Supervariable and BRST Approaches to a Reparameterization Invariant Nonrelativistic System

<jats:p>We exploit the theoretical strength of the supervariable and Becchi-Rouet-Stora-Tyutin (BRST) formalisms to derive the proper (i.e., off-shell nilpotent and absolutely anticommuting) (anti-)BRST symmetry transformations for the reparameterization invariant model of a nonrelativistic (NR) free particle whose space <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M1"> <mfenced open="(" close=")"> <mrow> <mi>x</mi> </mrow> </mfenced> </math> </jats:inline-formula> and time <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M2"> <mfenced open="(" close=")"> <mrow> <mi>t</mi> </mrow> </mfenced> </math> </jats:inline-formula> variables are a function of an evolution parameter <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M3"> <mfenced open="(" close=")"> <mrow> <mi>τ</mi> </mrow> </mfenced> </math> </jats:inline-formula>. The infinitesimal reparameterization (i.e., 1D diffeomorphism) symmetry transformation of our theory is defined w.r. t. this evolution parameter <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M4"> <mfenced open="(" close=")"> <mrow> <mi>τ</mi> </mrow> </mfenced> </math> </jats:inline-formula>. We apply the modified Bonora-Tonin (BT) supervariable approach (MBTSA) as well as the (anti)chiral supervariable approach (ACSA) to BRST formalism to discuss various aspects of our present system. For this purpose, our 1D ordinary theory (parameterized by <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <mi>τ</mi> </math> </jats:inline-formula>) is generalized onto a <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M6"> <mfenced open="(" close=")"> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </mfenced> </math> </jats:inline-formula>-dimensional supermanifold which is characterized by the superspace coordinates <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M7"> <msup> <mrow> <mi>Z</mi> </mrow> <mrow> <mi>M</mi> </mrow> </msup> <mo>=</mo> <mfenced open="(" close=")"> <mrow> <mi>τ</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mover accent="true"> <mi>θ</mi> <mo stretchy="true">¯</mo> </mover> </mrow> </mfenced> </math> </jats:inline-formula> where a pair of the Grassmannian variables satisfy the fermionic relationships: <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M8"> <msup> <mrow> <mi>θ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mover accent="true"> <mi>θ</mi> <mo stretchy="true">¯</mo> </mover> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </math> </jats:inline-formula>, <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M9"> <mi>θ</mi> <mtext> </mtext> <mover accent="true"> <mi>θ</mi> <mo stretchy="true">¯</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>θ</mi> <mo stretchy="true">¯</mo> </mover> <mtext> </mtext> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </math> </jats:inline-formula>, and <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M10"> <mi>τ</mi> </math> </jats:inline-formula> is the bosonic evolution parameter. In the context of ACSA, we take into account only the <jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M11"> <mfenced open="(" close=")"> <mrow> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </mfenced> </math> </jats:inline-formula>-dimensional (anti)chiral super submanifolds of the general<jats:inline-formula> <math xmlns="http://www.w3. org/1998/Math/MathML" id="M12"> <mfenced open="(" close=")"> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </mfenced> </math> </jats:inline-formula>-dimensional supermanifold. The derivation of the universal Curci-Ferrari- (CF-) type restriction, from various underlying theoretical methods, is a novel observation in our present endeavor. Furthermore, we note that the form of the gauge-fixing and Faddeev-Popov ghost terms for our NR and non-SUSY system is exactly the same as that of the reparameterization invariant SUSY (i.e., spinning) and non-SUSY (i.e., scalar) relativistic particles. This is a novel observation, too. </jats:p>