Graded identities for algebras with elementary gradings over an infinite field

Let $F$ be an infinite field of positive characteristic $p > 2$ and let $G$ be a group. In this paper, we study the graded identities satisfied by an associative algebra equipped with an elementary $G$ -grading. Let $E$ be the infinite-dimensional Grassmann algebra. For every $a$ , $b\in \mathbb {N}$ , we provide a basis for the graded polynomial identities, up to graded monomial identities, for the verbally prime algebras $M_{a,b}(E)$ , as well as their tensor products, with their elementary gradings. Moreover, we give an alternative proof of the fact that the tensor product $M_{a,b}(E)\otimes M_{r,s}(E)$ and $M_{ar+bs,as+br}(E)$ are $F$ -algebras which are not PI equivalent. Actually, we prove that the $T_{G}$ -ideal of the former algebra is contained in the $T$ -ideal of the latter. Furthermore, the inclusion is proper. Recall that it is well known that these algebras satisfy the same multilinear identities and hence in characteristic 0 they are PI equivalent.

From Classical to Quantum Fields. Free Fields

We apply the canonical and the path integral quantisation methods to scalar, spinor and vector fields. The scalar field is a generalisation to an infinite number of degrees of freedom of the single harmonic oscillator we studied in Chapter 9. For the spinor fields we show the need for anti-commutation relations and introduce the corresponding Grassmann algebra. The rules of Fermi statistics follow from these anti-commutation relations. The canonical quantisation method applied to the Maxwell field in a Lorentz covariant gauge requires the introduction of negative metric states in the Hilbert space. The power of the path integral quantisation is already manifest. In each case we expand the fields in creation and annihilation operators.

"An algorithm for automorphisms of infinite dimensional Grassmann algebras"

"Let G be the infinite dimensional Grassmann algebra. In this study, we determine a subgroup of the automorphism group Aut(G) of the algebra G which is of an importance in the description of the group Aut(G). We give an infinite generating set for this subgroup and suggest an algorithm which shows how to express each automorphism as compositions of generating elements."

Divergence-free quantum electrodynamics in locally conformally flat space–time

This paper describes an approach to quantum electrodynamics (QED) in curved space–time obtained by considering infinite-dimensional algebra bundles associated to a natural principal bundle [Formula: see text] associated with any locally conformally flat space–time, with typical fibers including the Fock space and a space of fermionic multiparticle states which forms a Grassmann algebra. Both these algebras are direct sums of generalized Hilbert spaces. The requirement of [Formula: see text] covariance associated with the geometry of space–time, where [Formula: see text] is the structure group of [Formula: see text], leads to the consideration of [Formula: see text] intertwining operators between various spaces. Scattering processes are associated with such operators and are encoded in an algebra of kernels. Intertwining kernels can be generated using [Formula: see text] covariant matrix-valued measures. Feynman propagators, fermion loops and the electron self-energy can be given well-defined interpretations as such measures. Divergence-free calculations in QED can be carried out by computing the spectra of these measures and kernels (a process called spectral regularization). As an example of the approach the precise Uehling potential function for the [Formula: see text] atom is calculated without requiring renormalization from which the Uehling contribution to the Lamb shift can be calculated exactly.

Fermionic QFT, Grassmann calculus, and combinatorics

In the first section of this chapter, we use Grassmann calculus, used in fermionic QFT, to give, first a reformulation of the Lingström–Gesse–Viennot lemma proof. We further show that this proof generalizes to graphs with cycles. We then use the same Grassmann calculus techniques to give new proofs of Stembridge's identities relating appropriate graph Pfaffians to sum over non-intersecting paths. The results presented here go further than the ones of Stembridge, because Grassmann algebra techniques naturally extend (without any cost!) to graphs with cycles. We thus obtain, instead of sums over non-intersecting paths, sums over non-intersecting paths and non-intersecting cycles. In the fifth section of the chapter, we give a generalization of these results. In the sixth section of this chapter we use Grassmann calculus to exhibit the relation between a multivariate version of Tutte polynomial and the Kirchhoff-Symanzik polynomials of the parametric representation of Feynman integrals, polynomials already introduced in Chapters 1 and Chapter 3.

Field theories with higher-group symmetry from composite currents

Higher-form symmetries are associated with transformations that only act on extended objects, not on point particles. Typically, higher-form symmetries live alongside ordinary, point-particle (0-form), symmetries and they can be jointly described in terms of a direct product symmetry group. However, when the actions of 0-form and higher-form symmetries become entangled, a more general mathematical structure is required, related to higher categorical groups. Systems with continuous higher-group symmetry were previously constructed in a top-down manner, descending from quantum field theories with a specific mixed ’t Hooft anomaly. I show that higher-group symmetry also naturally emerges from a bottom-up, low-energy perspective, when the physical system at hand contains at least two different given, spontaneously broken symmetries. This leads generically to a hierarchy of emergent higher-form symmetries, corresponding to the Grassmann algebra of topological currents of the theory, with an underlying higher-group structure. Examples of physical systems featuring such higher-group symmetry include superfluid mixtures and variants of axion electrodynamics.

Generalization of Geyer’s commutation relations with respect to the orthogonal group in even dimensions

A connection between the deformed Duffin–Kemmer–Petiau (DKP) algebra and an extended system of the parafermion trilinear commutation relations for the creation and annihilation operators $$a^{\pm }_{k}$$ a k ± and for an additional operator $$a_{0}$$ a 0 obeying para-Fermi statistics of order 2 based on the Lie algebra $${\mathfrak {s}}{\mathfrak {o}}(2M+2)$$ s o ( 2 M + 2 ) is established. An appropriate system of the parafermion coherent states as functions of para-Grassmann numbers is introduced. The representation for the operator $$a_{0}$$ a 0 in terms of generators of the orthogonal group SO(2M) correctly reproducing action of this operator on the state vectors of Fock space is obtained. A connection of the Geyer operator $$a_{0}^{2}$$ a 0 2 with the operator of so-called G-parity and with the CPT- operator $${\hat{\eta }}_{5}$$ η ^ 5 of the DKP-theory is established. In a para-Grassmann algebra a noncommutative, associative star product $$*$$ ∗ (the Moyal product) as a direct generalization of the star product in the algebra of Grassmann numbers is introduced. Two independent approaches to the calculation of the Moyal product $$*$$ ∗ are considered. It is shown that in calculating the matrix elements in the basis of parafermion coherent states of various operator expressions it should be taken into account constantly that we work in the so-called Ohnuki and Kamefuchi’s generalized state-vector space $${\mathfrak {U}}_{\;G}$$ U G , whose state vectors include para-Grassmann numbers $$\xi _{k}$$ ξ k in their definition, instead of the standard state-vector space $${\mathfrak {U}}$$ U (the Fock space).