A Specific N =2Supersymmetric Quantum Mechanical Model ... · modern physics because three out of four fundamental interactions of nature are governed by the gauge theory. TheBecchi-Rouet-Stora-Tyutin(BRST)formalismisoneof

Research ArticleA Specific N = 2 Supersymmetric Quantum Mechanical ModelSupervariable Approach

Aradhya Shukla

Indian Institute of Science Education and Research Kolkata Mohanpur 741246 India

Correspondence should be addressed to Aradhya Shukla ashukla038gmailcom

Received 6 August 2016 Accepted 5 January 2017 Published 19 February 2017

Academic Editor Shi-Hai Dong

Copyright copy 2017 Aradhya Shukla This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

By exploiting the supersymmetric invariant restrictions on the chiral and antichiral supervariables we derive the off-shell nilpotentsymmetry transformations for a specific (0 + 1)-dimensional N = 2 supersymmetric quantum mechanical model which isconsidered on a (1 2)-dimensional supermanifold (parametrized by a bosonic variable 119905 and a pair of Grassmannian variables(120579 120579)) We also provide the geometrical meaning to the symmetry transformations Finally we show that this specificN = 2 SUSYquantum mechanical model is a model for Hodge theory

1 Introduction

Gauge theory is one of the most important theories ofmodern physics because three out of four fundamentalinteractions of nature are governed by the gauge theoryThe Becchi-Rouet-Stora-Tyutin (BRST) formalism is one ofthe systematic approaches to covariantly quantize any 119901-form (119901 = 1 2 3 ) gauge theories where the local gaugesymmetry of a given theory is traded with the ldquoquantumrdquogauge (ie (anti-)BRST) symmetry transformations [1ndash4] Itis important to point out that the (anti-)BRST symmetriesare nilpotent and absolutely anticommuting in nature Oneof the unique elegant and geometrically rich methods toderive these (anti-)BRST transformations is the superfieldformalism where the horizontality condition (HC) plays animportant role [5ndash12] This HC is a useful tool to derive theBRST aswell as anti-BRST symmetry transformations for any(non-)Abelian 119901-form (119901 = 1 2 3 ) gauge theory whereno interaction between the gauge andmatter fields is present

For the derivation of a full set of (anti-)BRST symmetrytransformations in the case of interacting gauge theories apowerful method known as augmented version of superfieldformalism has been developed in a set of papers [13ndash16]In augmented superfield formalism some conditions namedgauge invariant restrictions (GIRs) in addition to the HC

have been imposed to obtain the off-shell nilpotent andabsolutely anticommuting (anti-)BRST transformations Itis worthwhile to mention here that this technique has alsobeen applied in case of some N = 2 supersymmetric(SUSY) quantum mechanical (QM) models to derive theoff-shell nilpotent SUSY symmetry transformations [17ndash20]These SUSY transformations have been derived by using thesupersymmetric invariant restrictions (SUSYIRs) and it hasbeen observed that the SUSYIRs are the generalizations of theGIRs in case ofN = 2 SUSY QM theory

The aim of present investigation is to explore and applythe augmented version of HC to a new N = 2 SUSY QMmodel which is different from the earlier models present inthe literature In our present endeavor we derive the off-shell nilpotent SUSY transformations for a specific N =2 SUSY QM model by exploiting the potential and powerof the SUSYIRs The additional reason behind our presentinvestigation is to take one more step forward in the direc-tion of the confirmation of SUSYIRs (ie generalization ofaugmented superfield formalism) as a powerful technique forthe derivation of SUSY transformations for any generalN = 2SUSY QM system

One of the key differences between the (anti-)BRST andSUSY symmetry transformations is that the (anti-)BRSTsymmetries are nilpotent as well as absolutely anticommuting

HindawiAdvances in High Energy PhysicsVolume 2017 Article ID 1403937 8 pageshttpsdoiorg10115520171403937

2 Advances in High Energy Physics

in nature whereas SUSY transformations are only nilpo-tent and the anticommutator of fermionic transformationsproduces an additional symmetry transformation of thetheory Due to this basic reason we are theoretically forcedto use the (anti)chiral supervariables generalized on the(1 1)-dimensional super-submanifolds of the full (1 2)-dimensional supermanifold The latter is parametrized bythe superspace coordinate 119885119872 = (119905 120579 120579) where 120579 120579 arethe Grassmannian variables and 119905 is the time-evolutionparameter

The contents of present investigation are organized asfollows In Section 2 we discuss the symmetry transforma-tions associated with the specific N = 2 SUSY QM modelIt is to be noted that there are three continuous symmetriesassociated with this particular model in which two of themare fermionic and one is bosonic in nature Section 3 isdevoted to the derivation of one of the fermionic transfor-mations by using the antichiral supervariable approach Wederive the second SUSY fermionic symmetry by exploitingthe chiral supervariable approach in Section 4 In Section 5the Lagrangian of the model is presented in terms of the(anti)chiral supervariables and the geometrical interpretationfor invariance of the Lagrangian in terms of the Grass-mannian derivatives (120597120579 and 120597120579) is explicated Furthermorewe also represent the charges corresponding to the con-tinuous symmetry transformations in terms of (anti)chiralsupervariables In Section 6 we show that the fermionicSUSY symmetry transformations satisfy the N = 2 SUSYalgebra which is identical to the Hodge algebra obeyed bythe cohomological operators of differential geometry Thuswe show that this particular N = 2 SUSY QM model is anexample of Hodge theory Finally we draw conclusions inSection 7 with remarks

2 Preliminaries A specific N = 2SUSY QM Model

We begin with the action of a specific (0 + 1)-dimensionalN = 2 QMmodel [21]

119878 = int1198891199051198710= int119889119905 [(119889120601119889119905 + 119904120597119881120597120601 )

2 minus 119894120595( 119889119889119905 + 119904 1205972119881120597120601120597120601)120595]

(1)

where the bosonic variable 120601 and fermionic variables 120595 120595are the functions of time-evolution parameter 119905 119881(120601) is ageneral potential function and 119904 is an independent constantparameter For algebraic convenience we linearize the firstterm in (1) by introducing an auxiliary variable 119860 As aconsequence the action can be written as (henceforth wedenote 120601 = 119889120601119889119905 1198811015840 = 120597119881120597120601 11988110158401015840 = 1205972119881120597120601120597120601 in the text)

119878 = int119889119905119871

= int119889119905 [119894 ( 120601 + 1199041198811015840)119860 + 11986022 minus 119894120595 ( + 11990412059511988110158401015840)]

(2)

where 119860 = minus119894( 120601 + 1199041198811015840) is the equation of motion Using thisexpression for 119860 in (2) one can recover the original action

For the present QM system we have the following off-shell nilpotent (11990421 = 0 11990422 = 0) SUSY transformations (wepoint out that the SUSY transformations in (3) differ by anoverall 119894-factor from [21])

1199041120601 = 1198941205951199041120595 = 01199041120595 = 1198941198601199041119860 = 01199042120601 = 1198941205951199042120595 = 01199042120595 = 119894119860 minus 211990411988110158401199042119860 = 211990412059511988110158401015840

(3)

Under the above symmetry transformations the Lagrangianin (2) transforms as

1199041119871 = 01199042119871 = 119889

119889119905 (minus119860120595) (4)

Thus the action integral remains invariant (ie 1199041119878 = 0 1199042119878 =0) According to Noetherrsquos theorem the continuous SUSYsymmetry transformations 1199041 and 1199042 lead to the followingconserved charges respectively

119876 = minus120595119860119876 = minus120595 [119860 + 21198941199041198811015840] (5)

The conservation of the SUSY charges (ie = 0 = 0)can be proven by exploiting the following Euler-Lagrangeequations of motion

= 11990411986011988110158401015840 minus 119904120595120595119881101584010158401015840 = minus11990412059511988110158401015840119860 = minus119894 ( 120601 + 1199041198811015840) = minus11990412059511988110158401015840

(6)

It turns out that these charges are the generators of SUSYtransformations (3) One can explicitly check that the follow-ing relations are true

1199041Φ = 119894 [Φ119876]plusmn 1199042Φ = 119894 [Φ119876]

plusmnΦ = 120601 119860 120595 120595

(7)

Advances in High Energy Physics 3

where the subscripts (plusmn) on the square brackets deal withthe (anti)commutator depending on the variables beingfermionicbosonic in nature

It is to be noted that the anticommutator of the fermionicSUSY transformations (1199041 and 1199042) leads to a bosonic symme-try (119904120596)

