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Advances in High Energy Physics
Volume 2017 (2017), Article ID 1403937, 8 pages
A Specific Supersymmetric Quantum Mechanical Model: Supervariable Approach
Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, India
Correspondence should be addressed to Aradhya Shukla
Received 6 August 2016; Accepted 5 January 2017; Published 19 February 2017
Academic Editor: Shi-Hai Dong
Copyright © 2017 Aradhya Shukla. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.
By exploiting the supersymmetric invariant restrictions on the chiral and antichiral supervariables, we derive the off-shell nilpotent symmetry transformations for a specific (0 + 1)-dimensional supersymmetric quantum mechanical model which is considered on a (1, 2)-dimensional supermanifold (parametrized by a bosonic variable and a pair of Grassmannian variables ()). We also provide the geometrical meaning to the symmetry transformations. Finally, we show that this specific SUSY quantum mechanical model is a model for Hodge theory.
Gauge theory is one of the most important theories of modern physics because three out of four fundamental interactions of nature are governed by the gauge theory. The Becchi-Rouet-Stora-Tyutin (BRST) formalism is one of the systematic approaches to covariantly quantize any -form gauge theories, where the local gauge symmetry of a given theory is traded with the “quantum” gauge (i.e., (anti-)BRST) symmetry transformations [1–4]. It is important to point out that the (anti-)BRST symmetries are nilpotent and absolutely anticommuting in nature. One of the unique, elegant, and geometrically rich methods to derive these (anti-)BRST transformations is the superfield formalism, where the horizontality condition (HC) plays an important role [5–12]. This HC is a useful tool to derive the BRST, as well as anti-BRST symmetry transformations for any (non-)Abelian -form gauge theory, where no interaction between the gauge and matter fields is present.
For the derivation of a full set of (anti-)BRST symmetry transformations in the case of interacting gauge theories, a powerful method known as augmented version of superfield formalism has been developed in a set of papers [13–16]. In augmented superfield formalism, some conditions named gauge invariant restrictions (GIRs), in addition to the HC, have been imposed to obtain the off-shell nilpotent and absolutely anticommuting (anti-)BRST transformations. It is worthwhile to mention here that this technique has also been applied in case of some supersymmetric (SUSY) quantum mechanical (QM) models to derive the off-shell nilpotent SUSY symmetry transformations [17–20]. These SUSY transformations have been derived by using the supersymmetric invariant restrictions (SUSYIRs) and it has been observed that the SUSYIRs are the generalizations of the GIRs, in case of SUSY QM theory.
The aim of present investigation is to explore and apply the augmented version of HC to a new SUSY QM model which is different from the earlier models present in the literature. In our present endeavor, we derive the off-shell nilpotent SUSY transformations for a specific SUSY QM model by exploiting the potential and power of the SUSYIRs. The additional reason behind our present investigation is to take one more step forward in the direction of the confirmation of SUSYIRs (i.e., generalization of augmented superfield formalism) as a powerful technique for the derivation of SUSY transformations for any general SUSY QM system.
One of the key differences between the (anti-)BRST and SUSY symmetry transformations is that the (anti-)BRST symmetries are nilpotent as well as absolutely anticommuting in nature, whereas SUSY transformations are only nilpotent and the anticommutator of fermionic transformations produces an additional symmetry transformation of the theory. Due to this basic reason, we are theoretically forced to use the (anti)chiral supervariables generalized on the (1, 1)-dimensional super-submanifolds of the full (1, 2)-dimensional supermanifold. The latter is parametrized by the superspace coordinate , where are the Grassmannian variables and is the time-evolution parameter.
