstates of G
and PT, non-conjugate bras and PT, non-conjugate bras uu
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Klein Gordon PT-SUPERSYMMETRY.
We may emphasize:We may emphasize:there are three levels of Hilbert there are three levels of Hilbert
spaces:spaces:level of action of level of action of D(L) and D(L) and D(R),D(R),plus their 2-component direct sum:plus their 2-component direct sum:level of action of level of action of H(L) and H(L) and H(R),H(R),plus their 4-component direct sum:plus their 4-component direct sum:SUSY level of the action of SUSY level of the action of GG
on all of them we have kets |uon all of them we have kets |u i.e., i.e., right eigenstates u = |u(1)> of right eigenstates u = |u(1)> of D,D,
FV 2-component right eigenstates of HFV 2-component right eigenstates of H SUSY 4-component right eigenstates of GSUSY 4-component right eigenstates of G
and PT, non-conjugate bras and PT, non-conjugate bras uu
seminar May 5, 2005 Physics Institute, Prague 48
Klein Gordon PT-SUPERSYMMETRY.
We may emphasize:We may emphasize:there are three levels of Hilbert there are three levels of Hilbert
spaces:spaces:level of action of level of action of D(L) and D(L) and D(R),D(R),plus their 2-component direct sum:plus their 2-component direct sum:level of action of level of action of H(L) and H(L) and H(R),H(R),plus their 4-component direct sum:plus their 4-component direct sum:SUSY level of the action of SUSY level of the action of GG
on all of them we have kets |uon all of them we have kets |u i.e., i.e., right eigenstates u = |u(1)> of right eigenstates u = |u(1)> of D,D,
FV 2-component right eigenstates of HFV 2-component right eigenstates of H SUSY 4-component right eigenstates of GSUSY 4-component right eigenstates of G
and PT, non-conjugate bras and PT, non-conjugate bras uu
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Klein Gordon PT-SUPERSYMMETRY.
In all these three cases,In all these three cases,the action of the action of GG may be both right and left: may be both right and left:
GG |u |u = E= E |u |uuuGG = = uuEE
As long as h.c.(G) = As long as h.c.(G) = G 1/ G 1/
we may assume Im E = 0 and get we may assume Im E = 0 and get
quasi-paritiesquasi-parities,,
|u|u = q= q . . |u|u
Insertions give relations between Insertions give relations between qqqqandandqq
seminar May 5, 2005 Physics Institute, Prague 50
Klein Gordon PT-SUPERSYMMETRY.
In all these three cases,In all these three cases,the action of the action of GG may be both right and left: may be both right and left:
GG |u |u = E= E |u |uuuGG = = uuEE
As long as h.c.(G) = As long as h.c.(G) = G 1/ G 1/
we may assume Im E = 0 and get we may assume Im E = 0 and get
quasi-paritiesquasi-parities,,
|u|u = q= q . . |u|u
Insertions give relations between Insertions give relations between qqqqandandqq
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Klein Gordon PT-SUPERSYMMETRY.
• bi-orthogonality bi-orthogonality
- - uuuu • overlaps overlaps
uuuu/R/R,, •completenesscompleteness
uu/R/Ruu
SE level SE level where where positive) positive) ::……
on the KG level (non-zero E):on the KG level (non-zero E):sign sign andand
on the SUSY level:on the SUSY level: L and R, bosons - fermionsL and R, bosons - fermions
seminar May 5, 2005 Physics Institute, Prague 52
Klein Gordon PT-SUPERSYMMETRY.
• bi-orthogonality bi-orthogonality
- - uuuu • overlaps overlaps
uuuu/R/R,, •completenesscompleteness
uu/R/Ruu
SE level SE level where where positive) positive) ::……
on the KG level (non-zero E):on the KG level (non-zero E):sign sign andand
on the SUSY level:on the SUSY level: L and R, bosons - fermionsL and R, bosons - fermions
seminar May 5, 2005 Physics Institute, Prague 53
Klein Gordon PT-SUPERSYMMETRY
IN MARTRIX FORM:.
sl(1|1)sl(1|1) generated by three 4 x 4 generated by three 4 x 4 items:items:
1. KG-super-Hamiltonian1. KG-super-Hamiltonian
| D(L) 0 | | | | 0 0 0 | G = | | | 0 0 D(R) | | | | 0 0 I 0 |
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Klein Gordon PT-SUPERSYMMETRY
MATRICES:.
2. KG-boson-annihilation supercharge2. KG-boson-annihilation supercharge
| 0 0 | | | | 0 0 0 0 | Q = | | | 0 0 0 | | | | 0 0 0 |
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Klein Gordon PT-SUPERSYMMETRY
MATRICES:.
3. its boson-creation3. its boson-creationpartnerpartner
| 0 0 | | | | 0 0 0 | P = | | | 0 0 0 | | | | 0 0 0 0 |
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Klein Gordon PT-SUPERSYMMETRY SUMMARY:
There exists a remarkable physical There exists a remarkable physical
interpretation of the interpretation of the exceptional E=0 ground-state.exceptional E=0 ground-state.
Due to our transition to relativistic Due to our transition to relativistic kinematics,kinematics,
it becomes unstable against decay!
OPEN QUESTION:DOES THIS EXPLAIN THE
EXPERIMENTALLY OBSERVED ABSENCE OF SUSY PARTNERS
IN PARTICLE PHYSICS?
seminar May 5, 2005 Physics Institute, Prague 57
V.Summary
• new ideas in SUSY:
natural -> auxiliary metric P in Hilbert space
Jordan blocks -> unavoided crossings of levels
quasi-parity -> C PT symmetry -> probability
• PT models in physics:
Winternitzian and Calogerian models anew:
non-equivalent Hermitian limits
new types of tunnelling
parallels between pseudo- and Hermitian SUSY
seminar May 5, 2005 Physics Institute, Prague 58
V.Summary
• new ideas in SUSY:
natural -> auxiliary metric P in Hilbert space
Jordan blocks -> unavoided crossings of levels
quasi-parity -> C PT symmetry -> probability
• PT models in physics:
Winternitzian and Calogerian models anew:
non-equivalent Hermitian limits
new types of tunnelling
parallels between pseudo- and Hermitian SUSY
seminar May 5, 2005 Physics Institute, Prague 59
V.Summary
• new ideas in SUSY:
natural -> auxiliary metric P in Hilbert space
Jordan blocks -> unavoided crossings of levels
quasi-parity -> C PT symmetry -> probability
• PT models in physics:
Winternitzian and Calogerian models anew:
non-equivalent Hermitian limits
new types of tunnelling
parallels between pseudo- and Hermitian SUSY
seminar May 5, 2005 Physics Institute, Prague 60
V.Summary
• new ideas in SUSY:
natural -> auxiliary metric P in Hilbert space
Jordan blocks -> unavoided crossings of levels
quasi-parity -> C PT symmetry -> probability
• PT models in physics:
Winternitzian and Calogerian models anew:
non-equivalent Hermitian limits
new types of tunnelling
parallels between pseudo- and Hermitian SUSY