119904120596120601 = minus2 (119860 + 1198941199041198811015840) 119904120596120595 = minus211989411990412059511988110158401015840119904120596120595 = 211989411990412059511988110158401015840119904120596119860 = 2119894119904 (11986011988110158401015840 minus 120595120595119881101584010158401015840)

(8)

The application of bosonic symmetry (119904120596) on the Lagrangianproduces total time derivative

119904120596119871 = (11990411199042 + 11990421199041) 119871 = 119889119889119905 (minus1198941198602) (9)

Thus according to Noetherrsquos theorem the above continuousbosonic symmetry leads to a bosonic conserved charge (119876120596)as follows

119876120596 = minus1198941198602 + 21199041198601198811015840 + 211990412059512059511988110158401015840

= 2119894 [Π120601Π1206012 minus 119904Π1206011198811015840 minus 119904120595Π12059511988110158401015840] equiv (2119894)119867(10)

where Π120601 = 119894119860Π120595 = 119894120595 are the canonical momentacorresponding to the variables 120601 120595 respectively It is clearthat the bosonic charge 119876120596 is the Hamiltonian119867 (modulo aconstant 2119894-factor) of our present model

One of the important features of SUSY transformations isthat the application of this bosonic symmetry must producethe time translation of the variable (modulo a constant 2119894-factor) which can be checked as

119904120596Φ = 1199041 1199042Φ = (2119894) Φ Φ = 120601 119860 120595 120595 (11)

where in order to prove the sanctity of this equation we haveused the equations of motion mentioned in (6)

3 Off-Shell Nilpotent SUSY TransformationsAntichiral Supervariable Approach

In order to derive the continuous transformation 1199041 weshall focus on the (1 1)-dimensional super-submanifold (ofgeneral (1 2)-dimensional supermanifold) parameterized bythe supervariable (119905 120579) For this purpose we impose super-symmetric invariant restrictions (SUSYIRs) on the antichiral

supervariablesWe then generalize the basic (explicit 119905 depen-dent) variables to their antichiral supervariable counterparts

120601 (119905) 997888rarr Φ (119905 120579 120579)10038161003816100381610038161003816120579=0 equiv Φ (119905 120579) = 120601 (119905) + 1205791198911 (119905) 120595 (119905) 997888rarr Ψ (119905 120579 120579)10038161003816100381610038161003816120579=0 equiv Ψ (119905 120579) = 120595 (119905) + 1198941205791198871 (119905) 120595 (119905) 997888rarr Ψ (119905 120579 120579)100381610038161003816100381610038161003816120579=0 equiv Ψ (119905 120579) = 120595 (119905) + 1198941205791198872 (119905) 119860 (119905) 997888rarr 119860(119905 120579 120579)10038161003816100381610038161003816120579=0 equiv 119860 (119905 120579) = 119860 (119905) + 120579119875 (119905)

(12)

where 1198911(119905) 119875(119905) and 1198871(119905) 1198872(119905) are the fermionic andbosonic secondary variables respectively

It is observed from (3) that 1199041(120595 119860) = 0 (ie both 120595 and119860 are invariant under 1199041) Therefore we demand that bothvariables should remain unchanged due to the presence ofGrassmannian variable 120579 As a result of the above restrictionswe obtain

Ψ (119905 120579 120579)10038161003816100381610038161003816120579=0 equiv Ψ (119905 120579) = 120595 (119905) 997904rArr 1198871 = 0119860 (119905 120579 120579)10038161003816100381610038161003816120579=0 equiv 119860 (119905 120579) = 119860 (119905) 997904rArr 119875 = 0 (13)

We further point out that 1199041(120601120595) = 0 and 1199041( 120601) = 0due to the fermionic nature of 120595 (ie 1205952 = 0) Thus theserestrictions yield

Φ (119905 120579) Ψ (119905 120579) = 120601 (119905) 120595 (119905) 997904rArr 1198911120595 = 0Φ (119905 120579) Ψ (119905 120579) = 120601 (119905) (119905) 997904rArr 1198911 = 0 (14)

The trivial solution for the above relationships is 1198911 prop 120595for algebraic convenience we choose 1198911 = 119894120595 Here the 119894-factor has been taken due to the convention we have adoptedfor the present SUSY QM theory Substituting the valuesof the secondary variables in the expansions of antichiralsupervariables (12) one obtains

Φ(119886119888) (119905 120579) = 120601 (119905) + 120579 (119894120595) equiv 120601 (119905) + 120579 (1199041120601 (119905)) Ψ(119886119888) (119905 120579) = 120595 (119905) + 120579 (0) equiv 120595 (119905) + 120579 (1199041120595 (119905)) 119860(119886119888) (119905 120579) = 119860 (119905) + 120579 (0) equiv 119860 (119905) + 120579 (1199041119860 (119905))

(15)

The superscript (119886119888) in the above represents the antichiralsupervariables obtained after the application of SUSIRsFurthermore we note that the 1D potential function 119881(120601)can be generalized to (Φ(119886119888)) onto (1 1)-dimensional super-submanifold as

119881 (120601) 997888rarr (Φ(119886119888)) = (119886119888) (120601 + 120579 (119894120595)) = 119881 (120601) + 120579 (1198941205951198811015840) equiv 119881 (120601) + 120579 (1199041119881 (120601)) (16)


where we have used the expression of antichiral supervariableΦ(119886119888)(119905 120579) as given in (15)

In order to find out the SUSY transformation for 120595 itcan be checked that the application of 1199041 on the followingvanishes

1199041 [119894 ( 120601 + 1199041198811015840)119860 minus 119894120595 ( + 11990412059511988110158401015840)] = 0 (17)

As a consequence the above can be used as a SUSYIRand we replace the ordinary variables by their antichiralsupervariables as

[119894 ( Φ(119886119888) + 1199041015840(119886119888))119860(119886119888)

minus 119894Ψ ( Ψ(119886119888) + 119904Ψ(119886119888)10158401015840(119886119888))] = [119894 ( 120601 + 1199041198811015840)119860minus 119894120595 ( + 11990412059511988110158401015840)]

(18)

After doing some trivial computations we obtain 1198872 =119860 Recollecting all the value of secondary variables andsubstituting them into (12) we finally obtain the followingantichiral supervariable expansions

Φ(119886119888) (119905 120579) = 120601 (119905) + 120579 (119894120595) equiv 120601 (119905) + 120579 (1199041120601 (119905)) Ψ(119886119888) (119905 120579) = 120595 (119905) + 120579 (0) equiv 120595 (119905) + 120579 (1199041120595 (119905)) Ψ(119886119888) (119905 120579) = 120595 (119905) + 120579 (119894119860) equiv 120595 (119905) + 120579 (1199041120595 (119905)) 119860(119886119888) (119905 120579) = 119860 (119905) + 120579 (0) equiv 119860 (119905) + 120579 (1199041119860 (119905))

(19)

Finally we have derived explicitly the SUSY transformation 1199041for all the variables by exploiting SUSY invariant restrictionson the antichiral supervariables These symmetry transfor-mations are

1199041120601 = 1198941205951199041120595 = 01199041120595 = 1198941198601199041119860 = 01199041119881 = 1198941205951198811015840

(20)

It is worthwhile to mention here that for the antichiralsupervariable expansions given in (11) we have the followingrelationship between the Grassmannian derivative 120597120579 andSUSY transformations 1199041

120597120597120579Ω(119886119888) (119905 120579 120579)10038161003816100381610038161003816120579=0 equiv 120597

120597120579Ω(119886119888) (119905 120579) = 1199041Ω (119905) (21)

where Ω(119886119888)(119905 120579) is the generic supervariable obtained byexploiting the SUSY invariant restriction on the antichiralsupervariables It is easy to check from the above equationthat the symmetry transformation (1199041) for any generic vari-able Ω(119905) is equal to the translation along the 120579-directionof the antichiral supervariable Furthermore it can also bechecked that nilpotency of the Grassmannian derivative 120597120579(ie 1205972

120579= 0) implies 11990421 = 0

4 Off-Shell Nilpotent SUSY TransformationsChiral Supervariable Approach

For the derivation of second fermionic SUSY transforma-tion 1199042 we concentrate on the chiral super-submanifoldparametrized by the supervariables (119905 120579) Now all the ordi-nary variables (depending explicitly on 119905) are generalized to a(1 1)-dimensional chiral super-submanifold as


(22)