The contents of present investigation are organized as follows. In Section 2, we discuss the symmetry transformations associated with the specific SUSY QM model. It is to be noted that there are three continuous symmetries associated with this particular model, in which two of them are fermionic and one is bosonic in nature. Section 3 is devoted to the derivation of one of the fermionic transformations by using the antichiral supervariable approach. We derive the second SUSY fermionic symmetry by exploiting the chiral supervariable approach in Section 4. In Section 5, the Lagrangian of the model is presented in terms of the (anti)chiral supervariables and the geometrical interpretation for invariance of the Lagrangian in terms of the Grassmannian derivatives ( and ) is explicated. Furthermore, we also represent the charges corresponding to the continuous symmetry transformations in terms of (anti)chiral supervariables. In Section 6, we show that the fermionic SUSY symmetry transformations satisfy the SUSY algebra, which is identical to the Hodge algebra obeyed by the cohomological operators of differential geometry. Thus, we show that this particular SUSY QM model is an example of Hodge theory. Finally, we draw conclusions in Section 7, with remarks.
2. Preliminaries: A specific SUSY QM Model
We begin with the action of a specific (0 + 1)-dimensional QM model :where the bosonic variable and fermionic variables are the functions of time-evolution parameter , is a general potential function, and is an independent constant parameter. For algebraic convenience, we linearize the first term in (1) by introducing an auxiliary variable . As a consequence, the action can be written as (henceforth, we denote in the text)where is the equation of motion. Using this expression for in (2), one can recover the original action.
For the present QM system, we have the following off-shell nilpotent SUSY transformations (we point out that the SUSY transformations in (3) differ by an overall -factor from ):Under the above symmetry transformations, the Lagrangian in (2) transforms asThus, the action integral remains invariant (i.e., ). According to Noether’s theorem, the continuous SUSY symmetry transformations and lead to the following conserved charges, respectively:The conservation of the SUSY charges (i.e., ) can be proven by exploiting the following Euler-Lagrange equations of motion:It turns out that these charges are the generators of SUSY transformations (3). One can explicitly check that the following relations are true: where the subscripts , on the square brackets, deal with the (anti)commutator depending on the variables being fermionic/bosonic in nature.
It is to be noted that the anticommutator of the fermionic SUSY transformations ( and ) leads to a bosonic symmetry :The application of bosonic symmetry on the Lagrangian produces total time derivative:Thus, according to Noether’s theorem, the above continuous bosonic symmetry leads to a bosonic conserved charge as follows:where are the canonical momenta corresponding to the variables , respectively. It is clear that the bosonic charge is the Hamiltonian (modulo a constant -factor) of our present model.
One of the important features of SUSY transformations is that the application of this bosonic symmetry must produce the time translation of the variable (modulo a constant -factor), which can be checked as where, in order to prove the sanctity of this equation, we have used the equations of motion mentioned in (6).
3. Off-Shell Nilpotent SUSY Transformations: Antichiral Supervariable Approach
In order to derive the continuous transformation , we shall focus on the (1, 1)-dimensional super-submanifold (of general (1, 2)-dimensional supermanifold) parameterized by the supervariable . For this purpose, we impose supersymmetric invariant restrictions (SUSYIRs) on the antichiral supervariables. We then generalize the basic (explicit dependent) variables to their antichiral supervariable counterparts:where and are the fermionic and bosonic secondary variables, respectively.
It is observed from (3) that (i.e., both and are invariant under ). Therefore, we demand that both variables should remain unchanged due to the presence of Grassmannian variable . As a result of the above restrictions, we obtainWe further point out that and due to the fermionic nature of (i.e., ). Thus, these restrictions yieldThe trivial solution for the above relationships is ; for algebraic convenience, we choose . Here, the -factor has been taken due to the convention we have adopted for the present SUSY QM theory. Substituting the values of the secondary variables in the expansions of antichiral supervariables (12), one obtainsThe superscript in the above represents the antichiral supervariables, obtained after the application of SUSIRs. Furthermore, we note that the 1D potential function can be generalized to onto (1, 1)-dimensional super-submanifold as where we have used the expression of antichiral supervariable as given in (15).
