Translational symmetries of quadratic Lagrangians

In this paper, we show that a quadratic Lagrangian, with no constraints, containing ordinary time derivatives up to the order [Formula: see text] of [Formula: see text] dynamical variables, has [Formula: see text] symmetries consisting in the translation of the variables with solutions of the equations of motion. We construct explicitly the generators of these transformations and prove that they satisfy the Heisenberg algebra. We also analyze other specific cases which are not included in our previous statement: the Klein–Gordon Lagrangian, [Formula: see text] Fermi oscillators and the Dirac Lagrangian. In the first case, the system is described by an equation involving partial derivatives, the second case is described by Grassmann variables and the third shows both features. Furthermore, the Fermi oscillator and the Dirac field Lagrangians lead to second class constraints. We prove that also in these last two cases there are translational symmetries and we construct the algebra of the generators. For the Klein–Gordon case we find a continuum version of the Heisenberg algebra, whereas in the other cases, the Grassmann generators satisfy, after quantization, the algebra of the Fermi creation and annihilation operators.

Calculus of Variations

This chapter presents an introduction to linear algebra. Classical mechanics is best understood in the language of differential geometry, which itself requires a working knowledge of the key concepts in linear algebra. This chapter walks through the required knowledge from this broad discipline and guides the reader towards the goal of the next chapter, differential geometry. Topics discussed include vector spaces, linear maps, basis sets, cobases, inner products, tensors, wedge products and exterior algebra, as well as the axioms of vector space geometry. The chapter concludes with a brief discussion of Grassmann variables, which tend to crop up when classical fermionic fields are defined.

Noether’s Theorem for Fields

This is a unique chapter that discusses classical path integrals in both configuration space and phase space. It examines both Lagrangian and Hamiltonian formulations before qualitatively discussing some interesting features of gauge fixing. This formulation is then linked to superspace and Grassmann variables for a fermionic field theory. The chapter then shows that the corresponding operatorial formulation is none other than the Koopman–von Neumann theory. In parallel to quantum theory, the classical propagator or the transition amplitude between two classical states is given exactly by the phase space partition function. The functional Dirac delta is discussed, and the chapter closes by briefly mentioning Faddeev–Popov ghosts, which were introduced earlier in the chapter.

General relativity for N properties

We determine the coefficients of the terms multiplying the gauge fields, gravitational field and cosmological term in a scheme whereby properties are characterized by N anticommuting scalar Grassmann variables. We do this for general N, using analytical methods; this obviates the need for our algebraic computing package which can become quite unwieldy as N is increased.

Exact canonically conjugate momenta approach to a one-dimensional neutron–proton system, I

Introducing collective variables, a collective description of nuclear surface oscillations has been developed with the first-quantized language, contrary to the second-quantized one in Sunakawa's approach for a Bose system. It overcomes difficulties remaining in the traditional theories of nuclear collective motions: Collective momenta are not exact canonically conjugate to collective coordinates and are not independent. On the contrary to such a description, Tomonaga first gave the basic idea to approach elementary excitations in a one-dimensional Fermi system. The Sunakawa's approach for a Fermi system is also expected to work well for such a problem. In this paper, on the isospin space, we define a density operator and further following Tomonaga, introduce a collective momentum. We propose an exact canonically momenta approach to a one-dimensional neutron–proton (N–P) system under the use of the Grassmann variables.