In the above secondary variables1198911(119905) 119875(119905) and 1198871(119905) 1198872(119905)are fermionic and bosonic variables respectively We canderive the values of these secondary variables in terms of thebasic variables by exploiting the power and potential of SUSYinvariant restrictions

It is to be noted from (3) that 120595 does not transform underSUSY transformations 1199042 (ie 1199042120595 = 0) so the variable 120595would remain unaffected by the presence of Grassmannianvariable 120579 As a consequence we have the following

Ψ (119905 120579 120579)100381610038161003816100381610038161003816120579=0 equiv Ψ (119905 120579) = 120595 (119905) 997904rArr 1198872 = 0 (23)

Furthermore we observe that 1199042(120601120595) = 0 and 1199042( 120601) = 0due to the fermionic nature of120595 Generalizing these invariantrestrictions to the chiral supersubmanifold we have thefollowing SUSYIRs in the following forms namely

Φ (119905 120579) Ψ (119905 120579) = 120601 (119905) 120595 (119905) Φ (119905 120579) Ψ (119905 120579) = 120601 (119905) (119905)

(24)

After putting the expansions for supervariable (22) in theabove we get

1198911120595 = 01 = 0 (25)

The solution for the above relationship is 1198911 = 119894120595 Substitut-ing the value of secondary variables in the chiral supervari-able expansions (21) we obtain the following expressions

Φ(119888) (119905 120579) = 120601 (119905) + 120579 (119894120595) equiv 120601 (119905) + 120579 (1199042120601 (119905)) Ψ(119888) (119905 120579) = 120595 (119905) + 119894120579 (0) equiv 120595 (119905) + 120579 (1199042120595 (119905))

(26)

where superscript (119888) represents the chiral supervariablesobtained after the applicationof SUSYIRs Using (26) one


can generalize 119881(120601) to (Φ(119888)) onto the (1 1)-dimensionalchiral super-submanifold as

119881 (120601) 997888rarr (Φ(119888)) = (119888) (120601 + 120579 (119894120595)) = 119881 (120601) + 120579 (1198941205951198811015840 (120601)) equiv 119881 (120601) + 1199042 (119881 (120601)) (27)

where we have used the expression of chiral supervariableΦ(119888)(119905 120579) given in (26)

We note that 1199042[119894119860minus21199041205951198811015840] = 0 because of the nilpotencyof 1199042 [cf (3)] Thus we have the following SUSYIR in ourpresent theory

119894119860(119888) minus 2119904Ψ1015840(119888) = [119894119860 minus 21199041205951198811015840] (28)

This restriction serves our purpose for the derivation of SUSYtransformation of 119860(119905) Exploiting the above restriction weget the value of secondary variable 119875(119905) in terms of basicvariables as 119875 = 211990412059511988110158401015840

It is important to note that the following sum of thecomposite variables is invariant under 1199042 namely

1199042 [1198941199041198811015840119860 + 121198602 minus 11989411990412059512059511988110158401015840] = 0 (29)

In order to calculate the fermionic symmetry transformationcorresponding to variable 120595 we use the above relationship asa SUSYIR

1198941199041015840(119888)119860(119888) + 121198602(119888) minus 119894119904Ψ(119888)Ψ10158401015840(119888)

= 1198941199041198811015840119860 + 121198602 minus 11989411990412059512059511988110158401015840

(30)

and after some computations one gets 1198871 = 119860 + 21198941199041198811015840Recollecting all the values of secondary variables and

substituting them into (21) we have the expansions of thechiral supervariables as

Φ(119888) (119905 120579) = 120601 (119905) + 120579 (119894120595) equiv 120601 (119905) + 120579 (1199042120601 (119905)) Ψ(119888) (119905 120579) = 120595 (119905) + 120579 (119894119860 minus 21199041198811015840)

equiv 120595 (119905) + 120579 (1199042120595 (119905)) Ψ(119888) (119905 120579) = 120595 (119905) + 119894120579 (0) equiv 120595 (119905) + 120579 (1199042120595 (119905)) 119860(119888) (119905 120579) = 119860 (119905) + 120579 (211990412059511988110158401015840) equiv 119860 (119905) + 120579 (1199042119860 (119905))

(31)

Finally the supersymmetric transformations (1199042) for all thebasic and auxiliary variables are listed as

1199042120601 = 1198941205951199042120595 = 01199042120595 = 119894119860 minus 211990411988110158401199042119881 = 11989412059511988110158401199042119860 = 211990412059511988110158401015840

(32)

It is important to point out here that we have the followingmapping between the Grassmannian derivative (120597120579) and thesymmetry transformation 1199042

120597120597120579Ω(119888) (119905 120579 120579)

10038161003816100381610038161003816120579=0 = 120597120597120579Ω(119888) (119905 120579) = 1199042Ω (119905) (33)

where Ω(119888)(119905 120579) is the generic chiral supervariables obtainedafter the application of supersymmetric invariant restrictionsand Ω(119905) denotes the basic variables of our present QMtheory The above equation captures the geometrical inter-pretation of transformation (1199042) in terms of the Grassman-nian derivative (120597120579) because of the fact that the translationalong 120579-direction of chiral supervariable is equivalent to thesymmetry transformation (1199042) of the same basic variable Weobserve from (33) that the nilpotency of SUSY transformation1199042 (ie 11990422 = 0) can be generalized in terms of Grassmannianderivative 1205972120579 = 05 Invariance and Off-Shell Nilpotency

Supervariable Approach

It is interesting to note that by exploiting the expansions ofsupervariables (11) the Lagrangian in (2) can be expressed interms of the antichiral supervariables as

119871 997904rArr 119871(119886119888) = 119894 ( Φ(119886119888) + 1199041015840(119886119888))119860(119886119888) + 12119860(119886119888)119860(119886119888) minus 119894Ψ(119886119888) ( Ψ(119886119888) + 119904Ψ(119886119888)10158401015840(119886119888)) (34)

In the earlier section we have shown that SUSY transforma-tion (1199041) and translational generator (120597120579) are geometricallyrelated to each other (ie 1199041 harr 120597120579) As a consequence one can

also capture the invariance of the Lagrangian in the followingfashion 120597

120597120579119871(119886119888) = 0 lArrrArr 1199041119871 = 0 (35)


Similarly Lagrangian (2) can also be written in terms of thechiral supervariables as

119871 997904rArr 119871(119888) = 119894 ( Φ(119888) + 1199041015840(119888))119860(119888) + 12119860(119888)119860(119888) minus 119894Ψ(119888) ( Ψ(119888) + 119904Ψ(119888)10158401015840(119888)) (36)

Since the fermionic symmetry (1199042) is geometrically connectedwith the translational generator (120597120579) therefore the invarianceof the Lagrangian can be geometrically interpreted as follows

120597120597120579119871(119888) =

119889119889119905 [minus120595119860] lArrrArr 1199042119871 = 119889

119889119905 [minus120595119860] (37)

As a result the action integral 119878 = int 119889119905119871(119888)|120579=0 remainsinvariant

We point out that the conserved charges 119876 and 119876corresponding to the continuous symmetry transformations1199041 and 1199042 can also be expressed as

119876 = 1199041 [minus119894120595120595] = minus120595119860 lArrrArr 119876 = 120597120597120579 [minus119894Ψ

(119886119888)Ψ(119886119888)] 119876 = 1199042 [119894120595120595] = minus120595 [119860 + 1198941199041198811015840] lArrrArr 119876 = 120597

120597120579 [119894Ψ(119888)Ψ(119888)]

(38)

The nilpotency properties of the above charges can be shownin a straightforward manner with the help of symmetryproperties

1199041119876 = +119894 119876 119876 = 0 997904rArr 1198762 = 01199042119876 = +119894 119876119876 = 0 997904rArr 1198762 = 0 (39)

In the language of translational generators these propertiescan be written as 120597120579119876 = 0 rArr 1198762 = 0 and 120597120579119876 = 0 rArr1198762 = 0 These relations hold due to the nilpotency of theGrassmannian derivatives (ie 1205972

120579= 0 1205972120579 = 0)

6 N = 2 SUSY Algebra and Its Interpretation

We observe that under the discrete symmetry

119905 997888rarr 119905120601 997888rarr 120601120595 997888rarr 120595120595 997888rarr 120595119904 997888rarr minus119904119860 997888rarr 119860 + 21198941199041198811015840

