Grassmann Calculus Research Articles

Z 2 × Z 2 generalizations of N=1 superconformal Galilei algebras and their representations

We introduce two classes of novel color superalgebras of Z2×Z2 grading. This is done by realizing members of each class within the universal enveloping algebra of the N=1 supersymmetric extension of the conformal Galilei algebra. This allows us to upgrade any representation of the super conformal Galilei algebras to a representation of the Z2×Z2 graded algebra. As an example, boson-fermion Fock space representation of one class is given. We also provide a vector field realization of members of the other class by using a generalization of the Grassmann calculus to Z2×Z2 graded setting.

Fermionic-mode entanglement in non-Markovian environment

We evaluate the non-Markovian effects on the entanglement dynamics of a fermionic system interacting with two dissipative vacuum reservoirs. The exact solution of density matrix is derived by utilizing the Feynman–Vernon influence functional theory in the fermionic coherent state representation and the Grassmann calculus, which are valid for both the fermionic and bosonic baths, and their difference lies in the dependence of the parity of the initial states. The fermionic entanglement dynamics is presented by adding an additional restriction to the density matrix known as the superselection rules. Our analysis shows that the usual decoherence suppression schemes implemented in qubits systems can also be achieved for systems of identical fermions, and the initial state proves its importance in the evolution of fermionic entanglement. Our results provide a potential way to decoherence controlling of identical fermions.

Some Indentities for the Gradient of the Principal Invariants, Traces and Determinants via Grassmann Calculus

Let A be a second order tensor in a finite dimensional space. In this work we determine the gradient of the principal invariants of A and obtain some trace and determinant identities using only some standard rigorous statements concerning Grassmann calculus. We recover some of the results of Dui et al. (J. Elast. 75:193–196, 2004) and of Truesdell and Noll (The Non-linear Field Theories of Mechanics, Springer, Berlin, 2002) and solve an old problem proposed in SIAM Review concerning a determinant identity from a new perspective in a concise and simple manner.

Grassmann calculus, pseudoclassical mechanics, and geometric algebra

A reformulation of Grassmann calculus is presented in terms of geometric algebra—a unified language for physics based on Clifford algebra. In this reformulation, Grassmann generators are replaced by vectors, so that every product of generators has a natural geometric interpretation. The calculus introduced by Berezin [The Method of Second Quantization (Academic, New York, 1966)] is shown to be unnecessary, amounting to no more than an algebraic contraction. This approach is not only conceptually clearer, but it is also computationally more efficient, as demonstrated by treatments of the ‘‘Grauss’’ integral and the Grassmann Fourier Transform. The reformulation is applied to pseudoclassical mechanics [Ann. Phys. 104, 336 (1977)], where it is shown to lead to a new concept, the multivector Lagrangian. To illustrate this idea, the three-dimensional Fermi oscillator is reformulated and solved, and its symmetry properties discussed. As a result, a new and highly compact formula for generating super-Lie algebras is revealed. The paper ends with a discussion of quantization, outlining a new approach to fermionic path integrals.

PARAGRASSMANN ANALYSIS AND QUANTUM GROUPS

Paragrassmann algebras with one and many paragrassmann variables are considered from the algebraic point of view without using the Green ansatz. A differential operator with respect to paragrassmann variable and a covariant para-super-derivative are introduced giving a natural generalization of the Grassmann calculus to a paragrassmann one. Deep relations between paragrassmann algebras and quantum groups with deformation parameters being root of unity are established.

A Grassmann calculus for infinite spin groups

The correlation functions of the two-dimensional Ising model satisfy nonlinear equations both on the lattice (discovered by McCoy and Wu [Nucl. Phys. B 180, 89 (1981)] and Perk [Phys. Lett. A1 79, 3 (1980)]) and in a continuum limit (discovered by Wu, McCoy, Tracy, and Barouch and generalized by Sato, Miwa, and Jimbo [Publ. RIMS 14, 223 (1978)]). In this paper results are presented for infinite spin groups that extend results of Sato, Miwa, and Jimbo [Publ. RIMS 14, 223 (1978)] and lead to normal ordered product formulas for the difference of adjacent Ising spin fields. These product formulas are shown to lead directly to the lattice difference equations for the correlations and are also a key ingredient in the scaling limit analysis.

Feynman path integral for fermions.

With Lagrangian quantum field theory as a starting point, the general quantum transition amplitude is derived as a sum of action exponentials over all paths connecting the initial and final fermion states. For bookkeeping convenience in intermediate stages anticommuting parameters are introduced. All integrals are determined by completeness and unitarity. Grassmann calculus is not postulated.

Základové geometrického počtu Grassmannova. [I.

Základové geometrického počtu Grassmannova. [I.