In order to find out the SUSY transformation for , it can be checked that the application of on the following vanishes:As a consequence, the above can be used as a SUSYIR and we replace the ordinary variables by their antichiral supervariables asAfter doing some trivial computations, we obtain . Recollecting all the value of secondary variables and substituting them into (12), we finally obtain the following antichiral supervariable expansions:Finally, we have derived explicitly the SUSY transformation for all the variables by exploiting SUSY invariant restrictions on the antichiral supervariables. These symmetry transformations are It is worthwhile to mention here that, for the antichiral supervariable expansions given in (11), we have the following relationship between the Grassmannian derivative and SUSY transformations : where is the generic supervariable obtained by exploiting the SUSY invariant restriction on the antichiral supervariables. It is easy to check from the above equation that the symmetry transformation for any generic variable is equal to the translation along the -direction of the antichiral supervariable. Furthermore, it can also be checked that nilpotency of the Grassmannian derivative (i.e., ) implies .
4. Off-Shell Nilpotent SUSY Transformations: Chiral Supervariable Approach
For the derivation of second fermionic SUSY transformation , we concentrate on the chiral super-submanifold parametrized by the supervariables . Now, all the ordinary variables (depending explicitly on ) are generalized to a (1, 1)-dimensional chiral super-submanifold asIn the above, secondary variables and are fermionic and bosonic variables, respectively. We can derive the values of these secondary variables in terms of the basic variables, by exploiting the power and potential of SUSY invariant restrictions.
It is to be noted from (3) that does not transform under SUSY transformations (i.e., ) so the variable would remain unaffected by the presence of Grassmannian variable . As a consequence, we have the following:Furthermore, we observe that and due to the fermionic nature of . Generalizing these invariant restrictions to the chiral supersubmanifold, we have the following SUSYIRs in the following forms; namely, After putting the expansions for supervariable (22) in the above, we get The solution for the above relationship is . Substituting the value of secondary variables in the chiral supervariable expansions (21), we obtain the following expressions: where superscript represents the chiral supervariables obtained after the application of SUSYIRs. Using (26), one can generalize to onto the (1, 1)-dimensional chiral super-submanifold as where we have used the expression of chiral supervariable given in (26).
We note that because of the nilpotency of [cf. (3)]. Thus, we have the following SUSYIR in our present theory: This restriction serves our purpose for the derivation of SUSY transformation of . Exploiting the above restriction, we get the value of secondary variable in terms of basic variables as .
It is important to note that the following sum of the composite variables is invariant under ; namely, In order to calculate the fermionic symmetry transformation corresponding to variable , we use the above relationship as a SUSYIR: and after some computations, one gets .
Recollecting all the values of secondary variables and substituting them into (21), we have the expansions of the chiral supervariables as Finally, the supersymmetric transformations () for all the basic and auxiliary variables are listed as It is important to point out here that we have the following mapping between the Grassmannian derivative and the symmetry transformation : where is the generic chiral supervariables obtained after the application of supersymmetric invariant restrictions and denotes the basic variables of our present QM theory. The above equation captures the geometrical interpretation of transformation in terms of the Grassmannian derivative because of the fact that the translation along -direction of chiral supervariable is equivalent to the symmetry transformation of the same basic variable. We observe from (33) that the nilpotency of SUSY transformation (i.e., ) can be generalized in terms of Grassmannian derivative .
5. Invariance and Off-Shell Nilpotency: Supervariable Approach
It is interesting to note that, by exploiting the expansions of supervariables (11), the Lagrangian in (2) can be expressed in terms of the antichiral supervariables as In the earlier section, we have shown that SUSY transformation () and translational generator are geometrically related to each other (i.e., ). As a consequence, one can also capture the invariance of the Lagrangian in the following fashion: Similarly, Lagrangian (2) can also be written in terms of the chiral supervariables as Since the fermionic symmetry is geometrically connected with the translational generator , therefore, the invariance of the Lagrangian can be geometrically interpreted as follows:As a result, the action integral remains invariant.
We point out that the conserved charges and corresponding to the continuous symmetry transformations and can also be expressed as The nilpotency properties of the above charges can be shown in a straightforward manner with the help of symmetry properties: In the language of translational generators, these properties can be written as and . These relations hold due to the nilpotency of the Grassmannian derivatives (i.e., ).