(40)

the Lagrangian (119871) transforms as 119871 rarr 119871+(119889119889119905)[119894120595120595minus2119904119881]Hence action integral (2) of the SUSY QM system remainsinvariant It is to be noted that the above discrete symmetrytransformations are important because they relate the twoSUSY transformations (1199041 1199042)

1199042Ω = plusmn lowast 1199041 lowast Ω (41)

where Ω is the generic variables present in the model It isto be noted that generally the (plusmn) signs are governed by thetwo successive operations of the discrete symmetry on thevariables as

lowast (lowastΩ) = plusmnΩ (42)

In the present case only the (+) sign will occur for all thevariables (ie Ω = 120601 120595 120595 119860) It can be easily seen thatrelationship (41) is analogous to the relationship 120575 = plusmn lowast119889lowast of differential geometry (where 119889 and 120575 are the exteriorand coexterior derivative resp and (lowast) is the Hodge dualityoperation)

We now focus on the physical identifications of the deRham cohomological operators of differential geometry interms of the symmetry transformations It can be explicitlychecked that the continuous symmetry transformationstogether with discrete symmetry for our SUSY QM modelsatisfy the following algebra [17ndash20 22ndash24]

11990421 = 011990422 = 0

1199041 1199042 = (1199041 + 1199042)2 = 119904120596[1199041 119904120596] = 0[1199042 119904120596] = 0

1199042 = plusmn lowast 1199041lowast

(43)

which is identical to the algebra obeyed by the de Rhamcohomological operators (119889 120575 Δ) [25ndash29]

1198892 = 01205752 = 0

119889 120575 = (119889 + 120575)2 = Δ[119889 Δ] = 0[120575 Δ] = 0

120575 = plusmn ⋆ 119889 ⋆

(44)

Here Δ is the Laplacian operator From (43) and (44) wecan identify the exterior derivative with 1199041 and coexteriorderivative 120575 with 1199042 The discrete symmetry (40) providesthe analogue of Hodge duality (⋆) operation of differentialgeometry In fact there is a one-to-one mapping between thesymmetry transformations and the de Rham cohomological


operators It is also clear from (43) and (44) that the bosonicsymmetry (119904120596) and Laplacian operator (Δ) are the Casimiroperators of the algebra given in (43) and (44) respectivelyThus our present N = 2 SUSY model provides a model forHodge theory Furthermore a similar algebra given in (44) isalso satisfied by the conserved charges 119876 119876 and 119876120596

1198762 = 01198762 = 0

119876 119876 = 119876120596 = 2119894119867[119876119867] = 0[119876119867] = 0

(45)

In the above we have used the canonical quantum(anti)commutation relations [120601 119860] = 1 and 120595 120595 = 1It is important to mention here that the bosonic charge (iethe Hamiltonian of the theory modulo a 2119894-factor) is theCasimir operator in algebra (45)

Some crucial properties related to the de Rham cohomo-logical operators (119889 120575 Δ) can be captured by these chargesFor instance we observe from (45) that the Hamiltonian isthe Casimir operator of the algebraThus it can be easily seenthat 119867119876 = 119876119867 implies that 119876119867minus1 = 119867minus1119876 if the inverseof the Hamiltonian exists Since we are dealing with the no-singular Hamiltonian we presume that the Casimir operatorhas its well-defined inverse value By exploiting (45) it can beseen that

[119876119876119867 119876] = 119876

[119876119876119867 119876] = minus119876

[119876119876119867 119876] = minus119876

[119876119876119867 119876] = 119876

(46)

Let us define an eigenvalue equation (119876119876119867)|120594⟩119901 = 119901|120594⟩119901where |120594⟩119901 is the quantumHilbert state with eigenvalue 119901 Byusing algebra (46) one can verify the following

(119876119876119867 )119876 1003816100381610038161003816120594⟩119901 = (119901 + 1) 1003816100381610038161003816120594⟩119901

(119876119876119867 )119876 1003816100381610038161003816120594⟩119901 = (119901 minus 1) 1003816100381610038161003816120594⟩119901

(119876119876119867 )119867 1003816100381610038161003816120594⟩119901 = 119901 1003816100381610038161003816120594⟩119901

(47)

As a consequence of (47) it is evident that 119876|120594⟩119901 119876|120594⟩119901and 119867|120594⟩119901 have the eigenvalues (119901 + 1) (119901 minus 1) and 119901respectively with respect to to the operator 119876119876119867

The above equation provides a connection between theconserved charges (119876 119876119867) and de Rham cohomologicaloperators (119889 120575 Δ) because as we know the action of 119889 ona given form increases the degree of the form by onewhereas application of 120575 decreases the degree by one unit andoperatorΔ keeps the degree of a form intactThese importantproperties can be realized by the charges (119876 119876 119876120596) wherethe eigenvalues and eigenfunctions play the key role [22]

7 Conclusions

In summary exploiting the supervariable approach we havederived the off-shell nilpotent symmetry transformations forthe N = 2 SUSY QM system This has been explicatedthrough the 1D SUSY invariant quantities which remainunaffected due to the presence of the Grassmannian vari-ables 120579 and 120579 Furthermore we have provided the geomet-rical interpretation of the SUSY transformations (1199041 and1199042) in terms of the translational generators (120597120579 and 120597120579)along the Grassmannian directions 120579 and 120579 respectivelyFurther we have expressed the Lagrangian in terms ofthe (anti)chiral supervariables and the invariance of theLagrangian under continuous transformations (1199041 1199042) hasbeen shown within the translations generators along (120579 120579)-directionsThe conserved SUSY charges corresponding to thefermionic symmetry transformations have been expressed interms of (anti)chiral supervariables and the Grassmannianderivatives The nilpotency of fermionic charges has beencaptured geometrically within the framework of supervari-able approach by the Grassmannian derivatives

Finally we have shown that the algebra satisfied by thecontinuous symmetry transformations 1199041 1199042 and 119904120596 (andcorresponding charges) is exactly analogous to the Hodgealgebra obeyed by the de Rham cohomological operators(119889 120575 andΔ) of differential geometryThe discrete symmetryof the theory provides physical realization of the Hodgeduality (lowast) operation Thus the present N = 2 SUSY QMmodel provides a model for Hodge theory

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

Acknowledgments

The author would like to thank Professor Prasanta K Pani-grahi for reading the manuscript as well as for offering thevaluable inputs on the topic of the present investigation Theauthor is also thankful to Dr Rohit Kumar for his importantcomments during the preparation of the manuscript

References

[1] C Becchi A Rouet and R Stora ldquoThe abelian Higgs Kibblemodel unitarity of the S-operatorrdquo Physics Letters B vol 52 no3 pp 344ndash346 1974


[2] C Becchi A Rouet and R Stora ldquoRenormalization of theabelianHiggs-KibblemodelrdquoCommunications inMathematicalPhysics vol 42 pp 127ndash162 1975

[3] C Becchi A Rouet and R Stora ldquoRenormalization of gaugetheoriesrdquo Annals of Physics vol 98 no 2 pp 287ndash321 1976

[4] I V Tyutin ldquoGauge invariance in field theory and statisticalphysics in operator formalismrdquo Tech Rep FIAN-39 LebedevInstitute 1975

[5] J Thierry-Mieg ldquoGeometrical reinterpretation of FaddeevndashPopov ghost particles and BRS transformationsrdquo Journal ofMathematical Physics vol 21 no 12 1980

[6] JThierry-Mieg ldquoExplicit classical construction of the Faddeev-Popov ghost fieldrdquo Il Nuovo Cimento A vol 56 article 396 1980

[7] M Quiros F J de Urries J Hoyos M L Mazon and ERodriguez ldquoGeometrical structure of FaddeevndashPopov fieldsand invariance properties of gauge theoriesrdquo Journal of Math-ematical Physics vol 22 no 8 pp 1767ndash1774 1981

[8] L Bonora and M Tonin ldquoSuperfield formulation of extendedBRS symmetryrdquo Physics Letters B vol 98 no 1-2 pp 48ndash501981

[9] L Bonora P Pasti and M Tonin ldquoGeometric description ofextended BRS symmetry in superfield formulationrdquo Il NuovoCimento A Serie 11 vol 63 no 3 pp 353ndash364 1981

[10] R Delbourgo and P D Jarvis ldquoExtended BRS invariance andosp(42) supersymmetryrdquo Journal of Physics A Mathematicaland General vol 15 p 611 1981

