seminar may 5, 2005physics institute, prague1 non-relativistic and relativistic susy constructions...

seminar May 5, 2005 Physics Institute, Prague 1

Non-relativistic and

relativisticSUSY constructions in QM


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)


TABLE OF CONTENTS

I. ZOO OF SYMMETRIES

II. SUSY PLUS PT-SYMMETRY

III. NON-RELATIVISTIC MODELS

IV. RELATIVISTIC WITTEN

V. CONCLUSIONS


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


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, ...


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


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


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


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


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


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


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 !)


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)


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)


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


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


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


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

•


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)


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)>


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,…)



• SUSY failure


E(L)=(-2,0,2,4,…)

E(R)=(4,8,12,…)




E(R)=(-2,2,4,6,…)


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)>


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


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)


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)


Non-relativistic application

HAMILTONIANS? PT COUPLED CHANNELS!

(complexified two-by-two matrices)

Example? Harmonic oscillator!

(with centrifugal singularity)

B. Beyond Witten


IV. Relativistic application:

Klein-Gordon field

HAMILTONIANS? FESHBACH - VILLARS!


Alternatively? Peano-Baker!

(stronger asymmetry)


Klein Gordon

FESHBACH - VILLARS HAMILTONIANS


POTENTIALS? SIMPLE MODELS m=m(x,E)

(exactly solvable examples)

interpretation: transitions to lower energies

(energy-dependence!)

meaning: non-Hermitian D admitted


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 |



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 |



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 |



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 |



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 |



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



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,


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



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



• 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


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 |



MATRICES:.

2. KG-boson-annihilation supercharge2. KG-boson-annihilation supercharge

| 0 0 | | | | 0 0 0 0 | Q = | | | 0 0 0 | | | | 0 0 0 |



MATRICES:.

3. its boson-creation3. its boson-creationpartnerpartner

| 0 0 | | | | 0 0 0 | P = | | | 0 0 0 | | | | 0 0 0 0 |


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?


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


V.Summary










seminar may 5, 2005physics institute, prague1 non-relativistic and relativistic susy constructions...

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