6. SUSY Algebra and Its Interpretation
We observe that, under the discrete symmetry the Lagrangian transforms as . Hence, action integral (2) of the SUSY QM system remains invariant. It is to be noted that the above discrete symmetry transformations are important because they relate the two SUSY transformations : where is the generic variables present in the model. It is to be noted that, generally, the () signs are governed by the two successive operations of the discrete symmetry on the variables as In the present case, only the sign will occur for all the variables (i.e., ). It can be easily seen that relationship (41) is analogous to the relationship of differential geometry (where and are the exterior and coexterior derivative, resp., and is the Hodge duality operation).
We now focus on the physical identifications of the de Rham cohomological operators of differential geometry in terms of the symmetry transformations. It can be explicitly checked that the continuous symmetry transformations, together with discrete symmetry for our SUSY QM model, satisfy the following algebra [17–20, 22–24]: which is identical to the algebra obeyed by the de Rham cohomological operators [25–29], Here, is the Laplacian operator. From (43) and (44), we can identify the exterior derivative with and coexterior derivative with . The discrete symmetry (40) provides the analogue of Hodge duality operation of differential geometry. In fact, there is a one-to-one mapping between the symmetry transformations and the de Rham cohomological operators. It is also clear from (43) and (44) that the bosonic symmetry () and Laplacian operator are the Casimir operators of the algebra given in (43) and (44), respectively. Thus, our present SUSY model provides a model for Hodge theory. Furthermore, a similar algebra given in (44) is also satisfied by the conserved charges , and : In the above, we have used the canonical quantum (anti)commutation relations and . It is important to mention here that the bosonic charge (i.e., the Hamiltonian of the theory modulo a -factor) is the Casimir operator in algebra (45).
Some crucial properties related to the de Rham cohomological operators can be captured by these charges. For instance, we observe from (45) that the Hamiltonian is the Casimir operator of the algebra. Thus, it can be easily seen that implies that , if the inverse of the Hamiltonian exists. Since we are dealing with the no-singular Hamiltonian, we presume that the Casimir operator has its well-defined inverse value. By exploiting (45), it can be seen that Let us define an eigenvalue equation , where is the quantum Hilbert state with eigenvalue . By using algebra (46), one can verify the following: As a consequence of (47), it is evident that , and have the eigenvalues , and , respectively, with respect to to the operator .
The above equation provides a connection between the conserved charges and de Rham cohomological operators because as we know the action of on a given form increases the degree of the form by one, whereas application of decreases the degree by one unit and operator keeps the degree of a form intact. These important properties can be realized by the charges , where the eigenvalues and eigenfunctions play the key role .
In summary, exploiting the supervariable approach, we have derived the off-shell nilpotent symmetry transformations for the SUSY QM system. This has been explicated through the 1D SUSY invariant quantities, which remain unaffected due to the presence of the Grassmannian variables and . Furthermore, we have provided the geometrical interpretation of the SUSY transformations ( and ) in terms of the translational generators ( and ) along the Grassmannian directions and , respectively. Further, we have expressed the Lagrangian in terms of the (anti)chiral supervariables and the invariance of the Lagrangian under continuous transformations has been shown within the translations generators along -directions. The conserved SUSY charges corresponding to the fermionic symmetry transformations have been expressed in terms of (anti)chiral supervariables and the Grassmannian derivatives. The nilpotency of fermionic charges has been captured geometrically, within the framework of supervariable approach by the Grassmannian derivatives.
Finally, we have shown that the algebra satisfied by the continuous symmetry transformations , and (and corresponding charges) is exactly analogous to the Hodge algebra obeyed by the de Rham cohomological operators (, and ) of differential geometry. The discrete symmetry of the theory provides physical realization of the Hodge duality () operation. Thus, the present SUSY QM model provides a model for Hodge theory.
The author declares that there are no competing interests regarding the publication of this paper.
The author would like to thank Professor Prasanta K. Panigrahi for reading the manuscript as well as for offering the valuable inputs on the topic of the present investigation. The author is also thankful to Dr. Rohit Kumar for his important comments during the preparation of the manuscript.
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