[11] R Delbourgo P D Jarvis and G Thompson ldquoLocal OSp(42)supersymmetry and extended BRS transformations for gravityrdquoPhysics Letters B vol 109 no 1-2 pp 25ndash27 1982

[12] D S Hwang and C Lee ldquoNon-Abelian topological mass gener-ation in four dimensionsrdquo Journal of Mathematical Physics vol38 no 1 1997

[13] R P Malik ldquoNilpotent symmetries for a spinning relativis-tic particlein augmented superfield formalismrdquo The EuropeanPhysical Journal C vol 45 no 2 pp 513ndash524 2006

[14] R Malik ldquoUnique nilpotent symmetry transformations formatter fields in QED augmented superfield formalismrdquo TheEuropean Physical Journal C vol 47 no 1 pp 227ndash234 2006

[15] R P Malik ldquoA generalization of the horizontality condition inthe superfield approach to nilpotent symmetries for QED withcomplex scalar fieldsrdquo Journal of Physics A Mathematical andTheoretical vol 40 no 18 pp 4877ndash4893 2007

[16] R Malik ldquoAn alternative to the horizontality condition inthe superfield approach to BRST symmetriesrdquo The EuropeanPhysical Journal C vol 51 no 1 pp 169ndash177 2007

[17] S Krishna A Shukla and R P Malik ldquoGeneral N=2 supersym-metric quantum mechanical model supervariable approach toits off-shell nilpotent symmetriesrdquoAnnals of Physics vol 351 pp558ndash570 2014

[18] S Krishna A Shukla and R P Malik ldquoSupervariable approachto nilpotent symmetries of a couple of N = 2 supersymmetricquantummechanical modelsrdquo Canadian Journal of Physics vol92 no 12 pp 1623ndash1631 2014

[19] S Krishna and R P Malik ldquoA free N = 2 supersymmetricsystem novel symmetriesrdquo Europhysics Letters vol 109 no 3Article ID 31001 2015

[20] S Krishna and R P Malik ldquoN=2 SUSY symmetries for amoving charged particle under influence of a magnetic fieldsupervariable approachrdquo Annals of Physics vol 355 pp 204ndash216 2015

[21] D Birmingham M Blau M Rakowski and G ThompsonldquoTopological field theoryrdquo Physics Reports A vol 209 no 4-5pp 129ndash340 1991

[22] R Kumar and R P Malik ldquoSupersymmetric oscillator novelsymmetriesrdquo EPL (Europhysics Letters) vol 98 no 1 Article ID11002 2012

[23] R Kumar and R P Malik ldquoNovel discrete symmetries in thegeneral N = 2 supersymmetric quantum mechanical modelrdquoThe European Physical Journal C vol 73 article 2514 2013

[24] R P Malik and A Khare ldquoNovel symmetries in N = 2 super-symmetric quantummechanical modelsrdquoAnnals of Physics vol334 pp 142ndash156 2013

[25] N Nakanishi and I Ojima Covariant Operator Formalism ofGauge Theory and Quantum Gravity World Scientifc Singa-pore 1990

[26] T Eguchi P B Gilkey and A J Hanson ldquoGravitation gaugetheories and differential geometryrdquo Physics Reports vol 66 no6 pp 213ndash393 1980

[27] S Mukhi and N Mukunda Introduction to Topology Differen-tial Geometry and Group Theory for Physicists Wiley EasternPrivate Limited New Delhi India 1990

[28] K Nishijima ldquoThe Casimir operator in the representations ofBRS algebrardquo Progress of Theoretical Physics vol 80 no 5 pp897ndash904 1988

[29] S Deser A Gomberoff M Henneaux and C TeitelboimldquoDuality self-duality sources and charge quantization inabelian N-form theoriesrdquo Physics Letters B vol 400 no 1-2 pp80ndash86 1997

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


in nature whereas SUSY transformations are only nilpo-tent and the anticommutator of fermionic transformationsproduces an additional symmetry transformation of thetheory Due to this basic reason we are theoretically forcedto use the (anti)chiral supervariables generalized on the(1 1)-dimensional super-submanifolds of the full (1 2)-dimensional supermanifold The latter is parametrized bythe superspace coordinate 119885119872 = (119905 120579 120579) where 120579 120579 arethe Grassmannian variables and 119905 is the time-evolutionparameter

The contents of present investigation are organized asfollows In Section 2 we discuss the symmetry transforma-tions associated with the specific N = 2 SUSY QM modelIt is to be noted that there are three continuous symmetriesassociated with this particular model in which two of themare fermionic and one is bosonic in nature Section 3 isdevoted to the derivation of one of the fermionic transfor-mations by using the antichiral supervariable approach Wederive the second SUSY fermionic symmetry by exploitingthe chiral supervariable approach in Section 4 In Section 5the Lagrangian of the model is presented in terms of the(anti)chiral supervariables and the geometrical interpretationfor invariance of the Lagrangian in terms of the Grass-mannian derivatives (120597120579 and 120597120579) is explicated Furthermorewe also represent the charges corresponding to the con-tinuous symmetry transformations in terms of (anti)chiralsupervariables In Section 6 we show that the fermionicSUSY symmetry transformations satisfy the N = 2 SUSYalgebra which is identical to the Hodge algebra obeyed bythe cohomological operators of differential geometry Thuswe show that this particular N = 2 SUSY QM model is anexample of Hodge theory Finally we draw conclusions inSection 7 with remarks

2 Preliminaries A specific N = 2SUSY QM Model

We begin with the action of a specific (0 + 1)-dimensionalN = 2 QMmodel [21]

119878 = int1198891199051198710= int119889119905 [(119889120601119889119905 + 119904120597119881120597120601 )

2 minus 119894120595( 119889119889119905 + 119904 1205972119881120597120601120597120601)120595]

(1)

where the bosonic variable 120601 and fermionic variables 120595 120595are the functions of time-evolution parameter 119905 119881(120601) is ageneral potential function and 119904 is an independent constantparameter For algebraic convenience we linearize the firstterm in (1) by introducing an auxiliary variable 119860 As aconsequence the action can be written as (henceforth wedenote 120601 = 119889120601119889119905 1198811015840 = 120597119881120597120601 11988110158401015840 = 1205972119881120597120601120597120601 in the text)

119878 = int119889119905119871

= int119889119905 [119894 ( 120601 + 1199041198811015840)119860 + 11986022 minus 119894120595 ( + 11990412059511988110158401015840)]

(2)

where 119860 = minus119894( 120601 + 1199041198811015840) is the equation of motion Using thisexpression for 119860 in (2) one can recover the original action

For the present QM system we have the following off-shell nilpotent (11990421 = 0 11990422 = 0) SUSY transformations (wepoint out that the SUSY transformations in (3) differ by anoverall 119894-factor from [21])

1199041120601 = 1198941205951199041120595 = 01199041120595 = 1198941198601199041119860 = 01199042120601 = 1198941205951199042120595 = 01199042120595 = 119894119860 minus 211990411988110158401199042119860 = 211990412059511988110158401015840

(3)

Under the above symmetry transformations the Lagrangianin (2) transforms as

1199041119871 = 01199042119871 = 119889

119889119905 (minus119860120595) (4)

Thus the action integral remains invariant (ie 1199041119878 = 0 1199042119878 =0) According to Noetherrsquos theorem the continuous SUSYsymmetry transformations 1199041 and 1199042 lead to the followingconserved charges respectively

119876 = minus120595119860119876 = minus120595 [119860 + 21198941199041198811015840] (5)

The conservation of the SUSY charges (ie = 0 = 0)can be proven by exploiting the following Euler-Lagrangeequations of motion

= 11990411986011988110158401015840 minus 119904120595120595119881101584010158401015840 = minus11990412059511988110158401015840119860 = minus119894 ( 120601 + 1199041198811015840) = minus11990412059511988110158401015840

(6)

It turns out that these charges are the generators of SUSYtransformations (3) One can explicitly check that the follow-ing relations are true

1199041Φ = 119894 [Φ119876]plusmn 1199042Φ = 119894 [Φ119876]

plusmnΦ = 120601 119860 120595 120595

(7)





(8)


119904120596119871 = (11990411199042 + 11990421199041) 119871 = 119889119889119905 (minus1198941198602) (9)


119876120596 = minus1198941198602 + 21199041198601198811015840 + 211990412059512059511988110158401015840




119904120596Φ = 1199041 1199042Φ = (2119894) Φ Φ = 120601 119860 120595 120595 (11)






(12)





