On varieties in multiple-projective spaces

AbstractIn this paper, mP will denote a projective space of dimension m, and (m,n)P will denote a doubly-projective space of dimension m+n, namely the space of all pairs of points (x¦y), where x varies in mP and y in nP. Just so, (m,n,s)P will denote a triply-projective space of dimension m+n+s, and so on.A variety V of dimension d in mP has just one degree g, namely the number of points of intersection of V with d generic linear hyperplanes (ux)=0, where (ux) means ∑uixi. On the other hand, a variety V' of dimension d in (m,n)P has several degree ga,b (a+b=d), defined as follows: ga,b is the number of points of intersection of V' with a hyperplanes (ux)=0 and b hyperplanes (vy)=0.Let xo, …, xm, yo, …, yn be the homogeneous coordinates of a point in m+n+1P. It sometimes happens that the equations of a variety V in m+n+1P are not only homogeneous in all variables x and y together, but even homogeneous in the x's and in the y's. In this case the same set of equations also defines a variety V' in the doubly-projective space (m+n)P. If d is the dimension of V', the dimension of V is d+1, for to every point (x¦y) of V' corresponds a whole straight line of points (xα, yβ) in V.In some cases it is easier to determine the degrees ga,b of V' than to determine the degree g of V. For this reason, it is desirable to have a rule that enables us to calculate g from the ga,b's. Such a rule will be proved here. It says:The degree g is the sum of all ga,b with a+b=d.In the case of multiply-projective spaces (m,n,…)P the same rule holds: g is the sum of the ga,b,… with a+b+…=d.Examples of applications of this rule will be given at the end

Why is CPT fundamental?

G. L\"uders and W. Pauli proved the $\mathcal{CPT}$ theorem based on Lagrangian quantum field theory almost half a century ago. R. Jost gave a more general proof based on ``axiomatic'' field theory nearly as long ago. The axiomatic point of view has two advantages over the Lagrangian one. First, the axiomatic point of view makes clear why $\mathcal{CPT}$ is fundamental--because it is intimately related to Lorentz invariance. Secondly, the axiomatic proof gives a simple way to calculate the $\mathcal{CPT}$ transform of any relativistic field without calculating $\mathcal{C}$, $\mathcal{P}$ and $\mathcal{T}$ separately and then multiplying them. The purpose of this pedagogical paper is to ``deaxiomatize'' the $\mathcal{CPT}$ theorem by explaining it in a few simple steps. We use theorems of distribution theory and of several complex variables without proof to make the exposition elementary.Comment: 17 pages, no figure

The fundamental solution and Strichartz estimates for the Schr\"odinger equation on flat euclidean cones

We study the Schr\"odinger equation on a flat euclidean cone $\mathbb{R}_+ \times \mathbb{S}^1_\rho$ of cross-sectional radius $\rho > 0$, developing asymptotics for the fundamental solution both in the regime near the cone point and at radial infinity. These asymptotic expansions remain uniform while approaching the intersection of the "geometric front", the part of the solution coming from formal application of the method of images, and the "diffractive front" emerging from the cone tip. As an application, we prove Strichartz estimates for the Schr\"odinger propagator on this class of cones.Comment: 21 pages, 4 figures. Minor typos corrected. To be published in Comm. Math. Phy

Glimpses of the Octonions and Quaternions History and Todays Applications in Quantum Physics

Before we dive into the accessibility stream of nowadays indicatory applications of octonions to computer and other sciences and to quantum physics let us focus for a while on the crucially relevant events for todays revival on interest to nonassociativity. Our reflections keep wandering back to the $Brahmagupta$ $Fibonacc$ two square identity and then via the $Euler$ four square identity up to the $Degen$ $Ggraves$ $Cayley$ eight square identity. These glimpses of history incline and invite us to retell the story on how about one month after quaternions have been carved on the $Broughamian$ bridge octonions were discovered by $John$ $Thomas$ $Ggraves$, jurist and mathematician, a friend of $William$ $Rowan$ $Hamilton$. As for today we just mention en passant quaternionic and octonionic quantum mechanics, generalization of $Cauchy$ $Riemann$ equations for octonions and triality principle and $G_2$ group in spinor language in a descriptive way in order not to daunt non specialists. Relation to finite geometries is recalled and the links to the 7stones of seven sphere, seven imaginary octonions units in out of the $Plato$ cave reality applications are appointed . This way we are welcomed back to primary ideas of $Heisenberg$, $Wheeler$ and other distinguished fathers of quantum mechanics and quantum gravity foundations.Comment: 26 pages, 7 figure