Φ (119905 120579) Ψ (119905 120579) = 120601 (119905) 120595 (119905) 997904rArr 1198911120595 = 0Φ (119905 120579) Ψ (119905 120579) = 120601 (119905) (119905) 997904rArr 1198911 = 0 (14)



(15)


119881 (120601) 997888rarr (Φ(119886119888)) = (119886119888) (120601 + 120579 (119894120595)) = 119881 (120601) + 120579 (1198941205951198811015840) equiv 119881 (120601) + 120579 (1199041119881 (120601)) (16)




1199041 [119894 ( 120601 + 1199041198811015840)119860 minus 119894120595 ( + 11990412059511988110158401015840)] = 0 (17)


[119894 ( Φ(119886119888) + 1199041015840(119886119888))119860(119886119888)

minus 119894Ψ ( Ψ(119886119888) + 119904Ψ(119886119888)10158401015840(119886119888))] = [119894 ( 120601 + 1199041198811015840)119860minus 119894120595 ( + 11990412059511988110158401015840)]

(18)



(19)


1199041120601 = 1198941205951199041120595 = 01199041120595 = 1198941198601199041119860 = 01199041119881 = 1198941205951198811015840

(20)


120597120597120579Ω(119886119888) (119905 120579 120579)10038161003816100381610038161003816120579=0 equiv 120597

120597120579Ω(119886119888) (119905 120579) = 1199041Ω (119905) (21)


120579= 0) implies 11990421 = 0




(22)



Ψ (119905 120579 120579)100381610038161003816100381610038161003816120579=0 equiv Ψ (119905 120579) = 120595 (119905) 997904rArr 1198872 = 0 (23)


Φ (119905 120579) Ψ (119905 120579) = 120601 (119905) 120595 (119905) Φ (119905 120579) Ψ (119905 120579) = 120601 (119905) (119905)

(24)


1198911120595 = 01 = 0 (25)


Φ(119888) (119905 120579) = 120601 (119905) + 120579 (119894120595) equiv 120601 (119905) + 120579 (1199042120601 (119905)) Ψ(119888) (119905 120579) = 120595 (119905) + 119894120579 (0) equiv 120595 (119905) + 120579 (1199042120595 (119905))

(26)




119881 (120601) 997888rarr (Φ(119888)) = (119888) (120601 + 120579 (119894120595)) = 119881 (120601) + 120579 (1198941205951198811015840 (120601)) equiv 119881 (120601) + 1199042 (119881 (120601)) (27)



119894119860(119888) minus 2119904Ψ1015840(119888) = [119894119860 minus 21199041205951198811015840] (28)



1199042 [1198941199041198811015840119860 + 121198602 minus 11989411990412059512059511988110158401015840] = 0 (29)


1198941199041015840(119888)119860(119888) + 121198602(119888) minus 119894119904Ψ(119888)Ψ10158401015840(119888)

= 1198941199041198811015840119860 + 121198602 minus 11989411990412059512059511988110158401015840

(30)



Φ(119888) (119905 120579) = 120601 (119905) + 120579 (119894120595) equiv 120601 (119905) + 120579 (1199042120601 (119905)) Ψ(119888) (119905 120579) = 120595 (119905) + 120579 (119894119860 minus 21199041198811015840)


(31)


1199042120601 = 1198941205951199042120595 = 01199042120595 = 119894119860 minus 211990411988110158401199042119881 = 11989412059511988110158401199042119860 = 211990412059511988110158401015840

(32)


120597120597120579Ω(119888) (119905 120579 120579)

10038161003816100381610038161003816120579=0 = 120597120597120579Ω(119888) (119905 120579) = 1199042Ω (119905) (33)




119871 997904rArr 119871(119886119888) = 119894 ( Φ(119886119888) + 1199041015840(119886119888))119860(119886119888) + 12119860(119886119888)119860(119886119888) minus 119894Ψ(119886119888) ( Ψ(119886119888) + 119904Ψ(119886119888)10158401015840(119886119888)) (34)



120597120579119871(119886119888) = 0 lArrrArr 1199041119871 = 0 (35)



119871 997904rArr 119871(119888) = 119894 ( Φ(119888) + 1199041015840(119888))119860(119888) + 12119860(119888)119860(119888) minus 119894Ψ(119888) ( Ψ(119888) + 119904Ψ(119888)10158401015840(119888)) (36)


120597120597120579119871(119888) =

119889119889119905 [minus120595119860] lArrrArr 1199042119871 = 119889

119889119905 [minus120595119860] (37)




(119886119888)Ψ(119886119888)] 119876 = 1199042 [119894120595120595] = minus120595 [119860 + 1198941199041198811015840] lArrrArr 119876 = 120597

120597120579 [119894Ψ(119888)Ψ(119888)]

(38)


1199041119876 = +119894 119876 119876 = 0 997904rArr 1198762 = 01199042119876 = +119894 119876119876 = 0 997904rArr 1198762 = 0 (39)


120579= 0 1205972120579 = 0)




(40)







11990421 = 011990422 = 0

1199041 1199042 = (1199041 + 1199042)2 = 119904120596[1199041 119904120596] = 0[1199042 119904120596] = 0


(43)


1198892 = 01205752 = 0

119889 120575 = (119889 + 120575)2 = Δ[119889 Δ] = 0[120575 Δ] = 0

120575 = plusmn ⋆ 119889 ⋆

(44)




1198762 = 01198762 = 0

119876 119876 = 119876120596 = 2119894119867[119876119867] = 0[119876119867] = 0

(45)



[119876119876119867 119876] = 119876

[119876119876119867 119876] = minus119876

[119876119876119867 119876] = minus119876

[119876119876119867 119876] = 119876

(46)


(119876119876119867 )119876 1003816100381610038161003816120594⟩119901 = (119901 + 1) 1003816100381610038161003816120594⟩119901

(119876119876119867 )119876 1003816100381610038161003816120594⟩119901 = (119901 minus 1) 1003816100381610038161003816120594⟩119901

(119876119876119867 )119867 1003816100381610038161003816120594⟩119901 = 119901 1003816100381610038161003816120594⟩119901

(47)



7 Conclusions



Competing Interests


Acknowledgments


References




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







(8)


119904120596119871 = (11990411199042 + 11990421199041) 119871 = 119889119889119905 (minus1198941198602) (9)


119876120596 = minus1198941198602 + 21199041198601198811015840 + 211990412059512059511988110158401015840




119904120596Φ = 1199041 1199042Φ = (2119894) Φ Φ = 120601 119860 120595 120595 (11)






(12)





Φ (119905 120579) Ψ (119905 120579) = 120601 (119905) 120595 (119905) 997904rArr 1198911120595 = 0Φ (119905 120579) Ψ (119905 120579) = 120601 (119905) (119905) 997904rArr 1198911 = 0 (14)



(15)


119881 (120601) 997888rarr (Φ(119886119888)) = (119886119888) (120601 + 120579 (119894120595)) = 119881 (120601) + 120579 (1198941205951198811015840) equiv 119881 (120601) + 120579 (1199041119881 (120601)) (16)




1199041 [119894 ( 120601 + 1199041198811015840)119860 minus 119894120595 ( + 11990412059511988110158401015840)] = 0 (17)


[119894 ( Φ(119886119888) + 1199041015840(119886119888))119860(119886119888)

minus 119894Ψ ( Ψ(119886119888) + 119904Ψ(119886119888)10158401015840(119886119888))] = [119894 ( 120601 + 1199041198811015840)119860minus 119894120595 ( + 11990412059511988110158401015840)]

(18)



(19)


1199041120601 = 1198941205951199041120595 = 01199041120595 = 1198941198601199041119860 = 01199041119881 = 1198941205951198811015840

(20)


120597120597120579Ω(119886119888) (119905 120579 120579)10038161003816100381610038161003816120579=0 equiv 120597

120597120579Ω(119886119888) (119905 120579) = 1199041Ω (119905) (21)


120579= 0) implies 11990421 = 0




(22)



Ψ (119905 120579 120579)100381610038161003816100381610038161003816120579=0 equiv Ψ (119905 120579) = 120595 (119905) 997904rArr 1198872 = 0 (23)


Φ (119905 120579) Ψ (119905 120579) = 120601 (119905) 120595 (119905) Φ (119905 120579) Ψ (119905 120579) = 120601 (119905) (119905)

(24)


1198911120595 = 01 = 0 (25)


Φ(119888) (119905 120579) = 120601 (119905) + 120579 (119894120595) equiv 120601 (119905) + 120579 (1199042120601 (119905)) Ψ(119888) (119905 120579) = 120595 (119905) + 119894120579 (0) equiv 120595 (119905) + 120579 (1199042120595 (119905))

(26)




119881 (120601) 997888rarr (Φ(119888)) = (119888) (120601 + 120579 (119894120595)) = 119881 (120601) + 120579 (1198941205951198811015840 (120601)) equiv 119881 (120601) + 1199042 (119881 (120601)) (27)



119894119860(119888) minus 2119904Ψ1015840(119888) = [119894119860 minus 21199041205951198811015840] (28)



1199042 [1198941199041198811015840119860 + 121198602 minus 11989411990412059512059511988110158401015840] = 0 (29)


1198941199041015840(119888)119860(119888) + 121198602(119888) minus 119894119904Ψ(119888)Ψ10158401015840(119888)

= 1198941199041198811015840119860 + 121198602 minus 11989411990412059512059511988110158401015840

(30)



Φ(119888) (119905 120579) = 120601 (119905) + 120579 (119894120595) equiv 120601 (119905) + 120579 (1199042120601 (119905)) Ψ(119888) (119905 120579) = 120595 (119905) + 120579 (119894119860 minus 21199041198811015840)


(31)


1199042120601 = 1198941205951199042120595 = 01199042120595 = 119894119860 minus 211990411988110158401199042119881 = 11989412059511988110158401199042119860 = 211990412059511988110158401015840

(32)


120597120597120579Ω(119888) (119905 120579 120579)

10038161003816100381610038161003816120579=0 = 120597120597120579Ω(119888) (119905 120579) = 1199042Ω (119905) (33)




119871 997904rArr 119871(119886119888) = 119894 ( Φ(119886119888) + 1199041015840(119886119888))119860(119886119888) + 12119860(119886119888)119860(119886119888) minus 119894Ψ(119886119888) ( Ψ(119886119888) + 119904Ψ(119886119888)10158401015840(119886119888)) (34)



120597120579119871(119886119888) = 0 lArrrArr 1199041119871 = 0 (35)



119871 997904rArr 119871(119888) = 119894 ( Φ(119888) + 1199041015840(119888))119860(119888) + 12119860(119888)119860(119888) minus 119894Ψ(119888) ( Ψ(119888) + 119904Ψ(119888)10158401015840(119888)) (36)


120597120597120579119871(119888) =

119889119889119905 [minus120595119860] lArrrArr 1199042119871 = 119889

119889119905 [minus120595119860] (37)




(119886119888)Ψ(119886119888)] 119876 = 1199042 [119894120595120595] = minus120595 [119860 + 1198941199041198811015840] lArrrArr 119876 = 120597

120597120579 [119894Ψ(119888)Ψ(119888)]

(38)


1199041119876 = +119894 119876 119876 = 0 997904rArr 1198762 = 01199042119876 = +119894 119876119876 = 0 997904rArr 1198762 = 0 (39)


120579= 0 1205972120579 = 0)




(40)







11990421 = 011990422 = 0

1199041 1199042 = (1199041 + 1199042)2 = 119904120596[1199041 119904120596] = 0[1199042 119904120596] = 0


(43)


1198892 = 01205752 = 0

119889 120575 = (119889 + 120575)2 = Δ[119889 Δ] = 0[120575 Δ] = 0

120575 = plusmn ⋆ 119889 ⋆

(44)




1198762 = 01198762 = 0

119876 119876 = 119876120596 = 2119894119867[119876119867] = 0[119876119867] = 0

(45)



[119876119876119867 119876] = 119876

[119876119876119867 119876] = minus119876

[119876119876119867 119876] = minus119876

[119876119876119867 119876] = 119876

(46)


(119876119876119867 )119876 1003816100381610038161003816120594⟩119901 = (119901 + 1) 1003816100381610038161003816120594⟩119901

(119876119876119867 )119876 1003816100381610038161003816120594⟩119901 = (119901 minus 1) 1003816100381610038161003816120594⟩119901

(119876119876119867 )119867 1003816100381610038161003816120594⟩119901 = 119901 1003816100381610038161003816120594⟩119901

(47)



7 Conclusions



Competing Interests


Acknowledgments


References




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






1199041 [119894 ( 120601 + 1199041198811015840)119860 minus 119894120595 ( + 11990412059511988110158401015840)] = 0 (17)


[119894 ( Φ(119886119888) + 1199041015840(119886119888))119860(119886119888)

minus 119894Ψ ( Ψ(119886119888) + 119904Ψ(119886119888)10158401015840(119886119888))] = [119894 ( 120601 + 1199041198811015840)119860minus 119894120595 ( + 11990412059511988110158401015840)]

(18)



(19)


1199041120601 = 1198941205951199041120595 = 01199041120595 = 1198941198601199041119860 = 01199041119881 = 1198941205951198811015840

(20)


120597120597120579Ω(119886119888) (119905 120579 120579)10038161003816100381610038161003816120579=0 equiv 120597

120597120579Ω(119886119888) (119905 120579) = 1199041Ω (119905) (21)


120579= 0) implies 11990421 = 0




(22)



Ψ (119905 120579 120579)100381610038161003816100381610038161003816120579=0 equiv Ψ (119905 120579) = 120595 (119905) 997904rArr 1198872 = 0 (23)


Φ (119905 120579) Ψ (119905 120579) = 120601 (119905) 120595 (119905) Φ (119905 120579) Ψ (119905 120579) = 120601 (119905) (119905)

(24)


1198911120595 = 01 = 0 (25)


Φ(119888) (119905 120579) = 120601 (119905) + 120579 (119894120595) equiv 120601 (119905) + 120579 (1199042120601 (119905)) Ψ(119888) (119905 120579) = 120595 (119905) + 119894120579 (0) equiv 120595 (119905) + 120579 (1199042120595 (119905))

(26)




119881 (120601) 997888rarr (Φ(119888)) = (119888) (120601 + 120579 (119894120595)) = 119881 (120601) + 120579 (1198941205951198811015840 (120601)) equiv 119881 (120601) + 1199042 (119881 (120601)) (27)



119894119860(119888) minus 2119904Ψ1015840(119888) = [119894119860 minus 21199041205951198811015840] (28)



1199042 [1198941199041198811015840119860 + 121198602 minus 11989411990412059512059511988110158401015840] = 0 (29)


1198941199041015840(119888)119860(119888) + 121198602(119888) minus 119894119904Ψ(119888)Ψ10158401015840(119888)

= 1198941199041198811015840119860 + 121198602 minus 11989411990412059512059511988110158401015840

(30)



Φ(119888) (119905 120579) = 120601 (119905) + 120579 (119894120595) equiv 120601 (119905) + 120579 (1199042120601 (119905)) Ψ(119888) (119905 120579) = 120595 (119905) + 120579 (119894119860 minus 21199041198811015840)


(31)


1199042120601 = 1198941205951199042120595 = 01199042120595 = 119894119860 minus 211990411988110158401199042119881 = 11989412059511988110158401199042119860 = 211990412059511988110158401015840

(32)


120597120597120579Ω(119888) (119905 120579 120579)

10038161003816100381610038161003816120579=0 = 120597120597120579Ω(119888) (119905 120579) = 1199042Ω (119905) (33)




119871 997904rArr 119871(119886119888) = 119894 ( Φ(119886119888) + 1199041015840(119886119888))119860(119886119888) + 12119860(119886119888)119860(119886119888) minus 119894Ψ(119886119888) ( Ψ(119886119888) + 119904Ψ(119886119888)10158401015840(119886119888)) (34)



120597120579119871(119886119888) = 0 lArrrArr 1199041119871 = 0 (35)



119871 997904rArr 119871(119888) = 119894 ( Φ(119888) + 1199041015840(119888))119860(119888) + 12119860(119888)119860(119888) minus 119894Ψ(119888) ( Ψ(119888) + 119904Ψ(119888)10158401015840(119888)) (36)


120597120597120579119871(119888) =

119889119889119905 [minus120595119860] lArrrArr 1199042119871 = 119889

119889119905 [minus120595119860] (37)




(119886119888)Ψ(119886119888)] 119876 = 1199042 [119894120595120595] = minus120595 [119860 + 1198941199041198811015840] lArrrArr 119876 = 120597

120597120579 [119894Ψ(119888)Ψ(119888)]

(38)


1199041119876 = +119894 119876 119876 = 0 997904rArr 1198762 = 01199042119876 = +119894 119876119876 = 0 997904rArr 1198762 = 0 (39)


120579= 0 1205972120579 = 0)




(40)







11990421 = 011990422 = 0

1199041 1199042 = (1199041 + 1199042)2 = 119904120596[1199041 119904120596] = 0[1199042 119904120596] = 0


(43)


1198892 = 01205752 = 0

119889 120575 = (119889 + 120575)2 = Δ[119889 Δ] = 0[120575 Δ] = 0

120575 = plusmn ⋆ 119889 ⋆

(44)




1198762 = 01198762 = 0

119876 119876 = 119876120596 = 2119894119867[119876119867] = 0[119876119867] = 0

(45)



[119876119876119867 119876] = 119876

[119876119876119867 119876] = minus119876

[119876119876119867 119876] = minus119876

[119876119876119867 119876] = 119876

(46)


(119876119876119867 )119876 1003816100381610038161003816120594⟩119901 = (119901 + 1) 1003816100381610038161003816120594⟩119901

(119876119876119867 )119876 1003816100381610038161003816120594⟩119901 = (119901 minus 1) 1003816100381610038161003816120594⟩119901

(119876119876119867 )119867 1003816100381610038161003816120594⟩119901 = 119901 1003816100381610038161003816120594⟩119901

(47)



7 Conclusions



Competing Interests


Acknowledgments


References




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119881 (120601) 997888rarr (Φ(119888)) = (119888) (120601 + 120579 (119894120595)) = 119881 (120601) + 120579 (1198941205951198811015840 (120601)) equiv 119881 (120601) + 1199042 (119881 (120601)) (27)



119894119860(119888) minus 2119904Ψ1015840(119888) = [119894119860 minus 21199041205951198811015840] (28)



1199042 [1198941199041198811015840119860 + 121198602 minus 11989411990412059512059511988110158401015840] = 0 (29)


1198941199041015840(119888)119860(119888) + 121198602(119888) minus 119894119904Ψ(119888)Ψ10158401015840(119888)

= 1198941199041198811015840119860 + 121198602 minus 11989411990412059512059511988110158401015840

(30)



Φ(119888) (119905 120579) = 120601 (119905) + 120579 (119894120595) equiv 120601 (119905) + 120579 (1199042120601 (119905)) Ψ(119888) (119905 120579) = 120595 (119905) + 120579 (119894119860 minus 21199041198811015840)


(31)


1199042120601 = 1198941205951199042120595 = 01199042120595 = 119894119860 minus 211990411988110158401199042119881 = 11989412059511988110158401199042119860 = 211990412059511988110158401015840

(32)


120597120597120579Ω(119888) (119905 120579 120579)

10038161003816100381610038161003816120579=0 = 120597120597120579Ω(119888) (119905 120579) = 1199042Ω (119905) (33)




119871 997904rArr 119871(119886119888) = 119894 ( Φ(119886119888) + 1199041015840(119886119888))119860(119886119888) + 12119860(119886119888)119860(119886119888) minus 119894Ψ(119886119888) ( Ψ(119886119888) + 119904Ψ(119886119888)10158401015840(119886119888)) (34)



120597120579119871(119886119888) = 0 lArrrArr 1199041119871 = 0 (35)



119871 997904rArr 119871(119888) = 119894 ( Φ(119888) + 1199041015840(119888))119860(119888) + 12119860(119888)119860(119888) minus 119894Ψ(119888) ( Ψ(119888) + 119904Ψ(119888)10158401015840(119888)) (36)


120597120597120579119871(119888) =

119889119889119905 [minus120595119860] lArrrArr 1199042119871 = 119889

119889119905 [minus120595119860] (37)




(119886119888)Ψ(119886119888)] 119876 = 1199042 [119894120595120595] = minus120595 [119860 + 1198941199041198811015840] lArrrArr 119876 = 120597

120597120579 [119894Ψ(119888)Ψ(119888)]

(38)


1199041119876 = +119894 119876 119876 = 0 997904rArr 1198762 = 01199042119876 = +119894 119876119876 = 0 997904rArr 1198762 = 0 (39)


120579= 0 1205972120579 = 0)




(40)







11990421 = 011990422 = 0

1199041 1199042 = (1199041 + 1199042)2 = 119904120596[1199041 119904120596] = 0[1199042 119904120596] = 0


(43)


1198892 = 01205752 = 0

119889 120575 = (119889 + 120575)2 = Δ[119889 Δ] = 0[120575 Δ] = 0

120575 = plusmn ⋆ 119889 ⋆

(44)




1198762 = 01198762 = 0

119876 119876 = 119876120596 = 2119894119867[119876119867] = 0[119876119867] = 0

(45)



[119876119876119867 119876] = 119876

[119876119876119867 119876] = minus119876

[119876119876119867 119876] = minus119876

[119876119876119867 119876] = 119876

(46)


(119876119876119867 )119876 1003816100381610038161003816120594⟩119901 = (119901 + 1) 1003816100381610038161003816120594⟩119901

(119876119876119867 )119876 1003816100381610038161003816120594⟩119901 = (119901 minus 1) 1003816100381610038161003816120594⟩119901

(119876119876119867 )119867 1003816100381610038161003816120594⟩119901 = 119901 1003816100381610038161003816120594⟩119901

(47)



7 Conclusions



Competing Interests


Acknowledgments


References




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119871 997904rArr 119871(119888) = 119894 ( Φ(119888) + 1199041015840(119888))119860(119888) + 12119860(119888)119860(119888) minus 119894Ψ(119888) ( Ψ(119888) + 119904Ψ(119888)10158401015840(119888)) (36)


120597120597120579119871(119888) =

119889119889119905 [minus120595119860] lArrrArr 1199042119871 = 119889

119889119905 [minus120595119860] (37)




(119886119888)Ψ(119886119888)] 119876 = 1199042 [119894120595120595] = minus120595 [119860 + 1198941199041198811015840] lArrrArr 119876 = 120597

120597120579 [119894Ψ(119888)Ψ(119888)]

(38)


1199041119876 = +119894 119876 119876 = 0 997904rArr 1198762 = 01199042119876 = +119894 119876119876 = 0 997904rArr 1198762 = 0 (39)


120579= 0 1205972120579 = 0)




(40)







11990421 = 011990422 = 0

1199041 1199042 = (1199041 + 1199042)2 = 119904120596[1199041 119904120596] = 0[1199042 119904120596] = 0


(43)


1198892 = 01205752 = 0

119889 120575 = (119889 + 120575)2 = Δ[119889 Δ] = 0[120575 Δ] = 0

120575 = plusmn ⋆ 119889 ⋆

(44)




1198762 = 01198762 = 0

119876 119876 = 119876120596 = 2119894119867[119876119867] = 0[119876119867] = 0

(45)



[119876119876119867 119876] = 119876

[119876119876119867 119876] = minus119876

[119876119876119867 119876] = minus119876

[119876119876119867 119876] = 119876

(46)


(119876119876119867 )119876 1003816100381610038161003816120594⟩119901 = (119901 + 1) 1003816100381610038161003816120594⟩119901

(119876119876119867 )119876 1003816100381610038161003816120594⟩119901 = (119901 minus 1) 1003816100381610038161003816120594⟩119901

(119876119876119867 )119867 1003816100381610038161003816120594⟩119901 = 119901 1003816100381610038161003816120594⟩119901

(47)



7 Conclusions



Competing Interests


Acknowledgments


References




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1198762 = 01198762 = 0

119876 119876 = 119876120596 = 2119894119867[119876119867] = 0[119876119867] = 0

(45)



[119876119876119867 119876] = 119876

[119876119876119867 119876] = minus119876

[119876119876119867 119876] = minus119876

[119876119876119867 119876] = 119876

(46)


(119876119876119867 )119876 1003816100381610038161003816120594⟩119901 = (119901 + 1) 1003816100381610038161003816120594⟩119901

(119876119876119867 )119876 1003816100381610038161003816120594⟩119901 = (119901 minus 1) 1003816100381610038161003816120594⟩119901

(119876119876119867 )119867 1003816100381610038161003816120594⟩119901 = 119901 1003816100381610038161003816120594⟩119901

(47)



7 Conclusions



Competing Interests


Acknowledgments


References




































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

A Specific N =2Supersymmetric Quantum Mechanical Model ... · modern physics because three out of four fundamental interactions of nature are governed by the gauge theory. TheBecchi-Rouet-Stora-Tyutin(BRST)formalismisoneof